Multiply and Divide Mixed Numbers and Complex Fractions

Multiply and Divide Mixed Numbers In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions. Multiply: 313·58 Solution 313·58 Convert 313 to an improper fraction. 103·58 Multiply. 10·53·8 Look for common factors. 2̸·5·53·2̸·4 Remove common factors. 5·53·4 Simplify. 2512

Notice that we left the answer as an improper fraction, 2512, and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between 2112 and 2·112. Multiply, and write your answer in simplified form: 523·617. 2 Multiply, and write your answer in simplified form: 37·514. 94 Multiply or divide mixed numbers.Convert the mixed numbers to improper fractions. Follow the rules for fraction multiplication or division. Simplify if possible. Multiply, and write your answer in simplified form: 245(−178). Solution 245(−178) Convert mixed numbers to improper fractions. 145(−158) Multiply. − 14·155·8 Look for common factors. − 2̸·7·5̸·35̸·2̸·4 Remove common factors. − 7·34 Simplify. − 214

Multiply, and write your answer in simplified form. 557(−258). −15 Multiply, and write your answer in simplified form. −325·416. −856 Divide, and write your answer in simplified form: 347÷5. Solution 347÷5 Convert mixed numbers to improper fractions. 257÷51 Multiply the first fraction by the reciprocal of the second. 257·15 Multiply. 25·17·5 Look for common factors. 5̸·5·17·5̸ Remove common factors. 5·17 Simplify. 57

Divide, and write your answer in simplified form: 438÷7. 58 Divide, and write your answer in simplified form: 258÷3. 78 Divide: 212÷114. Solution 212÷114 Convert mixed numbers to improper fractions. 52÷54 Multiply the first fraction by the reciprocal of the second. 52·45 Multiply. 5·42·5 Look for common factors. 5̸·2̸·22̸·1·5̸ Remove common factors. 21 Simplify. 2

Divide, and write your answer in simplified form: 223÷113. 2 Divide, and write your answer in simplified form: 334÷112. 52

Translate Phrases to Expressions with Fractions The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of a and b is the result you get from dividing a by b, or ab. Let’s practice translating some phrases into algebraic expressions using these terms. Translate the phrase into an algebraic expression: “the quotient of 3x and 8.” Solution The keyword is quotient; it tells us that the operation is division. Look for the words of and and to find the numbers to divide. The quotientof3xand8.This tells us that we need to divide 3x by 8. 3x8 Translate the phrase into an algebraic expression: the quotient of 9s and 14. 9s14 Translate the phrase into an algebraic expression: the quotient of 5y and 6. 5y6 Translate the phrase into an algebraic expression: the quotient of the difference of m and n, and p. Solution We are looking for the quotient of the difference of m and n, and p. This means we want to divide the difference of m and n by p. m−np Translate the phrase into an algebraic expression: the quotient of the difference of a and b, and cd. a−bcd Translate the phrase into an algebraic expression: the quotient of the sum of p and q, and r. p+qr

Simplify Complex Fractions Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.Some examples of complex fractions are: 6733458x256To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction 3458 can be written as 34÷58. Simplify: 3458. Solution 3458 Rewrite as division. 34÷58 Multiply the first fraction by the reciprocal of the second. 34·85 Multiply. 3·84·5 Look for common factors. 3·4̸·24̸·5 Remove common factors and simplify. 65

Simplify: 2356. 45 Simplify: 37611. 1114 Simplify a complex fraction.Rewrite the complex fraction as a division problem. Follow the rules for dividing fractions. Simplify if possible. Simplify: −673. Solution −673 Rewrite as division. −67÷3 Multiply the first fraction by the reciprocal of the second. −67·13 Multiply; the product will be negative. −6·17·3 Look for common factors. −3̸·2·17·3̸ Remove common factors and simplify. −27

Simplify: −874. −27 Simplify: −3910. −103 Simplify: x2xy6. Solution x2xy6 Rewrite as division. x2÷xy6 Multiply the first fraction by the reciprocal of the second. x2·6xy Multiply. x·62·xy Look for common factors. x̸·3·2̸2̸·x̸·y Remove common factors and simplify. 3y

Simplify: a8ab6. 34b Simplify: p2pq8. 4q Simplify: 23418. Solution 23418 Rewrite as division. 234÷18 Change the mixed number to an improper fraction. 114÷18 Multiply the first fraction by the reciprocal of the second. 114·81 Multiply. 11·84·1 Look for common factors. 11·4̸·24̸·1 Remove common factors and simplify. 22

Simplify: 57125. 2549. Simplify: 85315. 12

Simplify Expressions with a Fraction Bar Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction −13 could be the result of dividing −13, a negative by a positive, or of dividing 1−3, a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative. −1−3=13negativenegative=positive Placement of Negative Sign in a Fraction For any positive numbers a and b, −ab=a−b=−ab Which of the following fractions are equivalent to 7−8?−7−8,−78,78,−78 Solution The quotient of a positive and a negative is a negative, so 7−8 is negative. Of the fractions listed, −78 and −78 are also negative. Which of the following fractions are equivalent to −35? −3−5,35,−35,3−5 −35,3−5 Which of the following fractions are equivalent to −27? −2−7,−27,27,2−7 −27,2−7 Fraction bars act as grouping symbols. The expressions above and below the fraction bar should be treated as if they were in parentheses. For example, 4+85−3 means (4+8)÷(5−3). The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide. We’ll add fraction bars to our set of grouping symbols from Use the Language of Algebra to have a more complete set here.Grouping Symbols

Simplify an expression with a fraction bar.Simplify the numerator. Simplify the denominator. Simplify the fraction. Simplify: 4+85−3. Solution 4+85−3 Simplify the expression in the numerator. 125−3 Simplify the expression in the denominator. 122 Simplify the fraction. 6

Simplify: 4+611−2. 109 Simplify: 3+518−2. 12 Simplify: 4−2(3)22+2. Solution 4−2(3)22+2 Use the order of operations. Multiply in the numerator and use the exponent in the denominator. 4−64+2 Simplify the numerator and the denominator. −26 Simplify the fraction. -13

Simplify: 6−3(5)32+3. −34 Simplify: 4−4(6)33+3. −23 Simplify: (8−4)282−42. Solution (8−4)282−42 Use the order of operations (parentheses first, then exponents). (4)264−16 Simplify the numerator and denominator. 1648 Simplify the fraction. 13

Simplify: (11−7)2112−72. 29 Simplify: (6+2)262+22. 85 Simplify: 4(−3)+6(−2)−3(2)−2. Solution 4(−3)+6(−2)−3(2)−2 Multiply. −12+(−12)−6−2 Simplify. −24−8 Divide. 3

Simplify: 8(−2)+4(−3)−5(2)+3. 4 Simplify: 7(−1)+9(−3)−5(3)−2. 2 ACCESS ADDITIONAL ONLINE RESOURCES Division Involving Mixed Numbers Evaluate a Complex Fraction

Practice Makes PerfectMultiply and Divide Mixed Numbers In the following exercises, multiply and write the answer in simplified form. 438·710 249·67 4421 1522·335 2536·6310 358 423(−118) 225(−229) −163 −449·51316 −1720·21112 −6316 In the following exercises, divide, and write your answer in simplified form. 513÷4 1312÷9 32 −12÷3311 −7÷514 −43 638÷218 215÷1110 2 −935÷(−135) −1834÷(−334) 5 Translate Phrases to Expressions with Fractions In the following exercises, translate each English phrase into an algebraic expression. the quotient of 5u and 11 the quotient of 7v and 13 7v13 the quotient of p and q the quotient of a and b ab the quotient of r and the sum of s and 10 the quotient of A and the difference of 3 and B A3−B Simplify Complex Fractions In the following exercises, simplify the complex fraction. 2389 45815 32 −8211235 −9163340 −1522 −452 −9103 −310 258 5310 16 m3n2 r5s3 3r5s −x6−89 −38−y12 92y 245110 42316 28 79−245 38−634 −118 Simplify Expressions with a Fraction Bar In the following exercises, identify the equivalent fractions. Which of the following fractions are equivalent to 5−11?−5−11,−511,511,−511 Which of the following fractions are equivalent to −49? −4−9,−49,49,−49 −49,−49 Which of the following fractions are equivalent to −113?−113,113,−11−3,11−3 Which of the following fractions are equivalent to −136?136,13−6,−13−6,−136 13−6,−136 In the following exercises, simplify. 4+118 9+37 127 22+310 19−46 52 4824−15 464+4 234 −6+68+4 −6+317−8 −13 22−1419−13 15+918+12 45 5⋅8−10 3⋅4−24 −12 4⋅36⋅6 6⋅69⋅2 2 42−125 72+160 56 8⋅3+2⋅914+3 9⋅6−4⋅722+3 2625 15⋅5−522⋅10 12⋅9−323⋅18 116 5⋅6−3⋅44⋅5−2⋅3 8⋅9−7⋅65⋅6−9⋅2 52 52−323−5 62−424−6 −10 2+4(3)−3−22 7+3(5)−2−32 −2 7⋅4−2(8−5)9⋅3−3⋅5 9⋅7−3(12−8)8⋅7−6⋅6 5120 9(8−2)−3(15−7)6(7−1)−3(17−9) 8(9−2)−4(14−9)7(8−3)−3(16−9) 187

Everyday Math Baking A recipe for chocolate chip cookies calls for 214 cups of flour. Graciela wants to double the recipe. ⓐ How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number. ⓑ Measuring cups usually come in sets with cups for 18,14,13,12, and 1 cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe. Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains 223 cups of chocolate chips per pound. ⓐ How many cups of chocolate chips are in a half-pound of the fudge? ⓑ The owners of the booth make the fudge in 10-pound batches. How many chocolate chips do they need to make a 10-pound batch? Write your results as improper fractions and as a mixed numbers. ⓐ 43=113 cups ⓑ 803=2623 cups

Writing Exercises Explain how to find the reciprocal of a mixed number. Explain how to multiply mixed numbers. Answers will vary. Randy thinks that 312·514 is 1518. Explain what is wrong with Randy’s thinking. Explain why −12,−12, and 1−2 are equivalent. Answers will vary.

Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?