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4.3: Multiply and Divide Fractions

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Learning Objectives

By the end of this section, you will be able to:

  • Simplify fractions
  • Multiply fractions
  • Find reciprocals
  • Divide fractions

Be Prepared 4.3

Before you get started, take this readiness quiz.

Find the prime factorization of 48.
If you missed this problem, review Example 2.48.

Be Prepared 4.4

Draw a model of the fraction 34.
If you missed this problem, review Example 4.2.

Be Prepared 4.5

Find two fractions equivalent to 56.
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, 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,

  • 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.

Equivalent Fractions Property

If a,b,c are numbers where b0,c0, 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.

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.

Answer

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

Try It 4.37

Simplify: 812.

Try It 4.38

Simplify: 1216.

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.

Answer

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

Try It 4.39

Simplify: 2128.

Try It 4.40

Simplify: 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.

Answer

5632
Rewrite the numerator and denominator, showing the common factors, 8. .
Remove common factors. .
Simplify. 74

Try It 4.41

Simplify: 5442.

Try It 4.42

Simplify: 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.

Answer

Use factor trees to factor the numerator and denominator. 210385
.
Rewrite the numerator and denominator as the product of the primes. 210385=23575711
Remove the common factors. .
Simplify. 2311
Multiply any remaining factors. 611

Try It 4.43

Simplify: 69120.

Try It 4.44

Simplify: 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.

Answer

5xy15x
Rewrite numerator and denominator showing common factors. 5·x·y3·5·x
Remove common factors. 5·x·y3·5·x
Simplify. y3

Try It 4.45

Simplify: 7x7y.

Try It 4.46

Simplify: 9a9b.

Multiply Fractions

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.

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 68 is equivalent to 34. Taking half of the six 18 tiles gives us three 18 tiles, which is 38.

Therefore,

12·34=38

Manipulative Mathematics

Example 4.24

Use a diagram to model 12·34.

Answer

First shade in 34 of the rectangle.

A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded.

We will take 12 of this 34, so we heavily shade 12 of the shaded region.

A rectangle is shown, divided vertically into four equal pieces. Three of the pieces are shaded. The rectangle is divided by a horizontal line, creating eight equal pieces. Three of the eight pieces are darkly shaded.

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.

Try It 4.47

Use a diagram to model: 12·35.

Try It 4.48

Use a diagram to model: 12·56.

Look at the result we got from the model in Example 4.24. 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. 38

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, and d are numbers where b0 and d0, then

ab·cd=acbd

Example 4.25

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

Answer

34·15
Multiply the numerators; multiply the denominators. 3·14·5
Simplify. 320

There are no common factors, so the fraction is simplified.

Try It 4.49

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

Try It 4.50

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

Answer

58(23)
The signs are the same, so the product is positive. Multiply the numerators, multiply the denominators. 5283
Simplify. 1024
Look for common factors in the numerator and denominator. Rewrite showing common factors. 5b3ae236041febde02389f89a2b35985598acb3b
Remove common factors. 512

Another way to find this product involves removing common factors earlier.

58(23)
Determine the sign of the product. Multiply. 5283
Show common factors and then remove them. .
Multiply remaining factors. 512

We get the same result.

Try It 4.51

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

Try It 4.52

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

Example 4.27

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

Answer

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

Try It 4.53

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

Try It 4.54

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

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.

Example 4.28

Multiply, and write the answer in simplified form:

17·56

125(−20x)

Answer

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

Try It 4.55

Multiply, and write the answer in simplified form:

  1. 18·72
  2. 113(−9a)

Try It 4.56

Multiply, and write the answer in simplified form:

  1. 38·64
  2. 16x·1112

Find Reciprocals

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.

Reciprocal

The reciprocal of the fraction ab is ba, where a0 and b0,

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.

“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. Is there any number r so that 0·r=1? No. So, the number 0 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. 49
  2. 16
  3. 145
  4. 7
Answer

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. 4994
Multiply numerators and denominators. 3636
Simplify. 1
Find the reciprocal of -16-16. -61-61
Simplify. -6-6
Check: -16(-6)-16(-6)
11
Find the reciprocal of -145-145. -514-514
Check: -145(-514)-145(-514)
70707070
11
Find the reciprocal of 77.
Write 77 as a fraction. 7171
Write the reciprocal of 7171. 1717
Check: 7(17)7(17)
11

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      
Answer

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
3838 3838 3838 8383
1212 1212 1212 22
9595 9595 9595 5959
−5−5 55 55 1515

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 33s in 12.12. So, 12÷3=4.12÷3=4.

What about dividing fractions? Suppose we want to find the quotient: 12÷16.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

Example 4.31

Model: 14÷18.14÷18.

Answer

We want to determine how many 1818s are in 14.14. Start with one 1414 tile. Line up 1818 tiles underneath the 1414 tile.

A rectangle is shown, labeled one fourth. Below it is an identical rectangle split into two equal pieces, each labeled as one eighth.

There are two 1818s in 14.14.

So, 14÷18=2.14÷18=2.

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.

Answer

We are trying to determine how many 1414s there are in 2.2. We can model this as shown.

Two rectangles are shown, each labeled as 1. Below it are two identical rectangle, each split into four pieces. Each of the eight pieces is labeled as one fourth.

Because there are eight 1414s in 2,2÷14=8.2,2÷14=8.

Try It 4.63

Model: 2÷132÷13

Try It 4.64

Model: 3÷123÷12

Let’s use money to model 2÷14Figure 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,a,b,c, and dd are numbers where b0,c0,b0,c0, and d0,d0, 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,c0b0,c0 and d0d0 to be sure we don’t divide by zero.

Example 4.33

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

Answer

25÷(37)25÷(37)
Multiply the first fraction by the reciprocal of the second. 25(73)25(73)
Multiply. The product is negative. 14151415

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.

Answer

23÷n523÷n5
Multiply the first fraction by the reciprocal of the second. 23·5n23·5n
Multiply. 103n103n

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).

Answer

34÷(78)34÷(78)
Multiply the first fraction by the reciprocal of the second. 34·(87)34·(87)
Multiply. Remember to determine the sign first. 3·84·73·84·7
Rewrite to show common factors. 3·4·24·73·4·24·7
Remove common factors and simplify. 6767

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.

Answer

718÷1427718÷1427
Multiply the first fraction by the reciprocal of the second. 718·2714718·2714
Multiply. 7·2718·147·2718·14
Rewrite showing common factors. .
Remove common factors. 32·232·2
Simplify. 3434

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.

Media

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.

7 21 7 21

78.

8 24 8 24

79.

15 20 15 20

80.

12 18 12 18

81.

40 88 40 88

82.

63 99 63 99

83.

108 63 108 63

84.

104 48 104 48

85.

120 252 120 252

86.

182 294 182 294

87.

168 192 168 192

88.

140 224 140 224

89.

11 x 11 y 11 x 11 y

90.

15 a 15 b 15 a 15 b

91.

3 x 12 y 3 x 12 y

92.

4 x 32 y 4 x 32 y

93.

14 x 2 21 y 14 x 2 21 y

94.

24 a 32 b 2 24 a 32 b 2

Multiply Fractions

In the following exercises, use a diagram to model.

95.

1 2 · 2 3 1 2 · 2 3

96.

1 2 · 5 8 1 2 · 5 8

97.

1 3 · 5 6 1 3 · 5 6

98.

1 3 · 2 5 1 3 · 2 5

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

99.

2 5 · 1 3 2 5 · 1 3

100.

1 2 · 3 8 1 2 · 3 8

101.

3 4 · 9 10 3 4 · 9 10

102.

4 5 · 2 7 4 5 · 2 7

103.

2 3 ( 3 8 ) 2 3 ( 3 8 )

104.

3 4 ( 4 9 ) 3 4 ( 4 9 )

105.

5 9 · 3 10 5 9 · 3 10

106.

3 8 · 4 15 3 8 · 4 15

107.

7 12 ( 8 21 ) 7 12 ( 8 21 )

108.

5 12 ( 8 15 ) 5 12 ( 8 15 )

109.

( 14 15 ) ( 9 20 ) ( 14 15 ) ( 9 20 )

110.

( 9 10 ) ( 25 33 ) ( 9 10 ) ( 25 33 )

111.

( 63 84 ) ( 44 90 ) ( 63 84 ) ( 44 90 )

112.

( 33 60 ) ( 40 88 ) ( 33 60 ) ( 40 88 )

113.

4 · 5 11 4 · 5 11

114.

5 · 8 3 5 · 8 3

115.

3 7 · 21 n 3 7 · 21 n

116.

5 6 · 30 m 5 6 · 30 m

117.

−28 p ( 1 4 ) −28 p ( 1 4 )

118.

−51 q ( 1 3 ) −51 q ( 1 3 )

119.

−8 ( 17 4 ) −8 ( 17 4 )

120.

14 5 ( −15 ) 14 5 ( −15 )

121.

−1 ( 3 8 ) −1 ( 3 8 )

122.

( −1 ) ( 6 7 ) ( −1 ) ( 6 7 )

123.

( 2 3 ) 3 ( 2 3 ) 3

124.

( 4 5 ) 2 ( 4 5 ) 2

125.

( 6 5 ) 4 ( 6 5 ) 4

126.

( 4 7 ) 4 ( 4 7 ) 4

Find Reciprocals

In the following exercises, find the reciprocal.

127.

3 4 3 4

128.

2 3 2 3

129.

5 17 5 17

130.

6 19 6 19

131.

11 8 11 8

132.

−13 −13

133.

−19 −19

134.

−1 −1

135.

1 1

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.

1 2 ÷ 1 4 1 2 ÷ 1 4

139.

1 2 ÷ 1 8 1 2 ÷ 1 8

140.

2 ÷ 1 5 2 ÷ 1 5

141.

3 ÷ 1 4 3 ÷ 1 4

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

142.

1 2 ÷ 1 4 1 2 ÷ 1 4

143.

1 2 ÷ 1 8 1 2 ÷ 1 8

144.

3 4 ÷ 2 3 3 4 ÷ 2 3

145.

4 5 ÷ 3 4 4 5 ÷ 3 4

146.

4 5 ÷ 4 7 4 5 ÷ 4 7

147.

3 4 ÷ 3 5 3 4 ÷ 3 5

148.

7 9 ÷ ( 7 9 ) 7 9 ÷ ( 7 9 )

149.

5 6 ÷ ( 5 6 ) 5 6 ÷ ( 5 6 )

150.

3 4 ÷ x 11 3 4 ÷ x 11

151.

2 5 ÷ y 9 2 5 ÷ y 9

152.

5 8 ÷ a 10 5 8 ÷ a 10

153.

5 6 ÷ c 15 5 6 ÷ c 15

154.

5 18 ÷ ( 15 24 ) 5 18 ÷ ( 15 24 )

155.

7 18 ÷ ( 14 27 ) 7 18 ÷ ( 14 27 )

156.

7 p 12 ÷ 21 p 8 7 p 12 ÷ 21 p 8

157.

5 q 12 ÷ 15 q 8 5 q 12 ÷ 15 q 8

158.

8 u 15 ÷ 12 v 25 8 u 15 ÷ 12 v 25

159.

12 r 25 ÷ 18 s 35 12 r 25 ÷ 18 s 35

160.

−5 ÷ 1 2 −5 ÷ 1 2

161.

−3 ÷ 1 4 −3 ÷ 1 4

162.

3 4 ÷ ( −12 ) 3 4 ÷ ( −12 )

163.

2 5 ÷ ( −10 ) 2 5 ÷ ( −10 )

164.

−18 ÷ ( 9 2 ) −18 ÷ ( 9 2 )

165.

−15 ÷ ( 5 3 ) −15 ÷ ( 5 3 )

166.

1 2 ÷ ( 3 4 ) ÷ 7 8 1 2 ÷ ( 3 4 ) ÷ 7 8

167.

11 2 ÷ 7 8 · 2 11 11 2 ÷ 7 8 · 2 11

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 18,14,13,12,18,14,13,12, and 11 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 18,14,13,12,18,14,13,12, and 11 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 12·2312·23 is 13.13.

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?


This page titled 4.3: Multiply and Divide Fractions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.

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