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Prealgebra 2e

4.3 Multiply and Divide Mixed Numbers and Complex Fractions

Prealgebra 2e4.3 Multiply and Divide Mixed Numbers and Complex Fractions
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  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:
  • 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 aandbaandb is the result you get from dividing abyb,abyb, 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 aandb,aandb,

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 Access Additional Online Resources

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.

438·710438·710

177.

249·67249·67

178.

1522·3351522·335

179.

2536·63102536·6310

180.

423(−118)423(−118)

181.

225(−229)225(−229)

182.

−449·51316−449·51316

183.

−1720·21112−1720·21112

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

184.

513÷4513÷4

185.

1312÷91312÷9

186.

−12÷3311−12÷3311

187.

−7÷514−7÷514

188.

638÷218638÷218

189.

215÷1110215÷1110

190.

−935÷(−135)−935÷(−135)

191.

−1834÷(−334)−1834÷(−334)

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.

23892389

199.

4581545815

200.

82112358211235

201.

91633409163340

202.

452452

203.

91039103

204.

258258

205.

53105310

206.

m3n2m3n2

207.

r5s3r5s3

208.

x689x689

209.

38y1238y12

210.

245110245110

211.

4231642316

212.

79−24579−245

213.

38−63438−634

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+1184+118

219.

9+379+37

220.

22+31022+310

221.

19461946

222.

482415482415

223.

464+4464+4

224.

−6+68+4−6+68+4

225.

−6+3178−6+3178

226.

2214191322141913

227.

15+918+1215+918+12

228.

58−1058−10

229.

34−2434−24

230.

43664366

231.

66926692

232.

4212542125

233.

72+16072+160

234.

83+2914+383+2914+3

235.

964722+3964722+3

236.

1555221015552210

237.

1293231812932318

238.

5634452356344523

239.

8976569289765692

240.

523235523235

241.

624246624246

242.

2+4(3)−3222+4(3)−322

243.

7+3(5)−2327+3(5)−232

244.

742(85)9335742(85)9335

245.

973(128)8766973(128)8766

246.

9(82)−3(157)6(71)−3(179)9(82)−3(157)6(71)−3(179)

247.

8(92)−4(149)7(83)−3(169)8(92)−4(149)7(83)−3(169)

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,and118,14,13,12,and1 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?

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