Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo
Prealgebra 2e

4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Prealgebra 2e4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Learning Objectives

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

  • Multiply and divide mixed numbers
  • Translate phrases to expressions with fractions
  • Simplify complex fractions
  • Simplify expressions written with a fraction bar

Be Prepared 4.6

Before you get started, take this readiness quiz.

Divide and reduce, if possible: (4+5)÷(107).(4+5)÷(107).
If you missed this problem, review Example 3.21.

Be Prepared 4.7

Multiply and write the answer in simplified form: 18·2318·23.
If you missed this problem, review Example 4.25.

Be Prepared 4.8

Convert 235235 into an improper fraction.
If you missed this problem, review Example 4.11.

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.

Example 4.37

Multiply: 313·58313·58

Try It 4.73

Multiply, and write your answer in simplified form: 523·617.523·617.

Try It 4.74

Multiply, and write your answer in simplified form: 37·514.37·514.

How To

Multiply or divide mixed numbers.

  1. Step 1. Convert the mixed numbers to improper fractions.
  2. Step 2. Follow the rules for fraction multiplication or division.
  3. Step 3. Simplify if possible.

Example 4.38

Multiply, and write your answer in simplified form: 245(178).245(178).

Try It 4.75

Multiply, and write your answer in simplified form. 557(258).557(258).

Try It 4.76

Multiply, and write your answer in simplified form. −325·416.−325·416.

Example 4.39

Divide, and write your answer in simplified form: 347÷5.347÷5.

Try It 4.77

Divide, and write your answer in simplified form: 438÷7.438÷7.

Try It 4.78

Divide, and write your answer in simplified form: 258÷3.258÷3.

Example 4.40

Divide: 212÷114.212÷114.

Try It 4.79

Divide, and write your answer in simplified form: 223÷113.223÷113.

Try It 4.80

Divide, and write your answer in simplified form: 334÷112.334÷112.

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 aa and bb is the result you get from dividing aa by b,b, or ab.ab. Let’s practice translating some phrases into algebraic expressions using these terms.

Example 4.41

Translate the phrase into an algebraic expression: “the quotient of 3x3x and 8.”8.”

Try It 4.81

Translate the phrase into an algebraic expression: the quotient of 9s9s and 14.14.

Try It 4.82

Translate the phrase into an algebraic expression: the quotient of 5y5y and 6.6.

Example 4.42

Translate the phrase into an algebraic expression: the quotient of the difference of mm and n,n, and p.p.

Try It 4.83

Translate the phrase into an algebraic expression: the quotient of the difference of aa and b,b, and cd.cd.

Try It 4.84

Translate the phrase into an algebraic expression: the quotient of the sum of pp and q,q, and r.r.

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:

6733458x2566733458x256

To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction 34583458 can be written as 34÷58.34÷58.

Example 4.43

Simplify: 3458.3458.

Try It 4.85

Simplify: 2356.2356.

Try It 4.86

Simplify: 37611.37611.

How To

Simplify a complex fraction.

  1. Step 1. Rewrite the complex fraction as a division problem.
  2. Step 2. Follow the rules for dividing fractions.
  3. Step 3. Simplify if possible.

Example 4.44

Simplify: 673.673.

Try It 4.87

Simplify: 874.874.

Try It 4.88

Simplify: 3910.3910.

Example 4.45

Simplify: x2xy6.x2xy6.

Try It 4.89

Simplify: a8ab6.a8ab6.

Try It 4.90

Simplify: p2pq8.p2pq8.

Example 4.46

Simplify: 23418.23418.

Try It 4.91

Simplify: 57125.57125.

Try It 4.92

Simplify: 85315.85315.

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 1313 could be the result of dividing −13,−13, a negative by a positive, or of dividing 1−3,1−3, a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Negative 1 over positive 3 is equal to negative one third. Negative over positive equals negative. Positive 1 over negative 3 is equal to negative one third. Positive over negative equals 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−1−3=13negativenegative=positive

Placement of Negative Sign in a Fraction

For any positive numbers aa and b,b,

ab=ab=abab=ab=ab

Example 4.47

Which of the following fractions are equivalent to 7−8?7−8?

−7−8,−78,78,78−7−8,−78,78,78

Try It 4.93

Which of the following fractions are equivalent to −35?−35?

−3−5,35,35,3−5−3−5,35,35,3−5

Try It 4.94

Which of the following fractions are equivalent to 27?27?

−2−7,−27,27,2−7−2−7,−27,27,2−7

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+8534+853 means (4+8)÷(53).(4+8)÷(53). 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


Parentheses, brackets, braces, an absolute value sign, and a fraction bar are shown.

How To

Simplify an expression with a fraction bar.

  1. Step 1. Simplify the numerator.
  2. Step 2. Simplify the denominator.
  3. Step 3. Simplify the fraction.

Example 4.48

Simplify: 4+853.4+853.

Try It 4.95

Simplify: 4+6112.4+6112.

Try It 4.96

Simplify: 3+5182.3+5182.

Example 4.49

Simplify: 42(3)22+2.42(3)22+2.

Try It 4.97

Simplify: 63(5)32+3.63(5)32+3.

Try It 4.98

Simplify: 44(6)33+3.44(6)33+3.

Example 4.50

Simplify: (84)28242.(84)28242.

Try It 4.99

Simplify: (117)211272.(117)211272.

Try It 4.100

Simplify: (6+2)262+22.(6+2)262+22.

Example 4.51

Simplify: 4(−3)+6(−2)−3(2)−2.4(−3)+6(−2)−3(2)−2.

Try It 4.101

Simplify: 8(−2)+4(−3)−5(2)+3.8(−2)+4(−3)−5(2)+3.

Try It 4.102

Simplify: 7(−1)+9(−3)−5(3)−2.7(−1)+9(−3)−5(3)−2.

Media

Section 4.3 Exercises

Practice Makes Perfect

Multiply and Divide Mixed Numbers

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

176.

4 3 8 · 7 10 4 3 8 · 7 10

177.

2 4 9 · 6 7 2 4 9 · 6 7

178.

15 22 · 3 3 5 15 22 · 3 3 5

179.

25 36 · 6 3 10 25 36 · 6 3 10

180.

4 2 3 ( −1 1 8 ) 4 2 3 ( −1 1 8 )

181.

2 2 5 ( −2 2 9 ) 2 2 5 ( −2 2 9 )

182.

−4 4 9 · 5 13 16 −4 4 9 · 5 13 16

183.

−1 7 20 · 2 11 12 −1 7 20 · 2 11 12

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

184.

5 1 3 ÷ 4 5 1 3 ÷ 4

185.

13 1 2 ÷ 9 13 1 2 ÷ 9

186.

−12 ÷ 3 3 11 −12 ÷ 3 3 11

187.

−7 ÷ 5 1 4 −7 ÷ 5 1 4

188.

6 3 8 ÷ 2 1 8 6 3 8 ÷ 2 1 8

189.

2 1 5 ÷ 1 1 10 2 1 5 ÷ 1 1 10

190.

−9 3 5 ÷ ( −1 3 5 ) −9 3 5 ÷ ( −1 3 5 )

191.

−18 3 4 ÷ ( −3 3 4 ) −18 3 4 ÷ ( −3 3 4 )

Translate Phrases to Expressions with Fractions

In the following exercises, translate each English phrase into an algebraic expression.

192.

the quotient of 5u5u and 1111

193.

the quotient of 7v7v and 1313

194.

the quotient of pp and qq

195.

the quotient of aa and bb

196.

the quotient of rr and the sum of ss and 1010

197.

the quotient of AA and the difference of 33 and BB

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

198.

2 3 8 9 2 3 8 9

199.

4 5 8 15 4 5 8 15

200.

8 21 12 35 8 21 12 35

201.

9 16 33 40 9 16 33 40

202.

4 5 2 4 5 2

203.

9 10 3 9 10 3

204.

2 5 8 2 5 8

205.

5 3 10 5 3 10

206.

m 3 n 2 m 3 n 2

207.

r 5 s 3 r 5 s 3

208.

x 6 8 9 x 6 8 9

209.

3 8 y 12 3 8 y 12

210.

2 4 5 1 10 2 4 5 1 10

211.

4 2 3 1 6 4 2 3 1 6

212.

7 9 −2 4 5 7 9 −2 4 5

213.

3 8 −6 3 4 3 8 −6 3 4

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

214.

Which of the following fractions are equivalent to 5−11?5−11?
−5−11,−511,511,511−5−11,−511,511,511

215.

Which of the following fractions are equivalent to −49?−49?
−4−9,−49,49,49−4−9,−49,49,49

216.

Which of the following fractions are equivalent to 113?113?
−113,113,−11−3,11−3−113,113,−11−3,11−3

217.

Which of the following fractions are equivalent to 136?136?
136,13−6,−13−6,−136136,13−6,−13−6,−136

In the following exercises, simplify.

218.

4 + 11 8 4 + 11 8

219.

9 + 3 7 9 + 3 7

220.

22 + 3 10 22 + 3 10

221.

19 4 6 19 4 6

222.

48 24 15 48 24 15

223.

46 4 + 4 46 4 + 4

224.

−6 + 6 8 + 4 −6 + 6 8 + 4

225.

−6 + 3 17 8 −6 + 3 17 8

226.

22 14 19 13 22 14 19 13

227.

15 + 9 18 + 12 15 + 9 18 + 12

228.

5 8 −10 5 8 −10

229.

3 4 −24 3 4 −24

230.

4 3 6 6 4 3 6 6

231.

6 6 9 2 6 6 9 2

232.

4 2 1 25 4 2 1 25

233.

7 2 + 1 60 7 2 + 1 60

234.

8 3 + 2 9 14 + 3 8 3 + 2 9 14 + 3

235.

9 6 4 7 22 + 3 9 6 4 7 22 + 3

236.

15 5 5 2 2 10 15 5 5 2 2 10

237.

12 9 3 2 3 18 12 9 3 2 3 18

238.

5 6 3 4 4 5 2 3 5 6 3 4 4 5 2 3

239.

8 9 7 6 5 6 9 2 8 9 7 6 5 6 9 2

240.

5 2 3 2 3 5 5 2 3 2 3 5

241.

6 2 4 2 4 6 6 2 4 2 4 6

242.

2 + 4 ( 3 ) −3 2 2 2 + 4 ( 3 ) −3 2 2

243.

7 + 3 ( 5 ) −2 3 2 7 + 3 ( 5 ) −2 3 2

244.

7 4 2 ( 8 5 ) 9 3 3 5 7 4 2 ( 8 5 ) 9 3 3 5

245.

9 7 3 ( 12 8 ) 8 7 6 6 9 7 3 ( 12 8 ) 8 7 6 6

246.

9 ( 8 2 ) −3 ( 15 7 ) 6 ( 7 1 ) −3 ( 17 9 ) 9 ( 8 2 ) −3 ( 15 7 ) 6 ( 7 1 ) −3 ( 17 9 )

247.

8 ( 9 2 ) −4 ( 14 9 ) 7 ( 8 3 ) −3 ( 16 9 ) 8 ( 9 2 ) −4 ( 14 9 ) 7 ( 8 3 ) −3 ( 16 9 )

Everyday Math

248.

Baking A recipe for chocolate chip cookies calls for 214214 cups of flour. Graciela wants to double the recipe.

  1. How much flour will Graciela need? Show your calculation. Write your result as an improper fraction and as a mixed number.
  2. Measuring cups usually come in sets with cups for 18,14,13,12,18,14,13,12, and 11 cup. Draw a diagram to show two different ways that Graciela could measure out the flour needed to double the recipe.
249.

Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains 223223 cups of chocolate chips per pound.

  1. How many cups of chocolate chips are in a half-pound of the fudge?
  2. The owners of the booth make the fudge in 1010-pound batches. How many chocolate chips do they need to make a 1010-pound batch? Write your results as improper fractions and as a mixed numbers.

Writing Exercises

250.

Explain how to find the reciprocal of a mixed number.

251.

Explain how to multiply mixed numbers.

252.

Randy thinks that 312·514312·514 is 1518.1518. Explain what is wrong with Randy’s thinking.

253.

Explain why 12,−12,12,−12, and 1−21−2 are equivalent.

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?

Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction
Citation information

© Jan 23, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.