m81286 Multiply and Divide Fractions By the end of this section, you will be able to: Simplify fractions Multiply fractions Find reciprocals Divide fractions c4bc8665-2053-46a2-b4c6-da5277d1f63e Before you get started, take this readiness quiz. Find the prime factorization of 48. If you missed this problem, review . 2⋅2⋅2⋅2⋅3 Draw a model of the fraction 34. If you missed this problem, review . Find two fractions equivalent to 56. If you missed this problem, review . Answers may vary. Acceptable answers include 1012,1518,5060, etc.

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.A fraction is considered simplified if there are no common factors, other than 1, in the numerator and denominator. If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors. A fraction is considered simplified if there are no common factors in the numerator and denominator. For example, 23 is simplified because there are no common factors of 2 and 3. 1015 is not simplified because 5 is a common factor of 10 and 15. The process of simplifying a fraction is often called reducing the fraction. In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together. If a,b,c are numbers where b≠0,c≠0, then ab=a·cb·canda·cb·c=ab. Notice that c is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers. Simplify, using the equivalent fractions property, by removing common factors. Multiply any remaining factors. Simplify: 1015. To simplify the fraction, we look for any common factors in the numerator and the denominator. Notice that 5 is a factor of both 10 and 15. 1015 Factor the numerator and denominator. Remove the common factors. Simplify. 23 Simplify: 812. 23 Simplify: 1216. 34 To simplify a negative fraction, we use the same process as in . Remember to keep the negative sign. Simplify: −1824. We notice that 18 and 24 both have factors of 6. −1824 Rewrite the numerator and denominator showing the common factor. Remove common factors. Simplify. −34 Simplify: −2128. −34 Simplify: −1624. −23 After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator. When we simplify an improper fraction, there is no need to change it to a mixed number. Simplify: −5632. −5632 Rewrite the numerator and denominator, showing the common factors, 8. Remove common factors. Simplify. −74 Simplify: −5442. −97 Simplify: −8145. −95 Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers. Simplify, using the equivalent fractions property, by removing common factors. Multiply any remaining factorsSometimes it may not be easy to find common factors of the numerator and denominator. A good idea, then, is to factor the numerator and the denominator into prime numbers. (You may want to use the factor tree method to identify the prime factors.) Then divide out the common factors using the Equivalent Fractions Property. Simplify: 210385. Use factor trees to factor the numerator and denominator. 210385 Rewrite the numerator and denominator as the product of the primes. 210385=2⋅3⋅5⋅75⋅7⋅11 Remove the common factors. Simplify. 2⋅311 Multiply any remaining factors. 611 Simplify: 69120. 2340 Simplify: 120192. 58 We can also simplify fractions containing variables. If a variable is a common factor in the numerator and denominator, we remove it just as we do with an integer factor. Simplify: 5xy15x. 5xy15x Rewrite numerator and denominator showing common factors. 5·x·y3·5·x Remove common factors. 5·x·y3·5·x Simplify. y3 Simplify: 7x7y. xy Simplify: 9a9b. ab

A model may help you understand multiplication of fractions. We will use fraction tiles to model 12·34. To multiply 12 and 34, think 12 of 34. Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three 14 tiles evenly into two parts, we exchange them for smaller tiles. We see 68 is equivalent to 34. Taking half of the six 18 tiles gives us three 18 tiles, which is 38. Therefore, 12·34=38 Doing the Manipulative Mathematics activity "Model Fraction Multiplication" will help you develop a better understanding of how to multiply fractions. Use a diagram to model 12·34. First shade in 34 of the rectangle. We will take 12 of this 34, so we heavily shade 12 of the shaded region. Notice that 3 out of the 8 pieces are heavily shaded. This means that 38 of the rectangle is heavily shaded.Therefore, 12 of 34 is 38, or 12·34=38. Use a diagram to model: 12·35. 310 Use a diagram to model: 12·56. 512 Look at the result we got from the model in . We found that 12·34=38. Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators? 12·34 Multiply the numerators, and multiply the denominators. 12·34 Simplify. 38This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form. If a,b,c, and d are numbers where b≠0 and d≠0, then ab·cd=acbd Multiply, and write the answer in simplified form: 34·15. 34·15 Multiply the numerators; multiply the denominators. 3·14·5 Simplify. 320There are no common factors, so the fraction is simplified. Multiply, and write the answer in simplified form: 13·25. 215 Multiply, and write the answer in simplified form: 35·78. 2140 When multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step. In Example 4.26 we will multiply two negatives, so the product will be positive. Multiply, and write the answer in simplified form: −58(−23). −58(−23) The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators. 5⋅28⋅3 Simplify. 1024 Look for common factors in the numerator and denominator. Rewrite showing common factors. Remove common factors. 512Another way to find this product involves removing common factors earlier. −58(−23) Determine the sign of the product. Multiply. 5⋅28⋅3 Show common factors and then remove them. Multiply remaining factors. 512We get the same result. Multiply, and write the answer in simplified form: −47(−58). 514 Multiply, and write the answer in simplified form: −712(−89). 1427 Multiply, and write the answer in simplified form: −1415·2021. −1415·2021 Determine the sign of the product; multiply. −1415·2021 Are there any common factors in the numerator and the denominator?We know that 7 is a factor of 14 and 21, and 5 is a factor of 20 and 15. Rewrite showing common factors. Remove the common factors. −2·43·3 Multiply the remaining factors. −89 Multiply, and write the answer in simplified form: −1028·815. −421 Multiply, and write the answer in simplified form: −920·512. −316 When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as a1. So, 3=31, for example. Multiply, and write the answer in simplified form: ⓐ 17·56ⓑ 125(−20x) ⓐ 17·56 Write 56 as a fraction. 17·561 Determine the sign of the product; multiply. 567 Simplify. 8 ⓑ 125(−20x) Write −20x as a fraction. 125(−20x1) Determine the sign of the product; multiply. −12·20·x5·1 Show common factors and then remove them. Multiply remaining factors; simplify. −48x Multiply, and write the answer in simplified form: ⓐ 18·72ⓑ 113(−9a) ⓐ 9ⓑ −33a Multiply, and write the answer in simplified form: ⓐ 38·64 ⓑ 16x·1112 ⓐ 24 ⓑ 44x3

The fractions 23 and 32 are related to each other in a special way. So are −107 and −710. Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be 1. 23·32=1and−107(−710)=1Such pairs of numbers are called reciprocals. The reciprocal of the fraction ab is ba, where a≠0 and b≠0,A number and its reciprocal have a product of 1. ab·ba=1 To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator.To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign. To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to 1. Is there any number r so that 0·r=1? No. So, the number 0 does not have a reciprocal. Find the reciprocal of each number. Then check that the product of each number and its reciprocal is 1. ⓐ 49 ⓑ −16 ⓒ −145 ⓓ 7 To find the reciprocals, we keep the sign and invert the fractions. ⓐ Find the reciprocal of 49. The reciprocal of 49 is 94. Check: Multiply the number and its reciprocal. 49⋅94 Multiply numerators and denominators. 3636 Simplify. 1✓ ⓑ Find the reciprocal of -16. -61 Simplify. -6 Check: -16⋅(-6) 1✓ ⓒ Find the reciprocal of -145. -514 Check: -145⋅(-514) 7070 1✓ ⓓ Find the reciprocal of 7. Write 7 as a fraction. 71 Write the reciprocal of 71. 17 Check: 7⋅(17) 1✓ Find the reciprocal: ⓐ 57 ⓑ −18 ⓒ −114 ⓓ 14 ⓐ 75 ⓑ −8 ⓒ −411 ⓓ 114 Find the reciprocal: ⓐ 37 ⓑ −112 ⓒ −149 ⓓ 21 ⓐ 73 ⓑ −12 ⓒ −914 ⓓ 121 In a previous chapter, we worked with opposites and absolute values. compares opposites, absolute values, and reciprocals. Opposite Absolute Value Reciprocal has opposite sign is never negative has same sign, fraction inverts Fill in the chart for each fraction in the left column: Number Opposite Absolute Value Reciprocal −38 12 95 −5 To find the opposite, change the sign. To find the absolute value, leave the positive numbers the same, but take the opposite of the negative numbers. To find the reciprocal, keep the sign the same and invert the fraction. Number Opposite Absolute Value Reciprocal −38 38 38 −83 12 −12 12 2 95 −95 95 59 −5 5 5 −15 Fill in the chart for each number given: Number Opposite Absolute Value Reciprocal −58 14 83 −8 Fill in the chart for each number given: Number Opposite Absolute Value Reciprocal −47 18 94 −1

Why is 12÷3=4? We previously modeled this with counters. How many groups of 3 counters can be made from a group of 12 counters? There are 4 groups of 3 counters. In other words, there are four 3s in 12. So, 12÷3=4. What about dividing fractions? Suppose we want to find the quotient: 12÷16. We need to figure out how many 16s there are in 12. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in . Notice, there are three 16 tiles in 12, so 12÷16=3.

Doing the Manipulative Mathematics activity "Model Fraction Division" will help you develop a better understanding of dividing fractions. Model: 14÷18. We want to determine how many 18s are in 14. Start with one 14 tile. Line up 18 tiles underneath the 14 tile. There are two 18s in 14. So, 14÷18=2. Model: 13÷16. 2 Model: 12÷14. 2 Model: 2÷14. We are trying to determine how many 14s there are in 2. We can model this as shown. Because there are eight 14s in 2,2÷14=8. Model: 2÷13 6 Model: 3÷12 6 Let’s use money to model 2÷14 in another way. We often read 14 as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in . So we can think of 2÷14 as, “How many quarters are there in two dollars?” One dollar is 4 quarters, so 2 dollars would be 8 quarters. So again, 2÷14=8.

The U.S. coin called a quarter is worth one-fourth of a dollar.

Using fraction tiles, we showed that 12÷16=3. Notice that 12·61=3 also. How are 16 and 61 related? They are reciprocals. This leads us to the procedure for fraction division. If a,b,c, and d are numbers where b≠0,c≠0, and d≠0, then ab÷cd=ab·dc To divide fractions, multiply the first fraction by the reciprocal of the second. We need to say b≠0,c≠0 and d≠0 to be sure we don’t divide by zero. Divide, and write the answer in simplified form: 25÷(−37). 25÷(−37) Multiply the first fraction by the reciprocal of the second. 25(−73) Multiply. The product is negative. −1415 Divide, and write the answer in simplified form: 37÷(−23). −914 Divide, and write the answer in simplified form: 23÷(−75). −1021 Divide, and write the answer in simplified form: 23÷n5. 23÷n5 Multiply the first fraction by the reciprocal of the second. 23·5n Multiply. 103n Divide, and write the answer in simplified form: 35÷p7. 215p Divide, and write the answer in simplified form: 58÷q3. 158q Divide, and write the answer in simplified form: −34÷(−78). −34÷(−78) Multiply the first fraction by the reciprocal of the second. −34·(−87) Multiply. Remember to determine the sign first. 3·84·7 Rewrite to show common factors. 3·4·24·7 Remove common factors and simplify. 67 Divide, and write the answer in simplified form: −23÷(−56). 45 Divide, and write the answer in simplified form: −56÷(−23). 54 Divide, and write the answer in simplified form: 718÷1427. 718÷1427 Multiply the first fraction by the reciprocal of the second. 718·2714 Multiply. 7·2718·14 Rewrite showing common factors. Remove common factors. 32·2 Simplify. 34 Divide, and write the answer in simplified form: 727÷3536. 415 Divide, and write the answer in simplified form: 514÷1528. 23 Simplifying Fractions Multiplying Fractions (Positive Only) Multiplying Signed Fractions Dividing Fractions (Positive Only) Dividing Signed Fractions

Equivalent Fractions PropertyIf a, b, c are numbers where b≠0, c≠0, then ab=a⋅cb⋅c and a⋅cb⋅c=ab. Simplify a fraction.Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers. Simplify, using the equivalent fractions property, by removing common factors. Multiply any remaining factors. Fraction MultiplicationIf a, b, c, and d are numbers where b≠0and d≠0, then ab⋅cd=acbd. ReciprocalA number and its reciprocal have a product of 1. ab⋅ba=1 Simplify Fractions In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers. 721 13 824 1520 34 1218 −4088 −511 −6399 −10863 −127 −10448 120252 1021 182294 −168192 −78 −140224 11x11y xy 15a15b −3x12y −x4y −4x32y 14x221y 2x23y 24a32b2 Multiply Fractions In the following exercises, use a diagram to model. 12·23 13 12·58 13·56 518 13·25 In the following exercises, multiply, and write the answer in simplified form. 25·13 215 12·38 34·910 2740 45·27 −23(−38) 14 −34(−49) −59·310 −16 −38·415 712(−821) −29 512(−815) (−1415)(920) −2150 (−910)(2533) (−6384)(−4490) 1130 (−3360)(−4088) 4·511 2011 5·83 37·21n 9n 56·30m −28p(−14) 7p −51q(−13) −8(174) −34 145(−15) −1(−38) 38 (−1)(−67) (23)3 827 (45)2 (65)4 1296625 (47)4 Find Reciprocals In the following exercises, find the reciprocal. 34 43 23 −517 −175 −619 118 811 −13 −19 −119 −1 1 1 Fill in the chart. Opposite Absolute Value Reciprocal −711 45 107 −8 Fill in the chart. Opposite Absolute Value Reciprocal −313 914 157 −9 Divide Fractions In the following exercises, model each fraction division. 12÷14 12÷18 4 2÷15 3÷14 12 In the following exercises, divide, and write the answer in simplified form. 12÷14 12÷18 4 34÷23 45÷34 1615 −45÷47 −34÷35 −54 −79÷(−79) −56÷(−56) 1 34÷x11 25÷y9 185y 58÷a10 56÷c15 252c 518÷(−1524) 718÷(−1427) −34 7p12÷21p8 5q12÷15q8 29 8u15÷12v25 12r25÷18s35 14r15s −5÷12 −3÷14 −12 34÷(−12) 25÷(−10) −125 −18÷(−92) −15÷(−53) 9 12÷(−34)÷78 112÷78·211 87

Baking A recipe for chocolate chip cookies calls for 34 cup brown sugar. Imelda wants to double the recipe. ⓐ How much brown sugar will Imelda need? Show your calculation. Write your result as an improper fraction and as a mixed number.ⓑ Measuring cups usually come in sets of 18,14,13,12, and 1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe. Baking Nina is making 4 pans of fudge to serve after a music recital. For each pan, she needs 23 cup of condensed milk. ⓐ How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number. ⓑ Measuring cups usually come in sets of 18,14,13,12, and 1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs. ⓐ 4·23 cups=83 cups=223 cups ⓑ Answers will vary. Portions Don purchased a bulk package of candy that weighs 5 pounds. He wants to sell the candy in little bags that hold 14 pound. How many little bags of candy can he fill from the bulk package? Portions Kristen has 34 yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s 6 dolls. How long will each doll’s hair ribbon be? 18 yard

Explain how you find the reciprocal of a fraction. Explain how you find the reciprocal of a negative fraction. Answers will vary. Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 6 or 8 slices. Would he prefer 3 out of 6 slices or 4 out of 8 slices? Rafael replied that since he wasn’t very hungry, he would prefer 3 out of 6 slices. Explain what is wrong with Rafael’s reasoning. Give an example from everyday life that demonstrates how 12·23 is 13. Answers will vary.

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

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

reciprocalThe reciprocal of the fraction ab is ba where a≠0 and b≠0. simplified fractionA fraction is considered simplified if there are no common factors in the numerator and denominator.