m81298 Decimals and Fractions By the end of this section, you will be able to: Convert fractions to decimals Order decimals and fractions Simplify expressions using the order of operations Find the circumference and area of circles 69f66012-2d0d-4978-87c5-e6a4b16a8ec5 Before you get started, take this readiness quiz. Divide: 0.24÷8. If you missed this problem, review . 0.03 Order 0.64__0.6 using < or >. If you missed this problem, review . > Order −0.2__−0.1 using < or >. If you missed this problem, review . <

In Decimals, we learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar indicates division. So 45 can be written 4÷5 or 54. This means that we can convert a fraction to a decimal by treating it as a division problem. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. Write the fraction 34 as a decimal. A fraction bar means division, so we can write the fraction 34 using division. Divide. So the fraction 34 is equal to 0.75. Write the fraction as a decimal: 14. 0.25 Write the fraction as a decimal: 38. 0.375 Write the fraction −72 as a decimal. The value of this fraction is negative. After dividing, the value of the decimal will be negative. We do the division ignoring the sign, and then write the negative sign in the answer. −72 Divide 7 by 2. So, −72=−3.5. Write the fraction as a decimal: −94. −2.25 Write the fraction as a decimal: −112. −5.5

So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let’s see what happens when we convert the fraction 43 to a decimal. First, notice that 43 is an improper fraction. Its value is greater than 1. The equivalent decimal will also be greater than 1. We divide 4 by 3. No matter how many more zeros we write, there will always be a remainder of 1, and the threes in the quotient will go on forever. The number 1.333… is called a repeating decimal. Remember that the “…” means that the pattern repeats. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. How do you know how many ‘repeats’ to write? Instead of writing 1.333… we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal 1.333… is written 1.3–. The line above the 3 tells you that the 3 repeats endlessly. So 1.333…=1.3–For other decimals, two or more digits might repeat. shows some more examples of repeating decimals. 1.333…=1.3– 3 is the repeating digit 4.1666…=4.16– 6 is the repeating digit 4.161616…=4.16— 16 is the repeating block 0.271271271…=0.271––– 271 is the repeating block Write 4322 as a decimal. Divide 43 by 22. Notice that the differences of 120 and 100 repeat, so there is a repeat in the digits of the quotient; 54 will repeat endlessly. The first decimal place in the quotient, 9, is not part of the pattern. So, 4322=1.954— Write as a decimal: 2711. 2.45— Write as a decimal: 5122. 2.318— It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction. Simplify: 78+6.4. 78+6.4 Change 78 to a decimal. 0.875+6.4 Add. 7.275 Simplify: 38+4.9. 5.275 Simplify: 5.7+1320. 6.35

In Decimals, we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals. Order 38__0.4 using < or >. 38__0.4 Convert 38 to a decimal. 0.375__0.4 Compare 0.375 to 0.4 0.375<0.4 Rewrite with the original fraction. 38<0.4 Order each of the following pairs of numbers, using < or >. 1720__0.82 > Order each of the following pairs of numbers, using < or >. 34__0.785 < When ordering negative numbers, remember that larger numbers are to the right on the number line and any positive number is greater than any negative number. Order −0.5___−34 using < or >. −0.5___−34 Convert −34 to a decimal. −0.5___−0.75 Compare −0.5 to −0.75. −0.5>−0.75 Rewrite the inequality with the original fraction. −0.5>−34 Order each of the following pairs of numbers, using < or >: −58__−0.58 < Order each of the following pairs of numbers, using < or >: −0.53__−1120 > Write the numbers 1320,0.61,1116 in order from smallest to largest. 1320,0.61,1116 Convert the fractions to decimals. 0.65,0.61,0.6875 Write the smallest decimal number first. 0.61,____,_____ Write the next larger decimal number in the middle place. 0.61,0.65,_____ Write the last decimal number (the larger) in the third place. 0.61,0.65,0.6875 Rewrite the list with the original fractions. 0.61,1320,1116 Write each set of numbers in order from smallest to largest: 78,45,0.82. 45,0.82,78 Write each set of numbers in order from smallest to largest: 0.835,1316,34. 34,1316,0.835

The order of operations introduced in Use the Language of Algebra also applies to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for? Simplify the expressions: ⓐ 7(18.3−21.7) ⓑ 23(8.3−3.8) ⓐ 7(18.3−21.7) Simplify inside parentheses. 7(−3.4) Multiply. −23.8 ⓑ 23(8.3−3.8) Simplify inside parentheses. 23(4.5) Write 4.5 as a fraction. 23(4.51) Multiply. 93 Simplify. 3 Simplify: ⓐ 8(14.6−37.5) ⓑ 35(9.6−2.1). ⓐ −183.2 ⓑ 4.5 Simplify: ⓐ 25(25.69−56.74) ⓑ 27(11.9−4.2). ⓐ −776.25 ⓑ 2.2 Simplify each expression: ⓐ 6÷0.6+(0.2)4−(0.1)2 ⓑ (110)2+(3.5)(0.9) ⓐ 6÷0.6+(0.2)4−(0.1)2 Simplify exponents. 6÷0.6+(0.2)4−0.01 Divide. 10+(0.2)4−0.01 Multiply. 10+0.8−0.01 Add. 10.8−0.01 Subtract. 10.79 ⓑ (110)2+(3.5)(0.9) Simplify exponents. 1100+(3.5)(0.9) Multiply. 1100+3.15 Convert 1100 to a decimal. 0.01+3.15 Add. 3.16 Simplify: 9÷0.9+(0.4)3−(0.2)2. 11.16 Simplify: (12)2+(0.3)(4.2). 1.51

The properties of circles have been studied for over 2,000 years. All circles have exactly the same shape, but their sizes are affected by the length of the radius, a line segment from the center to any point on the circle. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius. See .The size of a circle can be measured in two ways. The distance around a circle is called its circumference.

Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter π (pronounced pie). However, the exact value of π cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.)Doing the Manipulative Mathematics activity Pi Lab will help you develop a better understanding of pi.If we want the exact circumference or area of a circle, we leave the symbol π in the answer. We can get an approximate answer by substituting 3.14 as the value of π. We use the symbol ≈ to show that the result is approximate, not exact. ris the length of the radius.dis the length of the diameter. The circumference is2πr.C=2πrThe area isπr2.A=πr2 Since the diameter is twice the radius, another way to find the circumference is to use the formula C=πd. Suppose we want to find the exact area of a circle of radius 10 inches. To calculate the area, we would evaluate the formula for the area when r=10 inches and leave the answer in terms of π. A=πr2A=π(102)A=π·100 We write π after the 100. So the exact value of the area is A=100π square inches. To approximate the area, we would substitute π≈3.14. A = 100 π ≈ 100 · 3.14 ≈ 314 square inches Remember to use square units, such as square inches, when you calculate the area. A circle has radius 10 centimeters. Approximate its ⓐ circumference and ⓑ area. ⓐ Find the circumference when r=10. Write the formula for circumference. C=2πr Substitute 3.14 for π and 10 for ,r. C≈2(3.14)(10) Multiply. C≈62.8centimeters ⓑ Find the area when r=10. Write the formula for area. A=πr2 Substitute 3.14 for π and 10 for r. A≈(3.14)(10)2 Multiply. A≈314square centimeters A circle has radius 50 inches. Approximate its ⓐ circumference and ⓑ area. ⓐ 314 in. ⓑ 7850 sq. in. A circle has radius 100 feet. Approximate its ⓐ circumference and ⓑ area. ⓐ 628 ft. ⓑ 31,400 sq. ft. A circle has radius 42.5 centimeters. Approximate its ⓐ circumference and ⓑ area. ⓐ Find the circumference when r=42.5. Write the formula for circumference. C=2πr Substitute 3.14 for π and 42.5 for r C≈2(3.14)(42.5) Multiply. C≈266.9centimeters ⓑ Find the area when r=42.5. Write the formula for area. A=πr2 Substitute 3.14 for π and 42.5 for r. A≈(3.14)(42.5)2 Multiply. A≈5671.625square centimeters A circle has radius 51.8 centimeters. Approximate its ⓐ circumference and ⓑ area. ⓐ 325.304 cm ⓑ 8425.3736 sq. cm A circle has radius 26.4 meters. Approximate its ⓐ circumference and ⓑ area. ⓐ 165.792 m ⓑ 2188.4544 sq. m

Convert the fraction 227 to a decimal. If you use your calculator, the decimal number will fill up the display and show 3.14285714. But if we round that number to two decimal places, we get 3.14, the decimal approximation of π. When we have a circle with radius given as a fraction, we can substitute 227 for π instead of 3.14. And, since 227 is also an approximation of π, we will use the ≈ symbol to show we have an approximate value. A circle has radius 1415 meter. Approximate its ⓐ circumference and ⓑ area. ⓐ Find the circumference when r=1415. Write the formula for circumference. C=2πr Substitute 227 for π and 1415 for r. C≈2(227)(1415) Multiply. C≈8815meters ⓑ Find the area when r=1415. Write the formula for area. A=πr2 Substitute 227 for π and 1415 for r. A≈(227)(1415)2 Multiply. A≈616225square meters A circle has radius 521 meters. Approximate its ⓐ circumference and ⓑ area. ⓐ 220147mⓑ 5503087sq. m A circle has radius 1033 inches. Approximate its ⓐ circumference and ⓑ area. ⓐ 4021in.ⓑ 200693sq.in. Converting a Fraction to a Decimal - Part 2 Convert a Fraction to a Decimal (repeating) Compare Fractions and Decimals using Inequality Symbols Determine the Area of a Circle Determine the Circumference of a Circle

Convert a Fraction to a Decimal To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction. Properties of Circles r is the length of the radius d is the length of the diameter The circumference is 2πr. C=2πr The area is πr2. A=πr2

Convert Fractions to Decimals In the following exercises, convert each fraction to a decimal. 25 0.4 45 −38 −0.375 −58 1720 0.85 1320 114 2.75 174 −31025 −12.4 −28425 59 0.5– 29 1511 1.36— 1811 15111 0.135— 25111 In the following exercises, simplify the expression. 12+6.5 7 14+10.75 2.4+58 3.025 3.9+920 9.73+1720 10.58 6.29+2140 Order Decimals and Fractions In the following exercises, order each pair of numbers, using < or >. 18___0.8 < 14___0.4 25___0.25 > 35___0.35 0.725___34 < 0.92___78 0.66___23 < 0.83___56 −0.75___−45 > −0.44___−920 −34___−0.925 > −23___−0.632 In the following exercises, write each set of numbers in order from least to greatest. 35,916,0.55 0.55,916,35 38,720,0.36 0.702,1320,58 58,1320,0.702 0.15,316,15 −0.3,−13,−720 −720,−13,−0.3 −0.2,−320,−16 −34,−79,−0.7 −79,−34,−0.7 −89,−45,−0.9 Simplify Expressions Using the Order of Operations In the following exercises, simplify. 10(25.1−43.8) −187 30(18.1−32.5) 62(9.75−4.99) 295.12 42(8.45−5.97) 34(12.4−4.2) 6.15 45(8.6+3.9) 512(30.58+17.9) 20.2 916(21.96−9.8) 10÷0.1+(1.8)4−(0.3)2 107.11 5÷0.5+(3.9)6−(0.7)2 (37.1+52.7)÷(12.5÷62.5) 449 (11.4+16.2)÷(18÷60) (15)2+(1.4)(6.5) 9.14 (12)2+(2.1)(8.3) −910·815+0.25 −0.23 −38·1415+0.72 Mixed Practice In the following exercises, simplify. Give the answer as a decimal. 314−6.5 −3.25 525−8.75 10.86÷23 16.29 5.79÷34 78(103.48)+112(361) 632.045 516(117.6)+213(699) 3.6(98−2.72) −5.742 5.1(125−3.91) Find the Circumference and Area of Circles In the following exercises, approximate the ⓐ circumference and ⓑ area of each circle. If measurements are given in fractions, leave answers in fraction form. radius=5 in. ⓐ 31.4 in ⓑ 78.5 sq.in. radius=20 in. radius=9 ft. ⓐ 56.52.ft. ⓑ 254.34 sq.ft. radius=4 ft. radius=46 cm ⓐ 288.88 cm ⓑ 6644.24 sq.cm radius=38 cm radius=18.6 m ⓐ 116.808 m ⓑ 1086.3144 sq.m radius=57.3 m radius=710mile ⓐ 225mileⓑ 7750sq.mile radius=711mile radius=38yard ⓐ 3314yardⓑ 99224sq.yard radius=512yard diameter=56m ⓐ 5521mⓑ 275504sq.m diameter=34m

Kelly wants to buy a pair of boots that are on sale for 23 of the original price. The original price of the boots is $84.99. What is the sale price of the shoes? $56.66 An architect is planning to put a circular mosaic in the entry of a new building. The mosaic will be in the shape of a circle with radius of 6 feet. How many square feet of tile will be needed for the mosaic? (Round your answer up to the next whole number.)

Is it easier for you to convert a decimal to a fraction or a fraction to a decimal? Explain. Answers will vary. Describe a situation in your life in which you might need to find the area or circumference of a circle.

ⓐ 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?

circumference of a circle The distance around a circle is called its circumference. diameter of a circle A diameter of a circle is a line segment that passes through a circle’s center connecting two points on the circle. radius of a circle A radius of a circle is a line segment from the center to any point on the circle. repeating decimal A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly.