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Prealgebra

4.2 Multiply and Divide Fractions

Prealgebra4.2 Multiply and Divide Fractions
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify fractions
  • Multiply fractions
  • Find reciprocals
  • Divide fractions
Be Prepared 4.2

Before you get started, take this readiness quiz.

  1. Find the prime factorization of 48.48.
    If you missed this problem, review Example 2.48.
  2. Draw a model of the fraction 34.34.
    If you missed this problem, review Example 4.2.
  3. Find two fractions equivalent to 56.56.
    Answers may vary. Acceptable answers include 1012,1518,5060,1012,1518,5060, etc.
    If you missed this problem, review Example 4.14.

Simplify Fractions

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 1,1, 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.

Simplified Fraction

A fraction is considered simplified if there are no common factors in the numerator and denominator.

For example,

  • 2323 is simplified because there are no common factors of 22 and 3.3.
  • 10151015 is not simplified because 55 is a common factor of 1010 and 15.15.

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.

Equivalent Fractions Property

If a,b,ca,b,c are numbers where b0,c0,b0,c0, then

ab=a·cb·canda·cb·c=ab.ab=a·cb·canda·cb·c=ab.

Notice that cc is a common factor in the numerator and denominator. Anytime we have a common factor in the numerator and denominator, it can be removed.

How To

Simplify a fraction.

  1. Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
  2. Step 2. Simplify, using the equivalent fractions property, by removing common factors.
  3. Step 3. Multiply any remaining factors.

Example 4.19

Simplify: 1015.1015.

Try It 4.37

Simplify: 812812.

Try It 4.38

Simplify: 12161216.

To simplify a negative fraction, we use the same process as in Example 4.19. Remember to keep the negative sign.

Example 4.20

Simplify: 1824.1824.

Try It 4.39

Simplify: 2128.2128.

Try It 4.40

Simplify: 1624.1624.

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.

Example 4.21

Simplify: 5632.5632.

Try It 4.41

Simplify: 5442.5442.

Try It 4.42

Simplify: 8145.8145.

How To

Simplify a fraction.

  1. Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
  2. Step 2. Simplify, using the equivalent fractions property, by removing common factors.
  3. Step 3. Multiply any remaining factors

Sometimes 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.

Example 4.22

Simplify: 210385.210385.

Try It 4.43

Simplify: 69120.69120.

Try It 4.44

Simplify: 120192.120192.

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.

Example 4.23

Simplify: 5xy15x.5xy15x.

Try It 4.45

Simplify: 7x7y.7x7y.

Try It 4.46

Simplify: 9a9b.9a9b.

Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model 12·34.12·34. To multiply 1212 and 34,34, think 1212 of 34.34.

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 1414 tiles evenly into two parts, we exchange them for smaller tiles.

A rectangle is divided vertically into three equal pieces. Each piece is labeled as one fourth. There is a an arrow pointing to an identical rectangle divided vertically into six equal pieces. Each piece is labeled as one eighth. There are braces showing that three of these rectangles represent three eighths.

We see 6868 is equivalent to 34.34. Taking half of the six 1818 tiles gives us three 1818 tiles, which is 38.38.

Therefore,

12·34=3812·34=38

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Model Fraction Multiplication" will help you develop a better understanding of how to multiply fractions.

Example 4.24

Use a diagram to model 12·34.12·34.

Try It 4.47

Use a diagram to model: 12·35.12·35.

Try It 4.48

Use a diagram to model: 12·56.12·56.

Look at the result we got from the model in Example 4.24. We found that 12·34=38.12·34=38. Do you notice that we could have gotten the same answer by multiplying the numerators and multiplying the denominators?

12·3412·34
Multiply the numerators, and multiply the denominators. 12·3412·34
Simplify. 3838

This 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.

Fraction Multiplication

If a,b,c,andda,b,c,andd are numbers where b0andd0,b0andd0, then

ab·cd=acbdab·cd=acbd

Example 4.25

Multiply, and write the answer in simplified form: 34·15.34·15.

Try It 4.49

Multiply, and write the answer in simplified form: 13·25.13·25.

Try It 4.50

Multiply, and write the answer in simplified form: 35·78.35·78.

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.

Example 4.26

Multiply, and write the answer in simplified form: 58(23).58(23).

Try It 4.51

Multiply, and write the answer in simplified form: 47(58).47(58).

Try It 4.52

Multiply, and write the answer in simplified form: 712(89).712(89).

Example 4.27

Multiply, and write the answer in simplified form: 1415·2021.1415·2021.

Try It 4.53

Multiply, and write the answer in simplified form: 1028·815.1028·815.

Try It 4.54

Multiply, and write the answer in simplified form: 920·512.920·512.

When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a,a, can be written as a1.a1. So, 3=31,3=31, for example.

Example 4.28

Multiply, and write the answer in simplified form:

17·5617·56

125(−20x)125(−20x)

Try It 4.55

Multiply, and write the answer in simplified form:

  1. 18·7218·72
  2. 113(−9a)113(−9a)
Try It 4.56

Multiply, and write the answer in simplified form:

  1. 38·6438·64
  2. 16x·111216x·1112

Find Reciprocals

The fractions 2323 and 3232 are related to each other in a special way. So are 107107 and 710.710. 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 1.1.

23·32=1and107(710)=123·32=1and107(710)=1

Such pairs of numbers are called reciprocals.

Reciprocal

The reciprocal of the fraction abab is ba,ba, where a0a0 and b0,b0,

A number and its reciprocal have a product of 1.1.

ab·ba=1ab·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.

“a” over “b” multiplied by “b” over “a” equals positive one.

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 1.1. Is there any number rr so that 0·r=1?0·r=1? No. So, the number 00 does not have a reciprocal.

Example 4.29

Find the reciprocal of each number. Then check that the product of each number and its reciprocal is 1.1.

  1. 4949
  2. 1616
  3. 145145
  4. 77
Try It 4.57

Find the reciprocal:

  1. 5757
  2. 1818
  3. 114114
  4. 1414
Try It 4.58

Find the reciprocal:

  1. 3737
  2. 112112
  3. 149149
  4. 2121

In a previous chapter, we worked with opposites and absolute values. Table 4.1 compares opposites, absolute values, and reciprocals.

Opposite Absolute Value Reciprocal
has opposite sign is never negative has same sign, fraction inverts
Table 4.1

Example 4.30

Fill in the chart for each fraction in the left column:

Number Opposite Absolute Value Reciprocal
3838
1212
9595
−5−5
Try It 4.59

Fill in the chart for each number given:

Number Opposite Absolute Value Reciprocal
5858
1414
8383
−8−8
Try It 4.60

Fill in the chart for each number given:

Number Opposite Absolute Value Reciprocal
4747
1818
9494
−1−1

Divide Fractions

Why is 12÷3=4?12÷3=4? We previously modeled this with counters. How many groups of 33 counters can be made from a group of 1212 counters?

Four red ovals are shown. Inside each oval are three grey circles.

There are 44 groups of 33 counters. In other words, there are four 3s3s in 12.12. So, 12÷3=4.12÷3=4.

What about dividing fractions? Suppose we want to find the quotient: 12÷16.12÷16. We need to figure out how many 16s16s there are in 12.12. We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown in Figure 4.5. Notice, there are three 1616 tiles in 12,12, so 12÷16=3.12÷16=3.

A rectangle is shown, labeled as one half. Below it is an identical rectangle split into three equal pieces, each labeled as one sixth.
Figure 4.5

Manipulative Mathematics

Doing the Manipulative Mathematics activity "Model Fraction Division" will help you develop a better understanding of dividing fractions.

Example 4.31

Model: 14÷18.14÷18.

Try It 4.61

Model: 13÷16.13÷16.

Try It 4.62

Model: 12÷14.12÷14.

Example 4.32

Model: 2÷14.2÷14.

Try It 4.63

Model: 2÷132÷13

Try It 4.64

Model: 3÷123÷12

Let’s use money to model 2÷142÷14 in another way. We often read 1414 as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in Figure 4.6. So we can think of 2÷142÷14 as, “How many quarters are there in two dollars?” One dollar is 44 quarters, so 22 dollars would be 88 quarters. So again, 2÷14=8.2÷14=8.

A picture of a United States quarter is shown.
Figure 4.6 The U.S. coin called a quarter is worth one-fourth of a dollar.

Using fraction tiles, we showed that 12÷16=3.12÷16=3. Notice that 12·61=312·61=3 also. How are 1616 and 6161 related? They are reciprocals. This leads us to the procedure for fraction division.

Fraction Division

If a,b,c,andda,b,c,andd are numbers where b0,c0,andd0,b0,c0,andd0, then

ab÷cd=ab·dcab÷cd=ab·dc

To divide fractions, multiply the first fraction by the reciprocal of the second.

We need to say b0,c0andd0b0,c0andd0 to be sure we don’t divide by zero.

Example 4.33

Divide, and write the answer in simplified form: 25÷(37).25÷(37).

Try It 4.65

Divide, and write the answer in simplified form: 37÷(23).37÷(23).

Try It 4.66

Divide, and write the answer in simplified form: 23÷(75).23÷(75).

Example 4.34

Divide, and write the answer in simplified form: 23÷n5.23÷n5.

Try It 4.67

Divide, and write the answer in simplified form: 35÷p7.35÷p7.

Try It 4.68

Divide, and write the answer in simplified form: 58÷q3.58÷q3.

Example 4.35

Divide, and write the answer in simplified form: 34÷(78).34÷(78).

Try It 4.69

Divide, and write the answer in simplified form: 23÷(56).23÷(56).

Try It 4.70

Divide, and write the answer in simplified form: 56÷(23).56÷(23).

Example 4.36

Divide, and write the answer in simplified form: 718÷1427.718÷1427.

Try It 4.71

Divide, and write the answer in simplified form: 727÷3536.727÷3536.

Try It 4.72

Divide, and write the answer in simplified form: 514÷1528.514÷1528.

Section 4.2 Exercises

Practice Makes Perfect

Simplify Fractions

In the following exercises, simplify each fraction. Do not convert any improper fractions to mixed numbers.

77.

721721

78.

824824

79.

15201520

80.

12181218

81.

40884088

82.

63996399

83.

1086310863

84.

1044810448

85.

120252120252

86.

182294182294

87.

168192168192

88.

140224140224

89.

11x11y11x11y

90.

15a15b15a15b

91.

3x12y3x12y

92.

4x32y4x32y

93.

14x221y14x221y

94.

24a32b224a32b2

Multiply Fractions

In the following exercises, use a diagram to model.

95.

12·2312·23

96.

12·5812·58

97.

13·5613·56

98.

13·2513·25

In the following exercises, multiply, and write the answer in simplified form.

99.

25·1325·13

100.

12·3812·38

101.

34·91034·910

102.

45·2745·27

103.

23(38)23(38)

104.

34(49)34(49)

105.

59·31059·310

106.

38·41538·415

107.

712(821)712(821)

108.

512(815)512(815)

109.

(1415)(920)(1415)(920)

110.

(910)(2533)(910)(2533)

111.

(6384)(4490)(6384)(4490)

112.

(3360)(4088)(3360)(4088)

113.

4·5114·511

114.

5·835·83

115.

37·21n37·21n

116.

56·30m56·30m

117.

−28p(14)−28p(14)

118.

−51q(13)−51q(13)

119.

−8(174)−8(174)

120.

145(−15)145(−15)

121.

−1(38)−1(38)

122.

(−1)(67)(−1)(67)

123.

(23)3(23)3

124.

(45)2(45)2

125.

(65)4(65)4

126.

(47)4(47)4

Find Reciprocals

In the following exercises, find the reciprocal.

127.

3434

128.

2323

129.

517517

130.

619619

131.

118118

132.

−13−13

133.

−19−19

134.

−1−1

135.

11

136.

Fill in the chart.

Opposite Absolute Value Reciprocal
711711
4545
107107
−8−8
137.

Fill in the chart.

Opposite Absolute Value Reciprocal
313313
914914
157157
−9−9

Divide Fractions

In the following exercises, model each fraction division.

138.

12÷1412÷14

139.

12÷1812÷18

140.

2÷152÷15

141.

3÷143÷14

In the following exercises, divide, and write the answer in simplified form.

142.

12÷1412÷14

143.

12÷1812÷18

144.

34÷2334÷23

145.

45÷3445÷34

146.

45÷4745÷47

147.

34÷3534÷35

148.

79÷(79)79÷(79)

149.

56÷(56)56÷(56)

150.

34÷x1134÷x11

151.

25÷y925÷y9

152.

58÷a1058÷a10

153.

56÷c1556÷c15

154.

518÷(1524)518÷(1524)

155.

718÷(1427)718÷(1427)

156.

7p12÷21p87p12÷21p8

157.

5q12÷15q85q12÷15q8

158.

8u15÷12v258u15÷12v25

159.

12r25÷18s3512r25÷18s35

160.

−5÷12−5÷12

161.

−3÷14−3÷14

162.

34÷(−12)34÷(−12)

163.

25÷(−10)25÷(−10)

164.

−18÷(92)−18÷(92)

165.

−15÷(53)−15÷(53)

166.

12÷(34)÷7812÷(34)÷78

167.

112÷78·211112÷78·211

Everyday Math

168.

Baking A recipe for chocolate chip cookies calls for 3434 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,and118,14,13,12,and1 cup. Draw a diagram to show two different ways that Imelda could measure the brown sugar needed to double the recipe.

169.

Baking Nina is making 44 pans of fudge to serve after a music recital. For each pan, she needs 2323 cup of condensed milk.

  1. How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number.
  2. Measuring cups usually come in sets of 18,14,13,12,and118,14,13,12,and1 cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.
170.

Portions Don purchased a bulk package of candy that weighs 55 pounds. He wants to sell the candy in little bags that hold 1414 pound. How many little bags of candy can he fill from the bulk package?

171.

Portions Kristen has 3434 yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s 66 dolls. How long will each doll’s hair ribbon be?

Writing Exercises

172.

Explain how you find the reciprocal of a fraction.

173.

Explain how you find the reciprocal of a negative fraction.

174.

Rafael wanted to order half a medium pizza at a restaurant. The waiter told him that a medium pizza could be cut into 66 or 88 slices. Would he prefer 33 out of 66 slices or 44 out of 88 slices? Rafael replied that since he wasn’t very hungry, he would prefer 33 out of 66 slices. Explain what is wrong with Rafael’s reasoning.

175.

Give an example from everyday life that demonstrates how 12·23is13.12·23is13.

Self Check

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?

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