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4.7 Solve Equations with Fractions

Clearing denominators by multiplying both sides by a reciprocal or the LCD, checking fractional solutions, and translating sentences into such equations.

Prealgebra 2e, 4.7 Solve Equations with Fractions

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133 exercises

4.7 Exercise 1

Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material.

Evaluate x+4x + 4 when x=−3x = −3
If you missed this problem, review the reference.

2.5

4.7 Exercise 2

Solve: 2y−3=9.2 y - 3 = 9 .
If you missed this problem, review the reference.

2.5

4.7 Exercise 3

Solve: y−3=−9y - 3 = −9
If you missed this problem, review the reference.

2.5

4.7 Exercise 5

Determine whether each number is a solution of the given equation.

x−23=16x - \frac{2}{3} = \frac{1}{6}:

  1. ⓐ x=1x = 1
  2. ⓑ x=56x = \frac{5}{6}
  3. ⓒ x=−56x = - \frac{5}{6}

2.5

4.7 Exercise 6

Determine whether each number is a solution of the given equation.

y−14=38y - \frac{1}{4} = \frac{3}{8}:

  1. ⓐ y=1y = 1
  2. ⓑ y=−58y = - \frac{5}{8}
  3. ⓒ y=58y = \frac{5}{8}

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4.7 Exercise 8

Solve: y+1112=512.y + \frac{11}{12} = \frac{5}{12} .

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4.7 Exercise 9

Solve: y+815=415.y + \frac{8}{15} = \frac{4}{15} .

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4.7 Exercise 11

Solve: a−35=−85.a - \frac{3}{5} = - \frac{8}{5} .

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4.7 Exercise 12

Solve: n−37=−97.n - \frac{3}{7} = - \frac{9}{7} .

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4.7 Exercise 13

Solve: 10q=44.10 q = 44 .

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4.7 Exercise 14

Solve: 12u=−76.12 u = −76 .

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4.7 Exercise 15

Solve: 8m=92.8 m = 92 .

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4.7 Exercise 17

Solve: f5=−25.\frac{f}{5} = −25 .

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4.7 Exercise 18

Solve: h9=−27.\frac{h}{9} = −27 .

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4.7 Exercise 20

Solve: c−7=−35.\frac{c}{−7} = −35 .

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4.7 Exercise 21

Solve: x−11=−12.\frac{x}{−11} = −12 .

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4.7 Exercise 23

Solve: −y=48.- y = 48 .

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4.7 Exercise 24

Solve: −c=−23.- c = −23 .

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4.7 Exercise 26

Solve: 25n=14.\frac{2}{5} n = 14 .

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4.7 Exercise 27

Solve: 56y=15.\frac{5}{6} y = 15 .

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4.7 Exercise 29

Solve: −47a=52.- \frac{4}{7} a = 52 .

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4.7 Exercise 30

Solve: −79w=84.- \frac{7}{9} w = 84 .

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4.7 Exercise 32

Translate and solve: nn divided by 77 is equal to −21.−21 .

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4.7 Exercise 33

Translate and solve: nn divided by 88 is equal to −56.−56 .

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4.7 Exercise 35

Translate and solve: The quotient of qq and −8−8 is 72.72 .

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4.7 Exercise 36

Translate and solve: The quotient of pp and −9−9 is 81.81 .

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4.7 Exercise 38

Translate and solve: Two-fifths of ff is 16.16 .

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4.7 Exercise 39

Translate and solve: Three-fourths of ff is 21.21 .

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4.7 Exercise 40

Translate and solve: The quotient of mm and 56\frac{5}{6} is 34.\frac{3}{4} .

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4.7 Exercise 41

Translate and solve. The quotient of nn and 23\frac{2}{3} is 512.\frac{5}{12} .

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4.7 Exercise 42

Translate and solve The quotient of cc and 38\frac{3}{8} is 49.\frac{4}{9} .

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4.7 Exercise 44

Translate and solve: The sum of five-eighths and xx is one-fourth.

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4.7 Exercise 45

Translate and solve: The difference of one-and-three-fourths and xx is five-sixths.

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4.7 Exercise 47

Determine whether each number is a solution of the given equation.

y−13=512y - \frac{1}{3} = \frac{5}{12}:

  1. ⓐ y=1y = 1
  2. ⓑ y=34y = \frac{3}{4}
  3. ⓒ y=−34y = - \frac{3}{4}

2.5

4.7 Exercise 49

Determine whether each number is a solution of the given equation.

k+25=56k + \frac{2}{5} = \frac{5}{6}:

  1. ⓐ k=1k = 1
  2. ⓑ k=1330k = \frac{13}{30}
  3. ⓒ k=−1330k = - \frac{13}{30}

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4.7 Exercise 51

Solve.

m+38=78m + \frac{3}{8} = \frac{7}{8}

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4.7 Exercise 53

Solve.

h+56=16h + \frac{5}{6} = \frac{1}{6}

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4.7 Exercise 55

Solve.

c−14=−54c - \frac{1}{4} = - \frac{5}{4}

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4.7 Exercise 57

Solve.

z−(−512)=−712z - \left(- \frac{5}{12}\right) = - \frac{7}{12}

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4.7 Exercise 59

Solve.

p−310=58p - \frac{3}{10} = \frac{5}{8}

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4.7 Exercise 61

Solve.

k+(−13)=−45k + \left(- \frac{1}{3}\right) = - \frac{4}{5}

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4.7 Exercise 63

Solve.

7k=187 k = 18

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4.7 Exercise 65

Solve.

−9v=33−9 v = 33

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4.7 Exercise 67

Solve.

b3=−9\frac{b}{3} = −9

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4.7 Exercise 69

Solve.

x8=−32\frac{x}{8} = −32

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4.7 Exercise 71

Solve.

q−4=−40\frac{q}{−4} = −40

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4.7 Exercise 73

Solve.

s−15=−3\frac{s}{−15} = −3

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4.7 Exercise 75

Solve.

−y=42- y = 42

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4.7 Exercise 77

Solve.

−k=−1720- k = - \frac{17}{20}

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4.7 Exercise 79

Solve.

310p=30\frac{3}{10} p = 30

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4.7 Exercise 81

Solve.

52m=−40\frac{5}{2} m = −40

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4.7 Exercise 83

Solve.

−37b=9- \frac{3}{7} b = 9

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4.7 Exercise 85

Solve.

−512v=−15- \frac{5}{12} v = −15

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4.7 Exercise 87

Solve.

8y=08 y = 0

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4.7 Exercise 89

Solve.

7g=797 g = \frac{7}{9}

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4.7 Exercise 91

Solve.

q+56=112q + \frac{5}{6} = \frac{1}{12}

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4.7 Exercise 93

Solve.

14n=710\frac{1}{4} n = \frac{7}{10}

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4.7 Exercise 95

Solve.

−23=y+38- \frac{2}{3} = y + \frac{3}{8}

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4.7 Exercise 97

Solve.

815=-d\frac{8}{15} = - \textit{d}

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4.7 Exercise 99

Translate to an algebraic equation and solve.

nn divided by six is −24.−24 .

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4.7 Exercise 101

Translate to an algebraic equation and solve.

mm divided by −7−7 is −8.−8 .

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4.7 Exercise 103

Translate to an algebraic equation and solve.

The quotient of ff and −4−4 is −20.−20 .

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4.7 Exercise 105

Translate to an algebraic equation and solve.

The quotient of gg and nine is 14.14 .

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4.7 Exercise 107

Translate to an algebraic equation and solve.

Two-fifths of qq is 20.20 .

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4.7 Exercise 109

Translate to an algebraic equation and solve.

Four-ninths of pp is −28.−28 .

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4.7 Exercise 111

Translate to an algebraic equation and solve.

The quotient of hh and 22 is 43.43 .

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4.7 Exercise 113

Translate to an algebraic equation and solve.

The quotient of aa and 23\frac{2}{3} is 34.\frac{3}{4} .

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4.7 Exercise 115

Translate to an algebraic equation and solve.

The sum of three-fourths and xx is 18.\frac{1}{8} .

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4.7 Exercise 117

Translate to an algebraic equation and solve.

The difference of yy and one-third is −16.- \frac{1}{6} .

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4.7 Exercise 119

Playhouse The table in a child’s playhouse is 35\frac{3}{5} of an adult-size table. The playhouse table is 1818 inches high. Find the height of an adult-size table by solving the equation 35h=18.\frac{3}{5} h = 18 .

2.5

4.7 Exercise 121

Richard thinks the solution to the equation 34x=24\frac{3}{4} x = 24 is 16.16 . Explain why Richard is wrong.

2.5

4.7 Exercise 127

Convert the improper fraction to a mixed number.

6311\frac{63}{11}

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4.7 Exercise 129

Convert the mixed number to an improper fraction.

9459 \frac{4}{5}

2.5

4.7 Exercise 131

Convert the mixed number to an improper fraction.

Find three fractions equivalent to −43.- \frac{4}{3} . Show your work, using figures or algebra.

2.5

4.7 Exercise 135

Order each pair of numbers, using << or >.>.

−212___−3−2 \frac{1}{2} ___ −3

2.5

4.7 Exercise 137

Simplify.

−90120- \frac{90}{120}

2.5

4.7 Exercise 139

Simplify.

−8x8y- \frac{8 x}{8 y}

2.5

4.7 Exercise 141

Multiply.

−13·127- \frac{1}{3} \cdot \frac{12}{7}

2.5

4.7 Exercise 143

Multiply.

6m·4116 m \cdot \frac{4}{11}

2.5

4.7 Exercise 145

Multiply.

165·158\frac{16}{5} \cdot \frac{15}{8}

2.5

4.7 Exercise 147

Find the reciprocal.

154\frac{15}{4}

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4.7 Exercise 149

Find the reciprocal.

−14- \frac{1}{4}

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4.7 Exercise 151

Divide.

23÷16\frac{2}{3} \div \frac{1}{6}

2.5

4.7 Exercise 153

Divide.

45÷3\frac{4}{5} \div 3

2.5

4.7 Exercise 155

Divide.

518÷(−b9)\frac{5}{18} \div \left(- \frac{b}{9}\right)

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4.7 Exercise 157

Perform the indicated operation.

−5712·4411−5 \frac{7}{12} \cdot 4 \frac{4}{11}

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4.7 Exercise 159

Perform the indicated operation.

823÷11128 \frac{2}{3} \div 1 \frac{1}{12}

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4.7 Exercise 161

Translate the English phrase into an algebraic expression.

the quotient of VV and the difference of hh and 66

2.5

4.7 Exercise 163

Simplify the complex fraction

89−4\frac{\frac{8}{9}}{−4}

2.5

4.7 Exercise 165

Simplify the complex fraction

−156−112\frac{−1 \frac{5}{6}}{- \frac{1}{12}}

2.5

4.7 Exercise 167

Simplify.

8·4−523·12\frac{8 \cdot 4 - 5^{2}}{3 \cdot 12}

2.5

4.7 Exercise 169

Add.

38+28\frac{3}{8} + \frac{2}{8}

2.5

4.7 Exercise 171

Add.

25+15\frac{2}{5} + \frac{1}{5}

2.5

4.7 Exercise 173

Add.

x10+710\frac{x}{10} + \frac{7}{10}

2.5

4.7 Exercise 175

Subtract.

1112−512\frac{11}{12} - \frac{5}{12}

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4.7 Exercise 177

Subtract.

−3130−730- \frac{31}{30} - \frac{7}{30}

2.5

4.7 Exercise 179

Subtract.

1115−515−(−215)\frac{11}{15} - \frac{5}{15} - \left(- \frac{2}{15}\right)

2.5

4.7 Exercise 181

Find the least common denominator.

13\frac{1}{3} and 45\frac{4}{5}

2.5

4.7 Exercise 183

Find the least common denominator.

34,16,\frac{3}{4} , \frac{1}{6} ,and510\frac{5}{10}

2.5

4.7 Exercise 185

Change to equivalent fractions using the given LCD.

38\frac{3}{8} and 56,\frac{5}{6} , LCD =24= 24

2.5

4.7 Exercise 187

Change to equivalent fractions using the given LCD.

13,34\frac{1}{3} , \frac{3}{4} and 45,\frac{4}{5} , LCD =60= 60

2.5

4.7 Exercise 189

Perform the indicated operations and simplify.

1112−23\frac{11}{12} - \frac{2}{3}

2.5

4.7 Exercise 191

Perform the indicated operations and simplify.

−1136−1120- \frac{11}{36} - \frac{11}{20}

2.5

4.7 Exercise 193

Perform the indicated operations and simplify.

y10−13\frac{y}{10} - \frac{1}{3}

2.5

4.7 Exercise 195

Perform the indicated operations and simplify.

411÷27d\frac{4}{11} \div \frac{2}{7 d}

2.5

4.7 Exercise 197

Perform the indicated operations and simplify.

(23)2(58)2\frac{\left(\frac{2}{3}\right)^{2}}{\left(\frac{5}{8}\right)^{2}}

2.5

4.7 Exercise 199

Evaluate.

y−45y - \frac{4}{5} when

  1. ⓐ y=−45y = - \frac{4}{5}
  2. ⓑ y=14y = \frac{1}{4}

2.5

4.7 Exercise 201

Perform the indicated operation.

413+9134 \frac{1}{3} + 9 \frac{1}{3}

2.5

4.7 Exercise 203

Perform the indicated operation.

5811+24115 \frac{8}{11} + 2 \frac{4}{11}

2.5

4.7 Exercise 205

Perform the indicated operation.

91320−411209 \frac{13}{20} - 4 \frac{11}{20}

2.5

4.7 Exercise 207

Perform the indicated operation.

21112−17122 \frac{11}{12} - 1 \frac{7}{12}

2.5

4.7 Exercise 209

Determine whether the each number is a solution of the given equation.

x−12=16x - \frac{1}{2} = \frac{1}{6}:

  1. ⓐ x=1x = 1
  2. ⓑ x=23x = \frac{2}{3}
  3. ⓒ x=−13x = - \frac{1}{3}

2.5

4.7 Exercise 211

Solve the equation.

n+911=411n + \frac{9}{11} = \frac{4}{11}

2.5

4.7 Exercise 213

Solve the equation.

h−(−78)=−25h - \left(- \frac{7}{8}\right) = - \frac{2}{5}

2.5

4.7 Exercise 215

Solve the equation.

−z=23- z = 23

2.5

4.7 Exercise 217

Translate and solve.

The difference of qq and one-tenth is 12.\frac{1}{2} .

2.5

4.7 Exercise 219

Translate and solve.

Three-eighths of yy is 24.24 .

2.5

4.7 Exercise 221

Convert the mixed number to an improper fraction.

3273 \frac{2}{7}

2.5

4.7 Exercise 223

Simplify.

520\frac{5}{20}

2.5

4.7 Exercise 225

Simplify.

13·34\frac{1}{3} \cdot \frac{3}{4}

2.5

4.7 Exercise 227

Simplify.

−36u(−49)−36 u \left(- \frac{4}{9}\right)

2.5

4.7 Exercise 229

Simplify.

−56÷512- \frac{5}{6} \div \frac{5}{12}

2.5

4.7 Exercise 231

Simplify.

9a10÷15a8\frac{9 a}{10} \div \frac{15 a}{8}

2.5

4.7 Exercise 233

Simplify.

(−1556)÷(−316)\left(−15 \frac{5}{6}\right) \div \left(−3 \frac{1}{6}\right)

2.5

4.7 Exercise 235

Simplify.

p2q5\frac{ \frac{p}{2} }{ \frac{q}{5} }

2.5

4.7 Exercise 237

Simplify.

92−429−4\frac{9^{2} - 4^{2}}{9 - 4}

2.5

4.7 Exercise 239

Simplify.

−313+(−413)- \frac{3}{13} + \left(- \frac{4}{13}\right)

2.5

4.7 Exercise 241

Simplify.

25+(−75)\frac{2}{5} + \left(- \frac{7}{5}\right)

2.5

4.7 Exercise 243

Simplify.

−34÷x3- \frac{3}{4} \div \frac{x}{3}

2.5

4.7 Exercise 245

Simplify.

514+18956\frac{\frac{5}{14} + \frac{1}{8}}{\frac{9}{56}}

2.5

4.7 Exercise 247

Solve the equation.

y+35=75y + \frac{3}{5} = \frac{7}{5}

2.5

4.7 Exercise 249

Solve the equation.

f+(−23)=512f + \left(- \frac{2}{3}\right) = \frac{5}{12}

2.5

4.7 Exercise 251

Solve the equation.

−23c=18- \frac{2}{3} c = 18

2.5

Adapted from Prealgebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Prealgebra 2e. Imported at revision 77a933c.