{ "schema_version": 1, "generated_from": { "claudiverse_commit": "4a88328", "generated_at": "2026-10-10T11:13:12Z" }, "node_types": [ "book", "person", "chapter", "exercise_set", "problem", "problem_form", "problem_shape", "concept", "method", "theorem", "law", "quantity", "unit", "instrument", "experiment", "excerpt", "equation", "capability" ], "edge_types": [ "written_by", "part_of", "taught_in", "practices", "quoted_from", "explains", "appears_in", "states", "relates", "instance_of", "needs", "prerequisite_of", "special_case_of", "generalizes", "uses", "inverse_of", "contrasts_with", "measures", "unit_of", "named_after", "discovered_by", "related_to" ], "book": { "id": "wentworth-first-steps-in-algebra-1894", "title": "The First Steps in Algebra", "authors": [ "George Albert Wentworth" ], "year": 1894, "edition": "Ginn & Company, 1894", "transcription": { "source": "Project Gutenberg eBook #36670", "url": "https://www.gutenberg.org/ebooks/36670", "released": "July 9, 2011", "licence_terms": "This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.net" }, "file": "books/wentworth-first-steps-in-algebra-1894.json" }, "chapters": [ { "id": "wentworth-first-steps-in-algebra-1894/ch-i", "number": "I", "title": "Introduction", "name": "Wentworth 1894, ch. 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III: Positive and Negative Numbers", "pages": [ "33", "46" ], "concepts": [ "concept/absolute-value", "concept/algebraic-number", "concept/algebraic-sum", "concept/integer", "concept/negative-number", "concept/number-line", "concept/positive-number", "concept/sign", "theorem/index-law-in-division", "theorem/index-law-in-multiplication", "theorem/law-of-signs-in-division", "theorem/law-of-signs-in-multiplication" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-2e84ad8721", "wentworth-first-steps-in-algebra-1894/x-26fc7e8d79", "wentworth-first-steps-in-algebra-1894/x-b4bec6cd19", "wentworth-first-steps-in-algebra-1894/x-d48699ca33", "wentworth-first-steps-in-algebra-1894/x-11cfed4918", "wentworth-first-steps-in-algebra-1894/x-fee9735e18" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-99af18c2ef", "wentworth-first-steps-in-algebra-1894/eq-f0c40bd36a", "wentworth-first-steps-in-algebra-1894/eq-ad9a39cd73", "wentworth-first-steps-in-algebra-1894/eq-ff681e42ac", "wentworth-first-steps-in-algebra-1894/eq-19bb597279", "wentworth-first-steps-in-algebra-1894/eq-1a36494be8", "wentworth-first-steps-in-algebra-1894/eq-851fe5e4f4", "wentworth-first-steps-in-algebra-1894/eq-0774b5a15a", "wentworth-first-steps-in-algebra-1894/eq-fb702170bc", "wentworth-first-steps-in-algebra-1894/eq-83a8b029bc", "wentworth-first-steps-in-algebra-1894/eq-d224a759c4" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-15", "wentworth-first-steps-in-algebra-1894/ex-16", "wentworth-first-steps-in-algebra-1894/ex-17" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-iv", "number": "IV", "title": "Addition and Subtraction", "name": "Wentworth 1894, ch. IV: Addition and Subtraction", "pages": [ "46", "53" ], "concepts": [ "concept/coefficient", "concept/integral-expression", "concept/like-terms", "concept/minuend", "concept/parenthesis", "concept/subtrahend", "method/addition", "method/arranging-in-columns", "method/collecting-like-terms", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus", "theorem/removing-parentheses-preceded-by-plus" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-a8557f7062", "wentworth-first-steps-in-algebra-1894/x-a174980315", "wentworth-first-steps-in-algebra-1894/x-bc46551e90", "wentworth-first-steps-in-algebra-1894/x-06d26340e8", "wentworth-first-steps-in-algebra-1894/x-8fbce39e17", "wentworth-first-steps-in-algebra-1894/x-d028ec758c", "wentworth-first-steps-in-algebra-1894/x-1038346704" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-a28804a446", "wentworth-first-steps-in-algebra-1894/eq-8f84c3b570", "wentworth-first-steps-in-algebra-1894/eq-e10ecaf800", "wentworth-first-steps-in-algebra-1894/eq-c55d9182a7", "wentworth-first-steps-in-algebra-1894/eq-670b148831", "wentworth-first-steps-in-algebra-1894/eq-03e187dbe8", "wentworth-first-steps-in-algebra-1894/eq-65d6f5b071", "wentworth-first-steps-in-algebra-1894/eq-df1a52b47d" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-18", "wentworth-first-steps-in-algebra-1894/ex-19", "wentworth-first-steps-in-algebra-1894/ex-20" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-v", "number": "V", "title": "Multiplication and Division", "name": "Wentworth 1894, ch. V: Multiplication and Division", "pages": [ "53", "64" ], "concepts": [ "concept/arrangement", "concept/dividend", "concept/divisor", "concept/like-terms", "concept/monomial", "concept/partial-product", "concept/polynomial", "concept/product", "concept/quotient", "concept/remainder", "method/collecting-like-terms", "method/dividing-a-polynomial-by-a-monomial", "method/dividing-a-polynomial-by-a-polynomial", "method/multiplying-a-compound-expression" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-7790d96b4a", "wentworth-first-steps-in-algebra-1894/x-0c005b237b", "wentworth-first-steps-in-algebra-1894/x-ef422e8f06", "wentworth-first-steps-in-algebra-1894/x-6bc3cb3111", "wentworth-first-steps-in-algebra-1894/x-090911c9f7", "wentworth-first-steps-in-algebra-1894/x-5e911c2a47" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-9c8f9bf8f6", "wentworth-first-steps-in-algebra-1894/eq-6682317b07", "wentworth-first-steps-in-algebra-1894/eq-e7d0036322", "wentworth-first-steps-in-algebra-1894/eq-3ab69dec11" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-21", "wentworth-first-steps-in-algebra-1894/ex-22", "wentworth-first-steps-in-algebra-1894/ex-23", "wentworth-first-steps-in-algebra-1894/ex-24", "wentworth-first-steps-in-algebra-1894/ex-25" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-vi", "number": "VI", "title": "Multiplication and Division", "name": "Wentworth 1894, ch. VI: Multiplication and Division", "pages": [ "64", "71" ], "concepts": [ "concept/divisibility", "method/multiplying-two-binomials", "method/writing-by-inspection", "theorem/difference-of-two-cubes", "theorem/difference-of-two-squares", "theorem/square-of-a-difference", "theorem/square-of-a-sum", "theorem/sum-of-two-cubes" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-e3797b8e51", "wentworth-first-steps-in-algebra-1894/x-5186a76e49", "wentworth-first-steps-in-algebra-1894/x-bca6da8048", "wentworth-first-steps-in-algebra-1894/x-cad1955f3c", "wentworth-first-steps-in-algebra-1894/x-803c638cbd", "wentworth-first-steps-in-algebra-1894/x-22060ce047" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-20f9664b9e", "wentworth-first-steps-in-algebra-1894/eq-acd11efd91", "wentworth-first-steps-in-algebra-1894/eq-9ea7d1a1f8", "wentworth-first-steps-in-algebra-1894/eq-9d1360e429" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-26", "wentworth-first-steps-in-algebra-1894/ex-27", "wentworth-first-steps-in-algebra-1894/ex-28", "wentworth-first-steps-in-algebra-1894/ex-29", "wentworth-first-steps-in-algebra-1894/ex-30" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-vii", "number": "VII", "title": "Factors", "name": "Wentworth 1894, ch. VII: Factors", "pages": [ "71", "84" ], "concepts": [ "concept/algebraic-sum", "concept/binomial", "concept/divisibility", "concept/factor", "concept/integral-expression", "concept/monomial", "concept/polynomial", "concept/rational-expression", "concept/root", "concept/trinomial", "method/factoring", "method/factoring-a-perfect-square-trinomial", "method/factoring-a-trinomial-of-the-form-x-2-ax-b", "method/factoring-by-grouping", "method/factoring-by-taking-out-a-common-factor", "method/writing-by-inspection", "theorem/difference-of-two-cubes", "theorem/difference-of-two-squares", "theorem/removing-parentheses-preceded-by-minus", "theorem/square-of-a-sum", "theorem/sum-of-two-cubes" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-7b54d1c573", "wentworth-first-steps-in-algebra-1894/x-828d6e999f", "wentworth-first-steps-in-algebra-1894/x-1253379b54", "wentworth-first-steps-in-algebra-1894/x-5e9b8437ff", "wentworth-first-steps-in-algebra-1894/x-55a6255f82", "wentworth-first-steps-in-algebra-1894/x-e44972fef4" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-e2f6bfbee5", "wentworth-first-steps-in-algebra-1894/eq-c17fe3aa07", "wentworth-first-steps-in-algebra-1894/eq-3df2422d67", "wentworth-first-steps-in-algebra-1894/eq-9ea7d1a1f8", "wentworth-first-steps-in-algebra-1894/eq-55d59c2fa2", "wentworth-first-steps-in-algebra-1894/eq-9d1360e429", "wentworth-first-steps-in-algebra-1894/eq-7b490e9a40", "wentworth-first-steps-in-algebra-1894/eq-d7d115c3f1", "wentworth-first-steps-in-algebra-1894/eq-7932ebc495" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-31", "wentworth-first-steps-in-algebra-1894/ex-32", "wentworth-first-steps-in-algebra-1894/ex-33", "wentworth-first-steps-in-algebra-1894/ex-34", "wentworth-first-steps-in-algebra-1894/ex-35", "wentworth-first-steps-in-algebra-1894/ex-36", "wentworth-first-steps-in-algebra-1894/ex-37", "wentworth-first-steps-in-algebra-1894/ex-38", "wentworth-first-steps-in-algebra-1894/ex-39" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-viii", "number": "VIII", "title": "Common Factors and Multiples", "name": "Wentworth 1894, ch. VIII: Common Factors and Multiples", "pages": [ "84", "89" ], "concepts": [ "concept/common-factor", "concept/common-multiple", "concept/highest-common-factor", "concept/lowest-common-multiple", "concept/relatively-prime", "method/finding-the-highest-common-factor", "method/finding-the-lowest-common-multiple" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-b53aea0c98", "wentworth-first-steps-in-algebra-1894/x-88e592b989", "wentworth-first-steps-in-algebra-1894/x-162a035eff", "wentworth-first-steps-in-algebra-1894/x-4b677a7bee", "wentworth-first-steps-in-algebra-1894/x-0b96e8bb4a" ], "equations": [], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-40", "wentworth-first-steps-in-algebra-1894/ex-41" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-ix", "number": "IX", "title": "Fractions", "name": "Wentworth 1894, ch. IX: Fractions", "pages": [ "89", "103" ], "concepts": [ "concept/algebraic-fraction", "concept/complex-fraction", "concept/denominator", "concept/lowest-common-denominator", "concept/lowest-terms", "concept/numerator", "concept/reciprocal", "method/adding-fractions", "method/dividing-by-a-fraction", "method/multiplying-fractions", "method/reducing-a-fraction-to-an-integral-or-mixed-expression", "method/reducing-a-fraction-to-lowest-terms", "method/reducing-a-mixed-expression-to-a-fraction", "method/reducing-fractions-to-lowest-common-denominator", "method/simplifying-a-complex-fraction", "theorem/fundamental-principle-of-fractions" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-aa3a64158f", "wentworth-first-steps-in-algebra-1894/x-865d75c450", "wentworth-first-steps-in-algebra-1894/x-0e0d7ec149", "wentworth-first-steps-in-algebra-1894/x-73b11d75ed", "wentworth-first-steps-in-algebra-1894/x-84d25a0a14", "wentworth-first-steps-in-algebra-1894/x-5fd066f232" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-6e032054d2", "wentworth-first-steps-in-algebra-1894/eq-312216bfb2", "wentworth-first-steps-in-algebra-1894/eq-2e70b5e188", "wentworth-first-steps-in-algebra-1894/eq-ff019a38b5", "wentworth-first-steps-in-algebra-1894/eq-da05d08912" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-42", "wentworth-first-steps-in-algebra-1894/ex-43", "wentworth-first-steps-in-algebra-1894/ex-44", "wentworth-first-steps-in-algebra-1894/ex-45", "wentworth-first-steps-in-algebra-1894/ex-46", "wentworth-first-steps-in-algebra-1894/ex-47", "wentworth-first-steps-in-algebra-1894/ex-48", "wentworth-first-steps-in-algebra-1894/ex-49", "wentworth-first-steps-in-algebra-1894/ex-50" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-x", "number": "X", "title": "Fractional Equations", "name": "Wentworth 1894, ch. X: Fractional Equations", "pages": [ "103", "122" ], "concepts": [ "concept/combined-rate", "concept/formula", "concept/fractional-equation", "concept/literal-equation", "concept/rule", "method/clearing-an-equation-of-fractions", "quantity/amount", "quantity/interest", "quantity/principal", "quantity/rate-of-interest", "quantity/time" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-1a53d6ab7a", "wentworth-first-steps-in-algebra-1894/x-31b78d1183", "wentworth-first-steps-in-algebra-1894/x-536d994939", "wentworth-first-steps-in-algebra-1894/x-3b627f7889", "wentworth-first-steps-in-algebra-1894/x-6892527ec1", "wentworth-first-steps-in-algebra-1894/x-04362a925d", "wentworth-first-steps-in-algebra-1894/x-a9e5a04d3b" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-bbce9cd735", "wentworth-first-steps-in-algebra-1894/eq-329d346ad7", "wentworth-first-steps-in-algebra-1894/eq-a67b1d4d0e", "wentworth-first-steps-in-algebra-1894/eq-267658dc32", "wentworth-first-steps-in-algebra-1894/eq-a7eb0e6569", "wentworth-first-steps-in-algebra-1894/eq-8664262bc6", "wentworth-first-steps-in-algebra-1894/eq-6a0a47b52c" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-51", "wentworth-first-steps-in-algebra-1894/ex-52", "wentworth-first-steps-in-algebra-1894/ex-53", "wentworth-first-steps-in-algebra-1894/ex-54", "wentworth-first-steps-in-algebra-1894/ex-55", "wentworth-first-steps-in-algebra-1894/ex-56", "wentworth-first-steps-in-algebra-1894/ex-57", "wentworth-first-steps-in-algebra-1894/ex-58", "wentworth-first-steps-in-algebra-1894/ex-59", "wentworth-first-steps-in-algebra-1894/ex-60", "wentworth-first-steps-in-algebra-1894/ex-61", "wentworth-first-steps-in-algebra-1894/ex-62", "wentworth-first-steps-in-algebra-1894/ex-63", "wentworth-first-steps-in-algebra-1894/ex-64" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-xi", "number": "XI", "title": "Simultaneous Equations of the First Degree", "name": "Wentworth 1894, ch. XI: Simultaneous Equations of the First Degree", "pages": [ "122", "132" ], "concepts": [ "concept/algebraic-fraction", "concept/coefficient", "concept/digit", "concept/independent-equations", "concept/linear-equation", "concept/quotient", "concept/remainder", "concept/simultaneous-equations", "concept/unknown", "method/addition", "method/clearing-an-equation-of-fractions", "method/elimination", "method/finding-the-lowest-common-multiple", "method/stating-a-problem-as-an-equation", "method/substitution", "method/subtraction", "quantity/amount", "quantity/interest", "quantity/principal", "quantity/rate-of-interest" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-659de2c6fc", "wentworth-first-steps-in-algebra-1894/x-9e99d9ed7d", "wentworth-first-steps-in-algebra-1894/x-8d45059f42", "wentworth-first-steps-in-algebra-1894/x-3a98fc05f3", "wentworth-first-steps-in-algebra-1894/x-331d88883d", "wentworth-first-steps-in-algebra-1894/x-8b2d75a1be", "wentworth-first-steps-in-algebra-1894/x-7ffed2b24a", "wentworth-first-steps-in-algebra-1894/x-03905975e9" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-ccad3db812", "wentworth-first-steps-in-algebra-1894/eq-250437650b", "wentworth-first-steps-in-algebra-1894/eq-a8d55786ea", "wentworth-first-steps-in-algebra-1894/eq-9df268b59e", "wentworth-first-steps-in-algebra-1894/eq-510c997133", "wentworth-first-steps-in-algebra-1894/eq-485175ce55", "wentworth-first-steps-in-algebra-1894/eq-67581977ce", "wentworth-first-steps-in-algebra-1894/eq-9127fb2210" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-65", "wentworth-first-steps-in-algebra-1894/ex-66", "wentworth-first-steps-in-algebra-1894/ex-67", "wentworth-first-steps-in-algebra-1894/ex-68", "wentworth-first-steps-in-algebra-1894/ex-69", "wentworth-first-steps-in-algebra-1894/ex-70" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-xii", "number": "XII", "title": "Quadratic Equations", "name": "Wentworth 1894, ch. XII: Quadratic Equations", "pages": [ "132", "142" ], "concepts": [ "concept/affected-quadratic-equation", "concept/coefficient", "concept/constant-term", "concept/fractional-equation", "concept/imaginary-root", "concept/literal-quadratic-equation", "concept/numerical-quadratic-equation", "concept/pure-quadratic-equation", "concept/quadratic-equation", "concept/radical-sign", "concept/rational-root", "concept/real-root", "concept/root", "concept/square", "concept/surd", "method/clearing-an-equation-of-fractions", "method/collecting-like-terms", "method/completing-the-square", "method/extracting-the-square-root", "method/solving-an-equation", "method/stating-a-problem-as-an-equation", "method/verification", "theorem/difference-of-two-squares", "theorem/square-of-a-difference", "theorem/square-of-a-sum" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-0811c5cd9f", "wentworth-first-steps-in-algebra-1894/x-d81f756db5", "wentworth-first-steps-in-algebra-1894/x-bcc79a35fd", "wentworth-first-steps-in-algebra-1894/x-3e85a54fb8", "wentworth-first-steps-in-algebra-1894/x-eb37734410", "wentworth-first-steps-in-algebra-1894/x-2b42154605" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-67a3f12238", "wentworth-first-steps-in-algebra-1894/eq-e6872b1c5d", "wentworth-first-steps-in-algebra-1894/eq-4acc1f6299", "wentworth-first-steps-in-algebra-1894/eq-d6cdd9ede5" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-71", "wentworth-first-steps-in-algebra-1894/ex-72", "wentworth-first-steps-in-algebra-1894/ex-73", "wentworth-first-steps-in-algebra-1894/ex-74" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-xiii", "number": "XIII", "title": "Arithmetical Progression", "name": "Wentworth 1894, ch. XIII: Arithmetical Progression", "pages": [ "142", "148" ], "concepts": [ "concept/arithmetical-mean", "concept/arithmetical-progression", "concept/common-difference", "method/inserting-arithmetical-means", "theorem/nth-term-of-an-arithmetical-progression", "theorem/sum-of-an-arithmetical-progression" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-a314bcd761", "wentworth-first-steps-in-algebra-1894/x-72a47cca8d", "wentworth-first-steps-in-algebra-1894/x-f5de054e82", "wentworth-first-steps-in-algebra-1894/x-2bb33fd0be", "wentworth-first-steps-in-algebra-1894/x-be59a84bc4", "wentworth-first-steps-in-algebra-1894/x-2b8527e620", "wentworth-first-steps-in-algebra-1894/x-82f185e255" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-e7134e170d", "wentworth-first-steps-in-algebra-1894/eq-acf2545185", "wentworth-first-steps-in-algebra-1894/eq-37f173344f", "wentworth-first-steps-in-algebra-1894/eq-8dd2f8b76d", "wentworth-first-steps-in-algebra-1894/eq-250f19306d", "wentworth-first-steps-in-algebra-1894/eq-2e58fe420b", "wentworth-first-steps-in-algebra-1894/eq-4b5542aa9c", "wentworth-first-steps-in-algebra-1894/eq-07772edd80" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-75", "wentworth-first-steps-in-algebra-1894/ex-76" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-xiv", "number": "XIV", "title": "Geometrical Progression", "name": "Wentworth 1894, ch. XIV: Geometrical Progression", "pages": [ "148", "152" ], "concepts": [ "concept/common-ratio", "concept/geometrical-mean", "concept/geometrical-progression", "theorem/nth-term-of-a-geometrical-progression", "theorem/sum-of-a-geometrical-progression" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-5c26feff53", "wentworth-first-steps-in-algebra-1894/x-27e69d9a62", "wentworth-first-steps-in-algebra-1894/x-0e19f333c9", "wentworth-first-steps-in-algebra-1894/x-e120fc3933", "wentworth-first-steps-in-algebra-1894/x-0fbd766b52", "wentworth-first-steps-in-algebra-1894/x-ca64b2c4cc" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-e920c0cf4e", "wentworth-first-steps-in-algebra-1894/eq-922e329311", "wentworth-first-steps-in-algebra-1894/eq-435d84bddc", "wentworth-first-steps-in-algebra-1894/eq-e6c973701d", "wentworth-first-steps-in-algebra-1894/eq-2093d86384" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-77" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ch-xv", "number": "XV", "title": "Square and Cube Roots", "name": "Wentworth 1894, ch. XV: Square and Cube Roots", "pages": [ "152", "165" ], "concepts": [ "concept/complete-divisor", "concept/perfect-cube", "concept/perfect-square", "concept/root", "concept/trial-divisor", "method/extracting-the-cube-root", "method/extracting-the-root-of-a-fraction", "method/extracting-the-square-root", "method/grouping-digits", "theorem/cube-of-a-sum" ], "excerpts": [ "wentworth-first-steps-in-algebra-1894/x-71c2354a6b", "wentworth-first-steps-in-algebra-1894/x-45040d228f", "wentworth-first-steps-in-algebra-1894/x-00c54e6160", "wentworth-first-steps-in-algebra-1894/x-6a7ca4662a", "wentworth-first-steps-in-algebra-1894/x-e263c7501e", "wentworth-first-steps-in-algebra-1894/x-268182ecbb", "wentworth-first-steps-in-algebra-1894/x-60e075fd53" ], "equations": [ "wentworth-first-steps-in-algebra-1894/eq-0795807a60", "wentworth-first-steps-in-algebra-1894/eq-f064f31fcf" ], "exercise_sets": [ "wentworth-first-steps-in-algebra-1894/ex-78", "wentworth-first-steps-in-algebra-1894/ex-79", "wentworth-first-steps-in-algebra-1894/ex-80", "wentworth-first-steps-in-algebra-1894/ex-81" ] } ], "excerpts": [ { "id": "wentworth-first-steps-in-algebra-1894/x-f810e54325", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "1", "location": "Introduction", "latex": "The principal definitions are put at the beginning of the book for convenient reference. They are not to be committed to memory. It is a good plan to have definitions and explanations read aloud in the class, and to encourage pupils to make comments upon them, and ask questions about them.", "markdown": "The principal definitions are put at the beginning of the book for convenient reference. They are not to be committed to memory. It is a good plan to have definitions and explanations read aloud in the class, and to encourage pupils to make comments upon them, and ask questions about them.", "why": "It tells the teacher how to use the definitions in class: read aloud, discuss, and question them rather than memorise them.", "use": [ "lesson" ], "concepts": [] }, { "id": "wentworth-first-steps-in-algebra-1894/x-55f57fabd2", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "1", "location": "Introduction", "latex": "In counting separate objects or in measuring magnitudes, the \\emph{standards} by which we count or measure are called \\Defn{units}.", "markdown": "In counting separate objects or in measuring magnitudes, the *standards* by which we count or measure are called **units**.", "why": "It gives learners the root idea of a unit as a standard against which counts and measures are made.", "use": [ "lesson" ], "concepts": [ "unit/unit" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-cb99abf9e0", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "4", "location": "Introduction", "latex": "The expression~$abc$ must not be confounded with $a + b + c$. $abc$~is a product; $a + b + c$ is a sum.", "markdown": "The expression $abc$ must not be confounded with $a + b + c$. $abc$ is a product; $a + b + c$ is a sum.", "why": "It warns learners that juxtaposed letters mean a product and not a sum, the most common early confusion.", "use": [ "lesson" ], "concepts": [ "concept/implied-multiplication", "method/addition" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-94cf2c7369", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "4", "location": "Introduction", "latex": "When a sign of operation is omitted in the notation of Arithmetic, it is always the \\emph{sign of addition}; but when a sign of operation is omitted in the notation of Algebra, it is always the \\emph{sign of multiplication}.", "markdown": "When a sign of operation is omitted in the notation of Arithmetic, it is always the *sign of addition*; but when a sign of operation is omitted in the notation of Algebra, it is always the *sign of multiplication*.", "why": "It contrasts the meaning of an omitted sign in arithmetic and in algebra, which learners often carry over wrongly.", "use": [ "lesson" ], "concepts": [ "concept/implied-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-273424057d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "5", "location": "Introduction", "latex": "The third power of a number is generally called the \\emph{cube} of that number; thus, $a^{3}$~is called the \\emph{cube} of~$a$, because if $a$~denotes the number of units of length in the edge of a cube, $a^{3}$~denotes the number of units of volume in the cube.", "markdown": "The third power of a number is generally called the *cube* of that number; thus, $a^{3}$ is called the *cube* of $a$, because if $a$ denotes the number of units of length in the edge of a cube, $a^{3}$ denotes the number of units of volume in the cube.", "why": "It links the third power to the volume of a cube, giving learners a picture of why the name is used.", "use": [ "history", "lesson" ], "concepts": [ "concept/cube", "concept/power" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-bd47485243", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "8", "location": "Introduction", "latex": "If a man has $10$~dollars and afterwards collects $3$~dollars and then $2$~dollars, it makes no difference whether he adds the $3$~dollars to his $10$~dollars, and then the $2$~dollars, or puts the $3$~and~$2$ dollars together and adds their sum to his $10$~dollars.", "markdown": "If a man has $10$ dollars and afterwards collects $3$ dollars and then $2$ dollars, it makes no difference whether he adds the $3$ dollars to his $10$ dollars, and then the $2$ dollars, or puts the $3$ and $2$ dollars together and adds their sum to his $10$ dollars.", "why": "It makes the rule for a parenthesis preceded by plus concrete through a money story before the symbols appear.", "use": [ "lesson" ], "concepts": [ "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-60a21c22c3", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "9", "location": "Introduction", "latex": "If a man has $10$~dollars consisting of $2$~five-dollar bills, and has a debt of $3$~dollars to pay, he can pay his debt by giving a five-dollar bill and receiving $2$~dollars.", "markdown": "If a man has $10$ dollars consisting of $2$ five-dollar bills, and has a debt of $3$ dollars to pay, he can pay his debt by giving a five-dollar bill and receiving $2$ dollars.", "why": "It explains why a minus sign in front of a bracket flips the signs inside, using change from a five-dollar bill.", "use": [ "lesson" ], "concepts": [ "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-9ce5e9ece4", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Introduction", "latex": "In finding the value of a compound expression the operations indicated \\emph{for each term} must be performed \\emph{before} the operation indicated by the sign prefixed to the term.", "markdown": "In finding the value of a compound expression the operations indicated *for each term* must be performed *before* the operation indicated by the sign prefixed to the term.", "why": "It gives the rule that prevents the most common error in evaluating an expression: working left to right across the whole thing.", "use": [ "lesson" ], "concepts": [ "concept/numerical-value", "method/order-of-operations" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-ffde3bb7bf", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "19", "location": "Simple Equations", "latex": "An equation is a statement in symbols that two expressions stand for the same number.", "markdown": "An equation is a statement in symbols that two expressions stand for the same number.", "why": "It gives the learner the plain meaning of an equation before any solving begins.", "use": [ "lesson" ], "concepts": [ "concept/equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-2304bd5252", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "19", "location": "Simple Equations", "latex": "Thus, $a + b = b + a$, which is true for \\emph{all values} of $a$~and~$b$, is an \\emph{identical equation}, and $3x + 2 = 8$, which is true only when $x$~stands for~$2$, is an \\emph{equation of condition}\\Add{.}", "markdown": "Thus, $a + b = b + a$, which is true for *all values* of $a$ and $b$, is an *identical equation*, and $3x + 2 = 8$, which is true only when $x$ stands for $2$, is an *equation of condition*", "why": "Two contrasting examples side by side show the learner how an identity differs from a conditional equation.", "use": [ "lesson" ], "concepts": [ "concept/equation", "concept/identity" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-7645706f2d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "22", "location": "Simple Equations", "latex": "Any term may be transposed from one side of an equation to the other, provided its sign is changed.", "markdown": "Any term may be transposed from one side of an equation to the other, provided its sign is changed.", "why": "It states the working rule for moving terms, which the learner will apply throughout the exercises.", "use": [ "lesson" ], "concepts": [ "method/transposing-the-terms", "theorem/transposition-rule" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-073f5c6e2e", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "22", "location": "Simple Equations", "latex": "When the root is substituted for its symbol in the given equation, and the equation reduces to an \\emph{identity}, the root is said to be \\Defn{verified}.", "markdown": "When the root is substituted for its symbol in the given equation, and the equation reduces to an *identity*, the root is said to be **verified**.", "why": "It shows the learner how to check an answer by substitution rather than trusting the arithmetic.", "use": [ "lesson" ], "concepts": [ "concept/root", "method/verification" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-582c9d5876", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "25", "location": "Simple Equations", "latex": "Remember that $x$~must not be put for money, length, time, weight,~etc., but for the \\textbf{required number} of \\emph{specified units} of money, length, time, weight,~etc.", "markdown": "Remember that $x$ must not be put for money, length, time, weight, etc., but for the **required number** of *specified units* of money, length, time, weight, etc.", "why": "It warns the beginner against the common error of letting the unknown stand for a quantity instead of a number.", "use": [ "lesson" ], "concepts": [ "concept/unknown", "method/stating-a-problem-as-an-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-2e0b7b6eda", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "26", "location": "Simple Equations", "latex": "Now, we know \\emph{what} James had. He had oranges, and we are to discover simply the \\emph{number} of oranges he had.", "markdown": "Now, we know *what* James had. He had oranges, and we are to discover simply the *number* of oranges he had.", "why": "It explains in plain terms why a problem's unknown must be a number, which is the main pitfall of word problems.", "use": [ "lesson" ], "concepts": [ "concept/unknown", "method/stating-a-problem-as-an-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-2e84ad8721", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "34", "location": "Positive and Negative Numbers", "latex": "If, now, we wish to subtract $7$~from~$4$, we begin at~$4$ in the positive series, count $7$~units in the \\emph{negative direction} (to the left), and arrive at~$-3$ in the negative series; that is, $4 - 7 = -3$.", "markdown": "If, now, we wish to subtract $7$ from $4$, we begin at $4$ in the positive series, count $7$ units in the *negative direction* (to the left), and arrive at $-3$ in the negative series; that is, $4 - 7 = -3$.", "why": "It shows a learner, with a concrete move along the line, why a result can fall below zero into the negative series.", "use": [ "lesson", "website" ], "concepts": [ "concept/negative-number", "concept/number-line", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-26fc7e8d79", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "33", "location": "Positive and Negative Numbers", "latex": "If we wish to subtract $5$~from~$2$, we cannot do it, because when we have counted backwards from~$2$ as far as~$0$, \\emph{the natural series of numbers comes to an end}.", "markdown": "If we wish to subtract $5$ from $2$, we cannot do it, because when we have counted backwards from $2$ as far as $0$, *the natural series of numbers comes to an end*.", "why": "Its plain statement of the natural series ending at zero motivates the need for negative numbers, and it reads well aloud.", "use": [ "history", "lesson" ], "concepts": [ "concept/negative-number", "concept/number-line" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-b4bec6cd19", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "34", "location": "Positive and Negative Numbers", "latex": "In order to subtract a greater number from a smaller, it is necessary to \\emph{assume} a new series of numbers, beginning at zero and extending to the left of zero.", "markdown": "In order to subtract a greater number from a smaller, it is necessary to *assume* a new series of numbers, beginning at zero and extending to the left of zero.", "why": "It presents the negative numbers as an assumption made for a purpose, which gives learners a sense of how the idea was introduced.", "use": [ "history" ], "concepts": [ "concept/negative-number", "concept/number-line" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-d48699ca33", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "35", "location": "Positive and Negative Numbers", "latex": "When no sign stands before a number, the sign~$+$ is always understood. Thus $4$~means the same as~$+4$, $a$~means the same as~$+a$. But \\emph{the sign~$-$ is never omitted}.", "markdown": "When no sign stands before a number, the sign $+$ is always understood. Thus $4$ means the same as $+4$, $a$ means the same as $+a$. But *the sign $-$ is never omitted*.", "why": "It warns learners that a missing plus is implied while a minus must always be written, a common source of error.", "use": [ "lesson" ], "concepts": [ "concept/positive-number", "concept/sign" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-11cfed4918", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "35", "location": "Positive and Negative Numbers", "latex": "In the first sense they are signs of \\emph{operations}, and are common to Arithmetic and Algebra; in the second sense they are signs of \\emph{opposition}, and are employed in Algebra alone.", "markdown": "In the first sense they are signs of *operations*, and are common to Arithmetic and Algebra; in the second sense they are signs of *opposition*, and are employed in Algebra alone.", "why": "It separates the two meanings of the plus and minus signs, so a learner knows when a sign names an operation and when it names a direction.", "use": [ "lesson" ], "concepts": [ "concept/sign", "method/addition", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-fee9735e18", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "41", "location": "Positive and Negative Numbers", "latex": "From these four cases it follows that in finding the product of two algebraic numbers, if the two numbers have \\emph{like} signs, the product will have the \\emph{plus} sign, and if \\emph{unlike} signs, the product will have the \\emph{minus} sign.", "markdown": "From these four cases it follows that in finding the product of two algebraic numbers, if the two numbers have *like* signs, the product will have the *plus* sign, and if *unlike* signs, the product will have the *minus* sign.", "why": "It states the law of signs in multiplication as a conclusion drawn from four worked cases, so learners see where the rule comes from.", "use": [ "lesson" ], "concepts": [ "method/multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-a8557f7062", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "46", "location": "Addition and Subtraction", "latex": "If an algebraic expression contains only \\emph{integral forms}, that is, contains \\emph{no letter in the denominator of any of its terms}, it is called an \\Defn{integral expression}.", "markdown": "If an algebraic expression contains only *integral forms*, that is, contains *no letter in the denominator of any of its terms*, it is called an **integral expression**.", "why": "It gives the learner a precise test for what counts as an integral expression, based on form alone.", "use": [ "lesson" ], "concepts": [ "concept/integral-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-a174980315", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "46", "location": "Addition and Subtraction", "latex": "Integral and fractional expressions are so named on account of the \\emph{form of the expressions}, and with no reference whatever to the \\emph{numerical value} of the expressions when definite numbers are put in place of the letters.", "markdown": "Integral and fractional expressions are so named on account of the *form of the expressions*, and with no reference whatever to the *numerical value* of the expressions when definite numbers are put in place of the letters.", "why": "It warns learners not to confuse the form of an expression with the number it takes when letters are given values.", "use": [ "lesson" ], "concepts": [ "concept/integral-expression", "concept/numerical-value" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-bc46551e90", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "47", "location": "Addition and Subtraction", "latex": "The coefficient of~$x^{2}$ in the result will be $6 + 1$, or~$7$; the coefficient of~$x$ will be $-4 + 5$, or~$1$; and the last term is $-5 + 4$, or~$-1$.", "markdown": "The coefficient of $x^{2}$ in the result will be $6 + 1$, or $7$; the coefficient of $x$ will be $-4 + 5$, or $1$; and the last term is $-5 + 4$, or $-1$.", "why": "It shows step by step how each set of like terms is collected into one term with a summed coefficient.", "use": [ "lesson" ], "concepts": [ "concept/coefficient", "concept/like-terms", "method/collecting-like-terms" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-06d26340e8", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "47", "location": "Addition and Subtraction", "latex": "When the coefficient of a term is~$1$, it is not written, but understood.", "markdown": "When the coefficient of a term is $1$, it is not written, but understood.", "why": "It tells learners that a coefficient of 1 is left out of the written term, which often causes confusion when collecting.", "use": [ "lesson" ], "concepts": [ "concept/coefficient" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-8fbce39e17", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "49", "location": "Addition and Subtraction", "latex": "This process is more easily performed by writing the subtrahend below the minuend, \\emph{mentally} changing the sign of each term in the subtrahend, and adding.", "markdown": "This process is more easily performed by writing the subtrahend below the minuend, *mentally* changing the sign of each term in the subtrahend, and adding.", "why": "It gives a compact column method for subtraction that turns the problem into an addition with changed signs.", "use": [ "lesson", "website" ], "concepts": [ "concept/minuend", "concept/subtrahend", "method/arranging-in-columns", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-d028ec758c", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "A parenthesis preceded by the sign~$-$ may be removed, \\emph{provided the sign of every term within the parenthesis is changed}, namely, $+$~to~$-$, and $-$~to~$+$; and any number of terms may be enclosed within a parenthesis preceded by the sign~$-$, \\emph{provided the sign of every term enclosed is changed}.", "markdown": "A parenthesis preceded by the sign $-$ may be removed, *provided the sign of every term within the parenthesis is changed*, namely, $+$ to $-$, and $-$ to $+$; and any number of terms may be enclosed within a parenthesis preceded by the sign $-$, *provided the sign of every term enclosed is changed*.", "why": "It states the sign-changing rule for removing a parenthesis that follows a minus sign, which is the step learners most often get wrong.", "use": [ "lesson" ], "concepts": [ "concept/parenthesis", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-1038346704", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "It is best to remove each parenthesis in succession, \\emph{beginning with the innermost}.", "markdown": "It is best to remove each parenthesis in succession, *beginning with the innermost*.", "why": "It gives learners a clear order of work for nested brackets, so each sign is applied in turn.", "use": [ "lesson" ], "concepts": [ "concept/parenthesis" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-7790d96b4a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Multiplication and Division", "latex": "\\Dictum{To multiply a polynomial by a monomial}, therefore, \\begin{Theorem} Multiply each term of the polynomial by the monomial, and add the partial products.", "markdown": "**To multiply a polynomial by a monomial**, therefore, Theorem Multiply each term of the polynomial by the monomial, and add the partial products.", "why": "States the rule for multiplying a polynomial by a monomial in one sentence a learner can apply directly.", "use": [ "lesson" ], "concepts": [ "concept/monomial", "concept/partial-product", "concept/polynomial", "method/multiplying-a-compound-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-0c005b237b", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "59", "location": "Multiplication and Division", "latex": "Arrange both the dividend and divisor in ascending or descending powers of some common letter.", "markdown": "Arrange both the dividend and divisor in ascending or descending powers of some common letter.", "why": "Gives the first step of polynomial long division, which every later step depends on.", "use": [ "lesson" ], "concepts": [ "concept/arrangement", "concept/dividend", "concept/divisor", "method/dividing-a-polynomial-by-a-polynomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-ef422e8f06", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "60", "location": "Multiplication and Division", "latex": "It is of fundamental importance to arrange the dividend and divisor \\emph{in the same order} with respect to a common letter, and \\emph{to keep this order throughout the operation}.", "markdown": "It is of fundamental importance to arrange the dividend and divisor *in the same order* with respect to a common letter, and *to keep this order throughout the operation*.", "why": "Names the most common source of error in long division, so the learner knows what to watch for.", "use": [ "lesson", "website" ], "concepts": [ "concept/arrangement", "method/dividing-a-polynomial-by-a-polynomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-6bc3cb3111", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Multiplication and Division", "latex": "The pupil should observe that, with a view to bringing like terms of the partial products in columns, the terms of the multiplicand and multiplier are arranged in the \\emph{same order}.", "markdown": "The pupil should observe that, with a view to bringing like terms of the partial products in columns, the terms of the multiplicand and multiplier are arranged in the *same order*.", "why": "Explains why both factors are ordered the same way, so like terms line up in columns.", "use": [ "lesson" ], "concepts": [ "concept/arrangement", "concept/like-terms", "method/multiplying-a-compound-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-090911c9f7", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "60", "location": "Multiplication and Division", "latex": "The pupil will notice that by this process we have in effect separated the dividend into two parts, $x^{2} + 7x$ and $11x + 77$, and divided each part by $x + 7$, and that the complete quotient is the sum of the partial quotients $x$~and~$11$.", "markdown": "The pupil will notice that by this process we have in effect separated the dividend into two parts, $x^{2} + 7x$ and $11x + 77$, and divided each part by $x + 7$, and that the complete quotient is the sum of the partial quotients $x$ and $11$.", "why": "Shows, with numbers, why the long-division steps work: each part of the dividend is divided separately and the quotients are added.", "use": [ "lesson" ], "concepts": [ "concept/dividend", "concept/quotient", "method/dividing-a-polynomial-by-a-polynomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-5e911c2a47", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "59", "location": "Multiplication and Division", "latex": "The first term of the dividend is~$an$; that is, the product of~$a$, the first term of the divisor, by~$n$, the first term of the quotient.", "markdown": "The first term of the dividend is $an$; that is, the product of $a$, the first term of the divisor, by $n$, the first term of the quotient.", "why": "Gives the reason the first quotient term is found by dividing the leading terms, which a learner can check for each step.", "use": [ "lesson" ], "concepts": [ "concept/divisor", "concept/quotient", "method/dividing-a-polynomial-by-a-polynomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-e3797b8e51", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "64", "location": "Multiplication and Division", "latex": "The square of the sum of two numbers is the sum of their squares plus twice their product.", "markdown": "The square of the sum of two numbers is the sum of their squares plus twice their product.", "why": "States the first special product in words a learner can check against the expansion just above it.", "use": [ "lesson" ], "concepts": [ "theorem/square-of-a-sum" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-5186a76e49", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "64", "location": "Multiplication and Division", "latex": "The square of the difference of two numbers is the sum of their squares minus twice their product.", "markdown": "The square of the difference of two numbers is the sum of their squares minus twice their product.", "why": "Shows that changing the sign inside the bracket changes only the sign of the middle term, which is the point to watch when squaring a difference.", "use": [ "lesson" ], "concepts": [ "theorem/square-of-a-difference" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-bca6da8048", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "66", "location": "Multiplication and Division", "latex": "The middle term of each result has for a coefficient the \\emph{algebraic sum} of the second terms of the binomials.", "markdown": "The middle term of each result has for a coefficient the *algebraic sum* of the second terms of the binomials.", "why": "Gives the pattern a learner needs to write the middle term of a product of two binomials without expanding it.", "use": [ "lesson" ], "concepts": [ "method/multiplying-two-binomials" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-cad1955f3c", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "66", "location": "Multiplication and Division", "latex": "The intermediate step given above may be omitted, and the products written at once by \\emph{inspection}.", "markdown": "The intermediate step given above may be omitted, and the products written at once by *inspection*.", "why": "Tells the learner that the pattern lets them skip the working and write the answer directly.", "use": [ "lesson", "website" ], "concepts": [ "method/multiplying-two-binomials", "method/writing-by-inspection" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-803c638cbd", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "67", "location": "Multiplication and Division", "latex": "(-8) + (-7) = -15,\\quad (-8)(-7) = +56.", "markdown": "(-8) + (-7) = -15, (-8)(-7) = +56.", "why": "Shows the sign rule at work: two negative second terms give a negative middle term and a positive last term.", "use": [ "lesson" ], "concepts": [ "method/multiplying-two-binomials" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-22060ce047", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "68", "location": "Multiplication and Division", "latex": "The difference of the squares of two numbers is divisible by the sum, and by the difference, of the numbers.", "markdown": "The difference of the squares of two numbers is divisible by the sum, and by the difference, of the numbers.", "why": "Links the product rule to division, so a learner sees why a^2 - b^2 divides evenly by a + b and by a - b.", "use": [ "lesson", "history" ], "concepts": [ "concept/divisibility", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-7b54d1c573", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "71", "location": "Factors", "latex": "The factors of a monomial may be found by inspection. Thus, the factors of~$21a^{2}b$ are $3$, $7$, $a$, $a$, and~$b$.", "markdown": "The factors of a monomial may be found by inspection. Thus, the factors of $21a^{2}b$ are $3$, $7$, $a$, $a$, and $b$.", "why": "It shows learners that a monomial's factors can be read straight off its form before any procedure is needed.", "use": [ "lesson" ], "concepts": [ "concept/factor", "concept/monomial", "method/writing-by-inspection" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-828d6e999f", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "74", "location": "Factors", "latex": "Take the square root of the first term and the square root of the second term. The sum of these roots will form the first factor; The difference of these roots will form the second factor.", "markdown": "Take the square root of the first term and the square root of the second term. The sum of these roots will form the first factor; The difference of these roots will form the second factor.", "why": "It gives the rule for factoring a difference of two squares in words a learner can apply to any binomial of that form.", "use": [ "lesson", "website" ], "concepts": [ "concept/root", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-1253379b54", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Factors", "latex": "The first two terms of $ac + ad + bc + bd$ are seen to have the common factor~$a$, and the last two terms, the common factor~$b$. Hence we bracket the first two terms and also the last two terms.", "markdown": "The first two terms of $ac + ad + bc + bd$ are seen to have the common factor $a$, and the last two terms, the common factor $b$. Hence we bracket the first two terms and also the last two terms.", "why": "It explains the idea behind factoring by grouping: bracket the terms so that each pair reveals a common factor.", "use": [ "lesson" ], "concepts": [ "method/factoring-by-grouping", "method/factoring-by-taking-out-a-common-factor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-5e9b8437ff", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "73", "location": "Factors", "latex": "Here the last two terms, $-bc - bd$, being put within a parenthesis preceded by the sign~$-$, have their signs changed to~$+$.", "markdown": "Here the last two terms, $-bc - bd$, being put within a parenthesis preceded by the sign $-$, have their signs changed to $+$.", "why": "It warns the learner about the sign change that happens when terms are placed inside a parenthesis preceded by a minus.", "use": [ "lesson" ], "concepts": [ "method/factoring-by-grouping", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-55a6255f82", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "81", "location": "Factors", "latex": "Since the product is~$+12$, the two numbers are \\emph{both positive} or \\emph{both negative}, and since their sum is~$-7$, they must both be negative.", "markdown": "Since the product is $+12$, the two numbers are *both positive* or *both negative*, and since their sum is $-7$, they must both be negative.", "why": "It models the sign reasoning used to find two numbers with a given product and sum, which is the core of factoring a trinomial of the form x^2+ax+b.", "use": [ "lesson" ], "concepts": [ "concept/algebraic-sum", "method/factoring-a-trinomial-of-the-form-x-2-ax-b" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-e44972fef4", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "76", "location": "Factors", "latex": "In like manner we can resolve into factors any expression which can be written as the difference of two cubes.", "markdown": "In like manner we can resolve into factors any expression which can be written as the difference of two cubes.", "why": "It tells learners that the cube formula applies to any expression whose form is a difference of two cubes, not only to plain numbers.", "use": [ "history", "lesson" ], "concepts": [ "theorem/difference-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-b53aea0c98", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "85", "location": "Common Factors and Multiples", "latex": "Resolve each expression into its simplest factors. Find the product of all the common factors, taking each factor the least number of times it occurs in any of the given expressions.", "markdown": "Resolve each expression into its simplest factors. Find the product of all the common factors, taking each factor the least number of times it occurs in any of the given expressions.", "why": "It gives the complete procedure for the highest common factor of expressions in two sentences a learner can follow step by step.", "use": [ "lesson" ], "concepts": [ "concept/highest-common-factor", "method/factoring", "method/finding-the-highest-common-factor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-88e592b989", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "85", "location": "Common Factors and Multiples", "latex": "We cannot apply the terms \\emph{greatest} and \\emph{least} to an algebraic expression in which particular values have not been given to the letters contained in the expression. Thus $a$~is \\emph{greater} than~$a^{2}$, if $a$~stands for~$\\frac{1}{4}$.", "markdown": "We cannot apply the terms *greatest* and *least* to an algebraic expression in which particular values have not been given to the letters contained in the expression. Thus $a$ is *greater* than $a^{2}$, if $a$ stands for $\\frac{1}{4}$.", "why": "It warns learners that 'greatest' only makes sense once the letters take definite values, which prevents a common misreading of the highest common factor.", "use": [ "lesson" ], "concepts": [ "concept/highest-common-factor", "concept/lowest-common-multiple" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-162a035eff", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "87", "location": "Common Factors and Multiples", "latex": "The \\LCM\\ must evidently contain each factor the greatest number of times that it occurs in any expression.", "markdown": "The L. C. M. must evidently contain each factor the greatest number of times that it occurs in any expression.", "why": "It explains the reasoning behind the lowest common multiple rule, so learners can see why each factor is taken the greatest number of times.", "use": [ "lesson" ], "concepts": [ "concept/lowest-common-multiple", "method/finding-the-lowest-common-multiple" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-4b677a7bee", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "85", "location": "Common Factors and Multiples", "latex": "The \\emph{highest common factor} in Algebra corresponds to the \\emph{greatest common measure}, or \\emph{greatest common divisor} in Arithmetic.", "markdown": "The *highest common factor* in Algebra corresponds to the *greatest common measure*, or *greatest common divisor* in Arithmetic.", "why": "It links the algebraic idea to the arithmetic greatest common divisor learners already know, which is a useful bridge for a lesson.", "use": [ "history", "lesson" ], "concepts": [ "concept/common-factor", "concept/highest-common-factor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-0b96e8bb4a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "84", "location": "Common Factors and Multiples", "latex": "The \\Defn{highest common factor} of two or more integral and rational \\emph{expressions} is an integral and rational expression of highest degree that will divide each of them without a remainder.", "markdown": "The **highest common factor** of two or more integral and rational *expressions* is an integral and rational expression of highest degree that will divide each of them without a remainder.", "why": "It defines the highest common factor for algebraic expressions by degree, extending the number case that learners already understand.", "use": [ "lesson", "website" ], "concepts": [ "concept/degree", "concept/highest-common-factor", "concept/integral-expression", "concept/rational-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-aa3a64158f", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "89", "location": "Fractions", "latex": "The introduction of the same factor into the dividend and divisor does not alter the value of the quotient, and the rejection of the same factor from the dividend and divisor does not alter the value of the quotient.", "markdown": "The introduction of the same factor into the dividend and divisor does not alter the value of the quotient, and the rejection of the same factor from the dividend and divisor does not alter the value of the quotient.", "why": "It states the principle that makes every cancellation and reduction in the chapter legitimate.", "use": [ "lesson" ], "concepts": [ "method/reducing-a-fraction-to-lowest-terms", "theorem/fundamental-principle-of-fractions" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-865d75c450", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Fractions", "latex": "The dividing line between the terms of a fraction has the force of a vinculum affecting the numerator.", "markdown": "The dividing line between the terms of a fraction has the force of a vinculum affecting the numerator.", "why": "It explains why a minus sign in front of a fraction bar must reach every term of the numerator, a common source of sign errors.", "use": [ "lesson", "website" ], "concepts": [ "concept/numerator", "method/reducing-a-mixed-expression-to-a-fraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-0e0d7ec149", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Fractions", "latex": "If, therefore, a \\emph{minus sign} precedes the dividing line, as in the preceding Example, and this line is removed, the numerator of the given fraction must be enclosed in a parenthesis preceded by the minus sign, or the sign of every term of the numerator must be changed.", "markdown": "If, therefore, a *minus sign* precedes the dividing line, as in the preceding Example, and this line is removed, the numerator of the given fraction must be enclosed in a parenthesis preceded by the minus sign, or the sign of every term of the numerator must be changed.", "why": "It gives the exact rule for removing a fraction bar that has a minus sign in front, which learners often get wrong.", "use": [ "lesson" ], "concepts": [ "concept/numerator", "method/reducing-a-mixed-expression-to-a-fraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-73b11d75ed", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Fractions", "latex": "By division, $\\dfrac{x^{3} - 1}{x + 1} = x^{2} - x + 1 - \\dfrac{2}{x + 1}$.", "markdown": "By division, $\\dfrac{x^{3} - 1}{x + 1} = x^{2} - x + 1 - \\dfrac{2}{x + 1}$.", "why": "This worked example shows a non-exact division ending in a remainder fraction, which makes the integral-or-mixed idea concrete.", "use": [ "lesson" ], "concepts": [ "method/division", "method/reducing-a-fraction-to-an-integral-or-mixed-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-84d25a0a14", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "98", "location": "Fractions", "latex": "The reciprocal of a fraction, therefore, is the fraction inverted.", "markdown": "The reciprocal of a fraction, therefore, is the fraction inverted.", "why": "One short sentence gives the rule that turns division by a fraction into multiplication.", "use": [ "lesson" ], "concepts": [ "concept/reciprocal", "method/dividing-by-a-fraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-5fd066f232", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "98", "location": "Fractions", "latex": "Every mixed expression should first be reduced to a fraction, and every integral expression should be written as a fraction having $1$~for the denominator.", "markdown": "Every mixed expression should first be reduced to a fraction, and every integral expression should be written as a fraction having $1$ for the denominator.", "why": "It tells learners how to put whole expressions into fraction form before multiplying or dividing, so the cancelling step works.", "use": [ "lesson" ], "concepts": [ "concept/algebraic-fraction", "method/dividing-by-a-fraction", "method/multiplying-fractions" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-1a53d6ab7a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "103", "location": "Fractional Equations", "latex": "If a fraction is preceded by a \\textbf{minus sign}, \\emph{the sign of every term of the numerator must be changed when the denominator is removed}.", "markdown": "If a fraction is preceded by a **minus sign**, *the sign of every term of the numerator must be changed when the denominator is removed*.", "why": "It warns the learner that a minus sign before a fraction changes every term of its numerator, the most common error when clearing fractions.", "use": [ "lesson" ], "concepts": [ "method/clearing-an-equation-of-fractions", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-31b78d1183", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "103", "location": "Fractional Equations", "latex": "Multiply each term by the \\LCM\\ of the denominators.", "markdown": "Multiply each term by the L. C. M. of the denominators.", "why": "It states the one-line procedure for clearing an equation of fractions, which the learner applies to every problem in the chapter.", "use": [ "lesson" ], "concepts": [ "concept/lowest-common-multiple", "method/clearing-an-equation-of-fractions" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-536d994939", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "107", "location": "Fractional Equations", "latex": "If the denominators contain both simple and compound expressions, it is generally best to remove the simple expressions first, and then the compound expressions. After each multiplication the result should be reduced to the simplest form.", "markdown": "If the denominators contain both simple and compound expressions, it is generally best to remove the simple expressions first, and then the compound expressions. After each multiplication the result should be reduced to the simplest form.", "why": "It gives a practical order of work for denominators of mixed difficulty, so the learner avoids needless algebra.", "use": [ "lesson" ], "concepts": [ "concept/lowest-common-denominator", "method/clearing-an-equation-of-fractions" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-3b627f7889", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "108", "location": "Fractional Equations", "latex": "Literal equations are equations in which some or all of the known numbers are represented by letters; the numbers regarded as known numbers are usually represented by the \\emph{first} letters of the alphabet.", "markdown": "Literal equations are equations in which some or all of the known numbers are represented by letters; the numbers regarded as known numbers are usually represented by the *first* letters of the alphabet.", "why": "It explains the convention that makes an equation with letters for numbers readable as an equation to be solved.", "use": [ "lesson", "history" ], "concepts": [ "concept/literal-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-6892527ec1", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Fractional Equations", "latex": "By working \\emph{$2$~days each} they build $\\frac{1}{12} + \\frac{1}{15} + \\frac{1}{20}$ of it.", "markdown": "By working *$2$ days each* they build $\\frac{1}{12} + \\frac{1}{15} + \\frac{1}{20}$ of it.", "why": "It shows the learner that the parts done by each worker in one unit of time simply add, which is the idea behind the combined-work equations.", "use": [ "lesson" ], "concepts": [ "concept/combined-rate" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-04362a925d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "118", "location": "Fractional Equations", "latex": "Such an expression is called a \\Defn{formula}, and the translation of this formula into words is called a \\Defn{rule}.", "markdown": "Such an expression is called a **formula**, and the translation of this formula into words is called a **rule**.", "why": "It separates a formula, a general algebraic result, from a rule, its statement in words, which the learner needs to read Section 143 onward.", "use": [ "lesson", "history" ], "concepts": [ "concept/formula", "concept/rule" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-a9e5a04d3b", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "115", "location": "Fractional Equations", "latex": "A hare takes $4$~leaps to a greyhound's~$3$; but $2$~of the greyhound's leaps are equivalent to $3$~of the hare's.", "markdown": "A hare takes $4$ leaps to a greyhound’s $3$; but $2$ of the greyhound’s leaps are equivalent to $3$ of the hare’s.", "why": "Its old-fashioned leaping race makes a memorable setting for turning a comparison of rates into an equation.", "use": [ "website", "history" ], "concepts": [ "method/stating-a-problem-as-an-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-659de2c6fc", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "Independent equations involving the \\emph{same} unknown numbers are called \\Defn{simultaneous equations}.", "markdown": "Independent equations involving the *same* unknown numbers are called **simultaneous equations**.", "why": "It gives the defining sentence for simultaneous equations, which the rest of the chapter depends on.", "use": [ "lesson" ], "concepts": [ "concept/independent-equations", "concept/simultaneous-equations" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-9e99d9ed7d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "If we have two unknown numbers and but one relation between them, we can find an unlimited number of pairs of values for which the given relation will hold true.", "markdown": "If we have two unknown numbers and but one relation between them, we can find an unlimited number of pairs of values for which the given relation will hold true.", "why": "It shows a learner why one equation in two unknowns cannot settle the values and why a second is needed.", "use": [ "lesson" ], "concepts": [ "concept/simultaneous-equations", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-8d45059f42", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "If we have two unknown numbers, and two independent equations involving them, there is but \\emph{one} pair of values which will hold true for both equations.", "markdown": "If we have two unknown numbers, and two independent equations involving them, there is but *one* pair of values which will hold true for both equations.", "why": "It states the central idea that two independent equations determine a single solution pair.", "use": [ "lesson", "website" ], "concepts": [ "concept/independent-equations", "concept/simultaneous-equations" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-3a98fc05f3", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Simultaneous Equations of the First Degree", "latex": "Multiply the equations by such numbers as will make the coefficients of one of the unknown numbers equal in the resulting equations.", "markdown": "Multiply the equations by such numbers as will make the coefficients of one of the unknown numbers equal in the resulting equations.", "why": "It gives the first step of elimination by addition or subtraction as a clear rule.", "use": [ "lesson" ], "concepts": [ "concept/coefficient", "method/elimination" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-331d88883d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Simultaneous Equations of the First Degree", "latex": "It is generally best to select the letter to be eliminated which requires the smallest multipliers to make its coefficients equal; and the smallest multiplier for each equation is found by dividing the \\LCM\\ of the coefficients of this letter by the given coefficient in that equation.", "markdown": "It is generally best to select the letter to be eliminated which requires the smallest multipliers to make its coefficients equal; and the smallest multiplier for each equation is found by dividing the L. C. M. of the coefficients of this letter by the given coefficient in that equation.", "why": "It explains how choosing the smallest multipliers with the LCM keeps the arithmetic short.", "use": [ "lesson" ], "concepts": [ "concept/coefficient", "method/elimination", "method/finding-the-lowest-common-multiple" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-8b2d75a1be", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Simultaneous Equations of the First Degree", "latex": "The expression $64$ means $60 + 4$, that is, $10$~\\emph{times} $6 + 4$, and has for its \\emph{digits} $6$~and~$4$.", "markdown": "The expression $64$ means $60 + 4$, that is, $10$ *times* $6 + 4$, and has for its *digits* $6$ and $4$.", "why": "It introduces place value so a learner can write a two-digit number with unknown digits.", "use": [ "lesson", "history" ], "concepts": [ "concept/digit" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-7ffed2b24a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "130", "location": "Simultaneous Equations of the First Degree", "latex": "The interest for one year is $\\dfrac{y}{100}$ of the principal; that is, $\\dfrac{xy}{100}$.", "markdown": "The interest for one year is $\\dfrac{y}{100}$ of the principal; that is, $\\dfrac{xy}{100}$.", "why": "It shows how a percentage rate turns into a simple algebraic expression for interest.", "use": [ "lesson" ], "concepts": [ "quantity/interest", "quantity/principal", "quantity/rate-of-interest" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-03905975e9", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "126", "location": "Simultaneous Equations of the First Degree", "latex": "To find $x$ in problem~26, add the equations; to find~$y$, subtract one from the other. Do not clear of fractions.", "markdown": "To find $x$ in problem 26, add the equations; to find $y$, subtract one from the other. Do not clear of fractions.", "why": "It warns that some systems are easier to eliminate directly than to clear of fractions first.", "use": [ "lesson" ], "concepts": [ "method/clearing-an-equation-of-fractions", "method/elimination" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-0811c5cd9f", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "132", "location": "Quadratic Equations", "latex": "An equation which contains the \\emph{square} of the unknown number, but no higher power, is called a \\Defn{quadratic equation}.", "markdown": "An equation which contains the *square* of the unknown number, but no higher power, is called a **quadratic equation**.", "why": "It gives the learner the exact boundary of what counts as a quadratic before any solving begins.", "use": [ "lesson" ], "concepts": [ "concept/quadratic-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-d81f756db5", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "133", "location": "Quadratic Equations", "latex": "The square root of any number is positive or negative. Hitherto we have given only the positive value. In this chapter we shall give both values.", "markdown": "The square root of any number is positive or negative. Hitherto we have given only the positive value. In this chapter we shall give both values.", "why": "It explains why a quadratic has two roots and why the positive-only habit from earlier chapters is being dropped.", "use": [ "lesson", "history" ], "concepts": [ "concept/root", "method/extracting-the-square-root" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-bcc79a35fd", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "135", "location": "Quadratic Equations", "latex": "This third term is the square of half the coefficient of~$x$.", "markdown": "This third term is the square of half the coefficient of $x$.", "why": "It gives the rule for the missing constant that makes the left side a perfect square.", "use": [ "lesson" ], "concepts": [ "method/completing-the-square" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-3e85a54fb8", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "136", "location": "Quadratic Equations", "latex": "Since the square root of a negative number cannot be taken, the coefficient of~$x^{2}$ must be changed to~$+$.", "markdown": "Since the square root of a negative number cannot be taken, the coefficient of $x^{2}$ must be changed to $+$.", "why": "It warns the learner to make the x^2 coefficient positive before completing the square, a step that is easy to skip.", "use": [ "lesson" ], "concepts": [ "method/completing-the-square", "method/extracting-the-square-root" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-eb37734410", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "133", "location": "Quadratic Equations", "latex": "The square root of~$-5$ differs from the square root of~$+5$ in that the latter can be found as accurately as we please, while the former cannot be found at all.", "markdown": "The square root of $-5$ differs from the square root of $+5$ in that the latter can be found as accurately as we please, while the former cannot be found at all.", "why": "It distinguishes a surd, which can be approximated, from an imaginary root, which cannot be found at all.", "use": [ "lesson" ], "concepts": [ "concept/imaginary-root", "concept/surd" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-2b42154605", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "138", "location": "Quadratic Equations", "latex": "The reason that every root of the equation will not always satisfy the conditions of the problem is that the problem may have certain restrictions, expressed or implied, that cannot be expressed in the equation.", "markdown": "The reason that every root of the equation will not always satisfy the conditions of the problem is that the problem may have certain restrictions, expressed or implied, that cannot be expressed in the equation.", "why": "It teaches learners to check each root against the wording of the problem rather than accept both automatically.", "use": [ "lesson", "website" ], "concepts": [ "concept/root", "method/stating-a-problem-as-an-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-a314bcd761", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "142", "location": "Arithmetical Progression", "latex": "A series of numbers is said to form an \\Defn{Arithmetical Progression} if the difference between any term and the preceding term is the same throughout the series.", "markdown": "A series of numbers is said to form an **Arithmetical Progression** if the difference between any term and the preceding term is the same throughout the series.", "why": "It gives the defining property of the progression in the book's own words, which a learner can test on any list of numbers.", "use": [ "lesson" ], "concepts": [ "concept/arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-72a47cca8d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "142", "location": "Arithmetical Progression", "latex": "the coefficient of~$d$ in each term being always less by~$1$ than the \\emph{number of the term}.", "markdown": "the coefficient of $d$ in each term being always less by $1$ than the *number of the term*.", "why": "It explains why the nth term has the factor (n - 1), a common place for learners to slip.", "use": [ "lesson" ], "concepts": [ "concept/common-difference", "theorem/nth-term-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-f5de054e82", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "145", "location": "Arithmetical Progression", "latex": "it is evident that the series beginning with the first term will be $a$, $a + d$, $a + 2d$,~etc., and beginning with the last term will be $l$, $l - d$, $l - 2d$,~etc.", "markdown": "it is evident that the series beginning with the first term will be $a$, $a + d$, $a + 2d$, etc., and beginning with the last term will be $l$, $l - d$, $l - 2d$, etc.", "why": "It shows the pairing of the series with its reverse that makes the sum formula work, which is the idea behind the derivation.", "use": [ "lesson" ], "concepts": [ "theorem/sum-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-2bb33fd0be", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "143", "location": "Arithmetical Progression", "latex": "Hence, the arithmetical mean of any two numbers is found by taking half their sum.", "markdown": "Hence, the arithmetical mean of any two numbers is found by taking half their sum.", "why": "It states the arithmetical mean rule in one line a learner can apply directly.", "use": [ "lesson" ], "concepts": [ "concept/arithmetical-mean" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-be59a84bc4", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "145", "location": "Arithmetical Progression", "latex": "Putting for $l$~its value $a + (n - 1)d$, in formula~(4), we have", "markdown": "Putting for $l$ its value $a + (n - 1)d$, in formula (4), we have", "why": "It shows substitution of a known formula into another, the step that turns the sum into a formula in a, d and n.", "use": [ "lesson" ], "concepts": [ "theorem/nth-term-of-an-arithmetical-progression", "theorem/sum-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-2b8527e620", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "142", "location": "Arithmetical Progression", "latex": "If $d$~is positive, the progression is an \\emph{increasing} series; if $d$~is negative, the progression is a \\emph{decreasing} series.", "markdown": "If $d$ is positive, the progression is an *increasing* series; if $d$ is negative, the progression is a *decreasing* series.", "why": "It links the sign of the common difference to whether the progression rises or falls, a useful check for learners.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetical-progression", "concept/common-difference" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-82f185e255", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "147", "location": "Arithmetical Progression", "latex": "A heavy body falling from a height falls $16.1$~feet the first second, and in each succeeding second $32.2$~feet more than in the second next preceding.", "markdown": "A heavy body falling from a height falls $16.1$ feet the first second, and in each succeeding second $32.2$ feet more than in the second next preceding.", "why": "It presents a physical problem that is an arithmetical progression, showing the idea outside pure number work.", "use": [ "website" ], "concepts": [ "concept/arithmetical-progression", "theorem/sum-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-5c26feff53", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "148", "location": "Geometrical Progression", "latex": "A series of numbers is said to be in \\Defn{Geometrical Progression} when the quotient of any term divided by the preceding term is the same throughout the series.", "markdown": "A series of numbers is said to be in **Geometrical Progression** when the quotient of any term divided by the preceding term is the same throughout the series.", "why": "It gives the defining idea of a geometrical progression as a constant quotient between successive terms, which is the foundation for every later formula.", "use": [ "lesson" ], "concepts": [ "concept/geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-27e69d9a62", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "148", "location": "Geometrical Progression", "latex": "Hence the $n$th~term will be~$ar^{n - 1}$.", "markdown": "Hence the $n$th term will be $ar^{n - 1}$.", "why": "It states the general term in one line, showing how the exponent on the ratio drops by one from the term's position.", "use": [ "lesson" ], "concepts": [ "theorem/nth-term-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-0e19f333c9", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "149", "location": "Geometrical Progression", "latex": "In formula~(1), put $5$~for~$n$, $3$~for~$a$, and $2$~for~$r$.", "markdown": "In formula (1), put $5$ for $n$, $3$ for $a$, and $2$ for $r$.", "why": "It shows a learner the substitution step that turns the general formula into a specific answer for a given term.", "use": [ "lesson" ], "concepts": [ "theorem/nth-term-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-e120fc3933", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "149", "location": "Geometrical Progression", "latex": "the geometrical mean of any two numbers is the square root of their product.", "markdown": "the geometrical mean of any two numbers is the square root of their product.", "why": "It links the geometrical mean to the square root, so learners see why the mean of two numbers involves a root.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometrical-mean" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-0fbd766b52", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "149", "location": "Geometrical Progression", "latex": "Therefore the series is $3$, $±6$, $12$, $±24$,~$\\dots$.", "markdown": "Therefore the series is $3$, $±6$, $12$, $±24$, $\\dots$.", "why": "It warns that taking a square root of the ratio gives two sign choices, so a learner must not forget the ± when solving for r.", "use": [ "lesson" ], "concepts": [ "concept/common-ratio", "concept/geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-ca64b2c4cc", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Geometrical Progression", "latex": "If a blacksmith uses seven nails in putting a shoe on a horse's foot, and receives $1$~cent for the first nail, $2$~cents for the second nail, and so on, what does he receive for putting on the shoe?", "markdown": "If a blacksmith uses seven nails in putting a shoe on a horse’s foot, and receives $1$ cent for the first nail, $2$ cents for the second nail, and so on, what does he receive for putting on the shoe?", "why": "It is a concrete doubling problem that shows why a geometrical progression sum models real accumulation, and it suits a website feature as a historical-style puzzle.", "use": [ "website", "history" ], "concepts": [ "concept/geometrical-progression", "theorem/sum-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-71c2354a6b", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "152", "location": "Square and Cube Roots", "latex": "Since the square of $a + b$ is $a^{2} + 2ab + b^{2}$, the square root of $a^{2} + 2ab + b^{2}$ is $a + b$. It is required to find a method of extracting the root $a + b$ when $a^{2} + 2ab + b^{2}$ is given.", "markdown": "Since the square of $a + b$ is $a^{2} + 2ab + b^{2}$, the square root of $a^{2} + 2ab + b^{2}$ is $a + b$. It is required to find a method of extracting the root $a + b$ when $a^{2} + 2ab + b^{2}$ is given.", "why": "It shows the learner that the square root of a perfect square trinomial is undone by the square of a binomial, which is the reason the method works.", "use": [ "lesson" ], "concepts": [ "concept/root", "method/extracting-the-square-root", "theorem/square-of-a-sum" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-45040d228f", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Square and Cube Roots", "latex": "The same method will apply to longer expressions, if care be taken to obtain the \\emph{trial-divisor} at each stage of the process, \\emph{by doubling the part of the root already found}, and to obtain the \\emph{complete divisor by annexing the new term of the root to the trial-divisor}.", "markdown": "The same method will apply to longer expressions, if care be taken to obtain the *trial-divisor* at each stage of the process, *by doubling the part of the root already found*, and to obtain the *complete divisor by annexing the new term of the root to the trial-divisor*.", "why": "It states the two-step bookkeeping rule that a learner must follow at every stage of a long square root.", "use": [ "lesson" ], "concepts": [ "concept/complete-divisor", "concept/trial-divisor", "method/extracting-the-square-root" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-00c54e6160", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "157", "location": "Square and Cube Roots", "latex": "Since the cube of $a + b$ is $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$, the cube root of a$^{3} + 3a^{2}b + 3ab^{2} + b^{3}$ is $a + b$.", "markdown": "Since the cube of $a + b$ is $a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$, the cube root of a$^{3} + 3a^{2}b + 3ab^{2} + b^{3}$ is $a + b$.", "why": "It gives the expansion that the cube-root procedure reverses, so the learner sees where the three-times-square divisor comes from.", "use": [ "lesson" ], "concepts": [ "method/extracting-the-cube-root", "theorem/cube-of-a-sum" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-6a7ca4662a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "154", "location": "Square and Cube Roots", "latex": "If, therefore, an integral square number is divided into groups of two figures each, from the right to the left, the number of figures in the root will be equal to the number of groups of figures.", "markdown": "If, therefore, an integral square number is divided into groups of two figures each, from the right to the left, the number of figures in the root will be equal to the number of groups of figures.", "why": "It gives the grouping rule that tells the learner how many digits the root must have before any digit is found.", "use": [ "lesson" ], "concepts": [ "concept/perfect-square", "method/grouping-digits" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-e263c7501e", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "155", "location": "Square and Cube Roots", "latex": "If the square root of a number has decimal places, the number itself will have \\emph{twice} as many.", "markdown": "If the square root of a number has decimal places, the number itself will have *twice* as many.", "why": "It explains why a decimal must be grouped in pairs from the point outward, which is a common place for learners to slip.", "use": [ "lesson", "history" ], "concepts": [ "method/extracting-the-square-root", "method/grouping-digits" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-268182ecbb", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "155", "location": "Square and Cube Roots", "latex": "If a number contain an \\emph{odd} number of decimal places, or if any number give a \\emph{remainder} when as many figures in the root have been obtained as the given number has groups, then its exact square root cannot be found.", "markdown": "If a number contain an *odd* number of decimal places, or if any number give a *remainder* when as many figures in the root have been obtained as the given number has groups, then its exact square root cannot be found.", "why": "It warns that some numbers have no exact root, so the learner should expect an approximation rather than a finite answer.", "use": [ "lesson" ], "concepts": [ "concept/perfect-square", "method/extracting-the-square-root" ] }, { "id": "wentworth-first-steps-in-algebra-1894/x-60e075fd53", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Square and Cube Roots", "latex": "It will be noticed that each successive trial-divisor may be obtained by taking the preceding complete divisor with its \\emph{last term doubled}.", "markdown": "It will be noticed that each successive trial-divisor may be obtained by taking the preceding complete divisor with its *last term doubled*.", "why": "It offers a shortcut that saves arithmetic and makes the pattern of the divisors visible to a learner.", "use": [ "lesson", "website" ], "concepts": [ "concept/complete-divisor", "concept/trial-divisor" ] } ], "equations": [ { "id": "wentworth-first-steps-in-algebra-1894/eq-53fd2dbd78", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "8", "location": "Introduction", "latex": "10 + (3 + 2) = 10 + 3 + 2", "name": null, "statement": "Removing a parenthesis preceded by + leaves the signs inside it unchanged, so 10 + (3 + 2) equals 10 + 3 + 2.", "kind": "identity", "symbols": [], "sympy": "Eq(10 + (3 + 2), 10 + 3 + 2)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/addition", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-4548443415", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "8", "location": "Introduction", "latex": "10 + (3 - 2) = 10 + 3 - 2", "name": null, "statement": "A parenthesis preceded by + can be removed without changing the sign of the term inside it, so 10 + (3 - 2) equals 10 + 3 - 2.", "kind": "identity", "symbols": [], "sympy": "Eq(10 + (3 - 2), 10 + 3 - 2)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/addition", "method/subtraction", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-4df56ad211", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "9", "location": "Introduction", "latex": "10 - (3 + 2) = 10 - 3 - 2", "name": null, "statement": "Subtracting a parenthesized sum is the same as subtracting each of its terms in turn, so 10 - (3 + 2) equals 10 - 3 - 2.", "kind": "identity", "symbols": [], "sympy": "Eq(10 - (3 + 2), 10 - 3 - 2)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-37c8da6320", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "9", "location": "Introduction", "latex": "10 &- (5 - 2) = 10 - 5 + 2", "name": null, "statement": "Subtracting a parenthesized difference flips the sign of each term inside, so 10 - (5 - 2) equals 10 - 5 + 2.", "kind": "identity", "symbols": [], "sympy": "Eq(10 - (5 - 2), 10 - 5 + 2)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-147f653b0d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "5", "location": "Introduction", "latex": "4a &= a + a + a + a", "name": null, "statement": "The coefficient 4 before a letter means the letter is taken as a term four times, so 4a means a + a + a + a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a general number" } ], "sympy": "Eq(4*a, a + a + a + a)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "method/addition", "method/multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e77a2e57b5", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "5", "location": "Introduction", "latex": "a^{4} &= a× a× a× a", "name": null, "statement": "The exponent 4 means the letter a is taken as a factor four times, so a^4 means a × a × a × a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a general number" } ], "sympy": "Eq(a**4, a*a*a*a)", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/power", "method/multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-332fdb61c5", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "11", "location": "Introduction", "latex": "a (b + c) &= ab + ac", "name": "distributive law", "statement": "Multiplying a sum by a number is the same as multiplying each term by that number and adding the products.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a general number" }, { "unit": null, "symbol": "b", "meaning": "a general number" }, { "unit": null, "symbol": "c", "meaning": "a general number" } ], "sympy": "Eq(a*(b + c), a*b + a*c)", "physics": false, "states": [ "law/distributive-law" ], "concepts": [ "method/addition", "method/multiplication", "method/multiplying-a-compound-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-6682317b07", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "11", "location": "Introduction", "latex": "a(b - c) &= ab - ac", "name": "distributive law", "statement": "Multiplying a difference by a number is the same as multiplying each term by that number and subtracting the products.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a general number" }, { "unit": null, "symbol": "b", "meaning": "a general number" }, { "unit": null, "symbol": "c", "meaning": "a general number" } ], "sympy": "Eq(a*(b - c), a*b - a*c)", "physics": false, "states": [ "law/distributive-law" ], "concepts": [ "concept/monomial", "concept/polynomial", "law/distributive-law", "method/multiplication", "method/multiplying-a-compound-expression", "method/subtraction" ], "pages": [ "11", "53" ], "chapters": [ "wentworth-first-steps-in-algebra-1894/ch-i", "wentworth-first-steps-in-algebra-1894/ch-v" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-a4c4b75016", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "19", "location": "Simple Equations", "latex": "a + b = b + a", "name": null, "statement": "Adding two numbers gives the same result in either order, so this equation holds for all values of a and b.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a number supposed known, given as an example of a general letter" }, { "unit": null, "symbol": "b", "meaning": "a number supposed known, given as an example of a general letter" } ], "sympy": "Eq(a + b, b + a)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/identity", "method/addition" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-b033e067ff", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "19", "location": "Simple Equations", "latex": "3x + 2 = 8", "name": null, "statement": "The statement that 3x + 2 and 8 stand for the same number, which is true only when x stands for 2, so it is an equation of condition.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" } ], "sympy": "Eq(3*x + 2, 8)", "physics": false, "states": [], "concepts": [ "concept/equality", "concept/equation", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-b0c3917b8d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "20", "location": "Simple Equations", "latex": "ax + b = c", "name": null, "statement": "The general form of a simple equation in x, in which a, b and c are known numbers and x is the unknown.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a known number, the coefficient of x" }, { "unit": null, "symbol": "b", "meaning": "a known number" }, { "unit": null, "symbol": "c", "meaning": "a known number" }, { "unit": null, "symbol": "x", "meaning": "the unknown number" } ], "sympy": "Eq(a*x + b, c)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant", "concept/linear-equation", "concept/unknown", "concept/variable" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-f9a831ae41", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "21", "location": "Simple Equations", "latex": "x = a - b", "name": null, "statement": "Solving x + b = a by subtracting b from both sides gives x equal to a minus b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" }, { "unit": null, "symbol": "a", "meaning": "a known number" }, { "unit": null, "symbol": "b", "meaning": "a known number" } ], "sympy": "Eq(x, a - b)", "physics": false, "states": [], "concepts": [ "concept/method-transposing-the-terms", "concept/unknown", "method/solving-an-equation", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-92239d6886", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "21", "location": "Simple Equations", "latex": "x = a + b", "name": null, "statement": "Solving x - b = a by adding b to both sides gives x equal to a plus b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" }, { "unit": null, "symbol": "a", "meaning": "a known number" }, { "unit": null, "symbol": "b", "meaning": "a known number" } ], "sympy": "Eq(x, a + b)", "physics": false, "states": [], "concepts": [ "concept/unknown", "method/addition", "method/solving-an-equation", "method/transposing-the-terms" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-99af18c2ef", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "35", "location": "Positive and Negative Numbers", "latex": "+a - a = 0", "name": null, "statement": "Adding a number to its opposite (equal absolute value, unlike sign) gives zero, so +a and -a cancel.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number (absolute value of an algebraic number)" } ], "sympy": "Eq(a - a, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-number", "concept/algebraic-sum", "concept/sign" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-f0c40bd36a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "41", "location": "Positive and Negative Numbers", "latex": "a^{m} × a^{n} = a^{m + n}", "name": null, "statement": "The product of two powers of the same number has the index equal to the sum of the indices of the factors.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number" }, { "unit": null, "symbol": "m", "meaning": "any integer (index of the first power)" }, { "unit": null, "symbol": "n", "meaning": "any integer (index of the second power)" } ], "sympy": "Eq(a**m * a**n, a**(m + n))", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/power", "concept/product", "theorem/index-law-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-ad9a39cd73", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "41", "location": "Positive and Negative Numbers", "latex": "(+a) × (+b) &= +ab", "name": null, "statement": "The product of two positive numbers is positive (like signs give plus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number" }, { "unit": null, "symbol": "b", "meaning": "any positive number" } ], "sympy": "Eq(a*b, a*b)", "physics": false, "states": [], "concepts": [ "concept/positive-number", "concept/product", "method/multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-ff681e42ac", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "41", "location": "Positive and Negative Numbers", "latex": "(+a) × (-b) &= -ab", "name": null, "statement": "A positive number times a negative number is negative (unlike signs give minus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number" }, { "unit": null, "symbol": "b", "meaning": "any positive number (absolute value of the negative factor)" } ], "sympy": "Eq(a*(-b), -a*b)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/positive-number", "concept/product", "method/multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-19bb597279", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "41", "location": "Positive and Negative Numbers", "latex": "(-a) × (+b) &= -ab", "name": null, "statement": "A negative number times a positive number is negative (unlike signs give minus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number (absolute value of the negative factor)" }, { "unit": null, "symbol": "b", "meaning": "any positive number" } ], "sympy": "Eq((-a)*b, -a*b)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/positive-number", "concept/product", "method/multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-1a36494be8", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "41", "location": "Positive and Negative Numbers", "latex": "(-a) × (-b) &= +ab", "name": null, "statement": "The product of two negative numbers is positive (like signs give plus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number (absolute value of the first negative factor)" }, { "unit": null, "symbol": "b", "meaning": "any positive number (absolute value of the second negative factor)" } ], "sympy": "Eq((-a)*(-b), a*b)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/product", "method/multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-851fe5e4f4", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "43", "location": "Positive and Negative Numbers", "latex": "+ab ÷ (+a) = +b", "name": null, "statement": "Dividing a positive product by a positive factor gives a positive quotient (like signs give plus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number" }, { "unit": null, "symbol": "b", "meaning": "any positive number" } ], "sympy": "Eq(a*b/a, b)", "physics": false, "states": [], "concepts": [ "concept/dividend", "concept/divisor", "concept/positive-number", "concept/quotient", "theorem/law-of-signs-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-0774b5a15a", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "43", "location": "Positive and Negative Numbers", "latex": "-ab ÷ (+a) = -b", "name": null, "statement": "Dividing a negative dividend by a positive divisor gives a negative quotient (unlike signs give minus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number" }, { "unit": null, "symbol": "b", "meaning": "any positive number" } ], "sympy": "Eq(-a*b/a, -b)", "physics": false, "states": [], "concepts": [ "concept/dividend", "concept/divisor", "concept/negative-number", "concept/quotient", "theorem/law-of-signs-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-fb702170bc", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "43", "location": "Positive and Negative Numbers", "latex": "-ab ÷ (-a) = +b", "name": null, "statement": "Dividing a negative dividend by a negative divisor gives a positive quotient (like signs give plus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number (absolute value of the divisor)" }, { "unit": null, "symbol": "b", "meaning": "any positive number" } ], "sympy": "Eq(-a*b/(-a), b)", "physics": false, "states": [], "concepts": [ "concept/dividend", "concept/divisor", "concept/negative-number", "concept/quotient", "theorem/law-of-signs-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-83a8b029bc", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "43", "location": "Positive and Negative Numbers", "latex": "+ab ÷ (-a) = -b", "name": null, "statement": "Dividing a positive dividend by a negative divisor gives a negative quotient (unlike signs give minus).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any positive number (absolute value of the divisor)" }, { "unit": null, "symbol": "b", "meaning": "any positive number" } ], "sympy": "Eq(a*b/(-a), -b)", "physics": false, "states": [], "concepts": [ "concept/dividend", "concept/divisor", "concept/positive-number", "concept/quotient", "theorem/law-of-signs-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-d224a759c4", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "44", "location": "Positive and Negative Numbers", "latex": "a^{m} - a^{n} = a^{m - n}", "name": null, "statement": "FLAG: as transcribed, the chapter prints a minus sign between the two powers, but the surrounding text (the index law in division, §78, and the worked quotients a^5/a^2 = a^(5-2) etc.) shows that the intended operation is division, a^m ÷ a^n = a^(m-n); the printed sign is most likely a transcription error in the source. Recorded verbatim here, not corrected; the sympy field follows the printed text. For m greater than n, the index of the quotient of two powers of the same number is m minus n.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number" }, { "unit": null, "symbol": "m", "meaning": "any integer greater than n (index of the dividend power)" }, { "unit": null, "symbol": "n", "meaning": "any integer (index of the divisor power)" } ], "sympy": "Eq(a**m - a**n, a**(m - n))", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/power", "concept/quotient", "theorem/index-law-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-a28804a446", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a + (b + c) &= a + b + c", "name": null, "statement": "A parenthesis preceded by + can be removed without changing the sign of any term inside it.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a + (b + c), a + b + c)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/addition", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-8f84c3b570", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a + b + c &= a + (b + c)", "name": null, "statement": "Any group of terms can be enclosed in a parenthesis preceded by + without changing the sign of any term.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a + b + c, a + (b + c))", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/addition", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e10ecaf800", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a + (b - c) &= a + b - c", "name": null, "statement": "A parenthesis preceded by + can be removed without changing the sign of the terms inside it, including a negative term.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a + (b - c), a + b - c)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/addition", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-c55d9182a7", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a + b - c &= a + (b - c)", "name": null, "statement": "A group of terms including a negative term can be enclosed in a parenthesis preceded by + without changing any sign.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a + b - c, a + (b - c))", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/addition", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-670b148831", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a - (b + c) &= a - b - c", "name": null, "statement": "A parenthesis preceded by - can be removed if the sign of every term inside it is changed.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a - (b + c), a - b - c)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-03e187dbe8", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a - b - c &= a - (b + c)", "name": null, "statement": "Terms can be enclosed in a parenthesis preceded by - provided the sign of every enclosed term is changed.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a - b - c, a - (b + c))", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-65d6f5b071", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a - (b - c) &= a - b + c", "name": null, "statement": "A parenthesis preceded by - containing a negative term can be removed by changing every sign inside it.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a - (b - c), a - b + c)", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-df1a52b47d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "51", "location": "Addition and Subtraction", "latex": "a - b + c &= a - (b - c)", "name": null, "statement": "Terms can be enclosed in a parenthesis preceded by - by changing the sign of every enclosed term, here turning a negative term into a positive one.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "b", "meaning": "number (positive or negative)" }, { "unit": null, "symbol": "c", "meaning": "number (positive or negative)" } ], "sympy": "Eq(a - b + c, a - (b - c))", "physics": false, "states": [], "concepts": [ "concept/parenthesis", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-9c8f9bf8f6", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Multiplication and Division", "latex": "a(b + c) &= ab + ac", "name": null, "statement": "A monomial times a sum equals the sum of the monomial times each term, for positive numbers and, the book says, for negative numbers too.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a general number (monomial multiplier)" }, { "unit": null, "symbol": "b", "meaning": "a general number (term of the sum)" }, { "unit": null, "symbol": "c", "meaning": "a general number (term of the sum)" } ], "sympy": "Eq(a*(b + c), a*b + a*c)", "physics": false, "states": [], "concepts": [ "concept/monomial", "concept/polynomial", "law/distributive-law", "method/multiplying-a-compound-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e7d0036322", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "54", "location": "Multiplication and Division", "latex": "M(m + n + p) = Mm + Mn + Mp", "name": null, "statement": "Multiplying a sum of terms m, n, p by a multiplier M gives the sum of the products of M with each term; this is the step used to multiply polynomials.", "kind": "result", "symbols": [ { "unit": null, "symbol": "M", "meaning": "the multiplier, a placeholder for a+b+c" }, { "unit": null, "symbol": "m", "meaning": "a term of the multiplicand" }, { "unit": null, "symbol": "n", "meaning": "a term of the multiplicand" }, { "unit": null, "symbol": "p", "meaning": "a term of the multiplicand" } ], "sympy": "Eq(M*(m + n + p), M*m + M*n + M*p)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/product", "law/distributive-law", "method/multiplying-a-compound-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-3ab69dec11", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "57", "location": "Multiplication and Division", "latex": "a(b + c - d) &= ab + ac - ad", "name": null, "statement": "A monomial times a polynomial equals the sum of the monomial times each term, with signs kept, which is the basis for dividing a polynomial by a monomial.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a general nonzero monomial" }, { "unit": null, "symbol": "b", "meaning": "a general term of the polynomial" }, { "unit": null, "symbol": "c", "meaning": "a general term of the polynomial" }, { "unit": null, "symbol": "d", "meaning": "a general term of the polynomial (subtracted)" } ], "sympy": "Eq(a*(b + c - d), a*b + a*c - a*d)", "physics": false, "states": [], "concepts": [ "concept/dividend", "concept/divisor", "concept/monomial", "concept/polynomial", "law/distributive-law" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-20f9664b9e", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "68", "location": "Multiplication and Division", "latex": "\\frac{a^{2} - b^{2}}{a + b} = a - b", "name": "Rule 1 (division)", "statement": "The difference of two squares divided by their sum equals their difference.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number" }, { "unit": null, "symbol": "b", "meaning": "any number" } ], "sympy": "Eq((a**2 - b**2)/(a + b), a - b)", "physics": false, "states": [ "theorem/difference-of-two-squares" ], "concepts": [ "concept/divisibility", "concept/quotient" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-acd11efd91", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "68", "location": "Multiplication and Division", "latex": "\\frac{a^{2} - b^{2}}{a - b} = a + b", "name": "Rule 1 (division)", "statement": "The difference of two squares divided by the difference of the numbers equals their sum.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number" }, { "unit": null, "symbol": "b", "meaning": "any number" } ], "sympy": "Eq((a**2 - b**2)/(a - b), a + b)", "physics": false, "states": [ "theorem/difference-of-two-squares" ], "concepts": [ "concept/divisibility", "concept/quotient" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-9ea7d1a1f8", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "69", "location": "Multiplication and Division", "latex": "\\frac{a^{3} - b^{3}}{a - b} = a^{2} + ab + b^{2}", "name": "Rule 2 (division)", "statement": "The difference of two cubes divided by the difference of the numbers equals the sum of their squares plus their product.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number" }, { "unit": null, "symbol": "b", "meaning": "any number" } ], "sympy": "Eq((a**3 - b**3)/(a - b), a**2 + a*b + b**2)", "physics": false, "states": [ "theorem/difference-of-two-cubes" ], "concepts": [ "concept/divisibility", "concept/factor", "concept/quotient", "concept/root", "method/factoring" ], "pages": [ "69", "76" ], "chapters": [ "wentworth-first-steps-in-algebra-1894/ch-vi", "wentworth-first-steps-in-algebra-1894/ch-vii" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-9d1360e429", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Multiplication and Division", "latex": "\\frac{a^{3} + b^{3}}{a + b} = a^{2} - ab + b^{2}", "name": "Rule 3 (division)", "statement": "The sum of two cubes divided by the sum of the numbers equals the sum of their squares minus their product.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "any number" }, { "unit": null, "symbol": "b", "meaning": "any number" } ], "sympy": "Eq((a**3 + b**3)/(a + b), a**2 - a*b + b**2)", "physics": false, "states": [ "theorem/sum-of-two-cubes" ], "concepts": [ "concept/divisibility", "concept/factor", "concept/quotient", "concept/root", "method/factoring" ], "pages": [ "70", "77" ], "chapters": [ "wentworth-first-steps-in-algebra-1894/ch-vi", "wentworth-first-steps-in-algebra-1894/ch-vii" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e2f6bfbee5", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "71", "location": "Factors", "latex": "3a^{2} - 6ab &= 3a(a - 2b)", "name": null, "statement": "A polynomial whose terms share the factor 3a is resolved by taking 3a out as a factor.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a letter standing for any number or expression" }, { "unit": null, "symbol": "b", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq(3*a**2 - 6*a*b, 3*a*(a - 2*b))", "physics": false, "states": [], "concepts": [ "concept/common-factor", "concept/factor", "method/factoring-by-taking-out-a-common-factor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-c17fe3aa07", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Factors", "latex": "4x^{3} + 12x^{2} - 8x &= 4x(x^{2} + 3x - 2)", "name": null, "statement": "The common factor 4x is taken out of each term of the polynomial, leaving the quotient in brackets.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq(4*x**3 + 12*x**2 - 8*x, 4*x*(x**2 + 3*x - 2))", "physics": false, "states": [], "concepts": [ "concept/common-factor", "concept/factor", "method/factoring-by-taking-out-a-common-factor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-3df2422d67", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "74", "location": "Factors", "latex": "(x + y)(x - y) = x^{2} - y^{2}", "name": "difference of two squares", "statement": "The product of the sum and the difference of two quantities equals the difference of their squares, so a difference of squares factors into a sum and a difference.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a letter standing for any number or expression" }, { "unit": null, "symbol": "y", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq((x + y)*(x - y), x**2 - y**2)", "physics": false, "states": [ "theorem/difference-of-two-squares" ], "concepts": [ "concept/factor", "concept/root", "method/factoring" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-55d59c2fa2", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "77", "location": "Factors", "latex": "a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})", "name": "difference of two cubes", "statement": "The difference of two cubes is the difference of their cube roots times the sum of the squares of the roots and their product.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a letter standing for any number or expression" }, { "unit": null, "symbol": "b", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq(a**3 - b**3, (a - b)*(a**2 + a*b + b**2))", "physics": false, "states": [ "theorem/difference-of-two-cubes" ], "concepts": [ "concept/factor", "concept/root", "method/factoring" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-7b490e9a40", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "78", "location": "Factors", "latex": "a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})", "name": "sum of two cubes", "statement": "The sum of two cubes is the sum of their cube roots times the square of the first root, minus their product, plus the square of the second root.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a letter standing for any number or expression" }, { "unit": null, "symbol": "b", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq(a**3 + b**3, (a + b)*(a**2 - a*b + b**2))", "physics": false, "states": [ "theorem/sum-of-two-cubes" ], "concepts": [ "concept/factor", "concept/root", "method/factoring" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-d7d115c3f1", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "79", "location": "Factors", "latex": "(x + y)^{2} = x^{2} + 2xy + y^{2}", "name": "square of a sum", "statement": "The square of a sum of two quantities is the square of the first, plus twice their product, plus the square of the second.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a letter standing for any number or expression" }, { "unit": null, "symbol": "y", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq((x + y)**2, x**2 + 2*x*y + y**2)", "physics": false, "states": [ "theorem/square-of-a-sum" ], "concepts": [ "concept/perfect-square", "concept/trinomial", "method/factoring-a-perfect-square-trinomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-7932ebc495", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "79", "location": "Factors", "latex": "(x - y)^{2} = x^{2} - 2xy + y^{2}", "name": "square of a difference", "statement": "The square of a difference of two quantities is the square of the first, minus twice their product, plus the square of the second.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a letter standing for any number or expression" }, { "unit": null, "symbol": "y", "meaning": "a letter standing for any number or expression" } ], "sympy": "Eq((x - y)**2, x**2 - 2*x*y + y**2)", "physics": false, "states": [ "theorem/square-of-a-difference" ], "concepts": [ "concept/perfect-square", "concept/trinomial", "method/factoring-a-perfect-square-trinomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-6e032054d2", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Fractions", "latex": "\\frac{a}{b} × \\frac{c}{d} = \\frac{ac}{bd}", "name": null, "statement": "The product of two fractions is the product of their numerators over the product of their denominators.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "numerator of the first fraction" }, { "unit": null, "symbol": "b", "meaning": "denominator of the first fraction" }, { "unit": null, "symbol": "c", "meaning": "numerator of the second fraction" }, { "unit": null, "symbol": "d", "meaning": "denominator of the second fraction" } ], "sympy": "Eq(a/b*(c/d), a*c/(b*d))", "physics": false, "states": [], "concepts": [ "concept/algebraic-fraction", "concept/method-multiplying-fractions", "concept/product" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-312216bfb2", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "98", "location": "Fractions", "latex": "\\frac{a}{b} × \\frac{c}{d} × \\frac{e}{f} = \\frac{ac}{bd} × \\frac{e}{f} = \\frac{ace}{bdf}", "name": null, "statement": "The product of three fractions is the product of all numerators over the product of all denominators, obtained by applying the two-fraction rule twice.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "numerator of the first fraction" }, { "unit": null, "symbol": "b", "meaning": "denominator of the first fraction" }, { "unit": null, "symbol": "c", "meaning": "numerator of the second fraction" }, { "unit": null, "symbol": "d", "meaning": "denominator of the second fraction" }, { "unit": null, "symbol": "e", "meaning": "numerator of the third fraction" }, { "unit": null, "symbol": "f", "meaning": "denominator of the third fraction" } ], "sympy": "Eq(a/b*(c/d)*(e/f), a*c*e/(b*d*f))", "physics": false, "states": [], "concepts": [ "concept/algebraic-fraction", "concept/method-multiplying-fractions", "concept/product" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-2e70b5e188", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "98", "location": "Fractions", "latex": "\\dfrac{b}{a} × \\dfrac{a}{b} = \\dfrac{ba}{ab} = 1", "name": null, "statement": "The fraction b/a multiplied by its inverse a/b equals 1, so b/a is the reciprocal of a/b.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "numerator of the given fraction, denominator of its reciprocal" }, { "unit": null, "symbol": "b", "meaning": "denominator of the given fraction, numerator of its reciprocal" } ], "sympy": "Eq(b/a*(a/b), 1)", "physics": false, "states": [], "concepts": [ "concept/algebraic-fraction", "concept/method-dividing-by-a-fraction", "concept/reciprocal" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-ff019a38b5", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Fractions", "latex": "\\dfrac{x^{3} - 1}{x - 1} = x^{2} + x + 1", "name": null, "statement": "Dividing x cubed minus one by x minus one gives x squared plus x plus one exactly, with no remainder.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" } ], "sympy": "Eq((x**3 - 1)/(x - 1), x**2 + x + 1)", "physics": false, "states": [], "concepts": [ "concept/method-dividing-a-polynomial-by-a-polynomial", "concept/quotient", "concept/remainder", "method/reducing-a-fraction-to-an-integral-or-mixed-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-da05d08912", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Fractions", "latex": "\\dfrac{x^{3} - 1}{x + 1} = x^{2} - x + 1 - \\dfrac{2}{x + 1}", "name": null, "statement": "Dividing x cubed minus one by x plus one gives an integral part x squared minus x plus one, plus a fraction whose numerator is the remainder 2 and whose denominator is the divisor.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" } ], "sympy": "Eq((x**3 - 1)/(x + 1), x**2 - x + 1 - 2/(x + 1))", "physics": false, "states": [], "concepts": [ "concept/method-dividing-a-polynomial-by-a-polynomial", "concept/mixed-number", "concept/quotient", "concept/remainder", "method/reducing-a-fraction-to-an-integral-or-mixed-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-bbce9cd735", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "119", "location": "Fractional Equations", "latex": "i = prt", "name": null, "statement": "The simple interest equals the principal times the rate times the time.", "kind": "formula", "symbols": [ { "unit": "dollar", "symbol": "i", "meaning": "the interest for the given time and rate" }, { "unit": "dollar", "symbol": "p", "meaning": "the principal" }, { "unit": null, "symbol": "r", "meaning": "the interest of $1 for 1 year, at the given rate" }, { "unit": "year", "symbol": "t", "meaning": "the time expressed in years" } ], "sympy": "Eq(i, p*r*t)", "physics": false, "states": [], "concepts": [ "concept/simple-interest", "quantity/interest", "quantity/principal", "quantity/rate-of-interest", "quantity/time" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-329d346ad7", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "120", "location": "Fractional Equations", "latex": "p = \\frac{i}{rt}", "name": null, "statement": "The principal is the interest divided by the product of the rate and the time.", "kind": "formula", "symbols": [ { "unit": "dollar", "symbol": "p", "meaning": "the principal" }, { "unit": "dollar", "symbol": "i", "meaning": "the interest for the given time and rate" }, { "unit": null, "symbol": "r", "meaning": "the interest of $1 for 1 year, at the given rate" }, { "unit": "year", "symbol": "t", "meaning": "the time expressed in years" } ], "sympy": "Eq(p, i/(r*t))", "physics": false, "states": [], "concepts": [ "concept/simple-interest", "quantity/interest", "quantity/principal", "quantity/rate-of-interest", "quantity/time" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-a67b1d4d0e", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "120", "location": "Fractional Equations", "latex": "p = \\frac{a}{1 + rt}", "name": null, "statement": "The principal that amounts to a given sum equals the amount divided by one plus the product of rate and time.", "kind": "formula", "symbols": [ { "unit": "dollar", "symbol": "p", "meaning": "the principal" }, { "unit": "dollar", "symbol": "a", "meaning": "the amount (sum of principal and interest)" }, { "unit": null, "symbol": "r", "meaning": "the interest of $1 for 1 year, at the given rate" }, { "unit": "year", "symbol": "t", "meaning": "the time expressed in years" } ], "sympy": "Eq(p, a/(1 + r*t))", "physics": false, "states": [], "concepts": [ "concept/simple-interest", "quantity/amount", "quantity/principal", "quantity/rate-of-interest", "quantity/time" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-267658dc32", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "120", "location": "Fractional Equations", "latex": "t = \\frac{a - p}{pr}", "name": null, "statement": "The time is the interest (amount minus principal) divided by the product of principal and rate.", "kind": "formula", "symbols": [ { "unit": "year", "symbol": "t", "meaning": "the time expressed in years" }, { "unit": "dollar", "symbol": "a", "meaning": "the amount (sum of principal and interest)" }, { "unit": "dollar", "symbol": "p", "meaning": "the principal" }, { "unit": null, "symbol": "r", "meaning": "the interest of $1 for 1 year, at the given rate" } ], "sympy": "Eq(t, (a - p)/(p*r))", "physics": false, "states": [], "concepts": [ "concept/simple-interest", "quantity/amount", "quantity/principal", "quantity/rate-of-interest", "quantity/time" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-a7eb0e6569", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "120", "location": "Fractional Equations", "latex": "r = \\frac{a - p}{pt}", "name": null, "statement": "The rate is the interest (amount minus principal) divided by the product of principal and time.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "the interest of $1 for 1 year, at the given rate" }, { "unit": "dollar", "symbol": "a", "meaning": "the amount (sum of principal and interest)" }, { "unit": "dollar", "symbol": "p", "meaning": "the principal" }, { "unit": "year", "symbol": "t", "meaning": "the time expressed in years" } ], "sympy": "Eq(r, (a - p)/(p*t))", "physics": false, "states": [], "concepts": [ "concept/simple-interest", "quantity/amount", "quantity/principal", "quantity/rate-of-interest", "quantity/time" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-8664262bc6", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "118", "location": "Fractional Equations", "latex": "x = \\frac{s - d}{2}", "name": null, "statement": "The smaller of two numbers equals half the difference of their sum and their difference.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the smaller number" }, { "unit": null, "symbol": "s", "meaning": "the sum of the two numbers" }, { "unit": null, "symbol": "d", "meaning": "the difference of the two numbers" } ], "sympy": "Eq(x, (s - d)/2)", "physics": false, "states": [], "concepts": [ "concept/algebraic-sum", "concept/difference", "concept/literal-equation", "concept/sum" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-6a0a47b52c", "chapter": "wentworth-first-steps-in-algebra-1894/ch-x", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "119", "location": "Fractional Equations", "latex": "x = \\frac{ab}{a + b}", "name": null, "statement": "Two agents working together need a time equal to the product of their individual times divided by their sum.", "kind": "result", "symbols": [ { "unit": "day", "symbol": "x", "meaning": "the required number of days for both together" }, { "unit": "day", "symbol": "a", "meaning": "the number of days A alone needs to do the work" }, { "unit": "day", "symbol": "b", "meaning": "the number of days B alone needs to do the work" } ], "sympy": "Eq(x, a*b/(a + b))", "physics": false, "states": [], "concepts": [ "concept/combined-rate", "concept/literal-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-ccad3db812", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "x + y = 10", "name": null, "statement": "One relation between two unknown numbers leaves infinitely many pairs of values that satisfy it.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number" }, { "unit": null, "symbol": "y", "meaning": "unknown number" } ], "sympy": "Eq(x + y, 10)", "physics": false, "states": [], "concepts": [ "concept/linear-equation", "concept/unknown", "concept/variable" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-250437650b", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "y = 10 - x", "name": null, "statement": "Solving x + y = 10 for y gives y as 10 minus x, so any chosen x determines y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number" }, { "unit": null, "symbol": "y", "meaning": "unknown number" } ], "sympy": "Eq(y, 10 - x)", "physics": false, "states": [], "concepts": [ "concept/linear-equation", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-a8d55786ea", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "x - y = 2", "name": null, "statement": "A second equation expressing a relation between x and y different from x + y = 10, so the two together are independent equations.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number" }, { "unit": null, "symbol": "y", "meaning": "unknown number" } ], "sympy": "Eq(x - y, 2)", "physics": false, "states": [], "concepts": [ "concept/independent-equations", "concept/simultaneous-equations", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-9df268b59e", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "122", "location": "Simultaneous Equations of the First Degree", "latex": "x + y &= 10", "name": null, "statement": "The sum of the two digits of the number is 10.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the tens' digit" }, { "unit": null, "symbol": "y", "meaning": "the units' digit" } ], "sympy": "Eq(x + y, 10)", "physics": false, "states": [], "concepts": [ "concept/digit", "concept/linear-equation", "concept/simultaneous-equations" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-510c997133", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Simultaneous Equations of the First Degree", "latex": "10x + y + 18 &= 10y + x", "name": null, "statement": "Adding 18 to the number 10x + y gives the number with its digits reversed, 10y + x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the tens' digit" }, { "unit": null, "symbol": "y", "meaning": "the units' digit" } ], "sympy": "Eq(10*x + y + 18, 10*y + x)", "physics": false, "states": [], "concepts": [ "concept/digit", "concept/linear-equation", "concept/simultaneous-equations" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-485175ce55", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Simultaneous Equations of the First Degree", "latex": "x - y &= -2", "name": null, "statement": "Dividing the reduced equation 9x - 9y = -18 by 9 gives x - y = -2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the tens' digit" }, { "unit": null, "symbol": "y", "meaning": "the units' digit" } ], "sympy": "Eq(x - y, -2)", "physics": false, "states": [], "concepts": [ "concept/digit", "concept/linear-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-67581977ce", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "126", "location": "Simultaneous Equations of the First Degree", "latex": "y + 10 &= 3(x - 10)", "name": null, "statement": "After A gives B $10, B's money equals three times A's money.", "kind": "formula", "symbols": [ { "unit": "dollar", "symbol": "x", "meaning": "the number of dollars A has" }, { "unit": "dollar", "symbol": "y", "meaning": "the number of dollars B has" } ], "sympy": "Eq(y + 10, 3*(x - 10))", "physics": false, "states": [], "concepts": [ "concept/linear-equation", "concept/simultaneous-equations", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-9127fb2210", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xi", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Simultaneous Equations of the First Degree", "latex": "x + 10 &= 2(y - 10)", "name": null, "statement": "After B gives A $10, A's money equals twice B's money.", "kind": "formula", "symbols": [ { "unit": "dollar", "symbol": "x", "meaning": "the number of dollars A has" }, { "unit": "dollar", "symbol": "y", "meaning": "the number of dollars B has" } ], "sympy": "Eq(x + 10, 2*(y - 10))", "physics": false, "states": [], "concepts": [ "concept/linear-equation", "concept/simultaneous-equations", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-67a3f12238", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "132", "location": "Quadratic Equations", "latex": "ax^{2} + bx + c = 0", "name": null, "statement": "Every quadratic equation can be collected into this standard form, where a, b and c are known numbers and x is the unknown.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "known number, coefficient of x squared" }, { "unit": null, "symbol": "b", "meaning": "known number, coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "known number, the constant term" }, { "unit": null, "symbol": "x", "meaning": "the unknown number" } ], "sympy": "Eq(a*x**2 + b*x + c, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant-term", "concept/quadratic-equation", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e6872b1c5d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "135", "location": "Quadratic Equations", "latex": "(x + b)^{2} = x^{2} + 2bx + b^{2}", "name": null, "statement": "The square of x plus b expands to x squared plus twice b times x plus b squared.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" }, { "unit": null, "symbol": "b", "meaning": "a number" } ], "sympy": "Eq((x + b)**2, x**2 + 2*b*x + b**2)", "physics": false, "states": [], "concepts": [ "concept/perfect-square", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-4acc1f6299", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "135", "location": "Quadratic Equations", "latex": "(x - b)^{2} = x^{2} - 2bx + b^{2}", "name": null, "statement": "The square of x minus b expands to x squared minus twice b times x plus b squared.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" }, { "unit": null, "symbol": "b", "meaning": "a number" } ], "sympy": "Eq((x - b)**2, x**2 - 2*b*x + b**2)", "physics": false, "states": [], "concepts": [ "concept/perfect-square", "concept/unknown" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-d6cdd9ede5", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "135", "location": "Quadratic Equations", "latex": "x^{2} + 2bx = c", "name": null, "statement": "Every affected quadratic can be brought to this form, with the x squared coefficient made 1, so that completing the square can be applied.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown number" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "a known number" } ], "sympy": "Eq(x**2 + 2*b*x, c)", "physics": false, "states": [], "concepts": [ "concept/affected-quadratic-equation", "concept/coefficient", "concept/unknown", "method/completing-the-square" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e7134e170d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "142", "location": "Arithmetical Progression", "latex": "l = a + (n - 1)d", "name": null, "statement": "The nth term l of an arithmetical progression equals the first term a plus (n - 1) times the common difference d.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "l", "meaning": "the nth term (last term)" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "n", "meaning": "the number of the term" }, { "unit": null, "symbol": "d", "meaning": "the common difference" } ], "sympy": "Eq(l, a + (n - 1)*d)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-progression", "concept/common-difference", "concept/term", "theorem/nth-term-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-acf2545185", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "143", "location": "Arithmetical Progression", "latex": "A = \\frac{a + b}{2}", "name": null, "statement": "If A is the arithmetical mean of a and b, then A is half their sum.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "the arithmetical mean of a and b" }, { "unit": null, "symbol": "a", "meaning": "the first of the two numbers" }, { "unit": null, "symbol": "b", "meaning": "the second of the two numbers" } ], "sympy": "Eq(A, (a + b)/2)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-37f173344f", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Arithmetical Progression", "latex": "m + 2 = n", "name": null, "statement": "The total number of terms n is the number of inserted means m plus the two given end numbers.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "m", "meaning": "the number of arithmetical means inserted" }, { "unit": null, "symbol": "n", "meaning": "the whole number of terms, including the two ends" } ], "sympy": "Eq(m + 2, n)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/number-of-terms", "method/inserting-arithmetical-means" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-8dd2f8b76d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Arithmetical Progression", "latex": "l = a + (m + 1)d", "name": null, "statement": "The last term l equals the first term a plus (m + 1) times the common difference, when m means are inserted.", "kind": "result", "symbols": [ { "unit": null, "symbol": "l", "meaning": "the last term" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "m", "meaning": "the number of arithmetical means inserted" }, { "unit": null, "symbol": "d", "meaning": "the common difference" } ], "sympy": "Eq(l, a + (m + 1)*d)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-progression", "concept/common-difference", "method/inserting-arithmetical-means", "theorem/nth-term-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-250f19306d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Arithmetical Progression", "latex": "\\frac{l - a}{m + 1} = d", "name": null, "statement": "The common difference when m arithmetical means are inserted between a and l is the difference l - a divided by m + 1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "l", "meaning": "the last term" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "m", "meaning": "the number of arithmetical means inserted" }, { "unit": null, "symbol": "d", "meaning": "the common difference" } ], "sympy": "Eq((l - a)/(m + 1), d)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-progression", "concept/common-difference", "method/inserting-arithmetical-means" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-2e58fe420b", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "145", "location": "Arithmetical Progression", "latex": "2s = n(a + l)", "name": null, "statement": "Twice the sum of the terms equals the number of terms times the sum of the first and last terms.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "the sum of the terms" }, { "unit": null, "symbol": "n", "meaning": "the number of terms" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "l", "meaning": "the last term" } ], "sympy": "Eq(2*s, n*(a + l))", "physics": false, "states": [], "concepts": [ "concept/arithmetical-progression", "concept/sum", "theorem/sum-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-4b5542aa9c", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "145", "location": "Arithmetical Progression", "latex": "s = \\frac{n}{2}(a + l)", "name": null, "statement": "The sum of n terms of an arithmetical progression is n over 2 times the sum of the first and last terms (Formula (4)).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "the sum of the terms" }, { "unit": null, "symbol": "n", "meaning": "the number of terms" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "l", "meaning": "the last term" } ], "sympy": "Eq(s, n/2*(a + l))", "physics": false, "states": [], "concepts": [ "concept/arithmetical-progression", "concept/sum", "theorem/sum-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-07772edd80", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiii", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "145", "location": "Arithmetical Progression", "latex": "s = \\frac{n}{2}\\bigl\\{a + a + (n - 1)d\\bigr\\}", "name": null, "statement": "The sum of n terms of an arithmetical progression, with l replaced by a + (n - 1)d, is n over 2 times the braced sum of a, a and (n - 1)d (Formula (5), first line).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "the sum of the terms" }, { "unit": null, "symbol": "n", "meaning": "the number of terms" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "d", "meaning": "the common difference" } ], "sympy": "Eq(s, n/2*(a + a + (n - 1)*d))", "physics": false, "states": [], "concepts": [ "concept/arithmetical-progression", "concept/common-difference", "concept/sum", "theorem/sum-of-an-arithmetical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e920c0cf4e", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "148", "location": "Geometrical Progression", "latex": "\\dfrac{b}{a} = \\dfrac{c}{b} = \\dfrac{d}{c}", "name": null, "statement": "Successive terms a, b, c, d are in geometrical progression when each term divided by the one before it gives the same quotient.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a term of the geometrical progression" }, { "unit": null, "symbol": "b", "meaning": "a term of the geometrical progression, following a" }, { "unit": null, "symbol": "c", "meaning": "a term of the geometrical progression, following b" }, { "unit": null, "symbol": "d", "meaning": "a term of the geometrical progression, following c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-922e329311", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "149", "location": "Geometrical Progression", "latex": "l = ar^{n - 1}", "name": null, "statement": "The nth term l of a geometrical progression equals the first term a times the common ratio r raised to the power n minus 1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "l", "meaning": "the nth term of the progression" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "r", "meaning": "the common ratio" }, { "unit": null, "symbol": "n", "meaning": "the number of the term in the series" } ], "sympy": "Eq(l, a*r**(n - 1))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/first-term", "concept/geometrical-progression", "theorem/nth-term-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-435d84bddc", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "149", "location": "Geometrical Progression", "latex": "G = ± \\sqrt{ab}", "name": null, "statement": "The geometrical mean G of two numbers a and b is plus or minus the square root of their product, which is equivalent to G squared equal to ab.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "G", "meaning": "the geometrical mean of a and b" }, { "unit": null, "symbol": "a", "meaning": "the first of the two numbers" }, { "unit": null, "symbol": "b", "meaning": "the second of the two numbers" } ], "sympy": "Eq(G**2, a*b)", "physics": false, "states": [], "concepts": [ "concept/geometrical-mean", "concept/geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-e6c973701d", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "150", "location": "Geometrical Progression", "latex": "s = \\frac{a(r^{n} - 1)}{r - 1}", "name": null, "statement": "The sum s of the first n terms of a geometrical progression equals a times (r to the power n minus 1) divided by (r minus 1).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "the sum of the n terms" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "r", "meaning": "the common ratio" }, { "unit": null, "symbol": "n", "meaning": "the number of terms" } ], "sympy": "Eq(s, a*(r**n - 1)/(r - 1))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/geometrical-progression", "theorem/sum-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-2093d86384", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xiv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "150", "location": "Geometrical Progression", "latex": "s = \\frac{a(1 - r^{n})}{1 - r}", "name": null, "statement": "For a common ratio r less than 1, the sum of n terms of a geometrical progression can be written as a times (1 minus r to the power n) divided by (1 minus r).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "the sum of the n terms" }, { "unit": null, "symbol": "a", "meaning": "the first term" }, { "unit": null, "symbol": "r", "meaning": "the common ratio, here less than 1" }, { "unit": null, "symbol": "n", "meaning": "the number of terms" } ], "sympy": "Eq(s, a*(1 - r**n)/(1 - r))", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/geometrical-progression", "theorem/sum-of-a-geometrical-progression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-0795807a60", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "152", "location": "Square and Cube Roots", "latex": "2ab + b^{2} = (2a + b)b", "name": null, "statement": "The remainder left after subtracting a squared from the square of a + b factors as (2a + b) times b, which is why the divisor is completed by adding the new root term b.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "the part of the square root already found (the first term of the root, taken as tens with respect to the next figure)" }, { "unit": null, "symbol": "b", "meaning": "the next term of the square root being found" } ], "sympy": "Eq(2*a*b + b**2, (2*a + b)*b)", "physics": false, "states": [], "concepts": [ "concept/complete-divisor", "concept/factor", "concept/root", "concept/trial-divisor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/eq-f064f31fcf", "chapter": "wentworth-first-steps-in-algebra-1894/ch-xv", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "157", "location": "Square and Cube Roots", "latex": "3a^{2}b + 3ab^{2} + b^{3} = (3a^{2} + 3ab + b^{2})b", "name": null, "statement": "The remainder left after subtracting a cubed from the cube of a + b factors as (3a^2 + 3ab + b^2) times b, which is why the complete divisor is formed by adding 3ab + b^2 to the trial divisor 3a^2.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "the part of the cube root already found (the first term of the root, taken as tens with respect to the next figure)" }, { "unit": null, "symbol": "b", "meaning": "the next term of the cube root being found" } ], "sympy": "Eq(3*a**2*b + 3*a*b**2 + b**3, (3*a**2 + 3*a*b + b**2)*b)", "physics": false, "states": [], "concepts": [ "concept/complete-divisor", "concept/factor", "concept/root", "concept/trial-divisor" ] } ], "exercise_sets": [ { "id": "wentworth-first-steps-in-algebra-1894/ex-1", "set": "1", "page": "10", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "concept/parenthesis", "method/addition", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-2", "set": "2", "page": "12", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "method/multiplication", "method/multiplying-a-compound-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3", "set": "3", "page": "12", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "concept/coefficient", "concept/exponent", "concept/implied-multiplication", "concept/numerical-value", "method/multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4", "set": "4", "page": "13", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "concept/exponent", "concept/numerical-value", "method/addition", "method/multiplying-a-compound-expression", "method/order-of-operations", "method/subtraction", "theorem/removing-parentheses-preceded-by-minus", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5", "set": "5", "page": "14", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "concept/exponent", "concept/power", "method/addition", "method/division", "method/multiplication", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6", "set": "6", "page": "15", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "method/addition", "method/division", "method/multiplication", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7", "set": "7", "page": "16", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "concept/equality", "concept/implied-multiplication", "concept/numerical-value", "method/addition", "method/multiplication", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8", "set": "8", "page": "17", "chapter": "wentworth-first-steps-in-algebra-1894/ch-i", "practices": [ "concept/algebraic-expression", "concept/square", "concept/variable", "method/addition", "method/division", "method/multiplication", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9", "set": "9", "page": "24", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "practices": [ "concept/like-terms", "concept/linear-equation", "concept/square", "method/addition", "method/solving-an-equation", "method/subtraction", "method/transposing-the-terms", "theorem/removing-parentheses-preceded-by-minus", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10", "set": "10", "page": "27", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "practices": [ "concept/linear-equation", "method/addition", "method/division", "method/multiplication", "method/stating-a-problem-as-an-equation", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11", "set": "11", "page": "29", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "practices": [ "concept/linear-equation", "method/addition", "method/division", "method/multiplication", "method/stating-a-problem-as-an-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12", "set": "12", "page": "30", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "practices": [ "concept/linear-equation", "method/addition", "method/division", "method/multiplication", "method/stating-a-problem-as-an-equation", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13", "set": "13", "page": "31", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "practices": [ "concept/linear-equation", "concept/parenthesis", "method/division", "method/multiplication", "method/stating-a-problem-as-an-equation" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14", "set": "14", "page": "32", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ii", "practices": [ "concept/linear-equation", "method/addition", "method/division", "method/multiplication", "method/stating-a-problem-as-an-equation", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15", "set": "15", "page": "39", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "practices": [ "concept/algebraic-sum", "concept/coefficient", "concept/like-terms", "concept/numerical-value", "method/addition", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16", "set": "16", "page": "42", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "practices": [ "concept/exponent", "concept/monomial", "concept/numerical-value", "method/multiplication", "theorem/index-law-in-multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17", "set": "17", "page": "45", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iii", "practices": [ "concept/exponent", "concept/monomial", "method/division", "theorem/index-law-in-division", "theorem/law-of-signs-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18", "set": "18", "page": "48", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "practices": [ "concept/coefficient", "concept/like-terms", "method/addition", "method/arranging-in-columns", "method/collecting-like-terms" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19", "set": "19", "page": "50", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "practices": [ "concept/coefficient", "concept/like-terms", "method/arranging-in-columns", "method/collecting-like-terms", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-20", "set": "20", "page": "52", "chapter": "wentworth-first-steps-in-algebra-1894/ch-iv", "practices": [ "concept/parenthesis", "method/collecting-like-terms", "theorem/removing-parentheses-preceded-by-minus", "theorem/removing-parentheses-preceded-by-plus" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21", "set": "21", "page": "53", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "practices": [ "concept/monomial", "concept/polynomial", "method/multiplying-a-compound-expression", "theorem/index-law-in-multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22", "set": "22", "page": "56", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "practices": [ "concept/like-terms", "concept/polynomial", "method/collecting-like-terms", "method/multiplying-a-compound-expression", "theorem/index-law-in-multiplication", "theorem/law-of-signs-in-multiplication" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23", "set": "23", "page": "58", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "practices": [ "concept/monomial", "concept/polynomial", "method/dividing-a-polynomial-by-a-monomial", "method/division", "theorem/index-law-in-division", "theorem/law-of-signs-in-division" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24", "set": "24", "page": "62", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "practices": [ "concept/arrangement", "concept/dividend", "concept/divisor", "concept/polynomial", "concept/quotient", "concept/remainder", "method/dividing-a-polynomial-by-a-polynomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25", "set": "25", "page": "63", "chapter": "wentworth-first-steps-in-algebra-1894/ch-v", "practices": [ "concept/minuend", "concept/numerical-value", "concept/parenthesis", "concept/subtrahend", "method/addition", "method/collecting-like-terms", "method/dividing-a-polynomial-by-a-polynomial", "method/multiplying-a-compound-expression", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-26", "set": "26", "page": "65", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "practices": [ "theorem/difference-of-two-squares", "theorem/square-of-a-difference", "theorem/square-of-a-sum" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-27", "set": "27", "page": "67", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "practices": [ "method/multiplying-two-binomials", "method/writing-by-inspection" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-28", "set": "28", "page": "68", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "practices": [ "concept/divisibility", "method/writing-by-inspection", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-29", "set": "29", "page": "69", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "practices": [ "method/writing-by-inspection", "theorem/difference-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30", "set": "30", "page": "70", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vi", "practices": [ "method/dividing-a-polynomial-by-a-polynomial", "method/writing-by-inspection", "theorem/sum-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31", "set": "31", "page": "72", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "method/factoring-by-taking-out-a-common-factor" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-32", "set": "32", "page": "73", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "method/factoring-by-grouping" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-33", "set": "33", "page": "74", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "concept/root", "method/factoring", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-34", "set": "34", "page": "75", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-35", "set": "35", "page": "77", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "theorem/difference-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-36", "set": "36", "page": "78", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "theorem/sum-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37", "set": "37", "page": "80", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "method/factoring-a-perfect-square-trinomial" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38", "set": "38", "page": "82", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "method/factoring-a-trinomial-of-the-form-x-2-ax-b" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-39", "set": "39", "page": "83", "chapter": "wentworth-first-steps-in-algebra-1894/ch-vii", "practices": [ "method/factoring", "method/factoring-a-perfect-square-trinomial", "method/factoring-a-trinomial-of-the-form-x-2-ax-b", "method/factoring-by-grouping", "method/factoring-by-taking-out-a-common-factor", "theorem/difference-of-two-cubes", "theorem/difference-of-two-squares", "theorem/sum-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40", "set": "40", "page": "86", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "practices": [ "concept/highest-common-factor", "method/factoring", "method/factoring-a-perfect-square-trinomial", "method/factoring-a-trinomial-of-the-form-x-2-ax-b", "method/factoring-by-grouping", "method/factoring-by-taking-out-a-common-factor", "method/finding-the-highest-common-factor", "theorem/difference-of-two-cubes", "theorem/difference-of-two-squares", "theorem/sum-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41", "set": "41", "page": "88", "chapter": "wentworth-first-steps-in-algebra-1894/ch-viii", "practices": [ "concept/lowest-common-multiple", "method/factoring", "method/factoring-a-perfect-square-trinomial", "method/factoring-a-trinomial-of-the-form-x-2-ax-b", "method/factoring-by-taking-out-a-common-factor", "method/finding-the-lowest-common-multiple", "theorem/difference-of-two-cubes", "theorem/difference-of-two-squares", "theorem/sum-of-two-cubes" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42", "set": "42", "page": "90", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/algebraic-fraction", "method/factoring", "method/factoring-a-perfect-square-trinomial", "method/factoring-a-trinomial-of-the-form-x-2-ax-b", "method/factoring-by-grouping", "method/factoring-by-taking-out-a-common-factor", "method/reducing-a-fraction-to-lowest-terms", "theorem/difference-of-two-cubes", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43", "set": "43", "page": "91", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/integral-expression", "concept/remainder", "method/dividing-a-polynomial-by-a-polynomial", "method/division", "method/reducing-a-fraction-to-an-integral-or-mixed-expression" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44", "set": "44", "page": "92", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/algebraic-fraction", "concept/integral-expression", "method/reducing-a-mixed-expression-to-a-fraction", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45", "set": "45", "page": "94", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/lowest-common-denominator", "method/factoring", "method/finding-the-lowest-common-multiple", "method/reducing-fractions-to-lowest-common-denominator", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46", "set": "46", "page": "95", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/lowest-common-denominator", "method/adding-fractions", "method/addition", "method/collecting-like-terms", "method/subtraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-47", "set": "47", "page": "96", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/lowest-common-denominator", "method/adding-fractions", "method/factoring", "method/reducing-fractions-to-lowest-common-denominator", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48", "set": "48", "page": "97", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/lowest-common-denominator", "method/adding-fractions", "method/factoring", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49", "set": "49", "page": "100", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/reciprocal", "method/dividing-by-a-fraction", "method/factoring", "method/multiplying-fractions", "method/reducing-a-fraction-to-lowest-terms", "theorem/difference-of-two-squares" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50", "set": "50", "page": "102", "chapter": "wentworth-first-steps-in-algebra-1894/ch-ix", "practices": [ "concept/complex-fraction", "concept/lowest-common-denominator", "method/reducing-a-fraction-to-lowest-terms", "method/simplifying-a-complex-fraction" ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-51", "set": 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"shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/15" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/21", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "10", "location": "Exercise 1, problem 21", "problem_latex": " $15 - (10 - 3 - 2)$.", "markdown": "$15 - (10 - 3 - 2)$.", "answer_latex": [ " $10$." ], "answer_markdown": [ "$10$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "15 - (10 - 3 - 2)", "answer_expr": "10" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 10" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/2" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/3", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "10", "location": "Exercise 1, problem 3", "problem_latex": " $7 + (5 + 1)$.", "markdown": "$7 + (5 + 1)$.", "answer_latex": [ " $13$." ], "answer_markdown": [ "$13$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "7 + (5 + 1)", "answer_expr": "13" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 13" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/5" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/4", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & 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"judge_why": null, "problem_expr": "6 + (4 + 3)", "answer_expr": "13" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 13" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/3" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/6", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "10", "location": "Exercise 1, problem 6", "problem_latex": " $6 + (4 - 3)$.", "markdown": "$6 + (4 - 3)$.", "answer_latex": [ " $7$." ], "answer_markdown": [ "$7$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "6 + (4 - 3)", "answer_expr": "7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 7" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/8" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/7", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "10", "location": "Exercise 1, problem 7", "problem_latex": " $3 + (8 - 2)$.", "markdown": "$3 + (8 - 2)$.", "answer_latex": [ " $9$." ], "answer_markdown": [ "$9$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3 + (8 - 2)", "answer_expr": "9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 9" ], "shape": [ "identity: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/8", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "10", "location": "Exercise 1, problem 8", "problem_latex": " $9 - (8 - 6)$.", "markdown": "$9 - (8 - 6)$.", "answer_latex": [ " $7$." ], "answer_markdown": [ "$7$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "9 - (8 - 6)", "answer_expr": "7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 7" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/6" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-1/9", "set": "wentworth-first-steps-in-algebra-1894/ex-1", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "10", "location": "Exercise 1, problem 9", "problem_latex": " $10 - (9 - 5)$.", "markdown": "$10 - (9 - 5)$.", "answer_latex": [ " $6$." ], "answer_markdown": [ "$6$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "10 - (9 - 5)", "answer_expr": "6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 6" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-1/12" ], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/1", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 1", "problem_latex": "If a number is multiplied by~$9$, the product is~$270$.\nFind the number.", "markdown": "If a number is multiplied by $9$, the product is $270$. Find the number.", "answer_latex": [ "$30$." ], "answer_markdown": [ "$30$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 30}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(9*x, 270)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "270 ENTER 9 ÷" ], "calculator_value": "+30E+0", "printed_value": "30", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/10", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 10", "problem_latex": "One number is $4$~times another, and their difference\nis~$30$. Find the numbers.", "markdown": "One number is $4$ times another, and their difference is $30$. Find the numbers.", "answer_latex": [ "$10$, $40$." ], "answer_markdown": [ "$10$, $40$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 40, y: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 4*b), Eq(a - b, 30))" ], "shape": [ "solve: (Eq(a, N*b), Eq(a - b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/11", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 11", "problem_latex": "The sum of two numbers is~$36$, and one of them\nexceeds twice the other by~$6$. Find the numbers.", "markdown": "The sum of two numbers is $36$, and one of them exceeds twice the other by $6$. Find the numbers.", "answer_latex": [ "$26$, $10$." ], "answer_markdown": [ "$26$, $10$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 26, y: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 2*b + 6), Eq(a + b, 36))" ], "shape": [ "solve: (Eq(a, N*b + N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/12", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 12", "problem_latex": "The sum of two numbers is~$40$, and $5$~times the\nsmaller exceeds $2$~times the greater by~$25$. Find the\nnumbers.", "markdown": "The sum of two numbers is $40$, and $5$ times the smaller exceeds $2$ times the greater by $25$. Find the numbers.", "answer_latex": [ "$25$, $15$." ], "answer_markdown": [ "$25$, $15$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 25, y: 15}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(-2*a + 5*b, 25), Eq(a + b, 40))" ], "shape": [ "solve: (Eq(N*a + N*b, N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/13", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 13", "problem_latex": "The number $30$ is divided into two parts such that\n$4$~times the greater part exceeds $5$~times the smaller part\nby~$30$. Find the parts.", "markdown": "The number $30$ is divided into two parts such that $4$ times the greater part exceeds $5$ times the smaller part by $30$. Find the parts.", "answer_latex": [ "$10$, $20$." ], "answer_markdown": [ "$10$, $20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20, y: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + b, 30), Eq(4*a - 5*b, 30))" ], "shape": [ "solve: (Eq(a + b, N), Eq(N*a + N*b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/14", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 14", "problem_latex": "The sum of two numbers is~$27$, and twice the greater\nnumber increased by $3$~times the less is~$61$. Find the\nnumbers.", "markdown": "The sum of two numbers is $27$, and twice the greater number increased by $3$ times the less is $61$. Find the numbers.", "answer_latex": [ "$7$, $20$." ], "answer_markdown": [ "$7$, $20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20, y: 7}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + b, 27), Eq(2*a + 3*b, 61))" ], "shape": [ "solve: (Eq(a + b, N), Eq(N*a + N*b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/15", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 15", "problem_latex": "The sum of two numbers is~$32$, and five times the\nsmaller is $3$~times the greater number. Find the numbers.", "markdown": "The sum of two numbers is $32$, and five times the smaller is $3$ times the greater number. Find the numbers.", "answer_latex": [ "$12$, $20$." ], "answer_markdown": [ "$12$, $20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20, y: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(5*b, 3*a), Eq(a + b, 32))" ], "shape": [ "solve: (Eq(N*b, N*a), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/2", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 2", "problem_latex": "If the sum of the ages of a father and son is $60$~years,\nand the father is $5$~times as old as the son, what is the\nage of each?", "markdown": "If the sum of the ages of a father and son is $60$ years, and the father is $5$ times as old as the son, what is the age of each?", "answer_latex": [ "Son, $10$~yr.; father, $50$~yr." ], "answer_markdown": [ "Son, $10$ yr.; father, $50$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{s: 10, f: 50}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 5*b), Eq(a + b, 60))" ], "shape": [ "solve: (Eq(a, N*b), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/3", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 3", "problem_latex": "The sum of two numbers is~$91$, and the greater is $6$~times\nthe less. Find the numbers.", "markdown": "The sum of two numbers is $91$, and the greater is $6$ times the less. Find the numbers.", "answer_latex": [ "$78$, $13$." ], "answer_markdown": [ "$78$, $13$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 78, y: 13}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 6*b), Eq(a + b, 91))" ], "shape": [ "solve: (Eq(a, N*b), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/4", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 4", "problem_latex": "A tree $90$~feet high was broken so that the part\nbroken off was $8$~times the length of the part left standing.\nFind the length of each part.", "markdown": "A tree $90$ feet high was broken so that the part broken off was $8$ times the length of the part left standing. Find the length of each part.", "answer_latex": [ "$80$~ft.\\ broken off; $10$~ft.\\ standing." ], "answer_markdown": [ "$80$ ft. broken off; $10$ ft. standing." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{b: 80, s: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 8*b), Eq(a + b, 90))" ], "shape": [ "solve: (Eq(a, N*b), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/5", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 5", "problem_latex": "The difference of two numbers is~$7$, and their sum is~$53$.\nFind the numbers.", "markdown": "The difference of two numbers is $7$, and their sum is $53$. Find the numbers.", "answer_latex": [ "$23$, $30$." ], "answer_markdown": [ "$23$, $30$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 30, y: 23}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a - b, 7), Eq(a + b, 53))" ], "shape": [ "solve: (Eq(a - b, N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/6", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 6", "problem_latex": "The difference of two numbers is~$12$, and their sum is~$84$.\nFind the numbers.", "markdown": "The difference of two numbers is $12$, and their sum is $84$. Find the numbers.", "answer_latex": [ "$36$, $48$." ], "answer_markdown": [ "$36$, $48$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 48, y: 36}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a - b, 12), Eq(a + b, 84))" ], "shape": [ "solve: (Eq(a - b, N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/7", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 7", "problem_latex": "Divide $35$ into two parts so that one part shall be\ngreater by~$5$ than the other part.", "markdown": "Divide $35$ into two parts so that one part shall be greater by $5$ than the other part.", "answer_latex": [ "$15$, $20$." ], "answer_markdown": [ "$15$, $20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20, y: 15}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a - b, 5), Eq(a + b, 35))" ], "shape": [ "solve: (Eq(a - b, N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/8", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 8", "problem_latex": "Three times a given number is equal to the number\nincreased by~$40$. Find the number.", "markdown": "Three times a given number is equal to the number increased by $40$. Find the number.", "answer_latex": [ "$20$." ], "answer_markdown": [ "$20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(3*x, x + 40)" ], "shape": [ "solve: Eq(N*x, N + x)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "40 ENTER 3 ENTER 1 −", "÷" ], "constants": [ { "value": "1", "source": "the coefficient of the unwritten x on the right side: 'the number increased by 40' is x + 40 = 1·x + 40, so moving that x across gives 3x − 1x = 40, i.e. x = 40 ÷ (3 − 1)" } ], "calculator_value": "+20E+0", "printed_value": "20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-10/9", "set": "wentworth-first-steps-in-algebra-1894/ex-10", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "27", "location": "Exercise 10, problem 9", "problem_latex": "Three times a given number diminished by~$24$ is\nequal to the given number. Find the number.", "markdown": "Three times a given number diminished by $24$ is equal to the given number. Find the number.", "answer_latex": [ "$12$." ], "answer_markdown": [ "$12$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(3*x - 24, x)" ], "shape": [ "solve: Eq(N*x + N, x)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "24 ENTER 3 ENTER 1 −", "÷" ], "constants": [ { "value": "3", "source": "the 3 in 'Three times' (3x)" }, { "value": "1", "source": "the coefficient of x on the right side (x = 1·x), moved across: 3x − x = 24, so 2x = 24 and x = 24 ÷ 2 (the 2 is computed as 3 − 1 on the calculator)" } ], "calculator_value": "+12E+0", "printed_value": "12", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/1", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 1", "problem_latex": "A farmer sold a horse and a cow for~\\$$210$. He\nsold the horse for four times as much as the cow. How\nmuch did he get for each?", "markdown": "A farmer sold a horse and a cow for $$210$. He sold the horse for four times as much as the cow. How much did he get for each?", "answer_latex": [ "Cow, \\$$42$; horse, \\$$168$." ], "answer_markdown": [ "Cow, $$42$; horse, $$168$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{c: 42, h: 168}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(b, 4*a), Eq(a + b, 210))" ], "shape": [ "solve: (Eq(b, N*a), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/10", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 10", "problem_latex": "A man $50$~years old has a son $10$~years old. In\nhow many years will the father be three times as old as\nthe son?", "markdown": "A man $50$ years old has a son $10$ years old. In how many years will the father be three times as old as the son?", "answer_latex": [ "$10$." ], "answer_markdown": [ "$10$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(50 + x, 3*(10 + x))", "answer_expr": "10" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x + 50, 3*x + 30)" ], "shape": [ "solve: Eq(N + x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=10", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "50 ENTER 3 ENTER 10 × −", "3 ENTER 1 −", "÷" ], "constants": [ { "value": "1", "source": "the coefficient of x on the left side (1·x), moved across so that 3x − x = 2x; the working is x = (50 − 3·10)/(3 − 1)" } ], "calculator_value": "+10E+0", "printed_value": "10", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/11", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 11", "problem_latex": "A~has~\\$$100$, and B~has~\\$$20$. How much must A\ngive B in order that they may each have the same sum?", "markdown": "A has $$100$, and B has $$20$. How much must A give B in order that they may each have the same sum?", "answer_latex": [ "\\$$40$." ], "answer_markdown": [ "$$40$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(100 - x, 20 + x)", "answer_expr": "40" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-x + 100, x + 20)" ], "shape": [ "solve: Eq(N - x, N + x)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=40", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "100 ENTER 20 −", "2 ÷" ], "constants": [ { "value": "2", "source": "the working: 100 − x = 20 + x gives 2x = 100 − 20, so x = (100 − 20)/2; the 2 is the two x's on the right-hand side (x + x)" } ], "calculator_value": "+40E+0", "printed_value": "40", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/12", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 12", "problem_latex": "A banker paid \\$$63$ in $5$-dollar bills and $2$-dollar\nbills. He paid just as many $5$-dollar bills as $2$-dollar bills.\nHow many bills of each kind did he pay?", "markdown": "A banker paid $$63$ in $5$-dollar bills and $2$-dollar bills. He paid just as many $5$-dollar bills as $2$-dollar bills. How many bills of each kind did he pay?", "answer_latex": [ "$9$." ], "answer_markdown": [ "$9$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(5*x + 2*x, 63)", "answer_expr": "9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(7*x, 63)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=9", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "63 ENTER 5 ENTER 2 + ÷" ], "constants": [ { "value": "5", "source": "the $5-dollar bills in the problem text (coefficient of x for the 5-dollar bills)" }, { "value": "2", "source": "the $2-dollar bills in the problem text (coefficient of x for the 2-dollar bills)" } ], "calculator_value": "+9E+0", "printed_value": "9", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/2", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 2", "problem_latex": "Three times the excess of a certain number over~$6$\nis equal to the number plus~$144$. Find the number.", "markdown": "Three times the excess of a certain number over $6$ is equal to the number plus $144$. Find the number.", "answer_latex": [ "$81$." ], "answer_markdown": [ "$81$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*(x - 6), x + 144)", "answer_expr": "81" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x - 18, x + 144)" ], "shape": [ "solve: Eq(N*x + N, N + x)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=81", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "144 ENTER 3 ENTER 6 × +", "3 ENTER 1 − ÷" ], "constants": [ { "value": "3", "source": "the 3 in 'Three times the excess' (coefficient of the left side, 3(x − 6))" }, { "value": "6", "source": "the 6 in 'excess of a certain number over 6'" }, { "value": "144", "source": "the 144 in 'the number plus 144'" }, { "value": "1", "source": "the implicit coefficient of x on the right side (x + 144); used in 3x − x = 2x, so 2 = 3 − 1" } ], "calculator_value": "+81E+0", "printed_value": "81", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/3", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 3", "problem_latex": "Thirty-one times a certain number is as much above~$40$\nas nine times the number is below~$40$. Find the number.", "markdown": "Thirty-one times a certain number is as much above $40$ as nine times the number is below $40$. Find the number.", "answer_latex": [ "$2$." ], "answer_markdown": [ "$2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(31*x - 40, 40 - 9*x)", "answer_expr": "2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(31*x - 40, -9*x + 40)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=2", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "40 ENTER 40 +", "31 ENTER 9 +", "÷" ], "calculator_value": "+2E+0", "printed_value": "2", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/4", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 4", "problem_latex": "Two numbers differ by~$10$, and their sum is equal to\nseven times their difference. Find the numbers.", "markdown": "Two numbers differ by $10$, and their sum is equal to seven times their difference. Find the numbers.", "answer_latex": [ "$30$, $40$." ], "answer_markdown": [ "$30$, $40$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 40, y: 30}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a - b, 10), Eq(a + b, 7*a - 7*b))" ], "shape": [ "solve: (Eq(a - b, N), Eq(a + b, N*a + N*b))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/5", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 5", "problem_latex": "Find three consecutive numbers, $x$, $x + 1$, and~$x + 2$,\nwhose sum is~$78$.", "markdown": "Find three consecutive numbers, $x$, $x + 1$, and $x + 2$, whose sum is $78$.", "answer_latex": [ "$25$, $26$, $27$." ], "answer_markdown": [ "$25$, $26$, $27$." ], "checks": [ { "task": "solve", "verdict": "FLAG-MISMATCH", "judge_why": "Eq(3*x + 3, 78) fails at {x: 26}: leaves 3.00000000000", "problem_expr": "Eq(x + (x + 1) + (x + 2), 78)", "answer_expr": "[25, 26, 27]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "solve: Eq(3*x + 3, 78)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/6", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 6", "problem_latex": "Find five consecutive numbers whose sum is~$35$.", "markdown": "Find five consecutive numbers whose sum is $35$.", "answer_latex": [ "$5$, $6$, $7$, $8$, $9$." ], "answer_markdown": [ "$5$, $6$, $7$, $8$, $9$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x + (x + 1) + (x + 2) + (x + 3) + (x + 4), 35)", "answer_expr": "5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(5*x + 10, 35)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/7", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 7", "problem_latex": "The sum of the ages of A~and~B is $40$~years, and $10$~years\nhence A~will be twice as old as~B\\@. Find their\npresent ages.", "markdown": "The sum of the ages of A and B is $40$ years, and $10$ years hence A will be twice as old as B. Find their present ages.", "answer_latex": [ "A, $30$~yr.; B, $10$~yr." ], "answer_markdown": [ "A, $30$ yr.; B, $10$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{A: 30, B: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a + 10, 2*b + 20), Eq(a + b, 40))" ], "shape": [ "solve: (Eq(N + a, N*b + N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/8", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 8", "problem_latex": "A father is four times as old as his son, and in $5$~years\nhe will be only three times as old. Find their present ages.", "markdown": "A father is four times as old as his son, and in $5$ years he will be only three times as old. Find their present ages.", "answer_latex": [ "Father, $40$~yr., son, $10$~yr." ], "answer_markdown": [ "Father, $40$ yr., son, $10$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{F: 40, S: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, 4*b), Eq(a + 5, 3*b + 15))" ], "shape": [ "solve: (Eq(a, N*b), Eq(N + a, N*b + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-11/9", "set": "wentworth-first-steps-in-algebra-1894/ex-11", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "29", "location": "Exercise 11, problem 9", "problem_latex": "One man is $60$~years old, and another man is $50$~years.\nHow many years ago was the first man twice as\nold as the second?", "markdown": "One man is $60$ years old, and another man is $50$ years. How many years ago was the first man twice as old as the second?", "answer_latex": [ "$40$." ], "answer_markdown": [ "$40$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(60 - x, 2*(50 - x))", "answer_expr": "40" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-x + 60, -2*x + 100)" ], "shape": [ "solve: Eq(N - x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=40", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 50 ×", "60 −" ], "constants": [ { "value": "2", "source": "the 'twice' in 'twice as old'" }, { "value": "50", "source": "the second man's age, '50 years'" }, { "value": "60", "source": "the first man's age, '60 years'" } ], "calculator_value": "+40E+0", "printed_value": "40", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/1", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 1", "problem_latex": "In a company of $90$~persons, composed of men,\nwomen, and children, there are three times as many children\nas men, and twice as many women as men. How many\nare there of each?", "markdown": "In a company of $90$ persons, composed of men, women, and children, there are three times as many children as men, and twice as many women as men. How many are there of each?", "answer_latex": [ "$15$~men; $30$~women; $45$~children." ], "answer_markdown": [ "$15$ men; $30$ women; $45$ children." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{m: 15, w: 30, c: 45}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 3*x), Eq(b, 2*x), Eq(a + b + x, 90))" ], "shape": [ "solve: (Eq(a, N*x), Eq(b, N*x), Eq(a + b + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/2", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 2", "problem_latex": "Find the number whose double exceeds~$70$ by as\nmuch as the number itself is less than~$80$.", "markdown": "Find the number whose double exceeds $70$ by as much as the number itself is less than $80$.", "answer_latex": [ "$50$." ], "answer_markdown": [ "$50$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(2*x - 70, 80 - x)", "answer_expr": "50" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2*x - 70, -x + 80)" ], "shape": [ "solve: Eq(N*x + N, N - x)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=50", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "80 ENTER 70 +", "2 ENTER 1 +", "÷" ], "constants": [ { "value": "80", "source": "the 80 in 'less than 80'" }, { "value": "70", "source": "the 70 in 'exceeds 70'" }, { "value": "2", "source": "the 2 in 'double' (the number's double is 2x)" }, { "value": "1", "source": "the implicit coefficient of x in 'the number itself' (x = 1x), added to the 2 to collect the x-terms" } ], "calculator_value": "+50E+0", "printed_value": "50", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/3", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 3", "problem_latex": "A farmer employed two men to build $112$~rods of\nwall. One of them built on the average $4$~rods a day, and\nthe other $3$~rods a day. How many days did they work?", "markdown": "A farmer employed two men to build $112$ rods of wall. One of them built on the average $4$ rods a day, and the other $3$ rods a day. How many days did they work?", "answer_latex": [ "$16$." ], "answer_markdown": [ "$16$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(4*x + 3*x, 112)", "answer_expr": "16" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(7*x, 112)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=16", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "112 ENTER 4 ENTER 3 + ÷" ], "calculator_value": "+16E+0", "printed_value": "16", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/4", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 4", "problem_latex": "Two men travel in \\emph{opposite} directions, one $30$~miles\na day, and the other $20$~miles a day. In how many days\nwill they be $350$~miles apart?", "markdown": "Two men travel in *opposite* directions, one $30$ miles a day, and the other $20$ miles a day. In how many days will they be $350$ miles apart?", "answer_latex": [ "$7$." ], "answer_markdown": [ "$7$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(30*x + 20*x, 350)", "answer_expr": "7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(50*x, 350)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=7", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/5", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 5", "problem_latex": "Two men travel in the same direction, one $30$~miles\na day, and the other $20$~miles a day. In how many days\nwill they be $350$~miles apart?", "markdown": "Two men travel in the same direction, one $30$ miles a day, and the other $20$ miles a day. In how many days will they be $350$ miles apart?", "answer_latex": [ "$35$." ], "answer_markdown": [ "$35$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(30*x - 20*x, 350)", "answer_expr": "35" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(10*x, 350)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=35", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "350 ENTER 30 ENTER 20 −", "÷" ], "calculator_value": "+35E+0", "printed_value": "35", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/6", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 6", "problem_latex": "A man bought $3$~equal lots of hay for~\\$$408$. For\nthe first lot he gave \\$$17$~a ton, for the second~\\$$16$, for the\nthird~\\$$18$. How many tons did he buy in all?", "markdown": "A man bought $3$ equal lots of hay for $$408$. For the first lot he gave $$17$ a ton, for the second $$16$, for the third $$18$. How many tons did he buy in all?", "answer_latex": [ "$24$." ], "answer_markdown": [ "$24$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(17*(t/3) + 16*(t/3) + 18*(t/3), 408)", "answer_expr": "24" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(17*x, 408)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=24", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/7", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 7", "problem_latex": "A farmer sold a quantity of wood for~\\$$84$, one half\nof it at \\$$3$~a cord, and the other half at \\$$4$~a cord. How\nmany cords did he sell?", "markdown": "A farmer sold a quantity of wood for $$84$, one half of it at $$3$ a cord, and the other half at $$4$ a cord. How many cords did he sell?", "answer_latex": [ "$24$." ], "answer_markdown": [ "$24$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(3*(t/2) + 4*(t/2), 84)", "answer_expr": "24" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(7*x/2, 84)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=24", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/8", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 8", "problem_latex": "If $2x - 3$ stands for~$29$, for what number will\n$4 + x$ stand?", "markdown": "If $2x - 3$ stands for $29$, for what number will $4 + x$ stand?", "answer_latex": [ "$20$." ], "answer_markdown": [ "$20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(2*(y - 4) - 3, 29)", "answer_expr": "20" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2*x - 11, 29)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": "X=20", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "29 ENTER 3 + 2 ÷", "4 +" ], "calculator_value": "+20E+0", "printed_value": "20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-12/9", "set": "wentworth-first-steps-in-algebra-1894/ex-12", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "30", "location": "Exercise 12, problem 9", "problem_latex": "At an election two opposing candidates received\ntogether $2044$~votes, and one received $104$~more votes than\nthe other. How many votes did each candidate receive?", "markdown": "At an election two opposing candidates received together $2044$ votes, and one received $104$ more votes than the other. How many votes did each candidate receive?", "answer_latex": [ "$970$; $1074$." ], "answer_markdown": [ "$970$; $1074$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 970, y: 1074}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a - x, 104), Eq(a + x, 2044))" ], "shape": [ "solve: (Eq(a - x, N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/1", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 1", "problem_latex": " A man walks $4$~miles an hour for $x$~hours, and\nanother man walks $3$~miles an hour for $x + 2$~hours. If\nthey each walk the same distance, how many miles does\neach walk?", "markdown": "A man walks $4$ miles an hour for $x$ hours, and another man walks $3$ miles an hour for $x + 2$ hours. If they each walk the same distance, how many miles does each walk?", "answer_latex": [ "$24$." ], "answer_markdown": [ "$24$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{d: 24}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 4*a), Eq(x, 3*a + 6))" ], "shape": [ "solve: (Eq(x, N*a), Eq(x, N*a + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 2 ×", "4 ENTER 3 − ÷", "4 ×" ], "constants": [ { "value": "3", "source": "the 3 in 'walks 3 miles an hour' (the coefficient in d = 3(x + 2))" }, { "value": "2", "source": "the 2 in 'x + 2 hours' (the +2 in d = 3(x + 2))" }, { "value": "4", "source": "the 4 in 'walks 4 miles an hour' (the coefficient in d = 4x); used in the subtraction 4 − 3 from 4x = 3(x + 2), and again as the final multiplier d = 4x" } ], "calculator_value": "+24E+0", "printed_value": "24", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/2", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 2", "problem_latex": " A has twice as much money as~B; but if A gives~B\n\\$$30$, it will take twice as much as A has left to equal~B's.\nHow much money has each?", "markdown": "A has twice as much money as B; but if A gives B $$30$, it will take twice as much as A has left to equal B’s. How much money has each?", "answer_latex": [ "A, \\$$60$; B, \\$$30$." ], "answer_markdown": [ "A, $$60$; B, $$30$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{a: 60, b: 30}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 2*a), Eq(a + 30, 2*x - 60))" ], "shape": [ "solve: (Eq(x, N*a), Eq(N + a, N*x + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/3", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 3", "problem_latex": " I have \\$$12.75$ in two-dollar bills and twenty-five\ncent pieces, and I have twice as many bills as twenty-five\ncent pieces. How many have I of each?", "markdown": "I have $$12.75$ in two-dollar bills and twenty-five cent pieces, and I have twice as many bills as twenty-five cent pieces. How many have I of each?", "answer_latex": [ "\\$$3$~quarters; $6$~bills." ], "answer_markdown": [ "$$3$ quarters; $6$ bills." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{q: 3, b: 6}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 2*x), Eq(2*a + x/4, 51/4))" ], "shape": [ "solve: (Eq(a, N*x), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/4", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 4", "problem_latex": " I have in mind a certain number. If this number\nis diminished by~$8$ and the remainder multiplied by~$8$, the\nresult is the same as if the number was diminished by~$6$\nand the remainder multiplied by~$6$. What is the number?", "markdown": "I have in mind a certain number. If this number is diminished by $8$ and the remainder multiplied by $8$, the result is the same as if the number was diminished by $6$ and the remainder multiplied by $6$. What is the number?", "answer_latex": [ "$14$." ], "answer_markdown": [ "$14$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(8*(n - 8), 6*(n - 6))", "answer_expr": "{n: 14}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(8*x - 64, 6*x - 36)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "8 GOLD x² 6 GOLD x² −", "8 ENTER 6 − ÷" ], "constants": [ { "value": "8", "source": "the 8 in 'diminished by 8' and 'multiplied by 8' in the problem" }, { "value": "6", "source": "the 6 in 'diminished by 6' and 'multiplied by 6' in the problem" } ], "calculator_value": "+14E+0", "printed_value": "14", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/5", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 5", "problem_latex": " I have five times as many half-dollars as quarters, and\nthe half-dollars and quarters amount to~\\$$11$. How\nmany of each have I?", "markdown": "I have five times as many half-dollars as quarters, and the half-dollars and quarters amount to $$11$. How many of each have I?", "answer_latex": [ "\\$$4$~quarters; $20$~half-dollars." ], "answer_markdown": [ "$$4$ quarters; $20$ half-dollars." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{q: 4, h: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 5*x), Eq(a/2 + x/4, 11))" ], "shape": [ "solve: (Eq(a, N*x), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/6", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 6", "problem_latex": " A man pays a debt of~\\$$91$ with ten-dollar bills and\none-dollar bills, paying three times as many one-dollar bills\nas ten-dollar bills. How many bills of each kind does he\npay?", "markdown": "A man pays a debt of $$91$ with ten-dollar bills and one-dollar bills, paying three times as many one-dollar bills as ten-dollar bills. How many bills of each kind does he pay?", "answer_latex": [ "\\$$7$~ten-dollar bills; $21$~one-dollar bills." ], "answer_markdown": [ "$$7$ ten-dollar bills; $21$ one-dollar bills." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{t: 7, o: 21}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 3*x), Eq(a + 10*x, 91))" ], "shape": [ "solve: (Eq(a, N*x), Eq(N*x + a, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/7", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 7", "problem_latex": " A father is four times as old as his son, but $4$~years\nhence he will be only three times as old as his son. How\nold is each?", "markdown": "A father is four times as old as his son, but $4$ years hence he will be only three times as old as his son. How old is each?", "answer_latex": [ "Father, $32$~yr.; son, $8$~yr." ], "answer_markdown": [ "Father, $32$ yr.; son, $8$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{f: 32, s: 8}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 4*a), Eq(x + 4, 3*a + 12))" ], "shape": [ "solve: (Eq(x, N*a), Eq(N + x, N*a + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-13/8", "set": "wentworth-first-steps-in-algebra-1894/ex-13", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "31", "location": "Exercise 13, problem 8", "problem_latex": " A workman was employed for $24$~days. For every\nday he worked he was to receive~\\$$1.50$, and for every day\nhe was idle he was to pay \\$$0.50$ for his board. At the end\nof the time he received~\\$$28$. How many days did he\nwork?", "markdown": "A workman was employed for $24$ days. For every day he worked he was to receive $$1.50$, and for every day he was idle he was to pay $$0.50$ for his board. At the end of the time he received $$28$. How many days did he work?", "answer_latex": [ "$20$." ], "answer_markdown": [ "$20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(Rational(3,2)*w - Rational(1,2)*(24 - w), 28)", "answer_expr": "{w: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2*x - 12, 28)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "0.5 ENTER 24 × 28 +", "1.5 ENTER 0.5 + ÷" ], "calculator_value": "+20E+0", "printed_value": "20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/1", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 1", "problem_latex": "A boy has $4$~hours at his disposal. How far can he\nride into the country at the rate of $9$~miles an hour and\nwalk back at the rate of $3$~miles an hour, if he returns just\non time?", "markdown": "A boy has $4$ hours at his disposal. How far can he ride into the country at the rate of $9$ miles an hour and walk back at the rate of $3$ miles an hour, if he returns just on time?", "answer_latex": [ "$9$~miles." ], "answer_markdown": [ "$9$ miles." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{d: 9}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 9*a), Eq(x, -3*a + 12))" ], "shape": [ "solve: (Eq(x, N*a), Eq(x, N*a + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 4 ×", "9 ENTER 3 + ÷", "9 ×" ], "constants": [ { "value": "3", "source": "the 3 miles an hour walking rate in the problem text (also the 3 in Eq(d, 3*(4 - x)))" }, { "value": "4", "source": "the 4 hours at his disposal in the problem text (also the 4 in Eq(d, 3*(4 - x)))" } ], "calculator_value": "+9E+0", "printed_value": "9", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/2", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 2", "problem_latex": "A has \\$$180$, and B has \\$$80$. How much must A\ngive B in order that six times B's money shall be equal to\n$7$~times~A's?", "markdown": "A has $$180$, and B has $$80$. How much must A give B in order that six times B’s money shall be equal to $7$ times A’s?", "answer_latex": [ "\\$$60$." ], "answer_markdown": [ "$$60$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 60}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(6*x + 480, -7*x + 1260)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/3", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 3", "problem_latex": "A grocer has two kinds of tea, one kind worth $45$~cents\na pound, and the other worth $65$~cents a pound.\nHow many pounds of each kind must he take to make $80$~pounds,\nworth $50$~cents a pound?", "markdown": "A grocer has two kinds of tea, one kind worth $45$ cents a pound, and the other worth $65$ cents a pound. How many pounds of each kind must he take to make $80$ pounds, worth $50$ cents a pound?", "answer_latex": [ "\\$$20$~lb.\\ at $65$~cts.;\n $60$~lb.\\ at $45$~cts." ], "answer_markdown": [ "$$20$ lb. at $65$ cts.; $60$ lb. at $45$ cts." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20, y: 60}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + x, 80), Eq(45*a + 65*x, 4000))" ], "shape": [ "solve: (Eq(a + x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/4", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 4", "problem_latex": "A tank holding $1200$~gallons has three pipes. The\nfirst lets in $8$~gallons a minute, the second $10$~gallons, and\nthe third $12$~gallons a minute. In how many minutes will\nthe tank be filled?", "markdown": "A tank holding $1200$ gallons has three pipes. The first lets in $8$ gallons a minute, the second $10$ gallons, and the third $12$ gallons a minute. In how many minutes will the tank be filled?", "answer_latex": [ "$40$." ], "answer_markdown": [ "$40$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{t: 40}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(30*x, 1200)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1200 ENTER 8 ENTER 10 + 12 + ÷" ], "constants": [ { "value": "8", "source": "the 8 in 'the first lets in 8 gallons a minute'" }, { "value": "10", "source": "the 10 in 'the second 10 gallons'" }, { "value": "12", "source": "the 12 in 'the third 12 gallons a minute'" } ], "calculator_value": "+40E+0", "printed_value": "40", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/5", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 5", "problem_latex": "The fore and hind wheels of a carriage are $10$~feet\nand $12$~feet respectively in circumference. How many\nfeet will the carriage have passed over when the fore wheel\nhas made $250$~revolutions more than the hind wheel?", "markdown": "The fore and hind wheels of a carriage are $10$ feet and $12$ feet respectively in circumference. How many feet will the carriage have passed over when the fore wheel has made $250$ revolutions more than the hind wheel?", "answer_latex": [ "$15,000$." ], "answer_markdown": [ "$15,000$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{D: 15000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x/10, x/12 + 250)" ], "shape": [ "solve: Eq(N*x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "250 ENTER 10 × 12 ×", "12 ENTER 10 −", "÷" ], "calculator_value": "+15000E+0", "printed_value": "15000", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/6", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 6", "problem_latex": "Divide a yard of tape into two parts so that one part\nshall be $6$~inches longer than the other part.", "markdown": "Divide a yard of tape into two parts so that one part shall be $6$ inches longer than the other part.", "answer_latex": [ "$15$~in.; $21$~in." ], "answer_markdown": [ "$15$ in.; $21$ in." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 15, y: 21}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, x + 6), Eq(a + x, 36))" ], "shape": [ "solve: (Eq(a, N + x), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/7", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 7", "problem_latex": "A boy bought $7$~dozen oranges for~\\$$1.50$. For a part\nhe paid $20$~cents a dozen; and for the remainder, $25$~cents\na dozen. How many dozen of each kind did he buy?", "markdown": "A boy bought $7$ dozen oranges for $$1.50$. For a part he paid $20$ cents a dozen; and for the remainder, $25$ cents a dozen. How many dozen of each kind did he buy?", "answer_latex": [ "\\$$2$~doz.\\ at $25$~cts.;\n $5$~doz.\\ at $20$~cts." ], "answer_markdown": [ "$$2$ doz. at $25$ cts.; $5$ doz. at $20$ cts." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + x, 7), Eq(25*a + 20*x, 150))" ], "shape": [ "solve: (Eq(a + x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-14/8", "set": "wentworth-first-steps-in-algebra-1894/ex-14", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "32", "location": "Exercise 14, problem 8", "problem_latex": "How can a bill of~\\$$3.30$ be paid in quarters and ten-cent\npieces so as to pay three times as many ten-cent\npieces as quarters?", "markdown": "How can a bill of $$3.30$ be paid in quarters and ten-cent pieces so as to pay three times as many ten-cent pieces as quarters?", "answer_latex": [ "\\$$6$~quarters, $18$~ten-cent pieces." ], "answer_markdown": [ "$$6$ quarters, $18$ ten-cent pieces." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{q: 6, t: 18}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 3*x), Eq(10*a + 25*x, 330))" ], "shape": [ "solve: (Eq(a, N*x), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/1", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 1", "problem_latex": " $5c$, $23c$, $c$, $11c$.", "markdown": "$5c$, $23c$, $c$, $11c$.", "answer_latex": [ " $40c$." ], "answer_markdown": [ "$40c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "5*c + 23*c + c + 11*c", "answer_expr": "40*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 40*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/10", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 10", "problem_latex": " $5d$, $-d$, $-4d$, $2d$.", "markdown": "$5d$, $-d$, $-4d$, $2d$.", "answer_latex": [ " $2d$." ], "answer_markdown": [ "$2d$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "5*d - d - 4*d + 2*d", "answer_expr": "2*d" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/11", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 11", "problem_latex": " $13n$, $13n$, $-11n$, $-6n$, $-9n$, $n$, $2n$, $-3n$.", "markdown": "$13n$, $13n$, $-11n$, $-6n$, $-9n$, $n$, $2n$, $-3n$.", "answer_latex": [ " $0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "13*n + 13*n - 11*n - 6*n - 9*n + n + 2*n - 3*n", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 0" ], "shape": [ "identity: 0" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-15/15", "wentworth-first-steps-in-algebra-1894/ex-15/17", "wentworth-first-steps-in-algebra-1894/ex-18/3" ], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/12", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 12", "problem_latex": " $5g$, $-3g$, $-6g$, $-4g$, $20g$, $-5g$, $-11g$, $-14g$.", "markdown": "$5g$, $-3g$, $-6g$, $-4g$, $20g$, $-5g$, $-11g$, $-14g$.", "answer_latex": [ " $-18g$." ], "answer_markdown": [ "$-18g$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "5*g - 3*g - 6*g - 4*g + 20*g - 5*g - 11*g - 14*g", "answer_expr": "-18*g" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -18*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/13", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 13", "problem_latex": " $-9a^{2}$, $5a^{2}$, $6a^{2}$, $a^{2}$, $2a^{2}$, $-a^{2}$, $-3a^{2}$.", "markdown": "$-9a^{2}$, $5a^{2}$, $6a^{2}$, $a^{2}$, $2a^{2}$, $-a^{2}$, $-3a^{2}$.", "answer_latex": [ " $a^{2}$." ], "answer_markdown": [ "$a^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-9*a**2 + 5*a**2 + 6*a**2 + a**2 + 2*a**2 - a**2 - 3*a**2", "answer_expr": "a**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2" ], "shape": [ "identity: x**N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-17/1" ], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/14", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 14", "problem_latex": " $3x^{3}$, $-2x^{3}$, $-5x^{3}$, $-7x^{3}$, $-x^{3}$, $2x^{3}$, $-10x^{3}$, $-x^{3}$.", "markdown": "$3x^{3}$, $-2x^{3}$, $-5x^{3}$, $-7x^{3}$, $-x^{3}$, $2x^{3}$, $-10x^{3}$, $-x^{3}$.", "answer_latex": [ " $-21x^{3}$." ], "answer_markdown": [ "$-21x^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3*x**3 - 2*x**3 - 5*x**3 - 7*x**3 - x**3 + 2*x**3 - 10*x**3 - x**3", "answer_expr": "-21*x**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -21*x**3" ], "shape": [ "identity: N*x**N" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/15", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 15", "problem_latex": " $4a^{2}b^{2}$, $-a^{2}b^{2}$, $-6a^{2}b^{2}$, $4a^{2}b^{2}$, $-2a^{2}b^{2}$, $a^{2}b^{2}$.", "markdown": "$4a^{2}b^{2}$, $-a^{2}b^{2}$, $-6a^{2}b^{2}$, $4a^{2}b^{2}$, $-2a^{2}b^{2}$, $a^{2}b^{2}$.", "answer_latex": [ " $0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "4*a**2*b**2 - a**2*b**2 - 6*a**2*b**2 + 4*a**2*b**2 - 2*a**2*b**2 + a**2*b**2", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 0" ], "shape": [ "identity: 0" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-15/11", "wentworth-first-steps-in-algebra-1894/ex-15/17", "wentworth-first-steps-in-algebra-1894/ex-18/3" ], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/16", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 16", "problem_latex": " $6mn$, $-5mn$, $mn$, $-3mn$, $4mn$.", "markdown": "$6mn$, $-5mn$, $mn$, $-3mn$, $4mn$.", "answer_latex": [ " $3mn$." ], "answer_markdown": [ "$3mn$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "6*m*n - 5*m*n + m*n - 3*m*n + 4*m*n", "answer_expr": "3*m*n" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a*x" ], "shape": [ "identity: N*a*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/17", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 17", "problem_latex": " $3xyz$, $-2xyz$, $5xyz$, $-7xyz$, $xyz$.", "markdown": "$3xyz$, $-2xyz$, $5xyz$, $-7xyz$, $xyz$.", "answer_latex": [ " $0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3*x*y*z - 2*x*y*z + 5*x*y*z - 7*x*y*z + x*y*z", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 0" ], "shape": [ "identity: 0" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-15/11", "wentworth-first-steps-in-algebra-1894/ex-15/15", "wentworth-first-steps-in-algebra-1894/ex-18/3" ], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/18", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 18", "problem_latex": " $5a^{3}b^{3}c^{3}$, $-7a^{3}b^{3}c^{3}$, $-3a^{3}b^{3}c^{3}$, $2a^{3}b^{3}c^{3}$.", "markdown": "$5a^{3}b^{3}c^{3}$, $-7a^{3}b^{3}c^{3}$, $-3a^{3}b^{3}c^{3}$, $2a^{3}b^{3}c^{3}$.", "answer_latex": [ " $-3a^{3}b^{3}c^{3}$." ], "answer_markdown": [ "$-3a^{3}b^{3}c^{3}$." ], "checks": [ { "task": "identity", "verdict": 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"shape": [ "identity: N*x" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-15/1", "boyden-first-book-in-algebra-1895/ex-46/17" ], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/20", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 20", "problem_latex": " Subtract $-a$ from $-b$, and find the value of the\nresult if $a = -4$, $b = -5$.", "markdown": "Subtract $-a$ from $-b$, and find the value of the result if $a = -4$, $b = -5$.", "answer_latex": [ " $1$." ], "answer_markdown": [ "$1$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.0, printed 1 (half-unit 0.5; correctly rounded at the printed digits: 1.0)", "problem_expr": "-b - (-a)", "answer_expr": "1" } ], "verdict": "verified", "judge": { 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"verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a - 2*b at a=4, b=-2, c=-3" ], "shape": [ "evaluate: N*a + N*b" ], "same_problem_in": [], "needs": [ "cas.subst", "core.arith" ], "expectation": "X#12,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/22", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 22", "problem_latex": " $a + (-b) + c$ and $a - (-b) + c$.", "markdown": "$a + (-b) + c$ and $a - (-b) + c$.", "answer_latex": [ " $4$." ], "answer_markdown": [ "$4$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 4.0, printed 4 (half-unit 0.5; correctly rounded at the printed digits: 4.0)", "problem_expr": "(a + (-b) + c) - (a - (-b) + c)", "answer_expr": "4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -2*a at a=4, b=-2, c=-3" ], "shape": [ "evaluate: N*a" ], "same_problem_in": [], "needs": [ "cas.subst", "core.arith" ], "expectation": "X#4,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 2 +/− +/− + 3 +/− +", "4 ENTER 2 +/− +/− − 3 +/− +", "−" ], "constants": [ { "value": "4", "source": "a = 4 in the problem's given values" }, { "value": "2", "source": "b = −2 in the problem; typed as 2 then +/− twice, giving −b = 2 from the given b" }, { "value": "3", "source": "c = −3 in the problem; typed as 3 then +/− to give c" } ], "calculator_value": "+4E+0", "printed_value": "4", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": 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digits: 10.0)", "problem_expr": "(a - b + (-c)) - (a - (-b) - (-c))", "answer_expr": "10" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -2*a - 2*b at a=4, b=-2, c=-3" ], "shape": [ "evaluate: N*a + N*b" ], "same_problem_in": [], "needs": [ "cas.subst", "core.arith" ], "expectation": "X#10,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 2 +/− − 3 +", "4 ENTER 2 − 3 −", "−" ], "constants": [ { "value": "4", "source": "a = 4 in the givens" }, { "value": "2", "source": "the magnitude of b = −2 in the givens; typed as 2 then +/− in step 1 to give b, and used unsigned in step 2 for −(−b) = 2" }, { "value": "3", "source": "the magnitude of c = −3 in the givens; −c = 3 in step 1, and −(−c) = −3 subtracted in step 2" } ], "calculator_value": "+10E+0", "printed_value": "10", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", 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"checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "7*x + 12*x + 11*x + 9*x", "answer_expr": "39*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 39*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/4", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 4", "problem_latex": " $6y$, $8y$, $2y$, $35y$.", "markdown": "$6y$, $8y$, $2y$, $35y$.", "answer_latex": [ " $51y$." ], "answer_markdown": [ "$51y$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "6*y + 8*y + 2*y + 35*y", "answer_expr": "51*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 51*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/5", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 5", "problem_latex": " $-3a$, $-5a$, $-18a$.", "markdown": "$-3a$, $-5a$, $-18a$.", "answer_latex": [ " $-26a$." ], "answer_markdown": [ "$-26a$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-3*a - 5*a - 18*a", "answer_expr": "-26*a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -26*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/6", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 6", "problem_latex": " $-5x$, $-6x$, $-18x$, $-11x$.", "markdown": "$-5x$, $-6x$, $-18x$, $-11x$.", "answer_latex": [ " $-40x$." ], "answer_markdown": [ "$-40x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-5*x - 6*x - 18*x - 11*x", "answer_expr": "-40*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -40*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/7", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 7", "problem_latex": " $-3b$, $-b$, $-9b$, $-4b$.", "markdown": "$-3b$, $-b$, $-9b$, $-4b$.", "answer_latex": [ " $-17b$." ], "answer_markdown": [ "$-17b$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-3*b - b - 9*b - 4*b", "answer_expr": "-17*b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -17*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/8", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 8", "problem_latex": " $-z$, $-2z$, $-10z$, $-53z$.", "markdown": "$-z$, $-2z$, $-10z$, $-53z$.", "answer_latex": [ " $-66z$." ], "answer_markdown": [ "$-66z$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-z - 2*z - 10*z - 53*z", "answer_expr": "-66*z" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -66*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-15/9", "set": "wentworth-first-steps-in-algebra-1894/ex-15", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "39", "location": "Exercise 15, problem 9", "problem_latex": " $-11m$, $-3m$, $-5m$, $-m$.", "markdown": "$-11m$, $-3m$, $-5m$, $-m$.", "answer_latex": [ " $-20m$." ], "answer_markdown": [ "$-20m$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "-11*m - 3*m - 5*m - m", "answer_expr": "-20*m" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -20*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.collect", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/1", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 1", "problem_latex": "$5a^{2}$ and $6a^{3}$.", "markdown": "$5a^{2}$ and $6a^{3}$.", "answer_latex": [ "$30a^{5}$." ], "answer_markdown": [ "$30a^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(5*a**2)*(6*a**3)", "answer_expr": "30*a**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 30*a**5" ], "shape": [ "identity: N*a**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/10", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 10", "problem_latex": "$-2x^{6}y^{3}z$ and $-6xy^{2}z$.", "markdown": "$-2x^{6}y^{3}z$ and $-6xy^{2}z$.", "answer_latex": [ "$12x^{7}y^{5}z^{2}$." ], "answer_markdown": [ "$12x^{7}y^{5}z^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-2*x**6*y**3*z)*(-6*x*y**2*z)", "answer_expr": "12*x**7*y**5*z**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 12*a**7*b**5*c**2" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/11", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 11", "problem_latex": "$3a^{2}b$, $-5ab^{2}$, and $-7a^{4}b^{2}$.", "markdown": "$3a^{2}b$, $-5ab^{2}$, and $-7a^{4}b^{2}$.", "answer_latex": [ "$105a^{7}b^{5}$." ], "answer_markdown": [ "$105a^{7}b^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**2*b)*(-5*a*b**2)*(-7*a**4*b**2)", "answer_expr": "105*a**7*b**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 105*a**7*b**5" ], "shape": [ "identity: N*a**N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/12", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 12", "problem_latex": "$2a^{2}bc^{3}$, $-3a^{3}b^{2}c$, and $-ab^{2}c^{3}$.", "markdown": "$2a^{2}bc^{3}$, $-3a^{3}b^{2}c$, and $-ab^{2}c^{3}$.", "answer_latex": [ "$6a^{6}b^{5}c^{7}$." ], "answer_markdown": [ "$6a^{6}b^{5}c^{7}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**2*b*c**3)*(-3*a**3*b**2*c)*(-a*b**2*c**3)", "answer_expr": "6*a**6*b**5*c**7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 6*a**6*b**5*c**7" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/13", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 13", "problem_latex": "$2b^{2}c^{2}x^{2}$, $2a^{2}b^{2}c^{3}$, and $-3a^{3}bx^{3}$.", "markdown": "$2b^{2}c^{2}x^{2}$, $2a^{2}b^{2}c^{3}$, and $-3a^{3}bx^{3}$.", "answer_latex": [ "$-12a^{5}b^{5}c^{5}x^{5}$." ], "answer_markdown": [ "$-12a^{5}b^{5}c^{5}x^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*b**2*c**2*x**2)*(2*a**2*b**2*c**3)*(-3*a**3*b*x**3)", "answer_expr": "-12*a**5*b**5*c**5*x**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -12*a**5*b**5*c**5*d**5" ], "shape": [ "identity: N*a**N*b**N*c**N*d**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/14", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 14", "problem_latex": "$2a^{3}b^{2}c$, $-3a^{2}b^{3}c$, and $-4a^{2}bc^{3}$.", "markdown": "$2a^{3}b^{2}c$, $-3a^{2}b^{3}c$, and $-4a^{2}bc^{3}$.", "answer_latex": [ "$24a^{7}b^{6}c^{5}$." ], "answer_markdown": [ "$24a^{7}b^{6}c^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**3*b**2*c)*(-3*a**2*b**3*c)*(-4*a**2*b*c**3)", "answer_expr": "24*a**7*b**6*c**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 24*a**7*b**6*c**5" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/15", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 15", "problem_latex": "$7am^{2}x^{3}$, $3a^{4}m^{2}x^{3}$, and $-2amx$.", "markdown": "$7am^{2}x^{3}$, $3a^{4}m^{2}x^{3}$, and $-2amx$.", "answer_latex": [ "$-42a^{6}m^{5}x^{7}$." ], "answer_markdown": [ "$-42a^{6}m^{5}x^{7}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(7*a*m**2*x**3)*(3*a**4*m**2*x**3)*(-2*a*m*x)", "answer_expr": "-42*a**6*m**5*x**7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -42*a**6*b**5*c**7" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/16", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 16", "problem_latex": "$-3x^{2}y^{2}z^{2}$, $2x^{2}yz^{3}$, and $-5x^{4}yz$.", "markdown": "$-3x^{2}y^{2}z^{2}$, $2x^{2}yz^{3}$, and $-5x^{4}yz$.", "answer_latex": [ "$30x^{8}y^{4}z^{6}$." ], "answer_markdown": [ "$30x^{8}y^{4}z^{6}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-3*x**2*y**2*z**2)*(2*x**2*y*z**3)*(-5*x**4*y*z)", "answer_expr": "30*x**8*y**4*z**6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 30*a**8*b**4*c**6" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/17", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 17", "problem_latex": "$2ab^{2} - 3bc^{2} + c$.", "markdown": "$2ab^{2} - 3bc^{2} + c$.", "answer_latex": [ "$-46$" ], "answer_markdown": [ "$-46$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -46.0, printed -46 (half-unit 0.5; correctly rounded at the printed digits: -46.0)", "problem_expr": "2*a*b**2 - 3*b*c**2 + c", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a*b**2 - 3*b*c**2 + c at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a*b**N + N*b*c**N + c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-46,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 2 +/− ×", "3 GOLD x² ×", "3 ENTER 3 × 1 +/− GOLD x² × −", "1 +/− +" ], "constants": [ { "value": "2", "source": "the coefficient 2 in '2ab^2' of the printed expression" }, { "value": "3", "source": "the coefficient 3 in '2ab^2' and '-3bc^2' of the printed expression" }, { "value": "1", "source": "the magnitude of the given c = -1 (typed as 1 then +/−)" } ], "calculator_value": "-46E+0", "printed_value": "-46", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/18", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 18", "problem_latex": "$4a^{2} - 2b^{2} - c^{2}$.", "markdown": "$4a^{2} - 2b^{2} - c^{2}$.", "answer_latex": [ "$-3$." ], "answer_markdown": [ "$-3$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -3.0, printed -3 (half-unit 0.5; correctly rounded at the printed digits: -3.0)", "problem_expr": "4*a**2 - 2*b**2 - c**2", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 4*a**2 - 2*b**2 - c**2 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a**N + N*b**N - c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-3,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 2 +/− GOLD x² ×", "2 ENTER 3 GOLD x² × −", "1 +/− GOLD x² −" ], "constants": [ { "value": "1", "source": "the magnitude of c = -1 in the givens, for c² = (-1)² = 1; the sign is applied with +/−" } ], "calculator_value": "-3E+0", "printed_value": "-3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/19", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 19", "problem_latex": "$5a + 2b - 4c^{4}$.", "markdown": "$5a + 2b - 4c^{4}$.", "answer_latex": [ "$-8$." ], "answer_markdown": [ "$-8$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -8.0, printed -8 (half-unit 0.5; correctly rounded at the printed digits: -8.0)", "problem_expr": "5*a + 2*b - 4*c**4", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5*a + 2*b - 4*c**4 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a + N*b + N*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-8,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 2 +/− ×", "2 ENTER 3 × +", "1 +/− ENTER 4 yˣ 4 × −" ], "constants": [ { "value": "2", "source": "the 2 in a = -2 (typed as 2, then +/− for the sign)" }, { "value": "1", "source": "the 1 in c = -1 (typed as 1, then +/− for the sign)" } ], "calculator_value": "-8E+0", "printed_value": "-8", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/2", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 2", "problem_latex": "$8ab$ and $5a^{3}b^{2}$.", "markdown": "$8ab$ and $5a^{3}b^{2}$.", "answer_latex": [ "$40a^{4}b^{3}$." ], "answer_markdown": [ "$40a^{4}b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(8*a*b)*(5*a**3*b**2)", "answer_expr": "40*a**4*b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 40*a**4*b**3" ], "shape": [ "identity: N*a**N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/20", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 20", "problem_latex": "$2a^{3} - 3b + 8c^{2}$.", "markdown": "$2a^{3} - 3b + 8c^{2}$.", "answer_latex": [ "$-17$." ], "answer_markdown": [ "$-17$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -17.0, printed -17 (half-unit 0.5; correctly rounded at the printed digits: -17.0)", "problem_expr": "2*a**3 - 3*b + 8*c**2", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a**3 - 3*b + 8*c**2 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a**N + N*b + N*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-17,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 +/− ENTER 3 yˣ 2 ×", "3 ENTER 3 × −", "1 +/− GOLD x² 8 × +" ], "constants": [ { "value": "2", "source": "the coefficient of a in '2a^3' (typed as 2 +/− for a = -2 from the givens)" }, { "value": "3", "source": "the exponent in '2a^3'" }, { "value": "2", "source": "the coefficient 2 in '2a^3' (second entry, multiplying the cube)" }, { "value": "3", "source": "the coefficient of b in '-3b'" }, { "value": "3", "source": "the given value b = 3" }, { "value": "1", "source": "the magnitude of the given value c = -1; the sign is applied by +/− and the square by GOLD x²" }, { "value": "8", "source": "the coefficient of c^2 in '8c^2'" } ], "calculator_value": "-17E+0", "printed_value": "-17", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/21", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 21", "problem_latex": "$-a + 3b - 2c^{2}$.", "markdown": "$-a + 3b - 2c^{2}$.", "answer_latex": [ "$9$." ], "answer_markdown": [ "$9$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 9.0, printed 9 (half-unit 0.5; correctly rounded at the printed digits: 9.0)", "problem_expr": "-a + 3*b - 2*c**2", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -a + 3*b - 2*c**2 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*b + N*c**N - a" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#9,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 3 ×", "2 ENTER 1 +/− GOLD x² ×", "−", "2 +/− +/−", "+" ], "constants": [ { "value": "3", "source": "the 3 in '3b' (b = 3), typed twice: once as the coefficient and once as b's value" }, { "value": "2", "source": "the 2 in '2c²' (the coefficient of c²)" }, { "value": "1", "source": "the 1 in 'c = -1' (its magnitude; the sign is applied with +/−)" }, { "value": "2", "source": "the 2 in 'a = -2'; the two +/− presses give −(−2) = 2, which is −a" } ], "calculator_value": "+9E+0", "printed_value": "9", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/22", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 22", "problem_latex": "$-a^{3} - 2b - 10c$.", "markdown": "$-a^{3} - 2b - 10c$.", "answer_latex": [ "$12$." ], "answer_markdown": [ "$12$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 12.0, printed 12 (half-unit 0.5; correctly rounded at the printed digits: 12.0)", "problem_expr": "-a**3 - 2*b - 10*c", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -a**3 - 2*b - 10*c at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*b + N*c - a**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#12,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 +/− ENTER 3 yˣ +/−", "2 ENTER 3 × −", "10 ENTER 1 +/− × −" ], "calculator_value": "+12E+0", "printed_value": "12", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/23", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 23", "problem_latex": "$3a^{3} - 3b^{3} - 3c^{3}$.", "markdown": "$3a^{3} - 3b^{3} - 3c^{3}$.", "answer_latex": [ "$-102$." ], "answer_markdown": [ "$-102$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -102.0, printed -102 (half-unit 0.5; correctly rounded at the printed digits: -102.0)", "problem_expr": "3*a**3 - 3*b**3 - 3*c**3", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a**3 - 3*b**3 - 3*c**3 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a**N + N*b**N + N*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-102,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 +/− ENTER 3 yˣ 3 ×", "3 ENTER 3 yˣ 3 × −", "1 +/− ENTER 3 yˣ 3 × −" ], "constants": [ { "value": "2", "source": "the 2 in a = -2, typed as 2 then +/− to make -2" }, { "value": "3", "source": "the exponent 3 and coefficient 3 in 3a^3 - 3b^3 - 3c^3; also b = 3" }, { "value": "1", "source": "the 1 in c = -1, typed as 1 then +/− to make -1" } ], "calculator_value": "-102E+0", "printed_value": "-102", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/24", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 24", "problem_latex": "$2ab^{2} - 3bc^{2} + 2ac$.", "markdown": "$2ab^{2} - 3bc^{2} + 2ac$.", "answer_latex": [ "$-41$." ], "answer_markdown": [ "$-41$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -41.0, printed -41 (half-unit 0.5; correctly rounded at the printed digits: -41.0)", "problem_expr": "2*a*b**2 - 3*b*c**2 + 2*a*c", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a*b**2 + 2*a*c - 3*b*c**2 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a*b**N + N*a*c + N*b*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-41,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 2 +/− × 3 GOLD x² ×", "3 ENTER 3 × 1 +/− GOLD x² × −", "2 ENTER 2 +/− × 1 +/− × +" ], "constants": [ { "value": "2", "source": "the coefficient 2 in 2ab² (and the 2 in 2ac)" }, { "value": "3", "source": "the 3 in 3bc² (and b = 3 given)" }, { "value": "1", "source": "the magnitude of c = -1 given; 1 +/− types c" } ], "calculator_value": "-41E+0", "printed_value": "-41", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/25", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 25", "problem_latex": "$3abc + 5a^{2}b^{2} - 2a^{2}b$.", "markdown": "$3abc + 5a^{2}b^{2} - 2a^{2}b$.", "answer_latex": [ "$174$." ], "answer_markdown": [ "$174$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 174.0, printed 174 (half-unit 0.5; correctly rounded at the printed digits: 174.0)", "problem_expr": "3*a*b*c + 5*a**2*b**2 - 2*a**2*b", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5*a**2*b**2 - 2*a**2*b + 3*a*b*c at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a*b*c + N*a**N*b + N*a**N*b**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#174,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 2 +/− × 3 × 1 +/− ×", "5 ENTER 2 +/− GOLD x² × 3 GOLD x² × +", "2 ENTER 2 +/− GOLD x² × 3 × −" ], "constants": [ { "value": "3", "source": "the coefficient 3 in '3abc'" }, { "value": "2", "source": "the 2 in a = -2 (the magnitude, entered as 2 then +/−)" }, { "value": "3", "source": "b = 3 in the given values" }, { "value": "1", "source": "the 1 in c = -1 (entered as 1 then +/−)" }, { "value": "5", "source": "the coefficient 5 in '5a²b²'" }, { "value": "2", "source": "the exponent 2 in a² (GOLD x²)" }, { "value": "2", "source": "the coefficient 2 in '2a²b'" } ], "calculator_value": "+174E+0", "printed_value": "174", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/26", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 26", "problem_latex": "$ab^{2}c^{2} + 2abc^{2} + a^{2}b^{2}c^{2}$.", "markdown": "$ab^{2}c^{2} + 2abc^{2} + a^{2}b^{2}c^{2}$.", "answer_latex": [ "$6$." ], "answer_markdown": [ "$6$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 6.0, printed 6 (half-unit 0.5; correctly rounded at the printed digits: 6.0)", "problem_expr": "a*b**2*c**2 + 2*a*b*c**2 + a**2*b**2*c**2", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a**2*b**2*c**2 + a*b**2*c**2 + 2*a*b*c**2 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a*b*c**N + a*b**N*c**N + a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#6,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 +/− ENTER 3 GOLD x² ×", "1 +/− GOLD x² ×", "2 ENTER 2 +/− × 3 × 1 +/− GOLD x² ×", "+", "2 +/− GOLD x² 3 GOLD x² × 1 +/− GOLD x² × +" ], "calculator_value": "+6E+0", "printed_value": "6", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/27", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 27", "problem_latex": "$2a^{2}bc + 3abc + a^{2}b^{2}c^{2}$.", "markdown": "$2a^{2}bc + 3abc + a^{2}b^{2}c^{2}$.", "answer_latex": [ "$30$." ], "answer_markdown": [ "$30$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 30.0, printed 30 (half-unit 0.5; correctly rounded at the printed digits: 30.0)", "problem_expr": "2*a**2*b*c + 3*a*b*c + a**2*b**2*c**2", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a**2*b**2*c**2 + 2*a**2*b*c + 3*a*b*c at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a*b*c + N*a**N*b*c + a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#30,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 2 +/− GOLD x² × 3 × 1 +/− ×", "3 ENTER 2 +/− × 3 × 1 +/− ×", "+", "2 +/− GOLD x² 3 GOLD x² × 1 +/− GOLD x² × +" ], "constants": [ { "value": "2", "source": "the 2 in '2a²bc' (coefficient of the first term) and the exponent 2 in a²" }, { "value": "3", "source": "the 3 in '3abc' (coefficient of the second term) and the given b = 3" }, { "value": "1", "source": "the 1 in the given c = -1, typed as 1 then +/− to make −1" } ], "calculator_value": "+30E+0", "printed_value": "30", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/28", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 28, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 28", "problem_latex": "$6a^{2} + 8a^{2}b^{2} - 5a^{2}bc$.", "markdown": "$6a^{2} + 8a^{2}b^{2} - 5a^{2}bc$.", "answer_latex": [ "$372$." ], "answer_markdown": [ "$372$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 372.0, printed 372 (half-unit 0.5; correctly rounded at the printed digits: 372.0)", "problem_expr": "6*a**2 + 8*a**2*b**2 - 5*a**2*b*c", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 8*a**2*b**2 - 5*a**2*b*c + 6*a**2 at a=-2, b=3, c=-1" ], "shape": [ "evaluate: N*a**N*b*c + N*a**N*b**N + N*a**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#372,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 2 +/− GOLD x² ×", "8 ENTER 2 +/− GOLD x² × 3 GOLD x² × +", "5 ENTER 2 +/− GOLD x² × 3 × 1 +/− × −" ], "constants": [ { "value": "2", "source": "the 2 in a² (exponent) and the magnitude of a = −2, entered with +/−" }, { "value": "1", "source": "the magnitude of c = −1 in the given values, entered with +/−" } ], "calculator_value": "+372E+0", "printed_value": "372", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/3", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 3", "problem_latex": "$9xy$ and $7xy$.", "markdown": "$9xy$ and $7xy$.", "answer_latex": [ "$63x^{2}y^{2}$." ], "answer_markdown": [ "$63x^{2}y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(9*x*y)*(7*x*y)", "answer_expr": "63*x**2*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 63*a**2*b**2" ], "shape": [ "identity: N*a**N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/4", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 4", "problem_latex": "$2a^{2}b$ and $a^{3}b^{4}c^{2}$.", "markdown": "$2a^{2}b$ and $a^{3}b^{4}c^{2}$.", "answer_latex": [ "$2a^{5}b^{5}c^{2}$." ], "answer_markdown": [ "$2a^{5}b^{5}c^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**2*b)*(a**3*b**4*c**2)", "answer_expr": "2*a**5*b**5*c**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**5*b**5*c**2" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/5", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 5", "problem_latex": "$3a^{3}b^{7}c^{8}$ and $3a^{4}b^{2}c$.", "markdown": "$3a^{3}b^{7}c^{8}$ and $3a^{4}b^{2}c$.", "answer_latex": [ "$9a^{7}b^{9}c^{9}$." ], "answer_markdown": [ "$9a^{7}b^{9}c^{9}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**3*b**7*c**8)*(3*a**4*b**2*c)", "answer_expr": "9*a**7*b**9*c**9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 9*a**7*b**9*c**9" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/6", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 6", "problem_latex": "$2a$ and $-5a$.", "markdown": "$2a$ and $-5a$.", "answer_latex": [ "$-10a^{2}$." ], "answer_markdown": [ "$-10a^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a)*(-5*a)", "answer_expr": "-10*a**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -10*a**2" ], "shape": [ "identity: N*a**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/7", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 7", "problem_latex": "$-3a$ and $-4b$.", "markdown": "$-3a$ and $-4b$.", "answer_latex": [ "$12ab$." ], "answer_markdown": [ "$12ab$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-3*a)*(-4*b)", "answer_expr": "12*a*b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 12*a*b" ], "shape": [ "identity: N*a*b" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/8", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 8", "problem_latex": "$-ab$ and $a^{3}b^{2}$.", "markdown": "$-ab$ and $a^{3}b^{2}$.", "answer_latex": [ "$-a^{4}b^{3}$." ], "answer_markdown": [ "$-a^{4}b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-a*b)*(a**3*b**2)", "answer_expr": "-a**4*b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a**4*b**3" ], "shape": [ "identity: -a**N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-16/9", "set": "wentworth-first-steps-in-algebra-1894/ex-16", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "42", "location": "Exercise 16, problem 9", "problem_latex": "$-2ab^{4}$ and $-5a^{4}bc$.", "markdown": "$-2ab^{4}$ and $-5a^{4}bc$.", "answer_latex": [ "$10a^{5}b^{5}c$." ], "answer_markdown": [ "$10a^{5}b^{5}c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-2*a*b**4)*(-5*a**4*b*c)", "answer_expr": "10*a**5*b**5*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 10*a**5*b**5*c" ], "shape": [ "identity: N*a**N*b**N*c" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/1", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 1", "problem_latex": "$x^{3}$~by~$x$.", "markdown": "$x^{3}$ by $x$.", "answer_latex": [ "$x^{2}$." ], "answer_markdown": [ "$x^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3)/(x)", "answer_expr": "x**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2" ], "shape": [ "identity: x**N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-15/13" ], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/10", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 10", "problem_latex": "$-25x^{4}y^{2}$~by~$-5x^{3}y^{2}$.", "markdown": "$-25x^{4}y^{2}$ by $-5x^{3}y^{2}$.", "answer_latex": [ "$5x$." ], "answer_markdown": [ "$5x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-25*x**4*y**2)/(-5*x**3*y**2)", "answer_expr": "5*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 5*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-46/16" ], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/11", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 11", "problem_latex": "$-51x^{2}y^{3}$~by~$-17x^{2}y$.", "markdown": "$-51x^{2}y^{3}$ by $-17x^{2}y$.", "answer_latex": [ "$3y^{2}$." ], "answer_markdown": [ "$3y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-51*x**2*y**3)/(-17*x**2*y)", "answer_expr": "3*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a**2" ], "shape": [ "identity: N*a**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/12", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 12", "problem_latex": "$-28a^{4}b^{3}$~by~$7a^{3}b$.", "markdown": "$-28a^{4}b^{3}$ by $7a^{3}b$.", "answer_latex": [ "$-4ab^{2}$." ], "answer_markdown": [ "$-4ab^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-28*a**4*b**3)/(7*a**3*b)", "answer_expr": "-4*a*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -4*a**2*x" ], "shape": [ "identity: N*a**N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/13", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 13", "problem_latex": "$-36x^{2}y^{6}$~by~$-3xy^{2}$.", "markdown": "$-36x^{2}y^{6}$ by $-3xy^{2}$.", "answer_latex": [ "$12xy^{4}$." ], "answer_markdown": [ "$12xy^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-36*x**2*y**6)/(-3*x*y**2)", "answer_expr": "12*x*y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 12*a**4*x" ], "shape": [ "identity: N*a**N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/14", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 14", "problem_latex": "$-3x^{4}y^{6}$~by~$-5xy^{3}$.", "markdown": "$-3x^{4}y^{6}$ by $-5xy^{3}$.", "answer_latex": [ "$\\dfrac{3x^{3}y^{3}}{5}$." ], "answer_markdown": [ "$\\dfrac{3x^{3}y^{3}}{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-3*x**4*y**6)/(-5*x*y**3)", "answer_expr": "3*x**3*y**3/5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a**3*x**3/5" ], "shape": [ "identity: N*a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/15", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 15", "problem_latex": "$-12a^{2}b^{3}$~by~$8ab^{3}$.", "markdown": "$-12a^{2}b^{3}$ by $8ab^{3}$.", "answer_latex": [ "$-\\dfrac{3a}{2}$." ], "answer_markdown": [ "$-\\dfrac{3a}{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-12*a**2*b**3)/(8*a*b**3)", "answer_expr": "-3*a/2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -3*x/2" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/16", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 16", "problem_latex": "$-abcd$~by~$ac$.", "markdown": "$-abcd$ by $ac$.", "answer_latex": [ "$-bd$." ], "answer_markdown": [ "$-bd$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-a*b*c*d)/(a*c)", "answer_expr": "-b*d" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a*b" ], "shape": [ "identity: -a*b" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/17", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 17", "problem_latex": "$-a^{2}b^{3}c^{4}d^{5}$~by~$-ab^{3}c^{3}d^{3}$.", "markdown": "$-a^{2}b^{3}c^{4}d^{5}$ by $-ab^{3}c^{3}d^{3}$.", "answer_latex": [ "$acd^{2}$." ], "answer_markdown": [ "$acd^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-a**2*b**3*c**4*d**5)/(-a*b**3*c**3*d**3)", "answer_expr": "a*c*d**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*b**2*x" ], "shape": [ "identity: a*b**N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/18", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 18", "problem_latex": "$2x^{2}y^{2}z^{3}$~by~$-3xyz^{3}$.", "markdown": "$2x^{2}y^{2}z^{3}$ by $-3xyz^{3}$.", "answer_latex": [ "$-\\dfrac{2xy}{3}$." ], "answer_markdown": [ "$-\\dfrac{2xy}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x**2*y**2*z**3)/(-3*x*y*z**3)", "answer_expr": "-2*x*y/3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -2*a*x/3" ], "shape": [ "identity: N*a*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/19", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 19", "problem_latex": "$-5a^{5}b^{3}c^{7}$~by~$-a^{4}b^{2}c^{7}$.", "markdown": "$-5a^{5}b^{3}c^{7}$ by $-a^{4}b^{2}c^{7}$.", "answer_latex": [ "$5ab$." ], "answer_markdown": [ "$5ab$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-5*a**5*b**3*c**7)/(-a**4*b**2*c**7)", "answer_expr": "5*a*b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 5*a*x" ], "shape": [ "identity: N*a*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/2", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 2", "problem_latex": "$21x^{5}$~by~$7x^{3}$.", "markdown": "$21x^{5}$ by $7x^{3}$.", "answer_latex": [ "$3x^{2}$." ], "answer_markdown": [ "$3x^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(21*x**5)/(7*x**3)", "answer_expr": "3*x**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*x**2" ], "shape": [ "identity: N*x**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/20", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 20", "problem_latex": "$52a^{2}m^{3}n^{4}$~by~$13a^{2}m^{2}n^{3}$.", "markdown": "$52a^{2}m^{3}n^{4}$ by $13a^{2}m^{2}n^{3}$.", "answer_latex": [ "$4mn$." ], "answer_markdown": [ "$4mn$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(52*a**2*m**3*n**4)/(13*a**2*m**2*n**3)", "answer_expr": "4*m*n" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a*b" ], "shape": [ "identity: N*a*b" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/21", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 21", "problem_latex": "$13xy^{2}z^{4}$~by~$39xyz$.", "markdown": "$13xy^{2}z^{4}$ by $39xyz$.", "answer_latex": [ "$\\dfrac{yz^{3}}{3}$." ], "answer_markdown": [ "$\\dfrac{yz^{3}}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(13*x*y**2*z**4)/(39*x*y*z)", "answer_expr": "y*z**3/3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*b**3/3" ], "shape": [ "identity: N*a*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/22", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 22", "problem_latex": "$68xc^{2}d^{3}$~by~$-4xcd^{2}$.", "markdown": "$68xc^{2}d^{3}$ by $-4xcd^{2}$.", "answer_latex": [ "$-17cd$." ], "answer_markdown": [ "$-17cd$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(68*x*c**2*d**3)/(-4*x*c*d**2)", "answer_expr": "-17*c*d" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -17*a*b" ], "shape": [ "identity: N*a*b" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/23", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 23", "problem_latex": "$-8m^{5}n^{3}p^{2}$~by~$-4m^{5}np$.", "markdown": "$-8m^{5}n^{3}p^{2}$ by $-4m^{5}np$.", "answer_latex": [ "$2n^{2}p$." ], "answer_markdown": [ "$2n^{2}p$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-8*m**5*n**3*p**2)/(-4*m**5*n*p)", "answer_expr": "2*n**2*p" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**2*b" ], "shape": [ "identity: N*a**N*b" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/24", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 24", "problem_latex": "$-6pqr^{3}$~by~$-2p^{2}qr$.", "markdown": "$-6pqr^{3}$ by $-2p^{2}qr$.", "answer_latex": [ "$\\dfrac{3r^{2}}{p}$." ], "answer_markdown": [ "$\\dfrac{3r^{2}}{p}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-6*p*q*r**3)/(-2*p**2*q*r)", "answer_expr": "3*r**2/p" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a**2/x" ], "shape": [ "identity: N*a**N/x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/25", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 25", "problem_latex": "$26a^{2}g^{2}t^{5}$~by~$-2agt^{4}$.", "markdown": "$26a^{2}g^{2}t^{5}$ by $-2agt^{4}$.", "answer_latex": [ "$-13agt$." ], "answer_markdown": [ "$-13agt$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(26*a**2*g**2*t**5)/(-2*a*g*t**4)", "answer_expr": "-13*a*g*t" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -13*a*b*x" ], "shape": [ "identity: N*a*b*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/26", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 26", "problem_latex": "$-a^{4}b^{2}c^{3}$~by~$-a^{5}b^{3}c^{4}$.", "markdown": "$-a^{4}b^{2}c^{3}$ by $-a^{5}b^{3}c^{4}$.", "answer_latex": [ "$\\dfrac{1}{abc}$." ], "answer_markdown": [ "$\\dfrac{1}{abc}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-a**4*b**2*c**3)/(-a**5*b**3*c**4)", "answer_expr": "1/(a*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 1/(a*b*x)" ], "shape": [ "identity: 1/(a*b*x)" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/27", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 27", "problem_latex": "$-3x^{2}y^{2}z^{2}$~by~$-2x^{3}y^{4}z^{5}$.", "markdown": "$-3x^{2}y^{2}z^{2}$ by $-2x^{3}y^{4}z^{5}$.", "answer_latex": [ "$\\dfrac{3}{2xy^{2}z^{3}}$." ], "answer_markdown": [ "$\\dfrac{3}{2xy^{2}z^{3}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-3*x**2*y**2*z**2)/(-2*x**3*y**4*z**5)", "answer_expr": "3/(2*x*y**2*z**3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3/(2*a**2*b**3*x)" ], "shape": [ "identity: N*a**N*b**N/x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/28", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 28, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 28", "problem_latex": "$-6mnp$~by~$-3m^{2}n^{2}p^{2}$.", "markdown": "$-6mnp$ by $-3m^{2}n^{2}p^{2}$.", "answer_latex": [ "$\\dfrac{2}{mnp}$." ], "answer_markdown": [ "$\\dfrac{2}{mnp}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-6*m*n*p)/(-3*m**2*n**2*p**2)", "answer_expr": "2/(m*n*p)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2/(a*b*x)" ], "shape": [ "identity: N/(a*b*x)" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/29", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 29, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 29", "problem_latex": "$-17a^{2}b^{3}c^{4}$~by~$51ab^{5}c^{4}$.", "markdown": "$-17a^{2}b^{3}c^{4}$ by $51ab^{5}c^{4}$.", "answer_latex": [ "$-\\dfrac{a}{3b^{2}}$." ], "answer_markdown": [ "$-\\dfrac{a}{3b^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-17*a**2*b**3*c**4)/(51*a*b**5*c**4)", "answer_expr": "-a/(3*b**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -x/(3*a**2)" ], "shape": [ "identity: N*a**N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/3", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 3", "problem_latex": "$35x^{2}$~by~$-5x^{2}$.", "markdown": "$35x^{2}$ by $-5x^{2}$.", "answer_latex": [ "$-7$." ], "answer_markdown": [ "$-7$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(35*x**2)/(-5*x**2)", "answer_expr": "-7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -7" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-17/4" ], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/30", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 30, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 30", "problem_latex": "$-19mg^{2}t^{3}$~by~$57m2^gt^{4}$.", "markdown": "$-19mg^{2}t^{3}$ by $57m2^gt^{4}$.", "answer_latex": [ "$-\\dfrac{g}{3mt}$." ], "answer_markdown": [ "$-\\dfrac{g}{3mt}$." ], "checks": [ { "task": "identity", "verdict": "FLAG-MISMATCH", "judge_why": [ "-0.36858435791959172217", "-0.30005782525079266074" ], "problem_expr": "(-19*m*g**2*t**3)/(57*m*2**g*t**4)", "answer_expr": "-g/(3*m*t)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "identity: -a**2/(3*2**a*b)" ], "shape": [ "identity: N*a**N/(N**a*b)" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/4", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 4", "problem_latex": "$-42x^{2}$~by~$6x^{2}$.", "markdown": "$-42x^{2}$ by $6x^{2}$.", "answer_latex": [ "$-7$." ], "answer_markdown": [ "$-7$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-42*x**2)/(6*x**2)", "answer_expr": "-7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -7" ], "shape": [ "identity: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-17/3" ], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/5", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 5", "problem_latex": "$-63x^{5}$~by~$-9x$.", "markdown": "$-63x^{5}$ by $-9x$.", "answer_latex": [ "$7x^{4}$." ], "answer_markdown": [ "$7x^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-63*x**5)/(-9*x)", "answer_expr": "7*x**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 7*x**4" ], "shape": [ "identity: N*x**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/6", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 6", "problem_latex": "$-72x^{3}$~by~$-8x^{2}$.", "markdown": "$-72x^{3}$ by $-8x^{2}$.", "answer_latex": [ "$9x$." ], "answer_markdown": [ "$9x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-72*x**3)/(-8*x**2)", "answer_expr": "9*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 9*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/7", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 7", "problem_latex": "$-32a^{2}b^{2}$~by~$8ab^{2}$.", "markdown": "$-32a^{2}b^{2}$ by $8ab^{2}$.", "answer_latex": [ "$-4a$." ], "answer_markdown": [ "$-4a$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-32*a**2*b**2)/(8*a*b**2)", "answer_expr": "-4*a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -4*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/8", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 8", "problem_latex": "$-16x^{3}y^{3}$~by~$-4xy$.", "markdown": "$-16x^{3}y^{3}$ by $-4xy$.", "answer_latex": [ "$4x^{2}y^{2}$." ], "answer_markdown": [ "$4x^{2}y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-16*x**3*y**3)/(-4*x*y)", "answer_expr": "4*x**2*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a**2*x**2" ], "shape": [ "identity: N*a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-17/9", "set": "wentworth-first-steps-in-algebra-1894/ex-17", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "45", "location": "Exercise 17, problem 9", "problem_latex": "$18x^{2}y$~by~$-2xy$.", "markdown": "$18x^{2}y$ by $-2xy$.", "answer_latex": [ "$-9x$." ], "answer_markdown": [ "$-9x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(18*x**2*y)/(-2*x*y)", "answer_expr": "-9*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -9*x" ], "shape": [ "identity: N*x" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/1", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 1", "problem_latex": "$a^{2} - ab + b^{2}$; $a^{2} + ab + b^{2}$.", "markdown": "$a^{2} - ab + b^{2}$; $a^{2} + ab + b^{2}$.", "answer_latex": [ "$2a^{2} + 2b^{2}$." ], "answer_markdown": [ "$2a^{2} + 2b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - a*b + b**2) + (a**2 + a*b + b**2)", "answer_expr": "2*a**2 + 2*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**2 + 2*b**2" ], "shape": [ "identity: N*a**N + N*b**N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-2/10" ], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/10", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 10", "problem_latex": "$3y^{2} - x^{2} - 3xy$; $5x^{2} + 6xy - 7y^{2}$; $x^{2} + 2y^{2}$.", "markdown": "$3y^{2} - x^{2} - 3xy$; $5x^{2} + 6xy - 7y^{2}$; $x^{2} + 2y^{2}$.", "answer_latex": [ "$5x^{2} + 3xy - 2y^{2}$." ], "answer_markdown": [ "$5x^{2} + 3xy - 2y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*y**2 - x**2 - 3*x*y) + (5*x**2 + 6*x*y - 7*y**2) + (x**2 + 2*y**2)", "answer_expr": "5*x**2 + 3*x*y - 2*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 5*a**2 + 3*a*b - 2*b**2" ], "shape": [ "identity: N*a*b + N*a**N + N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/11", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 11", "problem_latex": "$2a^{2} - 2ab + 3b^{2}$; $4b^{2} + 5ab - 2a^{2}$; $a^{2} - 3ab - 9b^{2}$.", "markdown": "$2a^{2} - 2ab + 3b^{2}$; $4b^{2} + 5ab - 2a^{2}$; $a^{2} - 3ab - 9b^{2}$.", "answer_latex": [ "$a^{2} - 2b^{2}$." ], "answer_markdown": [ "$a^{2} - 2b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**2 - 2*a*b + 3*b**2) + (4*b**2 + 5*a*b - 2*a**2) + (a**2 - 3*a*b - 9*b**2)", "answer_expr": "a**2 - 2*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**2 - 2*b**2" ], "shape": [ "identity: N*b**N + a**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/12", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 12", "problem_latex": "$a^{3} - a^{2} + a - 1$; $a^{2} - 2a + 2$; $3a^{3} + 7a + 1$.", "markdown": "$a^{3} - a^{2} + a - 1$; $a^{2} - 2a + 2$; $3a^{3} + 7a + 1$.", "answer_latex": [ "$4a^{3} + 6a + 2$." ], "answer_markdown": [ "$4a^{3} + 6a + 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 - a**2 + a - 1) + (a**2 - 2*a + 2) + (3*a**3 + 7*a + 1)", "answer_expr": "4*a**3 + 6*a + 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a**3 + 6*a + 2" ], "shape": [ "identity: N*a + N*a**N + N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/13", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 13", "problem_latex": "$2m^{3} - m^{2} - m$; $4m^{3} + 8m^{2} - 7$; $-3m^{3} + m + 9$.", "markdown": "$2m^{3} - m^{2} - m$; $4m^{3} + 8m^{2} - 7$; $-3m^{3} + m + 9$.", "answer_latex": [ "$3m^{3} + 7m^{2} + 2$." ], "answer_markdown": [ "$3m^{3} + 7m^{2} + 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*m**3 - m**2 - m) + (4*m**3 + 8*m**2 - 7) + (-3*m**3 + m + 9)", "answer_expr": "3*m**3 + 7*m**2 + 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a**3 + 7*a**2 + 2" ], "shape": [ "identity: 2*N*a**N + N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/14", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 14", "problem_latex": "$x^{3} - 3x + 6y$; $x^{2} + 2x - 5y$; $x^{3} - 3x^{2} + 5x$.", "markdown": "$x^{3} - 3x + 6y$; $x^{2} + 2x - 5y$; $x^{3} - 3x^{2} + 5x$.", "answer_latex": [ "$2x^{3} - 2x^{2} + 4x + y$." ], "answer_markdown": [ "$2x^{3} - 2x^{2} + 4x + y$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3 - 3*x + 6*y) + (x**2 + 2*x - 5*y) + (x**3 - 3*x**2 + 5*x)", "answer_expr": "2*x**3 - 2*x**2 + 4*x + y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**3 - 2*a**2 + 4*a + b" ], "shape": [ "identity: N*a + 2*N*a**N + b" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/15", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 15", "problem_latex": "$6x^{3} - 5x + 1$; $x^{3} + 3x + 4$; $7x^{2} + 2x - 3$.", "markdown": "$6x^{3} - 5x + 1$; $x^{3} + 3x + 4$; $7x^{2} + 2x - 3$.", "answer_latex": [ "$7x^{3} + 7x^{2} + 2$." ], "answer_markdown": [ "$7x^{3} + 7x^{2} + 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(6*x**3 - 5*x + 1) + (x**3 + 3*x + 4) + (7*x**2 + 2*x - 3)", "answer_expr": "7*x**3 + 7*x**2 + 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 7*a**3 + 7*a**2 + 2" ], "shape": [ "identity: 2*N*a**N + N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/16", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 16", "problem_latex": "$a^{3} + 3a^{2}b - 3ab^{2}$; $-3a^{2}b - 6ab^{2} - b^{3}$; $3a^{2}b + 4ab^{2}$.", "markdown": "$a^{3} + 3a^{2}b - 3ab^{2}$; $-3a^{2}b - 6ab^{2} - b^{3}$; $3a^{2}b + 4ab^{2}$.", "answer_latex": [ "$a^{3} + 3a^{2}b - 5ab^{2} - b^{3}$." ], "answer_markdown": [ "$a^{3} + 3a^{2}b - 5ab^{2} - b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 + 3*a**2*b - 3*a*b**2) + (-3*a**2*b - 6*a*b**2 - b**3) + (3*a**2*b + 4*a*b**2)", "answer_expr": "a**3 + 3*a**2*b - 5*a*b**2 - b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**3 + 3*a**2*b - 5*a*b**2 - b**3" ], "shape": [ "identity: N*a*b**N + N*a**N*b + a**N - b**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/17", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 17", "problem_latex": "$a^{3} - 2a^{2}b - 2ab^{2}$; $a^{2}b - 3ab^{2} - b^{3}$; $3ab^{2} - 2a^{3} - b^{3}$.", "markdown": "$a^{3} - 2a^{2}b - 2ab^{2}$; $a^{2}b - 3ab^{2} - b^{3}$; $3ab^{2} - 2a^{3} - b^{3}$.", "answer_latex": [ "$-a^{3} - a^{2}b - 2ab^{2} - 2b^{3}$." ], "answer_markdown": [ "$-a^{3} - a^{2}b - 2ab^{2} - 2b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 - 2*a**2*b - 2*a*b**2) + (a**2*b - 3*a*b**2 - b**3) + (3*a*b**2 - 2*a**3 - b**3)", "answer_expr": "-a**3 - a**2*b - 2*a*b**2 - 2*b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a**3 - a**2*b - 2*a*b**2 - 2*b**3" ], "shape": [ "identity: N*a*b**N + N*b**N - a**N*b - a**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/18", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 18", "problem_latex": "$7x^{3} - 2x^{2}y + 9xy^{2} + 13y^{3}$; $5x^{2}y - 4xy^{2} - 2x^{3} - 3y^{3}$;\n$y^{3} - x^{3} - 3x^{2}y - 5xy^{2}$; $2x^{2}y - 5y^{3} - 2x^{3} - xy^{2}$.", "markdown": "$7x^{3} - 2x^{2}y + 9xy^{2} + 13y^{3}$; $5x^{2}y - 4xy^{2} - 2x^{3} - 3y^{3}$; $y^{3} - x^{3} - 3x^{2}y - 5xy^{2}$; $2x^{2}y - 5y^{3} - 2x^{3} - xy^{2}$.", "answer_latex": [ "$2x^{3} + 2x^{2}y - xy^{2} + 6y^{3}$." ], "answer_markdown": [ "$2x^{3} + 2x^{2}y - xy^{2} + 6y^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(7*x**3 - 2*x**2*y + 9*x*y**2 + 13*y**3) + (5*x**2*y - 4*x*y**2 - 2*x**3 - 3*y**3) + (y**3 - x**3 - 3*x**2*y - 5*x*y**2) + (2*x**2*y - 5*y**3 - 2*x**3 - x*y**2)", "answer_expr": "2*x**3 + 2*x**2*y - x*y**2 + 6*y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**3 + 2*a**2*b - a*b**2 + 6*b**3" ], "shape": [ "identity: N*a**N*b + N*a**N + N*b**N - a*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/19", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 19", "problem_latex": "Show that $x + y + z = 0$, if $x = a - b - c$,\n$y = 2b + 2c - 3a$, and $z = 2a - b - c$.", "markdown": "Show that $x + y + z = 0$, if $x = a - b - c$, $y = 2b + 2c - 3a$, and $z = 2a - b - c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/2", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 2", "problem_latex": "$3a^{2} + 5a-7$; $6a^{2} - 7a + 13$.", "markdown": "$3a^{2} + 5a-7$; $6a^{2} - 7a + 13$.", "answer_latex": [ "$9a^{2} - 2a + 6$." ], "answer_markdown": [ "$9a^{2} - 2a + 6$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**2 + 5*a - 7) + (6*a**2 - 7*a + 13)", "answer_expr": "9*a**2 - 2*a + 6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 9*a**2 - 2*a + 6" ], "shape": [ "identity: N*a + N*a**N + N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/20", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 20", "problem_latex": "Show that $x + y = 3z$, if $x = 3a^{2} - 6a + 12$,\n$y = 9a^{2} + 12a - 21$, and $z = 4a^{2} + 2a - 3$.", "markdown": "Show that $x + y = 3z$, if $x = 3a^{2} - 6a + 12$, $y = 9a^{2} + 12a - 21$, and $z = 4a^{2} + 2a - 3$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/3", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 3", "problem_latex": "$x + 2y - 3z$; $-3x + y + 2z$; $2x - 3y + z$.", "markdown": "$x + 2y - 3z$; $-3x + y + 2z$; $2x - 3y + z$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 2*y - 3*z) + (-3*x + y + 2*z) + (2*x - 3*y + z)", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 0" ], "shape": [ "identity: 0" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-15/11", "wentworth-first-steps-in-algebra-1894/ex-15/15", "wentworth-first-steps-in-algebra-1894/ex-15/17" ], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/4", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 4", "problem_latex": "$3x + 2y - z$; $-x + 3y + 2z$; $2x - y + 3z$.", "markdown": "$3x + 2y - z$; $-x + 3y + 2z$; $2x - y + 3z$.", "answer_latex": [ "$4x + 4y + 4z$." ], "answer_markdown": [ "$4x + 4y + 4z$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x + 2*y - z) + (-x + 3*y + 2*z) + (2*x - y + 3*z)", "answer_expr": "4*x + 4*y + 4*z" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a + 4*b + 4*c" ], "shape": [ "identity: N*a + N*b + N*c" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-18/6" ], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/5", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 5", "problem_latex": "$-3a + 2b + c$; $a - 3b + 2c$; $2a + 3b - c$.", "markdown": "$-3a + 2b + c$; $a - 3b + 2c$; $2a + 3b - c$.", "answer_latex": [ "$2b + 2c$." ], "answer_markdown": [ "$2b + 2c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-3*a + 2*b + c) + (a - 3*b + 2*c) + (2*a + 3*b - c)", "answer_expr": "2*b + 2*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a + 2*b" ], "shape": [ "identity: N*a + N*b" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/6", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 6", "problem_latex": "$-a + 3b + 4c$; $3a - b + 2c$; $2a + 2b - 2c$.", "markdown": "$-a + 3b + 4c$; $3a - b + 2c$; $2a + 2b - 2c$.", "answer_latex": [ "$4a + 4b + 4c$." ], "answer_markdown": [ "$4a + 4b + 4c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-a + 3*b + 4*c) + (3*a - b + 2*c) + (2*a + 2*b - 2*c)", "answer_expr": "4*a + 4*b + 4*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a + 4*b + 4*c" ], "shape": [ "identity: N*a + N*b + N*c" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-18/4" ], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/7", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 7", "problem_latex": "$4a^{2} + 3a + 5$; $-2a^{2} + 3a - 8$; $a^{2} - a + 1$.", "markdown": "$4a^{2} + 3a + 5$; $-2a^{2} + 3a - 8$; $a^{2} - a + 1$.", "answer_latex": [ "$3a^{2} + 5a - 2$." ], "answer_markdown": [ "$3a^{2} + 5a - 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4*a**2 + 3*a + 5) + (-2*a**2 + 3*a - 8) + (a**2 - a + 1)", "answer_expr": "3*a**2 + 5*a - 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a**2 + 5*a - 2" ], "shape": [ "identity: N*a + N*a**N + N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/8", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 8", "problem_latex": "$5ab + 6bc - 7ac$; $3ab - 9bc + 4ac$; $3bc + 6ac$.", "markdown": "$5ab + 6bc - 7ac$; $3ab - 9bc + 4ac$; $3bc + 6ac$.", "answer_latex": [ "$8ab + 3ac$." ], "answer_markdown": [ "$8ab + 3ac$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(5*a*b + 6*b*c - 7*a*c) + (3*a*b - 9*b*c + 4*a*c) + (3*b*c + 6*a*c)", "answer_expr": "8*a*b + 3*a*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 8*a*b + 3*a*c" ], "shape": [ "identity: N*a*b + N*a*c" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-18/9", "set": "wentworth-first-steps-in-algebra-1894/ex-18", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "48", "location": "Exercise 18, problem 9", "problem_latex": "$x^{3} + x^{2} + x$; $2x^{3} + 3x^{2} - 2x$; $3x^{3} - 4x^{2} + x$.", "markdown": "$x^{3} + x^{2} + x$; $2x^{3} + 3x^{2} - 2x$; $3x^{3} - 4x^{2} + x$.", "answer_latex": [ "$6x^{3}$." ], "answer_markdown": [ "$6x^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3 + x**2 + x) + (2*x**3 + 3*x**2 - 2*x) + (3*x**3 - 4*x**2 + x)", "answer_expr": "6*x**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 6*a**3" ], "shape": [ "identity: N*a**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/1", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 1", "problem_latex": "$a - 2b + 3c$ from $2a - 3b + 4c$.", "markdown": "$a - 2b + 3c$ from $2a - 3b + 4c$.", "answer_latex": [ "$a - b + c$." ], "answer_markdown": [ "$a - b + c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a - 3*b + 4*c) - (a - 2*b + 3*c)", "answer_expr": "a - b + c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a - b + c" ], "shape": [ "identity: a - b + c" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/10", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 10", "problem_latex": "$x^{4} + x - 5x^{3} + 5$ from $7 - 2x^{2} - 3x^{3} + x^{4}$.", "markdown": "$x^{4} + x - 5x^{3} + 5$ from $7 - 2x^{2} - 3x^{3} + x^{4}$.", "answer_latex": [ "$2 - x - 2x^{2} + 2x^{3}$." ], "answer_markdown": [ "$2 - x - 2x^{2} + 2x^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(7 - 2*x**2 - 3*x**3 + x**4) - (x**4 + x - 5*x**3 + 5)", "answer_expr": "2 - x - 2*x**2 + 2*x**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**3 - 2*a**2 - a + 2" ], "shape": [ "identity: 2*N*a**N + N - a" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/11", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 11", "problem_latex": "$a^{3} + b^{3} + c^{3} - 3abc$ from $3abc + a^{3} - 2b^{3} - 3c^{3}$.", "markdown": "$a^{3} + b^{3} + c^{3} - 3abc$ from $3abc + a^{3} - 2b^{3} - 3c^{3}$.", "answer_latex": [ "$-3b^{3} - 4c^{3} + 6abc$." ], "answer_markdown": [ "$-3b^{3} - 4c^{3} + 6abc$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a*b*c + a**3 - 2*b**3 - 3*c**3) - (a**3 + b**3 + c**3 - 3*a*b*c)", "answer_expr": "-3*b**3 - 4*c**3 + 6*a*b*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 6*a*b*c - 3*b**3 - 4*c**3" ], "shape": [ "identity: N*a*b*c + N*b**N + N*c**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/12", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 12", "problem_latex": "$2x^{4} - 5x^{2} + 7x - 3$ from $x^{4} + 2 - 2x^{3} - x^{2}$.", "markdown": "$2x^{4} - 5x^{2} + 7x - 3$ from $x^{4} + 2 - 2x^{3} - x^{2}$.", "answer_latex": [ "$-x^{4} - 2x^{3} + 4x^{2} - 7x + 5$." ], "answer_markdown": [ "$-x^{4} - 2x^{3} + 4x^{2} - 7x + 5$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 + 2 - 2*x**3 - x**2) - (2*x**4 - 5*x**2 + 7*x - 3)", "answer_expr": "-x**4 - 2*x**3 + 4*x**2 - 7*x + 5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a**4 - 2*a**3 + 4*a**2 - 7*a + 5" ], "shape": [ "identity: N*a + 2*N*a**N + N - a**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/13", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 13", "problem_latex": "$1 - x^{5} - x + x^{4} - x^{3}$ from $x^{4} + 1 + x + x^{2}$.", "markdown": "$1 - x^{5} - x + x^{4} - x^{3}$ from $x^{4} + 1 + x + x^{2}$.", "answer_latex": [ "$2x + x^{2} + x^{3} + x^{5}$." ], "answer_markdown": [ "$2x + x^{2} + x^{3} + x^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 + 1 + x + x**2) - (1 - x**5 - x + x**4 - x**3)", "answer_expr": "2*x + x**2 + x**3 + x**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**5 + a**3 + a**2 + 2*a" ], "shape": [ "identity: N*a + 3*a**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/14", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 14", "problem_latex": "$a^{3} - b^{3} + 3a^{2}b - 3ab^{2}$ from $a^{3} + b^{3} - a^{2}b - ab^{2}$.", "markdown": "$a^{3} - b^{3} + 3a^{2}b - 3ab^{2}$ from $a^{3} + b^{3} - a^{2}b - ab^{2}$.", "answer_latex": [ "$2b^{3} - 4a^{2}b + 2ab^{2}$." ], "answer_markdown": [ "$2b^{3} - 4a^{2}b + 2ab^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 + b**3 - a**2*b - a*b**2) - (a**3 - b**3 + 3*a**2*b - 3*a*b**2)", "answer_expr": "2*b**3 - 4*a**2*b + 2*a*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -4*a**2*b + 2*a*b**2 + 2*b**3" ], "shape": [ "identity: N*a*b**N + N*a**N*b + N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/15", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 15", "problem_latex": "$a^{2} b - ab^{2} - 3a^{3} b^{3} - b^{4}$ from $b^{4} - 5a^{3} b^{3} - 2ab^{2} + a^{2} b$.", "markdown": "$a^{2} b - ab^{2} - 3a^{3} b^{3} - b^{4}$ from $b^{4} - 5a^{3} b^{3} - 2ab^{2} + a^{2} b$.", "answer_latex": [ "$2b^{4} - 2a^{3}b^{3} - ab^{2}$." ], "answer_markdown": [ "$2b^{4} - 2a^{3}b^{3} - ab^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(b**4 - 5*a**3*b**3 - 2*a*b**2 + a**2*b) - (a**2*b - a*b**2 - 3*a**3*b**3 - b**4)", "answer_expr": "2*b**4 - 2*a**3*b**3 - a*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -2*a**3*b**3 - a*b**2 + 2*b**4" ], "shape": [ "identity: N*a**N*b**N + N*b**N - a*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/16", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 16", "problem_latex": "$-x^{3} + 7x^{2} y - 2y^{3} + 3xy^{2}$ from $3x^{3} + 5y^{3} - xy^{2} + 4x^{2}y$.", "markdown": "$-x^{3} + 7x^{2} y - 2y^{3} + 3xy^{2}$ from $3x^{3} + 5y^{3} - xy^{2} + 4x^{2}y$.", "answer_latex": [ "$4x^{3} - 3x^{2}y - 4xy^{2} + 7y^{3}$." ], "answer_markdown": [ "$4x^{3} - 3x^{2}y - 4xy^{2} + 7y^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**3 + 5*y**3 - x*y**2 + 4*x**2*y) - (-x**3 + 7*x**2*y - 2*y**3 + 3*x*y**2)", "answer_expr": "4*x**3 - 3*x**2*y - 4*x*y**2 + 7*y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a**3 - 3*a**2*b - 4*a*b**2 + 7*b**3" ], "shape": [ "identity: N*a*b**N + N*a**N*b + N*a**N + N*b**N" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/2", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 2", "problem_latex": "$a - 3b - 5c$ from $3a - 5b + c$.", "markdown": "$a - 3b - 5c$ from $3a - 5b + c$.", "answer_latex": [ "$2a - 2b + 6c$." ], "answer_markdown": [ "$2a - 2b + 6c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a - 5*b + c) - (a - 3*b - 5*c)", "answer_expr": "2*a - 2*b + 6*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a - 2*b + 6*c" ], "shape": [ "identity: N*a + N*b + N*c" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/3", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 3", "problem_latex": "$2x - 4y + 6z$ from $4x - y - 2z$.", "markdown": "$2x - 4y + 6z$ from $4x - y - 2z$.", "answer_latex": [ "$2a + 3y -8z$." ], "answer_markdown": [ "$2a + 3y -8z$." ], "checks": [ { "task": "identity", "verdict": "FLAG-MISMATCH", "judge_why": [ "-1.4007972419737125619", "-5.3458521870286576169" ], "problem_expr": "(4*x - y - 2*z) - (2*x - 4*y + 6*z)", "answer_expr": "2*a + 3*y - 8*z" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "identity: 2*a + 3*b - 8*c" ], "shape": [ "identity: N*a + N*b + N*c" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/4", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 4", "problem_latex": "$5x - 11y - 3z$ from $6x - 7y + 2z$.", "markdown": "$5x - 11y - 3z$ from $6x - 7y + 2z$.", "answer_latex": [ "$x + 4y + 5z$." ], "answer_markdown": [ "$x + 4y + 5z$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(6*x - 7*y + 2*z) - (5*x - 11*y - 3*z)", "answer_expr": "x + 4*y + 5*z" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a + 4*b + 5*c" ], "shape": [ "identity: N*b + N*c + a" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-19/5", "set": "wentworth-first-steps-in-algebra-1894/ex-19", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "50", "location": "Exercise 19, problem 5", "problem_latex": "$ab - ac - bc + bd$ from $ab + ac + bc + bd$.", "markdown": "$ab - ac - bc + bd$ from $ab + ac + bc + bd$.", "answer_latex": [ "$2ac + 2bc$." ], "answer_markdown": [ "$2ac + 2bc$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a*b + a*c + b*c + b*d) - (a*b - a*c - b*c + b*d)", "answer_expr": "2*a*c + 2*b*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", 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"wentworth-first-steps-in-algebra-1894/ex-21/11", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 11", "problem_latex": "$x^{2} - 3y^{2}$ and $4y$.", "markdown": "$x^{2} - 3y^{2}$ and $4y$.", "answer_latex": [ "$4x^{2}y - 12y^{3}$." ], "answer_markdown": [ "$4x^{2}y - 12y^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 - 3*y**2)*(4*y)", "answer_expr": "4*x**2*y - 12*y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*b*(a**2 - 3*b**2)" ], "shape": [ "identity: N*b*(N*b**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/12", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 12, "part": null, 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"identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 + 2*a**2*b + 2*a*b**2)*(a**2)", "answer_expr": "a**5 + 2*a**4*b + 2*a**3*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**2*(a**3 + 2*a**2*b + 2*a*b**2)" ], "shape": [ "identity: a**N*(N*a*b**N + N*a**N*b + a**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/18", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 18", "problem_latex": "$a^{3} + 2a^{2}b + 2ab^{2}$ and $b^{3}$.", "markdown": "$a^{3} + 2a^{2}b + 2ab^{2}$ and $b^{3}$.", "answer_latex": [ "$a^{3}b^{3} + 2a^{2}b^{4} + 2ab^{5}$." ], "answer_markdown": [ "$a^{3}b^{3} + 2a^{2}b^{4} + 2ab^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 + 2*a**2*b + 2*a*b**2)*(b**3)", "answer_expr": "a**3*b**3 + 2*a**2*b**4 + 2*a*b**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: b**3*(a**3 + 2*a**2*b + 2*a*b**2)" ], "shape": [ "identity: b**N*(N*a*b**N + N*a**N*b + a**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/19", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 19", "problem_latex": "$4x^{2} - 6xy - 9y^{2}$ and $2x$.", "markdown": "$4x^{2} - 6xy - 9y^{2}$ and $2x$.", "answer_latex": [ "$8x^{3} - 12x^{2}y - 18xy^{2}$." ], "answer_markdown": [ "$8x^{3} - 12x^{2}y - 18xy^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4*x**2 - 6*x*y - 9*y**2)*(2*x)", "answer_expr": "8*x**3 - 12*x**2*y - 18*x*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a*(4*a**2 - 6*a*b - 9*b**2)" ], "shape": [ "identity: N*a*(N*a*b + N*a**N + N*b**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/2", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 2", "problem_latex": "$2x - 3y$ and $4x$.", "markdown": "$2x - 3y$ and $4x$.", "answer_latex": [ "$8x^{2} - 12xy$." ], "answer_markdown": [ "$8x^{2} - 12xy$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x - 3*y)*(4*x)", "answer_expr": "8*x**2 - 12*x*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a*(2*a - 3*b)" ], "shape": [ "identity: N*a*(N*a + N*b)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/20", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 20", "problem_latex": "$-x^{2} - 2xy + y^{2}$ and $-y$.", "markdown": "$-x^{2} - 2xy + y^{2}$ and $-y$.", "answer_latex": [ "$x^{2}y + 2xy^{2} - y^{3}$." ], "answer_markdown": [ "$x^{2}y + 2xy^{2} - y^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-x**2 - 2*x*y + y**2)*(-y)", "answer_expr": "x**2*y + 2*x*y**2 - y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -b*(-a**2 - 2*a*b + b**2)" ], "shape": [ "identity: -b*(N*a*b - a**N + b**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/21", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 21", "problem_latex": "$-a^{3} - a^{2}b^{2} - b^{3}$ and $-a^{2}$.", "markdown": "$-a^{3} - a^{2}b^{2} - b^{3}$ and $-a^{2}$.", "answer_latex": [ "$a^{5} + a^{4}b^{2} + a^{2}b^{3}$." ], "answer_markdown": [ "$a^{5} + a^{4}b^{2} + a^{2}b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-a**3 - a**2*b**2 - b**3)*(-a**2)", "answer_expr": "a**5 + a**4*b**2 + a**2*b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a**2*(-a**3 - a**2*b**2 - b**3)" ], "shape": [ "identity: -a**N*(-a**N*b**N - a**N - b**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-21/22", "set": "wentworth-first-steps-in-algebra-1894/ex-21", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "53", "location": "Exercise 21, problem 22", "problem_latex": "$-x^{2} + 2xy - y^{2}$ and $-y^{2}$.", "markdown": "$-x^{2} + 2xy - y^{2}$ and $-y^{2}$.", "answer_latex": [ "$x^{2}y^{2} - 2xy^{3} + y^{4}$." ], "answer_markdown": [ "$x^{2}y^{2} - 2xy^{3} + y^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-x**2 + 2*x*y - y**2)*(-y**2)", "answer_expr": "x**2*y**2 - 2*x*y**3 + y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -b**2*(-a**2 + 2*a*b - b**2)" ], "shape": [ "identity: -b**N*(N*a*b - a**N - b**N)" ], "same_problem_in": 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"wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 16", "problem_latex": "$a + b + c$ and $a - c$.", "markdown": "$a + b + c$ and $a - c$.", "answer_latex": [ "$a^{2} + ab - bc - c^{2}$." ], "answer_markdown": [ "$a^{2} + ab - bc - c^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a + b + c)*(a - c)", "answer_expr": "a**2 + a*b - b*c - c**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-b + x)*(a + b + x)" ], "shape": [ "identity: (-b + x)*(a + b + x)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/17", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 17", "problem_latex": "$a^{2} - ab + b^{2}$ and $a^{2} + b^{2}$.", "markdown": "$a^{2} - ab + b^{2}$ and $a^{2} + b^{2}$.", "answer_latex": [ "$a^{4} - a^{3}b + 2a^{2}b^{2} - ab^{3} + b^{4}$." ], "answer_markdown": [ "$a^{4} - a^{3}b + 2a^{2}b^{2} - ab^{3} + b^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - a*b + b**2)*(a**2 + b**2)", "answer_expr": "a**4 - a**3*b + 2*a**2*b**2 - a*b**3 + b**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + x**2)*(a**2 - a*x + x**2)" ], "shape": [ "identity: (a**N + x**N)*(-a*x + a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/18", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 18", "problem_latex": "$x^{3} - 3x^{2} + 7$ and $x^{2} - 3$.", "markdown": "$x^{3} - 3x^{2} + 7$ and $x^{2} - 3$.", "answer_latex": [ "$x^{5} - 3x^{4} - x^{3} + 16x^{2} - 21$." ], "answer_markdown": [ "$x^{5} - 3x^{4} - x^{3} + 16x^{2} - 21$." ], "checks": [ { "task": "identity", "verdict": "FLAG-MISMATCH", "judge_why": [ "-13.601711451380447852", "-12.673645928235873987" ], "problem_expr": "(x**3 - 3*x**2 + 7)*(x**2 - 3)", "answer_expr": "x**5 - 3*x**4 - x**3 + 16*x**2 - 21" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "identity: (x**2 - 3)*(x**3 - 3*x**2 + 7)" ], "shape": [ "identity: (N + x**N)*(N*x**N + N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/19", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 19", "problem_latex": "$a^{2} + ab + b^{2}$ and $a - b$.", "markdown": "$a^{2} + ab + b^{2}$ and $a - b$.", "answer_latex": [ "$a^{3} - b^{3}$." ], "answer_markdown": [ "$a^{3} - b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 + a*b + b**2)*(a - b)", "answer_expr": "a**3 - b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a + x)*(a**2 + a*x + x**2)" ], "shape": [ "identity: (-a + x)*(a*x + a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/2", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 2", "problem_latex": "$x - 7$ and $x + 6$.", "markdown": "$x - 7$ and $x + 6$.", "answer_latex": [ "$x^{2} - x - 42$." ], "answer_markdown": [ "$x^{2} - x - 42$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x - 7)*(x + 6)", "answer_expr": "x**2 - x - 42" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 7)*(x + 6)" ], "shape": [ "identity: (N + x)**2" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-20/7" ], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/20", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 20", "problem_latex": "$a^{2} - ab + b^{2}$ and $a + b$.", "markdown": "$a^{2} - ab + b^{2}$ and $a + b$.", "answer_latex": [ "$a^{3} + b^{3}$." ], "answer_markdown": [ "$a^{3} + b^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - a*b + b**2)*(a + b)", "answer_expr": "a**3 + b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a + x)*(a**2 - a*x + x**2)" ], "shape": [ "identity: (a + x)*(-a*x + a**N + x**N)" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-19/6" ], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/21", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 21", "problem_latex": "$x^{2} + 5x - 10$ and $2x^{2} + 3x - 4$.", "markdown": "$x^{2} + 5x - 10$ and $2x^{2} + 3x - 4$.", "answer_latex": [ "$2x^{4} + 13x^{3} - 9x^{2} - 50x + 40$." ], "answer_markdown": [ "$2x^{4} + 13x^{3} - 9x^{2} - 50x + 40$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + 5*x - 10)*(2*x**2 + 3*x - 4)", "answer_expr": "2*x**4 + 13*x**3 - 9*x**2 - 50*x + 40" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**2 + 5*x - 10)*(2*x**2 + 3*x - 4)" ], "shape": [ "identity: (N*x + N + x**N)*(N*x + N*x**N + N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/22", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 22", "problem_latex": "$3x^{3} - 2x^{2} + x$ and $3x^{2} + 2x - 2$.", "markdown": "$3x^{3} - 2x^{2} + x$ and $3x^{2} + 2x - 2$.", "answer_latex": [ "$9x^{5} - 7x^{3} + 6x^{2} - 2x$." ], "answer_markdown": [ "$9x^{5} - 7x^{3} + 6x^{2} - 2x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**3 - 2*x**2 + x)*(3*x**2 + 2*x - 2)", "answer_expr": "9*x**5 - 7*x**3 + 6*x**2 - 2*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (3*x**2 + 2*x - 2)*(3*x**3 - 2*x**2 + x)" ], "shape": [ "identity: (2*N*x**N + x)*(N*x + N*x**N + N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/23", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 23", "problem_latex": "$x^{3} + 2x^{2}y + 3xy^{2}$ and $x^{2} - 2xy + y^{2}$.", "markdown": "$x^{3} + 2x^{2}y + 3xy^{2}$ and $x^{2} - 2xy + y^{2}$.", "answer_latex": [ "$x^{5} - 4x^{2}y^{3} + 3xy^{4}$." ], "answer_markdown": [ "$x^{5} - 4x^{2}y^{3} + 3xy^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3 + 2*x**2*y + 3*x*y**2)*(x**2 - 2*x*y + y**2)", "answer_expr": "x**5 - 4*x**2*y**3 + 3*x*y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 - 2*a*x + x**2)*(3*a**2*x + 2*a*x**2 + x**3)" ], "shape": [ "identity: (N*a*x + a**N + x**N)*(N*a*x**N + N*a**N*x + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/24", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 24", "problem_latex": "$a^{2} - 3ab - b^{2}$ and $-a^{2} + ab + 2b^{2}$.", "markdown": "$a^{2} - 3ab - b^{2}$ and $-a^{2} + ab + 2b^{2}$.", "answer_latex": [ "$-a^{4} + 4a^{3}b - 7ab^{3} - 2b^{4}$." ], "answer_markdown": [ "$-a^{4} + 4a^{3}b - 7ab^{3} - 2b^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - 3*a*b - b**2)*(-a**2 + a*b + 2*b**2)", "answer_expr": "-a**4 + 4*a**3*b - 7*a*b**3 - 2*b**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**2 - 3*a*x + x**2)*(2*a**2 + a*x - x**2)" ], "shape": [ "identity: (N*a**N + a*x - x**N)*(N*a*x - a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/25", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 25", "problem_latex": "$3a^{2}b^{2} + 2 ab^{3} - 5a^{3}b$ and $5a^{2}b^{2} - ab^{3} - b^{4}$.", "markdown": "$3a^{2}b^{2} + 2 ab^{3} - 5a^{3}b$ and $5a^{2}b^{2} - ab^{3} - b^{4}$.", "answer_latex": [ "$-25a^{5}b^{3} + 20a^{4}b^{4} + 12a^{3}b^{5} - 5a^{2}b^{6} - 2ab^{7}$." ], "answer_markdown": [ "$-25a^{5}b^{3} + 20a^{4}b^{4} + 12a^{3}b^{5} - 5a^{2}b^{6} - 2ab^{7}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**2*b**2 + 2*a*b**3 - 5*a**3*b)*(5*a**2*b**2 - a*b**3 - b**4)", "answer_expr": "-25*a**5*b**3 + 20*a**4*b**4 + 12*a**3*b**5 - 5*a**2*b**6 - 2*a*b**7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**4 - a**3*x + 5*a**2*x**2)*(2*a**3*x + 3*a**2*x**2 - 5*a*x**3)" ], "shape": [ "identity: (N*a*x**N + N*a**N*x + N*a**N*x**N)*(N*a**N*x**N - a**N*x - a**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/26", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 26", "problem_latex": "$a^{2} - 2ab + b^{2}$ and $a^{2} + 2ab + b^{2}$.", "markdown": "$a^{2} - 2ab + b^{2}$ and $a^{2} + 2ab + b^{2}$.", "answer_latex": [ "$a^{4} - 2a^{2}b^{2} + b^{4}$." ], "answer_markdown": [ "$a^{4} - 2a^{2}b^{2} + b^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - 2*a*b + b**2)*(a**2 + 2*a*b + b**2)", "answer_expr": "a**4 - 2*a**2*b**2 + b**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 - 2*a*x + x**2)*(a**2 + 2*a*x + x**2)" ], "shape": [ "identity: (N*a*x + a**N + x**N)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/27", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 27", "problem_latex": "$ab + ac + cd$ and $ab - ac + cd$.", "markdown": "$ab + ac + cd$ and $ab - ac + cd$.", "answer_latex": [ "$a^{2}b^{2} + 2abcd - a^{2}c^{2} + c^{2}d^{2}$." ], "answer_markdown": [ "$a^{2}b^{2} + 2abcd - a^{2}c^{2} + c^{2}d^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a*b + a*c + c*d)*(a*b - a*c + c*d)", "answer_expr": "a**2*b**2 + 2*a*b*c*d - a**2*c**2 + c**2*d**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a*x + b*c - b*x)*(a*x + b*c + b*x)" ], "shape": [ "identity: (a*x + b*c - b*x)*(a*x + b*c + b*x)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/28", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 28, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 28", "problem_latex": "$3x^{2}y^{2} + xy^{3} - 2x^{3}y$ and $x^{2}y^{2} + xy^{3} - 3y^{4}$.", "markdown": "$3x^{2}y^{2} + xy^{3} - 2x^{3}y$ and $x^{2}y^{2} + xy^{3} - 3y^{4}$.", "answer_latex": [ "$-2x^{5}y^{3} + x^{4}y^{4} + 10x^{3}y^{5} - 8x^{2}y^{6} - 3xy^{7}$." ], "answer_markdown": [ "$-2x^{5}y^{3} + x^{4}y^{4} + 10x^{3}y^{5} - 8x^{2}y^{6} - 3xy^{7}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**2*y**2 + x*y**3 - 2*x**3*y)*(x**2*y**2 + x*y**3 - 3*y**4)", "answer_expr": "-2*x**5*y**3 + x**4*y**4 + 10*x**3*y**5 - 8*x**2*y**6 - 3*x*y**7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-3*a**4 + a**3*x + a**2*x**2)*(a**3*x + 3*a**2*x**2 - 2*a*x**3)" ], "shape": [ "identity: (N*a**N + a**N*x + a**N*x**N)*(N*a*x**N + N*a**N*x**N + a**N*x)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/29", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 29, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 29", "problem_latex": "$x^{2} + 2xy - y^{2}$ and $x^{2} - 2xy + y^{2}$.", "markdown": "$x^{2} + 2xy - y^{2}$ and $x^{2} - 2xy + y^{2}$.", "answer_latex": [ "$x^{4} - 4x^{2}y^{2} + 4xy^{3} - y^{4}$." ], "answer_markdown": [ "$x^{4} - 4x^{2}y^{2} + 4xy^{3} - y^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + 2*x*y - y**2)*(x**2 - 2*x*y + y**2)", "answer_expr": "x**4 - 4*x**2*y**2 + 4*x*y**3 - y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**2 + 2*a*x + x**2)*(a**2 - 2*a*x + x**2)" ], "shape": [ "identity: (N*a*x - a**N + x**N)*(N*a*x + a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/3", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 3", "problem_latex": "$x + 7$ and $x - 6$.", "markdown": "$x + 7$ and $x - 6$.", "answer_latex": [ "$x^{2} + x - 42$." ], "answer_markdown": [ "$x^{2} + x - 42$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 7)*(x - 6)", "answer_expr": "x**2 + x - 42" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 6)*(x + 7)" ], "shape": [ "identity: (N + x)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/30", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 30, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 30", "problem_latex": "$3x^{2} + xy - y^{2}$ and $x^{2} - 2xy - 3y^{2}$.", "markdown": "$3x^{2} + xy - y^{2}$ and $x^{2} - 2xy - 3y^{2}$.", "answer_latex": [ "$3x^{4} - 5x^{3}y - 12x^{2}y^{2} - xy^{3} + 3y^{4}$." ], "answer_markdown": [ "$3x^{4} - 5x^{3}y - 12x^{2}y^{2} - xy^{3} + 3y^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**2 + x*y - y**2)*(x**2 - 2*x*y - 3*y**2)", "answer_expr": "3*x**4 - 5*x**3*y - 12*x**2*y**2 - x*y**3 + 3*y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-3*a**2 - 2*a*x + x**2)*(-a**2 + a*x + 3*x**2)" ], "shape": [ "identity: (N*x**N + a*x - a**N)*(N*a*x + N*a**N + x**N)" ], 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"PASS" ] }, "form": [ "identity: (-a**2 - b**2 + x**2)*(a**2 + b**2 - b*x + x**2)" ], "shape": [ "identity: (-a**N - b**N + x**N)*(a**N - b*x + b**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/33", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 33, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 33", "problem_latex": "$a^{2} + 4abx - 4a^{2}b^{2}x^{2}$ and $a^{2} - 4abx + 4a^{2}b^{2}x^{2}$.", "markdown": "$a^{2} + 4abx - 4a^{2}b^{2}x^{2}$ and $a^{2} - 4abx + 4a^{2}b^{2}x^{2}$.", "answer_latex": [ "$a^{4} - 16a^{2}b^{2}x^{2} + 32a^{3}b^{3}x^{3} - 16a^{4}b^{4}x^{4}$." ], "answer_markdown": [ "$a^{4} - 16a^{2}b^{2}x^{2} + 32a^{3}b^{3}x^{3} - 16a^{4}b^{4}x^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 + 4*a*b*x - 4*a**2*b**2*x**2)*(a**2 - 4*a*b*x + 4*a**2*b**2*x**2)", "answer_expr": "a**4 - 16*a**2*b**2*x**2 + 32*a**3*b**3*x**3 - 16*a**4*b**4*x**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-4*a**2*b**2*x**2 + 4*a*b*x + x**2)*(4*a**2*b**2*x**2 - 4*a*b*x + x**2)" ], "shape": [ "identity: (N*a*b*x + N*a**N*b**N*x**N + x**N)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/34", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 34, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 34", "problem_latex": "$3 a^{2} - 2abx + b^{2}x^{2}$ and $2a^{2} + 3abx - 2b^{2}x^{2}$.", "markdown": "$3 a^{2} - 2abx + b^{2}x^{2}$ and $2a^{2} + 3abx - 2b^{2}x^{2}$.", "answer_latex": [ "$6a^{4} + 5a^{3}bx - 10a^{2}b^{2}x^{2} + 7ab^{3}x^{3} - 2b^{4}x^{4}$." ], "answer_markdown": [ "$6a^{4} + 5a^{3}bx - 10a^{2}b^{2}x^{2} + 7ab^{3}x^{3} - 2b^{4}x^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**2 - 2*a*b*x + b**2*x**2)*(2*a**2 + 3*a*b*x - 2*b**2*x**2)", "answer_expr": "6*a**4 + 5*a**3*b*x - 10*a**2*b**2*x**2 + 7*a*b**3*x**3 - 2*b**4*x**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-2*a**2*b**2 + 3*a*b*x + 2*x**2)*(a**2*b**2 - 2*a*b*x + 3*x**2)" ], "shape": [ "identity: (N*a*b*x + N*x**N + a**N*b**N)*(N*a*b*x + N*a**N*b**N + N*x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/35", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 35, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 35", "problem_latex": "$2x^{3}y + 4x^{2}y^{2} - 8xy^{3}$ and $2xy^{3} - 3x^{2}y^{2} + 5x^{3}y$.", "markdown": "$2x^{3}y + 4x^{2}y^{2} - 8xy^{3}$ and $2xy^{3} - 3x^{2}y^{2} + 5x^{3}y$.", "answer_latex": [ "$10x^{6}y^{2} + 14x^{5}y^{3} - 48x^{4}y^{4} + 32x^{3}y^{5} - 16x^{2}y^{6}$." ], "answer_markdown": [ "$10x^{6}y^{2} + 14x^{5}y^{3} - 48x^{4}y^{4} + 32x^{3}y^{5} - 16x^{2}y^{6}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x**3*y + 4*x**2*y**2 - 8*x*y**3)*(2*x*y**3 - 3*x**2*y**2 + 5*x**3*y)", "answer_expr": "10*x**6*y**2 + 14*x**5*y**3 - 48*x**4*y**4 + 32*x**3*y**5 - 16*x**2*y**6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-8*a**3*x + 4*a**2*x**2 + 2*a*x**3)*(2*a**3*x - 3*a**2*x**2 + 5*a*x**3)" ], "shape": [ "identity: (N*a*x**N + N*a**N*x + N*a**N*x**N)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/4", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 4", "problem_latex": "$x - 7$ and $x - 6$.", "markdown": "$x - 7$ and $x - 6$.", "answer_latex": [ "$x^{2} - 13x + 42$." ], "answer_markdown": [ "$x^{2} - 13x + 42$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x - 7)*(x - 6)", "answer_expr": "x**2 - 13*x + 42" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 7)*(x - 6)" ], "shape": [ "identity: (N + x)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/5", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 5", "problem_latex": "$x + 8$ and $x - 5$.", "markdown": "$x + 8$ and $x - 5$.", "answer_latex": [ "$x^{2} + 3x - 40$." ], "answer_markdown": [ "$x^{2} + 3x - 40$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 8)*(x - 5)", "answer_expr": "x**2 + 3*x - 40" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 5)*(x + 8)" ], "shape": [ "identity: (N + x)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/6", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 6", "problem_latex": "$2x + 3$ and $2x + 3$.", "markdown": "$2x + 3$ and $2x + 3$.", "answer_latex": [ "$4x^{2} + 12x + 9$." ], "answer_markdown": [ "$4x^{2} + 12x + 9$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x + 3)*(2*x + 3)", "answer_expr": "4*x**2 + 12*x + 9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*x + 3)**2" ], "shape": [ "identity: (N*x + N)**N" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/7", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 7", "problem_latex": "$2x - 3$ and $2x - 3$.", "markdown": "$2x - 3$ and $2x - 3$.", "answer_latex": [ "$4x^{2} - 12x + 9$." ], "answer_markdown": [ "$4x^{2} - 12x + 9$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x - 3)*(2*x - 3)", "answer_expr": "4*x**2 - 12*x + 9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*x - 3)**2" ], "shape": [ "identity: (N*x + N)**N" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/8", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 8", "problem_latex": "$2x + 3$ and $2x - 3$.", "markdown": "$2x + 3$ and $2x - 3$.", "answer_latex": [ "$4x^{2} - 9$." ], "answer_markdown": [ "$4x^{2} - 9$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x + 3)*(2*x - 3)", "answer_expr": "4*x**2 - 9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*x - 3)*(2*x + 3)" ], "shape": [ "identity: (N*x + N)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-22/9", "set": "wentworth-first-steps-in-algebra-1894/ex-22", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "56", "location": "Exercise 22, problem 9", "problem_latex": "$3x - 2$ and $2 - 3x$.", "markdown": "$3x - 2$ and $2 - 3x$.", "answer_latex": [ "$-9x^{2} + 12x - 4$." ], "answer_markdown": [ "$-9x^{2} + 12x - 4$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x - 2)*(2 - 3*x)", "answer_expr": "-9*x**2 + 12*x - 4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-3*x + 2)*(3*x - 2)" ], "shape": [ "identity: (N*x + N)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/1", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 1", "problem_latex": "$2a^{3} - a^{2}$ by $a$.", "markdown": "$2a^{3} - a^{2}$ by $a$.", "answer_latex": [ "$2a^{2} - a$." ], "answer_markdown": [ "$2a^{2} - a$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**3 - a**2)/(a)", "answer_expr": "2*a**2 - a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*a**3 - a**2)/a" ], "shape": [ "identity: (N*a**N - a**N)/a" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/10", "set": "wentworth-first-steps-in-algebra-1894/ex-23", 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"markdown": "$-x^{2}y - x^{2}y^{2}$ by $-xy$.", "answer_latex": [ "$x + xy$." ], "answer_markdown": [ "$x + xy$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-x**2*y - x**2*y**2)/(-x*y)", "answer_expr": "x + x*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -(-a**2*b**2 - a**2*b)/(a*b)" ], "shape": [ "identity: -(-a**N*b - a**N*b**N)/(a*b)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/13", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 13", "problem_latex": "$9a - 12b + 6c$ by $-3$.", "markdown": "$9a - 12b + 6c$ by $-3$.", "answer_latex": [ "$-3a + 4b - 2c$." ], "answer_markdown": [ "$-3a + 4b - 2c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(9*a - 12*b + 6*c)/(-3)", "answer_expr": "-3*a + 4*b - 2*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -3*a + 4*b - 2*c" ], "shape": [ "identity: N*a + N*b + N*c" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/14", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 14", "problem_latex": "$a^{3}b^{2} - a^{2}b^{5} - a^{4}b^{2}$ by $a^{2}b$.", "markdown": "$a^{3}b^{2} - a^{2}b^{5} - a^{4}b^{2}$ by $a^{2}b$.", "answer_latex": [ "$ab - b^{4} - a^{2}b$." ], "answer_markdown": [ "$ab - b^{4} - a^{2}b$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3*b**2 - a**2*b**5 - a**4*b**2)/(a**2*b)", "answer_expr": "a*b - b**4 - a**2*b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**4*b**2 + a**3*b**2 - a**2*b**5)/(a**2*b)" ], "shape": [ "identity: -a**(2*N)*b**N/b" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/15", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 15", "problem_latex": "$3x^{3} - 6x^{2}y - 9xy^{2}$ by $3x$.", "markdown": "$3x^{3} - 6x^{2}y - 9xy^{2}$ by $3x$.", "answer_latex": [ "$x^{2} - 2xy - 3y^{2}$." ], "answer_markdown": [ "$x^{2} - 2xy - 3y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**3 - 6*x**2*y - 9*x*y**2)/(3*x)", "answer_expr": "x**2 - 2*x*y - 3*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (3*a**3 - 6*a**2*b - 9*a*b**2)/(3*a)" ], "shape": [ "identity: N*(N*a*b**N + N*a**N*b + N*a**N)/a" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/16", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 16", "problem_latex": "$x^{2}y^{2} - x^{3}y - xy^{3}$ by $xy$.", "markdown": "$x^{2}y^{2} - x^{3}y - xy^{3}$ by $xy$.", "answer_latex": [ "$xy - x^{2} - y^{2}$." ], "answer_markdown": [ "$xy - x^{2} - y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2*y**2 - x**3*y - x*y**3)/(x*y)", "answer_expr": "x*y - x**2 - y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**3*b + a**2*b**2 - a*b**3)/(a*b)" ], "shape": [ "identity: (-a*b**N - a**N*b + a**N*b**N)/(a*b)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/17", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 17", "problem_latex": "$a^{3} - a^{2}b - ab^{2}$ by $-a$.", "markdown": "$a^{3} - a^{2}b - ab^{2}$ by $-a$.", "answer_latex": [ "$-a^{2} + ab + b^{2}$." ], "answer_markdown": [ "$-a^{2} + ab + b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 - a**2*b - a*b**2)/(-a)", "answer_expr": "-a**2 + a*b + b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -(a**3 - a**2*b - a*b**2)/a" ], "shape": [ "identity: -(-a*b**N - a**N*b + a**N)/a" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/18", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 18", "problem_latex": "$a^{2}b - ab + ab^{2}$ by $-ab$.", "markdown": "$a^{2}b - ab + ab^{2}$ by $-ab$.", "answer_latex": [ "$-a + 1 - b$." ], "answer_markdown": [ "$-a + 1 - b$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2*b - a*b + a*b**2)/(-a*b)", "answer_expr": "-a + 1 - b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -(a**2*b + a*b**2 - a*b)/(a*b)" ], "shape": [ "identity: -(-a*b + a*b**N + a**N*b)/(a*b)" ], "same_problem_in": [], "needs": 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"wentworth-first-steps-in-algebra-1894/ex-23/2", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 2", "problem_latex": "$42a^{5} - 6a^{2}$ by $6a$.", "markdown": "$42a^{5} - 6a^{2}$ by $6a$.", "answer_latex": [ "$7a^{4} - a$." ], "answer_markdown": [ "$7a^{4} - a$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(42*a**5 - 6*a**2)/(6*a)", "answer_expr": "7*a**4 - a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (42*a**5 - 6*a**2)/(6*a)" ], "shape": [ "identity: 2*N**2*a**N/a" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/20", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 20", "problem_latex": "$-x^{6} - 2x^{5} - x^{4}$ by $-x^{4}$.", "markdown": "$-x^{6} - 2x^{5} - x^{4}$ by $-x^{4}$.", "answer_latex": [ "$x^{2} + 2x + 1$." ], "answer_markdown": [ "$x^{2} + 2x + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-x**6 - 2*x**5 - x**4)/(-x**4)", "answer_expr": "x**2 + 2*x + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -(-a**6 - 2*a**5 - a**4)/a**4" ], "shape": [ "identity: -a**N*(N*a**N - 2*a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/21", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 21", "problem_latex": "$a^{2}x - abx - acx$ by $ax$.", "markdown": "$a^{2}x - abx - acx$ by $ax$.", "answer_latex": [ "$a - b - c$." ], "answer_markdown": [ "$a - b - c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2*x - a*b*x - a*c*x)/(a*x)", "answer_expr": "a - b - c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2*d - a*b*d - a*c*d)/(a*d)" ], "shape": [ "identity: (-a*b*d - a*c*d + a**N*d)/(a*d)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/22", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 22", "problem_latex": "$3x^{5}y^{2} - 3x^{4}y^{3} - 3x^{2}y^{4}$ by $3x^{2}y^{2}$.", "markdown": "$3x^{5}y^{2} - 3x^{4}y^{3} - 3x^{2}y^{4}$ by $3x^{2}y^{2}$.", "answer_latex": [ "$x^{3} - x^{2}y - y^{2}$." ], "answer_markdown": [ "$x^{3} - x^{2}y - y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**5*y**2 - 3*x**4*y**3 - 3*x**2*y**4)/(3*x**2*y**2)", "answer_expr": "x**3 - x**2*y - y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (3*a**5*b**2 - 3*a**4*b**3 - 3*a**2*b**4)/(3*a**2*b**2)" ], "shape": [ "identity: 3*N**2*a**(2*N)*b**(2*N)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/23", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 23", "problem_latex": "$a^{2}b^{2} - 2ab - 3ab^{3}$ by $ab$.", "markdown": "$a^{2}b^{2} - 2ab - 3ab^{3}$ by $ab$.", "answer_latex": [ "$ab - 2 - 3b^{2}$." ], "answer_markdown": [ "$ab - 2 - 3b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2*b**2 - 2*a*b - 3*a*b**3)/(a*b)", "answer_expr": "a*b - 2 - 3*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2*b**2 - 3*a*b**3 - 2*a*b)/(a*b)" ], "shape": [ "identity: (N*a*b + N*a*b**N + a**N*b**N)/(a*b)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/24", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 24", "problem_latex": "$3a^{3}c^{3} + 3a^{2}c - 3ac^{2}$ by $3ac$.", "markdown": "$3a^{3}c^{3} + 3a^{2}c - 3ac^{2}$ by $3ac$.", "answer_latex": [ "$a^{2}c^{2} + a - c$." ], "answer_markdown": [ "$a^{2}c^{2} + a - c$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**3*c**3 + 3*a**2*c - 3*a*c**2)/(3*a*c)", "answer_expr": "a**2*c**2 + a - c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (3*a**3*b**3 + 3*a**2*b - 3*a*b**2)/(3*a*b)" ], "shape": [ "identity: N*(N*a*b**N + N*a**N*b + N*a**N*b**N)/(a*b)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/3", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 3", "problem_latex": "$21x^{4} + 3x^{2}$ by $3x^{2}$.", "markdown": "$21x^{4} + 3x^{2}$ by $3x^{2}$.", "answer_latex": [ "$7x^{2} + 1$." ], "answer_markdown": [ "$7x^{2} + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(21*x**4 + 3*x**2)/(3*x**2)", "answer_expr": "7*x**2 + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (21*a**4 + 3*a**2)/(3*a**2)" ], "shape": [ "identity: 2*N**2*a**(2*N)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/4", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 4", "problem_latex": "$35m^{4} - 7p^{2}$ by $7$.", "markdown": "$35m^{4} - 7p^{2}$ by $7$.", "answer_latex": [ "$5m^{4} - p^{2}$." ], "answer_markdown": [ "$5m^{4} - p^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(35*m**4 - 7*p**2)/(7)", "answer_expr": "5*m**4 - p**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 5*a**4 - b**2" ], "shape": [ "identity: N*a**N - b**N" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/5", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 5", "problem_latex": "$27x^{5} - 45x^{4}$ by $9x^{2}$.", "markdown": "$27x^{5} - 45x^{4}$ by $9x^{2}$.", "answer_latex": [ "$3x^{3} - 5x^{2}$." ], "answer_markdown": [ "$3x^{3} - 5x^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(27*x**5 - 45*x**4)/(9*x**2)", "answer_expr": "3*x**3 - 5*x**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (27*a**5 - 45*a**4)/(9*a**2)" ], "shape": [ "identity: 2*N**2*a**(2*N)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-23/6", "set": "wentworth-first-steps-in-algebra-1894/ex-23", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "58", "location": "Exercise 23, problem 6", "problem_latex": "$24x^{6} - 8x^{3}$ by $-8x^{3}$.", "markdown": "$24x^{6} - 8x^{3}$ by $-8x^{3}$.", "answer_latex": [ "$-3x^{3} + 1$." ], "answer_markdown": [ "$-3x^{3} + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(24*x**6 - 8*x**3)/(-8*x**3)", "answer_expr": "-3*x**3 + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -(24*a**6 - 8*a**3)/(8*a**3)" ], "shape": [ "identity: 2*N**2*a**(2*N)" ], "same_problem_in": [], "needs": [ 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to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/24", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 24", "problem_latex": "$7 - 8c^{2} + 5c^{3} + 8c$ is divided by $5c - 3$.", "markdown": "$7 - 8c^{2} + 5c^{3} + 8c$ is divided by $5c - 3$.", "answer_latex": [ "$10$." ], "answer_markdown": [ "$10$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 10.0, printed 10 (half-unit 0.5; correctly rounded at the printed digits: 10.0)", "problem_expr": "7 - 8*c**2 + 5*c**3 + 8*c", "answer_expr": "10" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5*x**3 - 8*x**2 + 8*x + 7 at c=Rational(3, 5)" ], "shape": [ "evaluate: N*x + 2*N*x**N + N" ], "same_problem_in": [], "needs": [ "cas.subst" ], "expectation": "X#10,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ÷ ENTER ENTER ENTER 5 ×", "8 −", "×", "8 +", "×", "7 +" ], "constants": [ { "value": "3", "source": "the 3 in '5c - 3' (divisor); also the numerator of the given c = Rational(3, 5)" }, { "value": "5", "source": "the 5 in '5c - 3' (divisor, so c = 3/5); also the coefficient of c^3 in '5c**3'" }, { "value": "8", "source": "the 8 in '-8c**2'; the same 8 is the coefficient of c in '8*c', so it is keyed twice in the Horner evaluation" }, { "value": "7", "source": "the constant term '7' in '7 - 8*c**2 + 5*c**3 + 8*c'" } ], "calculator_value": "+10000E-3", "printed_value": "10", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/25", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 25", "problem_latex": "$3 + 11a^{3} + 30a^{4} - 82a^{2} - 5a$ is divided by $3a^{2} - 4 + 2a$.", "markdown": "$3 + 11a^{3} + 30a^{4} - 82a^{2} - 5a$ is divided by $3a^{2} - 4 + 2a$.", "answer_latex": [ "$7a - 45$." ], "answer_markdown": [ "$7a - 45$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(30*a**4 + 11*a**3 - 82*a**2 - 5*a + 3)/(3*a**2 + 2*a - 4)", "answer_expr": "7*a - 45" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/26", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 26", "problem_latex": "$2x^{3} - 16x + 10-39x^{2} + 17x^{4}$ is divided by $2 - 5x^{2} - 4x$.", "markdown": "$2x^{3} - 16x + 10-39x^{2} + 17x^{4}$ is divided by $2 - 5x^{2} - 4x$.", "answer_latex": [ "$2x^{4}$." ], "answer_markdown": [ "$2x^{4}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(17*x**4 + 2*x**3 - 39*x**2 - 16*x + 10)/(-5*x**2 - 4*x + 2)", "answer_expr": "2*x**4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/3", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 3", "problem_latex": "$x^{2} + x-56$ by $x - 7$.", "markdown": "$x^{2} + x-56$ by $x - 7$.", "answer_latex": [ "$x + 8$." ], "answer_markdown": [ "$x + 8$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + x - 56)/(x - 7)", "answer_expr": "x + 8" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**2 + x - 56)/(x - 7)" ], "shape": [ "identity: (N + x + x**N)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/4", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 4", "problem_latex": "$x^{2} - x-56$ by $x + 7$.", "markdown": "$x^{2} - x-56$ by $x + 7$.", "answer_latex": [ "$x - 8$." ], "answer_markdown": [ "$x - 8$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 - x - 56)/(x + 7)", "answer_expr": "x - 8" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**2 - x - 56)/(x + 7)" ], "shape": [ "identity: (N - x + x**N)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/5", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 5", "problem_latex": "$2a^{2} + 11a + 5$ by $2a + 1$.", "markdown": "$2a^{2} + 11a + 5$ by $2a + 1$.", "answer_latex": [ "$a + 5$." ], "answer_markdown": [ "$a + 5$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**2 + 11*a + 5)/(2*a + 1)", "answer_expr": "a + 5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*x**2 + 11*x + 5)/(2*x + 1)" ], "shape": [ "identity: (N*x + N*x**N + N)/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/6", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 6", "problem_latex": "$6a^{2} - 7a-3$ by $2a - 3$.", "markdown": "$6a^{2} - 7a-3$ by $2a - 3$.", "answer_latex": [ "$3a + 1$." ], "answer_markdown": [ "$3a + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(6*a**2 - 7*a - 3)/(2*a - 3)", "answer_expr": "3*a + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (6*x**2 - 7*x - 3)/(2*x - 3)" ], "shape": [ "identity: (N*x + N*x**N + N)/(N*x + N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/7", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 7", "problem_latex": "$4a^{2} + 23a + 15$ by $4a + 3$.", "markdown": "$4a^{2} + 23a + 15$ by $4a + 3$.", "answer_latex": [ "$a + 5$." ], "answer_markdown": [ "$a + 5$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4*a**2 + 23*a + 15)/(4*a + 3)", "answer_expr": "a + 5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (4*x**2 + 23*x + 15)/(4*x + 3)" ], "shape": [ "identity: (N*x + N*x**N + N)/(N*x + N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/8", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 8", "problem_latex": "$3a^{2} - 4a-4$ by $2 - a$.", "markdown": "$3a^{2} - 4a-4$ by $2 - a$.", "answer_latex": [ "$-3a - 2$." ], "answer_markdown": [ "$-3a - 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**2 - 4*a - 4)/(2 - a)", "answer_expr": "-3*a - 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (3*x**2 - 4*x - 4)/(-x + 2)" ], "shape": [ "identity: (N*x + N*x**N + N)/(N - x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-24/9", "set": "wentworth-first-steps-in-algebra-1894/ex-24", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "62", "location": "Exercise 24, problem 9", "problem_latex": "$x^{4} + x^{2} + 1$ by $x^{2} + x + 1$.", "markdown": "$x^{4} + x^{2} + 1$ by $x^{2} + x + 1$.", "answer_latex": [ "$x^{2} - x + 1$." ], "answer_markdown": [ "$x^{2} - x + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 + x**2 + 1)/(x**2 + x + 1)", "answer_expr": "x**2 - x + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**4 + x**2 + 1)/(x**2 + x + 1)" ], "shape": [ "identity: (2*x**N + 1)/(x + x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/1", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 1", "problem_latex": "Add $2a^{2} - 3ac - 3ab$; $2b^{2} + 3ac + a^{2}$; $-a^{2} - 2b^{2} + 3ab$.", "markdown": "Add $2a^{2} - 3ac - 3ab$; $2b^{2} + 3ac + a^{2}$; $-a^{2} - 2b^{2} + 3ab$.", "answer_latex": [ "$2a^{2}$." ], "answer_markdown": [ "$2a^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**2 - 3*a*c - 3*a*b) + (2*b**2 + 3*a*c + a**2) + (-a**2 - 2*b**2 + 3*a*b)", "answer_expr": "2*a**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*x**2" ], "shape": [ "identity: N*x**N" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/10", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 10", "problem_latex": "Add $-2a^{4} + 3a^{3}b - 4a^{2}b^{2}$; $2a^{3}b - 3a^{2}b^{2}$; $7a^{2}b^{2} + 2a^{4} - b^{4}$.", "markdown": "Add $-2a^{4} + 3a^{3}b - 4a^{2}b^{2}$; $2a^{3}b - 3a^{2}b^{2}$; $7a^{2}b^{2} + 2a^{4} - b^{4}$.", "answer_latex": [ "$5a^{3}b - b^{4}$." ], "answer_markdown": [ "$5a^{3}b - b^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(-2*a**4 + 3*a**3*b - 4*a**2*b**2) + (2*a**3*b - 3*a**2*b**2) + (7*a**2*b**2 + 2*a**4 - b**4)", "answer_expr": "5*a**3*b - b**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a**4 + 5*a*x**3" ], "shape": [ "identity: N*a*x**N - a**N" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/11", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 11", "problem_latex": "From $3x^{3} + 5x - 1$ take the sum of $x - 5 + 5x^{2}$ and\n$3 + 4x -3x^{2}$.", "markdown": "From $3x^{3} + 5x - 1$ take the sum of $x - 5 + 5x^{2}$ and $3 + 4x -3x^{2}$.", "answer_latex": [ "$3x^{3} - 2x^{2} + 1$." ], "answer_markdown": [ "$3x^{3} - 2x^{2} + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**3 + 5*x - 1) - ((x - 5 + 5*x**2) + (3 + 4*x - 3*x**2))", "answer_expr": "3*x**3 - 2*x**2 + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*x**3 - 2*x**2 + 1" ], "shape": [ "identity: 2*N*x**N + 1" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/12", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 12", "problem_latex": "The minuend is $9c^{2} + 11c - 5$, and the remainder is\n$6c^{2} - 13c + 7$. What is the subtrahend?", "markdown": "The minuend is $9c^{2} + 11c - 5$, and the remainder is $6c^{2} - 13c + 7$. What is the subtrahend?", "answer_latex": [ "$3c^{2} + 24c - 12$." ], "answer_markdown": [ "$3c^{2} + 24c - 12$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(9*c**2 + 11*c - 5) - (6*c**2 - 13*c + 7)", "answer_expr": "3*c**2 + 24*c - 12" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*x**2 + 24*x - 12" ], "shape": [ "identity: N*x + N*x**N + N" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/13", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 13", "problem_latex": "Find the remainder when $a^{4} + 6b^{4}$ is divided by\n$a^{2} + 2ab + 2b^{2}$.", "markdown": "Find the remainder when $a^{4} + 6b^{4}$ is divided by $a^{2} + 2ab + 2b^{2}$.", "answer_latex": [ "$2b^{4}$." ], "answer_markdown": [ "$2b^{4}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "2*b**4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/14", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 14", "problem_latex": "Multiply $2 - 5x^{2} - 4x$ by $5 + 2x - 3x^{2}$.", "markdown": "Multiply $2 - 5x^{2} - 4x$ by $5 + 2x - 3x^{2}$.", "answer_latex": [ "$10 - 16x - 39x^{2} + 2x^{3} + 15x^{4}$." ], "answer_markdown": [ "$10 - 16x - 39x^{2} + 2x^{3} + 15x^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2 - 5*x**2 - 4*x)*(5 + 2*x - 3*x**2)", "answer_expr": "10 - 16*x - 39*x**2 + 2*x**3 + 15*x**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-5*x**2 - 4*x + 2)*(-3*x**2 + 2*x + 5)" ], "shape": [ "identity: (N*x + N*x**N + N)**2" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/15", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 15", "problem_latex": "Divide $a^{6} + a^{5}x + a^{4}x^{2} - a^{3}x^{3} + x^{6}$ by $a^{2} + ax + x^{2}$.", "markdown": "Divide $a^{6} + a^{5}x + a^{4}x^{2} - a^{3}x^{3} + x^{6}$ by $a^{2} + ax + x^{2}$.", "answer_latex": [ "$a^{4} - ax^{3} + x^{4}$." ], "answer_markdown": [ "$a^{4} - ax^{3} + x^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**6 + a**5*x + a**4*x**2 - a**3*x**3 + x**6)/(a**2 + a*x + x**2)", "answer_expr": "a**4 - a*x**3 + x**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**6 - a**3*x**3 + a**2*x**4 + a*x**5 + x**6)/(a**2 + a*x + x**2)" ], "shape": [ "identity: (a*x**N + a**N + x**N)/(a*x + a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/16", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 16", "problem_latex": "$ax^{3} - cx + bx^{2} - bx^{3} + cx^{2} - x$.", "markdown": "$ax^{3} - cx + bx^{2} - bx^{3} + cx^{2} - x$.", "answer_latex": [ "$(a - b)x^{3} + (b + c)x^{2} - (c + 1)x$." ], "answer_markdown": [ "$(a - b)x^{3} + (b + c)x^{2} - (c + 1)x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "a*x**3 - c*x + b*x**2 - b*x**3 + c*x**2 - x", "answer_expr": "(a - b)*x**3 + (b + c)*x**2 - (c + 1)*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*x**3 - b*x**3 + b*x**2 + c*x**2 - c*x - x" ], "shape": [ "identity: a*x**N - c*x + c*x**N - x" ], "same_problem_in": [], "needs": [ "cas.collect" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/17", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 17", "problem_latex": "$ax^{4} - 2x + bx^{4} - cx - ax^{3} + bx^{3}$.", "markdown": "$ax^{4} - 2x + bx^{4} - cx - ax^{3} + bx^{3}$.", "answer_latex": [ "$(a + b)x^{4} - (a - b)x^{3} - (c + 2)x$." ], "answer_markdown": [ "$(a + b)x^{4} - (a - b)x^{3} - (c + 2)x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "a*x**4 - 2*x + b*x**4 - c*x - a*x**3 + b*x**3", "answer_expr": "(a + b)*x**4 - (a - b)*x**3 - (c + 2)*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*x**4 - a*x**3 + b*x**4 + b*x**3 - c*x - 2*x" ], "shape": [ "identity: N*x + 2*b*x**N - c*x" ], "same_problem_in": [], "needs": [ "cas.collect" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/18", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 18", "problem_latex": "$x^{3} - bx^{2} - cx + bx - cx^{2} + ax^{3}$.", "markdown": "$x^{3} - bx^{2} - cx + bx - cx^{2} + ax^{3}$.", "answer_latex": [ "$(a + 1)x^{3} - (b + c)x^{2} + (b - c)x$." ], "answer_markdown": [ "$(a + 1)x^{3} - (b + c)x^{2} + (b - c)x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x**3 - b*x**2 - c*x + b*x - c*x**2 + a*x**3", "answer_expr": "(a + 1)*x**3 - (b + c)*x**2 + (b - c)*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*x**3 - b*x**2 + b*x - c*x**2 - c*x + x**3" ], "shape": [ "identity: a*x**N + b*x - b*x**N - c*x - c*x**N + x**N" ], "same_problem_in": [], "needs": [ "cas.collect" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/2", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 2", "problem_latex": "Subtract $3a^{4} - 2 a^{3}b + 4 a^{2}b^{2}$ from $4b^{4} - 2 ab^{3} + 4 a^{2}b^{2}$.", "markdown": "Subtract $3a^{4} - 2 a^{3}b + 4 a^{2}b^{2}$ from $4b^{4} - 2 ab^{3} + 4 a^{2}b^{2}$.", "answer_latex": [ "$-3a^{4} + 2a^{3}b - 2ab^{3} + 4b^{4}$." ], "answer_markdown": [ "$-3a^{4} + 2a^{3}b - 2ab^{3} + 4b^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4*b**4 - 2*a*b**3 + 4*a**2*b**2) - (3*a**4 - 2*a**3*b + 4*a**2*b**2)", "answer_expr": "-3*a**4 + 2*a**3*b - 2*a*b**3 + 4*b**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a**4 - 2*a**3*x + 2*a*x**3 - 3*x**4" ], "shape": [ "identity: N*a*x**N + N*a**N*x + N*a**N + N*x**N" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/3", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 3", "problem_latex": "Simplify $x - y - \\{z - x - (y - x + z)\\}$.", "markdown": "Simplify $x - y - \\{z - x - (y - x + z)\\}$.", "answer_latex": [ "$x$." ], "answer_markdown": [ "$x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x - y - (z - x - (y - x + z))", "answer_expr": "x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x" ], "shape": [ "identity: x" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-75/11" ], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/4", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 4", "problem_latex": "Multiply $a^{2} + b^{2} + c^{2} - d^{2}$ by $a^{2} + b^{2} - c^{2} + d^{2}$.", "markdown": "Multiply $a^{2} + b^{2} + c^{2} - d^{2}$ by $a^{2} + b^{2} - c^{2} + d^{2}$.", "answer_latex": [ "$a^{4} + 2a^{2}b^{2} + b^{4} - c^{4} + 2c^{2}d^{2} - d^{4}$." ], "answer_markdown": [ "$a^{4} + 2a^{2}b^{2} + b^{4} - c^{4} + 2c^{2}d^{2} - d^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 + b**2 + c**2 - d**2)*(a**2 + b**2 - c**2 + d**2)", "answer_expr": "a**4 + 2*a**2*b**2 + b**4 - c**4 + 2*c**2*d**2 - d**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 - b**2 + c**2 + x**2)*(a**2 + b**2 - c**2 + x**2)" ], "shape": [ "identity: (a**N - b**N + c**N + x**N)*(a**N + b**N - c**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/5", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 5", "problem_latex": "Divide $10y^{6} + 2 - 12y^{5}$ by $1 + y^{2} - 2y$.", "markdown": "Divide $10y^{6} + 2 - 12y^{5}$ by $1 + y^{2} - 2y$.", "answer_latex": [ "$10y^{4} + 8y^{3} + 6y^{2} + 4y + 2$." ], "answer_markdown": [ "$10y^{4} + 8y^{3} + 6y^{2} + 4y + 2$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(10*y**6 + 2 - 12*y**5)/(1 + y**2 - 2*y)", "answer_expr": "10*y**4 + 8*y**3 + 6*y**2 + 4*y + 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (10*x**6 - 12*x**5 + 2)/(x**2 - 2*x + 1)" ], "shape": [ "identity: (2*N*x**N + N)/(N*x + x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/6", "set": 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"needs": [ "core.arith" ], "expectation": "X#0,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/7", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 7", "problem_latex": "Simplify $x - (y - z) - \\bigl\\{4y + [2y - (z - x)]\\bigr\\}$.", "markdown": "Simplify $x - (y - z) - \\bigl\\{4y + [2y - (z - x)]\\bigr\\}$.", "answer_latex": [ "$2z - 7y$." ], "answer_markdown": [ "$2z - 7y$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x - (y - z) - (4*y + (2*y - (z - x)))", "answer_expr": "2*z - 7*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -7*a + 2*b" ], "shape": [ "identity: N*a + N*b" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/8", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 8", "problem_latex": "Multiply $a^{2} + b^{2} + c^{2} - ab - ac - bc$ by $a + b + c$.", "markdown": "Multiply $a^{2} + b^{2} + c^{2} - ab - ac - bc$ by $a + b + c$.", "answer_latex": [ "$a^{3} - 3abc + b^{3} + c^{3}$." ], "answer_markdown": [ "$a^{3} - 3abc + b^{3} + c^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 + b**2 + c**2 - a*b - a*c - b*c)*(a + b + c)", "answer_expr": "a**3 - 3*a*b*c + b**3 + c**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a + b + x)*(a**2 - a*b - a*x + b**2 - b*x + x**2)" ], "shape": [ "identity: (a + b + x)*(-a*b - a*x + a**N - b*x + b**N + x**N)" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-19/12" ], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-25/9", "set": "wentworth-first-steps-in-algebra-1894/ex-25", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "63", "location": "Exercise 25, problem 9", "problem_latex": "Divide $16y^{4} - 21x^{2}y^{2} + 21x^{3}y - 10x^{4}$ by $4y^{2} - 5x^{2} + 3xy$.", "markdown": "Divide $16y^{4} - 21x^{2}y^{2} + 21x^{3}y - 10x^{4}$ by $4y^{2} - 5x^{2} + 3xy$.", "answer_latex": [ "$4y^{2} - 3xy + 2x^{2}$." ], "answer_markdown": [ "$4y^{2} - 3xy + 2x^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(16*y**4 - 21*x**2*y**2 + 21*x**3*y - 10*x**4)/(4*y**2 - 5*x**2 + 3*x*y)", "answer_expr": "4*y**2 - 3*x*y + 2*x**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-10*a**4 + 21*a**3*x - 21*a**2*x**2 + 16*x**4)/(-5*a**2 + 3*a*x + 4*x**2)" ], "shape": [ "identity: (N*a**N*x + N*a**N*x**N + N*a**N + N*x**N)/(N*a*x + N*a**N + N*x**N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-26/1", "set": "wentworth-first-steps-in-algebra-1894/ex-26", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "65", "location": "Exercise 26, problem 1", "problem_latex": "$(m + n)^{2}$.", "markdown": "$(m + n)^{2}$.", "answer_latex": [ "$m^{2} + 2mn + n^{2}$." ], "answer_markdown": [ "$m^{2} + 2mn + n^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(m + n)**2", "answer_expr": "m**2 + 2*m*n + n**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a + b)**2" ], "shape": [ "identity: (a + b)**N" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-22/5" ], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-26/10", "set": "wentworth-first-steps-in-algebra-1894/ex-26", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "65", "location": "Exercise 26, problem 10", "problem_latex": "$(2a - 5c)^{2}$.", "markdown": "$(2a - 5c)^{2}$.", "answer_latex": [ "$4a2 - 20ac + 25c^{2}$." ], "answer_markdown": [ "$4a2 - 20ac + 25c^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a - 5*c)**2", "answer_expr": "4*a**2 - 20*a*c + 25*c**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*a - 5*b)**2" ], "shape": [ "identity: (N*a + N*b)**N" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-26/11", "set": "wentworth-first-steps-in-algebra-1894/ex-26", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "65", "location": "Exercise 26, problem 11", "problem_latex": "$(x + y)(x - y)$.", "markdown": "$(x + y)(x - y)$.", "answer_latex": [ "$x^{2} - y^{2}$." ], "answer_markdown": [ "$x^{2} - y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + y)*(x - y)", "answer_expr": "x**2 - y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a - b)*(a + b)" ], "shape": [ "identity: (a - b)*(a + b)" ], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-26/12", "set": "wentworth-first-steps-in-algebra-1894/ex-26", "number": 12, "part": null, "book": 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"problem_latex": " $\\dfrac{c^{2} - 25}{c - 5}$.", "markdown": "$\\dfrac{c^{2} - 25}{c - 5}$.", "answer_latex": [ " $c + 5$." ], "answer_markdown": [ "$c + 5$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(c**2 - 25)/(c - 5)", "answer_expr": "c + 5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**2 - 25)/(x - 5)" ], "shape": [ "identity: (N + x**N)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-28/6", "set": "wentworth-first-steps-in-algebra-1894/ex-28", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "68", "location": "Exercise 28, problem 6", "problem_latex": " $\\dfrac{c^{2} - 25}{c + 5}$.", "markdown": "$\\dfrac{c^{2} - 25}{c + 5}$.", "answer_latex": [ " $c - 5$." ], "answer_markdown": [ "$c 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1}$.", "markdown": "$\\dfrac{x^{3}y^{3}z^{3} - 1}{xyz - 1}$.", "answer_latex": [ "$x^{2}y^{2}z^{2} + xyz + 1$." ], "answer_markdown": [ "$x^{2}y^{2}z^{2} + xyz + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3*y**3*z**3 - 1)/(x*y*z - 1)", "answer_expr": "x**2*y**2*z**2 + x*y*z + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3*b**3*x**3 - 1)/(a*b*x - 1)" ], "shape": [ "identity: (a**N*b**N*x**N - 1)/(a*b*x - 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-29/17", "set": "wentworth-first-steps-in-algebra-1894/ex-29", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "69", "location": "Exercise 29, problem 17", "problem_latex": "$\\dfrac{8a^{3}b^{3}c^{3} - 27}{2abc - 3}$.", "markdown": 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"problem_latex": "$\\dfrac{1 - 8a^{3}}{1 - 2a}$.", "markdown": "$\\dfrac{1 - 8a^{3}}{1 - 2a}$.", "answer_latex": [ "$1 + 2a + 4a^{2}$." ], "answer_markdown": [ "$1 + 2a + 4a^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1 - 8*a**3)/(1 - 2*a)", "answer_expr": "1 + 2*a + 4*a**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-8*x**3 + 1)/(-2*x + 1)" ], "shape": [ "identity: (N*x**N + 1)/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-29/3", "set": "wentworth-first-steps-in-algebra-1894/ex-29", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "69", "location": "Exercise 29, problem 3", "problem_latex": "$\\dfrac{1 - 27c^{3}}{1 - 2c}$.", "markdown": "$\\dfrac{1 - 27c^{3}}{1 - 2c}$.", "answer_latex": [ "$1 + 3c + 9c^{2}$." ], "answer_markdown": [ "$1 + 3c + 9c^{2}$." ], "checks": [ { "task": "identity", "verdict": "FLAG-MISMATCH", "judge_why": [ "165.72789115646258503", "104.11111111111111111" ], "problem_expr": "(1 - 27*c**3)/(1 - 2*c)", "answer_expr": "1 + 3*c + 9*c**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "identity: (-27*x**3 + 1)/(-2*x + 1)" ], "shape": [ "identity: (N*x**N + 1)/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-29/4", "set": "wentworth-first-steps-in-algebra-1894/ex-29", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "69", "location": "Exercise 29, problem 4", "problem_latex": "$\\dfrac{8a^{3} - b^{3}}{2a - b}$.", "markdown": "$\\dfrac{8a^{3} - b^{3}}{2a - b}$.", "answer_latex": [ "$4a^{2} + 2ab 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"PASS", "judge_why": null, "problem_expr": "(27*x**3 - 8*y**3)/(3*x - 2*y)", "answer_expr": "9*x**2 + 6*x*y + 4*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-8*a**3 + 27*x**3)/(-2*a + 3*x)" ], "shape": [ "identity: (N*a**N + N*x**N)/(N*a + N*x)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-24/22" ], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-29/7", "set": "wentworth-first-steps-in-algebra-1894/ex-29", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "69", "location": "Exercise 29, problem 7", "problem_latex": "$\\dfrac{x^{3}y^{3} - z^{3}}{xy - z}$.", "markdown": "$\\dfrac{x^{3}y^{3} - z^{3}}{xy - z}$.", "answer_latex": [ "$x^{2}y^{2} + xyz + z^{2}$." ], "answer_markdown": [ "$x^{2}y^{2} + xyz + z^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3*y**3 - z**3)/(x*y - z)", "answer_expr": "x**2*y**2 + x*y*z + z**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3*x**3 - b**3)/(a*x - b)" ], "shape": [ "identity: (a**N*x**N - b**N)/(a*x - b)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-29/8", "set": "wentworth-first-steps-in-algebra-1894/ex-29", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "69", "location": "Exercise 29, problem 8", "problem_latex": "$\\dfrac{a^{3}b^{3} - 8}{ab - 2}$.", "markdown": "$\\dfrac{a^{3}b^{3} - 8}{ab - 2}$.", "answer_latex": [ "$a^{2}b^{2} + 2ab + 4$." ], "answer_markdown": [ "$a^{2}b^{2} + 2ab + 4$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3*b**3 - 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"verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**3 + 125*x**3)/(-a + 5*x)" ], "shape": [ "identity: (N*x**N - a**N)/(N*x - a)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/1", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 1", "problem_latex": "$9a$.", "markdown": "$9a$.", "answer_latex": [ "$63$." ], "answer_markdown": [ "$63$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 63.0, printed 63 (half-unit 0.5; correctly rounded at the printed digits: 63.0)", "problem_expr": "9*a", "answer_expr": "63" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 9*a at a=7, b=5, c=3" ], "shape": [ 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"wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 11", "problem_latex": "$\\frac{1}{5}ab^{2}c$.", "markdown": "$\\frac{1}{5}ab^{2}c$.", "answer_latex": [ "$105$." ], "answer_markdown": [ "$105$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 105.0, printed 105 (half-unit 0.5; correctly rounded at the printed digits: 105.0)", "problem_expr": "Rational(1, 5)*a*b**2*c", "answer_expr": "105" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a*b**2*c/5 at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a*b**N*c" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#105,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 ENTER 5 ÷ 7 ×", "5 GOLD x² × 3 ×" ], "constants": [ { "value": "1", "source": "the 1 in the numerator of '1/5' in the problem" }, { "value": "5", "source": "the 5 in the denominator of '1/5' in the problem, and b = 5 in the given values (typed again for b²)" }, { "value": "7", "source": "a = 7 in the given values" }, { "value": "3", "source": "c = 3 in the given values" } ], "calculator_value": "+1050E-1", "printed_value": "105", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/12", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 12", "problem_latex": "$\\frac{1}{7}a^{2}bc$.", "markdown": "$\\frac{1}{7}a^{2}bc$.", "answer_latex": [ "$105$." ], "answer_markdown": [ "$105$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 105.0, printed 105 (half-unit 0.5; correctly rounded at the printed digits: 105.0)", "problem_expr": "Rational(1, 7)*a**2*b*c", "answer_expr": "105" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a**2*b*c/7 at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a**N*b*c" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#105,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 GOLD x²", "5 ×", "3 ×", "7 ÷" ], "constants": [ { "value": "7", "source": "the a = 7 given in the problem, squared with GOLD x²" }, { "value": "5", "source": "the b = 5 given in the problem" }, { "value": "3", "source": "the c = 3 given in the problem" }, { "value": "7", "source": "the denominator of the 1/7 in the expression, applied as a division after multiplying a²bc" } ], "calculator_value": "+105E+0", "printed_value": "105", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/13", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 13", "problem_latex": "$4acy^{2}$.", "markdown": "$4acy^{2}$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)", "problem_expr": "4*a*c*y**2", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 4*a*b*c**2 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#0,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 5 × 0 ×", "3 GOLD x² ×" ], "constants": [ { "value": "4", "source": "the coefficient 4 in 4acy^2 in the problem as printed" }, { "value": "5", "source": "a = 5 in the givens" }, { "value": "0", "source": "c = 0 in the givens" }, { "value": "3", "source": "y = 3 in the givens; the exponent 2 in y^2 is keyed as GOLD x²" } ], "calculator_value": "+0E-6176", "printed_value": "0", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/14", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 14", "problem_latex": "$3ax^{5}y^{2}$.", "markdown": "$3ax^{5}y^{2}$.", "answer_latex": [ "$135$." ], "answer_markdown": [ "$135$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 135.0, printed 135 (half-unit 0.5; correctly rounded at the printed digits: 135.0)", "problem_expr": "3*a*x**5*y**2", "answer_expr": "135" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a*b**5*c**2 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b**N*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#135,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ×", "1 ENTER 5 yˣ ×", "3 ENTER 2 yˣ ×" ], "constants": [ { "value": "3", "source": "the 3 coefficient in '3ax^5y^2'" }, { "value": "5", "source": "a = 5 in the givens" }, { "value": "1", "source": "x = 1 in the givens" }, { "value": "5", "source": "the exponent 5 on x in 'x^{5}'" }, { "value": "2", "source": "the exponent 2 on y in 'y^{2}'" } ], "calculator_value": "+135E+0", "printed_value": "135", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/15", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 15", "problem_latex": "$2ab^{2}y$.", "markdown": "$2ab^{2}y$.", "answer_latex": [ "$120$." ], "answer_markdown": [ "$120$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 120.0, printed 120 (half-unit 0.5; correctly rounded at the printed digits: 120.0)", "problem_expr": "2*a*b**2*y", "answer_expr": "120" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a*b**2*c at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b**N*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#120,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 ×", "2 GOLD x²", "×", "3 ×" ], "constants": [ { "value": "2", "source": "the 2 in '2ab²y' (the coefficient)" }, { "value": "5", "source": "a = 5 in the givens" }, { "value": "2", "source": "b = 2 in the givens, squared as the b² in '2ab²y'" }, { "value": "3", "source": "y = 3 in the givens" } ], "calculator_value": "+120E+0", "printed_value": "120", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/16", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 16", "problem_latex": "$2a^{2}b^{2}c^{2}y^{2}$.", "markdown": "$2a^{2}b^{2}c^{2}y^{2}$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)", "problem_expr": "2*a**2*b**2*c**2*y**2", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a**2*b**2*c**2*d**2 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a**N*b**N*c**N*d**N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-3/17" ], "needs": [ "core.arith" ], "expectation": "X#0,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 GOLD x² ×", "2 GOLD x² ×", "0 GOLD x² ×", "3 GOLD x² ×" ], "constants": [ { "value": "2", "source": "the coefficient 2 in 2a²b²c²y²" }, { "value": "2", "source": "the exponent 2 in a² (keyed as GOLD x²)" }, { "value": "5", "source": "a = 5 in the given values" }, { "value": "2", "source": "b = 2 in the given values, with the exponent 2 in b² (keyed as GOLD x²)" }, { "value": "0", "source": "c = 0 in the given values, with the exponent 2 in c²" }, { "value": "3", "source": "y = 3 in the given values, with the exponent 2 in y²" } ], "calculator_value": "+0E-6176", "printed_value": "0", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/17", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 17", "problem_latex": "$2a^{2}b^{2}x^{2}y^{2}$.", "markdown": "$2a^{2}b^{2}x^{2}y^{2}$.", "answer_latex": [ "$1800$." ], "answer_markdown": [ "$1800$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1800.0, printed 1800 (half-unit 0.5; correctly rounded at the printed digits: 1800.0)", "problem_expr": "2*a**2*b**2*x**2*y**2", "answer_expr": "1800" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a**2*b**2*c**2*d**2 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a**N*b**N*c**N*d**N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-3/16" ], "needs": [ "core.arith" ], "expectation": "X#1800,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 GOLD x² ×", "2 GOLD x² ×", "1 GOLD x² ×", "3 GOLD x² ×" ], "calculator_value": "+1800E+0", "printed_value": "1800", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/18", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 18", "problem_latex": "$2abx^{3}y^{3}$.", "markdown": "$2abx^{3}y^{3}$.", "answer_latex": [ "$540$." ], "answer_markdown": [ "$540$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 540.0, printed 540 (half-unit 0.5; correctly rounded at the printed digits: 540.0)", "problem_expr": "2*a*b*x**3*y**3", "answer_expr": "540" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a*b*c**3*d**3 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b*c**N*d**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#540,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 ×", "2 ×", "1 ENTER 3 yˣ ×", "3 ENTER 3 yˣ ×" ], "constants": [ { "value": "2", "source": "the coefficient 2 in '2abx^{3}y^{3}'" }, { "value": "5", "source": "a = 5 in the problem's givens" }, { "value": "2", "source": "b = 2 in the problem's givens" }, { "value": "1", "source": "x = 1 in the problem's givens (used as the base of x^3)" }, { "value": "3", "source": "the exponent 3 in 'x^{3}' and 'y^{3}'" }, { "value": "3", "source": "y = 3 in the problem's givens (used as the base of y^3)" } ], "calculator_value": "+540E+0", "printed_value": "540", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/19", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 19", "problem_latex": "$3abcxy$.", "markdown": "$3abcxy$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)", "problem_expr": "3*a*b*c*x*y", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a*b*c*d*e at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b*c*d*e" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#0,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ×", "2 ×", "0 ×", "1 ×", "3 ×" ], "calculator_value": "+0E-6176", "printed_value": "0", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/2", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 2", "problem_latex": "$8ab$.", "markdown": "$8ab$.", "answer_latex": [ "$280$." ], "answer_markdown": [ "$280$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 280.0, printed 280 (half-unit 0.5; correctly rounded at the printed digits: 280.0)", "problem_expr": "8*a*b", "answer_expr": "280" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 8*a*b at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a*b" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#280,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "8 ENTER 7 × 5 ×" ], "calculator_value": "+280E+0", "printed_value": "280", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/20", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 20", "problem_latex": "$3abx^{3}y^{2}$.", "markdown": "$3abx^{3}y^{2}$.", "answer_latex": [ "$270$." ], "answer_markdown": [ "$270$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 270.0, printed 270 (half-unit 0.5; correctly rounded at the printed digits: 270.0)", "problem_expr": "3*a*b*x**3*y**2", "answer_expr": "270" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a*b*c**3*d**2 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b*c**N*d**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#270,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ×", "2 ×", "1 ENTER 3 yˣ ×", "3 ENTER 2 yˣ ×" ], "calculator_value": "+270E+0", "printed_value": "270", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/21", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 21", "problem_latex": "$3ab^{2}xy^{2}$.", "markdown": "$3ab^{2}xy^{2}$.", "answer_latex": [ "$540$." ], "answer_markdown": [ "$540$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 540.0, printed 540 (half-unit 0.5; correctly rounded at the printed digits: 540.0)", "problem_expr": "3*a*b**2*x*y**2", "answer_expr": "540" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a*b**2*c*d**2 at a=5, b=2, c=0, x=1, y=3" ], "shape": [ "evaluate: N*a*b**N*c*d**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#540,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ×", "2 GOLD x² ×", "1 ×", "3 GOLD x² ×" ], "calculator_value": "+540E+0", "printed_value": "540", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/3", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 3", "problem_latex": "$4b^{2}c$.", "markdown": "$4b^{2}c$.", "answer_latex": [ "$300$." ], "answer_markdown": [ "$300$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 300.0, printed 300 (half-unit 0.5; correctly rounded at the printed digits: 300.0)", "problem_expr": "4*b**2*c", "answer_expr": "300" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 4*a**2*b at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a**N*b" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#300,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 5 GOLD x² × 3 ×" ], "constants": [ { "value": "4", "source": "the coefficient 4 in '4b²c' (the expression 4*b**2*c)" }, { "value": "5", "source": "given value b = 5 in 'If a = 7, b = 5, c = 3'" }, { "value": "3", "source": "given value c = 3 in 'If a = 7, b = 5, c = 3'" } ], "calculator_value": "+300E+0", "printed_value": "300", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/4", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 4", "problem_latex": "$2a^{2}$.", "markdown": "$2a^{2}$.", "answer_latex": [ "$98$." ], "answer_markdown": [ "$98$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 98.0, printed 98 (half-unit 0.5; correctly rounded at the printed digits: 98.0)", "problem_expr": "2*a**2", "answer_expr": "98" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a**2 at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#98,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 GOLD x² 2 ×" ], "calculator_value": "+98E+0", "printed_value": "98", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/5", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 5", "problem_latex": "$3c^{3}$.", "markdown": "$3c^{3}$.", "answer_latex": [ "$81$." ], "answer_markdown": [ "$81$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 81.0, printed 81 (half-unit 0.5; correctly rounded at the printed digits: 81.0)", "problem_expr": "3*c**3", "answer_expr": "81" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a**3 at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#81,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 3 yˣ", "3 ×" ], "constants": [ { "value": "3", "source": "the given value c = 3 (the base of c³)" }, { "value": "3", "source": "the exponent 3 in 'c^{3}'" }, { "value": "3", "source": "the coefficient 3 in '3c^{3}'" } ], "calculator_value": "+81E+0", "printed_value": "81", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/6", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 6", "problem_latex": "$2b^{4}$.", "markdown": "$2b^{4}$.", "answer_latex": [ "$1250$." ], "answer_markdown": [ "$1250$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1250.0, printed 1250 (half-unit 0.5; correctly rounded at the printed digits: 1250.0)", "problem_expr": "2*b**4", "answer_expr": "1250" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a**4 at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#1250,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 4 yˣ", "2 ×" ], "constants": [ { "value": "5", "source": "the given value b = 5" }, { "value": "4", "source": "the exponent in 2b^{4}" }, { "value": "2", "source": "the coefficient in 2b^{4}" } ], "calculator_value": "+1250E+0", "printed_value": "1250", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/7", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 7", "problem_latex": "$5ac$.", "markdown": "$5ac$.", "answer_latex": [ "$105$." ], "answer_markdown": [ "$105$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 105.0, printed 105 (half-unit 0.5; correctly rounded at the printed digits: 105.0)", "problem_expr": "5*a*c", "answer_expr": "105" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5*a*b at a=7, b=5, c=3" ], "shape": [ "evaluate: N*a*b" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#105,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 7 ×", "3 ×" ], "constants": [ { "value": "5", "source": "the coefficient 5 in 'find the value of 5ac'" }, { "value": "7", "source": "the given a = 7" }, { "value": "3", "source": "the given c = 3" } ], "calculator_value": "+105E+0", "printed_value": "105", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/8", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 8", "problem_latex": "$abc$.", "markdown": "$abc$.", "answer_latex": [ "$105$." ], "answer_markdown": [ "$105$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 105.0, printed 105 (half-unit 0.5; correctly rounded at the printed digits: 105.0)", "problem_expr": "a*b*c", "answer_expr": "105" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a*b*c at a=7, b=5, c=3" ], "shape": [ "evaluate: a*b*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#105,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 ENTER 5 ×", "3 ×" ], "constants": [ { "value": "7", "source": "given value a = 7 in the problem" }, { "value": "5", "source": "given value b = 5 in the problem" }, { "value": "3", "source": "given value c = 3 in the problem" } ], "calculator_value": "+105E+0", "printed_value": "105", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-3/9", "set": "wentworth-first-steps-in-algebra-1894/ex-3", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "12", "location": "Exercise 3, problem 9", "problem_latex": "$abc^{2}$.", "markdown": "$abc^{2}$.", "answer_latex": [ "$315$." ], "answer_markdown": [ "$315$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 315.0, printed 315 (half-unit 0.5; correctly rounded at the printed digits: 315.0)", "problem_expr": "a*b*c**2", "answer_expr": "315" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a*b*c**2 at a=7, b=5, c=3" ], "shape": [ "evaluate: a*b*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#315,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 ENTER 5 ×", "3 GOLD x² ×" ], "constants": [ { "value": "7", "source": "a = 7 in the problem statement" }, { "value": "5", "source": "b = 5 in the problem statement" }, { "value": "3", "source": "c = 3 in the problem statement" } ], "calculator_value": "+315E+0", "printed_value": "315", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/1", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 1", "problem_latex": "$\\dfrac{1 + x^{3}}{1 + x}$.", "markdown": "$\\dfrac{1 + x^{3}}{1 + x}$.", "answer_latex": [ "$1 - x + x^{2}$." ], "answer_markdown": [ "$1 - x + x^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1 + x**3)/(1 + x)", "answer_expr": "1 - x + x**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**3 + 1)/(x + 1)" ], "shape": [ "identity: (x**N + 1)/(x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/10", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 10", "problem_latex": "$\\dfrac{125a^{3} + b^{3}}{5a + b}$.", "markdown": "$\\dfrac{125a^{3} + b^{3}}{5a + b}$.", "answer_latex": [ "$25a^{2} - 5ab + b^{2}$." ], "answer_markdown": [ "$25a^{2} - 5ab + b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(125*a**3 + b**3)/(5*a + b)", "answer_expr": "25*a**2 - 5*a*b + b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3 + 125*x**3)/(a + 5*x)" ], "shape": [ "identity: (N*x**N + a**N)/(N*x + a)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/11", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 11", "problem_latex": "$\\dfrac{a^{3} + 8b^{3}}{a + 2b}$.", "markdown": "$\\dfrac{a^{3} + 8b^{3}}{a + 2b}$.", "answer_latex": [ "$a^{2} - 2ab + 4b^{2}$." ], "answer_markdown": [ "$a^{2} - 2ab + 4b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 + 8*b**3)/(a + 2*b)", "answer_expr": "a**2 - 2*a*b + 4*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (8*a**3 + x**3)/(2*a + x)" ], "shape": [ "identity: (N*a**N + x**N)/(N*a + x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/12", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 12", "problem_latex": "$\\dfrac{a^{6} + 64}{a^{2} + 4}$.", "markdown": "$\\dfrac{a^{6} + 64}{a^{2} + 4}$.", "answer_latex": [ "$a^{4} - 4a^{2} + 16$." ], "answer_markdown": [ "$a^{4} - 4a^{2} + 16$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**6 + 64)/(a**2 + 4)", "answer_expr": "a**4 - 4*a**2 + 16" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**6 + 64)/(x**2 + 4)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/13", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 13", "problem_latex": "$\\dfrac{a^{9} + 27}{a^{3} + 3}$.", "markdown": "$\\dfrac{a^{9} + 27}{a^{3} + 3}$.", "answer_latex": [ "$a^{6} - 3a^{3} + 9$." ], "answer_markdown": [ "$a^{6} - 3a^{3} + 9$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**9 + 27)/(a**3 + 3)", "answer_expr": "a**6 - 3*a**3 + 9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**9 + 27)/(x**3 + 3)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/14", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 14", "problem_latex": "$\\dfrac{8a^{6} + b^{3}}{2a^{2} + b}$.", "markdown": "$\\dfrac{8a^{6} + b^{3}}{2a^{2} + b}$.", "answer_latex": [ "$4a^{4} - 2a^{2}b + b^{2}$." ], "answer_markdown": [ "$4a^{4} - 2a^{2}b + b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(8*a**6 + b**3)/(2*a**2 + b)", "answer_expr": "4*a**4 - 2*a**2*b + b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3 + 8*x**6)/(a + 2*x**2)" ], "shape": [ "identity: (N*x**N + a**N)/(N*x**N + a)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/15", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 15", "problem_latex": "$\\dfrac{a^{12} + x^{6}y^{6}}{a^{4} + x^{2}y^{2}}$.", "markdown": "$\\dfrac{a^{12} + x^{6}y^{6}}{a^{4} + x^{2}y^{2}}$.", "answer_latex": [ "$a^{8} - a^{4}x^{2}y^{2} + x^{4}y^{4}$." ], "answer_markdown": [ "$a^{8} - a^{4}x^{2}y^{2} + x^{4}y^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**12 + x**6*y**6)/(a**4 + x**2*y**2)", "answer_expr": "a**8 - a**4*x**2*y**2 + x**4*y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**6*b**6 + x**12)/(a**2*b**2 + x**4)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/16", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 16", "problem_latex": "$\\dfrac{x^{15} + a^{9}b^{9}}{x^{5} + a^{3}b^{3}}$.", "markdown": "$\\dfrac{x^{15} + a^{9}b^{9}}{x^{5} + a^{3}b^{3}}$.", "answer_latex": [ "$x^{10} - x^{5}a^{3}b^{3} + a^{6}b^{6}$." ], "answer_markdown": [ "$x^{10} - x^{5}a^{3}b^{3} + a^{6}b^{6}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**15 + a**9*b**9)/(x**5 + a**3*b**3)", "answer_expr": "x**10 - x**5*a**3*b**3 + a**6*b**6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**9*b**9 + x**15)/(a**3*b**3 + x**5)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/17", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 17", "problem_latex": "$\\dfrac{27x^{3}y^{3} + z^{12}}{3xy + z^{4}}$.", "markdown": "$\\dfrac{27x^{3}y^{3} + z^{12}}{3xy + z^{4}}$.", "answer_latex": [ "$9x^{2}y^{2} - 3xyz^{4} + z^{8}$." ], "answer_markdown": [ "$9x^{2}y^{2} - 3xyz^{4} + z^{8}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(27*x**3*y**3 + z**12)/(3*x*y + z**4)", "answer_expr": "9*x**2*y**2 - 3*x*y*z**4 + z**8" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (27*a**3*x**3 + b**12)/(3*a*x + b**4)" ], "shape": [ "identity: (N*a**N*x**N + b**N)/(N*a*x + b**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/18", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 18", "problem_latex": "$\\dfrac{x^{3}y^{3}z^{3} + 1}{xyz + 1}$.", "markdown": "$\\dfrac{x^{3}y^{3}z^{3} + 1}{xyz + 1}$.", "answer_latex": [ "$x^{2}y^{2}z^{2} - xyz + 1$." ], "answer_markdown": [ "$x^{2}y^{2}z^{2} - xyz + 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3*y**3*z**3 + 1)/(x*y*z + 1)", "answer_expr": "x**2*y**2*z**2 - x*y*z + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3*b**3*x**3 + 1)/(a*b*x + 1)" ], "shape": [ "identity: (a**N*b**N*x**N + 1)/(a*b*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/19", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 19", "problem_latex": "$\\dfrac{8a^{3}b^{3}c^{3} + 27}{2abc + 3}$.", "markdown": "$\\dfrac{8a^{3}b^{3}c^{3} + 27}{2abc + 3}$.", "answer_latex": [ "$4a^{2}b^{2}c^{2} - 6abc + 9$." ], "answer_markdown": [ "$4a^{2}b^{2}c^{2} - 6abc + 9$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(8*a**3*b**3*c**3 + 27)/(2*a*b*c + 3)", "answer_expr": "4*a**2*b**2*c**2 - 6*a*b*c + 9" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (8*a**3*b**3*x**3 + 27)/(2*a*b*x + 3)" ], "shape": [ "identity: (N*a**N*b**N*x**N + N)/(N*a*b*x + N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/2", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 2", "problem_latex": "$\\dfrac{1 + 8a^{3}}{1 + 2a}$.", "markdown": "$\\dfrac{1 + 8a^{3}}{1 + 2a}$.", "answer_latex": [ "$1 - 2a + 4a^{2}$." ], "answer_markdown": [ "$1 - 2a + 4a^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1 + 8*a**3)/(1 + 2*a)", "answer_expr": "1 - 2*a + 4*a**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (8*x**3 + 1)/(2*x + 1)" ], "shape": [ "identity: (N*x**N + 1)/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/20", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 20", "problem_latex": "$\\dfrac{1 + 64x^{3}y^{3}z^{3}}{1 + 4xyz}$.", "markdown": "$\\dfrac{1 + 64x^{3}y^{3}z^{3}}{1 + 4xyz}$.", "answer_latex": [ "$1 - 4xyz + 16x^{2}y^{2}z^{2}$." ], "answer_markdown": [ "$1 - 4xyz + 16x^{2}y^{2}z^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1 + 64*x**3*y**3*z**3)/(1 + 4*x*y*z)", "answer_expr": "1 - 4*x*y*z + 16*x**2*y**2*z**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (64*a**3*b**3*x**3 + 1)/(4*a*b*x + 1)" ], "shape": [ "identity: (N*a**N*b**N*x**N + 1)/(N*a*b*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/21", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 21", "problem_latex": "$\\dfrac{1 + 27a^{6}b^{3}c^{3}}{1 + 3a^{2}bc}$.", "markdown": "$\\dfrac{1 + 27a^{6}b^{3}c^{3}}{1 + 3a^{2}bc}$.", "answer_latex": [ "$1 - 3a^{2}bc + 9a^{4}b^{2}c^{2}$." ], "answer_markdown": [ "$1 - 3a^{2}bc + 9a^{4}b^{2}c^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1 + 27*a**6*b**3*c**3)/(1 + 3*a**2*b*c)", "answer_expr": "1 - 3*a**2*b*c + 9*a**4*b**2*c**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (27*a**3*b**3*x**6 + 1)/(3*a*b*x**2 + 1)" ], "shape": [ "identity: (N*a**N*b**N*x**N + 1)/(N*a*b*x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/22", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 22", "problem_latex": "$\\dfrac{x^{4} - y^{4}}{x - y}$.", "markdown": "$\\dfrac{x^{4} - y^{4}}{x - y}$.", "answer_latex": [ "$x^{3} + x^{2}y + xy^{2} + y^{3}$." ], "answer_markdown": [ "$x^{3} + x^{2}y + xy^{2} + y^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 - y**4)/(x - y)", "answer_expr": "x**3 + x**2*y + x*y**2 + y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**4 + x**4)/(-a + x)" ], "shape": [ "identity: (-a**N + x**N)/(-a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/23", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 23", "problem_latex": "$\\dfrac{x^{4} - y^{4}}{x + y}$.", "markdown": "$\\dfrac{x^{4} - y^{4}}{x + y}$.", "answer_latex": [ "$x^{3} - x^{2}y + xy^{2} - y^{3}$." ], "answer_markdown": [ "$x^{3} - x^{2}y + xy^{2} - y^{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 - y**4)/(x + y)", "answer_expr": "x**3 - x**2*y + x*y**2 - y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**4 + x**4)/(a + x)" ], "shape": [ "identity: (-a**N + x**N)/(a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/24", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 24", "problem_latex": "$\\dfrac{x^{5} - y^{5}}{x - y}$.", "markdown": "$\\dfrac{x^{5} - y^{5}}{x - y}$.", "answer_latex": [ "$x^{4} + x^{3}y + x^{2}y^{2} + xy^{3} + y^{4}$." ], "answer_markdown": [ "$x^{4} + x^{3}y + x^{2}y^{2} + xy^{3} + y^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**5 - y**5)/(x - y)", "answer_expr": "x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**5 + x**5)/(-a + x)" ], "shape": [ "identity: (-a**N + x**N)/(-a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/25", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 25", "problem_latex": "$\\dfrac{x^{5} + y^{5}}{x + y}$.", "markdown": "$\\dfrac{x^{5} + y^{5}}{x + y}$.", "answer_latex": [ "$x^{4} - x^{3}y + x^{2}y^{2} - xy^{3} + y^{4}$." ], "answer_markdown": [ "$x^{4} - x^{3}y + x^{2}y^{2} - xy^{3} + y^{4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**5 + y**5)/(x + y)", "answer_expr": "x**4 - x**3*y + x**2*y**2 - x*y**3 + y**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**5 + x**5)/(a + x)" ], "shape": [ "identity: (a**N + x**N)/(a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/26", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 26", "problem_latex": "$\\dfrac{x^{6} - y^{6}}{x - y}$.", "markdown": "$\\dfrac{x^{6} - y^{6}}{x - y}$.", "answer_latex": [ "$x^{5} + x^{4}y + x^{3}y^{2} + x^{2}y^{3} + xy^{4} + y^{5}$." ], "answer_markdown": [ "$x^{5} + x^{4}y + x^{3}y^{2} + x^{2}y^{3} + xy^{4} + y^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**6 - y**6)/(x - y)", "answer_expr": "x**5 + x**4*y + x**3*y**2 + x**2*y**3 + x*y**4 + y**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**6 + x**6)/(-a + x)" ], "shape": [ "identity: (-a**N + x**N)/(-a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/27", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 27", "problem_latex": "$\\dfrac{x^{6} - y^{6}}{x + y}$.", "markdown": "$\\dfrac{x^{6} - y^{6}}{x + y}$.", "answer_latex": [ "$x^{5} - x^{4}y + x^{3}y^{2} - x^{2}y^{3} + xy^{4} - y^{5}$." ], "answer_markdown": [ "$x^{5} - x^{4}y + x^{3}y^{2} - x^{2}y^{3} + xy^{4} - y^{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**6 - y**6)/(x + y)", "answer_expr": "x**5 - x**4*y + x**3*y**2 - x**2*y**3 + x*y**4 - y**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**6 + x**6)/(a + x)" ], "shape": [ "identity: (-a**N + x**N)/(a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/3", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 3", "problem_latex": "$\\dfrac{1 + 27c^{3}}{1 + 3c}$.", "markdown": "$\\dfrac{1 + 27c^{3}}{1 + 3c}$.", "answer_latex": [ "$1 - 3c + 9c^{2}$." ], "answer_markdown": [ "$1 - 3c + 9c^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1 + 27*c**3)/(1 + 3*c)", "answer_expr": "1 - 3*c + 9*c**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (27*x**3 + 1)/(3*x + 1)" ], "shape": [ "identity: (N*x**N + 1)/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/4", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 4", "problem_latex": "$\\dfrac{8a^{3} + b^{3}}{2a + b}$.", "markdown": "$\\dfrac{8a^{3} + b^{3}}{2a + b}$.", "answer_latex": [ "$4a^{2} - 2ab + b^{2}$." ], "answer_markdown": [ "$4a^{2} - 2ab + b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(8*a**3 + b**3)/(2*a + b)", "answer_expr": "4*a**2 - 2*a*b + b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3 + 8*x**3)/(a + 2*x)" ], "shape": [ "identity: (N*x**N + a**N)/(N*x + a)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/5", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 5", "problem_latex": "$\\dfrac{64b^{3} + 27c^{3}}{4b + 3c}$.", "markdown": "$\\dfrac{64b^{3} + 27c^{3}}{4b + 3c}$.", "answer_latex": [ "$16b^{2} - 12bc + 9c^{2}$." ], "answer_markdown": [ "$16b^{2} - 12bc + 9c^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(64*b**3 + 27*c**3)/(4*b + 3*c)", "answer_expr": "16*b**2 - 12*b*c + 9*c**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (27*a**3 + 64*x**3)/(3*a + 4*x)" ], "shape": [ "identity: (N*a**N + N*x**N)/(N*a + N*x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/6", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 6", "problem_latex": "$\\dfrac{27x^{3} + 8y^{3}}{3x + 2y}$.", "markdown": "$\\dfrac{27x^{3} + 8y^{3}}{3x + 2y}$.", "answer_latex": [ "$9x^{2} - 6xy + 4y^{2}$." ], "answer_markdown": [ "$9x^{2} - 6xy + 4y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(27*x**3 + 8*y**3)/(3*x + 2*y)", "answer_expr": "9*x**2 - 6*x*y + 4*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (8*a**3 + 27*x**3)/(2*a + 3*x)" ], "shape": [ "identity: (N*a**N + N*x**N)/(N*a + N*x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/7", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 7", "problem_latex": "$\\dfrac{8x^{3} + 125y^{3}}{2x + 5y}$.", "markdown": "$\\dfrac{8x^{3} + 125y^{3}}{2x + 5y}$.", "answer_latex": [ "$4x^{2} - 10xy + 25y^{2}$." ], "answer_markdown": [ "$4x^{2} - 10xy + 25y^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(8*x**3 + 125*y**3)/(2*x + 5*y)", "answer_expr": "4*x**2 - 10*x*y + 25*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (125*a**3 + 8*x**3)/(5*a + 2*x)" ], "shape": [ "identity: (N*a**N + N*x**N)/(N*a + N*x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/8", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 8", "problem_latex": "$\\dfrac{x^{3}y^{3} + z^{3}}{xy + z}$.", "markdown": "$\\dfrac{x^{3}y^{3} + z^{3}}{xy + z}$.", "answer_latex": [ "$x^{2}y^{2} - xyz + z^{2}$." ], "answer_markdown": [ "$x^{2}y^{2} - xyz + z^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**3*y**3 + z**3)/(x*y + z)", "answer_expr": "x**2*y**2 - x*y*z + z**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3*x**3 + b**3)/(a*x + b)" ], "shape": [ "identity: (a**N*x**N + b**N)/(a*x + b)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-30/9", "set": "wentworth-first-steps-in-algebra-1894/ex-30", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "70", "location": "Exercise 30, problem 9", "problem_latex": "$\\dfrac{a^{3}b^{3} + 8}{ab + 2}$.", "markdown": "$\\dfrac{a^{3}b^{3} + 8}{ab + 2}$.", "answer_latex": [ "$a^{2}b^{2} - 2ab + 4$." ], "answer_markdown": [ "$a^{2}b^{2} - 2ab + 4$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3*b**3 + 8)/(a*b + 2)", "answer_expr": "a**2*b**2 - 2*a*b + 4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**3*x**3 + 8)/(a*x + 2)" ], "shape": [ "identity: (N + a**N*x**N)/(N + a*x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/1", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 1", "problem_latex": "$2x^{2} - 4x$.", "markdown": "$2x^{2} - 4x$.", "answer_latex": [ "$2x(x - 2)$." ], "answer_markdown": [ "$2x(x - 2)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "2*x**2 - 4*x", "answer_expr": "2*x*(x - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 2*x**2 - 4*x" ], "shape": [ "factor: N*x + N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/10", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 10", "problem_latex": "$3x^{3}y^{3} - 6x^{4}y^{4} - 9x^{2}y^{2}$.", "markdown": "$3x^{3}y^{3} - 6x^{4}y^{4} - 9x^{2}y^{2}$.", "answer_latex": [ "$3x^{2}y^{2}(xy - 2x^{2}y^{2} - 3)$." ], "answer_markdown": [ "$3x^{2}y^{2}(xy - 2x^{2}y^{2} - 3)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "3*x**3*y**3 - 6*x**4*y**4 - 9*x**2*y**2", "answer_expr": "3*x**2*y**2*(x*y - 2*x**2*y**2 - 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -6*a**4*x**4 + 3*a**3*x**3 - 9*a**2*x**2" ], "shape": [ "factor: 3*N*a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/2", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 2", "problem_latex": "$3a^{3} - 6a$.", "markdown": "$3a^{3} - 6a$.", "answer_latex": [ "$3a(a^{2} - 2)$." ], "answer_markdown": [ "$3a(a^{2} - 2)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "3*a**3 - 6*a", "answer_expr": "3*a*(a**2 - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 3*x**3 - 6*x" ], "shape": [ "factor: N*x + N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/3", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 3", "problem_latex": "$5a^{2}b^{2} - 10a^{3}b^{3}$.", "markdown": "$5a^{2}b^{2} - 10a^{3}b^{3}$.", "answer_latex": [ "$5a^{2}b^{2}(1 - 2ab)$." ], "answer_markdown": [ "$5a^{2}b^{2}(1 - 2ab)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "5*a**2*b**2 - 10*a**3*b**3", "answer_expr": "5*a**2*b**2*(1 - 2*a*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -10*a**3*x**3 + 5*a**2*x**2" ], "shape": [ "factor: 2*N*a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/4", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 4", "problem_latex": "$3x^{2}y + 4xy^{2}$.", "markdown": "$3x^{2}y + 4xy^{2}$.", "answer_latex": [ "$xy(3x + 4y)$." ], "answer_markdown": [ "$xy(3x + 4y)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "3*x**2*y + 4*x*y**2", "answer_expr": "x*y*(3*x + 4*y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 4*a**2*x + 3*a*x**2" ], "shape": [ "factor: N*a*x**N + N*a**N*x" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/5", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 5", "problem_latex": "$8a^{3}b^{2} + 4a^{2}b^{3}$.", "markdown": "$8a^{3}b^{2} + 4a^{2}b^{3}$.", "answer_latex": [ "$4a^{2}b^{2}(2a + b)$." ], "answer_markdown": [ "$4a^{2}b^{2}(2a + b)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "8*a**3*b**2 + 4*a**2*b**3", "answer_expr": "4*a**2*b**2*(2*a + b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 4*a**3*x**2 + 8*a**2*x**3" ], "shape": [ "factor: 2*N*a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/6", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 6", "problem_latex": "$3a^{4} - 12a^{2} - 6a^{3}$.", "markdown": "$3a^{4} - 12a^{2} - 6a^{3}$.", "answer_latex": [ "$3a^{2}(a^{2} - 4 - 2a)$." ], "answer_markdown": [ "$3a^{2}(a^{2} - 4 - 2a)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "3*a**4 - 12*a**2 - 6*a**3", "answer_expr": "3*a**2*(a**2 - 4 - 2*a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 3*x**4 - 6*x**3 - 12*x**2" ], "shape": [ "factor: 3*N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/7", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 7", "problem_latex": "$4x^{2} - 8x^{4} - 12x^{5}$.", "markdown": "$4x^{2} - 8x^{4} - 12x^{5}$.", "answer_latex": [ "$4x^{2}(1 - 2x^{2} - 3x^{3})$." ], "answer_markdown": [ "$4x^{2}(1 - 2x^{2} - 3x^{3})$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "4*x**2 - 8*x**4 - 12*x**5", "answer_expr": "4*x**2*(1 - 2*x**2 - 3*x**3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -12*x**5 - 8*x**4 + 4*x**2" ], "shape": [ "factor: 3*N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-31/8", "set": "wentworth-first-steps-in-algebra-1894/ex-31", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "72", "location": "Exercise 31, problem 8", "problem_latex": "$5 - 10x^{2}y^{2} + 15x^{2}y$.", "markdown": "$5 - 10x^{2}y^{2} 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"page": "75", "location": "Exercise 34, problem 13", "problem_latex": "$(a - 5b)^{2} - 9c^{2}$.", "markdown": "$(a - 5b)^{2} - 9c^{2}$.", "answer_latex": [ "$(a - 5b + 3c)(a - 5b - 3c)$." ], "answer_markdown": [ "$(a - 5b + 3c)(a - 5b - 3c)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "(a - 5*b)**2 - 9*c**2", "answer_expr": "(a - 5*b + 3*c)*(a - 5*b - 3*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -9*c**2 + (a - 5*b)**2" ], "shape": [ "factor: N*c**N + (N*b + a)**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-34/14", "set": "wentworth-first-steps-in-algebra-1894/ex-34", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "75", "location": "Exercise 34, problem 14", "problem_latex": "$16c^{2} - (a - 5b)^{2}$.", "markdown": "$16c^{2} - (a - 5b)^{2}$.", "answer_latex": [ "$(4c + a - 5b)(4c - a + 5b)$." ], "answer_markdown": [ "$(4c + a - 5b)(4c - a + 5b)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "16*c**2 - (a - 5*b)**2", "answer_expr": "(4*c + a - 5*b)*(4*c - a + 5*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 16*c**2 - (a - 5*b)**2" ], "shape": [ "factor: N*c**N - (N*b + a)**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-34/15", "set": "wentworth-first-steps-in-algebra-1894/ex-34", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "75", "location": "Exercise 34, problem 15", "problem_latex": "$4a^{2} - (x + y)^{2}$.", "markdown": "$4a^{2} - (x + y)^{2}$.", "answer_latex": [ "$(2a + x + y)(2a - x - y)$." ], 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"problem_expr": "b**2 - (a - 2*x)**2", "answer_expr": "(b + a - 2*x)*(b - a + 2*x)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: b**2 - (a - 2*c)**2" ], "shape": [ "factor: b**N - (N*c + a)**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-34/17", "set": "wentworth-first-steps-in-algebra-1894/ex-34", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "75", "location": "Exercise 34, problem 17", "problem_latex": "$4z^{2} - (x + 3y)^{2}$.", "markdown": "$4z^{2} - (x + 3y)^{2}$.", "answer_latex": [ "$(2z + x + 3y)(2z - x - 3y)$." ], "answer_markdown": [ "$(2z + x + 3y)(2z - x - 3y)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "4*z**2 - (x + 3*y)**2", "answer_expr": "(2*z + x + 3*y)*(2*z - x - 3*y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 4*c**2 - (a + 3*b)**2" ], "shape": [ "factor: N*c**N - (N*b + a)**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-34/18", "set": "wentworth-first-steps-in-algebra-1894/ex-34", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "75", "location": "Exercise 34, problem 18", "problem_latex": "$9 - (3a - 7b)^{2}$.", "markdown": "$9 - (3a - 7b)^{2}$.", "answer_latex": [ "$(3 + 3a - 7b)(3 - 3a + 7b)$." ], "answer_markdown": [ "$(3 + 3a - 7b)(3 - 3a + 7b)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "9 - (3*a - 7*b)**2", "answer_expr": "(3 + 3*a - 7*b)*(3 - 3*a + 7*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -(3*a - 7*b)**2 + 9" ], "shape": [ 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"Exercise 34, problem 23", "problem_latex": "$49m^{2} - (p + 2q)^{2}$.", "markdown": "$49m^{2} - (p + 2q)^{2}$.", "answer_latex": [ "$(7m + p + 2q)(7m - p - 2q)$." ], "answer_markdown": [ "$(7m + p + 2q)(7m - p - 2q)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "49*m**2 - (p + 2*q)**2", "answer_expr": "(7*m + p + 2*q)*(7*m - p - 2*q)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 49*a**2 - (b + 2*c)**2" ], "shape": [ "factor: N*a**N - (N*c + b)**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-34/24", "set": "wentworth-first-steps-in-algebra-1894/ex-34", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "75", "location": "Exercise 34, problem 24", "problem_latex": "$36n^{2} - (d - 2c)^{2}$.", "markdown": "$36n^{2} - (d - 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"commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 4*a**2 - 36*a*x + 81*x**2" ], "shape": [ "factor: N*a*x + N*a**N + N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/18", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 18", "problem_latex": " $m^{2}n^{2} + 14mnx^{2} + 49x^{2}$.", "markdown": "$m^{2}n^{2} + 14mnx^{2} + 49x^{2}$.", "answer_latex": [ " $(mn + 7x^{2})(mn + 7x^{2})$." ], "answer_markdown": [ "$(mn + 7x^{2})(mn + 7x^{2})$." ], "checks": [ { "task": "factor", "verdict": "FLAG-MISMATCH", "judge_why": [ "914.51033733446391851", "2994.2419277733202691" ], "problem_expr": "m**2*n**2 + 14*m*n*x**2 + 49*x**2", "answer_expr": "(m*n + 7*x**2)*(m*n + 7*x**2)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "factor: a**2*b**2 + 14*a*b*x**2 + 49*x**2" ], "shape": [ "factor: N*a*b*x**N + N*x**N + a**N*b**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/2", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 2", "problem_latex": " $x^{2} + 6xy + 9y^{2}$.", "markdown": "$x^{2} + 6xy + 9y^{2}$.", "answer_latex": [ " $(x + 3y)(x + 3y)$." ], "answer_markdown": [ "$(x + 3y)(x + 3y)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 + 6*x*y + 9*y**2", "answer_expr": "(x + 3*y)*(x + 3*y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 9*a**2 + 6*a*x + x**2" ], "shape": [ "factor: N*a*x + N*a**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/3", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 3", "problem_latex": " $x^{2} + 16x + 64$.", "markdown": "$x^{2} + 16x + 64$.", "answer_latex": [ " $(x + 8)(x + 8)$." ], "answer_markdown": [ "$(x + 8)(x + 8)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 + 16*x + 64", "answer_expr": "(x + 8)*(x + 8)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 + 16*x + 64" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/4", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 4", "problem_latex": " $x^{2} + 10ax + 25a^{2}$.", "markdown": "$x^{2} + 10ax + 25a^{2}$.", "answer_latex": [ " $(x + 5a)(x + 5a)$." ], "answer_markdown": [ "$(x + 5a)(x + 5a)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 + 10*a*x + 25*a**2", "answer_expr": "(x + 5*a)*(x + 5*a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 25*a**2 + 10*a*x + x**2" ], "shape": [ "factor: N*a*x + N*a**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/5", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 5", "problem_latex": " $a^{2} - 16a + 64$.", "markdown": "$a^{2} - 16a + 64$.", "answer_latex": [ " $(a - 8)(a - 8)$." ], "answer_markdown": [ "$(a - 8)(a - 8)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 16*a + 64", "answer_expr": "(a - 8)*(a - 8)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - 16*x + 64" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/6", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 6", "problem_latex": " $a^{2} - 10ab + 25b^{2}$.", "markdown": "$a^{2} - 10ab + 25b^{2}$.", "answer_latex": [ " $(a - 5b)(a - 5b)$." ], "answer_markdown": [ "$(a - 5b)(a - 5b)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 10*a*b + 25*b**2", "answer_expr": "(a - 5*b)*(a - 5*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 25*a**2 - 10*a*x + x**2" ], "shape": [ "factor: N*a*x + N*a**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/7", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 7", "problem_latex": " $c^{2} - 6cd + 9d^{2}$.", "markdown": "$c^{2} - 6cd + 9d^{2}$.", "answer_latex": [ " $(c - 3d)(c - 3d)$." ], "answer_markdown": [ "$(c - 3d)(c - 3d)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "c**2 - 6*c*d + 9*d**2", "answer_expr": "(c - 3*d)*(c - 3*d)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 9*a**2 - 6*a*x + x**2" ], "shape": [ "factor: N*a*x + N*a**N + x**N" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-36/5", "wentworth-first-steps-in-algebra-1894/ex-39/8" ], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-37/8", "set": "wentworth-first-steps-in-algebra-1894/ex-37", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "80", "location": "Exercise 37, problem 8", "problem_latex": " $4x^{2} - 4x + 1$.", "markdown": "$4x^{2} - 4x + 1$.", "answer_latex": [ " $(2x - 1)(2x - 1)$." ], "answer_markdown": [ "$(2x - 1)(2x - 1)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "4*x**2 - 4*x + 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"markdown": "$x^{2} - 9x + 14$.", "answer_latex": [ "$(x - 2)(x - 7)$." ], "answer_markdown": [ "$(x - 2)(x - 7)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 - 9*x + 14", "answer_expr": "(x - 2)*(x - 7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - 9*x + 14" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/13", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 13", "problem_latex": "$x^{2} - 5x - 14$.", "markdown": "$x^{2} - 5x - 14$.", "answer_latex": [ "$(x + 2)(x - 7)$." ], "answer_markdown": [ "$(x + 2)(x - 7)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 - 5*x - 14", "answer_expr": "(x + 2)*(x - 7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - 5*x - 14" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/14", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 14", "problem_latex": "$x^{2} - 9x + 20$.", "markdown": "$x^{2} - 9x + 20$.", "answer_latex": [ "$(x - 4)(x - 5)$." ], "answer_markdown": [ "$(x - 4)(x - 5)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 - 9*x + 20", "answer_expr": "(x - 4)*(x - 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, 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"markdown": "$x^{2} + 13x + 30$.", "answer_latex": [ "$(x + 3)(x + 10)$." ], "answer_markdown": [ "$(x + 3)(x + 10)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 + 13*x + 30", "answer_expr": "(x + 3)*(x + 10)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 + 13*x + 30" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/24", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 24", "problem_latex": "$x^{2} + 7x - 30$.", "markdown": "$x^{2} + 7x - 30$.", "answer_latex": [ "$(x - 3)(x + 10)$." ], "answer_markdown": [ "$(x - 3)(x + 10)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 + 7*x - 30", "answer_expr": "(x - 3)*(x + 10)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 + 7*x - 30" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/25", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 25", "problem_latex": "$x^{2} - 7x - 30$.", "markdown": "$x^{2} - 7x - 30$.", "answer_latex": [ "$(x + 3)(x - 10)$." ], "answer_markdown": [ "$(x + 3)(x - 10)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 - 7*x - 30", "answer_expr": "(x + 3)*(x - 10)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, 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null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 28", "problem_latex": "$a^{2} + 3ab - 4b^{2}$.", "markdown": "$a^{2} + 3ab - 4b^{2}$.", "answer_latex": [ "$(a - b)(a + 4b)$." ], "answer_markdown": [ "$(a - b)(a + 4b)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 + 3*a*b - 4*b**2", "answer_expr": "(a - b)*(a + 4*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -4*a**2 + 3*a*x + x**2" ], "shape": [ "factor: N*a*x + N*a**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/29", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 29, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 29", 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6y)(ax - 17y)$." ], "answer_markdown": [ "$(ax - 6y)(ax - 17y)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2*x**2 - 23*a*x*y + 102*y**2", "answer_expr": "(a*x - 6*y)*(a*x - 17*y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: a**2*x**2 - 23*a*b*x + 102*b**2" ], "shape": [ "factor: N*a*b*x + N*b**N + a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/38", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 38, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 38", "problem_latex": "$x^{4} - 9x^{2}y^{2} + 20y^{4}$.", "markdown": "$x^{4} - 9x^{2}y^{2} + 20y^{4}$.", "answer_latex": [ "$(x + 2y)(x - 2y)(x^{2} - 5y^{2})$." ], "answer_markdown": [ "$(x + 2y)(x - 2y)(x^{2} - 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13y^{2})$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**4*x**4 - 24*a**2*x**2*y**2 + 143*y**4", "answer_expr": "(a**2*x**2 - 11*y**2)*(a**2*x**2 - 13*y**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: a**4*x**4 - 24*a**2*b**2*x**2 + 143*b**4" ], "shape": [ "factor: N*a**N*b**N*x**N + N*b**N + a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/4", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 4", "problem_latex": "$a^{2} - 6a + 5$.", "markdown": "$a^{2} - 6a + 5$.", "answer_latex": [ "$(a - 1)(a - 5)$." ], "answer_markdown": [ "$(a - 1)(a - 5)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 6*a + 5", "answer_expr": "(a - 1)*(a - 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - 6*x + 5" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/40", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 40, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 40", "problem_latex": "$a^{6}b^{6} - 23a^{3}b^{3}c^{2} + 132c^{4}$.", "markdown": "$a^{6}b^{6} - 23a^{3}b^{3}c^{2} + 132c^{4}$.", "answer_latex": [ "$(a^{3}b^{3} - 11c^{2})(a^{3}b^{3} - 12c^{2})$." ], "answer_markdown": [ "$(a^{3}b^{3} - 11c^{2})(a^{3}b^{3} - 12c^{2})$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**6*b**6 - 23*a**3*b**3*c**2 + 132*c**4", "answer_expr": "(a**3*b**3 - 11*c**2)*(a**3*b**3 - 12*c**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: a**6*x**6 - 23*a**3*b**2*x**3 + 132*b**4" ], "shape": [ "factor: N*a**N*b**N*x**N + N*b**N + a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/41", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 41, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 41", "problem_latex": "$a^{2} - 20abc - 96b^{2}c^{2}$.", "markdown": "$a^{2} - 20abc - 96b^{2}c^{2}$.", "answer_latex": [ "$(a + 4bc)(a - 24bc)$." ], "answer_markdown": [ "$(a + 4bc)(a - 24bc)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 20*a*b*c - 96*b**2*c**2", "answer_expr": "(a + 4*b*c)*(a - 24*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -96*a**2*b**2 - 20*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/42", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 42, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 42", "problem_latex": "$a^{2} - 4abc - 96b^{2}c^{2}$.", "markdown": "$a^{2} - 4abc - 96b^{2}c^{2}$.", "answer_latex": [ "$(a + 8bc)(a - 12bc)$." ], "answer_markdown": [ "$(a + 8bc)(a - 12bc)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 4*a*b*c - 96*b**2*c**2", "answer_expr": "(a + 8*b*c)*(a - 12*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -96*a**2*b**2 - 4*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/43", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 43, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 43", "problem_latex": "$a^{2} - 10abc - 96b^{2}c^{2}$.", "markdown": "$a^{2} - 10abc - 96b^{2}c^{2}$.", "answer_latex": [ "$(a + 6bc)(a - 16bc)$." ], "answer_markdown": [ "$(a + 6bc)(a - 16bc)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 10*a*b*c - 96*b**2*c**2", "answer_expr": "(a + 6*b*c)*(a - 16*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -96*a**2*b**2 - 10*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/44", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 44, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 44", "problem_latex": "$a^{2} + 29abc - 96b^{2}c^{2}$.", "markdown": "$a^{2} + 29abc - 96b^{2}c^{2}$.", "answer_latex": [ "$(a - 3bc)(a + 32bc)$." ], "answer_markdown": [ "$(a - 3bc)(a + 32bc)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 + 29*a*b*c - 96*b**2*c**2", "answer_expr": "(a - 3*b*c)*(a + 32*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -96*a**2*b**2 + 29*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/45", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 45, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 45", "problem_latex": "$a^{2} - 46abc - 96b^{2}c^{2}$.", "markdown": "$a^{2} - 46abc - 96b^{2}c^{2}$.", "answer_latex": [ "$(a + 2bc)(a - 48bc)$." ], "answer_markdown": [ "$(a + 2bc)(a - 48bc)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 46*a*b*c - 96*b**2*c**2", "answer_expr": "(a + 2*b*c)*(a - 48*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -96*a**2*b**2 - 46*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/46", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 46, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 46", "problem_latex": "$a^{2} + 49abc + 48b^{2}c^{2}$.", "markdown": "$a^{2} + 49abc + 48b^{2}c^{2}$.", "answer_latex": [ "$(a + bc)(a + 48bc)$." ], "answer_markdown": [ "$(a + bc)(a + 48bc)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 + 49*a*b*c + 48*b**2*c**2", "answer_expr": "(a + b*c)*(a + 48*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: 48*a**2*b**2 + 49*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/47", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 47, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 47", "problem_latex": "$x^{2} - 18xyz - 243y^{2}z^{2}$.", "markdown": "$x^{2} - 18xyz - 243y^{2}z^{2}$.", "answer_latex": [ "$(x + 9yz)(x - 27yz)$." ], "answer_markdown": [ "$(x + 9yz)(x - 27yz)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 - 18*x*y*z - 243*y**2*z**2", "answer_expr": "(x + 9*y*z)*(x - 27*y*z)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: -243*a**2*b**2 - 18*a*b*x + x**2" ], "shape": [ "factor: N*a*b*x + N*a**N*b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/48", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 48, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 48", "problem_latex": "$x^{2}y^{2} - xyz - 182z^{2}$.", "markdown": "$x^{2}y^{2} - xyz - 182z^{2}$.", "answer_latex": [ "$(xy + 13z)(xy - 14z)$." ], "answer_markdown": [ "$(xy + 13z)(xy - 14z)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2*y**2 - x*y*z - 182*z**2", "answer_expr": "(x*y + 13*z)*(x*y - 14*z)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: a**2*x**2 - a*b*x - 182*b**2" ], "shape": [ "factor: N*b**N - a*b*x + a**N*x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/5", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 5", "problem_latex": "$a^{2} + 4a - 5$.", "markdown": "$a^{2} + 4a - 5$.", "answer_latex": [ "$(a - 1)(a + 5)$." ], "answer_markdown": [ "$(a - 1)(a + 5)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 + 4*a - 5", "answer_expr": "(a - 1)*(a + 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 + 4*x - 5" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-37/7" ], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/6", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 6", "problem_latex": "$a^{2} - 4a - 5$.", "markdown": "$a^{2} - 4a - 5$.", "answer_latex": [ "$(a + 1)(a - 5)$." ], "answer_markdown": [ "$(a + 1)(a - 5)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 - 4*a - 5", "answer_expr": "(a + 1)*(a - 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - 4*x - 5" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/7", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 7", "problem_latex": "$c^{2} - 9c + 18$.", "markdown": "$c^{2} - 9c + 18$.", "answer_latex": [ "$(c - 3)(c - 6)$." ], "answer_markdown": [ "$(c - 3)(c - 6)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "c**2 - 9*c + 18", "answer_expr": "(c - 3)*(c - 6)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - 9*x + 18" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/8", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 8", "problem_latex": "$c^{2} + 9c + 18$.", "markdown": "$c^{2} + 9c + 18$.", "answer_latex": [ "$(c + 3)(c + 6)$." ], "answer_markdown": [ "$(c + 3)(c + 6)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "c**2 + 9*c + 18", "answer_expr": "(c + 3)*(c + 6)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 + 9*x + 18" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-37/2" ], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-38/9", "set": "wentworth-first-steps-in-algebra-1894/ex-38", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "82", "location": "Exercise 38, problem 9", "problem_latex": "$c^{2} + 3c - 18$.", "markdown": "$c^{2} + 3c - 18$.", "answer_latex": [ "$(c - 3)(c + 6)$." ], "answer_markdown": [ "$(c - 3)(c + 6)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "c**2 + 3*c - 18", "answer_expr": "(c - 3)*(c + 6)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 + 3*x - 18" ], "shape": [ "factor: N*x + N + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-39/1", "set": "wentworth-first-steps-in-algebra-1894/ex-39", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "83", "location": "Exercise 39, problem 1", "problem_latex": "$a^{3} - 7a$.", "markdown": "$a^{3} - 7a$.", "answer_latex": [ "$a(a^{2} - 7)$." ], "answer_markdown": [ "$a(a^{2} - 7)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "a**3 - 7*a", "answer_expr": "a*(a**2 - 7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**3 - 7*x" ], "shape": [ "factor: N*x + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-39/10", "set": "wentworth-first-steps-in-algebra-1894/ex-39", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "83", "location": "Exercise 39, problem 10", "problem_latex": "$x^{2} - 2x - 3$.", "markdown": "$x^{2} - 2x - 3$.", 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"problem_latex": "$x^{2} - x -2$.", "markdown": "$x^{2} - x -2$.", "answer_latex": [ "$(x + 1)(x - 2)$." ], "answer_markdown": [ "$(x + 1)(x - 2)$." ], "checks": [ { "task": "factor", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2 - x - 2", "answer_expr": "(x + 1)*(x - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "factor: x**2 - x - 2" ], "shape": [ "factor: N - x + x**N" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/1", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 1", "problem_latex": "$9a - 2bc$.", "markdown": "$9a - 2bc$.", "answer_latex": [ "$21$." ], "answer_markdown": [ "$21$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 21.0, printed 21 (half-unit 0.5; correctly rounded at the printed digits: 21.0)", "problem_expr": "9*a - 2*b*c", "answer_expr": "21" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 9*a - 2*b*c at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a + N*b*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#21,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "9 ENTER 5 ×", "2 ENTER 4 × 3 ×", "−" ], "constants": [ { "value": "9", "source": "the 9 in '9a' in the expression 9a - 2bc" }, { "value": "5", "source": "a = 5 in the givens" }, { "value": "2", "source": "the 2 in '2bc' in the expression 9a - 2bc" }, { "value": "4", "source": "b = 4 in the givens" }, { "value": "3", "source": "c = 3 in the givens" } ], "calculator_value": "+21E+0", "printed_value": "21", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/10", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 10", "problem_latex": "$a^{2} - b^{2} - c^{2}$.", "markdown": "$a^{2} - b^{2} - c^{2}$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.0, printed 0 (half-unit 0.5; correctly rounded at the printed digits: 0.0)", "problem_expr": "a**2 - b**2 - c**2", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a**2 - b**2 - c**2 at a=5, b=4, c=3" ], "shape": [ "evaluate: a**N - b**N - c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#0,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 GOLD x²", "4 GOLD x² −", "3 GOLD x² −" ], "calculator_value": "+0E-6176", "printed_value": "0", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/11", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 11", "problem_latex": "$3(a - b + c)$.", "markdown": "$3(a - b + c)$.", "answer_latex": [ "$12$." ], "answer_markdown": [ "$12$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 12.0, printed 12 (half-unit 0.5; correctly rounded at the printed digits: 12.0)", "problem_expr": "3*(a - b + c)", "answer_expr": "12" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3*a - 3*b + 3*c at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a + N*b + N*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#12,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 4 − 3 + 3 ×" ], "constants": [ { "value": "5", "source": "a = 5 in the given values" }, { "value": "4", "source": "b = 4 in the given values" }, { "value": "3", "source": "c = 3 in the given values" }, { "value": "3", "source": "the coefficient 3 in 3(a − b + c)" } ], "calculator_value": "+12E+0", "printed_value": "12", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/12", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 12", "problem_latex": "$6ab - (bc + 8)$.", "markdown": "$6ab - (bc + 8)$.", "answer_latex": [ "$100$." ], "answer_markdown": [ "$100$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 100.0, printed 100 (half-unit 0.5; correctly rounded at the printed digits: 100.0)", "problem_expr": "6*a*b - (b*c + 8)", "answer_expr": "100" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 6*a*b - b*c - 8 at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a*b + N - b*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#100,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 5 × 4 ×", "4 ENTER 3 × 8 +", "−" ], "calculator_value": "+100E+0", "printed_value": "100", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/13", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 13", "problem_latex": "$7bc - c^{2} + a$.", "markdown": "$7bc - c^{2} + a$.", "answer_latex": [ "$80$." ], "answer_markdown": [ "$80$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 80.0, printed 80 (half-unit 0.5; correctly rounded at the printed digits: 80.0)", "problem_expr": "7*b*c - c**2 + a", "answer_expr": "80" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a + 7*b*c - c**2 at a=5, b=4, c=3" ], "shape": [ "evaluate: N*b*c + a - c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#80,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 ENTER 4 ×", "3 ×", "3 GOLD x² −", "5 +" ], "constants": [ { "value": "7", "source": "the 7 in '7bc' in the expression 7bc − c² + a" }, { "value": "4", "source": "given value b = 4" }, { "value": "3", "source": "given value c = 3 (typed twice: once for the factor c in 7bc, once for c² via GOLD x²)" }, { "value": "5", "source": "given value a = 5" } ], "calculator_value": "+80E+0", "printed_value": "80", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/14", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 14", "problem_latex": "$5ac - b^{2} + 3b$.", "markdown": "$5ac - b^{2} + 3b$.", "answer_latex": [ "$71$." ], "answer_markdown": [ "$71$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 71.0, printed 71 (half-unit 0.5; correctly rounded at the printed digits: 71.0)", "problem_expr": "5*a*c - b**2 + 3*b", "answer_expr": "71" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5*a*c - b**2 + 3*b at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a*c + N*b - b**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#71,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 5 × 3 ×", "4 GOLD x² −", "3 ENTER 4 × +" ], "constants": [ { "value": "5", "source": "the coefficient 5 in 5ac, from the problem's expression 5*a*c - b**2 + 3*b" }, { "value": "4", "source": "b = 4 from the given values; the exponent-2 on b is taken by GOLD x², and the 4 in 3b is the same given b" }, { "value": "3", "source": "the coefficient 3 in 3b, from the problem's expression; c = 3 from the given values feeds the factor 3 in 5ac" } ], "calculator_value": "+71E+0", "printed_value": "71", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/15", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 15", "problem_latex": "$4b^{2}c - 5c^{2} - 2b$.", "markdown": "$4b^{2}c - 5c^{2} - 2b$.", "answer_latex": [ "$139$." ], "answer_markdown": [ "$139$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 139.0, printed 139 (half-unit 0.5; correctly rounded at the printed digits: 139.0)", "problem_expr": "4*b**2*c - 5*c**2 - 2*b", "answer_expr": "139" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 4*a**2*b - 2*a - 5*b**2 at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a + N*a**N*b + N*b**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#139,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 4 GOLD x² × 3 ×", "5 ENTER 3 GOLD x² × −", "2 ENTER 4 × −" ], "constants": [ { "value": "4", "source": "the coefficient 4 in '4b^2c' (first 4, typed as the coefficient)" }, { "value": "4", "source": "the given value b = 4" }, { "value": "3", "source": "the given value c = 3" }, { "value": "5", "source": "the coefficient 5 in '5c^2'" }, { "value": "2", "source": "the coefficient 2 in '2b'" } ], "calculator_value": "+139E+0", "printed_value": "139", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/16", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 16", "problem_latex": "$2a + (b + c)$.", "markdown": "$2a + (b + c)$.", "answer_latex": [ "$17$." ], "answer_markdown": [ "$17$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 17.0, printed 17 (half-unit 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"problem_latex": "$2c - b(a - b)$.", "markdown": "$2c - b(a - b)$.", "answer_latex": [ "$2$." ], "answer_markdown": [ "$2$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 2.0, printed 2 (half-unit 0.5; correctly rounded at the printed digits: 2.0)", "problem_expr": "2*c - b*(a - b)", "answer_expr": "2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -b*(a - b) + 2*c at a=5, b=4, c=3" ], "shape": [ "evaluate: N*c - b*(a - b)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#2,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 3 ×", "4 ENTER 5 ENTER 4 − ×", "−" ], "constants": [ { "value": "2", "source": "the coefficient 2 in '2c' in the expression 2*c - b*(a - b)" }, { "value": "3", "source": "the given value c = 3" }, { "value": "4", "source": "the given value b = 4" }, { "value": "5", "source": "the given value a = 5" } ], 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"Exercise 4, problem 3", "problem_latex": "$abc+bc$.", "markdown": "$abc+bc$.", "answer_latex": [ "$72$." ], "answer_markdown": [ "$72$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 72.0, printed 72 (half-unit 0.5; correctly rounded at the printed digits: 72.0)", "problem_expr": "a*b*c + b*c", "answer_expr": "72" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a*b*c + b*c at a=5, b=4, c=3" ], "shape": [ "evaluate: a*b*c + b*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#72,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 4 × 3 ×", "4 ENTER 3 × +" ], "calculator_value": "+72E+0", "printed_value": "72", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/4", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 4", "problem_latex": "$5ac + 2a$.", "markdown": "$5ac + 2a$.", "answer_latex": [ "$85$." ], "answer_markdown": [ "$85$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 85.0, printed 85 (half-unit 0.5; correctly rounded at the printed digits: 85.0)", "problem_expr": "5*a*c + 2*a", "answer_expr": "85" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5*a*b + 2*a at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a*b + N*a" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#85,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 5 × 3 ×", "2 ENTER 5 × +" ], "constants": [ { "value": "5", "source": "the coefficient 5 in '5ac', and a = 5 from the given values" }, { "value": "3", "source": "c = 3 from the given values" }, { "value": "2", "source": "the coefficient 2 in '2a'" } ], "calculator_value": "+85E+0", "printed_value": "85", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/5", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 5", "problem_latex": "$2abc - 2ac^{2}$.", "markdown": "$2abc - 2ac^{2}$.", "answer_latex": [ "$30$." ], "answer_markdown": [ "$30$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 30.0, printed 30 (half-unit 0.5; correctly rounded at the printed digits: 30.0)", "problem_expr": "2*a*b*c - 2*a*c**2", "answer_expr": "30" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a*b*c - 2*a*c**2 at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a*b*c + N*a*c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#30,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 × 4 × 3 ×", "2 ENTER 5 × 3 GOLD x² ×", "−" ], "calculator_value": "+30E+0", "printed_value": "30", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/6", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 6", "problem_latex": "$ab + bc - ac$.", "markdown": "$ab + bc - ac$.", "answer_latex": [ "$17$." ], "answer_markdown": [ "$17$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 17.0, printed 17 (half-unit 0.5; correctly rounded at the printed digits: 17.0)", "problem_expr": "a*b + b*c - a*c", "answer_expr": "17" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a*b - a*c + b*c at a=5, b=4, c=3" ], "shape": [ "evaluate: a*b - a*c + b*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#17,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 4 ×", "4 ENTER 3 ×", "+", "5 ENTER 3 × −" ], "constants": [ { "value": "5", "source": "a = 5 in the givens" }, { "value": "4", "source": "b = 4 in the givens" }, { "value": "3", "source": "c = 3 in the givens" } ], "calculator_value": "+17E+0", "printed_value": "17", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/7", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 7", "problem_latex": "$ac - (b + c)$.", "markdown": "$ac - (b + c)$.", "answer_latex": [ "$8$." ], "answer_markdown": [ "$8$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 8.0, printed 8 (half-unit 0.5; correctly rounded at the printed digits: 8.0)", "problem_expr": "a*c - (b + c)", "answer_expr": "8" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a*c - b - c at a=5, b=4, c=3" ], "shape": [ "evaluate: a*c - b - c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#8,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 3 ×", "4 ENTER 3 + −" ], "constants": [ { "value": "5", "source": "given value a = 5" }, { "value": "3", "source": "given value c = 3 (used for both a·c and b + c)" }, { "value": "4", "source": "given value b = 4" } ], "calculator_value": "+8E+0", "printed_value": "8", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/8", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 8", "problem_latex": "$a^{2} + (b^{2} + c^{2})$\\Add{.}", "markdown": "$a^{2} + (b^{2} + c^{2})$", "answer_latex": [ "$50$." ], "answer_markdown": [ "$50$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 50.0, printed 50 (half-unit 0.5; correctly rounded at the printed digits: 50.0)", "problem_expr": "a**2 + (b**2 + c**2)", "answer_expr": "50" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: a**2 + b**2 + c**2 at a=5, b=4, c=3" ], "shape": [ "evaluate: a**N + b**N + c**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#50,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 GOLD x²", "4 GOLD x²", "3 GOLD x²", "+", "+" ], "constants": [ { "value": "5", "source": "a = 5 (given values)" }, { "value": "4", "source": "b = 4 (given values)" }, { "value": "3", "source": "c = 3 (given values)" } ], "calculator_value": "+50E+0", "printed_value": "50", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-4/9", "set": "wentworth-first-steps-in-algebra-1894/ex-4", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "13", "location": "Exercise 4, problem 9", "problem_latex": "$2a + (2b + 2c)$.", "markdown": "$2a + (2b + 2c)$.", "answer_latex": [ "$24$." ], "answer_markdown": [ "$24$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 24.0, printed 24 (half-unit 0.5; correctly rounded at the printed digits: 24.0)", "problem_expr": "2*a + (2*b + 2*c)", "answer_expr": "24" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*a + 2*b + 2*c at a=5, b=4, c=3" ], "shape": [ "evaluate: N*a + N*b + N*c" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#24,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 ×", "2 ENTER 4 ×", "2 ENTER 3 ×", "+", "+" ], "calculator_value": "+24E+0", "printed_value": "24", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/1", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 1", "problem_latex": "$330$ and $546$.", "markdown": "$330$ and $546$.", "answer_latex": [ "$6$." ], "answer_markdown": [ "$6$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (330, 546)" ], "shape": [ "hcf: (N, N)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X=6", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/10", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 10", "problem_latex": "$(x + y)^{2}$ and $x^{2} - y^{2}$.", "markdown": "$(x + y)^{2}$ and $x^{2} - y^{2}$.", "answer_latex": [ "$x + y$." ], "answer_markdown": [ "$x + y$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: ((a + b)**2, a**2 - b**2)" ], "shape": [ "hcf: ((a + b)**N, a**N - b**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/11", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 11", "problem_latex": "$a^{3} + a^{2}x$ and $a^{2} - x^{2}$.", "markdown": "$a^{3} + a^{2}x$ and $a^{2} - x^{2}$.", "answer_latex": [ "$a + x$." ], "answer_markdown": [ "$a + x$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "a + x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - b**2, a**3 + a**2*b)" ], "shape": [ "hcf: (a**N - b**N, a**N*b + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/12", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 12", "problem_latex": "$a^{2} - 4b^{2}$ and $a^{2} + 2ab$.", "markdown": "$a^{2} - 4b^{2}$ and $a^{2} + 2ab$.", "answer_latex": [ "$a + 2b$." ], "answer_markdown": [ "$a + 2b$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "a + 2*b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 4*b**2, a**2 + 2*a*b)" ], "shape": [ "hcf: (N*b**N + a**N, N*a*b + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/13", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 13", "problem_latex": "$x^{2} - 1$ and $x^{2} + 2x - 3$.", "markdown": "$x^{2} - 1$ and $x^{2} + 2x - 3$.", "answer_latex": [ "$x - 1$." ], "answer_markdown": [ "$x - 1$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 1, a**2 + 2*a - 3)" ], "shape": [ "hcf: (a**N - 1, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/14", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 14", "problem_latex": "$x^{2} + 5x + 6$ and $x^{2} + 4x + 3$.", "markdown": "$x^{2} + 5x + 6$ and $x^{2} + 4x + 3$.", "answer_latex": [ "$x + 3$." ], "answer_markdown": [ "$x + 3$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + 3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 + 4*a + 3, a**2 + 5*a + 6)" ], "shape": [ "hcf: (N*a + N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/15", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 15", "problem_latex": "$x^{2} - 9x + 18$ and $x^{2} - 10x + 24$.", "markdown": "$x^{2} - 9x + 18$ and $x^{2} - 10x + 24$.", "answer_latex": [ "$x - 6$." ], "answer_markdown": [ "$x - 6$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - 6" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 10*a + 24, a**2 - 9*a + 18)" ], "shape": [ "hcf: (N*a + N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/16", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 16", "problem_latex": "$x^{3} + 1$ and $x^{2} - x + 1$.", "markdown": "$x^{3} + 1$ and $x^{2} - x + 1$.", "answer_latex": [ "$x^{2} - x + 1$." ], "answer_markdown": [ "$x^{2} - x + 1$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x**2 - x + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**3 + 1, a**2 - a + 1)" ], "shape": [ "hcf: (a**N + 1, -a + a**N + 1)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/17", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 17", "problem_latex": "$x^{2} - 3x + 2$ and $x^{2} - 4x + 3$.", "markdown": "$x^{2} - 3x + 2$ and $x^{2} - 4x + 3$.", "answer_latex": [ "$x - 1$." ], "answer_markdown": [ "$x - 1$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 4*a + 3, a**2 - 3*a + 2)" ], "shape": [ "hcf: (N*a + N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/18", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 18", "problem_latex": "$x^{2} - 3xy + 2y^{2}$ and $x^{2} - 2xy + y^{2}$.", "markdown": "$x^{2} - 3xy + 2y^{2}$ and $x^{2} - 2xy + y^{2}$.", "answer_latex": [ "$x - y$." ], "answer_markdown": [ "$x - y$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 3*a*b + 2*b**2, a**2 - 2*a*b + b**2)" ], "shape": [ "hcf: (N*a*b + N*b**N + a**N, N*a*b + a**N + b**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/19", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 19", "problem_latex": "$x^{2} - 4x - 5$ and $x^{2} - 25$.", "markdown": "$x^{2} - 4x - 5$ and $x^{2} - 25$.", "answer_latex": [ "$x - 5$." ], "answer_markdown": [ "$x - 5$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - 5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 25, a**2 - 4*a - 5)" ], "shape": [ "hcf: (N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/2", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 2", "problem_latex": "$20x^{3}$ and $15x^{4}$.", "markdown": "$20x^{3}$ and $15x^{4}$.", "answer_latex": [ "$5x^{3}$." ], "answer_markdown": [ "$5x^{3}$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "5*x**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (20*a**3, 15*a**4)" ], "shape": [ "hcf: (N*a**N, N*a**N)" ], "same_problem_in": [], "needs": [ "cas.pgcd", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/20", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 20", "problem_latex": "$(a - b)^{2} - c^{2}$ and $ab - b^{2} - bc$.", "markdown": "$(a - b)^{2} - c^{2}$ and $ab - b^{2} - bc$.", "answer_latex": [ "$a - b - c$." ], "answer_markdown": [ "$a - b - c$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "a - b - c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (-c**2 + (a - b)**2, a*b - b**2 - b*c)" ], "shape": [ "hcf: (-c**N + (a - b)**N, a*b - b*c - b**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/21", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 21", "problem_latex": "$x^{2} + xy -2y^{2}$ and $x^{2} + 5xy + 6y^{2}$.", "markdown": "$x^{2} + xy -2y^{2}$ and $x^{2} + 5xy + 6y^{2}$.", "answer_latex": [ "$x + 2y$." ], "answer_markdown": [ "$x + 2y$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + 2*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 + a*b - 2*b**2, a**2 + 5*a*b + 6*b**2)" ], "shape": [ "hcf: (N*b**N + a*b + a**N, N*a*b + N*b**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/22", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 22", "problem_latex": "$x^{2} + 7xy + 12y^{2}$ and $x^{2} + 3xy - 4y^{2}$.", "markdown": "$x^{2} + 7xy + 12y^{2}$ and $x^{2} + 3xy - 4y^{2}$.", "answer_latex": [ "$x + 4y$." ], "answer_markdown": [ "$x + 4y$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + 4*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 + 3*a*b - 4*b**2, a**2 + 7*a*b + 12*b**2)" ], "shape": [ "hcf: (N*a*b + N*b**N + a**N, N*a*b + N*b**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/23", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 23", "problem_latex": "$x^{3} - 8y^{3}$ and $x^{2} + 2xy + 4y^{2}$.", "markdown": "$x^{3} - 8y^{3}$ and $x^{2} + 2xy + 4y^{2}$.", "answer_latex": [ "$x^{2} + 2xy + 4y^{2}$." ], "answer_markdown": [ "$x^{2} + 2xy + 4y^{2}$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x**2 + 2*x*y + 4*y**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**3 - 8*b**3, a**2 + 2*a*b + 4*b**2)" ], "shape": [ "hcf: (N*b**N + a**N, N*a*b + N*b**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/24", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 24", "problem_latex": "$x^{3} - 2x^{2} - x + 2$ and $x^{2} - 4x + 4$.", "markdown": "$x^{3} - 2x^{2} - x + 2$ and $x^{2} - 4x + 4$.", "answer_latex": [ "$x - 2$." ], "answer_markdown": [ "$x - 2$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - 2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 4*a + 4, a**3 - 2*a**2 - a + 2)" ], "shape": [ "hcf: (N*a + N + a**N, N*a**N + N - a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/25", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 25", "problem_latex": "$1 - 5a + 6a^{2}$ and $1 - 7a + 12a^{2}$.", "markdown": "$1 - 5a + 6a^{2}$ and $1 - 7a + 12a^{2}$.", "answer_latex": [ "$1 - 3a$." ], "answer_markdown": [ "$1 - 3a$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "1 - 3*a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (6*a**2 - 5*a + 1, 12*a**2 - 7*a + 1)" ], "shape": [ "hcf: (N*a + N*a**N + 1, N*a + N*a**N + 1)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/26", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 26", "problem_latex": "$x^{2} - 8xy + 7y^{2}$ and $x^{2} - 3xy - 28y^{2}$.", "markdown": "$x^{2} - 8xy + 7y^{2}$ and $x^{2} - 3xy - 28y^{2}$.", "answer_latex": [ "$x - 7y$." ], "answer_markdown": [ "$x - 7y$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x - 7*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 8*a*b + 7*b**2, a**2 - 3*a*b - 28*b**2)" ], "shape": [ "hcf: (N*a*b + N*b**N + a**N, N*a*b + N*b**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/27", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 27", "problem_latex": "$8a^{3} + b^{3}$ and $4a^{2} + 4ab + b^{2}$.", "markdown": "$8a^{3} + b^{3}$ and $4a^{2} + 4ab + b^{2}$.", "answer_latex": [ "$2a + b$." ], "answer_markdown": [ "$2a + b$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "2*a + b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (8*a**3 + b**3, 4*a**2 + 4*a*b + b**2)" ], "shape": [ "hcf: (N*a**N + b**N, N*a*b + N*a**N + b**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/28", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 28, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 28", "problem_latex": "$x^{2} - (y - z)^{2}$ and $(x + y)^{2} - z^{2}$.", "markdown": "$x^{2} - (y - z)^{2}$ and $(x + y)^{2} - z^{2}$.", "answer_latex": [ "$x + y - z$." ], "answer_markdown": [ "$x + y - z$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + y - z" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - (b - c)**2, -c**2 + (a + b)**2)" ], "shape": [ "hcf: (a**N - (b - c)**N, -c**N + (a + b)**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/3", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 3", "problem_latex": "$42ax^{2}$ and $60a^{2}x$.", "markdown": "$42ax^{2}$ and $60a^{2}x$.", "answer_latex": [ "$6ax$." ], "answer_markdown": [ "$6ax$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "6*a*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (42*a*b**2, 60*a**2*b)" ], "shape": [ "hcf: (N*a*b**N, N*a**N*b)" ], "same_problem_in": [], "needs": [ "cas.pgcd", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/4", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 4", "problem_latex": "$35a^{2}b^{2}$ and $49ab^{3}$.", "markdown": "$35a^{2}b^{2}$ and $49ab^{3}$.", "answer_latex": [ "$7ab^{2}$." ], "answer_markdown": [ "$7ab^{2}$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "7*a*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (49*a*b**3, 35*a**2*b**2)" ], "shape": [ "hcf: (N*a*b**N, N*a**N*b**N)" ], "same_problem_in": [], "needs": [ "cas.pgcd", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/5", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 5", "problem_latex": "$28x^{4}$ and $63y^{4}$.", "markdown": "$28x^{4}$ and $63y^{4}$.", "answer_latex": [ "$7$." ], "answer_markdown": [ "$7$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (28*a**4, 63*b**4)" ], "shape": [ "hcf: (N*a**N, N*b**N)" ], "same_problem_in": [], "needs": [ "cas.pgcd", "core.arith" ], "expectation": "X=7", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/6", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 6", "problem_latex": "$54a^{2}b^{2}$ and $56a^{3}b^{3}$.", "markdown": "$54a^{2}b^{2}$ and $56a^{3}b^{3}$.", "answer_latex": [ "$2a^{2}b^{2}$." ], "answer_markdown": [ "$2a^{2}b^{2}$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "2*a**2*b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (54*a**2*b**2, 56*a**3*b**3)" ], "shape": [ "hcf: (N*a**N*b**N, N*a**N*b**N)" ], "same_problem_in": [], "needs": [ "cas.pgcd", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/7", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 7", "problem_latex": "$x^{3} + 3x^{2}y$ and $x^{3} + 27y^{3}$.", "markdown": "$x^{3} + 3x^{2}y$ and $x^{3} + 27y^{3}$.", "answer_latex": [ "$x + 3y$." ], "answer_markdown": [ "$x + 3y$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + 3*y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**3 + 27*b**3, a**3 + 3*a**2*b)" ], "shape": [ "hcf: (N*b**N + a**N, N*a**N*b + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/8", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 8", "problem_latex": "$x^{2} + 3x$ and $x^{2} - 9$.", "markdown": "$x^{2} + 3x$ and $x^{2} - 9$.", "answer_latex": [ "$x + 3$." ], "answer_markdown": [ "$x + 3$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x + 3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (a**2 - 9, a**2 + 3*a)" ], "shape": [ "hcf: (N + a**N, N*a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-40/9", "set": "wentworth-first-steps-in-algebra-1894/ex-40", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "86", "location": "Exercise 40, problem 9", "problem_latex": "$2ax^{3} + x^{3}$ and $8a^{3} + 1$.", "markdown": "$2ax^{3} + x^{3}$ and $8a^{3} + 1$.", "answer_latex": [ "$2a + 1$." ], "answer_markdown": [ "$2a + 1$." ], "checks": [ { "task": "hcf", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "2*a + 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "hcf: (8*a**3 + 1, 2*a*b**3 + b**3)" ], "shape": [ "hcf: (N*a**N + 1, N*a*b**N + b**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/1", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 1", "problem_latex": "$9xy^{3}$ and $6x^{2}y$.", "markdown": "$9xy^{3}$ and $6x^{2}y$.", "answer_latex": [ "$18x^{2}y^{3}$." ], "answer_markdown": [ "$18x^{2}y^{3}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "18*x**2*y**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (9*a*b**3, 6*a**2*b)" ], "shape": [ "lcm: (N*a*b**N, N*a**N*b)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/10", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 10", "problem_latex": "$x^{2} - 1$ and $x^{2} - x$.", "markdown": "$x^{2} - 1$ and $x^{2} - x$.", "answer_latex": [ "$x(x + 1)(x - 1)$." ], "answer_markdown": [ "$x(x + 1)(x - 1)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x*(x + 1)*(x - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - 1, a**2 - a)" ], "shape": [ "lcm: (a**N - 1, -a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/11", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 11", "problem_latex": "$x^{2} + xy$ and $xy + y^{2}$.", "markdown": "$x^{2} + xy$ and $xy + y^{2}$.", "answer_latex": [ "$xy(x + y)$." ], "answer_markdown": [ "$xy(x + y)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x*y*(x + y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 + a*b, a*b + b**2)" ], "shape": [ "lcm: (a*b + a**N, a*b + b**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/12", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 12", "problem_latex": "$x^{2} + 2x$ and $(x + 2)^{2}$.", "markdown": "$x^{2} + 2x$ and $(x + 2)^{2}$.", "answer_latex": [ "$x(x + 2)^{2}$." ], "answer_markdown": [ "$x(x + 2)^{2}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x*(x + 2)**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: ((a + 2)**2, a**2 + 2*a)" ], "shape": [ "lcm: ((N + a)**N, N*a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/13", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 13", "problem_latex": "$a^{2} + 4a + 4$ and $a^{2} + 5a + 6$.", "markdown": "$a^{2} + 4a + 4$ and $a^{2} + 5a + 6$.", "answer_latex": [ "$(a + 2)(a + 2)(a + 3)$." ], "answer_markdown": [ "$(a + 2)(a + 2)(a + 3)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(a + 2)*(a + 2)*(a + 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 + 4*a + 4, a**2 + 5*a + 6)" ], "shape": [ "lcm: (N*a + N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/14", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 14", "problem_latex": "$c^{2} + c - 20$ and $c^{2} - c - 30$.", "markdown": "$c^{2} + c - 20$ and $c^{2} - c - 30$.", "answer_latex": [ "$(c - 4)(c + 5)(c - 6)$." ], "answer_markdown": [ "$(c - 4)(c + 5)(c - 6)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(c - 4)*(c + 5)*(c - 6)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - a - 30, a**2 + a - 20)" ], "shape": [ "lcm: (N - a + a**N, N + a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/15", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 15", "problem_latex": "$b^{2} + b - 42$ and $b^{2} - 11b + 30$.", "markdown": "$b^{2} + b - 42$ and $b^{2} - 11b + 30$.", "answer_latex": [ "$(b - 5)(b - 6)(b + 7)$." ], "answer_markdown": [ "$(b - 5)(b - 6)(b + 7)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(b - 5)*(b - 6)*(b + 7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - 11*a + 30, a**2 + a - 42)" ], "shape": [ "lcm: (N*a + N + a**N, N + a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/16", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 16", "problem_latex": "$y^{2} - 10y + 24$ and $y^{2} + y -20$.", "markdown": "$y^{2} - 10y + 24$ and $y^{2} + y -20$.", "answer_latex": [ "$(y - 4)(y + 5)(y - 6)$." ], "answer_markdown": [ "$(y - 4)(y + 5)(y - 6)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(y - 4)*(y + 5)*(y - 6)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - 10*a + 24, a**2 + a - 20)" ], "shape": [ "lcm: (N*a + N + a**N, N + a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/17", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 17", "problem_latex": "$z^{2} + 2z - 35$ and $z^{2} - 11z + 30$.", "markdown": "$z^{2} + 2z - 35$ and $z^{2} - 11z + 30$.", "answer_latex": [ "$(z - 5)(z - 6)(z + 7)$." ], "answer_markdown": [ "$(z - 5)(z - 6)(z + 7)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(z - 5)*(z - 6)*(z + 7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - 11*a + 30, a**2 + 2*a - 35)" ], "shape": [ "lcm: (N*a + N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/18", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 18", "problem_latex": "$x^{2} - 64; x^{3} - 64$; and $x + 8$.", "markdown": "$x^{2} - 64; x^{3} - 64$; and $x + 8$.", "answer_latex": [ "$(x - 4)(x + 8)(x - 8)(x^{2} + 4x + 16)$." ], "answer_markdown": [ "$(x - 4)(x + 8)(x - 8)(x^{2} + 4x + 16)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(x - 4)*(x + 8)*(x - 8)*(x**2 + 4*x + 16)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a + 8, a**2 - 64, a**3 - 64)" ], "shape": [ "lcm: (N + a, N + a**N, N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/19", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 19", "problem_latex": "$a^{2} - b^{2}; (a + b)^{2}$; and $(a - b)^{2}$.", "markdown": "$a^{2} - b^{2}; (a + b)^{2}$; and $(a - b)^{2}$.", "answer_latex": [ "$(a + b)(a + b)(a - b)(a - b)$." ], "answer_markdown": [ "$(a + b)(a + b)(a - b)(a - b)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(a + b)*(a + b)*(a - b)*(a - b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: ((a - b)**2, (a + b)**2, a**2 - b**2)" ], "shape": [ "lcm: ((a - b)**N, (a + b)**N, a**N - b**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/2", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 2", "problem_latex": "$3abc^{2}$ and $2a^{2}bc^{3}$.", "markdown": "$3abc^{2}$ and $2a^{2}bc^{3}$.", "answer_latex": [ "$6a^{2}bc^{3}$." ], "answer_markdown": [ "$6a^{2}bc^{3}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "6*a**2*b*c**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (3*a*b*c**2, 2*a**2*b*c**3)" ], "shape": [ "lcm: (N*a*b*c**N, N*a**N*b*c**N)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/20", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 20", "problem_latex": "$4ab(a + b)^{2}$ and $2a^{2}(a^{2} - b^{2})$.", "markdown": "$4ab(a + b)^{2}$ and $2a^{2}(a^{2} - b^{2})$.", "answer_latex": [ "$4a^{2}b(a + b)(a + b)(a - b)$." ], "answer_markdown": [ "$4a^{2}b(a + b)(a + b)(a - b)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "4*a**2*b*(a + b)*(a + b)*(a - b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (2*a**2*(a**2 - b**2), 4*a*b*(a + b)**2)" ], "shape": [ "lcm: (N*a**N*(a**N - b**N), N*a*b*(a + b)**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/21", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 21", "problem_latex": "$y^{2} + 7y + 12; y^{2} + 6y + 8$; and $y^{2} + 5y +6$.", "markdown": "$y^{2} + 7y + 12; y^{2} + 6y + 8$; and $y^{2} + 5y +6$.", "answer_latex": [ "$(y + 2)(y + 3)(y + 4)$." ], "answer_markdown": [ "$(y + 2)(y + 3)(y + 4)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(y + 2)*(y + 3)*(y + 4)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 + 5*a + 6, a**2 + 6*a + 8, a**2 + 7*a + 12)" ], "shape": [ "lcm: (N*a + N + a**N, N*a + N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/22", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 22", "problem_latex": "$x^{2} - 1; x^{3} + x^{2} + x + 1$; and $x^{3} - x^{2} + x - 1$.", "markdown": "$x^{2} - 1; x^{3} + x^{2} + x + 1$; and $x^{3} - x^{2} + x - 1$.", "answer_latex": [ "$(x + 1)(x - 1)(x^{2} + 1)$." ], "answer_markdown": [ "$(x + 1)(x - 1)(x^{2} + 1)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(x + 1)*(x - 1)*(x**2 + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - 1, a**3 - a**2 + a - 1, a**3 + a**2 + a + 1)" ], "shape": [ "lcm: (a**N - 1, a - 1, a + 2*a**N + 1)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/23", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 23", "problem_latex": "$1 - x^{2}; 1 - x^{3}$; and $1 + x$.", "markdown": "$1 - x^{2}; 1 - x^{3}$; and $1 + x$.", "answer_latex": [ "$(1 + x)(1 - x)(1 + x + x^{2})$." ], "answer_markdown": [ "$(1 + x)(1 - x)(1 + x + x^{2})$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(1 - x)*(1 + x)*(1 + x + x**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (-a**2 + 1, -a**3 + 1, a + 1)" ], "shape": [ "lcm: (-a**N + 1, -a**N + 1, a + 1)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/24", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 24", "problem_latex": "$x^{2} + 2xy + y^{2}; x^{2} - y^{2}$; and $x^{2} - 2xy + y^{2}$.", "markdown": "$x^{2} + 2xy + y^{2}; x^{2} - y^{2}$; and $x^{2} - 2xy + y^{2}$.", "answer_latex": [ "$(x + y)(x + y)(x - y)(x - y)$." ], "answer_markdown": [ "$(x + y)(x + y)(x - y)(x - y)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(x + y)*(x + y)*(x - y)*(x - y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - b**2, a**2 - 2*a*b + b**2, a**2 + 2*a*b + b**2)" ], "shape": [ "lcm: (a**N - b**N, N*a*b + a**N + b**N, N*a*b + a**N + b**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/25", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 25", "problem_latex": "$x^{3} - 27; x^{2} + 2x - 15; x^{2} + 5x$.", "markdown": "$x^{3} - 27; x^{2} + 2x - 15; x^{2} + 5x$.", "answer_latex": [ "$x(x - 3)(x + 5)(x^{2} + 3x + 9)$." ], "answer_markdown": [ "$x(x - 3)(x + 5)(x^{2} + 3x + 9)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x*(x - 3)*(x + 5)*(x**2 + 3*x + 9)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 + 5*a, a**3 - 27, a**2 + 2*a - 15)" ], "shape": [ "lcm: (N*a + a**N, N + a**N, N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/26", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 26", "problem_latex": "$(a + b)^{2} - c^{2}; (a + b + c)^{2}$; and $a + b - c$.", "markdown": "$(a + b)^{2} - c^{2}; (a + b + c)^{2}$; and $a + b - c$.", "answer_latex": [ "$(a + b + c)(a + b + c)(a + b - c)$." ], "answer_markdown": [ "$(a + b + c)(a + b + c)(a + b - c)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(a + b + c)*(a + b + c)*(a + b - c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (-c**2 + (a + b)**2, a + b - c, (a + b + c)**2)" ], "shape": [ "lcm: (-c**N + (a + b)**N, a + b - c, (a + b + c)**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/27", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 27", "problem_latex": "$x^{2} - (a + b)x + ab$ and $x^{2} - (a + c)x + ac$.", "markdown": "$x^{2} - (a + b)x + ab$ and $x^{2} - (a + c)x + ac$.", "answer_latex": [ "$(x - a)(x - b)(x - c)$." ], "answer_markdown": [ "$(x - a)(x - b)(x - c)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "(x - a)*(x - b)*(x - c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a*b + d**2 - d*(a + b), a*c + d**2 - d*(a + c))" ], "shape": [ "lcm: (a*b - d*(a + b) + d**N, a*c - d*(a + c) + d**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/28", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 28, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 28", "problem_latex": "$(a + b)^{2} - c^{2}$ and $a^{2} + ab + ac$.", "markdown": "$(a + b)^{2} - c^{2}$ and $a^{2} + ab + ac$.", "answer_latex": [ "$a(a + b + c)(a + b - c)$." ], "answer_markdown": [ "$a(a + b + c)(a + b - c)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "a*(a + b + c)*(a + b - c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (-c**2 + (a + b)**2, a**2 + a*b + a*c)" ], "shape": [ "lcm: (-c**N + (a + b)**N, a*b + a*c + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/3", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 3", "problem_latex": "$4a^{3}b$ and $10ab^{3}$.", "markdown": "$4a^{3}b$ and $10ab^{3}$.", "answer_latex": [ "$20a^{3}b^{3}$." ], "answer_markdown": [ "$20a^{3}b^{3}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "20*a**3*b**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (10*a*b**3, 4*a**3*b)" ], "shape": [ "lcm: (N*a*b**N, N*a**N*b)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/4", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 4", "problem_latex": "$6a^{3}b^{3}$ and $15a^{2}b^{4}$.", "markdown": "$6a^{3}b^{3}$ and $15a^{2}b^{4}$.", "answer_latex": [ "$30a^{3}b^{4}$." ], "answer_markdown": [ "$30a^{3}b^{4}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "30*a**3*b**4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (15*a**2*b**4, 6*a**3*b**3)" ], "shape": [ "lcm: (N*a**N*b**N, N*a**N*b**N)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/5", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 5", "problem_latex": "$21xy^{3}$ and $27x^{3}y^{5}$.", "markdown": "$21xy^{3}$ and $27x^{3}y^{5}$.", "answer_latex": [ "$189x^{3}y^{5}$." ], "answer_markdown": [ "$189x^{3}y^{5}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "189*x**3*y**5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (21*a*b**3, 27*a**3*b**5)" ], "shape": [ "lcm: (N*a*b**N, N*a**N*b**N)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/6", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 6", "problem_latex": "$xy^{3}z^{2}$ and $x^{2}y^{2}z^{3}$.", "markdown": "$xy^{3}z^{2}$ and $x^{2}y^{2}z^{3}$.", "answer_latex": [ "$x^{2}y^{3}z^{3}$." ], "answer_markdown": [ "$x^{2}y^{3}z^{3}$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x**2*y**3*z**3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a*b**3*c**2, a**2*b**2*c**3)" ], "shape": [ "lcm: (a*b**N*c**N, a**N*b**N*c**N)" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/7", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 7", "problem_latex": "$a^{2}$ and $a^{2} + a$.", "markdown": "$a^{2}$ and $a^{2} + a$.", "answer_latex": [ "$a^{2}(a + 1)$." ], "answer_markdown": [ "$a^{2}(a + 1)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "a**2*(a + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2, a**2 + a)" ], "shape": [ "lcm: (a**N, a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/8", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 8", "problem_latex": "$x^{2}$ and $x^{3} - 3x^{2}$.", "markdown": "$x^{2}$ and $x^{3} - 3x^{2}$.", "answer_latex": [ "$x^{2}(x - 3)$." ], "answer_markdown": [ "$x^{2}(x - 3)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x**2*(x - 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2, a**3 - 3*a**2)" ], "shape": [ "lcm: (a**N, N*a**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-41/9", "set": "wentworth-first-steps-in-algebra-1894/ex-41", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "88", "location": "Exercise 41, problem 9", "problem_latex": "$x^{2} - 1$ and $x^{2} + x$.", "markdown": "$x^{2} - 1$ and $x^{2} + x$.", "answer_latex": [ "$x(x + 1)(x - 1)$." ], "answer_markdown": [ "$x(x + 1)(x - 1)$." ], "checks": [ { "task": "lcm", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "x*(x + 1)*(x - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "lcm: (a**2 - 1, a**2 + a)" ], "shape": [ "lcm: (a**N - 1, a + a**N)" ], "same_problem_in": [], "needs": [ "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/1", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 1", "problem_latex": "$\\dfrac{2a}{6ab}$.", "markdown": "$\\dfrac{2a}{6ab}$.", "answer_latex": [ "$\\dfrac{1}{3b}$." ], "answer_markdown": [ "$\\dfrac{1}{3b}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "2*a/(6*a*b)", "answer_expr": "1/(3*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 1/(3*a)" ], "shape": [ "identity: N/a" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/10", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 10", "problem_latex": "$\\dfrac{abx - bx^{2}}{acx - cx^{2}}$.", "markdown": "$\\dfrac{abx - bx^{2}}{acx - cx^{2}}$.", "answer_latex": [ "$\\dfrac{b}{c}$." ], "answer_markdown": [ "$\\dfrac{b}{c}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a*b*x - b*x**2)/(a*c*x - c*x**2)", "answer_expr": "b/c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a*b*d - b*d**2)/(a*c*d - c*d**2)" ], "shape": [ "identity: (a*b*d - b*d**N)/(a*c*d - c*d**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/11", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 11", "problem_latex": "$\\dfrac{4a^{2} - 9b^{2}}{4a^{2} + 6ab}$.", "markdown": "$\\dfrac{4a^{2} - 9b^{2}}{4a^{2} + 6ab}$.", "answer_latex": [ "$\\dfrac{2a - 3b}{2a}$." ], "answer_markdown": [ "$\\dfrac{2a - 3b}{2a}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4*a**2 - 9*b**2)/(4*a**2 + 6*a*b)", "answer_expr": "(2*a - 3*b)/(2*a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (4*a**2 - 9*b**2)/(4*a**2 + 6*a*b)" ], "shape": [ "identity: (N*a**N + N*b**N)/(N*a*b + N*a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/12", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 12", "problem_latex": "$\\dfrac{3a^{2} + 6a}{a^{2} + 4a + 4}$.", "markdown": "$\\dfrac{3a^{2} + 6a}{a^{2} + 4a + 4}$.", "answer_latex": [ "$\\dfrac{3a}{a + 2}$." ], "answer_markdown": [ "$\\dfrac{3a}{a + 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*a**2 + 6*a)/(a**2 + 4*a + 4)", "answer_expr": "3*a/(a + 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (3*a**2 + 6*a)/(a**2 + 4*a + 4)" ], "shape": [ "identity: (N*a + N*a**N)/(N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/13", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 13", "problem_latex": "$\\dfrac{x^{2} + 5x}{x^{2} + 4x-5}$.", "markdown": "$\\dfrac{x^{2} + 5x}{x^{2} + 4x-5}$.", "answer_latex": [ "$\\dfrac{x}{x - 1}$." ], "answer_markdown": [ "$\\dfrac{x}{x - 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + 5*x)/(x**2 + 4*x - 5)", "answer_expr": "x/(x - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 5*a)/(a**2 + 4*a - 5)" ], "shape": [ "identity: (N*a + a**N)/(N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/14", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 14", "problem_latex": "$\\dfrac{xy - 3y^{2}}{x^{3} - 27y^{3}}$.", "markdown": "$\\dfrac{xy - 3y^{2}}{x^{3} - 27y^{3}}$.", "answer_latex": [ "$\\dfrac{y}{x^{2} + 3xy + 9y^{2}}$." ], "answer_markdown": [ "$\\dfrac{y}{x^{2} + 3xy + 9y^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x*y - 3*y**2)/(x**3 - 27*y**3)", "answer_expr": "y/(x**2 + 3*x*y + 9*y**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a*b - 3*b**2)/(a**3 - 27*b**3)" ], "shape": [ "identity: (N*b**N + a*b)/(N*b**N + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/15", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 15", "problem_latex": "$\\dfrac{x^{2} + 5x + 4}{x^{2} - x-20}$.", "markdown": "$\\dfrac{x^{2} + 5x + 4}{x^{2} - x-20}$.", "answer_latex": [ "$\\dfrac{x + 1}{x - 5}$." ], "answer_markdown": [ "$\\dfrac{x + 1}{x - 5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + 5*x + 4)/(x**2 - x - 20)", "answer_expr": "(x + 1)/(x - 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 5*a + 4)/(a**2 - a - 20)" ], "shape": [ "identity: (N*a + N + a**N)/(N - a + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/16", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 16", "problem_latex": "$\\dfrac{x^{2} + 2x + 1}{x^{2} - x-2}$.", "markdown": "$\\dfrac{x^{2} + 2x + 1}{x^{2} - x-2}$.", "answer_latex": [ "$\\dfrac{x + 1}{x - 2}$." ], "answer_markdown": [ "$\\dfrac{x + 1}{x - 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + 2*x + 1)/(x**2 - x - 2)", "answer_expr": "(x + 1)/(x - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 2*a + 1)/(a**2 - a - 2)" ], "shape": [ "identity: (N*a + a**N + 1)/(N - a + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/17", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 17", "problem_latex": "$\\dfrac{(a + b)^{2} - c^{2}}{a^{2} + ab-ac}$.", "markdown": "$\\dfrac{(a + b)^{2} - c^{2}}{a^{2} + ab-ac}$.", "answer_latex": [ "$\\dfrac{a + b + c}{a}$." ], "answer_markdown": [ "$\\dfrac{a + b + c}{a}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((a + b)**2 - c**2)/(a**2 + a*b - a*c)", "answer_expr": "(a + b + c)/a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-c**2 + (a + b)**2)/(a**2 + a*b - a*c)" ], "shape": [ "identity: (-c**N + (a + b)**N)/(a*b - a*c + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/18", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 18", "problem_latex": "$\\dfrac{x^{2} + 9x + 20}{x^{2} + 7x + 12}$.", "markdown": "$\\dfrac{x^{2} + 9x + 20}{x^{2} + 7x + 12}$.", "answer_latex": [ "$\\dfrac{x + 5}{x + 3}$." ], "answer_markdown": [ "$\\dfrac{x + 5}{x + 3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 + 9*x + 20)/(x**2 + 7*x + 12)", "answer_expr": "(x + 5)/(x + 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 9*a + 20)/(a**2 + 7*a + 12)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/19", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 19", "problem_latex": "$\\dfrac{x^{2} - 14x - 15}{x^{2} - 12x - 45}$.", "markdown": "$\\dfrac{x^{2} - 14x - 15}{x^{2} - 12x - 45}$.", "answer_latex": [ "$\\dfrac{x + 1}{x + 3}$." ], "answer_markdown": [ "$\\dfrac{x + 1}{x + 3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 - 14*x - 15)/(x**2 - 12*x - 45)", "answer_expr": "(x + 1)/(x + 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 - 14*a - 15)/(a**2 - 12*a - 45)" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/2", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 2", "problem_latex": "$\\dfrac{12m^{2}n}{15mn^{2}}$.", "markdown": "$\\dfrac{12m^{2}n}{15mn^{2}}$.", "answer_latex": [ "$\\dfrac{4m}{5n}$." ], "answer_markdown": [ "$\\dfrac{4m}{5n}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "12*m**2*n/(15*m*n**2)", "answer_expr": "4*m/(5*n)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*a/(5*b)" ], "shape": [ "identity: N*a/b" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/3", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 3", "problem_latex": "$\\dfrac{21m^{2}p^{2}}{28mp^{4}}$.", "markdown": "$\\dfrac{21m^{2}p^{2}}{28mp^{4}}$.", "answer_latex": [ "$\\dfrac{3m}{4p^{2}}$." ], "answer_markdown": [ "$\\dfrac{3m}{4p^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "21*m**2*p**2/(28*m*p**4)", "answer_expr": "3*m/(4*p**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*a/(4*b**2)" ], "shape": [ "identity: N*a*b**N" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/4", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 4", "problem_latex": "$\\dfrac{3x^{3}y^{2}z}{6xy^{3}z^{2}}$.", "markdown": "$\\dfrac{3x^{3}y^{2}z}{6xy^{3}z^{2}}$.", "answer_latex": [ "$\\dfrac{x^{2}}{2yz}$." ], "answer_markdown": [ "$\\dfrac{x^{2}}{2yz}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3*x**3*y**2*z/(6*x*y**3*z**2)", "answer_expr": "x**2/(2*y*z)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**2/(2*b*c)" ], "shape": [ "identity: N*a**N/(b*c)" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/5", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 5", "problem_latex": "$\\dfrac{5a^{3}b^{3}c^{3}}{15c^{5}}$.", "markdown": "$\\dfrac{5a^{3}b^{3}c^{3}}{15c^{5}}$.", "answer_latex": [ "$\\dfrac{a^{3}b^{3}}{3c^{2}}$." ], "answer_markdown": [ "$\\dfrac{a^{3}b^{3}}{3c^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "5*a**3*b**3*c**3/(15*c**5)", "answer_expr": "a**3*b**3/(3*c**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**3*b**3/(3*c**2)" ], "shape": [ "identity: N*a**N*b**N*c**N" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/6", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 6", "problem_latex": "$\\dfrac{34x^{3}y^{4}z^{5}}{51x^{2}y^{3}z^{5}}$.", "markdown": "$\\dfrac{34x^{3}y^{4}z^{5}}{51x^{2}y^{3}z^{5}}$.", "answer_latex": [ "$\\dfrac{2xy}{3}$." ], "answer_markdown": [ "$\\dfrac{2xy}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "34*x**3*y**4*z**5/(51*x**2*y**3*z**5)", "answer_expr": "2*x*y/3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a*b/3" ], "shape": [ "identity: N*a*b" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-49/2" ], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/7", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 7", "problem_latex": "$\\dfrac{46m^{2}np^{3}}{69mnp^{4}}$.", "markdown": "$\\dfrac{46m^{2}np^{3}}{69mnp^{4}}$.", "answer_latex": [ "$\\dfrac{2m}{3p}$." ], "answer_markdown": [ "$\\dfrac{2m}{3p}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "46*m**2*n*p**3/(69*m*n*p**4)", "answer_expr": "2*m/(3*p)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a/(3*b)" ], "shape": [ "identity: N*a/b" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-48/16" ], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/8", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 8", "problem_latex": "$\\dfrac{39a^{2}b^{3}c^{4}}{52a^{5}bc^{3}}$.", "markdown": "$\\dfrac{39a^{2}b^{3}c^{4}}{52a^{5}bc^{3}}$.", "answer_latex": [ "$\\dfrac{3b^{2}c}{4a^{3}}$." ], "answer_markdown": [ "$\\dfrac{3b^{2}c}{4a^{3}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "39*a**2*b**3*c**4/(52*a**5*b*c**3)", "answer_expr": "3*b**2*c/(4*a**3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*b**2*c/(4*a**3)" ], "shape": [ "identity: N*a**N*b**N*c" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-42/9", "set": "wentworth-first-steps-in-algebra-1894/ex-42", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "90", "location": "Exercise 42, problem 9", "problem_latex": "$\\dfrac{58xy^{4}z^{6}}{87xy^{2}z^{2}}$.", "markdown": "$\\dfrac{58xy^{4}z^{6}}{87xy^{2}z^{2}}$.", "answer_latex": [ "$\\dfrac{2y^{2}z^{4}}{3}$." ], "answer_markdown": [ "$\\dfrac{2y^{2}z^{4}}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "58*x*y**4*z**6/(87*x*y**2*z**2)", "answer_expr": "2*y**2*z**4/3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**2*b**4/3" ], "shape": [ "identity: N*a**N*b**N" ], "same_problem_in": [], "needs": [ "cas.cancel" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/1", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 1", "problem_latex": "$\\dfrac{a^{2} - b^{2} + 2}{a - b}$.", "markdown": "$\\dfrac{a^{2} - b^{2} + 2}{a - b}$.", "answer_latex": [ "$a + b + \\dfrac{2}{a - b}$." ], "answer_markdown": [ "$a + b + \\dfrac{2}{a - b}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - b**2 + 2)/(a - b)", "answer_expr": "a + b + 2/(a - b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**2 + x**2 + 2)/(-a + x)" ], "shape": [ "identity: (N - a**N + x**N)/(-a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/10", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 10", "problem_latex": "$\\dfrac{x^{5} - x^{4} + 1}{x^{2} - x - 1}$.", "markdown": "$\\dfrac{x^{5} - x^{4} + 1}{x^{2} - x - 1}$.", "answer_latex": [ "$x^{3} + x + 1 + \\dfrac{2x + 2}{x^{2} - x - 1}$." ], "answer_markdown": [ "$x^{3} + x + 1 + \\dfrac{2x + 2}{x^{2} - x - 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**5 - x**4 + 1)/(x**2 - x - 1)", "answer_expr": "x**3 + x + 1 + (2*x + 2)/(x**2 - x - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**5 - x**4 + 1)/(x**2 - x - 1)" ], "shape": [ "identity: 1/(-x + x**N - 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/2", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 2", "problem_latex": "$\\dfrac{a^{2} - b^{2} - 2}{a + b}$.", "markdown": "$\\dfrac{a^{2} - b^{2} - 2}{a + b}$.", "answer_latex": [ "$a - b - \\dfrac{2}{a + b}$." ], "answer_markdown": [ "$a - b - \\dfrac{2}{a + b}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - b**2 - 2)/(a + b)", "answer_expr": "a - b - 2/(a + b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-a**2 + x**2 - 2)/(a + x)" ], "shape": [ "identity: (N - a**N + x**N)/(a + x)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/3", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 3", "problem_latex": "$\\dfrac{a^{3} - 2a^{2} + 2a + 1}{a^{2} - a - 1}$.", "markdown": "$\\dfrac{a^{3} - 2a^{2} + 2a + 1}{a^{2} - a - 1}$.", "answer_latex": [ "$a - 1 + \\dfrac{2a}{a^{2} - a - 1}$." ], "answer_markdown": [ "$a - 1 + \\dfrac{2a}{a^{2} - a - 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 - 2*a**2 + 2*a + 1)/(a**2 - a - 1)", "answer_expr": "a - 1 + 2*a/(a**2 - a - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**3 - 2*x**2 + 2*x + 1)/(x**2 - x - 1)" ], "shape": [ "identity: (N*x + N*x**N + x**N + 1)/(-x + x**N - 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/4", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 4", "problem_latex": "$\\dfrac{2x^{2} - 2x + 1}{x + 1}$.", "markdown": "$\\dfrac{2x^{2} - 2x + 1}{x + 1}$.", "answer_latex": [ "$2x - 4 + \\dfrac{5}{x + 1}$." ], "answer_markdown": [ "$2x - 4 + \\dfrac{5}{x + 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x**2 - 2*x + 1)/(x + 1)", "answer_expr": "2*x - 4 + 5/(x + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*x**2 - 2*x + 1)/(x + 1)" ], "shape": [ "identity: (N*x + N*x**N + 1)/(x + 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/5", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 5", "problem_latex": "$\\dfrac{8x^{3}}{2x + 1}$.", "markdown": "$\\dfrac{8x^{3}}{2x + 1}$.", "answer_latex": [ "$4x^{2} - 2x + 1 - \\dfrac{1}{2x + 1}$." ], "answer_markdown": [ "$4x^{2} - 2x + 1 - \\dfrac{1}{2x + 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "8*x**3/(2*x + 1)", "answer_expr": "4*x**2 - 2*x + 1 - 1/(2*x + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 8*x**3/(2*x + 1)" ], "shape": [ "identity: N*x**N/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/6", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 6", "problem_latex": "$\\dfrac{5x^{3} + 9x^{2} + 3}{x^{2} + x - 1}$.", "markdown": "$\\dfrac{5x^{3} + 9x^{2} + 3}{x^{2} + x - 1}$.", "answer_latex": [ "$5x + 4 + \\dfrac{x + 7}{x^{2} + x - 1}$." ], "answer_markdown": [ "$5x + 4 + \\dfrac{x + 7}{x^{2} + x - 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(5*x**3 + 9*x**2 + 3)/(x**2 + x - 1)", "answer_expr": "5*x + 4 + (x + 7)/(x**2 + x - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (5*x**3 + 9*x**2 + 3)/(x**2 + x - 1)" ], "shape": [ "identity: (2*N*x**N + N)/(x + x**N - 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/7", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 7", "problem_latex": "$\\dfrac{a^{3} + a^{2} + 7a - 2}{a^{2} + a + 2}$.", "markdown": "$\\dfrac{a^{3} + a^{2} + 7a - 2}{a^{2} + a + 2}$.", "answer_latex": [ "$a + \\dfrac{5a - 2}{a^{2} + a + 2}$." ], "answer_markdown": [ "$a + \\dfrac{5a - 2}{a^{2} + a + 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**3 + a**2 + 7*a - 2)/(a**2 + a + 2)", "answer_expr": "a + (5*a - 2)/(a**2 + a + 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**3 + x**2 + 7*x - 2)/(x**2 + x + 2)" ], "shape": [ "identity: (N*x + N + 2*x**N)/(N + x + x**N)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/8", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 8", "problem_latex": "$\\dfrac{y^{4} + y^{2}x^{2} + x^{4}}{y^{2} + yx + x^{2}}$.", "markdown": "$\\dfrac{y^{4} + y^{2}x^{2} + x^{4}}{y^{2} + yx + x^{2}}$.", "answer_latex": [ "$y^{2} - yx + x^{2}$." ], "answer_markdown": [ "$y^{2} - yx + x^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(y**4 + y**2*x**2 + x**4)/(y**2 + y*x + x**2)", "answer_expr": "y**2 - y*x + x**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**4 + a**2*x**2 + x**4)/(a**2 + a*x + x**2)" ], "shape": [ "identity: (a**N*x**N + a**N + x**N)/(a*x + a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-43/9", "set": "wentworth-first-steps-in-algebra-1894/ex-43", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "91", "location": "Exercise 43, problem 9", "problem_latex": "$\\dfrac{x^{4} - 3x^{3} + x - 1}{x^{2} + x + 1}$.", "markdown": "$\\dfrac{x^{4} - 3x^{3} + x - 1}{x^{2} + x + 1}$.", "answer_latex": [ "$x^{2} - 4x + 3 + \\dfrac{2x - 4}{x^{2} + x + 1}$." ], "answer_markdown": [ "$x^{2} - 4x + 3 + \\dfrac{2x - 4}{x^{2} + x + 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**4 - 3*x**3 + x - 1)/(x**2 + x + 1)", "answer_expr": "x**2 - 4*x + 3 + (2*x - 4)/(x**2 + x + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**4 - 3*x**3 + x - 1)/(x**2 + x + 1)" ], "shape": [ "identity: (N*x**N + x + x**N - 1)/(x + x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/1", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 1", "problem_latex": " $x - y + \\dfrac{2xy}{x - y}$.", "markdown": "$x - y + \\dfrac{2xy}{x - y}$.", "answer_latex": [ " $\\dfrac{x^{2} + y^{2}}{x - y}$." ], "answer_markdown": [ "$\\dfrac{x^{2} + y^{2}}{x - y}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x - y + 2*x*y/(x - y)", "answer_expr": "(x**2 + y**2)/(x - y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a*x/(-a + x) - a + x" ], "shape": [ "identity: N*a*x/(-a + x) - a + x" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/10", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 10", "problem_latex": " $a^{2} + 2a - 5 - \\dfrac{2a - 1}{3a^{2} + 1}$.", "markdown": "$a^{2} + 2a - 5 - \\dfrac{2a - 1}{3a^{2} + 1}$.", "answer_latex": [ " $\\dfrac{3a^{4} + 6a^{3} - 14a^{2} - 4}{3a^{2} + 1}$." ], "answer_markdown": [ "$\\dfrac{3a^{4} + 6a^{3} - 14a^{2} - 4}{3a^{2} + 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2 + 2*a - 5 - (2*a - 1)/(3*a**2 + 1)", "answer_expr": "(3*a**4 + 6*a**3 - 14*a**2 - 4)/(3*a**2 + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2 + 2*x - (2*x - 1)/(3*x**2 + 1) - 5" ], "shape": [ "identity: N*x + N + x**N - (N*x - 1)/(N*x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/2", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 2", "problem_latex": " $x + y - \\dfrac{2xy}{x + y}$.", "markdown": "$x + y - \\dfrac{2xy}{x + y}$.", "answer_latex": [ " $\\dfrac{x^{2} + y^{2}}{x + y}$." ], "answer_markdown": [ "$\\dfrac{x^{2} + y^{2}}{x + y}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x + y - 2*x*y/(x + y)", "answer_expr": "(x**2 + y**2)/(x + y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -2*a*x/(a + x) + a + x" ], "shape": [ "identity: N*a*x/(a + x) + a + x" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/3", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 3", "problem_latex": " $1 - \\dfrac{x - y}{x + y}$.", "markdown": "$1 - \\dfrac{x - y}{x + y}$.", "answer_latex": [ " $\\dfrac{2y}{x + y}$." ], "answer_markdown": [ "$\\dfrac{2y}{x + y}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1 - (x - y)/(x + y)", "answer_expr": "2*y/(x + y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -(-a + x)/(a + x) + 1" ], "shape": [ "identity: -(-a + x)/(a + x) + 1" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/4", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 4", "problem_latex": " $a - x - \\dfrac{a^{2} + x^{2}}{a - x}$.", "markdown": "$a - x - \\dfrac{a^{2} + x^{2}}{a - x}$.", "answer_latex": [ " $-\\dfrac{2ax}{a - x}$." ], "answer_markdown": [ "$-\\dfrac{2ax}{a - x}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "a - x - (a**2 + x**2)/(a - x)", "answer_expr": "-2*a*x/(a - x)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a - x - (a**2 + x**2)/(a - x)" ], "shape": [ "identity: a - x - (a**N + x**N)/(a - x)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/5", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 5", "problem_latex": " $x + 2 - \\dfrac{x^{2} - 4}{x - 3}$.", "markdown": "$x + 2 - \\dfrac{x^{2} - 4}{x - 3}$.", "answer_latex": [ " $-\\dfrac{x + 2}{x - 3}$." ], "answer_markdown": [ "$-\\dfrac{x + 2}{x - 3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x + 2 - (x**2 - 4)/(x - 3)", "answer_expr": "-(x + 2)/(x - 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x + 2 - (x**2 - 4)/(x - 3)" ], "shape": [ "identity: N + x - (N + x**N)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/6", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 6", "problem_latex": " $\\dfrac{x - 3}{x - 2} - 2x + 1$.", "markdown": "$\\dfrac{x - 3}{x - 2} - 2x + 1$.", "answer_latex": [ " $-\\dfrac{2x^{2} - 6x + 5}{x - 2}$." ], "answer_markdown": [ "$-\\dfrac{2x^{2} - 6x + 5}{x - 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x - 3)/(x - 2) - 2*x + 1", "answer_expr": "-(2*x**2 - 6*x + 5)/(x - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -2*x + (x - 3)/(x - 2) + 1" ], "shape": [ "identity: N*x + 2" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/7", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 7", "problem_latex": " $\\dfrac{x + 3}{x + 2} + x^{2} - x - 1$.", "markdown": "$\\dfrac{x + 3}{x + 2} + x^{2} - x - 1$.", "answer_latex": [ " $\\dfrac{x^{3} + x^{2} - 2x + 1}{x + 2}$." ], "answer_markdown": [ "$\\dfrac{x^{3} + x^{2} - 2x + 1}{x + 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 3)/(x + 2) + x**2 - x - 1", "answer_expr": "(x**3 + x**2 - 2*x + 1)/(x + 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2 - x - 1 + (x + 3)/(x + 2)" ], "shape": [ "identity: -x + x**N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/8", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 8", "problem_latex": " $2a - 1 + \\dfrac{3 - 4a}{a - 3}$.", "markdown": "$2a - 1 + \\dfrac{3 - 4a}{a - 3}$.", "answer_latex": [ " $\\dfrac{2a^{2} - 11a + 6}{a - 3}$." ], "answer_markdown": [ "$\\dfrac{2a^{2} - 11a + 6}{a - 3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "2*a - 1 + (3 - 4*a)/(a - 3)", "answer_expr": "(2*a**2 - 11*a + 6)/(a - 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*x + (-4*x + 3)/(x - 3) - 1" ], "shape": [ "identity: N*x - 1 + (N*x + N)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-44/9", "set": "wentworth-first-steps-in-algebra-1894/ex-44", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "92", "location": "Exercise 44, problem 9", "problem_latex": " $1 - 2a^{2} - \\dfrac{a^{2} - a + 2}{a - 1}$.", "markdown": "$1 - 2a^{2} - \\dfrac{a^{2} - a + 2}{a - 1}$.", "answer_latex": [ " $\\dfrac{-2a^{3} + a^{2} + 2a-3}{a - 1}$." ], "answer_markdown": [ "$\\dfrac{-2a^{3} + a^{2} + 2a-3}{a - 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1 - 2*a**2 - (a**2 - a + 2)/(a - 1)", "answer_expr": "(-2*a**3 + a**2 + 2*a - 3)/(a - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -2*x**2 + 1 - (x**2 - x + 2)/(x - 1)" ], "shape": [ "identity: N*x**N + 1 - (N - x + x**N)/(x - 1)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/1a", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 1, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 1a", "problem_latex": "$\\dfrac{x}{x - a}$, $\\dfrac{x^{2}}{x^{2} - a^{2}}$.", "markdown": "$\\dfrac{x}{x - a}$, $\\dfrac{x^{2}}{x^{2} - a^{2}}$.", "answer_latex": [ "$\\dfrac{x(x + a)}{(x + a)(x - a)}$" ], "answer_markdown": [ "$\\dfrac{x(x + a)}{(x + a)(x - a)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x/(x - a)", "answer_expr": "x*(x + a)/((x + a)*(x - a))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x/(-a + x)" ], "shape": [ "identity: x/(-a + x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/1b", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 1, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 1b", "problem_latex": "$\\dfrac{x}{x - a}$, $\\dfrac{x^{2}}{x^{2} - a^{2}}$.", "markdown": "$\\dfrac{x}{x - a}$, $\\dfrac{x^{2}}{x^{2} - a^{2}}$.", "answer_latex": [ "$\\dfrac{x^{2}}{(x + a)(x - a)}$" ], "answer_markdown": [ "$\\dfrac{x^{2}}{(x + a)(x - a)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "x**2/(x**2 - a**2)", "answer_expr": "x**2/((x + a)*(x - a))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2/(-a**2 + x**2)" ], "shape": [ "identity: x**N/(-a**N + x**N)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-45/2b" ], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/2a", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 2, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 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$\\dfrac{a^{2}}{a^{2} - b^{2}}$.", "markdown": "$\\dfrac{a}{a + b}$, $\\dfrac{a^{2}}{a^{2} - b^{2}}$.", "answer_latex": [ "$\\dfrac{a^{2}}{(a + b)(a - b)}$" ], "answer_markdown": [ "$\\dfrac{a^{2}}{(a + b)(a - b)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2/(a**2 - b**2)", "answer_expr": "a**2/((a + b)*(a - b))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2/(-a**2 + x**2)" ], "shape": [ "identity: x**N/(-a**N + x**N)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-45/1b" ], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/3a", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 3, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 3a", "problem_latex": "$\\dfrac{1}{1 + 2a}$, $\\dfrac{1}{1 - 4a^{2}}$.", "markdown": "$\\dfrac{1}{1 + 2a}$, $\\dfrac{1}{1 - 4a^{2}}$.", "answer_latex": [ "$\\dfrac{1 - 2a}{(1 + 2a)(1 - 2a)}$" ], "answer_markdown": [ "$\\dfrac{1 - 2a}{(1 + 2a)(1 - 2a)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(1 + 2*a)", "answer_expr": "(1 - 2*a)/((1 + 2*a)*(1 - 2*a))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 1/(2*x + 1)" ], "shape": [ "identity: 1/(N*x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/3b", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 3, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 3b", "problem_latex": "$\\dfrac{1}{1 + 2a}$, $\\dfrac{1}{1 - 4a^{2}}$.", "markdown": "$\\dfrac{1}{1 + 2a}$, $\\dfrac{1}{1 - 4a^{2}}$.", "answer_latex": [ "$\\dfrac{1}{(1 + 2a)(1 - 2a)}$" ], "answer_markdown": [ "$\\dfrac{1}{(1 + 2a)(1 - 2a)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(1 - 4*a**2)", "answer_expr": "1/((1 + 2*a)*(1 - 2*a))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 1/(-4*x**2 + 1)" ], "shape": [ "identity: 1/(N*x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/4a", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 4, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 4a", "problem_latex": "$\\dfrac{9}{16 - x^{2}}$, $\\dfrac{4 - x}{4 + x}$.", "markdown": "$\\dfrac{9}{16 - x^{2}}$, $\\dfrac{4 - x}{4 + x}$.", "answer_latex": [ "$\\dfrac{9}{(4 + x)(4 - x)}$" ], "answer_markdown": [ "$\\dfrac{9}{(4 + x)(4 - x)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "9/(16 - x**2)", "answer_expr": "9/((4 + x)*(4 - x))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 9/(-x**2 + 16)" ], "shape": [ "identity: N/(N - x**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/4b", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 4, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 4b", "problem_latex": "$\\dfrac{9}{16 - x^{2}}$, $\\dfrac{4 - x}{4 + x}$.", "markdown": "$\\dfrac{9}{16 - x^{2}}$, $\\dfrac{4 - x}{4 + x}$.", "answer_latex": [ "$\\dfrac{(4 - x)^{2}}{(4 + x)(4 - x)}$" ], "answer_markdown": [ "$\\dfrac{(4 - x)^{2}}{(4 + x)(4 - x)}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4 - x)/(4 + x)", "answer_expr": "(4 - x)**2/((4 + x)*(4 - x))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-x + 4)/(x + 4)" ], "shape": [ "identity: (N - x)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/5a", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 5, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 5a", "problem_latex": "$\\dfrac{a^{2}}{27 - a^{3}}$, $\\dfrac{a}{3 - a}$.", "markdown": "$\\dfrac{a^{2}}{27 - a^{3}}$, $\\dfrac{a}{3 - a}$.", "answer_latex": [ "$\\dfrac{a^{2}}{(3 - a)(9 + 3a + a^{2})}$" ], "answer_markdown": [ "$\\dfrac{a^{2}}{(3 - a)(9 + 3a + a^{2})}$" ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "a**2/(27 - a**3)", "answer_expr": "a**2/((3 - a)*(9 + 3*a + a**2))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2/(-x**3 + 27)" ], "shape": [ "identity: x**N/(N - x**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-45/5b", "set": "wentworth-first-steps-in-algebra-1894/ex-45", "number": 5, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "94", "location": "Exercise 45, problem 5b", "problem_latex": "$\\dfrac{a^{2}}{27 - a^{3}}$, $\\dfrac{a}{3 - a}$.", "markdown": "$\\dfrac{a^{2}}{27 - a^{3}}$, $\\dfrac{a}{3 - a}$.", "answer_latex": [ "$\\dfrac{a(9 + 3a + a^{2})}{(3 - a)(9 + 3a + 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], "answer_markdown": [ "$\\dfrac{4x + 2}{5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 1)/2 + (x - 3)/5 + (x + 5)/10", "answer_expr": "(4*x + 2)/5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*x/5 + 2/5" ], "shape": [ "identity: N*x + N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/2", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 2", "problem_latex": "$\\dfrac{2x - 1}{3} + \\dfrac{x + 5}{4} + \\dfrac{x - 4}{6}$.", "markdown": "$\\dfrac{2x - 1}{3} + \\dfrac{x + 5}{4} + \\dfrac{x - 4}{6}$.", "answer_latex": [ "$\\dfrac{13x + 3}{12}$." ], "answer_markdown": [ "$\\dfrac{13x + 3}{12}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*x - 1)/3 + (x + 5)/4 + (x - 4)/6", "answer_expr": "(13*x + 3)/12" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 13*x/12 + 1/4" ], "shape": [ "identity: N*x + N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/3", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 3", "problem_latex": "$\\dfrac{7x - 1}{6} - \\dfrac{3x - 2}{7} + \\dfrac{x - 5}{3}$.", "markdown": "$\\dfrac{7x - 1}{6} - \\dfrac{3x - 2}{7} + \\dfrac{x - 5}{3}$.", "answer_latex": [ "$\\dfrac{5(9x - 13)}{42}$." ], "answer_markdown": [ "$\\dfrac{5(9x - 13)}{42}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(7*x - 1)/6 - (3*x - 2)/7 + (x - 5)/3", "answer_expr": "5*(9*x - 13)/42" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 15*x/14 - 65/42" ], "shape": [ "identity: N*x + N" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/4", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 4", "problem_latex": "$\\dfrac{3x - 2}{9} - \\dfrac{x - 2}{6} + \\dfrac{5x + 3}{4}$.", "markdown": "$\\dfrac{3x - 2}{9} - \\dfrac{x - 2}{6} + \\dfrac{5x + 3}{4}$.", "answer_latex": [ "$\\dfrac{51x + 31}{36}$." ], "answer_markdown": [ "$\\dfrac{51x + 31}{36}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x - 2)/9 - (x - 2)/6 + (5*x + 3)/4", "answer_expr": "(51*x + 31)/36" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 17*x/12 + 31/36" ], "shape": [ "identity: N*x + N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/5", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 5", "problem_latex": "$\\dfrac{x - 1}{6} - \\dfrac{x - 3}{3} + \\dfrac{x - 5}{2}$.", "markdown": "$\\dfrac{x - 1}{6} - \\dfrac{x - 3}{3} + \\dfrac{x - 5}{2}$.", "answer_latex": [ "$\\dfrac{x - 5}{3}$." ], "answer_markdown": [ "$\\dfrac{x - 5}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x - 1)/6 - (x - 3)/3 + (x - 5)/2", "answer_expr": "(x - 5)/3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x/3 - 5/3" ], "shape": [ "identity: N*x + N" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/6", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 6", "problem_latex": "$\\dfrac{x - 2y}{2x} + \\dfrac{x + 5y}{4x} - \\dfrac{x + 7y}{8x}$.", "markdown": "$\\dfrac{x - 2y}{2x} + \\dfrac{x + 5y}{4x} - \\dfrac{x + 7y}{8x}$.", "answer_latex": [ "$\\dfrac{5(x - y)}{8x}$." ], "answer_markdown": [ "$\\dfrac{5(x - y)}{8x}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x - 2*y)/(2*x) + (x + 5*y)/(4*x) - (x + 7*y)/(8*x)", "answer_expr": "5*(x - y)/(8*x)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (-2*a + x)/(2*x) + (5*a + x)/(4*x) - (7*a + x)/(8*x)" ], "shape": [ "identity: 3*N*(N*a + x)/x" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/7", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 7", "problem_latex": "$\\dfrac{5x - 11}{3} - \\dfrac{2x - 1}{10} - \\dfrac{11x - 5}{15}$.", "markdown": "$\\dfrac{5x - 11}{3} - \\dfrac{2x - 1}{10} - \\dfrac{11x - 5}{15}$.", "answer_latex": [ "$\\dfrac{22x - 97}{30}$." ], "answer_markdown": [ "$\\dfrac{22x - 97}{30}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(5*x - 11)/3 - (2*x - 1)/10 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3)/(3*x) - (x**2 - 6*x)/(5*x**2) - (7*x**2 - x**3)/(15*x**3)", "answer_expr": "(3*x - 4)/(15*x)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x - 3)/(3*x) - (x**2 - 6*x)/(5*x**2) - (-x**3 + 7*x**2)/(15*x**3)" ], "shape": [ "identity: N*x**N*(N*x + x**N) + N*x**N*(N*x**N - x**N) + N*(N + x)/x" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-46/9", "set": "wentworth-first-steps-in-algebra-1894/ex-46", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "95", "location": "Exercise 46, problem 9", "problem_latex": "$\\dfrac{ac - b^{2}}{ac} - \\dfrac{ab - c^{2}}{ab} + \\dfrac{a^{2} - bc}{bc}$.", "markdown": "$\\dfrac{ac - b^{2}}{ac} - \\dfrac{ab - c^{2}}{ab} + \\dfrac{a^{2} - bc}{bc}$.", "answer_latex": [ "$\\dfrac{a^{3} - b^{3} + c^{3} - 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"wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "96", "location": "Exercise 47, problem 13", "problem_latex": "$\\dfrac{x + 1}{x + 2} + \\dfrac{x - 2}{x - 3} + \\dfrac{2x + 7}{x^{2} - x - 6}$.", "markdown": "$\\dfrac{x + 1}{x + 2} + \\dfrac{x - 2}{x - 3} + \\dfrac{2x + 7}{x^{2} - x - 6}$.", "answer_latex": [ "$\\dfrac{2x^{2}}{(x + 2)(x - 3)}$." ], "answer_markdown": [ "$\\dfrac{2x^{2}}{(x + 2)(x - 3)}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 1)/(x + 2) + (x - 2)/(x - 3) + (2*x + 7)/(x**2 - x - 6)", "answer_expr": "2*x**2/((x + 2)*(x - 3))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x + 1)/(x + 2) + (2*x + 7)/(x**2 - x - 6) + (x - 2)/(x - 3)" ], "shape": [ "identity: (N*x + N)/(N - x + x**N) + 1 + (x + 1)/(N + x)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, 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"core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-47/7", "set": "wentworth-first-steps-in-algebra-1894/ex-47", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "96", "location": "Exercise 47, problem 7", "problem_latex": "$\\dfrac{7}{9 - a^{2}} - \\dfrac{1}{3 + a} - \\dfrac{1}{3 - a}$.", "markdown": "$\\dfrac{7}{9 - a^{2}} - \\dfrac{1}{3 + a} - \\dfrac{1}{3 - a}$.", "answer_latex": [ "$\\dfrac{1}{9 - a^{2}}$." ], "answer_markdown": [ "$\\dfrac{1}{9 - a^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "7/(9 - a**2) - 1/(3 + a) - 1/(3 - a)", "answer_expr": "1/(9 - a**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -1/(x + 3) + 7/(-x**2 + 9) - 1/(-x + 3)" ], "shape": [ "identity: N/(N - x**N) - 1/(N + x) - 1/(N - x)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-47/8", "set": "wentworth-first-steps-in-algebra-1894/ex-47", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "96", "location": "Exercise 47, problem 8", "problem_latex": "$\\dfrac{1}{a - b} - \\dfrac{1}{a + b} - \\dfrac{b}{a^{2} - b^{2}}$.", "markdown": "$\\dfrac{1}{a - b} - \\dfrac{1}{a + b} - \\dfrac{b}{a^{2} - b^{2}}$.", "answer_latex": [ "$\\dfrac{b}{a^{2} - b^{2}}$." ], "answer_markdown": [ "$\\dfrac{b}{a^{2} - b^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(a - b) - 1/(a + b) - b/(a**2 - b**2)", "answer_expr": "b/(a**2 - b**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a/(-a**2 + x**2) - 1/(a + x) + 1/(-a + x)" ], "shape": [ "identity: -a/(-a**N + x**N) - 1/(a + x) + 1/(-a + x)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-47/9", "set": "wentworth-first-steps-in-algebra-1894/ex-47", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "96", "location": "Exercise 47, problem 9", "problem_latex": "$\\dfrac{2}{x - 2} - \\dfrac{2}{x + 2} + \\dfrac{5x}{x^{2} - 4}$.", "markdown": "$\\dfrac{2}{x - 2} - \\dfrac{2}{x + 2} + \\dfrac{5x}{x^{2} - 4}$.", "answer_latex": [ "$\\dfrac{5x + 8}{x^{2} - 4}$." ], "answer_markdown": [ "$\\dfrac{5x + 8}{x^{2} - 4}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "2/(x - 2) - 2/(x + 2) + 5*x/(x**2 - 4)", "answer_expr": "(5*x + 8)/(x**2 - 4)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 5*x/(x**2 - 4) - 2/(x + 2) + 2/(x - 2)" ], "shape": [ "identity: N*x/(N + x**N) + 2*N/(N + x)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48/1", "set": "wentworth-first-steps-in-algebra-1894/ex-48", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Exercise 48, problem 1", "problem_latex": "$\\dfrac{1}{2x + 1} + \\dfrac{1}{2x - 1} - \\dfrac{4x}{4x^{2} - 1}$.", "markdown": "$\\dfrac{1}{2x + 1} + \\dfrac{1}{2x - 1} - \\dfrac{4x}{4x^{2} - 1}$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(2*x + 1) + 1/(2*x - 1) - 4*x/(4*x**2 - 1)", "answer_expr": "0" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -4*x/(4*x**2 - 1) + 1/(2*x + 1) + 1/(2*x - 1)" ], "shape": [ "identity: N*x/(N*x**N - 1) + 1/(N*x + 1) + 1/(N*x - 1)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48/2", "set": "wentworth-first-steps-in-algebra-1894/ex-48", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Exercise 48, problem 2", "problem_latex": "$\\dfrac{a^{2} + b^{2}}{a^{2} - b^{2}} + \\dfrac{a}{a + b} - \\dfrac{b}{a - b}$.", "markdown": "$\\dfrac{a^{2} + b^{2}}{a^{2} - b^{2}} + \\dfrac{a}{a + b} - \\dfrac{b}{a - b}$.", "answer_latex": [ "$\\dfrac{2a}{a + b}$." ], "answer_markdown": [ "$\\dfrac{2a}{a + b}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 + b**2)/(a**2 - b**2) + a/(a + b) - b/(a - b)", "answer_expr": "2*a/(a + b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -a/(-a + x) + x/(a + x) + (a**2 + x**2)/(-a**2 + x**2)" ], "shape": [ "identity: -a/(-a + x) + x/(a + x) + (a**N + x**N)/(-a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48/3", "set": "wentworth-first-steps-in-algebra-1894/ex-48", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Exercise 48, problem 3", "problem_latex": "$\\dfrac{3a}{1 - a^{2}} + \\dfrac{2}{1 - a} - \\dfrac{2}{1 + a}$.", "markdown": "$\\dfrac{3a}{1 - a^{2}} + \\dfrac{2}{1 - a} - \\dfrac{2}{1 + a}$.", "answer_latex": [ "$\\dfrac{7a}{1 - a^{2}}$." ], "answer_markdown": [ "$\\dfrac{7a}{1 - a^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3*a/(1 - a**2) + 2/(1 - a) - 2/(1 + a)", "answer_expr": "7*a/(1 - a**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 3*x/(-x**2 + 1) - 2/(x + 1) + 2/(-x + 1)" ], "shape": [ "identity: N*x/(-x**N + 1) + N/(x + 1) + N/(-x + 1)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48/4", "set": "wentworth-first-steps-in-algebra-1894/ex-48", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Exercise 48, problem 4", "problem_latex": "$\\dfrac{1}{2x + 5y} - \\dfrac{3x}{4x^{2} - 25y^{2}} + \\dfrac{1}{2x + 5y}$.", "markdown": "$\\dfrac{1}{2x + 5y} - \\dfrac{3x}{4x^{2} - 25y^{2}} + \\dfrac{1}{2x + 5y}$.", "answer_latex": [ "$\\dfrac{x - 10y}{4x^{2} - 25y^{2}}$." ], "answer_markdown": [ "$\\dfrac{x - 10y}{4x^{2} - 25y^{2}}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(2*x + 5*y) - 3*x/(4*x**2 - 25*y**2) + 1/(2*x + 5*y)", "answer_expr": "(x - 10*y)/(4*x**2 - 25*y**2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -3*x/(-25*a**2 + 4*x**2) + 2/(5*a + 2*x)" ], "shape": [ "identity: N*x/(N*a**N + N*x**N) + N/(N*a + N*x)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48/5", "set": "wentworth-first-steps-in-algebra-1894/ex-48", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Exercise 48, problem 5", "problem_latex": "$\\dfrac{1}{x + 4y} - \\dfrac{8y}{x^{2} - 16y^{2}} + \\dfrac{1}{x - 4y}$.", "markdown": "$\\dfrac{1}{x + 4y} - \\dfrac{8y}{x^{2} - 16y^{2}} + \\dfrac{1}{x - 4y}$.", "answer_latex": [ "$\\dfrac{2}{x + 4y}$." ], "answer_markdown": [ "$\\dfrac{2}{x + 4y}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(x + 4*y) - 8*y/(x**2 - 16*y**2) + 1/(x - 4*y)", "answer_expr": "2/(x + 4*y)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -8*a/(-16*a**2 + x**2) + 1/(4*a + x) + 1/(-4*a + x)" ], "shape": [ "identity: N*a/(N*a**N + x**N) + 2/(N*a + x)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-48/6", "set": "wentworth-first-steps-in-algebra-1894/ex-48", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "97", "location": "Exercise 48, problem 6", "problem_latex": "$\\dfrac{3}{2x - 3} - \\dfrac{2}{2x + 3} - \\dfrac{3}{4x^{2} - 9}$.", "markdown": "$\\dfrac{3}{2x - 3} - \\dfrac{2}{2x + 3} - \\dfrac{3}{4x^{2} - 9}$.", "answer_latex": [ "$\\dfrac{2(x + 6)}{4x^{2} - 9}$." ], "answer_markdown": [ "$\\dfrac{2(x + 6)}{4x^{2} - 9}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3/(2*x - 3) - 2/(2*x + 3) - 3/(4*x**2 - 9)", "answer_expr": "2*(x + 6)/(4*x**2 - 9)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: -3/(4*x**2 - 9) - 2/(2*x + 3) + 3/(2*x - 3)" ], "shape": [ "identity: N/(N*x**N + N) + 2*N/(N*x + N)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/1", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 1", "problem_latex": "$\\dfrac{15a^{2}}{7b^{2}} × \\dfrac{28ab}{9a^{3}c}$.", "markdown": "$\\dfrac{15a^{2}}{7b^{2}} × \\dfrac{28ab}{9a^{3}c}$.", "answer_latex": [ "$\\dfrac{20}{3bc}$." ], "answer_markdown": [ "$\\dfrac{20}{3bc}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(15*a**2/(7*b**2))*(28*a*b/(9*a**3*c))", "answer_expr": "20/(3*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 20/(3*a*b)" ], "shape": [ "identity: N/(a*b)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/10", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 10", "problem_latex": "$\\dfrac{a^{2} - 100}{a^{2} - 9} × \\dfrac{a - 3}{a - 10}$.", "markdown": "$\\dfrac{a^{2} - 100}{a^{2} - 9} × \\dfrac{a - 3}{a - 10}$.", "answer_latex": [ "$\\dfrac{a + 10}{a + 3}$." ], "answer_markdown": [ "$\\dfrac{a + 10}{a + 3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((a**2 - 100)/(a**2 - 9))*((a - 3)/(a - 10))", "answer_expr": "(a + 10)/(a + 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a - 3)*(a**2 - 100)/((a - 10)*(a**2 - 9))" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/11", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 11", "problem_latex": "$\\dfrac{9x^{2} - 4y^{2}}{x^{2} - 4} × \\dfrac{x + 2}{3x - 2y}$.", "markdown": "$\\dfrac{9x^{2} - 4y^{2}}{x^{2} - 4} × \\dfrac{x + 2}{3x - 2y}$.", "answer_latex": [ "$\\dfrac{3x + 2y}{x - 2}$." ], "answer_markdown": [ "$\\dfrac{3x + 2y}{x - 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((9*x**2 - 4*y**2)/(x**2 - 4))*((x + 2)/(3*x - 2*y))", "answer_expr": "(3*x + 2*y)/(x - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a + 2)*(9*a**2 - 4*b**2)/((3*a - 2*b)*(a**2 - 4))" ], "shape": [ "identity: (N + a)*(N*a**N + N*b**N)/((N + a**N)*(N*a + N*b))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/12", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 12", "problem_latex": "$\\dfrac{25a^{2} - b^{2}}{16a^{2} - 9b^{2}} ÷ \\dfrac{5a - b}{4a - 3b}$.", "markdown": "$\\dfrac{25a^{2} - b^{2}}{16a^{2} - 9b^{2}} ÷ \\dfrac{5a - b}{4a - 3b}$.", "answer_latex": [ "$\\dfrac{5a + b}{4a + 3b}$." ], "answer_markdown": [ "$\\dfrac{5a + b}{4a + 3b}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((25*a**2 - b**2)/(16*a**2 - 9*b**2))/((5*a - b)/(4*a - 3*b))", "answer_expr": "(5*a + b)/(4*a + 3*b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (4*a - 3*b)*(25*a**2 - b**2)/((5*a - b)*(16*a**2 - 9*b**2))" ], "shape": [ "identity: (N*a + N*b)*(N*a**N - b**N)/((N*a - b)*(N*a**N + N*b**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/13", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 13", "problem_latex": "$\\dfrac{x^{2} - 49}{(a + b)^{2} - c^{2}} ÷ \\dfrac{x + 7}{(a + b) - c}$.", "markdown": "$\\dfrac{x^{2} - 49}{(a + b)^{2} - c^{2}} ÷ \\dfrac{x + 7}{(a + b) - c}$.", "answer_latex": [ "$\\dfrac{x - 7}{a + b + c}$." ], "answer_markdown": [ "$\\dfrac{x - 7}{a + b + c}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((x**2 - 49)/((a + b)**2 - c**2))/((x + 7)/((a + b) - c))", "answer_expr": "(x - 7)/(a + b + c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (d**2 - 49)*(a + b - c)/((-c**2 + (a + b)**2)*(d + 7))" ], "shape": [ "identity: (N + d**N)*(a + b - c)/((N + d)*(-c**N + (a + b)**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/14", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 14", "problem_latex": "$\\dfrac{x^{2} + 2x + 1}{x^{2} - 25} ÷ \\dfrac{x + 1}{x^{2} + 5x}$.", "markdown": "$\\dfrac{x^{2} + 2x + 1}{x^{2} - 25} ÷ \\dfrac{x + 1}{x^{2} + 5x}$.", "answer_latex": [ "$\\dfrac{x(x + 1)}{x - 5}$." ], "answer_markdown": [ "$\\dfrac{x(x + 1)}{x - 5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((x**2 + 2*x + 1)/(x**2 - 25))/((x + 1)/(x**2 + 5*x))", "answer_expr": "x*(x + 1)/(x - 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 5*a)*(a**2 + 2*a + 1)/((a + 1)*(a**2 - 25))" ], "shape": [ "identity: (N*a + a**N)*(N*a + a**N + 1)/((N + a**N)*(a + 1))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/15", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 15", "problem_latex": "$\\dfrac{a^{2} + 3a + 2}{a^{2} + 5a + 6} × \\dfrac{a^{2} + 7a + 12}{a^{2} + 9a + 20}$.", "markdown": "$\\dfrac{a^{2} + 3a + 2}{a^{2} + 5a + 6} × \\dfrac{a^{2} + 7a + 12}{a^{2} + 9a + 20}$.", "answer_latex": [ "$\\dfrac{a + 1}{a + 5}$." ], "answer_markdown": [ "$\\dfrac{a + 1}{a + 5}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((a**2 + 3*a + 2)/(a**2 + 5*a + 6))*((a**2 + 7*a + 12)/(a**2 + 9*a + 20))", "answer_expr": "(a + 1)/(a + 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 3*a + 2)*(a**2 + 7*a + 12)/((a**2 + 5*a + 6)*(a**2 + 9*a + 20))" ], "shape": [ "identity: 1" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/16", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 16", "problem_latex": "$\\dfrac{y^{2} - y-30}{y^{2} - 36} × \\dfrac{y^{2} - y-2}{y^{2} + 3y-10} × \\dfrac{y^{2} + 6y}{y^{2} + y}$.", "markdown": "$\\dfrac{y^{2} - y-30}{y^{2} - 36} × \\dfrac{y^{2} - y-2}{y^{2} + 3y-10} × \\dfrac{y^{2} + 6y}{y^{2} + y}$.", "answer_latex": [ "$1$." ], "answer_markdown": [ "$1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((y**2 - y - 30)/(y**2 - 36))*((y**2 - y - 2)/(y**2 + 3*y - 10))*((y**2 + 6*y)/(y**2 + y))", "answer_expr": "1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 + 6*a)*(a**2 - a - 30)*(a**2 - a - 2)/((a**2 - 36)*(a**2 + a)*(a**2 + 3*a - 10))" ], "shape": [ "identity: (N*a + a**N)*(N - a + a**N)**2/((N + a**N)*(a + a**N)*(N*a + N + a**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/17", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 17", "problem_latex": "$\\dfrac{x^{2} - 2x + 1}{x^{2} - y^{2}} × \\dfrac{x^{2} + 2xy + y^{2}}{x - 1} ÷ \\dfrac{x^{2} - 1}{x^{2} - xy}$.", "markdown": "$\\dfrac{x^{2} - 2x + 1}{x^{2} - y^{2}} × \\dfrac{x^{2} + 2xy + y^{2}}{x - 1} ÷ \\dfrac{x^{2} - 1}{x^{2} - xy}$.", "answer_latex": [ "$\\dfrac{x(x + y)}{x + 1}$." ], "answer_markdown": [ "$\\dfrac{x(x + y)}{x + 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((x**2 - 2*x + 1)/(x**2 - y**2))*((x**2 + 2*x*y + y**2)/(x - 1))/((x**2 - 1)/(x**2 - x*y))", "answer_expr": "x*(x + y)/(x + 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a**2 - a*b)*(a**2 - 2*a + 1)*(a**2 + 2*a*b + b**2)/((a - 1)*(a**2 - 1)*(a**2 - b**2))" ], "shape": [ "identity: (-a*b + a**N)*(N*a + a**N + 1)*(N*a*b + a**N + b**N)/((a - 1)*(a**N - 1)*(a**N - b**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/18", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 18", "problem_latex": "$\\dfrac{a^{2} - b^{2}}{a^{2} - 3ab + 2b^{2}} × \\dfrac{ab - 2b^{2}}{a^{2} + ab} ÷ \\dfrac{(a - b)^{2}}{a(a - b)}$.", "markdown": "$\\dfrac{a^{2} - b^{2}}{a^{2} - 3ab + 2b^{2}} × \\dfrac{ab - 2b^{2}}{a^{2} + ab} ÷ \\dfrac{(a - b)^{2}}{a(a - b)}$.", "answer_latex": [ "$\\dfrac{b}{a - b}$." ], "answer_markdown": [ "$\\dfrac{b}{a - b}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((a**2 - b**2)/(a**2 - 3*a*b + 2*b**2))*((a*b - 2*b**2)/(a**2 + a*b))/((a - b)**2/(a*(a - b)))", "answer_expr": "b/(a - b)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*(a**2 - b**2)*(a*b - 2*b**2)/((a - b)*(a**2 + a*b)*(a**2 - 3*a*b + 2*b**2))" ], "shape": [ "identity: a*(a**N - b**N)*(N*b**N + a*b)/((a - b)*(a*b + a**N)*(N*a*b + N*b**N + a**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/19", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 19", "problem_latex": "$\\dfrac{(a + b)^{2} - c^{2}}{a^{2} + ab-ac} × \\dfrac{a^{2}b^{2}c^{2}}{a^{2} + ab + ac} ÷ \\dfrac{b^{2}c^{2}}{abc}$.", "markdown": "$\\dfrac{(a + b)^{2} - c^{2}}{a^{2} + ab-ac} × \\dfrac{a^{2}b^{2}c^{2}}{a^{2} + ab + ac} ÷ \\dfrac{b^{2}c^{2}}{abc}$.", "answer_latex": [ "$abc$." ], "answer_markdown": [ "$abc$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(((a + b)**2 - c**2)/(a**2 + a*b - a*c))*((a**2*b**2*c**2)/(a**2 + a*b + a*c))/((b**2*c**2)/(a*b*c))", "answer_expr": "a*b*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**3*b*c*(-c**2 + (a + b)**2)/((a**2 + a*b - a*c)*(a**2 + a*b + a*c))" ], "shape": [ "identity: a**N*b*c*(-c**N + (a + b)**N)/((a*b - a*c + a**N)*(a*b + a*c + a**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/2", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 2", "problem_latex": "$\\dfrac{3x^{2}y^{2}z^{3}}{4a^{2}b^{2}c^{2}} × \\dfrac{8a^{3}b^{2}c^{2}}{9x^{2}yz^{3}}$.", "markdown": "$\\dfrac{3x^{2}y^{2}z^{3}}{4a^{2}b^{2}c^{2}} × \\dfrac{8a^{3}b^{2}c^{2}}{9x^{2}yz^{3}}$.", "answer_latex": [ "$\\dfrac{2ay}{3}$." ], "answer_markdown": [ "$\\dfrac{2ay}{3}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(3*x**2*y**2*z**3/(4*a**2*b**2*c**2))*(8*a**3*b**2*c**2/(9*x**2*y*z**3))", "answer_expr": "2*a*y/3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a*b/3" ], "shape": [ "identity: N*a*b" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-42/6" ], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/20", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 20", "problem_latex": "$\\dfrac{x^{2} + 7xy + 10y^{2}}{x^{2} + 6xy + 5y^{2}} × \\dfrac{x + 1}{x^{2} + 4x + 4} ÷ \\dfrac{1}{x + 2}$.", "markdown": "$\\dfrac{x^{2} + 7xy + 10y^{2}}{x^{2} + 6xy + 5y^{2}} × \\dfrac{x + 1}{x^{2} + 4x + 4} ÷ \\dfrac{1}{x + 2}$.", "answer_latex": [ "$\\dfrac{(x + 2y)(x + 1)}{(x + y)(x + 2)}$." ], "answer_markdown": [ "$\\dfrac{(x + 2y)(x + 1)}{(x + y)(x + 2)}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((x**2 + 7*x*y + 10*y**2)/(x**2 + 6*x*y + 5*y**2))*((x + 1)/(x**2 + 4*x + 4))/(1/(x + 2))", "answer_expr": "(x + 2*y)*(x + 1)/((x + y)*(x + 2))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (a + 1)*(a + 2)*(a**2 + 7*a*b + 10*b**2)/((a**2 + 4*a + 4)*(a**2 + 6*a*b + 5*b**2))" ], "shape": [ "identity: (N + a)*(a + 1)/(N*a + N + a**N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/3", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 3", "problem_latex": "$\\dfrac{5m^{2}n^{2}p^{4}}{3x^{2}yz^{3}} × \\dfrac{21xyz^{2}}{20m^{2}n^{2}p^{2}}$.", "markdown": "$\\dfrac{5m^{2}n^{2}p^{4}}{3x^{2}yz^{3}} × \\dfrac{21xyz^{2}}{20m^{2}n^{2}p^{2}}$.", "answer_latex": [ "$\\dfrac{7p^{2}}{4xz}$." ], "answer_markdown": [ "$\\dfrac{7p^{2}}{4xz}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(5*m**2*n**2*p**4/(3*x**2*y*z**3))*(21*x*y*z**2/(20*m**2*n**2*p**2))", "answer_expr": "7*p**2/(4*x*z)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 7*a**2/(4*b*c)" ], "shape": [ "identity: N*a**N/(b*c)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/4", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 4", "problem_latex": "$\\dfrac{16a^{4}b^{2}c^{3}}{21m^{2}x^{3}y^{4}} × \\dfrac{3m^{3}x^{3}y^{4}}{8a^{2}b^{2}c^{2}}$.", "markdown": "$\\dfrac{16a^{4}b^{2}c^{3}}{21m^{2}x^{3}y^{4}} × \\dfrac{3m^{3}x^{3}y^{4}}{8a^{2}b^{2}c^{2}}$.", "answer_latex": [ "$\\dfrac{2a^{2}cm}{7}$." ], "answer_markdown": [ "$\\dfrac{2a^{2}cm}{7}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(16*a**4*b**2*c**3/(21*m**2*x**3*y**4))*(3*m**3*x**3*y**4/(8*a**2*b**2*c**2))", "answer_expr": "2*a**2*c*m/7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 2*a**2*b*c/7" ], "shape": [ "identity: N*a**N*b*c" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/5", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 5", "problem_latex": "$\\dfrac{2a}{bc} × \\dfrac{3b}{ac} × \\dfrac{5c}{ab}$.", "markdown": "$\\dfrac{2a}{bc} × \\dfrac{3b}{ac} × \\dfrac{5c}{ab}$.", "answer_latex": [ "$\\dfrac{30}{abc}$." ], "answer_markdown": [ "$\\dfrac{30}{abc}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a/(b*c))*(3*b/(a*c))*(5*c/(a*b))", "answer_expr": "30/(a*b*c)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 30/(a*b*c)" ], "shape": [ "identity: N/(a*b*c)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/6", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 6", "problem_latex": "$\\dfrac{2a^{3}}{3bc} × \\dfrac{3b^{3}}{5ac} × \\dfrac{5c^{3}}{2ab}$.", "markdown": "$\\dfrac{2a^{3}}{3bc} × \\dfrac{3b^{3}}{5ac} × \\dfrac{5c^{3}}{2ab}$.", "answer_latex": [ "$abc$." ], "answer_markdown": [ "$abc$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(2*a**3/(3*b*c))*(3*b**3/(5*a*c))*(5*c**3/(2*a*b))", "answer_expr": "a*b*c" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*b*c" ], "shape": [ "identity: a*b*c" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/7", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 7", "problem_latex": "$\\dfrac{5abc^{3}}{3x^{2}} ÷ \\dfrac{10ac^{3}}{6bx^{2}}$.", "markdown": "$\\dfrac{5abc^{3}}{3x^{2}} ÷ \\dfrac{10ac^{3}}{6bx^{2}}$.", "answer_latex": [ "$b^{2}$." ], "answer_markdown": [ "$b^{2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(5*a*b*c**3/(3*x**2))/(10*a*c**3/(6*b*x**2))", "answer_expr": "b**2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a**2" ], "shape": [ "identity: a**N" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/8", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 8", "problem_latex": "$\\dfrac{x^{2} - a^{2}}{x^{2} - 4a^{2}} × \\dfrac{x + 2a}{x - a}$.", "markdown": "$\\dfrac{x^{2} - a^{2}}{x^{2} - 4a^{2}} × \\dfrac{x + 2a}{x - a}$.", "answer_latex": [ "$\\dfrac{x + a}{x - 2a}$." ], "answer_markdown": [ "$\\dfrac{x + a}{x - 2a}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((x**2 - a**2)/(x**2 - 4*a**2))*((x + 2*a)/(x - a))", "answer_expr": "(x + a)/(x - 2*a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*a + b)*(-a**2 + b**2)/((-a + b)*(-4*a**2 + b**2))" ], "shape": [ "identity: (-a**N + b**N)*(N*a + b)/((-a + b)*(N*a**N + b**N))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-49/9", "set": "wentworth-first-steps-in-algebra-1894/ex-49", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "100", "location": "Exercise 49, problem 9", "problem_latex": "$\\dfrac{x^{2}y^{2} + 3xy}{4c^{2} - 1} × \\dfrac{2c + 1}{xy + 3}$.", "markdown": "$\\dfrac{x^{2}y^{2} + 3xy}{4c^{2} - 1} × \\dfrac{2c + 1}{xy + 3}$.", "answer_latex": [ "$\\dfrac{xy}{2c - 1}$." ], "answer_markdown": [ "$\\dfrac{xy}{2c - 1}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((x**2*y**2 + 3*x*y)/(4*c**2 - 1))*((2*c + 1)/(x*y + 3))", "answer_expr": "x*y/(2*c - 1)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (2*a + 1)*(b**2*c**2 + 3*b*c)/((4*a**2 - 1)*(b*c + 3))" ], "shape": [ "identity: (N*a + 1)*(N*b*c + b**N*c**N)/((N + b*c)*(N*a**N - 1))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/10", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 10", "problem_latex": "If one part of twenty-five is fifteen, what is the\nother part?", "markdown": "If one part of twenty-five is fifteen, what is the other part?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/11", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 11", "problem_latex": "If one part of~$35$ is~$x$, what is the other part?", "markdown": "If one part of $35$ is $x$, what is the other part?", "answer_latex": [ "$35 - x$." ], "answer_markdown": [ "$35 - x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "35 - x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/12", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 12", "problem_latex": "If one part of~$x$ is~$a$, what is the other part?", "markdown": "If one part of $x$ is $a$, what is the other part?", "answer_latex": [ "$x - a$." ], "answer_markdown": [ "$x - a$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x - a" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/13", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 13", "problem_latex": "How much does ten lack of being twelve?", "markdown": "How much does ten lack of being twelve?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/14", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 14", "problem_latex": "How much does $x$ lack of being fourteen?", "markdown": "How much does $x$ lack of being fourteen?", "answer_latex": [ "$14 - x$." ], "answer_markdown": [ "$14 - x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "14 - x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/15", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 15", "problem_latex": "How much does $x$ lack of being~$a$?", "markdown": "How much does $x$ lack of being $a$?", "answer_latex": [ "$a - x$." ], "answer_markdown": [ "$a - x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "a - x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/16", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 16", "problem_latex": "If a man walks four miles an hour, how many miles\nwill he walk in three hours?", "markdown": "If a man walks four miles an hour, how many miles will he walk in three hours?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/17", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 17", "problem_latex": "If a man walks $y$~miles an hour, how many miles\nwill he walk in $x$~hours?", "markdown": "If a man walks $y$ miles an hour, how many miles will he walk in $x$ hours?", "answer_latex": [ "$xy$." ], "answer_markdown": [ "$xy$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x*y" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/18", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 18", "problem_latex": "If a man walks $y$~miles an hour, how many hours\nwill it take him to walk $x$~miles?", "markdown": "If a man walks $y$ miles an hour, how many hours will it take him to walk $x$ miles?", "answer_latex": [ "$\\dfrac{x}{y}$." ], "answer_markdown": [ "$\\dfrac{x}{y}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x/y" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/1a", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 1, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 1a", "problem_latex": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "markdown": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "answer_latex": [ "$a$~plus~$b$;" ], "answer_markdown": [ "$a$ plus $b$;" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a + b", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/1b", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 1, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 1b", "problem_latex": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "markdown": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "answer_latex": [ "$a$~minus~$b$;" ], "answer_markdown": [ "$a$ minus $b$;" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a - b", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/1c", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 1, "part": "c", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 1c", "problem_latex": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "markdown": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "answer_latex": [ "$a$~times~$b$;" ], "answer_markdown": [ "$a$ times $b$;" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*b", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/1d", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 1, "part": "d", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 1d", "problem_latex": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "markdown": "Read $a + b$; $a - b$; $ab$; $a÷b$.", "answer_latex": [ "$a$~divided by~$b$." ], "answer_markdown": [ "$a$ divided by $b$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a/b", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/2", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 2", "problem_latex": "Write six increased by four.", "markdown": "Write six increased by four.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/3", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 3", "problem_latex": "Write $a$ increased by~$b$.", "markdown": "Write $a$ increased by $b$.", "answer_latex": [ "$a + b$." ], "answer_markdown": [ "$a + b$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "a + b" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/4", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 4", "problem_latex": "Write six diminished by four.", "markdown": "Write six diminished by four.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/5", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 5", "problem_latex": "Write $a$ diminished by~$b$.", "markdown": "Write $a$ diminished by $b$.", "answer_latex": [ "$a - b$." ], "answer_markdown": [ "$a - b$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "a - b" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/6", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 6", "problem_latex": "By how much does twenty-five exceed sixteen?", "markdown": "By how much does twenty-five exceed sixteen?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/7", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 7", "problem_latex": "By how much does $x$ exceed~$y$?", "markdown": "By how much does $x$ exceed $y$?", "answer_latex": [ "$x - y$." ], "answer_markdown": [ "$x - y$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x - y" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/8a", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 8, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 8a", "problem_latex": "Write four times three; the fourth power of three.", "markdown": "Write four times three; the fourth power of three.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/8b", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 8, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 8b", "problem_latex": "Write four times three; the fourth power of three.", "markdown": "Write four times three; the fourth power of three.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/9a", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 9, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 9a", "problem_latex": "Write four times~$x$; the fourth power of~$x$.", "markdown": "Write four times $x$; the fourth power of $x$.", "answer_latex": [ "$4x$;" ], "answer_markdown": [ "$4x$;" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "4*x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-5/9b", "set": "wentworth-first-steps-in-algebra-1894/ex-5", "number": 9, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "14", "location": "Exercise 5, problem 9b", "problem_latex": "Write four times~$x$; the fourth power of~$x$.", "markdown": "Write four times $x$; the fourth power of $x$.", "answer_latex": [ "$x^{4}$." ], "answer_markdown": [ "$x^{4}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x**4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50/1", "set": "wentworth-first-steps-in-algebra-1894/ex-50", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "102", "location": "Exercise 50, problem 1", "problem_latex": "$\\dfrac{\\dfrac{x}{b} + \\dfrac{y}{b}}{\\dfrac{z}{b}}$.", "markdown": "$\\dfrac{\\dfrac{x}{b} + \\dfrac{y}{b}}{\\dfrac{z}{b}}$.", "answer_latex": [ "$\\dfrac{x + y}{z}$." ], "answer_markdown": [ "$\\dfrac{x + y}{z}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x/b + y/b)/(z/b)", "answer_expr": "(x + y)/z" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: a*(b/a + x/a)/c" ], "shape": [ "identity: a*(b/a + x/a)/c" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50/10", "set": "wentworth-first-steps-in-algebra-1894/ex-50", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "102", "location": "Exercise 50, problem 10", "problem_latex": "$\\dfrac{x + 3 + \\dfrac{2}{x}}{1 + \\dfrac{3}{x} + \\dfrac{2}{x^{2}}}$.", "markdown": "$\\dfrac{x + 3 + \\dfrac{2}{x}}{1 + \\dfrac{3}{x} + \\dfrac{2}{x^{2}}}$.", "answer_latex": [ "$x$." ], "answer_markdown": [ "$x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x + 3 + 2/x)/(1 + 3/x + 2/x**2)", "answer_expr": "x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x + 3 + 2/x)/(1 + 3/x + 2/x**2)" ], "shape": [ "identity: (N + N/x + x)/(N*x**N + N/x + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50/11", "set": "wentworth-first-steps-in-algebra-1894/ex-50", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "102", "location": "Exercise 50, problem 11", "problem_latex": "$\\dfrac{\\dfrac{1}{x} - \\dfrac{2}{x^{2}} + \\dfrac{1}{x^{3}}}{\\dfrac{(1 - x)^{2}}{x^{2}}}$.", "markdown": "$\\dfrac{\\dfrac{1}{x} - \\dfrac{2}{x^{2}} + \\dfrac{1}{x^{3}}}{\\dfrac{(1 - x)^{2}}{x^{2}}}$.", "answer_latex": [ "$\\dfrac{1}{x}$." ], "answer_markdown": [ "$\\dfrac{1}{x}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(1/x - 2/x**2 + 1/x**3)/((1 - x)**2/x**2)", "answer_expr": "1/x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x**2*(1/x - 2/x**2 + x**(-3))/(-x + 1)**2" ], "shape": [ "identity: x**N*(-x + 1)**N*(N*x**N + x**N + 1/x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50/12", "set": "wentworth-first-steps-in-algebra-1894/ex-50", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "102", "location": "Exercise 50, problem 12", "problem_latex": "$\\dfrac{x^{2} - x-6}{1 - \\dfrac{4}{x^{2}}}$.", "markdown": "$\\dfrac{x^{2} - x-6}{1 - \\dfrac{4}{x^{2}}}$.", "answer_latex": [ "$\\dfrac{x^{2}(x - 3)}{x - 2}$." ], "answer_markdown": [ "$\\dfrac{x^{2}(x - 3)}{x - 2}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(x**2 - x - 6)/(1 - 4/x**2)", "answer_expr": "x**2*(x - 3)/(x - 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**2 - x - 6)/(1 - 4/x**2)" ], "shape": [ "identity: (N - x + x**N)/(N*x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50/13", "set": "wentworth-first-steps-in-algebra-1894/ex-50", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "102", "location": "Exercise 50, problem 13", "problem_latex": "$\\dfrac{a^{2} - a + \\dfrac{a - 1}{a + 1}}{a + \\dfrac{1}{a + 1}}$.", "markdown": "$\\dfrac{a^{2} - a + \\dfrac{a - 1}{a + 1}}{a + \\dfrac{1}{a + 1}}$.", "answer_latex": [ "$a - 1$." ], "answer_markdown": [ "$a - 1$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(a**2 - a + (a - 1)/(a + 1))/(a + 1/(a + 1))", "answer_expr": "a - 1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: (x**2 - x + (x - 1)/(x + 1))/(x + 1/(x + 1))" ], "shape": [ "identity: (-x + x**N + (x - 1)/(x + 1))/(x + 1/(x + 1))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.factor", "cas.simplify", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-50/14", "set": "wentworth-first-steps-in-algebra-1894/ex-50", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "102", "location": "Exercise 50, problem 14", "problem_latex": "$\\dfrac{\\dfrac{4a(a - x)}{a^{2} - x^{2}}}{\\dfrac{a - x}{a + x}}$.", "markdown": "$\\dfrac{\\dfrac{4a(a - x)}{a^{2} - x^{2}}}{\\dfrac{a - x}{a + x}}$.", "answer_latex": [ "$\\dfrac{4a}{a - x}$." ], "answer_markdown": [ "$\\dfrac{4a}{a - x}$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "(4*a*(a - x)/(a**2 - x**2))/((a - x)/(a + x))", "answer_expr": "4*a/(a - x)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 4*x*(a + x)/(-a**2 + x**2)" ], "shape": [ "identity: N*x*(a + x)/(-a**N + x**N)" ], "same_problem_in": [], "needs": [ 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"judge_why": null, "problem_expr": "Eq((x + 3)/4 + (7*x - 2)/5, (5*x - 1)/4 + (5*x + 4)/9)", "answer_expr": "1" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(33*x/20 + 7/20, 65*x/36 + 7/36)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 ENTER 4 ÷ 5 ENTER 4 ÷ − 7 ENTER 5 ÷ + 5 ENTER 9 ÷ −", "3 ENTER 4 ÷ 2 ENTER 5 ÷ − 1 ENTER 4 ÷ + 4 ENTER 9 ÷ −", "x↔y ÷ +/−" ], "constants": [ { "value": "1", "source": "the x in '(x + 3)/4' and '5x - 1' over 4, taken as the coefficient of x in x/4 (the 1 in 1/4)" }, { "value": "4", "source": "the denominator 4 in '(x + 3)/4' and '(5x - 1)/4'" }, { "value": "5", "source": "the coefficient 5 in '7x - 2' over 5 and '5x' over 4 and '5x + 4' over 9" }, { "value": "7", "source": "the 7 in '(7x - 2)/5'" }, { "value": "9", "source": 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"X#0.6666666666666666666666666666666667,1E-30", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-52/6", "set": "wentworth-first-steps-in-algebra-1894/ex-52", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "106", "location": "Exercise 52, problem 6", "problem_latex": "$\\dfrac{(2x - 1)(2 - x)}{2} + x^{2} - \\dfrac{1 + 3x}{2} = 0$.", "markdown": "$\\dfrac{(2x - 1)(2 - x)}{2} + x^{2} - \\dfrac{1 + 3x}{2} = 0$.", "answer_latex": [ "$1\\frac{1}{2}$." ], "answer_markdown": [ "$1\\frac{1}{2}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((2*x - 1)*(2 - x)/2 + x**2 - (1 + 3*x)/2, 0)", "answer_expr": "3/2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 3*x/2 + (-x + 2)*(2*x - 1)/2 - 1/2, 0)" ], "shape": [ "solve: Eq(N*x + N*(N - x)*(N*x - 1) + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=1.5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-52/7", "set": "wentworth-first-steps-in-algebra-1894/ex-52", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "106", "location": "Exercise 52, problem 7", "problem_latex": "$\\dfrac{6x - 11}{4} - \\dfrac{3 - 4x}{6} = \\dfrac{4}{3} - \\dfrac{x}{8}$.", "markdown": "$\\dfrac{6x - 11}{4} - \\dfrac{3 - 4x}{6} = \\dfrac{4}{3} - \\dfrac{x}{8}$.", "answer_latex": [ "$2$." ], "answer_markdown": [ "$2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((6*x - 11)/4 - (3 - 4*x)/6, Rational(4, 3) - x/8)", "answer_expr": "2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(13*x/6 - 13/4, -x/8 + 4/3)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=2", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 3 ÷ 11 ENTER 4 ÷ + 3 ENTER 6 ÷ +", "6 ENTER 4 ÷ 4 ENTER 6 ÷ + 1 ENTER 8 ÷ + ÷" ], "constants": [ { "value": "1", "source": "the x in 'x/8' has an implied coefficient of 1; it is the x-coefficient term 1/8, not a number from the working" } ], "calculator_value": "+2000000000000000000000000000000000E-33", "printed_value": "2", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-52/8", "set": "wentworth-first-steps-in-algebra-1894/ex-52", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "106", "location": "Exercise 52, problem 8", "problem_latex": "$\\dfrac{x + 6}{4} - \\dfrac{16 - 3x}{12} = 4\\frac{1}{6}$.", "markdown": "$\\dfrac{x + 6}{4} - \\dfrac{16 - 3x}{12} = 4\\frac{1}{6}$.", "answer_latex": [ "$8$." ], "answer_markdown": [ "$8$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x + 6)/4 - (16 - 3*x)/12, 4 + Rational(1, 6))", "answer_expr": "8" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/2 + 1/6, 25/6)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], 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"book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "106", "location": "Exercise 52, problem 9", "problem_latex": "$x - \\dfrac{x - 2}{3} = \\dfrac{x + 23}{4} - \\dfrac{10 + x}{5}$.", "markdown": "$x - \\dfrac{x - 2}{3} = \\dfrac{x + 23}{4} - \\dfrac{10 + x}{5}$.", "answer_latex": [ "$5$." ], "answer_markdown": [ "$5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x - (x - 2)/3, (x + 23)/4 - (10 + x)/5)", "answer_expr": "5" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x/3 + 2/3, x/20 + 15/4)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=5", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 3 ÷", "23 ENTER 4 ÷ −", "10 ENTER 5 ÷ +", "1 ENTER 3 ÷", "1 x↔y −", "1 ENTER 4 ÷ −", "1 ENTER 5 ÷ +", "+/− ÷" ], "constants": 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null, "problem_expr": "Eq(x**2 + b**2, (a - x)*(a - x))", "answer_expr": "(a**2 - b**2)/(2*a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(b**2 + x**2, (a - x)**2)" ], "shape": [ "solve: Eq(b**N + x**N, (a - x)**N)" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-54/8", "set": "wentworth-first-steps-in-algebra-1894/ex-54", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "108", "location": "Exercise 54, problem 8", "problem_latex": " $(a + b)(2 - x) = (a - b)(2 + x)$.", "markdown": "$(a + b)(2 - x) = (a - b)(2 + x)$.", "answer_latex": [ " $\\dfrac{2b}{a}$." ], "answer_markdown": [ "$\\dfrac{2b}{a}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((a + b)*(2 - x), (a - b)*(2 + x))", "answer_expr": "2*b/a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((-x + 2)*(a + b), (a - b)*(x + 2))" ], "shape": [ "solve: Eq((N - x)*(a + b), (N + x)*(a - b))" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-54/9", "set": "wentworth-first-steps-in-algebra-1894/ex-54", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "108", "location": "Exercise 54, problem 9", "problem_latex": " $(x - a)(2x - a) = 2(x - b)^{2}$.", "markdown": "$(x - a)(2x - a) = 2(x - b)^{2}$.", "answer_latex": [ " $\\dfrac{2b^{2} - a^{2}}{4b - 3a}$." ], "answer_markdown": [ "$\\dfrac{2b^{2} - a^{2}}{4b - 3a}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x - a)*(2*x - a), 2*(x - b)**2)", "answer_expr": "(2*b**2 - a**2)/(4*b - 3*a)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((-a + x)*(-a + 2*x), 2*(-b + x)**2)" ], "shape": [ "solve: Eq((-a + x)*(N*x - a), N*(-b + x)**N)" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-55/1", "set": "wentworth-first-steps-in-algebra-1894/ex-55", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "109", "location": "Exercise 55, problem 1", "problem_latex": "The difference between the fifth and seventh parts of\na certain number is~$2$. Find the number.", "markdown": "The difference between the fifth and seventh parts of a certain number is $2$. Find the number.", "answer_latex": [ "$35$." ], "answer_markdown": [ "$35$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 35}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2*x/35, 2)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 5 × 7 ×", "7 ENTER 5 − ÷" ], "constants": [ { "value": "2", "source": "the 2 in 'is~$2$' (right-hand side of Eq(x/5 - x/7, 2))" }, { "value": "5", "source": "the denominator 5 in 'fifth parts' (x/5)" }, { "value": "7", "source": "the denominator 7 in 'seventh part' (x/7)" } ], "calculator_value": "+35E+0", "printed_value": "35", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-55/2", "set": "wentworth-first-steps-in-algebra-1894/ex-55", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "109", "location": "Exercise 55, problem 2", "problem_latex": "One-half of a certain number exceeds the sum of its\nfifth and seventh parts by~$11$. Find the number.", "markdown": "One-half of a certain number exceeds the sum of its fifth and seventh parts by $11$. Find the number.", "answer_latex": [ "$70$." ], "answer_markdown": [ "$70$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 70}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(11*x/70, 11)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "11 ENTER", "2 1/x", "5 1/x −", "7 1/x −", "÷" ], "constants": [ { "value": "11", "source": "the 11 in 'exceeds the sum of its fifth and seventh parts by 11'" }, { "value": "2", "source": "the 2 in 'One-half'" }, { "value": "5", "source": "the 5 in 'its fifth ... parts'" }, { "value": "7", "source": "the 7 in 'its ... seventh parts'" } ], "calculator_value": "+7000000000000000000000000000000002E-32", "printed_value": "70", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-55/3", "set": "wentworth-first-steps-in-algebra-1894/ex-55", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "109", "location": "Exercise 55, problem 3", "problem_latex": "The sum of the third and sixth parts of a certain\nnumber exceeds the difference of its sixth and ninth parts\nby~$16$. Find the number.", "markdown": "The sum of the third and sixth parts of a certain number exceeds the difference of its sixth and ninth parts by $16$. Find the number.", "answer_latex": [ "$36$." ], "answer_markdown": [ "$36$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 36}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(4*x/9, 16)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "16 ENTER 3 1/x 9 1/x +", "÷" ], "constants": [ { "value": "16", "source": "the 16 in 'exceeds ... by 16' in the problem text" }, { "value": "3", "source": "the denominator in x/3 of the expression Eq(x/3 + x/6 - (x/6 - x/9), 16)" }, { "value": "9", "source": "the denominator in x/9 of the expression; after the x/6 terms cancel, the left side is x/3 + x/9" } ], "calculator_value": "+3600000000000000000000000000000000E-32", "printed_value": "36", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-55/4", "set": "wentworth-first-steps-in-algebra-1894/ex-55", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "109", "location": "Exercise 55, problem 4", "problem_latex": "There are two consecutive numbers, $x$~and~$x + 1$, such\nthat one-half the larger exceeds one-third the smaller\nnumber by~$10$. Find the numbers.", "markdown": "There are two consecutive numbers, $x$ and $x + 1$, such that one-half the larger exceeds one-third the smaller number by $10$. Find the numbers.", "answer_latex": [ "$57$, $58$." ], "answer_markdown": [ "$57$, $58$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 57}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x/6 + 1/2, 10)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "10 ENTER 1 ENTER 2 ÷ −", "2 ENTER 3 × ×" ], "constants": [ { "value": "10", "source": "the 10 in 'by 10'" }, { "value": "1", "source": "the 1 in 'x + 1' (the larger number)" }, { "value": "2", "source": "the 2 in 'one-half' and the denominator of (x + 1)/2" }, { "value": "3", "source": "the 3 in 'one-third' and the denominator of x/3" } ], "calculator_value": "+570E-1", "printed_value": "57", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-56/1", "set": "wentworth-first-steps-in-algebra-1894/ex-56", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "110", "location": "Exercise 56, problem 1", "problem_latex": "The sum of two numbers is~$100$, and if the greater is\ndivided by the smaller number, the quotient is~$4$ and the\nremainder~$5$. Find the numbers.", "markdown": "The sum of two numbers is $100$, and if the greater is divided by the smaller number, the quotient is $4$ and the remainder $5$. Find the numbers.", "answer_latex": [ "$81$, $19$." ], "answer_markdown": [ "$81$, $19$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 81, y: 19}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 4*a + 5), Eq(a + x, 100))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-56/2", "set": "wentworth-first-steps-in-algebra-1894/ex-56", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "110", "location": "Exercise 56, problem 2", "problem_latex": "The sum of two numbers is~$124$, and if the greater is\ndivided by the smaller number, the quotient is~$4$ and the\nremainder~$4$. Find the numbers.", "markdown": "The sum of two numbers is $124$, and if the greater is divided by the smaller number, the quotient is $4$ and the remainder $4$. Find the numbers.", "answer_latex": [ "$100$, $24$." ], "answer_markdown": [ "$100$, $24$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 100, y: 24}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 4*a + 4), Eq(a + x, 124))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-56/3", "set": "wentworth-first-steps-in-algebra-1894/ex-56", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "110", "location": "Exercise 56, problem 3", "problem_latex": "The difference of two numbers is~$49$, and if the greater\nis divided by the smaller, the quotient is~$4$ and the remainder~$4$.\nFind the numbers.", "markdown": "The difference of two numbers is $49$, and if the greater is divided by the smaller, the quotient is $4$ and the remainder $4$. Find the numbers.", "answer_latex": [ "$64$, $15$." ], "answer_markdown": [ "$64$, $15$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 64, y: 15}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 4*a + 4), Eq(-a + x, 49))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(-a + x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-56/4", "set": "wentworth-first-steps-in-algebra-1894/ex-56", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "110", "location": "Exercise 56, problem 4", "problem_latex": "The difference of two numbers is~$91$, and if the\ngreater is divided by the smaller, the quotient is~$8$ and the\nremainder~$7$. Find the numbers.", "markdown": "The difference of two numbers is $91$, and if the greater is divided by the smaller, the quotient is $8$ and the remainder $7$. Find the numbers.", "answer_latex": [ "$103$, $12$." ], "answer_markdown": [ "$103$, $12$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 103, y: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 8*a + 7), Eq(-a + x, 91))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(-a + x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-56/5", "set": "wentworth-first-steps-in-algebra-1894/ex-56", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "110", "location": "Exercise 56, problem 5", "problem_latex": "Divide $320$ into two parts such that the smaller part\nis contained in the larger part $11$~times, with a remainder\nof~$20$.", "markdown": "Divide $320$ into two parts such that the smaller part is contained in the larger part $11$ times, with a remainder of $20$.", "answer_latex": [ "$295$, $25$." ], "answer_markdown": [ "$295$, $25$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 295, y: 25}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 11*a + 20), Eq(a + x, 320))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-56/none", "set": "wentworth-first-steps-in-algebra-1894/ex-56", "number": null, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "110", "location": "Exercise 56, problem None", "problem_latex": "The sum of two numbers is~$63$, and if the greater\nis divided by the smaller number, the quotient is~$2$ and the\nremainder~$3$. Find the numbers.", "markdown": "The sum of two numbers is $63$, and if the greater is divided by the smaller number, the quotient is $2$ and the remainder $3$. Find the numbers.", "answer_latex": [ "x &= 43." ], "answer_markdown": [ "x &= 43." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x - 3)/(63 - x), 2)", "answer_expr": "43" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((x - 3)/(-x + 63), 2)" ], "shape": [ "solve: Eq((N + x)/(N - x), N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": "X=43", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "63 ENTER 2 × 3 +", "1 ENTER 2 + ÷" ], "constants": [ { "value": "1", "source": "the implicit coefficient of x in the working x - 3 = 2(63 - x), so that x + 2x = 3x; the 2 comes from 'quotient is 2' and the 3 from 'remainder 3'" } ], "calculator_value": "+43E+0", "printed_value": "43", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/1", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 1", "problem_latex": "A son is one-fourth as old as his father. In $24$~years\nhe will be one-half as old. Find the age of the son.", "markdown": "A son is one-fourth as old as his father. In $24$ years he will be one-half as old. Find the age of the son.", "answer_latex": [ "$12$~yr." ], "answer_markdown": [ "$12$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{s: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, a/4), Eq(x + 24, a/2 + 12))" ], "shape": [ "solve: (Eq(x, N*a), Eq(N + x, N*a + N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "24 ENTER 2 ÷" ], "constants": [ { "value": "24", "source": "the 24 in 'In 24 years'" }, { "value": "2", "source": "the 2 in the denominator of Eq(s + 24, (f + 24)/2), from 'one-half'" } ], "calculator_value": "+12E+0", "printed_value": "12", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/2", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 2", "problem_latex": "B's~age is one-sixth of A's~age. In $15$~years B's~age\nwill be one-third of A's~age. Find their ages.", "markdown": "B’s age is one-sixth of A’s age. In $15$ years B’s age will be one-third of A’s age. Find their ages.", "answer_latex": [ "A, $60$~yr.; B, $10$~yr." ], "answer_markdown": [ "A, $60$ yr.; B, $10$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{A: 60, B: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, x/6), Eq(a + 15, x/3 + 5))" ], "shape": [ "solve: (Eq(a, N*x), Eq(N + a, N*x + N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/3", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 3", "problem_latex": "The sum of the ages of A~and~B is $30$~years, and $5$~years\nhence B's~age will be one-third of~A's. Find their\nages.", "markdown": "The sum of the ages of A and B is $30$ years, and $5$ years hence B’s age will be one-third of A’s. Find their ages.", "answer_latex": [ "A, $25$~yr.; B, $5$~yr." ], "answer_markdown": [ "A, $25$ yr.; B, $5$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{A: 25, B: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + x, 30), Eq(a + 5, x/3 + 5/3))" ], "shape": [ "solve: (Eq(a + x, N), Eq(N + a, N*x + N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/4", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 4", "problem_latex": "A father is $35$~years old, and his son is one-fourth of\nthat age. In how many years will the son be half as old\nas his father?", "markdown": "A father is $35$ years old, and his son is one-fourth of that age. In how many years will the son be half as old as his father?", "answer_latex": [ "$17\\frac{1}{2}$~yr." ], "answer_markdown": [ "$17\\frac{1}{2}$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(35, 2)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 35), Eq(b, a/4), Eq(b + x, a/2 + x/2))" ], "shape": [ "solve: (Eq(a, N), Eq(b, N*a), Eq(b + x, N*a + N*x))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "35 ENTER 35 ENTER 4 ÷", "2 × −" ], "constants": [ { "value": "4", "source": "the 4 in 'one-fourth' (son's age is one-fourth of the father's)" }, { "value": "2", "source": "the 2 in 'half as old' (son is half as old as father, so 2(S + x) = F + x, which rearranges to x = F − 2S)" } ], "calculator_value": "+1750E-2", "printed_value": "35/2", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/5", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 5", "problem_latex": "A is $60$~years old, and B's~age is two-thirds of~A's.\nHow many years ago was B's~age one-fifth of~A's?", "markdown": "A is $60$ years old, and B’s age is two-thirds of A’s. How many years ago was B’s age one-fifth of A’s?", "answer_latex": [ "$35$~yr." ], "answer_markdown": [ "$35$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{k: 35}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 60), Eq(b, 2*a/3), Eq(b - x, a/5 - x/5))" ], "shape": [ "solve: (Eq(a, N), Eq(b, N*a), Eq(b - x, N*a + N*x))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "60 ENTER 2 × 3 ÷", "5 ×", "60 −", "4 ÷" ], "constants": [ { "value": "60", "source": "A is 60 years old, in the problem text and in Eq(A, 60)" }, { "value": "2", "source": "the 2 in 'two-thirds' (B = 2A/3)" }, { "value": "3", "source": "the 3 in 'two-thirds' (B = 2A/3)" }, { "value": "5", "source": "the 5 in 'one-fifth' (B − k = (A − k)/5, cleared by multiplying through by 5)" }, { "value": "4", "source": "the working: 5(B − k) = A − k gives 4k = 5B − A, so k = (5B − A)/4; the 4 is 5 − 1 from the k terms" } ], "calculator_value": "+35E+0", "printed_value": "35", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/6", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 6", "problem_latex": "A son is one-third as old as his father. Four years\nago he was only one-fourth as old as his father. What is\nthe age of each?", "markdown": "A son is one-third as old as his father. Four years ago he was only one-fourth as old as his father. What is the age of each?", "answer_latex": [ "Son, $12$~yr.; father, $36$~yr." ], "answer_markdown": [ "Son, $12$ yr.; father, $36$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{S: 12, F: 36}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, a/3), Eq(x - 4, a/4 - 1))" ], "shape": [ "solve: (Eq(x, N*a), Eq(N + x, N*a - 1))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/7", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 7", "problem_latex": "A is $50$~years old, and B~is half as old as~A\\@. In how\nmany years will B be two-thirds as old as~A?", "markdown": "A is $50$ years old, and B is half as old as A. In how many years will B be two-thirds as old as A?", "answer_latex": [ "$25$~yr." ], "answer_markdown": [ "$25$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{y: 25}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 50), Eq(b, a/2), Eq(b + x, 2*a/3 + 2*x/3))" ], "shape": [ "solve: (Eq(a, N), Eq(b, N*a), Eq(b + x, N*a + N*x))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "50 ENTER 2 ×", "50 ENTER 2 ÷", "3 ×", "−" ], "constants": [ { "value": "50", "source": "A is 50 years old (the problem as printed)" }, { "value": "2", "source": "the 2 in 'half' (B is half as old as A) and the 2 in 'two-thirds' (numerator)" }, { "value": "3", "source": "the 3 in 'two-thirds' (denominator)" } ], "calculator_value": "+25E+0", "printed_value": "25", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/8", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 8", "problem_latex": "B~is one-half as old as~A\\@. Ten years ago he was\none-fourth as old as~A\\@. What are their present ages?", "markdown": "B is one-half as old as A. Ten years ago he was one-fourth as old as A. What are their present ages?", "answer_latex": [ "A, $30$~yr.; B, $15$~yr." ], "answer_markdown": [ "A, $30$ yr.; B, $15$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{A: 30, B: 15}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, x/2), Eq(a - 10, x/4 - 5/2))" ], "shape": [ "solve: (Eq(a, N*x), Eq(N + a, N*x + N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/9", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem 9", "problem_latex": "The sum of the ages of a father and his son is $80$~years.\nThe son's age increased by $5$~years is one-fourth of\nthe father's age. Find their ages.", "markdown": "The sum of the ages of a father and his son is $80$ years. The son’s age increased by $5$ years is one-fourth of the father’s age. Find their ages.", "answer_latex": [ "Son, $12$~yr.; father, $68$~yr." ], "answer_markdown": [ "Son, $12$ yr.; father, $68$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{S: 12, F: 68}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + x, 80), Eq(x + 5, a/4))" ], "shape": [ "solve: (Eq(a + x, N), Eq(N + x, N*a))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-57/none", "set": "wentworth-first-steps-in-algebra-1894/ex-57", "number": null, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "111", "location": "Exercise 57, problem None", "problem_latex": "Eight years ago a boy was one-fourth as old as he\nwill be one year hence. How old is he now?", "markdown": "Eight years ago a boy was one-fourth as old as he will be one year hence. How old is he now?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "11" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x - 8, x/4 + 1/4)" ], "shape": [ "solve: Eq(N + x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=11", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "8 ENTER 4 × 1 +", "4 ENTER 1 − ÷" ], "constants": [ { "value": "8", "source": "the 8 in 'Eight years ago'" }, { "value": "4", "source": "the 4 in 'one-fourth'" }, { "value": "1", "source": "the 1 in 'one year hence' (the +1 in (x + 1)/4)" } ], "calculator_value": "+11E+0", "printed_value": "11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-58/1", "set": "wentworth-first-steps-in-algebra-1894/ex-58", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Exercise 58, problem 1", "problem_latex": " A~can do a piece of work in $3$~days, B~in $5$~days, and\nC~in $6$~days. How long will it take them to do it working\ntogether?", "markdown": "A can do a piece of work in $3$ days, B in $5$ days, and C in $6$ days. How long will it take them to do it working together?", "answer_latex": [ " $1\\frac{3}{7}$~dy." ], "answer_markdown": [ "$1\\frac{3}{7}$ dy." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(1/x, Rational(1,3) + Rational(1,5) + Rational(1,6))", "answer_expr": "Rational(10,7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(1/x, 7/10)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 1/x 5 1/x + 6 1/x + 1/x" ], "constants": [ { "value": "3", "source": "the 3 in 'A can do a piece of work in 3 days'" }, { "value": "5", "source": "the 5 in 'B in 5 days'" }, { "value": "6", "source": "the 6 in 'C in 6 days'" } ], "calculator_value": "+1428571428571428571428571428571429E-33", "printed_value": "10/7", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-58/2", "set": "wentworth-first-steps-in-algebra-1894/ex-58", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Exercise 58, problem 2", "problem_latex": " A~can do a piece of work in $5$~days, B~in $4$~days, and\nC~in $3$~days. How long will it take them together to do\nthe work?", "markdown": "A can do a piece of work in $5$ days, B in $4$ days, and C in $3$ days. How long will it take them together to do the work?", "answer_latex": [ " $1\\frac{13}{47}$~dy." ], "answer_markdown": [ "$1\\frac{13}{47}$ dy." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(1/x, Rational(1,5) + Rational(1,4) + Rational(1,3))", "answer_expr": "Rational(60,47)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(1/x, 47/60)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 1/x", "4 1/x +", "3 1/x +", "1/x" ], "calculator_value": "+1276595744680851063829787234042553E-33", "printed_value": "60/47", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-58/3", "set": "wentworth-first-steps-in-algebra-1894/ex-58", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Exercise 58, problem 3", "problem_latex": " A~can do a piece of work in $2\\frac{1}{2}$~days, B~in $3\\frac{1}{2}$~days,\nand C~in $3\\frac{3}{4}$~days. How long will it take them together to\ndo the work?", "markdown": "A can do a piece of work in $2\\frac{1}{2}$ days, B in $3\\frac{1}{2}$ days, and C in $3\\frac{3}{4}$ days. How long will it take them together to do the work?", "answer_latex": [ " $1\\frac{1}{20}$~dy." ], "answer_markdown": [ "$1\\frac{1}{20}$ dy." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(1/x, 1/Rational(5,2) + 1/Rational(7,2) + 1/Rational(15,4))", "answer_expr": "Rational(21,20)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(1/x, 20/21)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 1 ENTER 2 ÷ + 1/x", "3 ENTER 1 ENTER 2 ÷ + 1/x +", "3 ENTER 3 ENTER 4 ÷ + 1/x +", "1/x" ], "calculator_value": "+1050000000000000000000000000000000E-33", "printed_value": "21/20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-58/4", "set": "wentworth-first-steps-in-algebra-1894/ex-58", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Exercise 58, problem 4", "problem_latex": " A~can do a piece of work in $10$~days, B~in $12$~days;\nA~and~B together, with the help of~C, can do the work in\n$4$~days. How long will it take C~alone to do the work?", "markdown": "A can do a piece of work in $10$ days, B in $12$ days; A and B together, with the help of C, can do the work in $4$ days. How long will it take C alone to do the work?", "answer_latex": [ " $15$~dy." ], "answer_markdown": [ "$15$ dy." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(Rational(1,10) + Rational(1,12) + 1/x, Rational(1,4))", "answer_expr": "15" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(11/60 + 1/x, 1/4)" ], "shape": [ "solve: Eq(N + 1/x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=15", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 ENTER 4 ÷ 1 ENTER 10 ÷ −", "1 ENTER 12 ÷ − 1/x" ], "calculator_value": "+1500000000000000000000000000000000E-32", "printed_value": "15", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-58/5", "set": "wentworth-first-steps-in-algebra-1894/ex-58", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Exercise 58, problem 5", "problem_latex": " A~and~B together can mow a field in $10$~hours, A~and~C\nin $12$~hours, and A~alone in $20$~hours. In what time\ncan B~and~C together mow the field?", "markdown": "A and B together can mow a field in $10$ hours, A and C in $12$ hours, and A alone in $20$ hours. In what time can B and C together mow the field?", "answer_latex": [ " $12$~hr." ], "answer_markdown": [ "$12$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{a: Rational(1,20), b: Rational(1,20), c: Rational(1,30), t: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 1/20), Eq(a + b, 1/10), Eq(a + c, 1/12), Eq(b + c, 1/x))" ], "shape": [ "solve: (Eq(a, N), Eq(a + b, N), Eq(a + c, N), Eq(b + c, 1/x))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-58/6", "set": "wentworth-first-steps-in-algebra-1894/ex-58", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "112", "location": "Exercise 58, problem 6", "problem_latex": " A~and~B together can build a wall in $12$~days, A~and~C\nin $15$~days, B~and~C in $20$~days. In what time can they\nbuild the wall if they all work together?", "markdown": "A and B together can build a wall in $12$ days, A and C in $15$ days, B and C in $20$ days. In what time can they build the wall if they all work together?", "answer_latex": [ " $10$~dy." ], "answer_markdown": [ "$10$ dy." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{a: Rational(1,20), b: Rational(1,30), c: Rational(1,60), t: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + b, 1/12), Eq(a + c, 1/15), Eq(b + c, 1/20), Eq(a + b + c, 1/x))" ], "shape": [ "solve: (Eq(a + b, N), Eq(a + c, N), Eq(b + c, N), Eq(a + b + c, 1/x))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-59/1", "set": "wentworth-first-steps-in-algebra-1894/ex-59", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "113", "location": "Exercise 59, problem 1", "problem_latex": "A cistern can be filled by three pipes in $16$, $24$, and\n$32$~hours, respectively. In what time will it be filled by\nall the pipes together?", "markdown": "A cistern can be filled by three pipes in $16$, $24$, and $32$ hours, respectively. In what time will it be filled by all the pipes together?", "answer_latex": [ "$7\\frac{5}{13}$~hr." ], "answer_markdown": [ "$7\\frac{5}{13}$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 96/13}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(1/x, 13/96)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-59/2", "set": "wentworth-first-steps-in-algebra-1894/ex-59", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "113", "location": "Exercise 59, problem 2", "problem_latex": "A tank can be filled by two pipes in $3$~hours and $4$~hours,\nrespectively, and can be emptied by a third pipe in\n$6$~hours. In what time will the cistern be filled if the pipes\nare all running together?", "markdown": "A tank can be filled by two pipes in $3$ hours and $4$ hours, respectively, and can be emptied by a third pipe in $6$ hours. In what time will the cistern be filled if the pipes are all running together?", "answer_latex": [ "$2\\frac{2}{5}$~hr." ], "answer_markdown": [ "$2\\frac{2}{5}$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 12/5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(1/x, 5/12)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 1/x 4 1/x +", "6 1/x −", "1/x" ], "calculator_value": "+2400000000000000000000000000000000E-33", "printed_value": "12/5", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-59/3", "set": "wentworth-first-steps-in-algebra-1894/ex-59", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "113", "location": "Exercise 59, problem 3", "problem_latex": "A tank can be filled by three pipes in $1$~hour and $40$\nminutes, $3$~hours and $20$ minutes, and $5$~hours, respectively.\nIn what time will the tank be filled if all three pipes are\nrunning together?", "markdown": "A tank can be filled by three pipes in $1$ hour and $40$ minutes, $3$ hours and $20$ minutes, and $5$ hours, respectively. In what time will the tank be filled if all three pipes are running together?", "answer_latex": [ "$\\frac{10}{11}$~hr." ], "answer_markdown": [ "$\\frac{10}{11}$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 10/11}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(1/x, 11/10)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num", "core.units" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ÷", "3 ENTER 10 ÷", "+", "1 ENTER 5 ÷", "+", "1/x" ], "constants": [ { "value": "3", "source": "the 3 in '3/5' in the expression Eq(1/x, 3/5 + 3/10 + 1/5); the first pipe's rate of 3/5 of the tank per hour, from 1 hour 40 minutes = 5/3 hours" }, { "value": "5", "source": "the 5 in '3/5' and '1/5' in the expression; the first pipe's 5/3 hours and the third pipe's 5 hours" }, { "value": "10", "source": "the 10 in '3/10' in the expression; the second pipe's rate of 3/10 per hour, from 3 hours 20 minutes = 10/3 hours" }, { "value": "1", "source": "the 1 in '1/5' in the expression (the third pipe's rate); the 1 in '1/x' in the expression, which is the combined rate 1/x" } ], "calculator_value": "+9090909090909090909090909090909091E-34", "printed_value": "10/11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-59/4", "set": "wentworth-first-steps-in-algebra-1894/ex-59", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "113", "location": "Exercise 59, problem 4", "problem_latex": "A cistern can be filled by three pipes in $2\\frac{1}{3}$~hours, $3\\frac{1}{2}$~hours,\nand $4\\frac{2}{3}$~hours, respectively. In what time will the\ncistern be filled if all the pipes are running together?", "markdown": "A cistern can be filled by three pipes in $2\\frac{1}{3}$ hours, $3\\frac{1}{2}$ hours, and $4\\frac{2}{3}$ hours, respectively. In what time will the cistern be filled if all the pipes are running together?", "answer_latex": [ "$1\\frac{1}{13}$~hr." ], "answer_markdown": [ "$1\\frac{1}{13}$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 14/13}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(1/x, 13/14)" ], "shape": [ "solve: Eq(1/x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 ENTER 3 ÷ 2 + 1/x", "1 ENTER 2 ÷ 3 + 1/x +", "2 ENTER 3 ÷ 4 + 1/x +", "1/x" ], "constants": [ { "value": "1", "source": "the 1 in the numerator of 2\\frac{1}{3}" }, { "value": "3", "source": "the 3 in the denominator of 2\\frac{1}{3}" }, { "value": "2", "source": "the 2 in 2\\frac{1}{3}" }, { "value": "4", "source": "the 4 in 4\\frac{2}{3}" } ], "calculator_value": "+1076923076923076923076923076923077E-33", "printed_value": "14/13", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-59/5", "set": "wentworth-first-steps-in-algebra-1894/ex-59", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "113", "location": "Exercise 59, problem 5", "problem_latex": "A cistern has three pipes. The first pipe will fill the\ncistern in $12$~hours, the second in $20$~hours, and all three\npipes together will fill it in $6$~hours. How long will it take\nthe third pipe alone to fill it?", "markdown": "A cistern has three pipes. The first pipe will fill the cistern in $12$ hours, the second in $20$ hours, and all three pipes together will fill it in $6$ hours. How long will it take the third pipe alone to fill it?", "answer_latex": [ "$30$~hr." ], "answer_markdown": [ "$30$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 30}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2/15 + 1/x, 1/6)" ], "shape": [ "solve: Eq(N + 1/x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 1/x", "12 1/x −", "20 1/x −", "1/x" ], "calculator_value": "+2999999999999999999999999999999997E-32", "printed_value": "30", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/1", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 1", "problem_latex": "If the dividend is twenty and the quotient five,\nwhat is the divisor?", "markdown": "If the dividend is twenty and the quotient five, what is the divisor?", "answer_latex": [ "$\\frac{20}{5}$." ], "answer_markdown": [ "$\\frac{20}{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{d: 20/5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(5*x, 20)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "20 ENTER 5 ÷" ], "calculator_value": "+4E+0", "printed_value": "4", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/10", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 10", "problem_latex": "If the difference of two numbers is five, and the\nsmaller number is fifteen, what is the greater number?", "markdown": "If the difference of two numbers is five, and the smaller number is fifteen, what is the greater number?", "answer_latex": [ "$15 + 5$." ], "answer_markdown": [ "$15 + 5$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{g: 15 + 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x - 15, 5)" ], "shape": [ "solve: Eq(N + x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 15 +" ], "calculator_value": "+20E+0", "printed_value": "20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/11", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 11", "problem_latex": "If the difference of two numbers is eight, and the\nsmaller number is~$x$, what is the greater number?", "markdown": "If the difference of two numbers is eight, and the smaller number is $x$, what is the greater number?", "answer_latex": [ "$x + 8$." ], "answer_markdown": [ "$x + 8$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{g: x + 8}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(-a + x, 8)" ], "shape": [ "solve: Eq(-a + x, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/12", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 12", "problem_latex": "If the sum of two numbers is~$30$, and one of them\nis~$20$, what is the other?", "markdown": "If the sum of two numbers is $30$, and one of them is $20$, what is the other?", "answer_latex": [ "$30 - 20$." ], "answer_markdown": [ "$30 - 20$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{o: 30 - 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x + 20, 30)" ], "shape": [ "solve: Eq(N + x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "30 ENTER 20 −" ], "calculator_value": "+10E+0", "printed_value": "10", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/13", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 13", "problem_latex": "If the sum of two numbers is~$x$, and one of them is~$10$,\nwhat is the other?", "markdown": "If the sum of two numbers is $x$, and one of them is $10$, what is the other?", "answer_latex": [ "$x - 10$." ], "answer_markdown": [ "$x - 10$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{o: x - 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x + 10, a)" ], "shape": [ "solve: Eq(N + x, a)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/14", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 14", "problem_latex": "If $100$ contains~$x$ ten times, what is the value of~$x$?", "markdown": "If $100$ contains $x$ ten times, what is the value of $x$?", "answer_latex": [ "$10$." ], "answer_markdown": [ "$10$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 10}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(10*x, 100)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/2", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 2", "problem_latex": "If the dividend is~$a$ and the quotient~$b$, what is the\ndivisor?", "markdown": "If the dividend is $a$ and the quotient $b$, what is the divisor?", "answer_latex": [ "$\\dfrac{a}{b}$." ], "answer_markdown": [ "$\\dfrac{a}{b}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{d: a/b}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(b*x, a)" ], "shape": [ "solve: Eq(b*x, a)" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-42/18" ], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/3a", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 3, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 3a", "problem_latex": "If John is twenty years old to-day, how old was he\nfour years ago? How old will he be five years hence?", "markdown": "If John is twenty years old to-day, how old was he four years ago? How old will he be five years hence?", "answer_latex": [ "$20 - 4$" ], "answer_markdown": [ "$20 - 4$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 16.0, printed 20 - 4 (half-unit 0.5; correctly rounded at the printed digits: 16.0)", "problem_expr": "20 - 4", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 16" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#20 - 4,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/3b", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 3, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 3b", "problem_latex": "If John is twenty years old to-day, how old was he\nfour years ago? How old will he be five years hence?", "markdown": "If John is twenty years old to-day, how old was he four years ago? How old will he be five years hence?", "answer_latex": [ "$20 + 5$." ], "answer_markdown": [ "$20 + 5$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 25.0, printed 20 + 5 (half-unit 0.5; correctly rounded at the printed digits: 25.0)", "problem_expr": "20 + 5", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 25" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#20 + 5,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/4a", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 4, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 4a", "problem_latex": "If James is $x$~years old to-day, how old was he three\nyears ago? How old will he be seven years hence?", "markdown": "If James is $x$ years old to-day, how old was he three years ago? How old will he be seven years hence?", "answer_latex": [ "$(x - 3)$~yr." ], "answer_markdown": [ "$(x - 3)$ yr." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/4b", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 4, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 4b", "problem_latex": "If James is $x$~years old to-day, how old was he three\nyears ago? How old will he be seven years hence?", "markdown": "If James is $x$ years old to-day, how old was he three years ago? How old will he be seven years hence?", "answer_latex": [ "$(x + 7)$~yr." ], "answer_markdown": [ "$(x + 7)$ yr." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/5", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 5", "problem_latex": "Write four times the expression seven minus five.", "markdown": "Write four times the expression seven minus five.", "answer_latex": [ "$4(7 - 5)$." ], "answer_markdown": [ "$4(7 - 5)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/6", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 6", "problem_latex": "Write seven times the expression $2x$~minus~$y$.", "markdown": "Write seven times the expression $2x$ minus $y$.", "answer_latex": [ "$7(2x - y)$." ], "answer_markdown": [ "$7(2x - y)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/7", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 7", "problem_latex": "Write the next integral number above four.", "markdown": "Write the next integral number above four.", "answer_latex": [ "$4 + 1$." ], "answer_markdown": [ "$4 + 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/8a", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 8, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 8a", "problem_latex": "If $x$~is an integral number, write the next integral\nnumber above it; the next integral number below it.", "markdown": "If $x$ is an integral number, write the next integral number above it; the next integral number below it.", "answer_latex": [ "$x + 1$" ], "answer_markdown": [ "$x + 1$" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/8b", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 8, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 8b", "problem_latex": "If $x$~is an integral number, write the next integral\nnumber above it; the next integral number below it.", "markdown": "If $x$ is an integral number, write the next integral number above it; the next integral number below it.", "answer_latex": [ "$x - 1$." ], "answer_markdown": [ "$x - 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-6/9", "set": "wentworth-first-steps-in-algebra-1894/ex-6", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "15", "location": "Exercise 6, problem 9", "problem_latex": "What number is less than $20$ by~$d$?", "markdown": "What number is less than $20$ by $d$?", "answer_latex": [ "$20 - d$." ], "answer_markdown": [ "$20 - d$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-60/1", "set": "wentworth-first-steps-in-algebra-1894/ex-60", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "114", "location": "Exercise 60, problem 1", "problem_latex": "A sets out from Boston and walks towards Portland\nat the rate of $3$~miles an hour. Three hours afterward B\nsets out from the same place and walks in the same direction\nat the rate of $4$~miles an hour. How far from Boston\nwill B~overtake~A?", "markdown": "A sets out from Boston and walks towards Portland at the rate of $3$ miles an hour. Three hours afterward B sets out from the same place and walks in the same direction at the rate of $4$ miles an hour. How far from Boston will B overtake A?", "answer_latex": [ "$36$~mi." ], "answer_markdown": [ "$36$ mi." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x/3, x/4 + 3)", "answer_expr": "{x: 36}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/3, x/4 + 3)" ], "shape": [ "solve: Eq(N*x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 3 ×", "4 ×", "4 ENTER 3 −", "÷" ], "calculator_value": "+36E+0", "printed_value": "36", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-60/2", "set": "wentworth-first-steps-in-algebra-1894/ex-60", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "114", "location": "Exercise 60, problem 2", "problem_latex": "A courier who goes at the rate of $6\\frac{1}{2}$~miles an hour is\nfollowed, after $4$~hours, by another who goes at the rate of\n$7\\frac{1}{2}$~miles an hour. In how many hours will the second\novertake the first?", "markdown": "A courier who goes at the rate of $6\\frac{1}{2}$ miles an hour is followed, after $4$ hours, by another who goes at the rate of $7\\frac{1}{2}$ miles an hour. In how many hours will the second overtake the first?", "answer_latex": [ "$26$~hr." ], "answer_markdown": [ "$26$ hr." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(Rational(15, 2)*x, Rational(13, 2)*(x + 4))", "answer_expr": "{x: 26}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(15*x/2, 13*x/2 + 26)" ], "shape": [ "solve: Eq(N*x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 ENTER 1 ENTER 2 ÷ + 6 ENTER 1 ENTER 2 ÷ + −", "6 ENTER 1 ENTER 2 ÷ + 4 × x↔y ÷" ], "constants": [ { "value": "7", "source": "the 7 in '7½ miles an hour'" }, { "value": "1", "source": "the 1 in the fraction ½ in '7½ miles an hour'" }, { "value": "2", "source": "the 2 in the fraction ½ in '7½ miles an hour'" }, { "value": "6", "source": "the 6 in '6½ miles an hour'" }, { "value": "1", "source": "the 1 in the fraction ½ in '6½ miles an hour'" }, { "value": "2", "source": "the 2 in the fraction ½ in '6½ miles an hour'" }, { "value": "4", "source": "the 4 in 'followed, after 4 hours'" } ], "calculator_value": "+26E+0", "printed_value": "26", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-60/3", "set": "wentworth-first-steps-in-algebra-1894/ex-60", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "114", "location": "Exercise 60, problem 3", "problem_latex": "A person walks to the top of a mountain at the rate\nof two miles an hour, and down the same way at the rate\nof $4$~miles an hour. If he is out $6$~hours, how far is it to\nthe top of the mountain?", "markdown": "A person walks to the top of a mountain at the rate of two miles an hour, and down the same way at the rate of $4$ miles an hour. If he is out $6$ hours, how far is it to the top of the mountain?", "answer_latex": [ "$8$~mi." ], "answer_markdown": [ "$8$ mi." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x/2 + x/4, 6)", "answer_expr": "{x: 8}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x/4, 6)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 2 1/x 4 1/x + ÷" ], "calculator_value": "+8E+0", "printed_value": "8", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-60/4", "set": "wentworth-first-steps-in-algebra-1894/ex-60", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "114", "location": "Exercise 60, problem 4", "problem_latex": "In going a certain distance, a train travelling at the\nrate of $40$~miles an hour takes $2$~hours less than a train\ntravelling $30$~miles an hour. Find the distance.", "markdown": "In going a certain distance, a train travelling at the rate of $40$ miles an hour takes $2$ hours less than a train travelling $30$ miles an hour. Find the distance.", "answer_latex": [ "$240$~mi." ], "answer_markdown": [ "$240$ mi." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(d/40, d/30 - 2)", "answer_expr": "{d: 240}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/40, x/30 - 2)" ], "shape": [ "solve: Eq(N*x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 30 1/x", "40 1/x −", "÷" ], "constants": [ { "value": "2", "source": "the 2 in 'takes 2 hours less'" }, { "value": "30", "source": "the 30 in 'a train travelling 30 miles an hour'" }, { "value": "40", "source": "the 40 in 'a train travelling at the rate of 40 miles an hour'" } ], "calculator_value": "+2400000000000000000000000000000001E-31", "printed_value": "240", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-60/none", "set": "wentworth-first-steps-in-algebra-1894/ex-60", "number": null, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "114", "location": "Exercise 60, problem None", "problem_latex": "A courier who travels $6$~miles an hour is followed,\nafter $2$~hours, by a second courier who travels $7\\frac{1}{2}$~miles an\nhour. In how many hours will the second courier overtake\nthe first?", "markdown": "A courier who travels $6$ miles an hour is followed, after $2$ hours, by a second courier who travels $7\\frac{1}{2}$ miles an hour. In how many hours will the second courier overtake the first?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(6*x, (x - 2)*Rational(15, 2))", "answer_expr": "10" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(6*x, 15*x/2 - 15)" ], "shape": [ "solve: Eq(N*x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=10", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "7 ENTER 1 ENTER 2 ÷ +", "ENTER ENTER 6 −", "x↔y 2 ×", "x↔y ÷" ], "constants": [ { "value": "7", "source": "the 7 in '7½ miles an hour' (second courier's speed)" }, { "value": "1", "source": "the 1 in '7½' (numerator of the fraction 1/2)" }, { "value": "2", "source": "the 2 in '7½' (denominator of the fraction 1/2)" }, { "value": "6", "source": "the 6 in '6 miles an hour' (first courier's speed)" }, { "value": "2", "source": "the 2 in 'after 2 hours' (first courier's head start, the x − 2 in the equation)" } ], "calculator_value": "+10E+0", "printed_value": "10", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-61/1", "set": "wentworth-first-steps-in-algebra-1894/ex-61", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "115", "location": "Exercise 61, problem 1", "problem_latex": " A hound makes $3$~leaps while a rabbit makes~$5$; but\n$1$~of the hound's leaps is equivalent to $2$~of the rabbit's.\nThe rabbit has a start of $120$~leaps. How many leaps\nwill the rabbit take before she is caught?", "markdown": "A hound makes $3$ leaps while a rabbit makes $5$; but $1$ of the hound’s leaps is equivalent to $2$ of the rabbit’s. The rabbit has a start of $120$ leaps. How many leaps will the rabbit take before she is caught?", "answer_latex": [ "$600$." ], "answer_markdown": [ "$600$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{R: 600}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(2*a, b + 120), Eq(5*a, 3*b))" ], "shape": [ "solve: (Eq(N*a, N + b), Eq(N*a, N*b))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "120 ENTER 5 ×", "2 ENTER 3 × 5 − ÷" ], "constants": [ { "value": "120", "source": "the rabbit has a start of 120 leaps" }, { "value": "5", "source": "the rabbit makes 5 leaps in the same time the hound makes 3" }, { "value": "2", "source": "1 of the hound's leaps is equivalent to 2 of the rabbit's (the 2 in the 6 = 2 × 3 and in the 2H = 120 + R relation)" }, { "value": "3", "source": "the hound makes 3 leaps while the rabbit makes 5 (the 3 in 6 = 2 × 3 and in 5H = 3R)" } ], "calculator_value": "+600E+0", "printed_value": "600", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-61/2", "set": "wentworth-first-steps-in-algebra-1894/ex-61", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "115", "location": "Exercise 61, problem 2", "problem_latex": " A rabbit takes $6$~leaps to a dog's~$5$, and $7$~of the dog's\nleaps are equivalent to $9$~of the rabbit's. The rabbit has\na start of~$60$ of her own leaps. How many leaps must the\ndog take to catch the rabbit?", "markdown": "A rabbit takes $6$ leaps to a dog’s $5$, and $7$ of the dog’s leaps are equivalent to $9$ of the rabbit’s. The rabbit has a start of $60$ of her own leaps. How many leaps must the dog take to catch the rabbit?", "answer_latex": [ "$700$." ], "answer_markdown": [ "$700$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{D: 700}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(9*a/7, b + 60), Eq(5*b, 6*a))" ], "shape": [ "solve: (Eq(N*a, N + b), Eq(N*b, N*a))" ], "same_problem_in": [], "needs": [ "core.frac", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "60 ENTER 9 ENTER 7 ÷ 6 ENTER 5 ÷ − ÷" ], "calculator_value": "+6999999999999999999999999999999977E-31", "printed_value": "700", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-61/3", "set": "wentworth-first-steps-in-algebra-1894/ex-61", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "115", "location": "Exercise 61, problem 3", "problem_latex": " A dog makes $4$~leaps while a rabbit makes~$5$; but $3$~of\nthe dog's leaps are equivalent to $4$~of the rabbit's. The\nrabbit has a start of $90$~of the \\emph{dog's leaps}. How many\nleaps will each take before the rabbit is caught?", "markdown": "A dog makes $4$ leaps while a rabbit makes $5$; but $3$ of the dog’s leaps are equivalent to $4$ of the rabbit’s. The rabbit has a start of $90$ of the *dog’s leaps*. How many leaps will each take before the rabbit is caught?", "answer_latex": [ "Dog, $1440$; rabbit, $1800$." ], "answer_markdown": [ "Dog, $1440$; rabbit, $1800$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{D: 1440, R: 1800}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 3*b/4 + 90), Eq(5*a, 4*b))" ], "shape": [ "solve: (Eq(a, N*b + N), Eq(N*a, N*b))" ], "same_problem_in": [], "needs": [ "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-62/1", "set": "wentworth-first-steps-in-algebra-1894/ex-62", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "116", "location": "Exercise 62, problem 1", "problem_latex": "Find the time between $5$ and $6$~o'clock when the\nhands of a clock are together.", "markdown": "Find the time between $5$ and $6$ o’clock when the hands of a clock are together.", "answer_latex": [ "$27\\frac{3}{11}$~min.\\ past 5~o'clock." ], "answer_markdown": [ "$27\\frac{3}{11}$ min. past 5 o’clock." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(300, 11)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, x/12 + 25)" ], "shape": [ "solve: Eq(x, N*x + N)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-62/2" ], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "25 ENTER 12 ×", "12 ENTER 1 − ÷" ], "constants": [ { "value": "1", "source": "the working: multiply x = 25 + x/12 by 12 to get 12x = 300 + x, so 11x = 300; the 1 is the coefficient of x moved across (12 − 1 = 11)" } ], "calculator_value": "+2727272727272727272727272727272727E-32", "printed_value": "300/11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-62/2", "set": "wentworth-first-steps-in-algebra-1894/ex-62", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "116", "location": "Exercise 62, problem 2", "problem_latex": "Find the time between $2$ and $3$~o'clock when the\nhands of a clock are at right angles to each other.", "markdown": "Find the time between $2$ and $3$ o’clock when the hands of a clock are at right angles to each other.", "answer_latex": [ "$27\\frac{3}{11}$~min.\\ past 2~o'clock." ], "answer_markdown": [ "$27\\frac{3}{11}$ min. past 2 o’clock." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(300, 11)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, x/12 + 25)" ], "shape": [ "solve: Eq(x, N*x + N)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-62/1" ], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "25 ENTER 1 ENTER 12 1/x −", "÷" ], "constants": [ { "value": "1", "source": "the coefficient of x in the equation Eq(x, 25 + x/12), written as x - x/12 = 25 so that x(1 - 1/12) = 25" }, { "value": "12", "source": "the 12 in 'x/12' in the mathematics Eq(x, 25 + x/12)" }, { "value": "25", "source": "the 25 in 'Eq(x, 25 + x/12)' from the mathematics" } ], "calculator_value": "+2727272727272727272727272727272727E-32", "printed_value": "300/11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-62/3", "set": "wentworth-first-steps-in-algebra-1894/ex-62", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "116", "location": "Exercise 62, problem 3", "problem_latex": "Find the time between $2$ and $3$~o'clock when the\nhands of a clock point in opposite directions.", "markdown": "Find the time between $2$ and $3$ o’clock when the hands of a clock point in opposite directions.", "answer_latex": [ "$43\\frac{7}{11}$~min.\\ past 2~o'clock." ], "answer_markdown": [ "$43\\frac{7}{11}$ min. past 2 o’clock." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(480, 11)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, x/12 + 40)" ], "shape": [ "solve: Eq(x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-62/4", "set": "wentworth-first-steps-in-algebra-1894/ex-62", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "116", "location": "Exercise 62, problem 4", "problem_latex": "Find the time between $1$ and $2$~o'clock when the\nhands of a clock are at right angles to each other.", "markdown": "Find the time between $1$ and $2$ o’clock when the hands of a clock are at right angles to each other.", "answer_latex": [ "$21\\frac{9}{11}$~min.\\ past 1~o'clock." ], "answer_markdown": [ "$21\\frac{9}{11}$ min. past 1 o’clock." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(240, 11)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, x/12 + 20)" ], "shape": [ "solve: Eq(x, N*x + N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "20 ENTER 1 ENTER 12 1/x −", "÷" ], "constants": [ { "value": "20", "source": "the 20 in 'Eq(x, 20 + x/12)'" }, { "value": "1", "source": "the coefficient 1 of x on the left: moving x/12 across gives x − x/12 = 20, so x(1 − 1/12) = 20" }, { "value": "12", "source": "the 12 in 'x/12' of Eq(x, 20 + x/12)" } ], "calculator_value": "+2181818181818181818181818181818182E-32", "printed_value": "240/11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-62/5", "set": "wentworth-first-steps-in-algebra-1894/ex-62", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "116", "location": "Exercise 62, problem 5", "problem_latex": "Find the time between $1$ and $2$~o'clock when the\nhands of a clock point in opposite directions.", "markdown": "Find the time between $1$ and $2$ o’clock when the hands of a clock point in opposite directions.", "answer_latex": [ "$38\\frac{2}{11}$~min.\\ past 1~o'clock." ], "answer_markdown": [ "$38\\frac{2}{11}$ min. past 1 o’clock." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(420, 11)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, x/12 + 35)" ], "shape": [ "solve: Eq(x, N*x + N)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-62/6" ], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "35 ENTER 1 ENTER 12 1/x − ÷" ], "constants": [ { "value": "1", "source": "the coefficient of x in x − x/12 = 35, from moving the x/12 term across the equation in Eq(x, 35 + x/12)" } ], "calculator_value": "+3818181818181818181818181818181818E-32", "printed_value": "420/11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-62/6", "set": "wentworth-first-steps-in-algebra-1894/ex-62", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "116", "location": "Exercise 62, problem 6", "problem_latex": "At what time between $7$ and $8$~o'clock are the hands\nof a watch together?", "markdown": "At what time between $7$ and $8$ o’clock are the hands of a watch together?", "answer_latex": [ "$38\\frac{2}{11}$~min.\\ past 7~o'clock." ], "answer_markdown": [ "$38\\frac{2}{11}$ min. past 7 o’clock." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(420, 11)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x, x/12 + 35)" ], "shape": [ "solve: Eq(x, N*x + N)" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-62/5" ], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "35 ENTER 12 × 12 ENTER 12 ÷ 12 − +/− ÷" ], "calculator_value": "+3818181818181818181818181818181818E-32", "printed_value": "420/11", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-63/1", "set": "wentworth-first-steps-in-algebra-1894/ex-63", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "117", "location": "Exercise 63, problem 1", "problem_latex": "A rectangle has its length and breadth respectively $7$~feet\nlonger and $6$~feet shorter than the side of the equivalent\nsquare. Find its area.", "markdown": "A rectangle has its length and breadth respectively $7$ feet longer and $6$ feet shorter than the side of the equivalent square. Find its area.", "answer_latex": [ "$1764$~sq.~ft." ], "answer_markdown": [ "$1764$ sq. ft." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 42, A: 1764}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, (x - 6)*(x + 7)), Eq((x - 6)*(x + 7), x**2))" ], "shape": [ "solve: (Eq(a, (N + x)**2), Eq((N + x)**2, x**N))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-63/2", "set": "wentworth-first-steps-in-algebra-1894/ex-63", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "117", "location": "Exercise 63, problem 2", "problem_latex": "The length of a floor exceeds the breadth by $5$~feet.\nIf each dimension were $1$~foot more, the area of the floor\nwould be $42$~sq.~ft.\\ more. Find its dimensions.", "markdown": "The length of a floor exceeds the breadth by $5$ feet. If each dimension were $1$ foot more, the area of the floor would be $42$ sq. ft. more. Find its dimensions.", "answer_latex": [ "$18$~ft.\\ by $23$~ft." ], "answer_markdown": [ "$18$ ft. by $23$ ft." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{B: 18, L: 23}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, x + 5), Eq((a + 1)*(x + 1), a*x + 42))" ], "shape": [ "solve: (Eq(a, N + x), Eq((a + 1)*(x + 1), N + a*x))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-63/3", "set": "wentworth-first-steps-in-algebra-1894/ex-63", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "117", "location": "Exercise 63, problem 3", "problem_latex": "A rectangle whose length is $6$~feet more than its\nbreadth, would have its area $35$~sq.~ft.\\ more, if each dimension\nwere $1$~foot more. Find its dimensions.", "markdown": "A rectangle whose length is $6$ feet more than its breadth, would have its area $35$ sq. ft. more, if each dimension were $1$ foot more. Find its dimensions.", "answer_latex": [ "$14$~ft.\\ by $20$~ft." ], "answer_markdown": [ "$14$ ft. by $20$ ft." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{B: 14, L: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, x + 6), Eq((a + 1)*(x + 1), a*x + 35))" ], "shape": [ "solve: (Eq(a, N + x), Eq((a + 1)*(x + 1), N + a*x))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-63/4", "set": "wentworth-first-steps-in-algebra-1894/ex-63", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "117", "location": "Exercise 63, problem 4", "problem_latex": "The length of a rectangle exceeds its width by $3$~feet.\nIf the length is increased by $3$~feet and the width diminished\nby $2$~feet, the area will not be altered. Find its\ndimensions.", "markdown": "The length of a rectangle exceeds its width by $3$ feet. If the length is increased by $3$ feet and the width diminished by $2$ feet, the area will not be altered. Find its dimensions.", "answer_latex": [ "$12$~ft.\\ by $15$~ft." ], "answer_markdown": [ "$12$ ft. by $15$ ft." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{W: 12, L: 15}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, x + 3), Eq((a + 3)*(x - 2), a*x))" ], "shape": [ "solve: (Eq(a, N + x), Eq((N + a)*(N + x), a*x))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-63/5", "set": "wentworth-first-steps-in-algebra-1894/ex-63", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "117", "location": "Exercise 63, problem 5", "problem_latex": "The length of a floor exceeds its width by $10$~feet\nIf each dimension were $2$~feet more, the area would be $144$~sq.~ft.\\ \nmore. Find its dimensions.", "markdown": "The length of a floor exceeds its width by $10$ feet If each dimension were $2$ feet more, the area would be $144$ sq. ft. more. Find its dimensions.", "answer_latex": [ "$30$~ft.\\ by $40$~ft." ], "answer_markdown": [ "$30$ ft. by $40$ ft." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{W: 30, L: 40}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, x + 10), Eq((a + 2)*(x + 2), a*x + 144))" ], "shape": [ "solve: (Eq(a, N + x), Eq((N + a)*(N + x), N + a*x))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/1", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 1", "problem_latex": "The sum of two angles is $120°\\, 30'\\, 30''$ and their difference\n$59°\\, 30'\\, 30''$. Find the angles.", "markdown": "The sum of two angles is $120°\\, 30'\\, 30''$ and their difference $59°\\, 30'\\, 30''$. Find the angles.", "answer_latex": [ "$90°\\,0'\\,30''$; $30°\\,30'$." ], "answer_markdown": [ "$90°\\,0'\\,30''$; $30°\\,30'$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: Rational(324030, 3600), y: Rational(61, 2)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(-a + x, 7141/120), Eq(a + x, 14461/120))" ], "shape": [ "solve: (Eq(-a + x, N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/10", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 10", "problem_latex": "Find the time required for \\$$160$ to amount to \\$$250$\nat~$6$\\%.", "markdown": "Find the time required for $$160$ to amount to $$250$ at $6$%.", "answer_latex": [ "$9\\frac{3}{8}$~yr." ], "answer_markdown": [ "$9\\frac{3}{8}$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{t: Rational(75, 8)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(48*x/5 + 160, 250)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "250 ENTER 160 −", "160 ENTER 6 × 100 ÷", "÷" ], "constants": [ { "value": "100", "source": "the 100 implied by '6%' (6 per cent = 6/100)" } ], "calculator_value": "+9375E-3", "printed_value": "75/8", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/11", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 11", "problem_latex": "How much money must be invested at~$5$\\% to yield\nan annual income of~\\$$1250$?", "markdown": "How much money must be invested at $5$% to yield an annual income of $$1250$?", "answer_latex": [ "\\$$25,000$." ], "answer_markdown": [ "$$25,000$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{P: 25000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x/20, 1250)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1250 ENTER 5 ENTER 100 ÷ ÷" ], "constants": [ { "value": "5", "source": "the 5 in 'at 5%'" }, { "value": "100", "source": "the 100 in Rational(5, 100), the denominator of the percent" } ], "calculator_value": "+250E+2", "printed_value": "25000", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/12", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 12", "problem_latex": "Find the principal that will produce \\$$100$ a month\nif invested at $6$\\%~per annum.", "markdown": "Find the principal that will produce $$100$ a month if invested at $6$% per annum.", "answer_latex": [ "\\$$20,000$." ], "answer_markdown": [ "$$20,000$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{P: 20000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x/200, 100)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "100 ENTER 12 ×", "100 ×", "6 ÷" ], "constants": [ { "value": "100", "source": "the $100 a month in the problem text, the right-hand side of Eq(P*Rational(6, 100)/12, 100)" }, { "value": "12", "source": "the /12 in Eq(P*Rational(6, 100)/12, 100), which converts the annual rate to a monthly one; the problem says 'a month' and 'per annum'" }, { "value": "100", "source": "the 100 in Rational(6, 100), which converts the 6% to a fraction" }, { "value": "6", "source": "the 6 in 'at 6%' in the problem text, and the numerator in Rational(6, 100)" } ], "calculator_value": "+20000E+0", "printed_value": "20000", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/13", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 13", "problem_latex": "Find the rate if the interest on \\$$1000$ for $8$~months\nis~\\$$40$.", "markdown": "Find the rate if the interest on $$1000$ for $8$ months is $$40$.", "answer_latex": [ "\\$$6$\\%." ], "answer_markdown": [ "$$6$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{r: Rational(3, 50)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(2000*x/3, 40)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/14", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 14", "problem_latex": "Find the time for a sum of money on interest at~$5$\\%\nto double itself.", "markdown": "Find the time for a sum of money on interest at $5$% to double itself.", "answer_latex": [ "$20$~yr." ], "answer_markdown": [ "$20$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{t: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(a*(x/20 + 1), 2*a)" ], "shape": [ "solve: Eq(a*(N*x + 1), N*a)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 100 ÷", "1/x" ], "calculator_value": "+2E+1", "printed_value": "20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/2", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 2", "problem_latex": "Find the interest of \\$$1000$ for $3$~years and $4$~months\nat~$4$\\%.", "markdown": "Find the interest of $$1000$ for $3$ years and $4$ months at $4$%.", "answer_latex": [ "\\$$133\\frac{1}{3}$." ], "answer_markdown": [ "$$133\\frac{1}{3}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 133.333333333, printed 400/3 (half-unit 1.0e-40; correctly rounded at the printed digits: 133.333333333)", "problem_expr": "1000*Rational(4, 100)*Rational(10, 3)", "answer_expr": "Rational(400, 3)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 400/3" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#133\\frac{1}{3},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/3", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 3", "problem_latex": "Find the principal that will amount to \\$$2280$ in $3$~years\nand $6$~months at~$4$\\%.", "markdown": "Find the principal that will amount to $$2280$ in $3$ years and $6$ months at $4$%.", "answer_latex": [ "\\$$2000$." ], "answer_markdown": [ "$$2000$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{P: 2000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(57*x/50, 2280)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 12 ÷ 3 +", "4 ENTER 100 ÷ ×", "1 +", "2280 x↔y ÷" ], "constants": [ { "value": "12", "source": "the 12 months in a year, used to convert 6 months to 6/12 of a year (not printed in the problem text)" }, { "value": "100", "source": "the 100 in 4% (4 per cent = 4/100)" }, { "value": "1", "source": "the 1 in the SymPy expression P*(1 + ...), i.e. amount = principal × (1 + rate × time)" } ], "calculator_value": "+2E+3", "printed_value": "2000", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/4", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 4", "problem_latex": "Find the principal that will produce \\$$280$ interest in\n$2$~years and $4$~months at~$3$\\%.", "markdown": "Find the principal that will produce $$280$ interest in $2$ years and $4$ months at $3$%.", "answer_latex": [ "\\$$4000$." ], "answer_markdown": [ "$$4000$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{P: 4000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(7*x/100, 280)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 12 ÷ 2 +", "3 × 100 ÷", "280 x↔y ÷" ], "constants": [ { "value": "4", "source": "the 4 in '4 months'" }, { "value": "12", "source": "months per year, the conversion needed to express 4 months in years (not printed in the problem; standard unit constant)" }, { "value": "2", "source": "the 2 in '2 years'" }, { "value": "3", "source": "the 3 in '3%'" }, { "value": "100", "source": "the divisor in 'per cent' (3% = 3/100)" } ], "calculator_value": "+4000000000000000000000000000000001E-30", "printed_value": "4000", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/5", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 5", "problem_latex": "Find the principal that will produce \\$$270$ interest\nin $1$~year and $6$~months at~$6$\\%.", "markdown": "Find the principal that will produce $$270$ interest in $1$ year and $6$ months at $6$%.", "answer_latex": [ "\\$$3000$." ], "answer_markdown": [ "$$3000$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{P: 3000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(9*x/100, 270)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "270 ENTER 6 ENTER 100 ÷ ÷", "3 ENTER 2 ÷ ÷" ], "constants": [ { "value": "100", "source": "per cent means per hundred: the 6% in 'at 6%' is 6/100, so 100 is the divisor that converts it" }, { "value": "3 ENTER 2 ÷ (i.e. 3/2)", "source": "the 3/2 in the SymPy expression, from '1 year 6 months' = 1 1/2 years = 3/2 year" } ], "calculator_value": "+3E+3", "printed_value": "3000", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": 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"answer_markdown": [ "$$500$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{P: 500}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(59*x/50, 590)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 1 ENTER 2 ÷ +", "100 ÷", "4 ×", "1 +", "590 x↔y ÷" ], "constants": [ { "value": "4", "source": "the 4 in '4 years' (also the whole part of '4½%')" }, { "value": "1", "source": "the numerator 1 of '½' in '4½%'" }, { "value": "2", "source": "the denominator 2 of '½' in '4½%'" }, { "value": "100", "source": "the per cent sign '%' in '4½%' means divide by 100" } ], "calculator_value": "+5E+2", "printed_value": "500", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/7", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 7", "problem_latex": "Find the rate if the amount of \\$$250$ for $4$~years\nis~\\$$300$.", "markdown": "Find the rate if the amount of $$250$ for $4$ years is $$300$.", "answer_latex": [ "$5$\\%." ], "answer_markdown": [ "$5$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{r: Rational(1, 20)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(1000*x + 250, 300)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "300 ENTER 250 ÷", "1 −", "4 ÷" ], "constants": [ { "value": "300", "source": "the amount $300 in the problem text and in Eq(250*(1 + r*4), 300)" }, { "value": "250", "source": "the principal $250 in the problem text and in Eq(250*(1 + r*4), 300)" }, { "value": "1", "source": "the 1 in the formula A = P(1 + rt), as written in Eq(250*(1 + r*4), 300)" }, { "value": "4", "source": "the 4 years in the problem text and the t = 4 in Eq(250*(1 + r*4), 300)" } ], "calculator_value": "+5E-2", "printed_value": "1/20", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/8", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 8", "problem_latex": "Find the rate if \\$$1000$ amounts to \\$$2000$ in $16$~years\nand $8$~months.", "markdown": "Find the rate if $$1000$ amounts to $$2000$ in $16$ years and $8$ months.", "answer_latex": [ "$6$\\%." ], "answer_markdown": [ "$6$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{r: Rational(3, 50)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(50000*x/3 + 1000, 2000)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-64/9", "set": "wentworth-first-steps-in-algebra-1894/ex-64", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "121", "location": "Exercise 64, problem 9", "problem_latex": "Find the time required for the interest on \\$$400$ to\nbe \\$$54$ at~$4\\frac{1}{2}$\\%.", "markdown": "Find the time required for the interest on $$400$ to be $$54$ at $4\\frac{1}{2}$%.", "answer_latex": [ "$3$~yr." ], "answer_markdown": [ "$3$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{t: 3}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(18*x, 54)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 1 ENTER 2 ÷ +", "100 ÷", "400 ×", "54 x↔y ÷" ], "constants": [ { "value": "4", "source": "the 4 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records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/1", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 1", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 5x &+ 4y &&= 14 \\\\\n17x &- 3y &&= 31\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 5x &+ 4y &&= 14 \\\\ 17x &- 3y &&= 31 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 2$, $y = 1$." ], "answer_markdown": [ "$x = 2$, $y = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 2, y: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(4*a + 5*x, 14), Eq(-3*a + 17*x, 31))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], 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-1))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, -1))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/11", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 11", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 7x &+ \\Z y &&= 265 \\\\\n 3x &- 5y &&= \\Z\\Z5\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 7x &+ \\Z y &&= 265 \\\\ 3x &- 5y &&= \\Z\\Z5 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 35$, $y = 20$." ], "answer_markdown": [ "$x = 35$, $y = 20$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 35, y: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-5*a + 3*x, 5), Eq(a + 7*x, 265))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*x + a, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/12", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 12", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 2x &+ 3y &&= \\Z7 \\\\\n 8x &- 5y &&= 11\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 2x &+ 3y &&= \\Z7 \\\\ 8x &- 5y &&= 11 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 2$, $y = 1$." ], "answer_markdown": [ "$x = 2$, $y = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 2, y: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(3*a + 2*x, 7), Eq(-5*a + 8*x, 11))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/13", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 13", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 5x &+ 7y &&= 19 \\\\\n 7x &+ 4y &&= 15\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 5x &+ 7y &&= 19 \\\\ 7x &+ 4y &&= 15 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 1$, $y = 2$." ], "answer_markdown": [ "$x = 1$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 1, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(7*a + 5*x, 19), Eq(4*a + 7*x, 15))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/14", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 14", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n11x &- 12y &&= \\Z9 \\\\\n 4x &+ \\Z5y &&= 22\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 11x &- 12y &&= \\Z9 \\\\ 4x &+ \\Z5y &&= 22 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 3$, $y = 2$." ], "answer_markdown": [ "$x = 3$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(5*a + 4*x, 22), Eq(-12*a + 11*x, 9))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/15", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 15", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n x &+ 8y &&= 17 \\\\\n 7x &- 3y &&= \\Z1\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} x &+ 8y &&= 17 \\\\ 7x &- 3y &&= \\Z1 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 1$, $y = 2$." ], "answer_markdown": [ "$x = 1$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 1, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(8*a + x, 17), Eq(-3*a + 7*x, 1))" ], "shape": [ "solve: (Eq(N*a + x, N), Eq(N*a + N*x, 1))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/16", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 16", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 4x &+ 3y &&= 25 \\\\\n 5x &- 4y &&= \\Z8\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 4x &+ 3y &&= 25 \\\\ 5x &- 4y &&= \\Z8 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 4$, $y = 3$." ], "answer_markdown": [ "$x = 4$, $y = 3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 4, y: 3}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(3*a + 4*x, 25), Eq(-4*a + 5*x, 8))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/17", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 17", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n \\frac{2x}{3} &- \\frac{5y}{4} &&= 3 \\\\\n \\frac{7x}{4} &- \\frac{5y}{3} &&= \\frac{43}{3}\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} \\frac{2x}{3} &- \\frac{5y}{4} &&= 3 \\\\ \\frac{7x}{4} &- \\frac{5y}{3} &&= \\frac{43}{3} \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 12$, $y = 4$." ], "answer_markdown": [ "$x = 12$, $y = 4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 12, y: 4}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-5*a/4 + 2*x/3, 3), Eq(-5*a/3 + 7*x/4, 43/3))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/18", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 18", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n \\frac{7x}{6} &+ \\frac{6y}{7} &&= 32 \\\\\n \\frac{5x}{4} &- \\frac{2y}{3} &&= 1\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} \\frac{7x}{6} &+ \\frac{6y}{7} &&= 32 \\\\ \\frac{5x}{4} &- \\frac{2y}{3} &&= 1 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 12$, $y = 21$." ], "answer_markdown": [ "$x = 12$, $y = 21$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 12, y: 21}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(6*a/7 + 7*x/6, 32), Eq(-2*a/3 + 5*x/4, 1))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, 1))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/19", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 19", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{aligned}\n \\frac{x + y}{4} - \\frac{7x - 5y}{11} &= 3 \\\\\n \\frac{x}{5} - \\frac{2y}{7} + 1 &= 0\n\\end{aligned}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{aligned} \\frac{x + y}{4} - \\frac{7x - 5y}{11} &= 3 \\\\ \\frac{x}{5} - \\frac{2y}{7} + 1 &= 0 \\end{aligned}\\right\\}$", "answer_latex": [ " $x = 5$, $y = 7$." ], "answer_markdown": [ "$x = 5$, $y = 7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 7}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(31*a/44 - 17*x/44, 3), Eq(-2*a/7 + x/5 + 1, 0))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x + 1, 0))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/2", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 2", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 3x &- 2y &&= \\Z5 \\\\\n 2x &+ 5y &&= 16\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 3x &- 2y &&= \\Z5 \\\\ 2x &+ 5y &&= 16 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 3$, $y = 2$." ], "answer_markdown": [ "$x = 3$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(5*a + 2*x, 16), Eq(-2*a + 3*x, 5))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/20", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 20", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{aligned}\n \\frac{ 6x + 7y}{2} &= 22 \\\\\n \\frac{55y - 2x}{5} &= 20\n\\end{aligned}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{aligned} \\frac{ 6x + 7y}{2} &= 22 \\\\ \\frac{55y - 2x}{5} &= 20 \\end{aligned}\\right\\}$", "answer_latex": [ " $x = 5$, $y = 2$." ], "answer_markdown": [ "$x = 5$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(11*a - 2*x/5, 20), Eq(7*a/2 + 3*x, 22))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/21", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 21", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n \\frac{x + y}{2} &- \\frac{x - y}{3} &&= \\Z8 \\\\\n \\frac{x + y}{3} &+ \\frac{x - y}{4} &&= 11\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} \\frac{x + y}{2} &- \\frac{x - y}{3} &&= \\Z8 \\\\ \\frac{x + y}{3} &+ \\frac{x - y}{4} &&= 11 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 18$, $y = 6$." ], "answer_markdown": [ "$x = 18$, $y = 6$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 18, y: 6}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(5*a/6 + x/6, 8), Eq(a/12 + 7*x/12, 11))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/22", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 22", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{aligned}\n \\frac{8x - 5y}{7} + \\frac{11y - 4x}{5} &= 4 \\\\\n \\frac{17x - 13y}{5} + \\frac{2x}{3} &= 7\n\\end{aligned}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{aligned} \\frac{8x - 5y}{7} + \\frac{11y - 4x}{5} &= 4 \\\\ \\frac{17x - 13y}{5} + \\frac{2x}{3} &= 7 \\end{aligned}\\right\\}$", "answer_latex": [ " $x = 3$, $y = 2$." ], "answer_markdown": [ "$x = 3$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(52*a/35 + 12*x/35, 4), Eq(-13*a/5 + 61*x/15, 7))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/23", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 23", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n \\frac{ 5x - 3y}{3} &+ \\frac{7x - 5y}{11} &&= 4 \\\\\n \\frac{15y - 3x}{7} &+ \\frac{7y - 3x}{5} &&= 4\n \\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} \\frac{ 5x - 3y}{3} &+ \\frac{7x - 5y}{11} &&= 4 \\\\ \\frac{15y - 3x}{7} &+ \\frac{7y - 3x}{5} &&= 4 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 3$, $y = 2$." ], "answer_markdown": [ "$x = 3$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(124*a/35 - 36*x/35, 4), Eq(-16*a/11 + 76*x/33, 4))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/24", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 24", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{aligned}\n \\frac{2x - 3}{4} - \\frac{y - 8}{5} &= \\frac{y + 3}{4} \\\\\n \\frac{x - 7}{3} + \\frac{4y + 1}{11} &= 3\n \\end{aligned}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{aligned} \\frac{2x - 3}{4} - \\frac{y - 8}{5} &= \\frac{y + 3}{4} \\\\ \\frac{x - 7}{3} + \\frac{4y + 1}{11} &= 3 \\end{aligned}\\right\\}$", "answer_latex": [ " $x = 7$, $y = 8$." ], "answer_markdown": [ "$x = 7$, $y = 8$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 7, y: 8}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(4*a/11 + x/3 - 74/33, 3), Eq(-a/5 + x/2 + 17/20, a/4 + 3/4))" ], "shape": [ "solve: (Eq(N*a + N*x + N, N), Eq(N*a + N*x + N, N*a + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/25", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 25", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n \\frac{x - 2y}{6} &- \\frac{x + 3y}{4} &&= \\frac{3}{2} \\\\\n \\frac{2x - y}{6} &- \\frac{3x + y}{4} &&= \\frac{5y}{4}\n \\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} \\frac{x - 2y}{6} &- \\frac{x + 3y}{4} &&= \\frac{3}{2} \\\\ \\frac{2x - y}{6} &- \\frac{3x + y}{4} &&= \\frac{5y}{4} \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 8$, $y = -2$." ], "answer_markdown": [ "$x = 8$, $y = -2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 8, y: -2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-5*a/12 - 5*x/12, 5*a/4), Eq(-13*a/12 - x/12, 3/2))" ], "shape": [ "solve: (Eq(N*a + N*x, N*a), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/26", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 26", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n \\frac{x}{a + b} &+ \\frac{y}{a - b} &&= \\frac{1}{a - b} \\\\\n \\frac{x}{a + b} &- \\frac{y}{a - b} &&= \\frac{1}{a + b}\n \\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} \\frac{x}{a + b} &+ \\frac{y}{a - b} &&= \\frac{1}{a - b} \\\\ \\frac{x}{a + b} &- \\frac{y}{a - b} &&= \\frac{1}{a + b} \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = \\dfrac{a}{(a - b)}$, $y = \\dfrac{b}{(a + b)}$." ], "answer_markdown": [ "$x = \\dfrac{a}{(a - b)}$, $y = \\dfrac{b}{(a + b)}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: a/(a - b), y: b/(a + b)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-c/(a - b) + x/(a + b), 1/(a + b)), Eq(c/(a - b) + x/(a + b), 1/(a - b)))" ], "shape": [ "solve: (Eq(-c/(a - b) + x/(a + b), 1/(a + b)), Eq(c/(a - b) + x/(a + b), 1/(a - b)))" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/3", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 3", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 2x &- 3y &&= \\Z7 \\\\\n 5x &+ 2y &&= 27\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 2x &- 3y &&= \\Z7 \\\\ 5x &+ 2y &&= 27 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 5$, $y = 1$." ], "answer_markdown": [ "$x = 5$, $y = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-3*a + 2*x, 7), Eq(2*a + 5*x, 27))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/4", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 4", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 7x &+ 6y &&= 20 \\\\\n 2x &+ 5y &&= \\Z9\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 7x &+ 6y &&= 20 \\\\ 2x &+ 5y &&= \\Z9 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 2$, $y = 1$." ], "answer_markdown": [ "$x = 2$, $y = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 2, y: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(5*a + 2*x, 9), Eq(6*a + 7*x, 20))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/5", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 5", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n x &+ 5y &&= 11 \\\\\n 3x &+ 2y &&= \\Z7\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} x &+ 5y &&= 11 \\\\ 3x &+ 2y &&= \\Z7 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 1$, $y = 2$." ], "answer_markdown": [ "$x = 1$, $y = 2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 1, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(5*a + x, 11), Eq(2*a + 3*x, 7))" ], "shape": [ "solve: (Eq(N*a + x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/6", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 6", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 3x &- 5y &&= 13 \\\\\n 4x &- 7y &&= 17\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 3x &- 5y &&= 13 \\\\ 4x &- 7y &&= 17 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 6$, $y = 1$." ], "answer_markdown": [ "$x = 6$, $y = 1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 6, y: 1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-5*a + 3*x, 13), Eq(-7*a + 4*x, 17))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/7", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 7", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 8x &- \\Z y &&= \\Z3 \\\\\n 7x &+ 2y &&= 63\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 8x &- \\Z y &&= \\Z3 \\\\ 7x &+ 2y &&= 63 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 3$, $y = 21$." ], "answer_markdown": [ "$x = 3$, $y = 21$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 21}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(2*a + 7*x, 63), Eq(-a + 8*x, 3))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*x - a, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/8", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 8", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n 5x &- 4y &&= \\Z7 \\\\\n 7x &+ 3y &&= 70\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} 5x &- 4y &&= \\Z7 \\\\ 7x &+ 3y &&= 70 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 7$, $y = 7$." ], "answer_markdown": [ "$x = 7$, $y = 7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 7, y: 7}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(-4*a + 5*x, 7), Eq(3*a + 7*x, 70))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-65/9", "set": "wentworth-first-steps-in-algebra-1894/ex-65", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "124", "location": "Exercise 65, problem 9", "problem_latex": " \\raisebox{-0.5\\baselineskip}{$\\left.\\begin{alignedat}{2}\n x &+ 21y &&= \\Z2 \\\\\n 2x &+ 27y &&= 19\n\\end{alignedat}\\right\\}$}", "markdown": "-0.5$\\left.\\begin{alignedat}{2} x &+ 21y &&= \\Z2 \\\\ 2x &+ 27y &&= 19 \\end{alignedat}\\right\\}$", "answer_latex": [ " $x = 23$, $y = -1$." ], "answer_markdown": [ "$x = 23$, $y = -1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 23, y: -1}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(21*a + x, 2), Eq(27*a + 2*x, 19))" ], "shape": [ "solve: (Eq(N*a + x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/1", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 1", "problem_latex": " If A gives B \\$$200$, A~will then have half as much\nmoney as~B; but if B gives A \\$$200$, B~will have one-third\nas much as A\\@. How much has each?", "markdown": "If A gives B $$200$, A will then have half as much money as B; but if B gives A $$200$, B will have one-third as much as A. How much has each?", "answer_latex": [ " A, \\$$520$; B, \\$$440$." ], "answer_markdown": [ "A, $$520$; B, $$440$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{a: 520, b: 440}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x - 200, a/2 + 100), Eq(a - 200, x/3 + 200/3))" ], "shape": [ "solve: (Eq(N + x, N*a + N), Eq(N + a, N*x + N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/2", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 2", "problem_latex": " Half the sum of two numbers is~$20$, and three times\ntheir difference is~$18$. Find the numbers.", "markdown": "Half the sum of two numbers is $20$, and three times their difference is $18$. Find the numbers.", "answer_latex": [ " $23$~and~$17$." ], "answer_markdown": [ "$23$ and $17$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 23, y: 17}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a/2 + x/2, 20), Eq(-3*a + 3*x, 18))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/3", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 3", "problem_latex": " The sum of two numbers is~$36$, and their difference\nis equal to one-eighth of the smaller number increased by~$2$.\nFind the numbers.", "markdown": "The sum of two numbers is $36$, and their difference is equal to one-eighth of the smaller number increased by $2$. Find the numbers.", "answer_latex": [ " $20$~and~$16$." ], "answer_markdown": [ "$20$ and $16$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 20, y: 16}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(-a + x, a/8 + 2), Eq(a + x, 36))" ], "shape": [ "solve: (Eq(-a + x, N*a + N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/4", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 4", "problem_latex": " If $4$~yards of velvet and $3$~yards of silk are sold for~\\$$33$,\nand $5$~yards of velvet and $6$~yards of silk for~\\$$48$,\nwhat is the price per yard of the velvet and of the silk?", "markdown": "If $4$ yards of velvet and $3$ yards of silk are sold for $$33$, and $5$ yards of velvet and $6$ yards of silk for $$48$, what is the price per yard of the velvet and of the silk?", "answer_latex": [ " Velvet, \\$$6$; silk, \\$$3$." ], "answer_markdown": [ "Velvet, $$6$; silk, $$3$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{v: 6, s: 3}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(3*a + 4*x, 33), Eq(6*a + 5*x, 48))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/5", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 5", "problem_latex": " If $7$~bushels of wheat and $10$~of rye are sold for~\\$$15$,\nand $4$~bushels of wheat and $5$~of rye are sold for~\\$$8$, what\nis the price per bushel of the wheat and of the rye?", "markdown": "If $7$ bushels of wheat and $10$ of rye are sold for $$15$, and $4$ bushels of wheat and $5$ of rye are sold for $$8$, what is the price per bushel of the wheat and of the rye?", "answer_latex": [ " Wheat, \\$$1$; rye, \\$$\\frac{4}{5}$." ], "answer_markdown": [ "Wheat, $$1$; rye, $$\\frac{4}{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{w: 1, r: Rational(4, 5)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(5*a + 4*x, 8), Eq(10*a + 7*x, 15))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/6", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 6", "problem_latex": " If $12$~pounds of tea and $4$~pounds of coffee cost~\\$$7$,\nand $4$~pounds of tea and $12$~pounds of coffee cost~\\$$5$, what is\nthe price per pound of tea and of coffee?", "markdown": "If $12$ pounds of tea and $4$ pounds of coffee cost $$7$, and $4$ pounds of tea and $12$ pounds of coffee cost $$5$, what is the price per pound of tea and of coffee?", "answer_latex": [ " Tea, \\$$\\frac{1}{2}$; coffee, \\$$\\frac{1}{4}$." ], "answer_markdown": [ "Tea, $$\\frac{1}{2}$; coffee, $$\\frac{1}{4}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{t: Rational(1, 2), c: Rational(1, 4)}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(4*a + 12*x, 7), Eq(12*a + 4*x, 5))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-66/7", "set": "wentworth-first-steps-in-algebra-1894/ex-66", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "127", "location": "Exercise 66, problem 7", "problem_latex": " Six horses and $7$~cows can be bought for~\\$$1000$,\nand $11$~horses and $13$~cows for~\\$$1844$. Find the value of\na horse and of a cow.", "markdown": "Six horses and $7$ cows can be bought for $$1000$, and $11$ horses and $13$ cows for $$1844$. Find the value of a horse and of a cow.", "answer_latex": [ " Horse, \\$$92$; cow, \\$$64$." ], "answer_markdown": [ "Horse, $$92$; cow, $$64$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{h: 92, c: 64}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(7*a + 6*x, 1000), Eq(13*a + 11*x, 1844))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-67/1", "set": "wentworth-first-steps-in-algebra-1894/ex-67", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "128", "location": "Exercise 67, problem 1", "problem_latex": "If the numerator of a certain fraction is increased\nby~$2$ and its denominator diminished by~$2$, its value will\nbe~$1$. If the numerator is increased by the denominator\nand the denominator is diminished by~$5$, its value will be~$5$.\nFind the fraction.", "markdown": "If the numerator of a certain fraction is increased by $2$ and its denominator diminished by $2$, its value will be $1$. If the numerator is increased by the denominator and the denominator is diminished by $5$, its value will be $5$. Find the fraction.", "answer_latex": [ "$\\frac{3}{7}$." ], "answer_markdown": [ "$\\frac{3}{7}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 7}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq((x + 2)/(a - 2), 1), Eq((a + x)/(a - 5), 5))" ], "shape": [ "solve: (Eq((N + x)/(N + a), 1), Eq((a + x)/(N + a), N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-67/2", "set": "wentworth-first-steps-in-algebra-1894/ex-67", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "128", "location": "Exercise 67, problem 2", "problem_latex": "If $1$~is added to the denominator of a fraction, its\nvalue will be~$\\frac{1}{2}$. If $2$~is added to its numerator, its value\nwill be~$\\frac{3}{5}$. Find the fraction.", "markdown": "If $1$ is added to the denominator of a fraction, its value will be $\\frac{1}{2}$. If $2$ is added to its numerator, its value will be $\\frac{3}{5}$. Find the fraction.", "answer_latex": [ "$\\frac{13}{25}$." ], "answer_markdown": [ "$\\frac{13}{25}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 13, y: 25}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(x/(a + 1), 1/2), Eq((x + 2)/a, 3/5))" ], "shape": [ "solve: (Eq(x/(a + 1), N), Eq((N + x)/a, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-67/3", "set": "wentworth-first-steps-in-algebra-1894/ex-67", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "128", "location": "Exercise 67, problem 3", "problem_latex": "If $1$~is added to the numerator of a fraction, its value\nwill be~$\\frac{1}{5}$. If $1$~is added to its denominator, its value will\nbe~$\\frac{1}{7}$. Find the fraction.", "markdown": "If $1$ is added to the numerator of a fraction, its value will be $\\frac{1}{5}$. If $1$ is added to its denominator, its value will be $\\frac{1}{7}$. Find the fraction.", "answer_latex": [ "$\\frac{3}{20}$." ], "answer_markdown": [ "$\\frac{3}{20}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 20}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(x/(a + 1), 1/7), Eq((x + 1)/a, 1/5))" ], "shape": [ "solve: (Eq(x/(a + 1), N), Eq((x + 1)/a, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-67/4", "set": "wentworth-first-steps-in-algebra-1894/ex-67", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "128", "location": "Exercise 67, problem 4", "problem_latex": "If the numerator of a fraction is doubled and its\ndenominator diminished by~$1$, its value will be~$\\frac{1}{2}$. If its\ndenominator is doubled and its numerator increased by~$1$,\nits value will be~$\\frac{1}{7}$. Find the fraction.", "markdown": "If the numerator of a fraction is doubled and its denominator diminished by $1$, its value will be $\\frac{1}{2}$. If its denominator is doubled and its numerator increased by $1$, its value will be $\\frac{1}{7}$. Find the fraction.", "answer_latex": [ "$\\frac{5}{21}$." ], "answer_markdown": [ "$\\frac{5}{21}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 5, y: 21}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(2*x/(a - 1), 1/2), Eq((x + 1)/(2*a), 1/7))" ], "shape": [ "solve: (Eq(N*x/(a - 1), N), Eq(N*(x + 1)/a, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-67/5", "set": "wentworth-first-steps-in-algebra-1894/ex-67", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "128", "location": "Exercise 67, problem 5", "problem_latex": "In a certain proper fraction the difference between\nthe numerator and the denominator is~$15$. If the numerator\nis multiplied by~$4$ and the denominator increased by~$6$,\nits value will be~$1$. Find the fraction.", "markdown": "In a certain proper fraction the difference between the numerator and the denominator is $15$. If the numerator is multiplied by $4$ and the denominator increased by $6$, its value will be $1$. Find the fraction.", "answer_latex": [ "$\\frac{7}{22}$." ], "answer_markdown": [ "$\\frac{7}{22}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 7, y: 22}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(4*x/(a + 6), 1), Eq(a - x, 15))" ], "shape": [ "solve: (Eq(N*x/(N + a), 1), Eq(a - x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-67/none", "set": "wentworth-first-steps-in-algebra-1894/ex-67", "number": null, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "128", "location": "Exercise 67, problem None", "problem_latex": "A certain fraction becomes equal to~$\\frac{1}{2}$ if $2$~is added\nto its numerator, and equal to~$\\frac{1}{3}$ if $3$~is added to its denominator.\nFind the fraction.", "markdown": "A certain fraction becomes equal to $\\frac{1}{2}$ if $2$ is added to its numerator, and equal to $\\frac{1}{3}$ if $3$ is added to its denominator. Find the fraction.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 7, y: 18}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(x/(a + 3), 1/3), Eq((x + 2)/a, 1/2))" ], "shape": [ "solve: (Eq(x/(N + a), N), Eq((N + x)/a, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-68/1", "set": "wentworth-first-steps-in-algebra-1894/ex-68", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Exercise 68, problem 1", "problem_latex": "The sum of the two digits of a number is~$9$, and if $9$\nis added to the number, the digits will be reversed. Find\nthe number.", "markdown": "The sum of the two digits of a number is $9$, and if $9$ is added to the number, the digits will be reversed. Find the number.", "answer_latex": [ "$45$." ], "answer_markdown": [ "$45$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{N: 45, x: 4, y: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(x, 10*a + b), Eq(a + b, 9), Eq(10*a + b + 9, a + 10*b))" ], "shape": [ "solve: (Eq(x, N*a + b), Eq(a + b, N), Eq(N*a + N + b, N*b + a))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-68/2", "set": "wentworth-first-steps-in-algebra-1894/ex-68", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Exercise 68, problem 2", "problem_latex": "A certain number of two digits is equal to eight times\nthe sum of its digits, and if $45$~is subtracted from the\nnumber, the digits will be reversed. Find the number.", "markdown": "A certain number of two digits is equal to eight times the sum of its digits, and if $45$ is subtracted from the number, the digits will be reversed. Find the number.", "answer_latex": [ "$72$." ], "answer_markdown": [ "$72$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{N: 72, x: 7, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(x, 8*a + 8*b), Eq(x, 10*a + b), Eq(x - 45, a + 10*b))" ], "shape": [ "solve: (Eq(x, N*a + N*b), Eq(x, N*a + b), Eq(N + x, N*b + a))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-68/3", "set": "wentworth-first-steps-in-algebra-1894/ex-68", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Exercise 68, problem 3", "problem_latex": "The sum of a certain number of two digits and the\nnumber formed by reversing the digits is~$132$, and the\ndifference of these numbers is~$18$. Find the numbers.", "markdown": "The sum of a certain number of two digits and the number formed by reversing the digits is $132$, and the difference of these numbers is $18$. Find the numbers.", "answer_latex": [ "$75$~and~$57$." ], "answer_markdown": [ "$75$ and $57$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{N: 75, M: 57, x: 7, y: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(a, b + 10*c), Eq(x, 10*b + c), Eq(-a + x, 18), Eq(a + x, 132))" ], "shape": [ "solve: (Eq(a, N*c + b), Eq(x, N*b + c), Eq(-a + x, N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-68/4", "set": "wentworth-first-steps-in-algebra-1894/ex-68", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "129", "location": "Exercise 68, problem 4", "problem_latex": "The sum of the two digits of a number is~$9$, and if\nthe number is divided by the sum of its digits, the quotient\nis~$6$. Find the number.", "markdown": "The sum of the two digits of a number is $9$, and if the number is divided by the sum of its digits, the quotient is $6$. Find the number.", "answer_latex": [ "$54$." ], "answer_markdown": [ "$54$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": null, "answer_expr": "{N: 54, x: 5, y: 4}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: (Eq(x, 10*a + b), Eq(x/(a + b), 6), Eq(a + b, 9))" ], "shape": [ "solve: (Eq(x, N*a + b), Eq(x/(a + b), N), Eq(a + b, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.arith", "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-69/1", "set": "wentworth-first-steps-in-algebra-1894/ex-69", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "130", "location": "Exercise 69, problem 1", "problem_latex": " A sum of money, at simple interest, amounted in $5$~years\n to~\\$$3000$, and in $6$~years to~\\$$3100$. Find the sum\n and the rate of interest.", "markdown": "A sum of money, at simple interest, amounted in $5$ years to $$3000$, and in $6$ years to $$3100$. Find the sum and the rate of interest.", "answer_latex": [ "\\$$2500$ at $4$\\%." ], "answer_markdown": [ "$$2500$ at $4$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 2500, y: 4}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a*x/20 + x, 3000), Eq(3*a*x/50 + x, 3100))" ], "shape": [ "solve: (Eq(N*a*x + x, N), Eq(N*a*x + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-69/2", "set": "wentworth-first-steps-in-algebra-1894/ex-69", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "130", "location": "Exercise 69, problem 2", "problem_latex": " A sum of money, at simple interest, amounted in $10$~months\n to~\\$$1680$, and in $18$~months to~\\$$1744$. Find the\n sum and the rate of interest.", "markdown": "A sum of money, at simple interest, amounted in $10$ months to $$1680$, and in $18$ months to $$1744$. Find the sum and the rate of interest.", "answer_latex": [ "\\$$1600$ at $6$\\%." ], "answer_markdown": [ "$$1600$ at $6$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 1600, y: 6}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a*x/120 + x, 1680), Eq(3*a*x/200 + x, 1744))" ], "shape": [ "solve: (Eq(N*a*x + x, N), Eq(N*a*x + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-69/3", "set": "wentworth-first-steps-in-algebra-1894/ex-69", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "130", "location": "Exercise 69, problem 3", "problem_latex": "A man has \\$$10,000$ invested, a part at~$4$\\%, and the\nremainder at~$5$\\%. The annual income from his $4$\\%~investment\nis \\$$40$~more than from his $5$\\%~investment. Find the\nsum invested at~$4$\\% and at~$5$\\%.", "markdown": "A man has $$10,000$ invested, a part at $4$%, and the remainder at $5$%. The annual income from his $4$% investment is $$40$ more than from his $5$% investment. Find the sum invested at $4$% and at $5$%.", "answer_latex": [ "\\$$6000$ at $4$\\%; \\$$4000$ at $5$\\%." ], "answer_markdown": [ "$$6000$ at $4$%; $$4000$ at $5$%." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 6000, y: 4000}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x/25, a/20 + 40), Eq(a + x, 10000))" ], "shape": [ "solve: (Eq(N*x, N*a + N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-69/none", "set": "wentworth-first-steps-in-algebra-1894/ex-69", "number": null, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "130", "location": "Exercise 69, problem None", "problem_latex": " A sum of money, at simple interest, amounted to\n\\$$2480$ in $4$~years, and to \\$$2600$ in $5$~years. Find the sum\nand the rate of interest.", "markdown": "A sum of money, at simple interest, amounted to $$2480$ in $4$ years, and to $$2600$ in $5$ years. Find the sum and the rate of interest.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/1", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 1", "problem_latex": " In $x$~years a man will be $40$~years old; what is his\npresent age?", "markdown": "In $x$ years a man will be $40$ years old; what is his present age?", "answer_latex": [ "$(40 - x)$~yr." ], "answer_markdown": [ "$(40 - x)$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(P + x, 40)", "answer_expr": "40 - x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(a + x, 40)" ], "shape": [ "solve: Eq(a + x, N)" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-16/29" ], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/10", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 10", "problem_latex": "If $a$~stands for~$10$, and $b$~for~$2$, find the value of\n$2(a - 2b)$.", "markdown": "If $a$ stands for $10$, and $b$ for $2$, find the value of $2(a - 2b)$.", "answer_latex": [ "$12$." ], "answer_markdown": [ "$12$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 12.0, printed 12 (half-unit 0.5; correctly rounded at the printed digits: 12.0)", "problem_expr": "2*(10 - 2*2)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 12" ], "shape": [ "evaluate: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-76/15" ], "needs": [ "core.arith" ], "expectation": "X#12,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/11", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 11", "problem_latex": "How many cents in $a$~dollars, $b$~quarters, and $c$~dimes?", "markdown": "How many cents in $a$ dollars, $b$ quarters, and $c$ dimes?", "answer_latex": [ "$100a + 25b + 10c$." ], "answer_markdown": [ "$100a + 25b + 10c$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/12", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 12", "problem_latex": "A book-shelf contains French, Latin, and Greek\nbooks. There are $100$~books in all, and there are $x$~Latin\nand $y$~Greek books. How many French books are there?", "markdown": "A book-shelf contains French, Latin, and Greek books. There are $100$ books in all, and there are $x$ Latin and $y$ Greek books. How many French books are there?", "answer_latex": [ "$100 - x - y$." ], "answer_markdown": [ "$100 - x - y$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(F + x + y, 100)", "answer_expr": "100 - x - y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(a + b + x, 100)" ], "shape": [ "solve: Eq(a + b + x, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/13", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 13", "problem_latex": "A regiment of men is drawn up in $10$~ranks of $80$~men\neach, and there are $15$~men over. How many men\nare there in the regiment?", "markdown": "A regiment of men is drawn up in $10$ ranks of $80$ men each, and there are $15$ men over. How many men are there in the regiment?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/14", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 14", "problem_latex": "A regiment of men is drawn up in $x$~ranks of $y$~men\neach, and there are $c$~men over. How many men are there\nin the regiment?", "markdown": "A regiment of men is drawn up in $x$ ranks of $y$ men each, and there are $c$ men over. How many men are there in the regiment?", "answer_latex": [ "$xy + c$." ], "answer_markdown": [ "$xy + c$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/2", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 2", "problem_latex": " How old will a man be in $y$~years, if his present age\nis $a$~years?", "markdown": "How old will a man be in $y$ years, if his present age is $a$ years?", "answer_latex": [ "$(a + y)$~yr." ], "answer_markdown": [ "$(a + y)$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(F - y, a)", "answer_expr": "a + y" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(-b + x, a)" ], "shape": [ "solve: Eq(-b + x, a)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/3", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 3", "problem_latex": "What is the value of~$x$ if $7x$~equals~$28$?", "markdown": "What is the value of $x$ if $7x$ equals $28$?", "answer_latex": [ "$4$." ], "answer_markdown": [ "$4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(7*x, 28)", "answer_expr": "4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(7*x, 28)" ], "shape": [ "solve: Eq(N*x, N)" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": "X=4", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "28 ENTER 7 ÷" ], "calculator_value": "+4E+0", "printed_value": "4", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/4", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 4", "problem_latex": "If it takes $3$~men $4$~days to reap a field, how many\ndays will it take one man to reap it?", "markdown": "If it takes $3$ men $4$ days to reap a field, how many days will it take one man to reap it?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/5", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 5", "problem_latex": "If it takes $a$~men $b$~days to reap a field, how many\ndays will it take one man to reap it?", "markdown": "If it takes $a$ men $b$ days to reap a field, how many days will it take one man to reap it?", "answer_latex": [ "$ab$." ], "answer_markdown": [ "$ab$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(a*b, D)", "answer_expr": "a*b" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(a*b, x)" ], "shape": [ "solve: Eq(a*b, x)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/6", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 6", "problem_latex": "What is the excess of~$5x$ over~$3x$?", "markdown": "What is the excess of $5x$ over $3x$?", "answer_latex": [ "$5x - 3x$." ], "answer_markdown": [ "$5x - 3x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/7", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 7", "problem_latex": "By how much does $20 - 3$ exceed $(10 + 1)$?", "markdown": "By how much does $20 - 3$ exceed $(10 + 1)$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/8", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 8", "problem_latex": "By how much does $2x - 3$ exceed $(x + 1)$?", "markdown": "By how much does $2x - 3$ exceed $(x + 1)$?", "answer_latex": [ "$2x - 3 - (x + 1)$." ], "answer_markdown": [ "$2x - 3 - (x + 1)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-7/9", "set": "wentworth-first-steps-in-algebra-1894/ex-7", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "16", "location": "Exercise 7, problem 9", "problem_latex": "If $x$~stands for~$10$, find the value of~$4(3x - 20)$.", "markdown": "If $x$ stands for $10$, find the value of $4(3x - 20)$.", "answer_latex": [ "$40$." ], "answer_markdown": [ "$40$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 40.0, printed 40 (half-unit 0.5; correctly rounded at the printed digits: 40.0)", "problem_expr": "4*(3*10 - 20)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 40" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#40,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/1", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 1", "problem_latex": "Half the sum of two numbers is~$20$; and $5$~times\ntheir difference is~$20$. Find the numbers.", "markdown": "Half the sum of two numbers is $20$; and $5$ times their difference is $20$. Find the numbers.", "answer_latex": [ "$22$~and~$18$." ], "answer_markdown": [ "$22$ and $18$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "x": 22, "y": 18 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a/2 + x/2, 20), Eq(-5*a + 5*x, 20))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/2", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 2", "problem_latex": "A certain number when divided by a second number\ngives $7$ for a quotient and $4$ for a remainder. If three\ntimes the first number is divided by twice the second\nnumber, the quotient is~$11$ and the remainder~$4$. Find\nthe numbers.", "markdown": "A certain number when divided by a second number gives $7$ for a quotient and $4$ for a remainder. If three times the first number is divided by twice the second number, the quotient is $11$ and the remainder $4$. Find the numbers.", "answer_latex": [ "$60$~and~$8$." ], "answer_markdown": [ "$60$ and $8$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "x": 60, "y": 8 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 7*a + 4), Eq(3*x, 22*a + 4))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(N*x, N*a + N))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/3", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 3", "problem_latex": "A fraction becomes $\\frac{4}{5}$ in value by the addition of~$2$ to\nits numerator and $3$~to its denominator. If $2$~is subtracted\nfrom its numerator and $1$~from its denominator, the value\nof the fraction is~$\\frac{3}{4}$. Find the fraction.", "markdown": "A fraction becomes $\\frac{4}{5}$ in value by the addition of $2$ to its numerator and $3$ to its denominator. If $2$ is subtracted from its numerator and $1$ from its denominator, the value of the fraction is $\\frac{3}{4}$. Find the fraction.", "answer_latex": [ "$\\frac{14}{17}$." ], "answer_markdown": [ "$\\frac{14}{17}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "x": 14, "y": 17 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq((x - 2)/(a - 1), 3/4), Eq((x + 2)/(a + 3), 4/5))" ], "shape": [ "solve: (Eq((N + x)/(a - 1), N), Eq((N + x)/(N + a), N))" ], "same_problem_in": [], "needs": [ "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/4", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 4", "problem_latex": "A farmer sold $50$~bushels of wheat and $30$~of barley\nfor $74$~dollars; and at the same prices he sold $30$~bushels\nof wheat and $50$~bushels of barley for $70$~dollars. What\nwas the price of the wheat and of the barley per bushel?", "markdown": "A farmer sold $50$ bushels of wheat and $30$ of barley for $74$ dollars; and at the same prices he sold $30$ bushels of wheat and $50$ bushels of barley for $70$ dollars. What was the price of the wheat and of the barley per bushel?", "answer_latex": [ "Wheat, \\$$1$; barley, \\$$\\frac{4}{5}$." ], "answer_markdown": [ "Wheat, $$1$; barley, $$\\frac{4}{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "b": "4/5", "w": 1 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(30*a + 50*x, 74), Eq(50*a + 30*x, 70))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.frac", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/5", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 5", "problem_latex": "If A gave \\$$10$ to~B, he would then have three times\nas much money as~B; but if B gave \\$$5$ to~A, A~would\nhave four times as much as~B\\@. How much has each?", "markdown": "If A gave $$10$ to B, he would then have three times as much money as B; but if B gave $$5$ to A, A would have four times as much as B. How much has each?", "answer_latex": [ "A, \\$$235$; B, \\$$65$." ], "answer_markdown": [ "A, $$235$; B, $$65$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "A": 235, "B": 65 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x - 10, 3*a + 30), Eq(x + 5, 4*a - 20))" ], "shape": [ "solve: (Eq(N + x, N*a + N), Eq(N + x, N*a + N))" ], "same_problem_in": [], "needs": [ "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/6", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 6", "problem_latex": "A and B have together \\$$100$. If A~were to spend\none-half of his money, and B~one-third of his, they would\nthen have only \\$$55$ between them. How much money\nhas each?", "markdown": "A and B have together $$100$. If A were to spend one-half of his money, and B one-third of his, they would then have only $$55$ between them. How much money has each?", "answer_latex": [ "A, \\$$70$; B, \\$$30$." ], "answer_markdown": [ "A, $$70$; B, $$30$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "A": 70, "B": 30 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(2*a/3 + x/2, 55), Eq(a + x, 100))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "core.frac", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/7", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 7", "problem_latex": "A fruit-dealer sold $6$~lemons and $3$~oranges for $21$~cents,\nand $3$~lemons and $8$~oranges for $30$~cents. What\nwas the price of each?", "markdown": "A fruit-dealer sold $6$ lemons and $3$ oranges for $21$ cents, and $3$ lemons and $8$ oranges for $30$ cents. What was the price of each?", "answer_latex": [ "Lemon, $2$~cts.; orange, $3$~cts." ], "answer_markdown": [ "Lemon, $2$ cts.; orange, $3$ cts." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "L": 2, "O": 3 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(8*a + 3*x, 30), Eq(3*a + 6*x, 21))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-70/8", "set": "wentworth-first-steps-in-algebra-1894/ex-70", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "131", "location": "Exercise 70, problem 8", "problem_latex": "If A gives me $10$~apples, he will have just twice as\nmany as~B\\@. If he gives the $10$~apples to~$B$ instead of to\nme, A~and~B will each have the same number. How\nmany apples has each?", "markdown": "If A gives me $10$ apples, he will have just twice as many as B. If he gives the $10$ apples to $B$ instead of to me, A and B will each have the same number. How many apples has each?", "answer_latex": [ "A, $30$~apples; B, $10$~apples." ], "answer_markdown": [ "A, $30$ apples; B, $10$ apples." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "A": 30, "B": 10 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x - 10, 2*a), Eq(x - 10, a + 10))" ], "shape": [ "solve: (Eq(N + x, N*a), Eq(N + x, N + a))" ], "same_problem_in": [], "needs": [ "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/1", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 1", "problem_latex": " $5x^{2} - 2 = 3x^{2} + 6$.", "markdown": "$5x^{2} - 2 = 3x^{2} + 6$.", "answer_latex": [ " $±2$." ], "answer_markdown": [ "$±2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(5*x**2 - 2, 3*x**2 + 6)", "answer_expr": "[-2, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(5*x**2 - 2, 3*x**2 + 6)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/10", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 10", "problem_latex": " $86 - 52x = 2(8 - x)(2 - 3x)$.", "markdown": "$86 - 52x = 2(8 - x)(2 - 3x)$.", "answer_latex": [ " $±3$." ], "answer_markdown": [ "$±3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(86 - 52*x, 2*(8 - x)*(2 - 3*x))", "answer_expr": "[-3, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-52*x + 86, (-3*x + 2)*(-2*x + 16))" ], "shape": [ "solve: Eq(N*x + N, (N*x + N)**2)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/11", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 11", "problem_latex": " Find two numbers that are to each other as $3$~to~$4$,\nand the difference of whose squares is~$112$.", "markdown": "Find two numbers that are to each other as $3$ to $4$, and the difference of whose squares is $112$.", "answer_latex": [ " $12$~and~$16$." ], "answer_markdown": [ "$12$ and $16$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{s: 12, l: 16}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(4*x, 3*a), Eq(a**2 - x**2, 112))" ], "shape": [ "solve: (Eq(N*x, N*a), Eq(a**N - x**N, N))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/12", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 12", "problem_latex": " A boy bought a number of oranges for $36$~cents.\nThe price of an orange was to the number bought as $1$~to~$4$.\nHow many oranges did he buy, and how many cents\ndid each orange cost?", "markdown": "A boy bought a number of oranges for $36$ cents. The price of an orange was to the number bought as $1$ to $4$. How many oranges did he buy, and how many cents did each orange cost?", "answer_latex": [ " $12$~oranges at $3$~cts." ], "answer_markdown": [ "$12$ oranges at $3$ cts." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{n: 12, p: 3}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 4*a), Eq(a*x, 36))" ], "shape": [ "solve: (Eq(x, N*a), Eq(a*x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/13", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 13", "problem_latex": " A certain street contains $144$~square rods, and the\nlength is $16$~times the width. Find the width.", "markdown": "A certain street contains $144$ square rods, and the length is $16$ times the width. Find the width.", "answer_latex": [ " $3$~rods." ], "answer_markdown": [ "$3$ rods." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{w: 3}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 16*x), Eq(a*x, 144))" ], "shape": [ "solve: (Eq(a, N*x), Eq(a*x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.linsys" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "144 ENTER 16 ÷", "√x" ], "calculator_value": "+3E+0", "printed_value": "3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/14", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 14", "problem_latex": " Find the number of rods in the length, and in the\nwidth of a rectangular field containing $3\\frac{3}{5}$~acres, if the\nlength is $4$~times the width.", "markdown": "Find the number of rods in the length, and in the width of a rectangular field containing $3\\frac{3}{5}$ acres, if the length is $4$ times the width.", "answer_latex": [ " Width, $12$~rd.; length, $48$~rd." ], "answer_markdown": [ "Width, $12$ rd.; length, $48$ rd." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{w: 12, L: 48}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 4*x), Eq(a*x, 576))" ], "shape": [ "solve: (Eq(a, N*x), Eq(a*x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.frac", "core.linsys", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/2", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 2", "problem_latex": " $3x^{2} + 1 = 2x^{2} + 10$.", "markdown": "$3x^{2} + 1 = 2x^{2} + 10$.", "answer_latex": [ " $±3$." ], "answer_markdown": [ "$±3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*x**2 + 1, 2*x**2 + 10)", "answer_expr": "[-3, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x**2 + 1, 2*x**2 + 10)" ], "shape": [ "solve: Eq(N*x**N + 1, N*x**N + N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/3", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 3", "problem_latex": " $4x^{2} - 50 = x^{2} + 25$.", "markdown": "$4x^{2} - 50 = x^{2} + 25$.", "answer_latex": [ " $±5$." ], "answer_markdown": [ "$±5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*x**2 - 50, x**2 + 25)", "answer_expr": "[-5, 5]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(4*x**2 - 50, x**2 + 25)" ], "shape": [ "solve: Eq(N*x**N + N, N + x**N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/4", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 4", "problem_latex": " $(x - 6)(x + 6) = 28$.", "markdown": "$(x - 6)(x + 6) = 28$.", "answer_latex": [ " $±8$." ], "answer_markdown": [ "$±8$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x - 6)*(x + 6), 28)", "answer_expr": "[-8, 8]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((x - 6)*(x + 6), 28)" ], "shape": [ "solve: Eq((N + x)**2, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/5", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 5", "problem_latex": " $(x - 5)(x + 5) = 24$.", "markdown": "$(x - 5)(x + 5) = 24$.", "answer_latex": [ " $±7$." ], "answer_markdown": [ "$±7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x - 5)*(x + 5), 24)", "answer_expr": "[-7, 7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((x - 5)*(x + 5), 24)" ], "shape": [ "solve: Eq((N + x)**2, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/6", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 6", "problem_latex": " $3(x^{2} - 11) + 2(x^{2} - 5) = 82$.", "markdown": "$3(x^{2} - 11) + 2(x^{2} - 5) = 82$.", "answer_latex": [ " $±5$." ], "answer_markdown": [ "$±5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*(x**2 - 11) + 2*(x**2 - 5), 82)", "answer_expr": "[-5, 5]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(5*x**2 - 43, 82)" ], "shape": [ "solve: Eq(N*x**N + N, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/7", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 7", "problem_latex": " $11(x^{2} + 5) + 6(3 - x^{2}) = 198$.", "markdown": "$11(x^{2} + 5) + 6(3 - x^{2}) = 198$.", "answer_latex": [ " $±5$." ], "answer_markdown": [ "$±5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(11*(x**2 + 5) + 6*(3 - x**2), 198)", "answer_expr": "[-5, 5]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(5*x**2 + 73, 198)" ], "shape": [ "solve: Eq(N*x**N + N, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/8", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 8", "problem_latex": " $5x^{2} + 3 - 2(17 - x^{2}) = 32$.", "markdown": "$5x^{2} + 3 - 2(17 - x^{2}) = 32$.", "answer_latex": [ " $±3$." ], "answer_markdown": [ "$±3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(5*x**2 + 3 - 2*(17 - x**2), 32)", "answer_expr": "[-3, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(7*x**2 - 31, 32)" ], "shape": [ "solve: Eq(N*x**N + N, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-71/9", "set": "wentworth-first-steps-in-algebra-1894/ex-71", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "134", "location": "Exercise 71, problem 9", "problem_latex": " $4(x + 1) - 4(x - 1) = x^{2} - 1$.", "markdown": "$4(x + 1) - 4(x - 1) = x^{2} - 1$.", "answer_latex": [ " $±3$." ], "answer_markdown": [ "$±3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*(x + 1) - 4*(x - 1), x**2 - 1)", "answer_expr": "[-3, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(8, x**2 - 1)" ], "shape": [ "solve: Eq(N, x**N - 1)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/1", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 1", "problem_latex": "$x^{2} - 12x + 27 = 0$.", "markdown": "$x^{2} - 12x + 27 = 0$.", "answer_latex": [ "$9$~or~$3$." ], "answer_markdown": [ "$9$ or $3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 12*x + 27, 0)", "answer_expr": "[9, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 12*x + 27, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/10", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 10", "problem_latex": "$3x^{2} - 10x + 3 = 0$.", "markdown": "$3x^{2} - 10x + 3 = 0$.", "answer_latex": [ "$3$~or~$\\frac{1}{3}$." ], "answer_markdown": [ "$3$ or $\\frac{1}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*x**2 - 10*x + 3, 0)", "answer_expr": "[3, Rational(1, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x**2 - 10*x + 3, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/11", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 11", "problem_latex": "$x^{2} - 14x - 51 = 0$.", "markdown": "$x^{2} - 14x - 51 = 0$.", "answer_latex": [ "$17$~or~$-3$." ], "answer_markdown": [ "$17$ or $-3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 14*x - 51, 0)", "answer_expr": "[17, -3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 14*x - 51, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/12", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 12", "problem_latex": "$34x - x^{2} - 225 = 0$.", "markdown": "$34x - x^{2} - 225 = 0$.", "answer_latex": [ "$25$~or~$9$." ], "answer_markdown": [ "$25$ or $9$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(34*x - x**2 - 225, 0)", "answer_expr": "[25, 9]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-x**2 + 34*x - 225, 0)" ], "shape": [ "solve: Eq(N*x + N - x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/13", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 13", "problem_latex": "$x^{2} + x - 20 = 0$.", "markdown": "$x^{2} + x - 20 = 0$.", "answer_latex": [ "$4$~or~$-5$." ], "answer_markdown": [ "$4$ or $-5$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 + x - 20, 0)", "answer_expr": "[4, -5]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 + x - 20, 0)" ], "shape": [ "solve: Eq(N + x + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/14", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 14", "problem_latex": "$x^{2} - x - 12 = 0$.", "markdown": "$x^{2} - x - 12 = 0$.", "answer_latex": [ "$4$~or~$-3$." ], "answer_markdown": [ "$4$ or $-3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - x - 12, 0)", "answer_expr": "[4, -3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - x - 12, 0)" ], "shape": [ "solve: Eq(N - x + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/15", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 15", "problem_latex": "$2x^{2} - 12x = - 10$.", "markdown": "$2x^{2} - 12x = - 10$.", "answer_latex": [ "$5$~or~$1$." ], "answer_markdown": [ "$5$ or $1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2*x**2 - 12*x, -10)", "answer_expr": "[5, 1]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x**2 - 12*x, -10)" ], "shape": [ "solve: Eq(N*x + N*x**N, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/16", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 16", "problem_latex": "$3x^{2} + 12x - 36 = 0$.", "markdown": "$3x^{2} + 12x - 36 = 0$.", "answer_latex": [ "$2$~or~$-6$." ], "answer_markdown": [ "$2$ or $-6$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*x**2 + 12*x - 36, 0)", "answer_expr": "[2, -6]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x**2 + 12*x - 36, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/17", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 17", "problem_latex": "$(2x - 1)^{2} + 9 = 6(2x - 1)$.", "markdown": "$(2x - 1)^{2} + 9 = 6(2x - 1)$.", "answer_latex": [ "$2$~or~$2$." ], "answer_markdown": [ "$2$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((2*x - 1)**2 + 9, 6*(2*x - 1))", "answer_expr": "[2, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((2*x - 1)**2 + 9, 12*x - 6)" ], "shape": [ "solve: Eq(N + (N*x - 1)**N, N*x + N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/18", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 18", "problem_latex": "$6(9x^{2} - x) = 55(x^{2} - 1)$.", "markdown": "$6(9x^{2} - x) = 55(x^{2} - 1)$.", "answer_latex": [ "$5$~or~$-11$." ], "answer_markdown": [ "$5$ or $-11$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(6*(9*x**2 - x), 55*(x**2 - 1))", "answer_expr": "[5, -11]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(54*x**2 - 6*x, 55*x**2 - 55)" ], "shape": [ "solve: Eq(N*x + N*x**N, N*x**N + N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/19", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 19", "problem_latex": "$32 - 3x^{2} - 10x = 0$.", "markdown": "$32 - 3x^{2} - 10x = 0$.", "answer_latex": [ "$2$~or~$-5\\frac{1}{3}$." ], "answer_markdown": [ "$2$ or $-5\\frac{1}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(32 - 3*x**2 - 10*x, 0)", "answer_expr": "[2, Rational(-16, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-3*x**2 - 10*x + 32, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/2", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 2", "problem_latex": "$x^{2} - 6x + 8 = 0$.", "markdown": "$x^{2} - 6x + 8 = 0$.", "answer_latex": [ "$4$~or~$2$." ], "answer_markdown": [ "$4$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 6*x + 8, 0)", "answer_expr": "[4, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 6*x + 8, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/20", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 20", "problem_latex": "$9x^{2} - 6x - 143 = 0$.", "markdown": "$9x^{2} - 6x - 143 = 0$.", "answer_latex": [ "$4\\frac{1}{3}$~or~$-3\\frac{2}{3}$." ], "answer_markdown": [ "$4\\frac{1}{3}$ or $-3\\frac{2}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(9*x**2 - 6*x - 143, 0)", "answer_expr": "[Rational(13, 3), Rational(-11, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(9*x**2 - 6*x - 143, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/21", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 21", "problem_latex": "$\\dfrac{x}{x - 1} - \\dfrac{x - 1}{x} = \\dfrac{3}{2}$.", "markdown": "$\\dfrac{x}{x - 1} - \\dfrac{x - 1}{x} = \\dfrac{3}{2}$.", "answer_latex": [ "$2$~or~$\\frac{1}{3}$." ], "answer_markdown": [ "$2$ or $\\frac{1}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x/(x - 1) - (x - 1)/x, Rational(3, 2))", "answer_expr": "[2, Rational(1, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/(x - 1) - (x - 1)/x, 3/2)" ], "shape": [ "solve: Eq(x/(x - 1) - (x - 1)/x, N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/22", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 22", "problem_latex": "$\\dfrac{1}{x - 2} + \\dfrac{2}{x + 2} = \\dfrac{5}{6}$.", "markdown": "$\\dfrac{1}{x - 2} + \\dfrac{2}{x + 2} = \\dfrac{5}{6}$.", "answer_latex": [ "$4$~or~$-\\frac{2}{5}$." ], "answer_markdown": [ "$4$ or $-\\frac{2}{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(1/(x - 2) + 2/(x + 2), Rational(5, 6))", "answer_expr": "[4, Rational(-2, 5)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2/(x + 2) + 1/(x - 2), 5/6)" ], "shape": [ "solve: Eq(N/(N + x) + 1/(N + x), N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/23", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 23", "problem_latex": "$\\dfrac{5x + 7}{x - 1} = 3x + 11$.", "markdown": "$\\dfrac{5x + 7}{x - 1} = 3x + 11$.", "answer_latex": [ "$2$~or~$-3$." ], "answer_markdown": [ "$2$ or $-3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((5*x + 7)/(x - 1), 3*x + 11)", "answer_expr": "[2, -3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((5*x + 7)/(x - 1), 3*x + 11)" ], "shape": [ "solve: Eq((N*x + N)/(x - 1), N*x + N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/24", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 24", "problem_latex": "$\\dfrac{7}{x + 4} - \\dfrac{1}{4 - x} = \\dfrac{2}{3}$.", "markdown": "$\\dfrac{7}{x + 4} - \\dfrac{1}{4 - x} = \\dfrac{2}{3}$.", "answer_latex": [ "$10$~or~$2$." ], "answer_markdown": [ "$10$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(7/(x + 4) - 1/(4 - x), Rational(2, 3))", "answer_expr": "[10, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(7/(x + 4) - 1/(-x + 4), 2/3)" ], "shape": [ "solve: Eq(N/(N + x) - 1/(N - x), N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/25", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 25, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 25", "problem_latex": "$\\dfrac{2}{x + 3} + \\dfrac{x + 3}{2} = \\dfrac{10}{3}$.", "markdown": "$\\dfrac{2}{x + 3} + \\dfrac{x + 3}{2} = \\dfrac{10}{3}$.", "answer_latex": [ "$3$~or~$-2\\frac{1}{3}$." ], "answer_markdown": [ "$3$ or $-2\\frac{1}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2/(x + 3) + (x + 3)/2, Rational(10, 3))", "answer_expr": "[3, Rational(-7, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/2 + 3/2 + 2/(x + 3), 10/3)" ], "shape": [ "solve: Eq(N*x + N + N/(N + x), N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/26", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 26, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 26", "problem_latex": "$\\dfrac{2x}{x + 2} + \\dfrac{x + 2}{2x} = 2$.", "markdown": "$\\dfrac{2x}{x + 2} + \\dfrac{x + 2}{2x} = 2$.", "answer_latex": [ "$2$~or~$2$." ], "answer_markdown": [ "$2$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2*x/(x + 2) + (x + 2)/(2*x), 2)", "answer_expr": "[2, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x/(x + 2) + (x + 2)/(2*x), 2)" ], "shape": [ "solve: Eq(N*x/(N + x) + N*(N + x)/x, N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/27", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 27, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 27", "problem_latex": "$\\dfrac{3(x - 1)}{x + 1} - \\dfrac{2(x + 1)}{x - 1} = 5$.", "markdown": "$\\dfrac{3(x - 1)}{x + 1} - \\dfrac{2(x + 1)}{x - 1} = 5$.", "answer_latex": [ "$\\frac{1}{2}$~or~$-3$." ], "answer_markdown": [ "$\\frac{1}{2}$ or $-3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*(x - 1)/(x + 1) - 2*(x + 1)/(x - 1), 5)", "answer_expr": "[Rational(1, 2), -3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((3*x - 3)/(x + 1) - (2*x + 2)/(x - 1), 5)" ], "shape": [ "solve: Eq((N*x + N)/(x + 1) - (N*x + N)/(x - 1), N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/28", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 28, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 28", "problem_latex": "$\\dfrac{2x + 5}{2x - 5} = \\dfrac{7x - 5}{2x}$.", "markdown": "$\\dfrac{2x + 5}{2x - 5} = \\dfrac{7x - 5}{2x}$.", "answer_latex": [ "$5$~or~$\\frac{1}{2}$." ], "answer_markdown": [ "$5$ or $\\frac{1}{2}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((2*x + 5)/(2*x - 5), (7*x - 5)/(2*x))", "answer_expr": "[5, Rational(1, 2)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((2*x + 5)/(2*x - 5), (7*x - 5)/(2*x))" ], "shape": [ "solve: Eq(1, N*(N*x + N)/x)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/29", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 29, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 29", "problem_latex": "$\\dfrac{3x - 1}{4x + 7} = \\dfrac{x + 1}{x + 7}$.", "markdown": "$\\dfrac{3x - 1}{4x + 7} = \\dfrac{x + 1}{x + 7}$.", "answer_latex": [ "$7$~or~$2$." ], "answer_markdown": [ "$7$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((3*x - 1)/(4*x + 7), (x + 1)/(x + 7))", "answer_expr": "[7, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((3*x - 1)/(4*x + 7), (x + 1)/(x + 7))" ], "shape": [ "solve: Eq((N*x - 1)/(N*x + N), (x + 1)/(N + x))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/3", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 3", "problem_latex": "$x^{2} - 4 = 4x - 7$.", "markdown": "$x^{2} - 4 = 4x - 7$.", "answer_latex": [ "$3$~or~$1$." ], "answer_markdown": [ "$3$ or $1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 - 4, 4*x - 7)", "answer_expr": "[3, 1]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2 - 4, 4*x - 7)" ], "shape": [ "solve: Eq(N + x**N, N*x + N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/30", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 30, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 30", "problem_latex": "$\\dfrac{2x - 1}{x + 3} = \\dfrac{x + 3}{2x - 1}$.", "markdown": "$\\dfrac{2x - 1}{x + 3} = \\dfrac{x + 3}{2x - 1}$.", "answer_latex": [ "$4$~or~$-\\frac{2}{3}$." ], "answer_markdown": [ "$4$ or $-\\frac{2}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((2*x - 1)/(x + 3), (x + 3)/(2*x - 1))", "answer_expr": "[4, Rational(-2, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq((2*x - 1)/(x + 3), (x + 3)/(2*x - 1))" ], "shape": [ "solve: Eq((N*x - 1)/(N + x), (N + x)/(N*x - 1))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/31", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 31, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 31", "problem_latex": "$\\dfrac{x + 4}{x - 4} - \\dfrac{x + 2}{x - 3} = 1$.", "markdown": "$\\dfrac{x + 4}{x - 4} - \\dfrac{x + 2}{x - 3} = 1$.", "answer_latex": [ "$8$~or~$2$." ], "answer_markdown": [ "$8$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x + 4)/(x - 4) - (x + 2)/(x - 3), 1)", "answer_expr": "[8, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-(x + 2)/(x - 3) + (x + 4)/(x - 4), 1)" ], "shape": [ "solve: False" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/32", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 32, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 32", "problem_latex": "$\\dfrac{4}{x - 1} - \\dfrac{5}{x + 2} = \\dfrac{1}{2}$.", "markdown": "$\\dfrac{4}{x - 1} - \\dfrac{5}{x + 2} = \\dfrac{1}{2}$.", "answer_latex": [ "$4$~or~$-7$." ], "answer_markdown": [ "$4$ or $-7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4/(x - 1) - 5/(x + 2), Rational(1, 2))", "answer_expr": "[4, -7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-5/(x + 2) + 4/(x - 1), 1/2)" ], "shape": [ "solve: Eq(N/(x - 1) + N/(N + x), N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/33", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 33, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 33", "problem_latex": "$\\dfrac{2}{x - 1} = \\dfrac{3}{x - 2} + \\dfrac{2}{x - 4}$.", "markdown": "$\\dfrac{2}{x - 1} = \\dfrac{3}{x - 2} + \\dfrac{2}{x - 4}$.", "answer_latex": [ "$0$~or~$3$." ], "answer_markdown": [ "$0$ or $3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2/(x - 1), 3/(x - 2) + 2/(x - 4))", "answer_expr": "[0, 3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2/(x - 1), 3/(x - 2) + 2/(x - 4))" ], "shape": [ "solve: Eq(N/(x - 1), 2*N/(N + x))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/34", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 34, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 34", "problem_latex": "$\\dfrac{5}{x - 2} - \\dfrac{3}{x - 1} = \\dfrac{1}{2}$.", "markdown": "$\\dfrac{5}{x - 2} - \\dfrac{3}{x - 1} = \\dfrac{1}{2}$.", "answer_latex": [ "$0$~or~$7$." ], "answer_markdown": [ "$0$ or $7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(5/(x - 2) - 3/(x - 1), Rational(1, 2))", "answer_expr": "[0, 7]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(-3/(x - 1) + 5/(x - 2), 1/2)" ], "shape": [ "solve: Eq(N/(x - 1) + N/(N + x), N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/35", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 35, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 35", "problem_latex": "$\\dfrac{x}{7 - x} + \\dfrac{7 - x}{x} = \\dfrac{29}{10}$.", "markdown": "$\\dfrac{x}{7 - x} + \\dfrac{7 - x}{x} = \\dfrac{29}{10}$.", "answer_latex": [ "$5$~or~$2$." ], "answer_markdown": [ "$5$ or $2$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x/(7 - x) + (7 - x)/x, Rational(29, 10))", "answer_expr": "[5, 2]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/(-x + 7) + (-x + 7)/x, 29/10)" ], "shape": [ "solve: Eq(x/(N - x) + (N - x)/x, N)" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/36", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 36, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 36", "problem_latex": "$\\dfrac{2x - 1}{x - 1} + \\dfrac{1}{6} = \\dfrac{2x - 3}{x - 2}$.", "markdown": "$\\dfrac{2x - 1}{x - 1} + \\dfrac{1}{6} = \\dfrac{2x - 3}{x - 2}$.", "answer_latex": [ "$4$~or~$-1$." ], "answer_markdown": [ "$4$ or $-1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((2*x - 1)/(x - 1) + Rational(1, 6), (2*x - 3)/(x - 2))", "answer_expr": "[4, -1]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(1/6 + (2*x - 1)/(x - 1), (2*x - 3)/(x - 2))" ], "shape": [ "solve: Eq(N + (N*x - 1)/(x - 1), (N*x + N)/(N + x))" ], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/4", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 4", "problem_latex": "$5x^{2} - 4x-1 = 0$.", "markdown": "$5x^{2} - 4x-1 = 0$.", "answer_latex": [ "$1$~or~$-\\frac{1}{5}$." ], "answer_markdown": [ "$1$ or $-\\frac{1}{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(5*x**2 - 4*x - 1, 0)", "answer_expr": "[1, Rational(-1, 5)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(5*x**2 - 4*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/5", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 5", "problem_latex": "$4x - 3 = 2x - x^{2}$.", "markdown": "$4x - 3 = 2x - x^{2}$.", "answer_latex": [ "$1$~or~$-3$." ], "answer_markdown": [ "$1$ or $-3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*x - 3, 2*x - x**2)", "answer_expr": "[1, -3]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(4*x - 3, -x**2 + 2*x)" ], "shape": [ "solve: Eq(N*x + N, N*x - x**N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/6", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 6", "problem_latex": "$9x^{2} - 24x + 16 = 0$.", "markdown": "$9x^{2} - 24x + 16 = 0$.", "answer_latex": [ "$\\frac{4}{3}$~or~$\\frac{4}{3}$." ], "answer_markdown": [ "$\\frac{4}{3}$ or $\\frac{4}{3}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(9*x**2 - 24*x + 16, 0)", "answer_expr": "[Rational(4, 3), Rational(4, 3)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(9*x**2 - 24*x + 16, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/7", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 7", "problem_latex": "$6x^{2} - 5x-1 = 0$.", "markdown": "$6x^{2} - 5x-1 = 0$.", "answer_latex": [ "$1$~or~$-\\frac{1}{6}$." ], "answer_markdown": [ "$1$ or $-\\frac{1}{6}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(6*x**2 - 5*x - 1, 0)", "answer_expr": "[1, Rational(-1, 6)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(6*x**2 - 5*x - 1, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N - 1, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/8", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 8", "problem_latex": "$4x + 3 = x^{2} + 2x$.", "markdown": "$4x + 3 = x^{2} + 2x$.", "answer_latex": [ "$3$~or~$-1$." ], "answer_markdown": [ "$3$ or $-1$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(4*x + 3, x**2 + 2*x)", "answer_expr": "[3, -1]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(4*x + 3, x**2 + 2*x)" ], "shape": [ "solve: Eq(N*x + N, N*x + x**N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-72/9", "set": "wentworth-first-steps-in-algebra-1894/ex-72", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "137", "location": "Exercise 72, problem 9", "problem_latex": "$16x^{2} - 16x + 3 = 0$.", "markdown": "$16x^{2} - 16x + 3 = 0$.", "answer_latex": [ "$\\frac{3}{4}$~or~$\\frac{1}{4}$." ], "answer_markdown": [ "$\\frac{3}{4}$ or $\\frac{1}{4}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(16*x**2 - 16*x + 3, 0)", "answer_expr": "[Rational(3, 4), Rational(1, 4)]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(16*x**2 - 16*x + 3, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.factor", "cas.solve.poly", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/1", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 1", "problem_latex": "Find two numbers whose sum is~$11$, and whose\nproduct is~$30$.", "markdown": "Find two numbers whose sum is $11$, and whose product is $30$.", "answer_latex": [ "$6$~and~$5$." ], "answer_markdown": [ "$6$ and $5$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 6, y: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a*x, 30), Eq(a + x, 11))" ], "shape": [ "solve: (Eq(a*x, N), Eq(a + x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/10", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 10", "problem_latex": "The combined ages of a father and son amount to\n$64$~years. Twice the father's age exceeds the square of the\nson's age by $8$~years. Find their respective ages.", "markdown": "The combined ages of a father and son amount to $64$ years. Twice the father’s age exceeds the square of the son’s age by $8$ years. Find their respective ages.", "answer_latex": [ "Son, $10$~yr.; father, $54$~yr." ], "answer_markdown": [ "Son, $10$ yr.; father, $54$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{s: 10, f: 54}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a + x, 64), Eq(2*a - x**2, 8))" ], "shape": [ "solve: (Eq(a + x, N), Eq(N*a - x**N, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/2", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 2", "problem_latex": "Find two numbers whose difference is~$10$, and the\nsum of whose squares is~$250$.", "markdown": "Find two numbers whose difference is $10$, and the sum of whose squares is $250$.", "answer_latex": [ "$5$~and~$15$." ], "answer_markdown": [ "$5$ and $15$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 15, y: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(-a + x, 10), Eq(a**2 + x**2, 250))" ], "shape": [ "solve: (Eq(-a + x, N), Eq(a**N + x**N, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/3", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 3", "problem_latex": "A man is five times as old as his son, and the square\nof the son's age diminished by the father's age is~$24$. Find\ntheir ages.", "markdown": "A man is five times as old as his son, and the square of the son’s age diminished by the father’s age is $24$. Find their ages.", "answer_latex": [ "Son, $8$~yr.; father, $40$~yr." ], "answer_markdown": [ "Son, $8$ yr.; father, $40$ yr." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{s: 8, f: 40}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, 5*x), Eq(-a + x**2, 24))" ], "shape": [ "solve: (Eq(a, N*x), Eq(-a + x**N, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/4", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 4", "problem_latex": "A number increased by its square is equal to nine\ntimes the next higher number. Find the number.", "markdown": "A number increased by its square is equal to nine times the next higher number. Find the number.", "answer_latex": [ "$9$." ], "answer_markdown": [ "$9$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{n: 9}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(x**2 + x, 9*x + 9)" ], "shape": [ "solve: Eq(x + x**N, N*x + N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/5", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 5", "problem_latex": "The square of the sum of any two consecutive numbers\nlacks~$1$ of being twice the sum of the squares of the\nnumbers. Show that this statement is true.", "markdown": "The square of the sum of any two consecutive numbers lacks $1$ of being twice the sum of the squares of the numbers. Show that this statement is true.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/6", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 6", "problem_latex": "The length of a rectangular court exceeds its breadth\nby $2$~rods. If the length and breadth were each increased\nby $3$~rods, the area of the court would be $80$~square rods.\nFind the dimensions of the court.", "markdown": "The length of a rectangular court exceeds its breadth by $2$ rods. If the length and breadth were each increased by $3$ rods, the area of the court would be $80$ square rods. Find the dimensions of the court.", "answer_latex": [ "$5$~rd.\\ by $7$~rd." ], "answer_markdown": [ "$5$ rd. by $7$ rd." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{l: 7, b: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, a + 2), Eq((a + 3)*(x + 3), 80))" ], "shape": [ "solve: (Eq(x, N + a), Eq((N + a)*(N + x), N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/7", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 7", "problem_latex": "The area of a certain square will be doubled, if its\ndimensions are increased by $6$~feet and $4$~feet respectively.\nFind its dimensions.", "markdown": "The area of a certain square will be doubled, if its dimensions are increased by $6$ feet and $4$ feet respectively. Find its dimensions.", "answer_latex": [ "$12$~ft." ], "answer_markdown": [ "$12$ ft." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{s: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq((x + 4)*(x + 6), 2*x**2)" ], "shape": [ "solve: Eq((N + x)**2, N*x**N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 4 ×", "4 ×", "6 ENTER 4 + GOLD x²", "+", "√x", "6 ENTER 4 + +", "2 ÷" ], "constants": [ { "value": "6", "source": "the 6 in 'increased by 6 feet' (also the 6 in the expansion (s + 6)(s + 4))" }, { "value": "4", "source": "the 4 in 'by 4 feet' (also the 4 in 4ac of the quadratic formula, with c = 24 from 6 × 4)" }, { "value": "2", "source": "the 2 in '2*s**2' (the doubled area), which is also the 2 of 2a in the quadratic formula with a = 1 from s²" } ], "calculator_value": "+12E+0", "printed_value": "12", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/8", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 8", "problem_latex": "The perimeter of a rectangular floor is $76$~feet and\nthe area of the floor is $360$~square feet. Find the dimensions\nof the floor.", "markdown": "The perimeter of a rectangular floor is $76$ feet and the area of the floor is $360$ square feet. Find the dimensions of the floor.", "answer_latex": [ "$20$~ft.\\ by $18$~ft." ], "answer_markdown": [ "$20$ ft. by $18$ ft." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{l: 20, b: 18}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a*x, 360), Eq(2*a + 2*x, 76))" ], "shape": [ "solve: (Eq(a*x, N), Eq(N*a + N*x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-73/9", "set": "wentworth-first-steps-in-algebra-1894/ex-73", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "140", "location": "Exercise 73, problem 9", "problem_latex": "The length of a rectangular court exceeds its breadth\nby $2$~rods, and its area is $120$~square rods. Find the\ndimensions of the court.", "markdown": "The length of a rectangular court exceeds its breadth by $2$ rods, and its area is $120$ square rods. Find the dimensions of the court.", "answer_latex": [ "$10$~rd.\\ by $12$~rd." ], "answer_markdown": [ "$10$ rd. by $12$ rd." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{b: 10, l: 12}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a, x + 2), Eq(a*x, 120))" ], "shape": [ "solve: (Eq(a, N + x), Eq(a*x, N))" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-74/1", "set": "wentworth-first-steps-in-algebra-1894/ex-74", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "141", "location": "Exercise 74, problem 1", "problem_latex": "A boat sails $30$~miles at a uniform rate. If the rate\nhad been $1$~mile an hour less, the time of the sailing would\nhave been $1$~hour more. Find the rate of the sailing.", "markdown": "A boat sails $30$ miles at a uniform rate. If the rate had been $1$ mile an hour less, the time of the sailing would have been $1$ hour more. Find the rate of the sailing.", "answer_latex": [ "$6$~miles an hour." ], "answer_markdown": [ "$6$ miles an hour." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 6}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(30/(x - 1) - 30/x, 1)" ], "shape": [ "solve: Eq(N/(x - 1) + N/x, 1)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "30 ENTER 4 × 1 + √x", "1 + 2 ÷" ], "constants": [ { "value": "30", "source": "the 30 miles in 'sails 30 miles'; also the 30 in Eq(30/(x - 1) - 30/x, 1)" }, { "value": "4", "source": "the 4 in the quadratic formula 4ac, with a = 1 and c = −30 from the cleared equation x² − x − 30 = 0 (30x − 30(x − 1) = x(x − 1))" }, { "value": "1", "source": "the 1 in '1 mile an hour less', also −b = 1 from the cleared equation x² − x − 30 = 0" }, { "value": "2", "source": "the 2 in the quadratic formula's denominator 2a, with a = 1 from x² − x − 30 = 0" } ], "calculator_value": "+6E+0", "printed_value": "6", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-74/2", "set": "wentworth-first-steps-in-algebra-1894/ex-74", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "141", "location": "Exercise 74, problem 2", "problem_latex": "A laborer built $35$~rods of stone wall. If he had built $2$~rods less each day, it would have taken him $2$~days\nlonger. How many rods did he build a day on the\naverage?", "markdown": "A laborer built $35$ rods of stone wall. If he had built $2$ rods less each day, it would have taken him $2$ days longer. How many rods did he build a day on the average?", "answer_latex": [ "$7$." ], "answer_markdown": [ "$7$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 7}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(35/(x - 2) - 35/x, 2)" ], "shape": [ "solve: Eq(N/(N + x) + N/x, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 GOLD x² ENTER 35 × 2 GOLD x² +", "√x", "2 + 2 ÷" ], "constants": [ { "value": "2", "source": "the 2 in \"2 rods less each day\"; it is the coefficient b in x² − 2x − 35 = 0, obtained by clearing the denominators of 35/(x−2) − 35/x = 2 (giving 70 = 2x² − 4x)" }, { "value": "35", "source": "\"35 rods of stone wall\" in the problem; also the constant term c = −35 of x² − 2x − 35 = 0" }, { "value": "2", "source": "the same 2 from \"2 rods less\", used as the 2a divisor and the + in x = (−b ± √(b² − 4ac)) / 2a, with a = 1" } ], "calculator_value": "+7E+0", "printed_value": "7", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-74/3", "set": "wentworth-first-steps-in-algebra-1894/ex-74", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "141", "location": "Exercise 74, problem 3", "problem_latex": "A man bought flour for~\\$$30$. Had he bought $1$~barrel\nmore for the same sum, the flour would have cost\nhim \\$$1$~less per barrel. How many barrels did he buy?", "markdown": "A man bought flour for $$30$. Had he bought $1$ barrel more for the same sum, the flour would have cost him $$1$ less per barrel. How many barrels did he buy?", "answer_latex": [ "$5$." ], "answer_markdown": [ "$5$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{n: 5}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(-30/(x + 1) + 30/x, 1)" ], "shape": [ "solve: Eq(N/(x + 1) + N/x, 1)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 30 × 1 + √x", "1 −", "2 ÷" ], "constants": [ { "value": "4", "source": "the 4 in the quadratic formula's b² − 4ac, with a = 1 and c = −30, from clearing denominators in 30/n − 30/(n+1) = 1 to get n² + n − 30 = 0" }, { "value": "30", "source": "the $30 the man paid for the flour (the sum in 'bought flour for $30')" }, { "value": "1", "source": "the 1 in 'the 1 barrel more' (the 1 in n + 1), which is also b = 1 in n² + n − 30 = 0" }, { "value": "2", "source": "the 2 in the quadratic formula's 2a, with a = 1" } ], "calculator_value": "+5E+0", "printed_value": "5", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-74/4", "set": "wentworth-first-steps-in-algebra-1894/ex-74", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "141", "location": "Exercise 74, problem 4", "problem_latex": "A man bought some knives for~\\$$6$. Had he bought\n$2$~less for the same money, he would have paid $25$~cents\nmore for each knife. How many knives did he buy?", "markdown": "A man bought some knives for $$6$. Had he bought $2$ less for the same money, he would have paid $25$ cents more for each knife. How many knives did he buy?", "answer_latex": [ "$8$." ], "answer_markdown": [ "$8$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{k: 8}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(6/(x - 2) - 6/x, 1/4)" ], "shape": [ "solve: Eq(N/(N + x) + N/x, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "4 ENTER 6 × 2 × 1 +", "√x 1 +" ], "constants": [ { "value": "1", "source": "the 1 in 'Eq(6/(k - 2) - 6/k, 1/4)' (the numerator of 1/4); it also gives k = 1 + √(1 + c) after halving the −2 in k² − 2k − 48 = 0" } ], "calculator_value": "+8E+0", "printed_value": "8", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-74/5", "set": "wentworth-first-steps-in-algebra-1894/ex-74", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "141", "location": "Exercise 74, problem 5", "problem_latex": "What number exceeds its square root by~$30$?", "markdown": "What number exceeds its square root by $30$?", "answer_latex": [ "$36$." ], "answer_markdown": [ "$36$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{N: 36}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(-sqrt(x) + x, 30)" ], "shape": [ "solve: Eq(x - x**N, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.arith", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "30 ENTER 4 ×", "1 +", "√x", "1 +", "2 ÷", "GOLD x²" ], "constants": [ { "value": "4", "source": "the 4 in the quadratic formula's 4ac term, with a = 1 from substituting s = √N, which gives s² − s − 30 = 0" }, { "value": "1", "source": "the 1 in the quadratic formula's −b term: the coefficient of s in s² − s − 30 = 0 (from N − √N = 30 with s = √N)" }, { "value": "2", "source": "the 2 in the quadratic formula's 2a denominator, with a = 1" } ], "calculator_value": "+36E+0", "printed_value": "36", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/1", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 1", "problem_latex": "Find the 25th~term in the series $3$, $6$, $9$,~$\\dots$.", "markdown": "Find the 25th term in the series $3$, $6$, $9$, $\\dots$.", "answer_latex": [ "$75$." ], "answer_markdown": [ "$75$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 75.0, printed 75 (half-unit 0.5; correctly rounded at the printed digits: 75.0)", "problem_expr": "3 + (25 - 1)*3", "answer_expr": "75" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 75" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X=75", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/10", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 10", "problem_latex": "What is the arithmetical mean of $20$~and~$32$?", "markdown": "What is the arithmetical mean of $20$ and $32$?", "answer_latex": [ "$26$." ], "answer_markdown": [ "$26$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 26.0, printed 26 (half-unit 0.5; correctly rounded at the printed digits: 26.0)", "problem_expr": "(20 + 32)/2", "answer_expr": "26" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 26" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X=26", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/11", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 11", "problem_latex": "What is the arithmetical mean of $a + b$ and $a - b$?", "markdown": "What is the arithmetical mean of $a + b$ and $a - b$?", "answer_latex": [ "$a$." ], "answer_markdown": [ "$a$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "((a + b) + (a - b))/2", "answer_expr": "a" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: x" ], "shape": [ "identity: x" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-25/3" ], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/12", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 12", "problem_latex": "Insert $8$ arithmetical means between $20$~and~$29$.", "markdown": "Insert $8$ arithmetical means between $20$ and $29$.", "answer_latex": [ "$21$, $22$,~etc." ], "answer_markdown": [ "$21$, $22$, etc." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/2", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 2", "problem_latex": "Find the 13th~term in the series $50$, $49$, $48$,~$\\dots$.", "markdown": "Find the 13th term in the series $50$, $49$, $48$, $\\dots$.", "answer_latex": [ "$38$." ], "answer_markdown": [ "$38$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 38.0, printed 38 (half-unit 0.5; correctly rounded at the printed digits: 38.0)", "problem_expr": "50 - (13 - 1)*1", "answer_expr": "38" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 38" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X=38", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/3", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 3", "problem_latex": "Find the 15th~term in the series $\\frac{1}{7}$, $\\frac{3}{7}$, $\\frac{5}{7}$,~$\\dots$.", "markdown": "Find the 15th term in the series $\\frac{1}{7}$, $\\frac{3}{7}$, $\\frac{5}{7}$, $\\dots$.", "answer_latex": [ "$4\\frac{1}{7}$." ], "answer_markdown": [ "$4\\frac{1}{7}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 4.14285714286, printed 29/7 (half-unit 1.0e-40; correctly rounded at the printed digits: 4.14285714286)", "problem_expr": "Rational(1, 7) + (15 - 1)*Rational(2, 7)", "answer_expr": "Rational(29, 7)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 29/7" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/4", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 4", "problem_latex": "Find the 19th~term in the series $\\frac{1}{4}$, $-\\frac{1}{4}$, $-\\frac{3}{4}$,~$\\dots$.", "markdown": "Find the 19th term in the series $\\frac{1}{4}$, $-\\frac{1}{4}$, $-\\frac{3}{4}$, $\\dots$.", "answer_latex": [ "$-8\\frac{3}{4}$." ], "answer_markdown": [ "$-8\\frac{3}{4}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -8.75, printed -35/4 (half-unit 1.0e-40; correctly rounded at the printed digits: -8.75)", "problem_expr": "Rational(1, 4) + (19 - 1)*Rational(-1, 2)", "answer_expr": "Rational(-35, 4)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -35/4" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/5", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 5", "problem_latex": "Find the 10th~term in an arithmetical progression\nwhose 1st~term is~$5$ and 3d~term~$9$.", "markdown": "Find the 10th term in an arithmetical progression whose 1st term is $5$ and 3d term $9$.", "answer_latex": [ "$23$." ], "answer_markdown": [ "$23$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "T": 23 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 9*a + 5), Eq(2*a + 5, 9))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(N*a + N, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/6", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 6", "problem_latex": "Find the 11th~term in an arithmetical progression\nwhose 1st~term is~$10$ and whose 6th~term is~$5$.", "markdown": "Find the 11th term in an arithmetical progression whose 1st term is $10$ and whose 6th term is $5$.", "answer_latex": [ "$0$." ], "answer_markdown": [ "$0$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "T": 0 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, 10*a + 10), Eq(5*a + 10, 5))" ], "shape": [ "solve: (Eq(x, N*a + N), Eq(N*a + N, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/7", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 7", "problem_latex": "If the 3d~term of an arithmetical progression is~$20$\nand the 13th~term is~$100$, what is the 20th~term?", "markdown": "If the 3d term of an arithmetical progression is $20$ and the 13th term is $100$, what is the 20th term?", "answer_latex": [ "$156$." ], "answer_markdown": [ "$156$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": { "T": 156 } } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(x, a + 19*b), Eq(a + 2*b, 20), Eq(a + 12*b, 100))" ], "shape": [ "solve: (Eq(x, N*b + a), Eq(N*b + a, N), Eq(N*b + a, N))" ], "same_problem_in": [], "needs": [ "core.arith", "core.eqn", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/8", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 8", "problem_latex": "Which term of the series $5$, $7$, $9$, $11$,~$\\dots$, is~$43$?", "markdown": "Which term of the series $5$, $7$, $9$, $11$, $\\dots$, is $43$?", "answer_latex": [ "$20$th." ], "answer_markdown": [ "$20$th." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(5 + (n - 1)*2, 43)", "answer_expr": "20" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x + 3, 43)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.solve.num" ], "expectation": "X=20", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-75/9", "set": "wentworth-first-steps-in-algebra-1894/ex-75", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "144", "location": "Exercise 75, problem 9", "problem_latex": "Which term of the series $\\frac{4}{3}$, $\\frac{3}{2}$, $\\frac{5}{3}$,~$\\dots$, is~$18$?", "markdown": "Which term of the series $\\frac{4}{3}$, $\\frac{3}{2}$, $\\frac{5}{3}$, $\\dots$, is $18$?", "answer_latex": [ "$101$st." ], "answer_markdown": [ "$101$st." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(Rational(4, 3) + (n - 1)*Rational(1, 6), 18)", "answer_expr": "101" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x/6 + 7/6, 18)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.solve.num" ], "expectation": "X=101", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/1", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 1", "problem_latex": "Find the sum of $3$, $5$, $7$, $\\dots$, to $20$~terms.", "markdown": "Find the sum of $3$, $5$, $7$, $\\dots$, to $20$ terms.", "answer_latex": [ "$440$." ], "answer_markdown": [ "$440$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 440.0, printed 440 (half-unit 0.5; correctly rounded at the printed digits: 440.0)", "problem_expr": "Rational(20,2)*(2*3+(20-1)*2)", "answer_expr": "440" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 440" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#440,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 3 ×", "20 ENTER 1 −", "2 ×", "+", "20 ENTER 2 ÷", "×" ], "constants": [ { "value": "2", "source": "the 2 in 2a of the arithmetic-series formula S = n/2 · (2a + (n−1)d); it is the formula's factor, as in the SymPy expression, not a number printed in the problem" }, { "value": "1", "source": "the 1 in (n−1) of the arithmetic-series formula, from the number of common differences between n terms; not printed in the problem" }, { "value": "2", "source": "the 2 in 'Rational(20,2)' of the SymPy expression, the n/2 halving in the formula; not printed in the problem" } ], "calculator_value": "+440E+0", "printed_value": "440", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/10", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 10", "problem_latex": "In a potato race each man picked up $50$~potatoes\nplaced in line a yard apart, and the first potato one yard\nfrom the basket, picking up one potato at a time and bringing\nit to the basket. How many yards did each man run,\nthe start being made from the basket?", "markdown": "In a potato race each man picked up $50$ potatoes placed in line a yard apart, and the first potato one yard from the basket, picking up one potato at a time and bringing it to the basket. How many yards did each man run, the start being made from the basket?", "answer_latex": [ "$2550$~yd." ], "answer_markdown": [ "$2550$ yd." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 2550.0, printed 2550 (half-unit 0.5; correctly rounded at the printed digits: 2550.0)", "problem_expr": "2*Rational(50,2)*(1+50)", "answer_expr": "2550" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2550" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#2550,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "50 ENTER 1 +", "50 ×" ], "constants": [ { "value": "1", "source": "the 'one yard' from the basket to the first potato, the first term of the run distances 1, 2, ..., 50" } ], "calculator_value": "+2550E+0", "printed_value": "2550", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/11", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 11", "problem_latex": "A heavy body falling from a height falls $16.1$~feet\nthe first second, and in each succeeding second $32.2$~feet\nmore than in the second next preceding. How far will a\nbody fall in $19$~seconds?", "markdown": "A heavy body falling from a height falls $16.1$ feet the first second, and in each succeeding second $32.2$ feet more than in the second next preceding. How far will a body fall in $19$ seconds?", "answer_latex": [ "$5812.1$~ft." ], "answer_markdown": [ "$5812.1$ ft." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 5812.1, printed 5812.1 (half-unit 0.05; correctly rounded at the printed digits: 5812.1)", "problem_expr": "Rational(19,2)*(2*Rational(161,10)+18*Rational(322,10))", "answer_expr": "Rational(58121, 10)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 58121/10" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#5812.1,5E-2", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "16.1 ENTER 2 ×", "18 ENTER 32.2 ×", "+", "19 ENTER 2 ÷", "×" ], "constants": [ { "value": "2", "source": "the 2 in 2*Rational(161,10) in the expression: the 2a term, twice the first second's 16.1 ft" }, { "value": "18", "source": "the 18 in 18*Rational(322,10) in the expression: the number of additional seconds (19 − 1) carrying the 32.2 ft increase" } ], "calculator_value": "+581210E-2", "printed_value": "5812.1", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/12", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 12", "problem_latex": "A stone dropped from a bridge reached the water in\njust $3$~seconds. Find the height of the bridge. (See Ex.~11.)", "markdown": "A stone dropped from a bridge reached the water in just $3$ seconds. Find the height of the bridge. (See Ex. 11.)", "answer_latex": [ "$144.9$~ft." ], "answer_markdown": [ "$144.9$ ft." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 144.9, printed 144.9 (half-unit 0.05; correctly rounded at the printed digits: 144.9)", "problem_expr": "Rational(3,2)*(2*Rational(161,10)+2*Rational(322,10))", "answer_expr": "Rational(1449, 10)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 1449/10" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#144.9,5E-2", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 2 ÷", "16.1 ENTER 2 × 32.2 ENTER 2 × + ×" ], "constants": [ { "value": "3", "source": "the 3 in 'in just 3 seconds'" }, { "value": "2", "source": "the 2 in the halving of the free-fall formula s = ½gt² (the 3/2 and the factors of 2 in the SymPy expression); not printed in this problem's text" }, { "value": "16.1", "source": "half of g = 32.2 ft/s², from the book's Ex. 11 formula (cited as '(See Ex. 11)'); NOT read from the source copy, so this constant is unverified and must be checked against Ex. 11 before the record is accepted" }, { "value": "32.2", "source": "g = 32.2 ft/s², the acceleration of gravity in Ex. 11 as the SymPy expression uses it; NOT read from the source copy, so unverified" } ], "calculator_value": "+14490E-2", "printed_value": "144.9", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/13", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 13", "problem_latex": "The arithmetical mean between two numbers is~$13$,\nand the mean between the double of the first and the triple\nof the second is~$33\\frac{1}{2}$. Find the numbers.", "markdown": "The arithmetical mean between two numbers is $13$, and the mean between the double of the first and the triple of the second is $33\\frac{1}{2}$. Find the numbers.", "answer_latex": [ "$11$, $15$." ], "answer_markdown": [ "$11$, $15$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{a: 11, b: 15}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(a/2 + x/2, 13), Eq(3*a/2 + x, 67/2))" ], "shape": [ "solve: (Eq(N*a + N*x, N), Eq(N*a + x, N))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/14", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 14", "problem_latex": "Find three numbers of an arithmetical series whose\nsum shall be~$27$, and the sum of the first and second shall\nbe~$frac{4}{5}$ of the sum of the second and third.", "markdown": "Find three numbers of an arithmetical series whose sum shall be $27$, and the sum of the first and second shall be $frac{4}{5}$ of the sum of the second and third.", "answer_latex": [ "$7$, $9$, $11$." ], "answer_markdown": [ "$7$, $9$, $11$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 9, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(3*x, 27), Eq(-a + 2*x, 4*a/5 + 8*x/5))" ], "shape": [ "solve: (Eq(N*x, N), Eq(N*x - a, N*a + N*x))" ], "same_problem_in": [], "needs": [ "core.eqn", "core.frac", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/15", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 15", "problem_latex": "A travels uniformly $20$~miles a day; B~travels $8$~miles\nthe first day, $12$~the second, and so on, in arithmetical\nprogression. If they start Monday morning from the same\nplace and travel in the same direction, how far apart will\nthey be Saturday night?", "markdown": "A travels uniformly $20$ miles a day; B travels $8$ miles the first day, $12$ the second, and so on, in arithmetical progression. If they start Monday morning from the same place and travel in the same direction, how far apart will they be Saturday night?", "answer_latex": [ "$12$~miles." ], "answer_markdown": [ "$12$ miles." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 12.0, printed 12 (half-unit 0.5; correctly rounded at the printed digits: 12.0)", "problem_expr": "6*20 - Rational(6,2)*(2*8+(6-1)*4)", "answer_expr": "12" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 12" ], "shape": [ "evaluate: N" ], "same_problem_in": [ "wentworth-first-steps-in-algebra-1894/ex-7/10" ], "needs": [ "core.arith" ], "expectation": "X#12,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "20 ENTER 6 ×", "12 ENTER 8 −", "6 ENTER 1 − ×", "8 ENTER 2 × +", "6 ENTER 2 ÷ ×", "−" ], "constants": [ { "value": "20", "source": "'A travels uniformly 20 miles a day'" }, { "value": "6", "source": "the days Monday through Saturday, counted from 'start Monday morning' to 'Saturday night' (6 days of travel)" }, { "value": "8", "source": "'B travels 8 miles the first day'" }, { "value": "12", "source": "'12 the second' (B's second day)" }, { "value": "1", "source": "the 1 in (n − 1) for the terms after the first, n = 6 days, from 'and so on' in arithmetical progression" }, { "value": "2", "source": "the 2 in the arithmetic-progression sum formula S = (n/2)(2a + (n − 1)d), the standard formula for summing an arithmetic progression, not a number printed in the problem" } ], "calculator_value": "+12E+0", "printed_value": "12", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/16", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 16", "problem_latex": "The sum of three terms of an arithmetical progression\nis~$36$, and the square of the mean exceeds the product of\nthe other two terms by~$49$. Find the numbers.", "markdown": "The sum of three terms of an arithmetical progression is $36$, and the square of the mean exceeds the product of the other two terms by $49$. Find the numbers.", "answer_latex": [ "$5$, $12$, $19$." ], "answer_markdown": [ "$5$, $12$, $19$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 12, y: 7}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(3*x, 36), Eq(x**2 - (-a + x)*(a + x), 49))" ], "shape": [ "solve: (Eq(N*x, N), Eq(x**N - (-a + x)*(a + x), N))" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.linsys", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/2", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 2", "problem_latex": "Find the sum of $14$, $14\\frac{1}{2}$, $15$, $\\dots$, to $12$~terms.", "markdown": "Find the sum of $14$, $14\\frac{1}{2}$, $15$, $\\dots$, to $12$ terms.", "answer_latex": [ "$201$." ], "answer_markdown": [ "$201$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 201.0, printed 201 (half-unit 0.5; correctly rounded at the printed digits: 201.0)", "problem_expr": "Rational(12,2)*(2*14+(12-1)*Rational(1,2))", "answer_expr": "201" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 201" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#201,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "12 ENTER 1 − 2 ÷", "14 ENTER +", "+", "12 × 2 ÷" ], "constants": [ { "value": "1", "source": "the 1 in (n − 1) = (12 − 1), the number of common differences in 12 terms" }, { "value": "2", "source": "the 2 in the common difference d = 1/2 (from 14½ − 14 = 1/2, so multiplying by 1/2 is dividing by 2), and the 2 in the sum formula S = n/2 · (2a + (n − 1)d)" } ], "calculator_value": "+2010E-1", "printed_value": "201", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/3", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 3", "problem_latex": "Find the sum of $\\frac{7}{6}$, $1$, $\\frac{5}{6}$, $\\dots$, to $10$~terms.", "markdown": "Find the sum of $\\frac{7}{6}$, $1$, $\\frac{5}{6}$, $\\dots$, to $10$ terms.", "answer_latex": [ "$4frac{1}{6}$." ], "answer_markdown": [ "$4frac{1}{6}$." ], "checks": [ { "task": "evaluate", "verdict": "FLAG-PARSE", "judge_why": "evaluate: SyntaxError: invalid syntax (, line 1)", "problem_expr": "Rational(10,2)*(2*Rational(7,6)+(10-1)*(-Rational(1,6)))", "answer_expr": "Rational(25, 6)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "evaluate: 25/6" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/4", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 4", "problem_latex": "Find the sum of $-7$, $-5$, $-3$, $\\dots$, to $16$~terms.", "markdown": "Find the sum of $-7$, $-5$, $-3$, $\\dots$, to $16$ terms.", "answer_latex": [ "$128$." ], "answer_markdown": [ "$128$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 128.0, printed 128 (half-unit 0.5; correctly rounded at the printed digits: 128.0)", "problem_expr": "Rational(16,2)*(2*(-7)+(16-1)*2)", "answer_expr": "128" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 128" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#128,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 7 +/− ×", "16 ENTER 1 − 2 × +", "16 ENTER 2 ÷ ×" ], "constants": [ { "value": "2", "source": "the common difference, -5 − (−7) = 2 from the printed terms; also the 2 in the SymPy expression Rational(16,2)*(2*(-7)+(16-1)*2)" }, { "value": "1", "source": "the 1 in (16-1) of the SymPy expression: the number of terms minus one, for the 15 steps from the first term to the 16th" } ], "calculator_value": "+128E+0", "printed_value": "128", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/5", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 5", "problem_latex": "Find the sum of $12$, $9$, $6$, $\\dots$, to $21$~terms.", "markdown": "Find the sum of $12$, $9$, $6$, $\\dots$, to $21$ terms.", "answer_latex": [ "$-378$." ], "answer_markdown": [ "$-378$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed -378.0, printed -378 (half-unit 0.5; correctly rounded at the printed digits: -378.0)", "problem_expr": "Rational(21,2)*(2*12+(21-1)*(-3))", "answer_expr": "-378" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: -378" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#-378,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "9 ENTER 12 −", "21 ENTER 1 −", "×", "12 ENTER 2 × +", "21 × 2 ÷" ], "constants": [ { "value": "2", "source": "the 2 in the sum formula S = n/2 (2a + (n−1)d): doubles the first term, and halves the product at the end" }, { "value": "1", "source": "the 1 in (n−1) of the sum formula, with n = 21 terms from the problem" } ], "calculator_value": "-378E+0", "printed_value": "-378", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/6", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 6", "problem_latex": "Find the sum of $-10\\frac{1}{2}$, $-9$, $-7\\frac{1}{2}$, $\\dots$, to $25$~terms.", "markdown": "Find the sum of $-10\\frac{1}{2}$, $-9$, $-7\\frac{1}{2}$, $\\dots$, to $25$ terms.", "answer_latex": [ "$187\\frac{1}{2}$." ], "answer_markdown": [ "$187\\frac{1}{2}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 187.5, printed 375/2 (half-unit 1.0e-40; correctly rounded at the printed digits: 187.5)", "problem_expr": "Rational(25,2)*(2*Rational(-21,2)+(25-1)*Rational(3,2))", "answer_expr": "Rational(375, 2)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 375/2" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#187\\frac{1}{2},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/7", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 7", "problem_latex": "The sum of three numbers in arithmetical progression\nis~$9$, and the sum of their squares is~$35$. Find the numbers.", "markdown": "The sum of three numbers in arithmetical progression is $9$, and the sum of their squares is $35$. Find the numbers.", "answer_latex": [ "$1$, $3$, $5$." ], "answer_markdown": [ "$1$, $3$, $5$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": null, "answer_expr": "{x: 3, y: 2}" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: (Eq(3*x, 9), Eq(x**2 + (-a + x)**2 + (a + x)**2, 35))" ], "shape": [ "solve: (Eq(N*x, N), Eq(x**N + (-a + x)**N + (a + x)**N, N))" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/8", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 8", "problem_latex": "A common clock strikes the hours from $1$ to~$12$.\nHow many times does it strike every $24$~hours?", "markdown": "A common clock strikes the hours from $1$ to $12$. How many times does it strike every $24$ hours?", "answer_latex": [ "$156$." ], "answer_markdown": [ "$156$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 156.0, printed 156 (half-unit 0.5; correctly rounded at the printed digits: 156.0)", "problem_expr": "2*Rational(12,2)*(1+12)", "answer_expr": "156" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 156" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#156,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "12 ENTER 2 ÷", "12 ENTER 1 + ×", "2 ×" ], "constants": [ { "value": "2", "source": "the 2 in the SymPy expression 2*Rational(12,2)*(1+12): the clock runs through the 12 hours twice in every 24 hours" } ], "calculator_value": "+156E+0", "printed_value": "156", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-76/9", "set": "wentworth-first-steps-in-algebra-1894/ex-76", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "146", "location": "Exercise 76, problem 9", "problem_latex": "The Greenwich clock strikes the hours from $1$ to~$24$.\nHow many times does it strike in $24$~hours?", "markdown": "The Greenwich clock strikes the hours from $1$ to $24$. How many times does it strike in $24$ hours?", "answer_latex": [ "$300$." ], "answer_markdown": [ "$300$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 300.0, printed 300 (half-unit 0.5; correctly rounded at the printed digits: 300.0)", "problem_expr": "Rational(24,2)*(1+24)", "answer_expr": "300" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 300" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#300,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1 ENTER 24 +", "24 ENTER 2 ÷", "×" ], "constants": [ { "value": "2", "source": "the divisor in Rational(24,2) of the SymPy expression, from the arithmetic-series sum n(n+1)/2; it is not printed in the problem text" } ], "calculator_value": "+300E+0", "printed_value": "300", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/1", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 1", "problem_latex": "Find the 5th~term of $3$, $9$, $27$\\Add{,}~$\\dots$.", "markdown": "Find the 5th term of $3$, $9$, $27$ $\\dots$.", "answer_latex": [ "$243$." ], "answer_markdown": [ "$243$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 243.0, printed 243 (half-unit 0.5; correctly rounded at the printed digits: 243.0)", "problem_expr": "3*3**4", "answer_expr": "243" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 243" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#243,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 5 ENTER 1 − yˣ", "3 ×" ], "constants": [ { "value": "3", "source": "the first term 3 in 'Find the 5th term of 3, 9, 27, ...'; also the common ratio, since each term is 3 times the one before" }, { "value": "5", "source": "the 5 in '5th term'" }, { "value": "1", "source": "the position of the first term (1st), not printed in the problem text; used with the 5 to count the 4 common-ratio steps from the 1st term to the 5th" } ], "calculator_value": "+243E+0", "printed_value": "243", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/10", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 10", "problem_latex": "$8$, $4$, $2$, $\\dots$ to $8$~terms.", "markdown": "$8$, $4$, $2$, $\\dots$ to $8$ terms.", "answer_latex": [ "$15\\frac{15}{16}$." ], "answer_markdown": [ "$15\\frac{15}{16}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 15.9375, printed 255/16 (half-unit 1.0e-40; correctly rounded at the printed digits: 15.9375)", "problem_expr": "8*(1 - Rational(1, 2)**8)/(1 - Rational(1, 2))", "answer_expr": "Rational(255, 16)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 255/16" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#15\\frac{15}{16},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/11", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 11", "problem_latex": "$64$, $32$, $16$, $\\dots$ to $9$~terms.", "markdown": "$64$, $32$, $16$, $\\dots$ to $9$ terms.", "answer_latex": [ "$127\\frac{3}{4}$." ], "answer_markdown": [ "$127\\frac{3}{4}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 127.75, printed 511/4 (half-unit 1.0e-40; correctly rounded at the printed digits: 127.75)", "problem_expr": "64*(1 - Rational(1, 2)**9)/(1 - Rational(1, 2))", "answer_expr": "Rational(511, 4)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 511/4" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#127\\frac{3}{4},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/12", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 12", "problem_latex": "$64$, $-32$, $16$, $\\dots$ to $5$~terms.", "markdown": "$64$, $-32$, $16$, $\\dots$ to $5$ terms.", "answer_latex": [ "$44$." ], "answer_markdown": [ "$44$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 44.0, printed 44 (half-unit 0.5; correctly rounded at the printed digits: 44.0)", "problem_expr": "64*(1 - (-Rational(1, 2))**5)/(1 - (-Rational(1, 2)))", "answer_expr": "44" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 44" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#44,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/13", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 13", "problem_latex": "$\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{2}{9}$, $\\dots$ to $4$~terms.", "markdown": "$\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{2}{9}$, $\\dots$ to $4$ terms.", "answer_latex": [ "$1\\frac{11}{54}$." ], "answer_markdown": [ "$1\\frac{11}{54}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.2037037037, printed 65/54 (half-unit 1.0e-40; correctly rounded at the printed digits: 1.2037037037)", "problem_expr": "Rational(1, 2)*(1 - Rational(2, 3)**4)/(1 - Rational(2, 3))", "answer_expr": "Rational(65, 54)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 65/54" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#1\\frac{11}{54},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/14", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 14", "problem_latex": "If a blacksmith uses seven nails in putting a shoe on\na horse's foot, and receives $1$~cent for the first nail, $2$~cents\nfor the second nail, and so on, what does he receive for\nputting on the shoe?", "markdown": "If a blacksmith uses seven nails in putting a shoe on a horse’s foot, and receives $1$ cent for the first nail, $2$ cents for the second nail, and so on, what does he receive for putting on the shoe?", "answer_latex": [ "\\$$1.27$." ], "answer_markdown": [ "$$1.27$." ], "checks": [ { "task": "evaluate", "verdict": "FLAG-PARSE", "judge_why": "evaluate: TokenError: ('unexpected character after line continuation character', (1, 6))", "problem_expr": "(2**7 - 1)/100", "answer_expr": "Rational(127, 100)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-PARSE" ] }, "form": [ "evaluate: 127/100" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/15", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 15", "problem_latex": "If a boy receives $2$~cents for his first day's work,\n$4$~cents for his second day, $8$~cents for the third day, and\nso on for $12$~days, what will his wages amount to?", "markdown": "If a boy receives $2$ cents for his first day’s work, $4$ cents for his second day, $8$ cents for the third day, and so on for $12$ days, what will his wages amount to?", "answer_latex": [ "\\$$81.90$." ], "answer_markdown": [ "$$81.90$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 81.9, printed 81.90 (half-unit 0.005; correctly rounded at the printed digits: 81.9)", "problem_expr": "Rational(2*(2**12 - 1), 100)", "answer_expr": "Rational(8190, 100)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 819/10" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.units" ], "expectation": "X#81.90,5E-3", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/16", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 16", "problem_latex": "If the population of a city is $10,000$, and increases\n$10$\\%~a year for four years, what will be its population at\nthe end of the four years? (Here $l = ar^{4}$.)", "markdown": "If the population of a city is $10,000$, and increases $10$% a year for four years, what will be its population at the end of the four years? (Here $l = ar^{4}$.)", "answer_latex": [ "$14,641$." ], "answer_markdown": [ "$14,641$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 14641.0, printed 14641 (half-unit 0.5; correctly rounded at the printed digits: 14641.0)", "problem_expr": "10000*Rational(11, 10)**4", "answer_expr": "14641" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 14641" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#14,641,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/2", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 2", "problem_latex": "Find the 7th~term of $3$, $6$, $12$\\Add{,}~$\\dots$.", "markdown": "Find the 7th term of $3$, $6$, $12$ $\\dots$.", "answer_latex": [ "$192$." ], "answer_markdown": [ "$192$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 192.0, printed 192 (half-unit 0.5; correctly rounded at the printed digits: 192.0)", "problem_expr": "3*2**6", "answer_expr": "192" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 192" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#192,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 3 ÷ ENTER 7 ENTER 1 − yˣ 3 ×" ], "constants": [ { "value": "6", "source": "the second term in the printed sequence '3, 6, 12'" }, { "value": "3", "source": "the first term in the printed sequence '3, 6, 12'" }, { "value": "7", "source": "the '7th' in 'Find the 7th term'" }, { "value": "1", "source": "the 1 in 7 − 1: the first term is term 1, so the 7th term is 7 − 1 = 6 common-ratio steps past it; the 7 − 1 is keyed, not typed as 6" } ], "calculator_value": "+192E+0", "printed_value": "192", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/3", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 3", "problem_latex": "Find the 8th~term of $6$, $3$, $\\frac{3}{2}$\\Add{,}~$\\dots$.", "markdown": "Find the 8th term of $6$, $3$, $\\frac{3}{2}$ $\\dots$.", "answer_latex": [ "$\\frac{3}{64}$." ], "answer_markdown": [ "$\\frac{3}{64}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.046875, printed 3/64 (half-unit 1.0e-40; correctly rounded at the printed digits: 0.046875)", "problem_expr": "6*Rational(1, 2)**7", "answer_expr": "Rational(3, 64)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3/64" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#\\frac{3}{64},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/4", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 4", "problem_latex": "Find the 9th~term of $1$, $-2$, $4$\\Add{,}~$\\dots$.", "markdown": "Find the 9th term of $1$, $-2$, $4$ $\\dots$.", "answer_latex": [ "$256$." ], "answer_markdown": [ "$256$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 256.0, printed 256 (half-unit 0.5; correctly rounded at the printed digits: 256.0)", "problem_expr": "1*(-2)**8", "answer_expr": "256" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 256" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#256,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 +/− ENTER 9 ENTER 1 − yˣ" ], "constants": [ { "value": "2", "source": "the 2 in the sequence '1, −2, 4': the common ratio is −2, and the sign is entered with +/−" }, { "value": "9", "source": "the 9 in '9th term'" }, { "value": "1", "source": "the 1 in '9th term': from the first term to the 9th term takes 9 − 1 = 8 steps of the ratio, so the exponent is n − 1; the 9 − 1 is computed on the keys" } ], "calculator_value": "+256E+0", "printed_value": "256", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/5", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 5", "problem_latex": "Find the geometrical mean between $2$~and~$8$.", "markdown": "Find the geometrical mean between $2$ and $8$.", "answer_latex": [ "$±4$." ], "answer_markdown": [ "$±4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2, 2*8)", "answer_expr": "[-4, 4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x**2, 16)" ], "shape": [ "solve: Eq(x**N, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/6", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 6", "problem_latex": "Find the common ratio if the 1st~and 3d~terms are\n$2$~and~$32$.", "markdown": "Find the common ratio if the 1st and 3d terms are $2$ and $32$.", "answer_latex": [ "$±4$." ], "answer_markdown": [ "$±4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2*r**2, 32)", "answer_expr": "[-4, 4]" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x**2, 32)" ], "shape": [ "solve: Eq(N*x**N, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/7", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 7", "problem_latex": "$3$, $9$, $27$, $\\dots$ to $6$~terms.", "markdown": "$3$, $9$, $27$, $\\dots$ to $6$ terms.", "answer_latex": [ "$1092$." ], "answer_markdown": [ "$1092$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1092.0, printed 1092 (half-unit 0.5; correctly rounded at the printed digits: 1092.0)", "problem_expr": "3*(3**6 - 1)/(3 - 1)", "answer_expr": "1092" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 1092" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#1092,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 6 yˣ 1 −", "3 ×", "3 ENTER 1 −", "÷" ], "constants": [ { "value": "1", "source": "the 1 in the geometric-series formula S = a(rⁿ − 1)/(r − 1); it is structural, not printed in the problem text" } ], "calculator_value": "+1092E+0", "printed_value": "1092", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/8", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 8", "problem_latex": "$3$, $6$, $12$, $\\dots$ to $8$~terms.", "markdown": "$3$, $6$, $12$, $\\dots$ to $8$ terms.", "answer_latex": [ "$765$." ], "answer_markdown": [ "$765$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 765.0, printed 765 (half-unit 0.5; correctly rounded at the printed digits: 765.0)", "problem_expr": "3*(2**8 - 1)/(2 - 1)", "answer_expr": "765" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 765" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#765,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "6 ENTER 3 ÷", "ENTER ENTER 8 yˣ 1 −", "x↔y 1 − ÷", "3 ×" ], "constants": [ { "value": "1", "source": "the −1 in the geometric sum formula a(rⁿ − 1)/(r − 1), used in both the numerator and the denominator" } ], "calculator_value": "+765E+0", "printed_value": "765", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-77/9", "set": "wentworth-first-steps-in-algebra-1894/ex-77", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "151", "location": "Exercise 77, problem 9", "problem_latex": "$6$, $3$, $\\frac{3}{2}$, $\\dots$ to $7$~terms.", "markdown": "$6$, $3$, $\\frac{3}{2}$, $\\dots$ to $7$ terms.", "answer_latex": [ "$11\\frac{29}{32}$." ], "answer_markdown": [ "$11\\frac{29}{32}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 11.90625, printed 381/32 (half-unit 1.0e-40; correctly rounded at the printed digits: 11.90625)", "problem_expr": "6*(1 - Rational(1, 2)**7)/(1 - Rational(1, 2))", "answer_expr": "Rational(381, 32)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 381/32" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#11\\frac{29}{32},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-78/1", "set": "wentworth-first-steps-in-algebra-1894/ex-78", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Exercise 78, problem 1", "problem_latex": "$a^{2} + 2ab + 2ac + b^{2} + 2bc + c^{2}$.", "markdown": "$a^{2} + 2ab + 2ac + b^{2} + 2bc + c^{2}$.", "answer_latex": [ "$a + b + c$." ], "answer_markdown": [ "$a + b + c$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a**2 + 2*a*b + 2*a*c + b**2 + 2*b*c + c**2", "answer_expr": "a + b + c" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "other:polynomial square root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-78/2", "set": "wentworth-first-steps-in-algebra-1894/ex-78", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Exercise 78, problem 2", "problem_latex": "$x^{4} + 2x^{3} + 3x^{2} + 2x + 1$.", "markdown": "$x^{4} + 2x^{3} + 3x^{2} + 2x + 1$.", "answer_latex": [ "$x^{2} + x + 1$." ], "answer_markdown": [ "$x^{2} + x + 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 + 2*x**3 + 3*x**2 + 2*x + 1", "answer_expr": "x**2 + x + 1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "other:polynomial square root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-78/3", "set": "wentworth-first-steps-in-algebra-1894/ex-78", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Exercise 78, problem 3", "problem_latex": "$x^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + y^{4}$.", "markdown": "$x^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + y^{4}$.", "answer_latex": [ "$x^{2} - 2xy + y^{2}$." ], "answer_markdown": [ "$x^{2} - 2xy + y^{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**4 - 4*x**3*y + 6*x**2*y**2 - 4*x*y**3 + y**4", "answer_expr": "x**2 - 2*x*y + y**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "other:polynomial square root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-78/4", "set": "wentworth-first-steps-in-algebra-1894/ex-78", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Exercise 78, problem 4", "problem_latex": "$4a^{4} - 12a^{3}b + 29a^{2}b^{2} - 30ab^{3} + 25b^{4}$.", "markdown": "$4a^{4} - 12a^{3}b + 29a^{2}b^{2} - 30ab^{3} + 25b^{4}$.", "answer_latex": [ "$2a^{2} - 3ab + 5b^{2}$." ], "answer_markdown": [ "$2a^{2} - 3ab + 5b^{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "4*a**4 - 12*a**3*b + 29*a**2*b**2 - 30*a*b**3 + 25*b**4", "answer_expr": "2*a**2 - 3*a*b + 5*b**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "other:polynomial square root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-78/5", "set": "wentworth-first-steps-in-algebra-1894/ex-78", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Exercise 78, problem 5", "problem_latex": "$16x^{6} + 24x^{5}y + 9x^{4}y^{2} - 16x^{3}y^{3} - 12x^{2}y^{4} + 4y^{6}$.", "markdown": "$16x^{6} + 24x^{5}y + 9x^{4}y^{2} - 16x^{3}y^{3} - 12x^{2}y^{4} + 4y^{6}$.", "answer_latex": [ "$4x^{3} + 3x^{2}y - 2y^{3}$." ], "answer_markdown": [ "$4x^{3} + 3x^{2}y - 2y^{3}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "16*x**6 + 24*x**5*y + 9*x**4*y**2 - 16*x**3*y**3 - 12*x**2*y**4 + 4*y**6", "answer_expr": "4*x**3 + 3*x**2*y - 2*y**3" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "other:polynomial square root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-78/6", "set": "wentworth-first-steps-in-algebra-1894/ex-78", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "153", "location": "Exercise 78, problem 6", "problem_latex": "$4x^{6} - 4x^{4}y^{2} + 12x^{3}y^{3} + x^{2}y^{4} - 6xy^{5} + 9y^{6}$.", "markdown": "$4x^{6} - 4x^{4}y^{2} + 12x^{3}y^{3} + x^{2}y^{4} - 6xy^{5} + 9y^{6}$.", "answer_latex": [ "$2x^{3} - xy^{2} + 3y^{3}$." ], "answer_markdown": [ "$2x^{3} - xy^{2} + 3y^{3}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "4*x**6 - 4*x**4*y**2 + 12*x**3*y**3 + x**2*y**4 - 6*x*y**5 + 9*y**6", "answer_expr": "2*x**3 - x*y**2 + 3*y**3" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "other:polynomial square root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/1", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 1", "problem_latex": "$324$.", "markdown": "$324$.", "answer_latex": [ "$18$." ], "answer_markdown": [ "$18$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 18.0, printed 18 (half-unit 0.5; correctly rounded at the printed digits: 18.0)", "problem_expr": "sqrt(324)", "answer_expr": "18" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 18" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#18,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "324 √x" ], "calculator_value": "+18E+0", "printed_value": "18", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/10", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 10", "problem_latex": "$346,921$.", "markdown": "$346,921$.", "answer_latex": [ "$589$." ], "answer_markdown": [ "$589$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 589.0, printed 589 (half-unit 0.5; correctly rounded at the printed digits: 589.0)", "problem_expr": "sqrt(346921)", "answer_expr": "589" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 589" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#589,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "346921 √x" ], "calculator_value": "+589E+0", "printed_value": "589", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/11", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 11", "problem_latex": "$31,371,201$.", "markdown": "$31,371,201$.", "answer_latex": [ "$5601$." ], "answer_markdown": [ "$5601$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 5601.0, printed 5601 (half-unit 0.5; correctly rounded at the printed digits: 5601.0)", "problem_expr": "sqrt(31371201)", "answer_expr": "5601" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5601" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#5601,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/12", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 12", "problem_latex": "$1,522,756$.", "markdown": "$1,522,756$.", "answer_latex": [ "$1234$." ], "answer_markdown": [ "$1234$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1234.0, printed 1234 (half-unit 0.5; correctly rounded at the printed digits: 1234.0)", "problem_expr": "sqrt(1522756)", "answer_expr": "1234" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 1234" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#1234,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1522756 √x" ], "calculator_value": "+1234E+0", "printed_value": "1234", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/13", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 13", "problem_latex": "$2$.", "markdown": "$2$.", "answer_latex": [ "$1.4142\\dots$" ], "answer_markdown": [ "$1.4142\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.41421356237, printed 1.4142 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.4142)", "problem_expr": "sqrt(2)", "answer_expr": "1.4142" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: sqrt(2)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#1.4142,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/14", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 14", "problem_latex": "$3$.", "markdown": "$3$.", "answer_latex": [ "$1.7320\\dots$" ], "answer_markdown": [ "$1.7320\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 1.73205080757, printed 1.7320 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.7321): one unit off in the last printed place", "problem_expr": "sqrt(3)", "answer_expr": "1.7320" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: sqrt(3)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#1.7320,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/15", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 15", "problem_latex": "$5$.", "markdown": "$5$.", "answer_latex": [ "$2.2360\\dots$" ], "answer_markdown": [ "$2.2360\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 2.2360679775, printed 2.2360 (half-unit 5.0e-5; correctly rounded at the printed digits: 2.2361): one unit off in the last printed place", "problem_expr": "sqrt(5)", "answer_expr": "2.2360" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: sqrt(5)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#2.2360,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/16", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 16", "problem_latex": "$6$.", "markdown": "$6$.", "answer_latex": [ "$2.4494\\dots$" ], "answer_markdown": [ "$2.4494\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 2.44948974278, printed 2.4494 (half-unit 5.0e-5; correctly rounded at the printed digits: 2.4495): one unit off in the last printed place", "problem_expr": "sqrt(6)", "answer_expr": "2.4494" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: sqrt(6)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#2.4494,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/17", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 17", "problem_latex": "$0.5$.", "markdown": "$0.5$.", "answer_latex": [ "$0.7071\\dots$" ], "answer_markdown": [ "$0.7071\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.707106781187, printed 0.7071 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.7071)", "problem_expr": "sqrt(0.5)", "answer_expr": "0.7071" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: sqrt(2)/2" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#0.7071,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/18", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 18", "problem_latex": "$0.9$.", "markdown": "$0.9$.", "answer_latex": [ "$0.9486\\dots$" ], "answer_markdown": [ "$0.9486\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 0.948683298051, printed 0.9486 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.9487): one unit off in the last printed place", "problem_expr": "sqrt(0.9)", "answer_expr": "0.9486" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: 3*sqrt(10)/10" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#0.9486,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/19", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 19", "problem_latex": "$\\frac{2}{3}$.", "markdown": "$\\frac{2}{3}$.", "answer_latex": [ "$0.8164\\dots$" ], "answer_markdown": [ "$0.8164\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 0.816496580928, printed 0.8164 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.8165): one unit off in the last printed place", "problem_expr": "sqrt(Rational(2, 3))", "answer_expr": "0.8164" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: sqrt(6)/3" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#0.8164,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/2", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 2", "problem_latex": "$441$.", "markdown": "$441$.", "answer_latex": [ "$21$." ], "answer_markdown": [ "$21$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 21.0, printed 21 (half-unit 0.5; correctly rounded at the printed digits: 21.0)", "problem_expr": "sqrt(441)", "answer_expr": "21" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 21" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#21,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "441 √x" ], "calculator_value": "+21E+0", "printed_value": "21", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/20", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 20", "problem_latex": "$\\frac{3}{4}$.", "markdown": "$\\frac{3}{4}$.", "answer_latex": [ "$0.8660\\dots$" ], "answer_markdown": [ "$0.8660\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.866025403784, printed 0.8660 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.866)", "problem_expr": "sqrt(Rational(3, 4))", "answer_expr": "0.8660" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: sqrt(3)/2" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#0.8660,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/21", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 21", "problem_latex": "$\\frac{4}{5}$.", "markdown": "$\\frac{4}{5}$.", "answer_latex": [ "$0.8944\\dots$" ], "answer_markdown": [ "$0.8944\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.894427191, printed 0.8944 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.8944)", "problem_expr": "sqrt(Rational(4, 5))", "answer_expr": "0.8944" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2*sqrt(5)/5" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#0.8944,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/22", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 22", "problem_latex": "$\\frac{5}{8}$.", "markdown": "$\\frac{5}{8}$.", "answer_latex": [ "$0.7905\\dots$" ], "answer_markdown": [ "$0.7905\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 0.790569415042, printed 0.7905 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.7906): one unit off in the last printed place", "problem_expr": "sqrt(Rational(5, 8))", "answer_expr": "0.7905" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: sqrt(10)/4" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": "X#0.7905,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/3", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 3", "problem_latex": "$529$.", "markdown": "$529$.", "answer_latex": [ "$23$." ], "answer_markdown": [ "$23$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 23.0, printed 23 (half-unit 0.5; correctly rounded at the printed digits: 23.0)", "problem_expr": "sqrt(529)", "answer_expr": "23" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 23" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#23,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "529 √x" ], "calculator_value": "+23E+0", "printed_value": "23", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/4", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 4", "problem_latex": "$961$.", "markdown": "$961$.", "answer_latex": [ "$31$." ], "answer_markdown": [ "$31$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 31.0, printed 31 (half-unit 0.5; correctly rounded at the printed digits: 31.0)", "problem_expr": "sqrt(961)", "answer_expr": "31" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 31" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#31,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "961 √x" ], "calculator_value": "+31E+0", "printed_value": "31", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/5", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 5", "problem_latex": "$10.24$.", "markdown": "$10.24$.", "answer_latex": [ "$3.2$." ], "answer_markdown": [ "$3.2$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 3.2, printed 3.2 (half-unit 0.05; correctly rounded at the printed digits: 3.2)", "problem_expr": "sqrt(10.24)", "answer_expr": "3.2" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 16/5" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#3.2,5E-2", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/6", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 6", "problem_latex": "$53.29$.", "markdown": "$53.29$.", "answer_latex": [ "$7.3$." ], "answer_markdown": [ "$7.3$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 7.3, printed 7.3 (half-unit 0.05; correctly rounded at the printed digits: 7.3)", "problem_expr": "sqrt(53.29)", "answer_expr": "7.3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 73/10" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#7.3,5E-2", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "53.29 √x" ], "calculator_value": "+73E-1", "printed_value": "7.3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/7", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 7", "problem_latex": "$53,824$.", "markdown": "$53,824$.", "answer_latex": [ "$232$." ], "answer_markdown": [ "$232$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 232.0, printed 232 (half-unit 0.5; correctly rounded at the printed digits: 232.0)", "problem_expr": "sqrt(53824)", "answer_expr": "232" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 232" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#232,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/8", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 8", "problem_latex": "$616,225$.", "markdown": "$616,225$.", "answer_latex": [ "$785$." ], "answer_markdown": [ "$785$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 785.0, printed 785 (half-unit 0.5; correctly rounded at the printed digits: 785.0)", "problem_expr": "sqrt(616225)", "answer_expr": "785" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 785" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#785,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "616225 √x" ], "constants": [ { "value": "616225", "source": "the number in 'Find the square root of: 616,225'" } ], "calculator_value": "+785E+0", "printed_value": "785", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-79/9", "set": "wentworth-first-steps-in-algebra-1894/ex-79", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "156", "location": "Exercise 79, problem 9", "problem_latex": "$1,500,625$.", "markdown": "$1,500,625$.", "answer_latex": [ "$1225$." ], "answer_markdown": [ "$1225$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1225.0, printed 1225 (half-unit 0.5; correctly rounded at the printed digits: 1225.0)", "problem_expr": "sqrt(1500625)", "answer_expr": "1225" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 1225" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#1225,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1500625 √x" ], "calculator_value": "+1225E+0", "printed_value": "1225", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/1", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 1", "problem_latex": "A room is $10$~yards long and $8$~yards wide. In the\nmiddle there is a carpet $6$~yards square. How many\nsquare yards of oilcloth will be required to cover the rest\nof the floor?", "markdown": "A room is $10$ yards long and $8$ yards wide. In the middle there is a carpet $6$ yards square. How many square yards of oilcloth will be required to cover the rest of the floor?", "answer_latex": [ "$10 × 8 - 6^{2}$." ], "answer_markdown": [ "$10 × 8 - 6^{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "10*8 - 6**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/10a", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 10, "part": "a", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 10a", "problem_latex": "Write an expression for the sum, and also for the\nproduct, of three consecutive numbers of which the least is~$n$.", "markdown": "Write an expression for the sum, and also for the product, of three consecutive numbers of which the least is $n$.", "answer_latex": [ "$n + (n + 1) + (n + 2)$" ], "answer_markdown": [ "$n + (n + 1) + (n + 2)$" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "n + (n + 1) + (n + 2)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/10b", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 10, "part": "b", "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 10b", "problem_latex": "Write an expression for the sum, and also for the\nproduct, of three consecutive numbers of which the least is~$n$.", "markdown": "Write an expression for the sum, and also for the product, of three consecutive numbers of which the least is $n$.", "answer_latex": [ "$n(n + 1)(n + 2)$." ], "answer_markdown": [ "$n(n + 1)(n + 2)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "n*(n + 1)*(n + 2)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/11", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 11", "problem_latex": "The product of two factors is~$36$; if one of the\nfactors is~$x$, what is the other factor?", "markdown": "The product of two factors is $36$; if one of the factors is $x$, what is the other factor?", "answer_latex": [ "$\\dfrac{36}{x}$." ], "answer_markdown": [ "$\\dfrac{36}{x}$." ], "checks": [ { "task": "solve", "verdict": "PASS-INTERPRETED", "judge_why": null, "problem_expr": "Eq(x*y, 36)", "answer_expr": "36/x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-INTERPRETED" ] }, "form": [ "solve: Eq(a*x, 36)" ], "shape": [ "solve: Eq(a*x, N)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/12", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 12", "problem_latex": "If $d$~is the divisor and $q$~the quotient, what is the\ndividend?", "markdown": "If $d$ is the divisor and $q$ the quotient, what is the dividend?", "answer_latex": [ "$qd$." ], "answer_markdown": [ "$qd$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "q*d" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/13", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 13", "problem_latex": "If $d$~is the divisor, $q$~the quotient, and $r$~the remainder,\nwhat is the dividend?", "markdown": "If $d$ is the divisor, $q$ the quotient, and $r$ the remainder, what is the dividend?", "answer_latex": [ "$qd + r$." ], "answer_markdown": [ "$qd + r$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "q*d + r" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/14", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 14", "problem_latex": "If $x$~oranges can be bought for $50$~cents, how many\noranges can be bought for $100$~cents?", "markdown": "If $x$ oranges can be bought for $50$ cents, how many oranges can be bought for $100$ cents?", "answer_latex": [ "$2x$." ], "answer_markdown": [ "$2x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "2*x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/15", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 15", "problem_latex": "What is the price in cents of $x$~apples, if they are\nten cents a dozen?", "markdown": "What is the price in cents of $x$ apples, if they are ten cents a dozen?", "answer_latex": [ "$\\dfrac{10x}{12}$." ], "answer_markdown": [ "$\\dfrac{10x}{12}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "10*x/12" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/16", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 16", "problem_latex": "If $b$~oranges cost $6$~cents, what will $a$~oranges cost?", "markdown": "If $b$ oranges cost $6$ cents, what will $a$ oranges cost?", "answer_latex": [ "$\\dfrac{6a}{b}$." ], "answer_markdown": [ "$\\dfrac{6a}{b}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "6*a/b" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/17", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 17", "problem_latex": "How many miles between two places, if a train\ntravelling $m$~miles an hour requires $4$~hours to make the\njourney?", "markdown": "How many miles between two places, if a train travelling $m$ miles an hour requires $4$ hours to make the journey?", "answer_latex": [ "$4m$." ], "answer_markdown": [ "$4m$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "4*m" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.units", "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/18", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 18", "problem_latex": "If a man was $x$~years old $10$~years ago, how many\nyears old will he be $7$~years hence?", "markdown": "If a man was $x$ years old $10$ years ago, how many years old will he be $7$ years hence?", "answer_latex": [ "$x + 17$." ], "answer_markdown": [ "$x + 17$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x + 17" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/19", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 19", "problem_latex": "If a man was $x$~years old $y$~years ago, how many\nyears old will he be $c$~years hence?", "markdown": "If a man was $x$ years old $y$ years ago, how many years old will he be $c$ years hence?", "answer_latex": [ "$x + y + c$." ], "answer_markdown": [ "$x + y + c$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x + y + c" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/2", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 2", "problem_latex": "A room is $x$~yards long and $y$~yards wide. In the\nmiddle there is a carpet $a$~yards square. How many\nsquare yards of oilcloth will be required to cover the rest\nof the floor?", "markdown": "A room is $x$ yards long and $y$ yards wide. In the middle there is a carpet $a$ yards square. How many square yards of oilcloth will be required to cover the rest of the floor?", "answer_latex": [ "$xy - a^{2}$." ], "answer_markdown": [ "$xy - a^{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "x*y - a**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/20", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 20", "problem_latex": "If a floor is $3x$~yards long and $12$~yards wide, how\nmany square yards does the floor contain?", "markdown": "If a floor is $3x$ yards long and $12$ yards wide, how many square yards does the floor contain?", "answer_latex": [ "$36x$." ], "answer_markdown": [ "$36x$." ], "checks": [ { "task": "identity", "verdict": "PASS", "judge_why": null, "problem_expr": "3*x*12", "answer_expr": "36*x" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "identity: 36*a" ], "shape": [ "identity: N*a" ], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/21", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 21", "problem_latex": "How many hours will it take to walk $c$~miles, at\nthe rate of one mile in $15$~minutes?", "markdown": "How many hours will it take to walk $c$ miles, at the rate of one mile in $15$ minutes?", "answer_latex": [ "$\\dfrac{c}{4}$." ], "answer_markdown": [ "$\\dfrac{c}{4}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "c/4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "core.units" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/22", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 22", "problem_latex": "Write three consecutive numbers of which $x$~is the\nmiddle number.", "markdown": "Write three consecutive numbers of which $x$ is the middle number.", "answer_latex": [ "$x - 1$, $x$, $x + 1$." ], "answer_markdown": [ "$x - 1$, $x$, $x + 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "[x - 1, x, x + 1]" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/23", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 23", "problem_latex": "If an odd number is represented by~$2n + 1$, what\nwill represent the next odd number?", "markdown": "If an odd number is represented by $2n + 1$, what will represent the next odd number?", "answer_latex": [ "$2n + 3$." ], "answer_markdown": [ "$2n + 3$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "2*n + 3" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/3", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 3", "problem_latex": "How many rolls of paper $g$~feet long and $k$~feet\nwide will be required to paper a room, the perimeter of\nwhich, after proper allowance is made for doors and windows,\nis $p$~feet and the height $h$~feet?", "markdown": "How many rolls of paper $g$ feet long and $k$ feet wide will be required to paper a room, the perimeter of which, after proper allowance is made for doors and windows, is $p$ feet and the height $h$ feet?", "answer_latex": [ "$\\dfrac{ph}{gk}$." ], "answer_markdown": [ "$\\dfrac{ph}{gk}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "p*h/(g*k)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/4", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 4", "problem_latex": "Write six times the square of~$m$, plus five~$c$ times\nthe expression $d$~plus $b$~minus~$a$.", "markdown": "Write six times the square of $m$, plus five $c$ times the expression $d$ plus $b$ minus $a$.", "answer_latex": [ "$6m^{2} + 5c(d + b - a)$." ], "answer_markdown": [ "$6m^{2} + 5c(d + b - a)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "6*m**2 + 5*c*(d + b - a)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/5", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 5", "problem_latex": "Write five times the expression two~$n$ plus one,\ndiminished by six times the expression $c$~minus $a$~plus~$b$.", "markdown": "Write five times the expression two $n$ plus one, diminished by six times the expression $c$ minus $a$ plus $b$.", "answer_latex": [ "$5(2n + 1) - 6(c - a + b)$." ], "answer_markdown": [ "$5(2n + 1) - 6(c - a + b)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "5*(2*n + 1) - 6*(c - a + b)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/6", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 6", "problem_latex": "A lady bought a dress for $a$~dollars, a cloak for $b$~dollars,\ntwo pairs of gloves for $c$~dollars a pair. She gave\na hundred-dollar bill in payment. How much money\nshould be returned to her?", "markdown": "A lady bought a dress for $a$ dollars, a cloak for $b$ dollars, two pairs of gloves for $c$ dollars a pair. She gave a hundred-dollar bill in payment. How much money should be returned to her?", "answer_latex": [ "\\$$100 - \\text{\\$}(a + b + 2c)$." ], "answer_markdown": [ "$$100 - \\text{\\$(a + b + 2c)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "100 - (a + b + 2*c)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/7", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 7", "problem_latex": "If a man can perform a piece of work in $4$~days, how\nmuch of it can he do in one day?", "markdown": "If a man can perform a piece of work in $4$ days, how much of it can he do in one day?", "answer_latex": [ "$\\frac{1}{4}$." ], "answer_markdown": [ "$\\frac{1}{4}$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.25, printed 1/4 (half-unit 1.0e-40; correctly rounded at the printed digits: 0.25)", "problem_expr": "1/4", "answer_expr": "1/4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 1/4" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.frac" ], "expectation": "X#\\frac{1}{4},5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/8", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 8", "problem_latex": "If a man can perform a piece of work in $x$~days,\nhow much of it can he do in one day?", "markdown": "If a man can perform a piece of work in $x$ days, how much of it can he do in one day?", "answer_latex": [ "$\\dfrac{1}{x}$." ], "answer_markdown": [ "$\\dfrac{1}{x}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "1/x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-8/9", "set": "wentworth-first-steps-in-algebra-1894/ex-8", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "17", "location": "Exercise 8, problem 9", "problem_latex": "If A~can do a piece of work in $x$~days, B~in $y$~days,\nC~in $z$~days, how much of it can they all do in one day,\nworking together?", "markdown": "If A can do a piece of work in $x$ days, B in $y$ days, C in $z$ days, how much of it can they all do in one day, working together?", "answer_latex": [ "$\\dfrac{1}{x} + \\dfrac{1}{y} + \\dfrac{1}{z}$." ], "answer_markdown": [ "$\\dfrac{1}{x} + \\dfrac{1}{y} + \\dfrac{1}{z}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "1/x + 1/y + 1/z" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:word-to-expression" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-80/1", "set": "wentworth-first-steps-in-algebra-1894/ex-80", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "159", "location": "Exercise 80, problem 1", "problem_latex": "$x^{3} + 3x^{2} y + 3xy^{2} + y^{3}$\\Add{.}", "markdown": "$x^{3} + 3x^{2} y + 3xy^{2} + y^{3}$", "answer_latex": [ "$x + y$." ], "answer_markdown": [ "$x + y$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**3 + 3*x**2*y + 3*x*y**2 + y**3", "answer_expr": "x + y" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:polynomial_cube_root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-80/2", "set": "wentworth-first-steps-in-algebra-1894/ex-80", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "159", "location": "Exercise 80, problem 2", "problem_latex": "$8x^{3} - 12x^{2} + 6x - 1$.", "markdown": "$8x^{3} - 12x^{2} + 6x - 1$.", "answer_latex": [ "$2x - 1$." ], "answer_markdown": [ "$2x - 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "8*x**3 - 12*x**2 + 6*x - 1", "answer_expr": "2*x - 1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:polynomial_cube_root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-80/3", "set": "wentworth-first-steps-in-algebra-1894/ex-80", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "159", "location": "Exercise 80, problem 3", "problem_latex": "$8x^{3} - 36x^{2} y + 54xy^{2} - 27y^{3}$.", "markdown": "$8x^{3} - 36x^{2} y + 54xy^{2} - 27y^{3}$.", "answer_latex": [ "$2x - 3y$." ], "answer_markdown": [ "$2x - 3y$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "8*x**3 - 36*x**2*y + 54*x*y**2 - 27*y**3", "answer_expr": "2*x - 3*y" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:polynomial_cube_root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-80/4", "set": "wentworth-first-steps-in-algebra-1894/ex-80", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "159", "location": "Exercise 80, problem 4", "problem_latex": "$64a^{3} - 144a^{2} x + 108ax^{2} - 27x^{3}$\\Add{.}", "markdown": "$64a^{3} - 144a^{2} x + 108ax^{2} - 27x^{3}$", "answer_latex": [ "$4a - 3x$." ], "answer_markdown": [ "$4a - 3x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "64*a**3 - 144*a**2*x + 108*a*x**2 - 27*x**3", "answer_expr": "4*a - 3*x" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:polynomial_cube_root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-80/5", "set": "wentworth-first-steps-in-algebra-1894/ex-80", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "159", "location": "Exercise 80, problem 5", "problem_latex": "$1 + 3x + 6x^{2} + 7x^{3} + 6x^{4} + 3x^{5} + x^{6}$.", "markdown": "$1 + 3x + 6x^{2} + 7x^{3} + 6x^{4} + 3x^{5} + x^{6}$.", "answer_latex": [ "$1 + x + x^{2}$." ], "answer_markdown": [ "$1 + x + x^{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "1 + 3*x + 6*x**2 + 7*x**3 + 6*x**4 + 3*x**5 + x**6", "answer_expr": "1 + x + x**2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:polynomial_cube_root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-80/6", "set": "wentworth-first-steps-in-algebra-1894/ex-80", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "159", "location": "Exercise 80, problem 6", "problem_latex": "$x^{6} - 3x^{5} + 6x^{4} - 7x^{3} + 6x^{2} - 3x + 1$.", "markdown": "$x^{6} - 3x^{5} + 6x^{4} - 7x^{3} + 6x^{2} - 3x + 1$.", "answer_latex": [ "$x^{2} - x + 1$." ], "answer_markdown": [ "$x^{2} - x + 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**6 - 3*x**5 + 6*x**4 - 7*x**3 + 6*x**2 - 3*x + 1", "answer_expr": "x**2 - x + 1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "other:polynomial_cube_root" ], "expectation": null, "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/1", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 1", "problem_latex": "$46,656$.", "markdown": "$46,656$.", "answer_latex": [ "$36$." ], "answer_markdown": [ "$36$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 36.0, printed 36 (half-unit 0.5; correctly rounded at the printed digits: 36.0)", "problem_expr": "Rational(46656)**Rational(1, 3)", "answer_expr": "36" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 36" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#36,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "46656 ENTER 3 1/x yˣ" ], "constants": [ { "value": "3", "source": "the 3 in 'cube root' (a cube root is the 3rd root, i.e. the power 1/3)" } ], "calculator_value": "+3599999999999999999999999999999999E-32", "printed_value": "36", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/10", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 10", "problem_latex": "$385,828,352$.", "markdown": "$385,828,352$.", "answer_latex": [ "$728$." ], "answer_markdown": [ "$728$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 728.0, printed 728 (half-unit 0.5; correctly rounded at the printed digits: 728.0)", "problem_expr": "Rational(385828352)**Rational(1, 3)", "answer_expr": "728" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 728" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#728,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "385828352 ENTER 3 GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the 3 in 'cube root' (the root index, x in the x-th root of y)" } ], "calculator_value": "+7280E-1", "printed_value": "728", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/11", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 11", "problem_latex": "$1879.080904$.", "markdown": "$1879.080904$.", "answer_latex": [ "$12.34$." ], "answer_markdown": [ "$12.34$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 12.34, printed 12.34 (half-unit 0.005; correctly rounded at the printed digits: 12.34)", "problem_expr": "Rational(1879080904, 1000000)**Rational(1, 3)", "answer_expr": "12.34" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 617/50" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#12.34,5E-3", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/12", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 12, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 12", "problem_latex": "$1838.265625$.", "markdown": "$1838.265625$.", "answer_latex": [ "$12.25$." ], "answer_markdown": [ "$12.25$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 12.25, printed 12.25 (half-unit 0.005; correctly rounded at the printed digits: 12.25)", "problem_expr": "Rational(1838265625, 1000000)**Rational(1, 3)", "answer_expr": "12.25" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 49/4" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#12.25,5E-3", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "1838.265625 ENTER 3", "GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the 3 in 'cube root' (the root index, from the expression's exponent Rational(1,3))" } ], "calculator_value": "+12250E-3", "printed_value": "12.25", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/13", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 13, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 13", "problem_latex": "$0.01$.", "markdown": "$0.01$.", "answer_latex": [ "$0.2154\\dots$" ], "answer_markdown": [ "$0.2154\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.215443469003, printed 0.2154 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.2154)", "problem_expr": "Rational(1, 100)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 10**(1/3)/10" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#0.2154,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/14", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 14, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 14", "problem_latex": "$0.05$.", "markdown": "$0.05$.", "answer_latex": [ "$0.3684\\dots$" ], "answer_markdown": [ "$0.3684\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.368403149864, printed 0.3684 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.3684)", "problem_expr": "Rational(1, 20)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 50**(1/3)/10" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#0.3684,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/15", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 15, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 15", "problem_latex": "$0.2$.", "markdown": "$0.2$.", "answer_latex": [ "$0.5848\\dots$" ], "answer_markdown": [ "$0.5848\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 0.584803547643, printed 0.5848 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.5848)", "problem_expr": "Rational(1, 5)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 5**(2/3)/5" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#0.5848,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/16", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 16", "problem_latex": "$4$.", "markdown": "$4$.", "answer_latex": [ "$1.5874\\dots$" ], "answer_markdown": [ "$1.5874\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.58740105197, printed 1.5874 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.5874)", "problem_expr": "Rational(4)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2**(2/3)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#1.5874,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/17", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 17", "problem_latex": "$10$.", "markdown": "$10$.", "answer_latex": [ "$2.1544\\dots$" ], "answer_markdown": [ "$2.1544\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 2.15443469003, printed 2.1544 (half-unit 5.0e-5; correctly rounded at the printed digits: 2.1544)", "problem_expr": "Rational(10)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 10**(1/3)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#2.1544,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/18", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 18", "problem_latex": "$87$.", "markdown": "$87$.", "answer_latex": [ "$4.4310\\dots$" ], "answer_markdown": [ "$4.4310\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 4.43104762169, printed 4.4310 (half-unit 5.0e-5; correctly rounded at the printed digits: 4.431)", "problem_expr": "Rational(87)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 87**(1/3)" ], "shape": [ "evaluate: N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#4.4310,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/19", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 19", "problem_latex": "$2.5$.", "markdown": "$2.5$.", "answer_latex": [ "$1.3572\\dots$" ], "answer_markdown": [ "$1.3572\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.3572088083, printed 1.3572 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.3572)", "problem_expr": "Rational(5, 2)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2**(2/3)*5**(1/3)/2" ], "shape": [ "evaluate: N*N**(2*N)" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#1.3572,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/2", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 2, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 2", "problem_latex": "$42,875$.", "markdown": "$42,875$.", "answer_latex": [ "$35$." ], "answer_markdown": [ "$35$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 35.0, printed 35 (half-unit 0.5; correctly rounded at the printed digits: 35.0)", "problem_expr": "Rational(42875)**Rational(1, 3)", "answer_expr": "35" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 35" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#35,5E-1", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/20", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 20, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 20", "problem_latex": "$2.05$.", "markdown": "$2.05$.", "answer_latex": [ "$1.2703\\dots$" ], "answer_markdown": [ "$1.2703\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.27033409277, printed 1.2703 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.2703)", "problem_expr": "Rational(41, 20)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 2050**(1/3)/10" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#1.2703,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/21", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 21, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 21", "problem_latex": "$3.02$.", "markdown": "$3.02$.", "answer_latex": [ "$1.4454\\dots$" ], "answer_markdown": [ "$1.4454\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 1.44544747339, printed 1.4454 (half-unit 5.0e-5; correctly rounded at the printed digits: 1.4454)", "problem_expr": "Rational(151, 50)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 3020**(1/3)/10" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.solve.num" ], "expectation": "X#1.4454,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/22", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 22, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 22", "problem_latex": "$\\frac{2}{3}$.", "markdown": "$\\frac{2}{3}$.", "answer_latex": [ "$0.8735\\dots$" ], "answer_markdown": [ "$0.8735\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 0.873580464736, printed 0.8735 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.8736): one unit off in the last printed place", "problem_expr": "Rational(2, 3)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: 2**(1/3)*3**(2/3)/3" ], "shape": [ "evaluate: N*N**(2*N)" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": "X#0.8735,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/23", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 23, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 23", "problem_latex": "$\\frac{3}{4}$.", "markdown": "$\\frac{3}{4}$.", "answer_latex": [ "$0.9085\\dots$" ], "answer_markdown": [ "$0.9085\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 0.908560296416, printed 0.9085 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.9086): one unit off in the last printed place", "problem_expr": "Rational(3, 4)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: 6**(1/3)/2" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": "X#0.9085,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/24", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 24, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 24", "problem_latex": "$\\frac{9}{11}$.", "markdown": "$\\frac{9}{11}$.", "answer_latex": [ "$0.9352\\dots$" ], "answer_markdown": [ "$0.9352\\dots$" ], "checks": [ { "task": "evaluate", "verdict": "PASS-LOOSE", "judge_why": "computed 0.935297861646, printed 0.9352 (half-unit 5.0e-5; correctly rounded at the printed digits: 0.9353): one unit off in the last printed place", "problem_expr": "Rational(9, 11)**Rational(1, 3)", "answer_expr": null } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS-LOOSE" ] }, "form": [ "evaluate: 33**(2/3)/11" ], "shape": [ "evaluate: N*N**N" ], "same_problem_in": [], "needs": [ "core.frac", "core.solve.num" ], "expectation": "X#0.9352,5E-5", "keys": [] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/3", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 3", "problem_latex": "$91,125$.", "markdown": "$91,125$.", "answer_latex": [ "$45$." ], "answer_markdown": [ "$45$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 45.0, printed 45 (half-unit 0.5; correctly rounded at the printed digits: 45.0)", "problem_expr": "Rational(91125)**Rational(1, 3)", "answer_expr": "45" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 45" ], "shape": [ "evaluate: N" ], "same_problem_in": [ "boyden-first-book-in-algebra-1895/ex-30/1" ], "needs": [ "core.arith" ], "expectation": "X#45,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "91125 ENTER 3", "GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the index 3 in 'cube root' (the problem asks for the cube root, the 3rd root of 91125; the word 'cube' gives the index)" } ], "calculator_value": "+450E-1", "printed_value": "45", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/4", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 4, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 4", "problem_latex": "$274,625$.", "markdown": "$274,625$.", "answer_latex": [ "$65$." ], "answer_markdown": [ "$65$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 65.0, printed 65 (half-unit 0.5; correctly rounded at the printed digits: 65.0)", "problem_expr": "Rational(274625)**Rational(1, 3)", "answer_expr": "65" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 65" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#65,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "274625 ENTER 3 GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the word 'cube' in 'Find the cube root of' (the root index)" } ], "calculator_value": "+650E-1", "printed_value": "65", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/5", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 5, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 5", "problem_latex": "$110,592$.", "markdown": "$110,592$.", "answer_latex": [ "$48$." ], "answer_markdown": [ "$48$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 48.0, printed 48 (half-unit 0.5; correctly rounded at the printed digits: 48.0)", "problem_expr": "Rational(110592)**Rational(1, 3)", "answer_expr": "48" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 48" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#48,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "110592 ENTER 3 GOLD ˣ√y" ], "constants": [ { "value": "110592", "source": "the number 110,592 in the problem text 'Find the cube root of: 110,592' (comma dropped)" }, { "value": "3", "source": "the 3 implied by 'cube' in 'cube root' in the problem text" } ], "calculator_value": "+480E-1", "printed_value": "48", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/6", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 6, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 6", "problem_latex": "$258,474,853$.", "markdown": "$258,474,853$.", "answer_latex": [ "$637$." ], "answer_markdown": [ "$637$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 637.0, printed 637 (half-unit 0.5; correctly rounded at the printed digits: 637.0)", "problem_expr": "Rational(258474853)**Rational(1, 3)", "answer_expr": "637" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 637" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#637,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "258474853 ENTER 3 1/x yˣ" ], "calculator_value": "+6369999999999999999999999999999996E-31", "printed_value": "637", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/7", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 7, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 7", "problem_latex": "$109,215,352$.", "markdown": "$109,215,352$.", "answer_latex": [ "$478$." ], "answer_markdown": [ "$478$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 478.0, printed 478 (half-unit 0.5; correctly rounded at the printed digits: 478.0)", "problem_expr": "Rational(109215352)**Rational(1, 3)", "answer_expr": "478" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 478" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#478,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "109215352 ENTER 3 GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the 'cube' in 'cube root' of the problem (root index 3)" } ], "calculator_value": "+4780E-1", "printed_value": "478", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/8", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 8, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 8", "problem_latex": "$259,694,072$.", "markdown": "$259,694,072$.", "answer_latex": [ "$638$." ], "answer_markdown": [ "$638$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 638.0, printed 638 (half-unit 0.5; correctly rounded at the printed digits: 638.0)", "problem_expr": "Rational(259694072)**Rational(1, 3)", "answer_expr": "638" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 638" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#638,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "259694072 ENTER 3 GOLD ˣ√y" ], "constants": [ { "value": "3", "source": "the denominator of the exponent Rational(1, 3) in the SymPy expression, i.e. the cube root index" } ], "calculator_value": "+6380E-1", "printed_value": "638", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-81/9", "set": "wentworth-first-steps-in-algebra-1894/ex-81", "number": 9, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "163", "location": "Exercise 81, problem 9", "problem_latex": "$127,263,527$.", "markdown": "$127,263,527$.", "answer_latex": [ "$503$." ], "answer_markdown": [ "$503$." ], "checks": [ { "task": "evaluate", "verdict": "PASS", "judge_why": "computed 503.0, printed 503 (half-unit 0.5; correctly rounded at the printed digits: 503.0)", "problem_expr": "Rational(127263527)**Rational(1, 3)", "answer_expr": "503" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "evaluate: 503" ], "shape": [ "evaluate: N" ], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": "X#503,5E-1", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "127263527 ENTER 3 1/x yˣ" ], "constants": [ { "value": "3", "source": "the cube in 'cube root' of the problem text (the root index); 1/3 is formed by the 1/x key" } ], "calculator_value": "+5029999999999999999999999999999997E-31", "printed_value": "503", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/1", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 1, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 1", "problem_latex": "$3x = x + 8$.", "markdown": "$3x = x + 8$.", "answer_latex": [ "$4$." ], "answer_markdown": [ "$4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*x, x + 8)", "answer_expr": "4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x, x + 8)" ], "shape": [ "solve: Eq(N*x, N + x)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.solve.num" ], "expectation": "X=4", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "8 ENTER 3 ENTER 1 −", "÷" ], "constants": [ { "value": "1", "source": "the coefficient of x on the right side (x + 8 is 1x + 8), needed to move x across: 3x − x = 2x, so 2x = 8" } ], "calculator_value": "+4E+0", "printed_value": "4", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/10", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 10, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 10", "problem_latex": "$3(x - 2) = 2(x - 1)$.", "markdown": "$3(x - 2) = 2(x - 1)$.", "answer_latex": [ "$4$." ], "answer_markdown": [ "$4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*(x - 2), 2*(x - 1))", "answer_expr": "4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x - 6, 2*x - 2)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.solve.num" ], "expectation": "X=4", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "3 ENTER 2 × 2 ENTER 1 × −", "3 ENTER 2 − ÷" ], "calculator_value": "+4E+0", "printed_value": "4", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/11", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 11, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 11", "problem_latex": "$2x + 3 = 16 - (2x - 3)$.", "markdown": "$2x + 3 = 16 - (2x - 3)$.", "answer_latex": [ "$4$." ], "answer_markdown": [ "$4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2*x + 3, 16 - (2*x - 3))", "answer_expr": "4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x + 3, 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"mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 7 × 3 +", "19 ENTER 2 −", "÷" ], "constants": [ { "value": "2", "source": "the 2 in '2(7 + x)', the coefficient that multiplies the bracket" }, { "value": "7", "source": "the 7 inside '(7 + x)'" }, { "value": "3", "source": "the 3 in '19x - 3', the constant term moved across the equals sign" }, { "value": "19", "source": "the 19 in '19x - 3', the coefficient of x" } ], "calculator_value": "+1E+0", "printed_value": "1", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py 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"STU", "entry": "rpn", "steps": [ "70 ENTER 20 −", "7 ENTER 5 − ÷" ], "constants": [ { "value": "7", "source": "the 7 in 7x of the equation 7x − 70 = 5x − 20" }, { "value": "5", "source": "the 5 in 5x of the equation 7x − 70 = 5x − 20" }, { "value": "70", "source": "the 70 in 7x − 70 of the equation" }, { "value": "20", "source": "the 20 in 5x − 20 of the equation" } ], "calculator_value": "+25E+0", "printed_value": "25", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied 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"keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 2 +", "108 ENTER 22 +", "x↔y ÷" ], "constants": [ { "value": "2", "source": "the coefficient 2 of x in 2x - 22 = 108 - 2x; the second 2 is the same coefficient, added to the first to make 4x" }, { "value": "22", "source": "the 22 in 2x - 22 = 108 - 2x" }, { "value": "108", "source": "the 108 in 2x - 22 = 108 - 2x" } ], "calculator_value": "+325E-1", "printed_value": "65/2", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied 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"entry": "rpn", "steps": [ "32 ENTER 2 ENTER 5 × 5 ENTER 4 × − −", "2 ENTER 5 + ÷" ], "constants": [ { "value": "2", "source": "the 2 in '2(x + 5)'" }, { "value": "5", "source": "the 5 in '(x + 5)' and the 5 in '5(x - 4)'" }, { "value": "4", "source": "the 4 in '(x - 4)'" }, { "value": "32", "source": "the right-hand side '= 32'" } ], "calculator_value": "+6E+0", "printed_value": "6", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/16", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 16, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 16", "problem_latex": "$2(3x - 25) = 10$.", "markdown": "$2(3x - 25) = 10$.", "answer_latex": [ "$10$." ], "answer_markdown": [ "$10$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(2*(3*x - 25), 10)", "answer_expr": "10" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(6*x - 50, 10)" ], "shape": [ "solve: Eq(N*x + N, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.solve.num" ], "expectation": "X=10", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "10 ENTER 2 ÷", "25 +", "3 ÷" ], 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digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/17", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 17, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 17", "problem_latex": "$33x - 70 = 3x + 20$.", "markdown": "$33x - 70 = 3x + 20$.", "answer_latex": [ "$3$." ], "answer_markdown": [ "$3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(33*x - 70, 3*x + 20)", "answer_expr": "3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(33*x - 70, 3*x + 20)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "core.eqn", "core.solve.num" ], "expectation": "X=3", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "70 ENTER 20 +", "33 ENTER 3 −", "÷" ], "calculator_value": "+3E+0", "printed_value": "3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/18", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 18, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 18", 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"printed_value": "1", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/19", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 19, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 19", 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"markdown": "$(x^{2} - 9) - (x^{2} - 16) + x = 10$.", "answer_latex": [ "$3$." ], "answer_markdown": [ "$3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq((x**2 - 9) - (x**2 - 16) + x, 10)", "answer_expr": "3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(x + 7, 10)" ], "shape": [ "solve: Eq(N + x, N)" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn", "core.solve.num" ], "expectation": "X=3", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "16 ENTER 9 −", "10 x↔y −" ], "constants": [ { "value": "16", "source": "the 16 in '(x² − 16)' in the problem; subtracting it from the other bracket leaves +16" }, { "value": "9", "source": "the 9 in '(x² − 9)' in the problem" }, { "value": "10", "source": "the right-hand side '= 10' of the problem" } ], "calculator_value": "+3E+0", "printed_value": "3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/3", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 3, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 3", "problem_latex": "$3x + 4 = x + 10$.", "markdown": "$3x + 4 = x + 10$.", "answer_latex": [ "$3$." ], "answer_markdown": [ "$3$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(3*x + 4, x + 10)", "answer_expr": "3" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x + 4, x + 10)" ], "shape": [ "solve: Eq(N*x + N, N + x)" ], "same_problem_in": [], "needs": [ "core.eqn", "core.solve.num" ], "expectation": "X=3", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "10 ENTER 4 −", "3 ENTER 1 −", "÷" ], "constants": [ { "value": "1", "source": "the coefficient of x in 'x + 10' (that x is 1x); used to get 3x − x = 2x" } ], "calculator_value": "+3E+0", "printed_value": "3", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/30", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 30, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 30", "problem_latex": "$x^{2} + 8x - (x^{2} - x - 2) = 5(x + 3) + 3$.", "markdown": "$x^{2} + 8x - (x^{2} - x - 2) = 5(x + 3) + 3$.", "answer_latex": [ "$4$." ], "answer_markdown": [ "$4$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 + 8*x - (x**2 - x - 2), 5*(x + 3) + 3)", "answer_expr": "4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(9*x + 2, 5*x + 18)" ], "shape": [ "solve: True" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.solve.poly", "core.eqn", "core.solve.num" ], "expectation": "X=4", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "5 ENTER 3 × 3 + 2 −", "8 ENTER 1 + 5 − ÷" ], "constants": [ { "value": "5", "source": "the 5 in 5(x + 3) + 3" }, { "value": "3", "source": "the 3 in (x + 3) and the 3 in + 3 on the right side" }, { "value": "2", "source": "the 2 in 'x^2 - x - 2'; the sign flip from subtracting (x^2 - x - 2) makes it +2 on the left, so the constant difference is 18 - 2" }, { "value": "8", "source": "the 8 in 8x on the left side" }, { "value": "1", "source": "the implied coefficient of x in 'x^2 - x - 2' (the -x inside the subtracted bracket becomes +x, coefficient 1)" } ], "calculator_value": "+4E+0", "printed_value": "4", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/31", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 31, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 31", "problem_latex": "$x^{2} + x - 2 + x^{2} + 2x - 3 = 2x^{2} - 7x - 1$.", "markdown": "$x^{2} + x - 2 + x^{2} + 2x - 3 = 2x^{2} - 7x - 1$.", "answer_latex": [ "$\\frac{2}{5}$." ], "answer_markdown": [ "$\\frac{2}{5}$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(x**2 + x - 2 + x**2 + 2*x - 3, 2*x**2 - 7*x - 1)", "answer_expr": "Rational(2, 5)" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(2*x**2 + 3*x - 5, 2*x**2 - 7*x - 1)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, N*x + N*x**N - 1)" ], "same_problem_in": [], "needs": [ "cas.collect", "cas.solve.poly", "core.eqn", "core.frac", "core.solve.num" ], "expectation": null, "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "2 ENTER 3 + 1 −", "3 ENTER 7 + ÷" ], "constants": [ { "value": "2", "source": "the constant −2 on the left side: x² + x − 2 + x² + 2x − 3" }, { "value": "3", "source": "the constant −3 on the left side (x² + x − 2 + x² + 2x − 3); the same 3 is the x-coefficient sum from x + 2x" }, { "value": "1", "source": "the constant −1 on the right side: 2x² − 7x − 1" }, { "value": "7", "source": "the coefficient of x on the right side: −7x in 2x² − 7x − 1" } ], "calculator_value": "+4E-1", "printed_value": "2/5", "core_pins": { "firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits 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"firmware": "628c96c8634194ec8196315d8aa75f88f04d49e6", "casim": "dbb6d4c6f1014957b14a1711d2369aa70cb63b24", "stu32-tutor": "1fe14f61238f3b829624546f726f74b24a0b7b9a", "intel-dfp-sha256": "85dafd70f0fe2a8da218ade4233fca9d3228b0b04cd6d8527f7499926be01037", "builder": "gcc@sha256:9188ac751ca24431dc43dbd142a223c98ea74f01d2858e84d30ba342a0d67844", "base": "gcr.io/distroless/cc-debian13@sha256:e792ab3d241a468a4fd7519ddbbebe66b49b5f365771716ea688ad40b6c6f1c2" }, "records_checker": "stu32-tutor tools/records.py at fa8658e (vendor/records.py, copied unmodified)", "checked_by": "stu32-calc ran the keys; mpmath at 50 digits agrees to 1E-30; tutor's records.py accepted them" } ] }, { "id": "wentworth-first-steps-in-algebra-1894/ex-9/34", "set": "wentworth-first-steps-in-algebra-1894/ex-9", "number": 34, "part": null, "book": "wentworth-first-steps-in-algebra-1894", "edition": "Ginn & Company, 1894", "page": "24", "location": "Exercise 9, problem 34", "problem_latex": "$6x + 3 - (3x + 2) = (2x - 1) + 9$.", "markdown": "$6x + 3 - (3x + 2) = (2x - 1) + 9$.", "answer_latex": [ "$7$." ], "answer_markdown": [ "$7$." ], "checks": [ { "task": "solve", "verdict": "PASS", "judge_why": null, "problem_expr": "Eq(6*x + 3 - (3*x + 2), (2*x - 1) + 9)", "answer_expr": "7" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "solve: Eq(3*x + 1, 2*x + 8)" ], "shape": [ "solve: Eq(N*x + 1, N*x + N)" ], "same_problem_in": [], "needs": [ "cas.expand", "core.eqn", "core.solve.num" ], "expectation": "X=7", "keys": [ { "kind": "keys", "mode": "STU", "entry": "rpn", "steps": [ "9 ENTER 1 − 3 ENTER 2 − −", "6 ENTER 3 − 2 − ÷" ], "constants": [ { "value": "9", "source": "the 9 in '+ 9' on the right side of the equation" }, { "value": "1", "source": "the 1 in '(2x − 1)' on the right side of the equation" }, { "value": "3", "source": "the 3 in '+ 3' on the left side, and the 3 in '3x' inside the parentheses" }, { "value": "2", "source": "the 2 in '(3x + 2)' on 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"wentworth-first-steps-in-algebra-1894/T003", "kind": "transcriber-marked", "page": null, "printed": ":", "corrected": ",", "class": "spacing-or-punctuation", "context": "ook uses commas elsewhere] \\Dictum{To Add Algebraic Numbers}", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T004", "kind": "transcriber-marked", "page": null, "printed": ",", "corrected": ".", "class": "spacing-or-punctuation", "context": "ivide $a^{5}$~by~$a^{2}$, $a^{6}$~by~$a^{4}$, $a^{4}$~by~$a$", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T005", "kind": "transcriber-marked", "page": null, "printed": ",", "corrected": ";", "class": "spacing-or-punctuation", "context": "}{a^{4}} &= \\frac{aaaaaa}{aaaa} &&= aa &&= a^{2} &&= a^{6-4}", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T006", "kind": "transcriber-marked", "page": null, "printed": ",", "corrected": ".", "class": "spacing-or-punctuation", "context": "ac{a^{4}}{a} &= \\frac{aaaa}{a} &&= aaa &&= a^{3} &&= a^{4-1}", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T007", "kind": "transcriber-marked", "page": null, "printed": ",", "corrected": "", "class": "spacing-or-punctuation", "context": "cient of~$ax^{4}$ will be $-5 + 2$, or~$-3$; the coefficient", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T008", "kind": "transcriber-marked", "page": null, "printed": ".", "corrected": "", "class": "spacing-or-punctuation", "context": " + b + c) p \\\\ &= am + bm + cm + an + bn + cn + ap + bp + cp", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T009", "kind": "transcriber-marked", "page": null, "printed": "", "corrected": "\\therefore", "class": "content", "context": ". \\begin{align*} &+6 - 5 = 1,\\quad 6y × (-5y) = -30y^{2}. \\\\", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T010", "kind": "transcriber-marked", "page": null, "printed": "", "corrected": "Resolve into factors:", "class": "content", "context": "{2} = (a + 3b - 5c)(a - 3b + 5c)$. \\end{Soln} \\Exercise{34.}", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T011", "kind": "transcriber-marked", "page": null, "printed": ",", "corrected": ".", "class": "spacing-or-punctuation", "context": "um of the first $16$~terms of the series $5$, $7$, $9$,~$11$", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/T012", "kind": "transcriber-marked", "page": null, "printed": "the", "corrected": "then", "class": "content", "context": "y $3(x + y)^{2}$; and if the third term of the root be~$+z$,", "found_by": "Project Gutenberg Distributed Proofreaders (marked in the source)", "status": "transcriber_marked" }, { "id": "wentworth-first-steps-in-algebra-1894/G001", "kind": "transcription", "status": "confirmed", "page": "45", "location": "Exercise 17, problem 30 (Divide)", "printed": "$-19mg^{2}t^{3}$~by~$57m2^gt^{4}$", "corrected": "$-19mg^{2}t^{3}$~by~$57m^{2}gt^{4}$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 45 (leaf n56), reads \"30. -19mg^2t^3 by 57m^2gt^4.\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G002", "kind": "transcription", "status": "confirmed", "page": "169", "location": "Answers, Exercise 19, answer 3", "printed": "$2a + 3y -8z$", "corrected": "$2x + 3y - 8z$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 169 (leaf n180), reads \"3. 2x + 3y - 8z.\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G003", "kind": "transcription", "status": "confirmed", "page": "169", "location": "Answers, Exercise 20, answer 13", "printed": "$6z - 2y-z$", "corrected": "$6x - 2y - z$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 169 (leaf n180), reads \"13. 6x - 2y - z.\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G004", "kind": "transcription", "status": "confirmed", "page": "170", "location": "Answers, Exercise 22, answer 18", "printed": "$x^{5} - 3x^{4} - x^{3} + 16x^{2} - 21$", "corrected": "$x^{5} - 3x^{4} - 3x^{3} + 16x^{2} - 21$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 170 (leaf n181), reads \"18. x^5 - 3x^4 - 3x^3 + 16x^2 - 21.\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G005", "kind": "transcription", "status": "confirmed", "page": "69", "location": "Exercise 29, problem 3", "printed": "\\dfrac{1 - 27c^{3}}{1 - 2c}", "corrected": "\\dfrac{1 - 27c^{3}}{1 - 3c}", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 69 (leaf n80), reads \"3. (1 - 27c^3)/(1 - 3c).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G006", "kind": "transcription", "status": "confirmed", "page": "172", "location": "Answers, Exercise 29, answer 17", "printed": "$4a^{2}b^{2}c^{2} - 6abc + 9$", "corrected": "$4a^{2}b^{2}c^{2} + 6abc + 9$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 172 (leaf n183), reads \"17. 4a^2b^2c^2 + 6abc + 9.\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G007", "kind": "transcription", "status": "confirmed", "page": "174", "location": "Answers, Exercise 34, answer 20", "printed": "$(5c + 3a - 2x)(4c - 3a + 2x)$", "corrected": "$(5c + 3a - 2x)(5c - 3a + 2x)$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 174 (leaf n185), reads \"20. (5c + 3a - 2x)(5c - 3a + 2x).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G008", "kind": "transcription", "status": "confirmed", "page": "174", "location": "Answers, Exercise 35, answer 16", "printed": "$(4x^{2} - 3y^{5})(16x^{8} + 12x^{4}y^{5} + 9y^{10})$", "corrected": "$(4x^{4} - 3y^{5})(16x^{8} + 12x^{4}y^{5} + 9y^{10})$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 174 (leaf n185), reads \"16. (4x^4 - 3y^5)(16x^8 + 12x^4y^5 + 9y^10).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G009", "kind": "transcription", "status": "confirmed", "page": "175", "location": "Answers, Exercise 36, answer 1", "printed": "$(x + 1)(a^{2} - x + 1)$", "corrected": "$(x + 1)(x^{2} - x + 1)$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 175 (leaf n186), reads \"1. (x + 1)(x^2 - x + 1).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G010", "kind": "transcription", "status": "confirmed", "page": "80", "location": "Exercise 37, problem 18", "printed": "$m^{2}n^{2} + 14mnx^{2} + 49x^{2}$", "corrected": "$m^{2}n^{2} + 14mnx^{2} + 49x^{4}$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 80 (leaf n91), reads \"18. m^2n^2 + 14mnx^2 + 49x^4.\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G011", "kind": "transcription", "status": "confirmed", "page": "176", "location": "Answers, Exercise 39, answer 21", "printed": "$(x - 8)(x + 11)$", "corrected": "$(x - 8)(x - 11)$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 176 (leaf n187), reads \"21. (x - 8)(x - 11).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G012", "kind": "transcription", "status": "confirmed", "page": "178", "location": "Answers, Exercise 47, answer 4", "printed": "$\\dfrac{4ax}{a^{3} - x^{2}}$", "corrected": "$\\dfrac{4ax}{a^{2} - x^{2}}$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 178 (leaf n189), reads \"4. 4ax/(a^2 - x^2).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G013", "kind": "transcription", "status": "confirmed", "page": "105", "location": "Exercise 51, problem 11", "printed": "+ 1 - \\frac{1}{4} = 0", "corrected": "+ 1\\frac{1}{4} = 0", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 105 (leaf n116), reads \"11. (2x + 1)/4 - (4x - 1)/10 + 1 1/4 = 0 (a mixed number).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/G014", "kind": "transcription", "status": "confirmed", "page": "183", "location": "Answers, Exercise 76, answer 3", "printed": "$4frac{1}{6}$", "corrected": "$4\\frac{1}{6}$", "class": "transcription (the 1894 page is right; the 2011 transcription differs)", "evidence": "Status: confirmed against a scan. The Internet Archive scan bwb_P9-CSD-871 (Ginn & Company, 1894), printed page 183 (leaf n194), reads \"3. 4 1/6 (the backslash is missing, so the transcription renders as 4frac16).\", read from the page image on 2026-10-10. Flagged by the claudiverse shelf: the judge found the problem and its printed answer inconsistent, and the Opus adjudicator named the likely misreading.", "found_by": "claudiverse shelf (Haiku 5.5 extractor, judge.py, Opus 5.5 adjudicator); settled against the scan by qwenniverse-fermion", "date": "2026-10-10" }, { "id": "wentworth-first-steps-in-algebra-1894/N001", "kind": "note", "status": "note", "page": "45", "location": "Worked example 5 before Exercise 17 (Divide)", "printed": "\\Item{5.} Divide $-15a^{3}b^{2}c^{3}$~by~$-60a^{2}bc^{3}$.", "corrected": null, "class": "note (no correction proposed)", "evidence": "Status: the scan and the transcription differ; the 1894 page is itself inconsistent. The Internet Archive scan bwb_P9-CSD-871, printed page 45 (leaf n56), reads \"5. Divide -15a^3b^2c^3 by -10a^2bc^3.\" with the working below it dividing by -60a^2bc^3 to get ab/4. So the 1894 printing contradicts itself, and the transcription reads -60 in both places, either from a later corrected printing or a silent correction. Recorded so a reader comparing a copy is not surprised; the transcription's -60 is the self-consistent reading.", "found_by": "qwenniverse-fermion, while reading page 45 for G001", "date": "2026-10-10" } ] }