Content check — Prealgebra, Chapter 4: Fractions

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 4: Fractions

588 problems · 102 figures

Be Prepared 4.1 §4.1 source ↗

Simplify:

5 \cdot 2 + 1 .

If you missed this problem, review Example 2.8.

Example 2.8 in the book ↗

Answer

11

Be Prepared 4.2 §4.1 source ↗

Fill in the blank with

<

>: −2 \underset{__}{} −5

If you missed this problem, review Example 3.2.

Example 3.2 in the book ↗

Answer

- 2 > - 5

Try It 4.1 §4.1 source ↗

Name the fraction of the shape that is shaded in each figure:

In part “a”, a circle is divided into eight equal wedges. Three of the wedges are shaded. In part “b”, a square is divided into nine equal pieces. Four of the pieces are shaded.

Answer

ⓐ $\frac{3}{8}$

ⓑ $\frac{4}{9}$

Try It 4.2 §4.1 source ↗

Name the fraction of the shape that is shaded in each figure:

Answer

ⓐ $\frac{3}{5}$

ⓑ $\frac{3}{4}$

Try It 4.3 §4.1 source ↗

Shade $\frac{6}{8}$ of the circle.

Answer

Try It 4.4 §4.1 source ↗

Shade $\frac{2}{5}$ of the rectangle.

Answer

Try It 4.5 §4.1 source ↗

Use fraction circles to make wholes with the following pieces:

3

thirds.

Answer

Try It 4.6 §4.1 source ↗

Use fraction circles to make wholes with the following pieces:

8

eighths.

Answer

Try It 4.7 §4.1 source ↗

Use fraction circles to make wholes with the following pieces:

5

thirds.

Answer

Try It 4.8 §4.1 source ↗

Use fraction circles to make wholes with the following pieces:

5

halves.

Answer

Three circles are shown. Each is divided into two sections. The first two circles are completely shaded. Half of the third circle is shaded.

Try It 4.9 §4.1 source ↗

Name the improper fraction. Then write it as a mixed number.

Answer

\frac{5}{3} = 1 \frac{2}{3}

Try It 4.10 §4.1 source ↗

Name the improper fraction. Then write it as a mixed number.

Answer

\frac{13}{8} = 1 \frac{5}{8}

Try It 4.11 §4.1 source ↗

Draw a figure to model

\frac{7}{6} .

Answer

Try It 4.12 §4.1 source ↗

Draw a figure to model

\frac{6}{5} .

Answer

Try It 4.13 §4.1 source ↗

Use a model to rewrite the improper fraction as a mixed number:

\frac{9}{7} .

Answer

$1 \frac{2}{7}$

Try It 4.14 §4.1 source ↗

Use a model to rewrite the improper fraction as a mixed number:

\frac{7}{4} .

Answer

$1 \frac{3}{4}$

Try It 4.15 §4.1 source ↗

Use a model to rewrite the mixed number as an improper fraction:

1 \frac{3}{8} .

Answer

$\frac{11}{8}$

Try It 4.16 §4.1 source ↗

Use a model to rewrite the mixed number as an improper fraction:

1 \frac{5}{6} .

Answer

$\frac{11}{6}$

Try It 4.17 §4.1 source ↗

Convert the improper fraction to a mixed number:

\frac{13}{7} .

Answer

1 \frac{6}{7} .

Try It 4.18 §4.1 source ↗

Convert the improper fraction to a mixed number:

\frac{14}{9} .

Answer

1 \frac{5}{9}

Try It 4.19 §4.1 source ↗

Convert the improper fraction to a mixed number:

\frac{23}{7} .

Answer

3 \frac{2}{7}

Try It 4.20 §4.1 source ↗

Convert the improper fraction to a mixed number:

\frac{48}{11} .

Answer

4 \frac{4}{11}

Try It 4.21 §4.1 source ↗

Convert the mixed number to an improper fraction:

3 \frac{5}{7} .

Answer

\frac{26}{7}

Try It 4.22 §4.1 source ↗

Convert the mixed number to an improper fraction:

2 \frac{7}{8} .

Answer

\frac{23}{8}

Try It 4.23 §4.1 source ↗

Convert the mixed number to an improper fraction:

4 \frac{6}{11} .

Answer

\frac{50}{11}

Try It 4.24 §4.1 source ↗

Convert the mixed number to an improper fraction:

11 \frac{1}{3} .

Answer

\frac{34}{3}

Try It 4.25 §4.1 source ↗

Use fraction tiles to find equivalent fractions: How many eighths equal one-fourth?

Answer

Try It 4.26 §4.1 source ↗

Use fraction tiles to find equivalent fractions: How many twelfths equal one-fourth?

Answer

Try It 4.27 §4.1 source ↗

Find three fractions equivalent to

\frac{3}{5} .

Answer

Correct answers include

\frac{6}{10}, \frac{9}{15},

and

\frac{12}{20} .

Try It 4.28 §4.1 source ↗

Find three fractions equivalent to

\frac{4}{5} .

Answer

Correct answers include

\frac{8}{10}, \frac{12}{15},

and

\frac{16}{20} .

Try It 4.29 §4.1 source ↗

Find a fraction with a denominator of

21

that is equivalent to

\frac{6}{7} .

Answer

\frac{18}{21}

Try It 4.30 §4.1 source ↗

Find a fraction with a denominator of

100

that is equivalent to

\frac{3}{10} .

Answer

\frac{30}{100}

Try It 4.31 §4.1 source ↗

Locate and label the following on a number line:

\frac{1}{3},

\frac{5}{4},

\frac{7}{4},

2 \frac{3}{5},

\frac{9}{2} .

Answer

Try It 4.32 §4.1 source ↗

Locate and label the following on a number line:

\frac{2}{3},

\frac{5}{2},

\frac{9}{4},

\frac{11}{4},

3 \frac{2}{5} .

Answer

Try It 4.33 §4.1 source ↗

Locate and label each of the given fractions on a number line:

$\frac{2}{3},$ $-$ $\frac{2}{3},$ $2 \frac{1}{4},$ $−2 \frac{1}{4},$ $\frac{3}{2},$ $-$ $\frac{3}{2}$

Answer

Try It 4.34 §4.1 source ↗

Locate and label each of the given fractions on a number line:

$\frac{3}{4},$ $-$ $\frac{3}{4},$ $1 \frac{1}{2},$ $−1 \frac{1}{2},$ $\frac{7}{3},$ $-$ $\frac{7}{3}$

Answer

Try It 4.35 §4.1 source ↗

Order each of the following pairs of numbers, using $<$ or $>:$

ⓐ $- \frac{1}{3} \underset{__}{} - 1$

ⓑ $−1 \frac{1}{2} \underset{__}{} - 2$

ⓓ $−3 \underset{__}{} - \frac{7}{3}$

Answer

ⓐ >

ⓑ >

ⓒ <

ⓓ <

Try It 4.36 §4.1 source ↗

Order each of the following pairs of numbers, using $<$ or $>:$

ⓐ $−3__- \frac{17}{5}$

ⓑ $−2 \frac{1}{4}__−2$

ⓓ $−4 \underset{__}{} - \frac{10}{3}$

Answer

ⓐ >

ⓑ <

ⓒ >

ⓓ <

4.1-1 §4.1 source ↗

In the following exercises, name the fraction of each figure that is shaded.

In part “a”, a circle is divided into 4 equal pieces. 1 piece is shaded. In part “b”, a circle is divided into 4 equal pieces. 3 pieces are shaded. In part “c”, a circle is divided into 8 equal pieces. 3 pieces are shaded. In part “d”, a circle is divided into 8 equal pieces. 5 pieces are shaded.

Answer

ⓐ $\frac{1}{4}$

ⓑ $\frac{3}{4}$

ⓓ $\frac{5}{8}$

4.1-3 §4.1 source ↗

In the following exercises, shade parts of circles or squares to model the following fractions.

$\frac{1}{2}$

Answer

4.1-5 §4.1 source ↗

In the following exercises, shade parts of circles or squares to model the following fractions.

$\frac{3}{4}$

Answer

4.1-7 §4.1 source ↗

In the following exercises, shade parts of circles or squares to model the following fractions.

$\frac{5}{6}$

Answer

4.1-9 §4.1 source ↗

In the following exercises, shade parts of circles or squares to model the following fractions.

$\frac{5}{8}$

Answer

4.1-11 §4.1 source ↗

In the following exercises, use fraction circles to make wholes using the following pieces.

$3$ thirds

Answer

4.1-13 §4.1 source ↗

In the following exercises, use fraction circles to make wholes using the following pieces.

$7$ sixths

Answer

4.1-15 §4.1 source ↗

In the following exercises, use fraction circles to make wholes using the following pieces.

$7$ fifths

Answer

4.1-17 §4.1 source ↗

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

In part “a”, two circles are shown. Each is divided into 4 equal pieces. The circle on the left has all 4 pieces shaded. The circle on the right has 1 piece shaded. In part “b”, two circles are shown. Each is divided into 4 equal pieces. The circle on the left has all 4 pieces shaded. The circle on the right has 3 pieces shaded. In part “c”, two circles are shown. Each is divided into 8 equal pieces. The circle on the left has all 8 pieces shaded. The circle on the right has 3 pieces shaded.

Answer

ⓐ $\frac{5}{4} = 1 \frac{1}{4}$

ⓑ $\frac{7}{4} = 1 \frac{3}{4}$

4.1-19 §4.1 source ↗

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

In part “a”, 3 circles are shown. Each is divided into 4 equal pieces. The first two circles have all 4 pieces shaded. The third circle has 3 pieces shaded. In part “b”, 3 circles are shown. Each is divided into 8 equal pieces. The first two circles have all 8 pieces shaded. The third circle has 3 pieces shaded.

Answer

ⓐ $\frac{11}{4} = 2 \frac{3}{4}$

ⓑ $\frac{19}{8} = 2 \frac{3}{8}$

4.1-21 §4.1 source ↗

In the following exercises, draw fraction circles to model the given fraction.

$\frac{4}{4}$

Answer

4.1-23 §4.1 source ↗

In the following exercises, draw fraction circles to model the given fraction.

$\frac{5}{3}$

Answer

4.1-25 §4.1 source ↗

In the following exercises, draw fraction circles to model the given fraction.

$\frac{13}{8}$

Answer

4.1-27 §4.1 source ↗

In the following exercises, draw fraction circles to model the given fraction.

$\frac{9}{4}$

Answer

Three circles are shown. Each is divided into 4 equal pieces. All 4 pieces are shaded in the two circles on the left. 1 piece is shaded in the circle on the right.

4.1-29 §4.1 source ↗

In the following exercises, rewrite the improper fraction as a mixed number.

$\frac{5}{3}$

Answer

1 \frac{2}{3}

4.1-31 §4.1 source ↗

In the following exercises, rewrite the improper fraction as a mixed number.

$\frac{13}{5}$

Answer

2 \frac{3}{5}

4.1-33 §4.1 source ↗

In the following exercises, rewrite the improper fraction as a mixed number.

$\frac{28}{9}$

Answer

3 \frac{1}{9}

4.1-35 §4.1 source ↗

In the following exercises, rewrite the improper fraction as a mixed number.

$\frac{47}{15}$

Answer

3 \frac{2}{15}

4.1-37 §4.1 source ↗

In the following exercises, rewrite the mixed number as an improper fraction.

$1 \frac{2}{5}$

Answer

\frac{7}{5}

4.1-39 §4.1 source ↗

In the following exercises, rewrite the mixed number as an improper fraction.

$2 \frac{5}{6}$

Answer

\frac{17}{6}

4.1-41 §4.1 source ↗

In the following exercises, rewrite the mixed number as an improper fraction.

$2 \frac{5}{7}$

Answer

\frac{19}{7}

4.1-43 §4.1 source ↗

In the following exercises, rewrite the mixed number as an improper fraction.

$3 \frac{5}{9}$

Answer

\frac{32}{9}

4.1-45 §4.1 source ↗

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

How many twelfths equal one-third?

Answer

4.1-47 §4.1 source ↗

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

How many twelfths equal three-fourths?

Answer

4.1-49 §4.1 source ↗

In the following exercises, use fraction tiles or draw a figure to find equivalent fractions.

How many sixths equal three-halves?

Answer

4.1-51 §4.1 source ↗

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

$\frac{1}{3}$

Answer

Answers may vary. Correct answers include

\frac{2}{6}, \frac{3}{9}, \frac{4}{12} .

4.1-53 §4.1 source ↗

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

$\frac{5}{6}$

Answer

Answers may vary. Correct answers include

\frac{10}{12},

\frac{15}{18},

\frac{20}{24} .

4.1-55 §4.1 source ↗

In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.

$\frac{5}{9}$

Answer

Answers may vary. Correct answers include

\frac{10}{18},

\frac{15}{27},

\frac{20}{36} .

4.1-57 §4.1 source ↗

In the following exercises, plot the numbers on a number line.

$\frac{1}{3}, \frac{7}{4}, \frac{13}{5}$

Answer

4.1-59 §4.1 source ↗

In the following exercises, plot the numbers on a number line.

$\frac{7}{10},$ $\frac{5}{2},$ $\frac{13}{8},$ $3$

Answer

4.1-61 §4.1 source ↗

In the following exercises, plot the numbers on a number line.

$1 \frac{3}{4}, −1 \frac{3}{5}$

Answer

4.1-63 §4.1 source ↗

In the following exercises, plot the numbers on a number line.

$\frac{2}{5},$ $-$ $\frac{2}{5},$ $1 \frac{3}{4},$ $−1 \frac{3}{4},$ $\frac{8}{3},$ $-$ $\frac{8}{3}$

Answer

4.1-65 §4.1 source ↗

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

$−1 \underset{__}{} - \frac{1}{3}$

Answer

4.1-67 §4.1 source ↗

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

$−1 \frac{3}{4} \underset{__}{} - 2$

Answer

4.1-69 §4.1 source ↗

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

$- \frac{9}{10} \underset{__}{} - \frac{3}{10}$

Answer

4.1-71 §4.1 source ↗

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

$−4 \underset{__}{} - \frac{23}{6}$

Answer

4.1-73 §4.1 source ↗

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

Music Measures Fractions are used often in music. In $\frac{4}{4}$ time, there are four quarter notes in one measure.

ⓐ How many measures would eight quarter notes make?

ⓑ The song “Happy Birthday to You” has $25$ quarter notes. How many measures are there in “Happy Birthday to You?”

Answer

ⓐ 2

ⓑ $6 \frac{1}{4}$

4.1-75 §4.1 source ↗

In the following exercises, order each of the following pairs of numbers, using $<$ or $>.$

Give an example from your life experience (outside of school) where it was important to understand fractions.

Answer

Answers will vary.

Be Prepared 4.3 §4.2 source ↗

Find the prime factorization of

48 .

If you missed this problem, review Example 2.48.

Example 2.48 in the book ↗

Answer

2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

Be Prepared 4.4 §4.2 source ↗

Draw a model of the fraction

\frac{3}{4} .

If you missed this problem, review Example 4.2.

Example 4.2 in the book ↗

Answer

Be Prepared 4.5 §4.2 source ↗

Find two fractions equivalent to

\frac{5}{6} .

If you missed this problem, review Example 4.14.

Example 4.14 in the book ↗

Answer

Answers may vary. Acceptable answers include

\frac{10}{12},

\frac{15}{18},

\frac{50}{60},

etc.

Try It 4.37 §4.2 source ↗

Simplify:

\frac{8}{12}

Answer

\frac{2}{3}

Try It 4.38 §4.2 source ↗

Simplify:

\frac{12}{16}

Answer

\frac{3}{4}

Try It 4.39 §4.2 source ↗

Simplify:

- \frac{21}{28} .

Answer

- \frac{3}{4}

Try It 4.40 §4.2 source ↗

Simplify:

- \frac{16}{24} .

Answer

- \frac{2}{3}

Try It 4.41 §4.2 source ↗

Simplify:

- \frac{54}{42} .

Answer

- \frac{9}{7}

Try It 4.42 §4.2 source ↗

Simplify:

- \frac{81}{45} .

Answer

- \frac{9}{5}

Try It 4.43 §4.2 source ↗

Simplify:

\frac{69}{120} .

Answer

\frac{23}{40}

Try It 4.44 §4.2 source ↗

Simplify:

\frac{120}{192} .

Answer

\frac{5}{8}

Try It 4.45 §4.2 source ↗

Simplify:

\frac{7 x}{7 y} .

Answer

\frac{x}{y}

Try It 4.46 §4.2 source ↗

Simplify:

\frac{9 a}{9 b} .

Answer

\frac{a}{b}

Try It 4.47 §4.2 source ↗

Use a diagram to model:

\frac{1}{2} \cdot \frac{3}{5} .

Answer

$\frac{3}{10}$

Try It 4.48 §4.2 source ↗

Use a diagram to model:

\frac{1}{2} \cdot \frac{5}{6} .

Answer

$\frac{5}{12}$

Try It 4.49 §4.2 source ↗

Multiply, and write the answer in simplified form:

\frac{1}{3} \cdot \frac{2}{5} .

Answer

\frac{2}{15}

Try It 4.50 §4.2 source ↗

Multiply, and write the answer in simplified form:

\frac{3}{5} \cdot \frac{7}{8} .

Answer

\frac{21}{40}

Try It 4.51 §4.2 source ↗

Multiply, and write the answer in simplified form:

- \frac{4}{7} (- \frac{5}{8}) .

Answer

\frac{5}{14}

Try It 4.52 §4.2 source ↗

Multiply, and write the answer in simplified form:

- \frac{7}{12} (- \frac{8}{9}) .

Answer

\frac{14}{27}

Try It 4.53 §4.2 source ↗

Multiply, and write the answer in simplified form:

- \frac{10}{28} \cdot \frac{8}{15} .

Answer

- \frac{4}{21}

Try It 4.54 §4.2 source ↗

Multiply, and write the answer in simplified form:

- \frac{9}{20} \cdot \frac{5}{12} .

Answer

- \frac{3}{16}

Try It 4.55 §4.2 source ↗

Multiply, and write the answer in simplified form:

ⓐ $\frac{1}{8} \cdot 72$

ⓑ $\frac{11}{3} (−9 a)$

Answer

ⓐ 9

ⓑ −33a

Try It 4.56 §4.2 source ↗

Multiply, and write the answer in simplified form:

ⓐ $\frac{3}{8} \cdot 64$

ⓑ $16 x \cdot \frac{11}{12}$

Answer

ⓐ $24$

ⓑ $\frac{44 x}{3}$

Try It 4.57 §4.2 source ↗

Find the reciprocal:

ⓐ $\frac{5}{7}$

ⓑ $- \frac{1}{8}$

ⓓ $14$

Answer

ⓐ $\frac{7}{5}$

ⓑ $−8$

ⓓ $\frac{1}{14}$

Try It 4.58 §4.2 source ↗

Find the reciprocal:

ⓐ $\frac{3}{7}$

ⓑ $- \frac{1}{12}$

ⓓ $21$

Answer

ⓐ $\frac{7}{3}$

ⓑ $- 12$

ⓓ $\frac{1}{21}$

Try It 4.59 §4.2 source ↗

Fill in the chart for each number given:

Number	Opposite	Absolute Value	Reciprocal
$- \frac{5}{8}$
$\frac{1}{4}$
$\frac{8}{3}$
$−8$

Answer

The image is of a table with four columns and 5 rows. The top header row reads: number, opposite, absolute value, and reciprocal. The rest of the cells contain answers to the problem.

Try It 4.60 §4.2 source ↗

Fill in the chart for each number given:

Number	Opposite	Absolute Value	Reciprocal
$- \frac{4}{7}$
$\frac{1}{8}$
$\frac{9}{4}$
$−1$

Answer

$This table demonstrates how to find the opposite, absolute value, and reciprocal for various numbers, including fractions and integers, as fundamental mathematical concepts.$

Try It 4.61 §4.2 source ↗

Model:

\frac{1}{3} \div \frac{1}{6} .

Answer

Try It 4.62 §4.2 source ↗

Model:

\frac{1}{2} \div \frac{1}{4} .

Answer

Try It 4.63 §4.2 source ↗

Model:

2 \div \frac{1}{3}

Answer

Try It 4.64 §4.2 source ↗

Model:

3 \div \frac{1}{2}

Answer

Try It 4.65 §4.2 source ↗

Divide, and write the answer in simplified form:

\frac{3}{7} \div (- \frac{2}{3}) .

Answer

- \frac{9}{14}

Try It 4.66 §4.2 source ↗

Divide, and write the answer in simplified form:

\frac{2}{3} \div (- \frac{7}{5}) .

Answer

- \frac{10}{21}

Try It 4.67 §4.2 source ↗

Divide, and write the answer in simplified form:

\frac{3}{5} \div \frac{p}{7} .

Answer

\frac{21}{5 p}

Try It 4.68 §4.2 source ↗

Divide, and write the answer in simplified form:

\frac{5}{8} \div \frac{q}{3} .

Answer

\frac{15}{8 q}

Try It 4.69 §4.2 source ↗

Divide, and write the answer in simplified form:

- \frac{2}{3} \div (- \frac{5}{6}) .

Answer

\frac{4}{5}

Try It 4.70 §4.2 source ↗

Divide, and write the answer in simplified form:

- \frac{5}{6} \div (- \frac{2}{3}) .

Answer

\frac{5}{4}

Try It 4.71 §4.2 source ↗

Divide, and write the answer in simplified form:

\frac{7}{27} \div \frac{35}{36} .

Answer

\frac{4}{15}

Try It 4.72 §4.2 source ↗

Divide, and write the answer in simplified form:

\frac{5}{14} \div \frac{15}{28} .

Answer

\frac{2}{3}

4.2-77 §4.2 source ↗

Simplify Fractions

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

$\frac{7}{21}$

Answer

\frac{1}{3}

4.2-79 §4.2 source ↗

Simplify Fractions

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

$\frac{15}{20}$

Answer

\frac{3}{4}

4.2-81 §4.2 source ↗

Simplify Fractions

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

$- \frac{40}{88}$

Answer

- \frac{5}{11}

4.2-83 §4.2 source ↗

Simplify Fractions

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

$- \frac{108}{63}$

Answer

- \frac{12}{7}

4.2-85 §4.2 source ↗

Simplify Fractions

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

$\frac{120}{252}$

Answer

\frac{10}{21}

4.2-87 §4.2 source ↗

Simplify Fractions

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

$- \frac{168}{192}$

Answer

- \frac{7}{8}

4.2-89 §4.2 source ↗

Simplify Fractions

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

$\frac{11 x}{11 y}$

Answer

\frac{x}{y}

4.2-91 §4.2 source ↗

Simplify Fractions

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

$- \frac{3 x}{12 y}$

Answer

- \frac{x}{4 y}

4.2-93 §4.2 source ↗

Simplify Fractions

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

$\frac{14 x^{2}}{21 y}$

Answer

\frac{2 x^{2}}{3 y}

4.2-95 §4.2 source ↗

Multiply Fractions

In the following exercises, use a diagram to model.

$\frac{1}{2} \cdot \frac{2}{3}$

Answer

$\frac{1}{3}$

$The image shows green blocks to demonstrate the fraction problem given.$

4.2-97 §4.2 source ↗

Multiply Fractions

In the following exercises, use a diagram to model.

$\frac{1}{3} \cdot \frac{5}{6}$

Answer

$\frac{5}{18}$

$The image shows green blocks to demonstrate the fraction problem given.$

4.2-99 §4.2 source ↗

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

$\frac{2}{5} \cdot \frac{1}{3}$

Answer

\frac{2}{15}

4.2-101 §4.2 source ↗

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

$\frac{3}{4} \cdot \frac{9}{10}$

Answer

\frac{27}{40}

4.2-103 §4.2 source ↗

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

$- \frac{2}{3} (- \frac{3}{8})$

Answer

\frac{1}{4}

4.2-105 §4.2 source ↗

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

$- \frac{5}{9} \cdot \frac{3}{10}$

Answer

- \frac{1}{6}

4.2-107 §4.2 source ↗

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

$\frac{7}{12} (- \frac{8}{21})$

Answer

- \frac{2}{9}

4.2-109 §4.2 source ↗

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

$(- \frac{14}{15}) (\frac{9}{20})$

Answer

- \frac{21}{50}

4.2-111 §4.2 source ↗

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

$(- \frac{63}{84}) (- \frac{44}{90})$

Answer

\frac{11}{30}

4.2-113 §4.2 source ↗

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

$4 \cdot \frac{5}{11}$

Answer

\frac{20}{11}

4.2-115 §4.2 source ↗

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

$\frac{3}{7} \cdot 21 n$

Answer

4.2-117 §4.2 source ↗

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

$−28 p (- \frac{1}{4})$

Answer

4.2-119 §4.2 source ↗

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

$−8 (\frac{17}{4})$

Answer

−34

4.2-121 §4.2 source ↗

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

$−1 (- \frac{3}{8})$

Answer

\frac{3}{8}

4.2-123 §4.2 source ↗

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

${(\frac{2}{3})}^{3}$

Answer

\frac{8}{27}

4.2-125 §4.2 source ↗

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

${(\frac{6}{5})}^{4}$

Answer

\frac{1296}{625}

4.2-127 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

$\frac{3}{4}$

Answer

\frac{4}{3}

4.2-129 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

$- \frac{5}{17}$

Answer

- \frac{17}{5}

4.2-131 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

$\frac{11}{8}$

Answer

\frac{8}{11}

4.2-133 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

$−19$

Answer

- \frac{1}{19}

4.2-135 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

$1$

Answer

4.2-136 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

Fill in the chart.

	Opposite	Absolute Value	Reciprocal
$- \frac{7}{11}$
$\frac{4}{5}$
$\frac{10}{7}$
$−8$

Answer

$This table demonstrates how to find the opposite, absolute value, and reciprocal for various numbers, including fractions and integers, as fundamental mathematical concepts.$

4.2-137 §4.2 source ↗

Find Reciprocals

In the following exercises, find the reciprocal.

Fill in the chart.

	Opposite	Absolute Value	Reciprocal
$- \frac{3}{13}$
$\frac{9}{14}$
$\frac{15}{7}$
$−9$

Answer

4.2-139 §4.2 source ↗

Divide Fractions

In the following exercises, model each fraction division.

$\frac{1}{2} \div \frac{1}{8}$

Answer

$The image shows blocks to demonstrate the fraction division problem given.$

4.2-141 §4.2 source ↗

Divide Fractions

In the following exercises, model each fraction division.

$3 \div \frac{1}{4}$

Answer

$The image shows blocks to demonstrate the fraction division problem given.$

4.2-143 §4.2 source ↗

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

$\frac{1}{2} \div \frac{1}{8}$

Answer

4.2-145 §4.2 source ↗

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

$\frac{4}{5} \div \frac{3}{4}$

Answer

\frac{16}{15}

4.2-147 §4.2 source ↗

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

$- \frac{3}{4} \div \frac{3}{5}$

Answer

- \frac{5}{4}

4.2-149 §4.2 source ↗

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

$- \frac{5}{6} \div (- \frac{5}{6})$

Answer

4.2-151 §4.2 source ↗

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

$\frac{2}{5} \div \frac{y}{9}$

Answer

\frac{18}{5 y}

4.2-153 §4.2 source ↗

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

$\frac{5}{6} \div \frac{c}{15}$

Answer

\frac{25}{2 c}

4.2-155 §4.2 source ↗

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

$\frac{7}{18} \div (- \frac{14}{27})$

Answer

- \frac{3}{4}

4.2-157 §4.2 source ↗

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

$\frac{5 q}{12} \div \frac{15 q}{8}$

Answer

\frac{2}{9}

4.2-159 §4.2 source ↗

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

$\frac{12 r}{25} \div \frac{18 s}{35}$

Answer

\frac{14 r}{15 s}

4.2-161 §4.2 source ↗

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

$−3 \div \frac{1}{4}$

Answer

−12

4.2-163 §4.2 source ↗

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

$\frac{2}{5} \div (−10)$

Answer

- \frac{1}{25}

4.2-165 §4.2 source ↗

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

$−15 \div (- \frac{5}{3})$

Answer

4.2-167 §4.2 source ↗

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

$\frac{11}{2} \div \frac{7}{8} \cdot \frac{2}{11}$

Answer

\frac{8}{7}

4.2-169 §4.2 source ↗

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

Baking Nina is making $4$ pans of fudge to serve after a music recital. For each pan, she needs $\frac{2}{3}$ cup of condensed milk.

ⓐ How much condensed milk will Nina need? Show your calculation. Write your result as an improper fraction and as a mixed number.

ⓑ Measuring cups usually come in sets of $\frac{1}{8},$ $\frac{1}{4},$ $\frac{1}{3},$ $\frac{1}{2},$ and $1$ cup. Draw a diagram to show two different ways that Nina could measure the condensed milk she needs.

Answer

ⓐ $4 \cdot \frac{2}{3}$ cups $= \frac{8}{3}$ cups $= 2 \frac{2}{3}$ cups

ⓑ Answers will vary.

4.2-171 §4.2 source ↗

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

Portions Kristen has $\frac{3}{4}$ yards of ribbon. She wants to cut it into equal parts to make hair ribbons for her daughter’s $6$ dolls. How long will each doll’s hair ribbon be?

Answer

\frac{1}{8}

yard

4.2-173 §4.2 source ↗

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

Explain how you find the reciprocal of a negative fraction.

Answer

Answers will vary.

4.2-175 §4.2 source ↗

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

Give an example from everyday life that demonstrates how $\frac{1}{2} \cdot \frac{2}{3}$ is $\frac{1}{3} .$

Answer

Answers will vary.

Be Prepared 4.6 §4.3 source ↗

Divide and reduce, if possible:

(4 + 5) \div (10 - 7) .

If you missed this problem, review Example 3.21.

Example 3.21 in the book ↗

Answer

3

Be Prepared 4.7 §4.3 source ↗

Multiply and write the answer in simplified form:

\frac{1}{8} \cdot \frac{2}{3}

.
If you missed this problem, review Example 4.25.

Example 4.25 in the book ↗

Answer

\frac{1}{12}

Be Prepared 4.8 §4.3 source ↗

Convert

2 \frac{3}{5}

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

Example 4.11 in the book ↗

Answer

\frac{13}{5}

Try It 4.73 §4.3 source ↗

Multiply, and write your answer in simplified form:

5 \frac{2}{3} \cdot \frac{6}{17} .

Answer

Try It 4.74 §4.3 source ↗

Multiply, and write your answer in simplified form:

\frac{3}{7} \cdot 5 \frac{1}{4} .

Answer

\frac{9}{4}

Try It 4.75 §4.3 source ↗

Multiply, and write your answer in simplified form.

5 \frac{5}{7} (- 2 \frac{5}{8}) .

Answer

−15

Try It 4.76 §4.3 source ↗

Multiply, and write your answer in simplified form.

−3 \frac{2}{5} \cdot 4 \frac{1}{6} .

Answer

- \frac{85}{6}

Try It 4.77 §4.3 source ↗

Divide, and write your answer in simplified form:

4 \frac{3}{8} \div 7 .

Answer

\frac{5}{8}

Try It 4.78 §4.3 source ↗

Divide, and write your answer in simplified form:

2 \frac{5}{8} \div 3 .

Answer

\frac{7}{8}

Try It 4.79 §4.3 source ↗

Divide, and write your answer in simplified form:

2 \frac{2}{3} \div 1 \frac{1}{3} .

Answer

Try It 4.80 §4.3 source ↗

Divide, and write your answer in simplified form:

3 \frac{3}{4} \div 1 \frac{1}{2} .

Answer

\frac{5}{2}

Try It 4.81 §4.3 source ↗

Translate the phrase into an algebraic expression: the quotient of

9 s

and

14 .

Answer

\frac{9 s}{14}

Try It 4.82 §4.3 source ↗

Translate the phrase into an algebraic expression: the quotient of

5 y

and

6 .

Answer

\frac{5 y}{6}

Try It 4.83 §4.3 source ↗

Translate the phrase into an algebraic expression: the quotient of the difference of

a

and

b,

and

c d .

Answer

\frac{a - b}{c d}

Try It 4.84 §4.3 source ↗

Translate the phrase into an algebraic expression: the quotient of the sum of

p

and

q,

and

r .

Answer

\frac{p + q}{r}

Try It 4.85 §4.3 source ↗

Simplify:

\frac{\frac{2}{3}}{\frac{5}{6}} .

Answer

\frac{4}{5}

Try It 4.86 §4.3 source ↗

Simplify:

\frac{\frac{3}{7}}{\frac{6}{11}} .

Answer

\frac{11}{14}

Try It 4.87 §4.3 source ↗

Simplify:

\frac{- \frac{8}{7}}{4} .

Answer

- \frac{2}{7}

Try It 4.88 §4.3 source ↗

Simplify:

- \frac{3}{\frac{9}{10}} .

Answer

- \frac{10}{3}

Try It 4.89 §4.3 source ↗

Simplify:

\frac{\frac{a}{8}}{\frac{a b}{6}} .

Answer

\frac{3}{4 b}

Try It 4.90 §4.3 source ↗

Simplify:

\frac{\frac{p}{2}}{\frac{p q}{8}} .

Answer

\frac{4}{q}

Try It 4.91 §4.3 source ↗

Simplify:

\frac{\frac{5}{7}}{1 \frac{2}{5}} .

Answer

\frac{25}{49} .

Try It 4.92 §4.3 source ↗

Simplify:

\frac{\frac{8}{5}}{3 \frac{1}{5}} .

Answer

\frac{1}{2}

Try It 4.93 §4.3 source ↗

Which of the following fractions are equivalent to $\frac{−3}{5} ?$

$\frac{−3}{−5},$ $\frac{3}{5},$ $-$ $\frac{3}{5},$ $\frac{3}{−5}$

Answer

- \frac{3}{5}, \frac{3}{−5}

Try It 4.94 §4.3 source ↗

Which of the following fractions are equivalent to $- \frac{2}{7} ?$

$\frac{−2}{−7},$ $\frac{−2}{7},$ $\frac{2}{7},$ $\frac{2}{−7}$

Answer

\frac{−2}{7}, \frac{2}{−7}

Try It 4.95 §4.3 source ↗

Simplify:

\frac{4 + 6}{11 - 2} .

Answer

\frac{10}{9}

Try It 4.96 §4.3 source ↗

Simplify:

\frac{3 + 5}{18 - 2} .

Answer

\frac{1}{2}

Try It 4.97 §4.3 source ↗

Simplify:

\frac{6 - 3 (5)}{3^{2} + 3} .

Answer

\frac{−3}{4}

Try It 4.98 §4.3 source ↗

Simplify:

\frac{4 - 4 (6)}{3^{3} + 3} .

Answer

- \frac{2}{3}

Try It 4.99 §4.3 source ↗

Simplify:

\frac{{(11 - 7)}^{2}}{11^{2} - 7^{2}} .

Answer

\frac{2}{9}

Try It 4.100 §4.3 source ↗

Simplify:

\frac{{(6 + 2)}^{2}}{6^{2} + 2^{2}} .

Answer

\frac{8}{5}

Try It 4.101 §4.3 source ↗

Simplify:

\frac{8 (−2) + 4 (−3)}{−5 (2) + 3} .

Answer

Try It 4.102 §4.3 source ↗

Simplify:

\frac{7 (−1) + 9 (−3)}{−5 (3) −2} .

Answer

4.3-177 §4.3 source ↗

Multiply and Divide Mixed Numbers

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

$2 \frac{4}{9} \cdot \frac{6}{7}$

Answer

\frac{44}{21}

4.3-179 §4.3 source ↗

Multiply and Divide Mixed Numbers

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

$\frac{25}{36} \cdot 6 \frac{3}{10}$

Answer

\frac{35}{8}

4.3-181 §4.3 source ↗

Multiply and Divide Mixed Numbers

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

$2 \frac{2}{5} (−2 \frac{2}{9})$

Answer

- \frac{16}{3}

4.3-183 §4.3 source ↗

Multiply and Divide Mixed Numbers

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

$−1 \frac{7}{20} \cdot 2 \frac{11}{12}$

Answer

- \frac{63}{16}

4.3-185 §4.3 source ↗

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

$13 \frac{1}{2} \div 9$

Answer

\frac{3}{2}

4.3-187 §4.3 source ↗

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

$−7 \div 5 \frac{1}{4}$

Answer

- \frac{4}{3}

4.3-189 §4.3 source ↗

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

$2 \frac{1}{5} \div 1 \frac{1}{10}$

Answer

4.3-191 §4.3 source ↗

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

$−18 \frac{3}{4} \div (−3 \frac{3}{4})$

Answer

4.3-193 §4.3 source ↗

Translate Phrases to Expressions with Fractions

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

the quotient of $7 v$ and $13$

Answer

\frac{7 v}{13}

4.3-195 §4.3 source ↗

Translate Phrases to Expressions with Fractions

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

the quotient of $a$ and $b$

Answer

\frac{a}{b}

4.3-197 §4.3 source ↗

Translate Phrases to Expressions with Fractions

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

the quotient of $A$ and the difference of $3$ and $B$

Answer

\frac{A}{3 - B}

4.3-199 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{\frac{4}{5}}{\frac{8}{15}}$

Answer

\frac{3}{2}

4.3-201 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{- \frac{9}{16}}{\frac{33}{40}}$

Answer

- \frac{15}{22}

4.3-203 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{- \frac{9}{10}}{3}$

Answer

- \frac{3}{10}

4.3-205 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{\frac{5}{3}}{10}$

Answer

\frac{1}{6}

4.3-207 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{\frac{r}{5}}{\frac{s}{3}}$

Answer

\frac{3 r}{5 s}

4.3-209 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{- \frac{3}{8}}{- \frac{y}{12}}$

Answer

\frac{9}{2 y}

4.3-211 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{4 \frac{2}{3}}{\frac{1}{6}}$

Answer

4.3-213 §4.3 source ↗

Simplify Complex Fractions

In the following exercises, simplify the complex fraction.

$\frac{\frac{3}{8}}{−6 \frac{3}{4}}$

Answer

- \frac{1}{18}

4.3-215 §4.3 source ↗

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to $\frac{−4}{9} ?$
$\frac{−4}{−9},$ $\frac{−4}{9},$ $\frac{4}{9},$ $-$ $\frac{4}{9}$

Answer

\frac{−4}{9}, - \frac{4}{9}

4.3-217 §4.3 source ↗

Simplify Expressions with a Fraction Bar

In the following exercises, identify the equivalent fractions.

Which of the following fractions are equivalent to $- \frac{13}{6} ?$
$\frac{13}{6},$ $\frac{13}{−6},$ $\frac{−13}{−6},$ $\frac{−13}{6}$

Answer

\frac{13}{−6}, \frac{−13}{6}

4.3-219 §4.3 source ↗

In the following exercises, simplify.

$\frac{9 + 3}{7}$

Answer

\frac{12}{7}

4.3-221 §4.3 source ↗

In the following exercises, simplify.

$\frac{19 - 4}{6}$

Answer

\frac{5}{2}

4.3-223 §4.3 source ↗

In the following exercises, simplify.

$\frac{46}{4 + 4}$

Answer

\frac{23}{4}

4.3-225 §4.3 source ↗

In the following exercises, simplify.

$\frac{−6 + 3}{17 - 8}$

Answer

- \frac{1}{3}

4.3-227 §4.3 source ↗

In the following exercises, simplify.

$\frac{15 + 9}{18 + 12}$

Answer

\frac{4}{5}

4.3-229 §4.3 source ↗

In the following exercises, simplify.

$\frac{3 \cdot 4}{−24}$

Answer

- \frac{1}{2}

4.3-231 §4.3 source ↗

In the following exercises, simplify.

$\frac{6 \cdot 6}{9 \cdot 2}$

Answer

4.3-233 §4.3 source ↗

In the following exercises, simplify.

$\frac{7^{2} + 1}{60}$

Answer

\frac{5}{6}

4.3-235 §4.3 source ↗

In the following exercises, simplify.

$\frac{9 \cdot 6 - 4 \cdot 7}{22 + 3}$

Answer

\frac{26}{25}

4.3-237 §4.3 source ↗

In the following exercises, simplify.

$\frac{12 \cdot 9 - 3^{2}}{3 \cdot 18}$

Answer

\frac{11}{6}

4.3-239 §4.3 source ↗

In the following exercises, simplify.

$\frac{8 \cdot 9 - 7 \cdot 6}{5 \cdot 6 - 9 \cdot 2}$

Answer

\frac{5}{2}

4.3-241 §4.3 source ↗

In the following exercises, simplify.

$\frac{6^{2} - 4^{2}}{4 - 6}$

Answer

−10

4.3-243 §4.3 source ↗

In the following exercises, simplify.

$\frac{7 + 3 (5)}{−2 - 3^{2}}$

Answer

−2

4.3-245 §4.3 source ↗

In the following exercises, simplify.

$\frac{9 \cdot 7 - 3 (12 - 8)}{8 \cdot 7 - 6 \cdot 6}$

Answer

\frac{51}{20}

4.3-247 §4.3 source ↗

In the following exercises, simplify.

$\frac{8 (9 - 2) −4 (14 - 9)}{7 (8 - 3) −3 (16 - 9)}$

Answer

\frac{18}{7}

4.3-249 §4.3 source ↗

In the following exercises, simplify.

Baking A booth at the county fair sells fudge by the pound. Their award winning “Chocolate Overdose” fudge contains $2 \frac{2}{3}$ cups of chocolate chips per pound.

ⓐ How many cups of chocolate chips are in a half-pound of the fudge?

ⓑ The owners of the booth make the fudge in $10$ -pound batches. How many chocolate chips do they need to make a $10$ -pound batch? Write your results as improper fractions and as a mixed numbers.

Answer

ⓐ $\frac{4}{3} = 1 \frac{1}{3}$ cups

ⓑ $\frac{80}{3} = 26 \frac{2}{3}$ cups

4.3-251 §4.3 source ↗

In the following exercises, simplify.

Explain how to multiply mixed numbers.

Answer

Answers will vary.

4.3-253 §4.3 source ↗

In the following exercises, simplify.

Explain why $- \frac{1}{2}, \frac{−1}{2},$ and $\frac{1}{−2}$ are equivalent.

Answer

Answers will vary.

Be Prepared 4.9 §4.4 source ↗

Simplify:

2 x + 9 + 3 x - 4 .

If you missed this problem, review Example 2.22.

Example 2.22 in the book ↗

Answer

5 x + 5

Be Prepared 4.10 §4.4 source ↗

Draw a model of the fraction

\frac{3}{4} .

If you missed this problem, review Example 4.2.

Example 4.2 in the book ↗

Answer

Be Prepared 4.11 §4.4 source ↗

Simplify:

\frac{3 + 2}{6} .

If you missed this problem, review Example 4.48.

Example 4.48 in the book ↗

Answer

\frac{5}{6}

Try It 4.103 §4.4 source ↗

Use a model to find each sum. Show a diagram to illustrate your model.

$\frac{1}{8} + \frac{4}{8}$

Answer

$\frac{5}{8}$

Try It 4.104 §4.4 source ↗

Use a model to find each sum. Show a diagram to illustrate your model.

$\frac{1}{6} + \frac{4}{6}$

Answer

$\frac{5}{6}$

Try It 4.105 §4.4 source ↗

Find each sum:

\frac{3}{6} + \frac{2}{6} .

Answer

\frac{5}{6}

Try It 4.106 §4.4 source ↗

Find each sum:

\frac{3}{10} + \frac{7}{10} .

Answer

Try It 4.107 §4.4 source ↗

Find the sum:

\frac{x}{4} + \frac{3}{4} .

Answer

\frac{x + 3}{4}

Try It 4.108 §4.4 source ↗

Find the sum:

\frac{y}{8} + \frac{5}{8} .

Answer

\frac{y + 5}{8}

Try It 4.109 §4.4 source ↗

Find the sum:

- \frac{7}{d} + \frac{8}{d} .

Answer

\frac{1}{d}

Try It 4.110 §4.4 source ↗

Find the sum:

- \frac{6}{m} + \frac{9}{m} .

Answer

\frac{3}{m}

Try It 4.111 §4.4 source ↗

Find the sum:

\frac{3 p}{8} + \frac{6 p}{8} .

Answer

\frac{9 p}{8}

Try It 4.112 §4.4 source ↗

Find the sum:

\frac{2 q}{5} + \frac{7 q}{5} .

Answer

\frac{9 q}{5}

Try It 4.113 §4.4 source ↗

Find each sum:

-

\frac{4}{15} +

(- \frac{6}{15}) .

Answer

- \frac{2}{3}

Try It 4.114 §4.4 source ↗

Find each sum:

-

\frac{5}{21} +

(- \frac{9}{21}) .

Answer

- \frac{2}{3}

Try It 4.115 §4.4 source ↗

Use a model to find each difference. Show a diagram to illustrate your model.

$\frac{7}{8} - \frac{4}{8}$

Answer

\frac{3}{8}

, models may differ.

Try It 4.116 §4.4 source ↗

Use a model to find each difference. Show a diagram to illustrate your model.

$\frac{5}{6} - \frac{4}{6}$

Answer

\frac{1}{6}

, models may differ

Try It 4.117 §4.4 source ↗

Find the difference:

\frac{19}{28} - \frac{7}{28} .

Answer

\frac{3}{7}

Try It 4.118 §4.4 source ↗

Find the difference:

\frac{27}{32} - \frac{11}{32} .

Answer

\frac{1}{2}

Try It 4.119 §4.4 source ↗

Find the difference:

\frac{x}{7} - \frac{2}{7} .

Answer

\frac{x - 2}{7}

Try It 4.120 §4.4 source ↗

Find the difference:

\frac{y}{14} - \frac{13}{14} .

Answer

\frac{y - 13}{14}

Try It 4.121 §4.4 source ↗

Find the difference:

- \frac{9}{x} - \frac{7}{x} .

Answer

- \frac{16}{x}

Try It 4.122 §4.4 source ↗

Find the difference:

- \frac{17}{a} - \frac{5}{a} .

Answer

- \frac{22}{a}

Try It 4.123 §4.4 source ↗

Simplify:

\frac{2}{5} +

(- \frac{4}{5}) -

\frac{3}{5} .

Answer

−1

Try It 4.124 §4.4 source ↗

Simplify:

\frac{5}{9} +

(- \frac{4}{9}) -

\frac{7}{9} .

Answer

- \frac{2}{3}

4.4-255 §4.4 source ↗

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

$\frac{3}{10} + \frac{4}{10}$

Answer

$\frac{7}{10}$

4.4-257 §4.4 source ↗

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

$\frac{3}{8} + \frac{3}{8}$

Answer

$\frac{3}{4}$

4.4-259 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{2}{9} + \frac{5}{9}$

Answer

\frac{7}{9}

4.4-261 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{9}{15} + \frac{7}{15}$

Answer

\frac{16}{15}

4.4-263 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{y}{3} + \frac{2}{3}$

Answer

\frac{y + 2}{3}

4.4-265 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{8}{q} + \frac{6}{q}$

Answer

\frac{14}{q}

4.4-267 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{5 a}{7} + \frac{4 a}{7}$

Answer

\frac{9 a}{7}

4.4-269 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{−11 x}{5} + \frac{7 x}{5}$

Answer

\frac{−4 x}{5}

4.4-271 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$- \frac{1}{8} + (- \frac{5}{8})$

Answer

- \frac{3}{4}

4.4-273 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$- \frac{5}{16} + (- \frac{9}{16})$

Answer

- \frac{7}{8}

4.4-275 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$- \frac{9}{19} + \frac{17}{19}$

Answer

\frac{8}{19}

4.4-277 §4.4 source ↗

Add Fractions with a Common Denominator

In the following exercises, find each sum.

$\frac{5}{12} +$ $(- \frac{7}{12}) +$ $(- \frac{11}{12})$

Answer

- \frac{13}{12}

4.4-279 §4.4 source ↗

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

$\frac{5}{6} - \frac{2}{6}$

Answer

$\frac{1}{2}$

4.4-281 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{4}{5} - \frac{3}{5}$

Answer

\frac{1}{5}

4.4-283 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{9}{13} - \frac{4}{13}$

Answer

\frac{5}{13}

4.4-285 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{7}{12} - \frac{5}{12}$

Answer

\frac{1}{6}

4.4-287 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$- \frac{8}{9} - \frac{16}{9}$

Answer

- \frac{8}{3}

4.4-289 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{x}{19} - \frac{8}{19}$

Answer

\frac{x - 8}{19}

4.4-291 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{11 z}{13} - \frac{8}{13}$

Answer

\frac{11 z - 8}{13}

4.4-293 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$- \frac{7}{c} - \frac{7}{c}$

Answer

- \frac{14}{c}

4.4-295 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$- \frac{29}{v} - \frac{26}{v}$

Answer

- \frac{55}{v}

4.4-297 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{12 d}{11} - \frac{9 d}{11}$

Answer

\frac{3 d}{11}

4.4-299 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$\frac{−7 s}{3} - \frac{7 s}{3}$

Answer

- \frac{14 s}{3}

4.4-301 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$- \frac{3}{7} - (- \frac{5}{7})$

Answer

\frac{2}{7}

4.4-303 §4.4 source ↗

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

$- \frac{8}{11} - (- \frac{5}{11})$

Answer

- \frac{3}{11}

4.4-305 §4.4 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

$- \frac{3}{14} \cdot \frac{7}{12}$

Answer

- \frac{1}{8}

4.4-307 §4.4 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

$\frac{6}{11} - \frac{s}{11}$

Answer

\frac{6 - s}{11}

4.4-309 §4.4 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

$- \frac{5}{18} + \frac{1}{18}$

Answer

- \frac{2}{9}

4.4-311 §4.4 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

$\frac{7}{12} \div \frac{9}{28}$

Answer

\frac{49}{27}

4.4-313 §4.4 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

Baking Janet needs $\frac{5}{8}$ of a cup of flour for a recipe she is making. She only has $\frac{3}{8}$ of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?

Answer

\frac{1}{4}

cup

4.4-315 §4.4 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

After a party, Lupe has $\frac{5}{12}$ of a cheese pizza, $\frac{4}{12}$ of a pepperoni pizza, and $\frac{4}{12}$ of a veggie pizza left. Will all the slices fit into $1$ pizza box? Explain your reasoning.

Answer

No, adding up the number of pieces gives

\frac{13}{12}

, which is greater than 1. (Answers may vary.)

Be Prepared 4.12 §4.5 source ↗

Find two fractions equivalent to

\frac{5}{6} .

If you missed this problem, review Example 4.14.

Example 4.14 in the book ↗

Answer

\frac{10}{12}, \frac{15}{18}

Be Prepared 4.13 §4.5 source ↗

Simplify:

\frac{1 + 5 \cdot 3}{2^{2} + 4} .

If you missed this problem, review Example 4.48.

Example 4.48 in the book ↗

Answer

2

Try It 4.125 §4.5 source ↗

Find the least common denominator for the fractions:

\frac{7}{12}

and

\frac{11}{15} .

Answer

Try It 4.126 §4.5 source ↗

Find the least common denominator for the fractions:

\frac{13}{15}

and

\frac{17}{5} .

Answer

Try It 4.127 §4.5 source ↗

Find the least common denominator for the fractions:

\frac{13}{24}

and

\frac{17}{32} .

Answer

Try It 4.128 §4.5 source ↗

Find the least common denominator for the fractions:

\frac{9}{28}

and

\frac{21}{32} .

Answer

224

Try It 4.129 §4.5 source ↗

Change to equivalent fractions with the LCD:

$\frac{3}{4}$ and $\frac{5}{6},$ LCD $= 12$

Answer

\frac{9}{12}, \frac{10}{12}

Try It 4.130 §4.5 source ↗

Change to equivalent fractions with the LCD:

$- \frac{7}{12}$ and $\frac{11}{15},$ LCD $= 60$

Answer

- \frac{35}{60}, \frac{44}{60}

Try It 4.131 §4.5 source ↗

Change to equivalent fractions with the LCD:

$\frac{13}{24}$ and $\frac{17}{32},$ LCD $96$

Answer

\frac{52}{96}, \frac{51}{96}

Try It 4.132 §4.5 source ↗

Change to equivalent fractions with the LCD:

$\frac{9}{28}$ and $\frac{27}{32},$ LCD $224$

Answer

\frac{72}{224},

\frac{189}{224}

Try It 4.133 §4.5 source ↗

Add:

\frac{1}{4} + \frac{1}{3} .

Answer

\frac{7}{12}

Try It 4.134 §4.5 source ↗

Add:

\frac{1}{2} + \frac{1}{5} .

Answer

\frac{7}{10}

Try It 4.135 §4.5 source ↗

Simplify:

\frac{1}{2} - (- \frac{1}{8}) .

Answer

\frac{5}{8}

Try It 4.136 §4.5 source ↗

Simplify:

\frac{1}{3} - (- \frac{1}{6}) .

Answer

\frac{1}{2}

Try It 4.137 §4.5 source ↗

Add:

\frac{7}{12} + \frac{11}{15} .

Answer

\frac{79}{60}

Try It 4.138 §4.5 source ↗

Add:

\frac{13}{15} + \frac{17}{20} .

Answer

\frac{103}{60}

Try It 4.139 §4.5 source ↗

Subtract:

\frac{13}{24} - \frac{17}{32} .

Answer

\frac{1}{96}

Try It 4.140 §4.5 source ↗

Subtract:

\frac{21}{32} - \frac{9}{28} .

Answer

\frac{75}{224}

Try It 4.141 §4.5 source ↗

Add:

- \frac{13}{42} + \frac{17}{35} .

Answer

\frac{37}{210}

Try It 4.142 §4.5 source ↗

Add:

- \frac{19}{24} + \frac{17}{32} .

Answer

- \frac{25}{96}

Try It 4.143 §4.5 source ↗

Add:

\frac{y}{6} + \frac{7}{9} .

Answer

\frac{3 y + 14}{18}

Try It 4.144 §4.5 source ↗

Add:

\frac{x}{6} + \frac{7}{15} .

Answer

\frac{5 x + 14}{30}

Try It 4.145 §4.5 source ↗

Simplify each expression:

ⓐ $- \frac{3}{4} - \frac{1}{6}$

ⓑ $- \frac{3}{4} \cdot \frac{1}{6}$

Answer

ⓐ $- \frac{11}{12}$

ⓑ $- \frac{1}{8}$

Try It 4.146 §4.5 source ↗

Simplify each expression:

ⓐ $\frac{5}{6} \div (- \frac{1}{4})$

ⓑ $\frac{5}{6} - (- \frac{1}{4})$

Answer

ⓐ $- \frac{10}{3}$

ⓑ $\frac{13}{12}$

Try It 4.147 §4.5 source ↗

Simplify:

ⓐ $\frac{(27 a - 32)}{36}$

ⓑ $\frac{2 a}{3}$

Answer

ⓐ $\frac{(27 a - 32)}{36}$

ⓑ $\frac{2 a}{3}$

Try It 4.148 §4.5 source ↗

Simplify:

ⓐ $\frac{(24 k + 25)}{30}$

ⓑ $\frac{24 k}{5}$

Answer

ⓐ $\frac{(24 k + 25)}{30}$

ⓑ $\frac{24 k}{5}$

Try It 4.149 §4.5 source ↗

Simplify:

\frac{{(\frac{1}{3})}^{2}}{2^{3} + 2}

Answer

\frac{1}{90}

Try It 4.150 §4.5 source ↗

Simplify:

\frac{1 + 4^{2}}{{(\frac{1}{4})}^{2}}

Answer

272

Try It 4.151 §4.5 source ↗

Simplify:

\frac{\frac{1}{3} + \frac{1}{2}}{\frac{3}{4} - \frac{1}{3}}

Answer

Try It 4.152 §4.5 source ↗

Simplify:

\frac{\frac{2}{3} - \frac{1}{2}}{\frac{1}{4} + \frac{1}{3}}

Answer

\frac{2}{7}

Try It 4.153 §4.5 source ↗

Evaluate: $x + \frac{3}{4}$ when

ⓐ $x = - \frac{7}{4}$

ⓑ $x = - \frac{5}{4}$

Answer

ⓐ $−1$

ⓑ $- \frac{1}{2}$

Try It 4.154 §4.5 source ↗

Evaluate: $y + \frac{1}{2}$ when

ⓐ $y = \frac{2}{3}$

ⓑ $y = - \frac{3}{4}$

Answer

ⓐ $\frac{7}{6}$

ⓑ $- \frac{1}{4}$

Try It 4.155 §4.5 source ↗

Evaluate:

y - \frac{1}{2}

when

y = - \frac{1}{4} .

Answer

- \frac{3}{4}

Try It 4.156 §4.5 source ↗

Evaluate:

x - \frac{3}{8}

when

x = - \frac{5}{2} .

Answer

- \frac{23}{8}

Try It 4.157 §4.5 source ↗

Evaluate.

3 a b^{2}

when

a = - \frac{2}{3}

and

b = - \frac{1}{2} .

Answer

- \frac{1}{2}

Try It 4.158 §4.5 source ↗

Evaluate.

4 c^{3} d

when

c = - \frac{1}{2}

and

d = - \frac{4}{3} .

Answer

\frac{2}{3}

Try It 4.159 §4.5 source ↗

Evaluate:

\frac{a + b}{c}

when

a = −8, b = −7,

and

c = 6 .

Answer

- \frac{5}{2}

Try It 4.160 §4.5 source ↗

Evaluate:

\frac{x + y}{z}

when

x = 9, y = −18,

and

z = −6 .

Answer

\frac{3}{2}

4.5-317 §4.5 source ↗

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

$\frac{3}{4}$ and $\frac{2}{5}$

Answer

4.5-319 §4.5 source ↗

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

$\frac{9}{16}$ and $\frac{7}{12}$

Answer

4.5-321 §4.5 source ↗

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

$\frac{23}{30}$ and $\frac{5}{48}$

Answer

240

4.5-323 §4.5 source ↗

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

$\frac{18}{35}$ and $\frac{33}{49}$

Answer

245

4.5-325 §4.5 source ↗

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

$\frac{2}{3}, \frac{1}{4},$ and $\frac{3}{5}$

Answer

4.5-327 §4.5 source ↗

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

$\frac{1}{4}$ and $\frac{1}{5},$ LCD $= 20$

Answer

\frac{5}{20}, \frac{4}{20}

4.5-329 §4.5 source ↗

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

$\frac{7}{12}$ and $\frac{5}{8},$ LCD $= 24$

Answer

\frac{14}{24}, \frac{15}{24}

4.5-331 §4.5 source ↗

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

$\frac{11}{16}$ and $- \frac{5}{12},$ LCD $= 48$

Answer

\frac{33}{48}, - \frac{20}{48}

4.5-333 §4.5 source ↗

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

$\frac{1}{3}, \frac{3}{4},$ and $\frac{3}{5},$ LCD $= 60$

Answer

\frac{20}{60},

\frac{45}{60},

\frac{36}{60}

4.5-335 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{1}{4} + \frac{1}{5}$

Answer

\frac{9}{20}

4.5-337 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{1}{3} + \frac{1}{8}$

Answer

\frac{11}{24}

4.5-339 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{1}{4} - (- \frac{1}{8})$

Answer

\frac{3}{8}

4.5-341 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{1}{2} - (- \frac{1}{6})$

Answer

\frac{2}{3}

4.5-343 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{3}{4} + \frac{2}{5}$

Answer

\frac{23}{20}

4.5-345 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{5}{12} + \frac{3}{8}$

Answer

\frac{19}{24}

4.5-347 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{7}{16} - \frac{5}{12}$

Answer

\frac{1}{48}

4.5-349 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{5}{8} - \frac{7}{12}$

Answer

\frac{1}{24}

4.5-351 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{5}{6} - \frac{3}{4}$

Answer

\frac{1}{12}

4.5-353 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$- \frac{9}{20} + \frac{17}{30}$

Answer

\frac{7}{60}

4.5-355 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$- \frac{23}{30} + \frac{5}{48}$

Answer

- \frac{53}{80}

4.5-357 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$- \frac{33}{49} - \frac{18}{35}$

Answer

- \frac{291}{245}

4.5-359 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$- \frac{3}{4} - (- \frac{4}{5})$

Answer

\frac{1}{20}

4.5-361 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$- \frac{7}{20} - (- \frac{5}{8})$

Answer

\frac{11}{40}

4.5-363 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$1 + \frac{5}{6}$

Answer

\frac{11}{6}

4.5-365 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$1 - \frac{3}{10}$

Answer

\frac{7}{10}

4.5-367 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{y}{2} + \frac{2}{3}$

Answer

\frac{3 y + 4}{6}

4.5-369 §4.5 source ↗

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

$\frac{x}{5} - \frac{1}{4}$

Answer

\frac{4 x - 5}{20}

4.5-371 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

ⓐ $\frac{2}{3} + \frac{1}{6}$

ⓑ $\frac{2}{3} \div \frac{1}{6}$

Answer

ⓐ $\frac{5}{6}$

ⓑ $4$

4.5-373 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

ⓐ $- \frac{4}{5} - \frac{1}{8}$

ⓑ $- \frac{4}{5} \cdot \frac{1}{8}$

Answer

ⓐ $- \frac{37}{40}$

ⓑ $- \frac{1}{10}$

4.5-375 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

ⓐ $\frac{3 a}{8} \div \frac{7}{12}$

ⓑ $\frac{3 a}{8} - \frac{7}{12}$

Answer

ⓐ $\frac{9 a}{14}$

ⓑ $\frac{9 a - 14}{24}$

4.5-377 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

ⓐ $\frac{4}{15} \cdot (- \frac{5 q}{9})$

ⓑ $\frac{4}{15} + (- \frac{5 q}{9})$

Answer

ⓐ $- \frac{4 q}{27}$

ⓑ $\frac{12 - 25 q}{45}$

4.5-379 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$- \frac{5}{12} \div (- \frac{5}{9})$

Answer

\frac{3}{4}

4.5-381 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$- \frac{1}{8} + \frac{7}{12}$

Answer

\frac{11}{24}

4.5-383 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$\frac{5}{9} - \frac{1}{6}$

Answer

\frac{7}{18}

4.5-385 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$\frac{7}{12} \cdot (- \frac{8}{35})$

Answer

- \frac{2}{15}

4.5-387 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$- \frac{3}{8} - \frac{x}{11}$

Answer

\frac{−33 - 8 x}{88}

4.5-389 §4.5 source ↗

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

$\frac{10 y}{13} \cdot \frac{8}{15 y}$

Answer

\frac{16}{39}

4.5-391 §4.5 source ↗

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{{(\frac{1}{3})}^{2}}{5 + 2^{2}}$

Answer

\frac{1}{81}

4.5-393 §4.5 source ↗

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{3^{3} - 3^{2}}{{(\frac{3}{4})}^{2}}$

Answer

4.5-395 §4.5 source ↗

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{{(\frac{3}{4})}^{2}}{{(\frac{5}{8})}^{2}}$

Answer

\frac{36}{25}

4.5-397 §4.5 source ↗

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{5}{\frac{1}{4} + \frac{1}{3}}$

Answer

\frac{60}{7}

4.5-399 §4.5 source ↗

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{\frac{3}{4} + \frac{1}{2}}{\frac{5}{6} - \frac{2}{3}}$

Answer

\frac{15}{2}

4.5-401 §4.5 source ↗

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

$\frac{\frac{3}{4} - \frac{3}{5}}{\frac{1}{4} + \frac{2}{5}}$

Answer

\frac{3}{13}

4.5-403 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$\frac{1}{3} + \frac{2}{5} \cdot \frac{3}{4}$

Answer

\frac{19}{30}

4.5-405 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$1 - \frac{5}{6} \div \frac{1}{12}$

Answer

−9

4.5-407 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$\frac{2}{3} + \frac{1}{4} + \frac{3}{5}$

Answer

\frac{91}{60}

4.5-409 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$\frac{2}{5} + \frac{5}{8} - \frac{3}{4}$

Answer

\frac{11}{40}

4.5-411 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$8 (\frac{15}{16} - \frac{5}{6})$

Answer

\frac{5}{6}

4.5-413 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$\frac{\frac{1}{6} + \frac{3}{10}}{\frac{14}{30}}$

Answer

4.5-415 §4.5 source ↗

Mixed Practice

In the following exercises, simplify.

$(\frac{3}{4} + \frac{1}{6}) \div (\frac{5}{8} - \frac{1}{3})$

Answer

\frac{22}{7}

4.5-417 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$x + \frac{2}{3}$ when

ⓐ $x = - \frac{1}{6}$

ⓑ $x = - \frac{5}{3}$

Answer

ⓐ $\frac{1}{2}$

ⓑ $−1$

4.5-419 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$x + (- \frac{11}{12})$ when

ⓐ $x = \frac{11}{12}$

ⓑ $x = \frac{3}{4}$

Answer

ⓐ $0$

ⓑ $- \frac{1}{6}$

4.5-421 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$x - \frac{1}{3}$ when

ⓐ $x = \frac{2}{3}$

ⓑ $x = - \frac{2}{3}$

Answer

ⓐ $\frac{1}{3}$

ⓑ $−1$

4.5-423 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$\frac{5}{12} - w$ when

ⓐ $w = \frac{1}{4}$

ⓑ $w = - \frac{1}{4}$

Answer

ⓐ $\frac{1}{6}$

ⓑ $\frac{2}{3}$

4.5-425 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$5 m^{2} n$ when $m = - \frac{2}{5}$ and $n = \frac{1}{3}$

Answer

\frac{4}{15}

4.5-427 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$8 u^{2} v^{3}$ when $u = - \frac{3}{4}$ and $v = - \frac{1}{2}$

Answer

- \frac{9}{16}

4.5-429 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$\frac{m + n}{p}$ when $m =$ $−6,$ $n =$ $−2,$ $p =$ $4$

Answer

−2

4.5-431 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

$\frac{r - s}{r + s}$ when $r = 10, s = −5$

Answer

4.5-433 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs $1 \frac{1}{4}$ cups of sugar for the chocolate chip cookies, and $1 \frac{1}{8}$ cups for the oatmeal cookies How much sugar does she need altogether?

Answer

She needs

2 \frac{3}{8}

cups

4.5-435 §4.5 source ↗

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

Explain how to find the LCD of two fractions.

Answer

Answers will vary.

Be Prepared 4.14 §4.6 source ↗

Draw a model of the fraction

\frac{7}{3} .

If you missed this problem, review Example 4.6.

Example 4.6 in the book ↗

Answer

Be Prepared 4.15 §4.6 source ↗

Change

\frac{11}{4}

to a mixed number.
If you missed this problem, review Example 4.9.

Example 4.9 in the book ↗

Answer

2 \frac{3}{4}

Be Prepared 4.16 §4.6 source ↗

Change

3 \frac{1}{2}

to an improper fraction.
If you missed this problem, review Example 4.11.

Example 4.11 in the book ↗

Answer

\frac{7}{2}

Try It 4.161 §4.6 source ↗

Use a model to add the following. Draw a picture to illustrate your model.

$1 \frac{2}{5} + 3 \frac{3}{5}$

Answer

The figure shows five circles. The last one is divided into five equal sections.

Try It 4.162 §4.6 source ↗

Use a model to add the following. Draw a picture to illustrate your model.

$2 \frac{1}{6} + 2 \frac{5}{6}$

Answer

The figure shows five circles. The last one is divided into six equal sections.

Try It 4.163 §4.6 source ↗

Model, and give the sum as a mixed number. Draw a picture to illustrate your model.

$2 \frac{5}{6} + 1 \frac{5}{6}$

Answer

$4 \frac{2}{3}$

The figure shows five circles. The last two are divided into six equal sections. Two sections of the last circle are white.

Try It 4.164 §4.6 source ↗

Model, and give the sum as a mixed number. Draw a picture to illustrate your model.

$1 \frac{5}{8} + 1 \frac{7}{8}$

Answer

$3 \frac{1}{2}$

The figure shows four circles. The last two are divided into eight equal sections. Four sections of the last circle are white.

Try It 4.165 §4.6 source ↗

Find the sum:

4 \frac{4}{7} + 1 \frac{2}{7} .

Answer

5 \frac{6}{7}

Try It 4.166 §4.6 source ↗

Find the sum:

2 \frac{3}{11} + 5 \frac{6}{11} .

Answer

7 \frac{9}{11}

Try It 4.167 §4.6 source ↗

Find the sum:

8 \frac{7}{8} + 7 \frac{5}{8} .

Answer

16 \frac{1}{2}

Try It 4.168 §4.6 source ↗

Find the sum:

6 \frac{7}{9} + 8 \frac{5}{9} .

Answer

15 \frac{1}{3}

Try It 4.169 §4.6 source ↗

Find the sum by converting the mixed numbers to improper fractions:

$5 \frac{5}{9} + 3 \frac{7}{9} .$

Answer

9 \frac{1}{3}

Try It 4.170 §4.6 source ↗

Find the sum by converting the mixed numbers to improper fractions:

$3 \frac{7}{10} + 2 \frac{9}{10} .$

Answer

6 \frac{3}{5}

Try It 4.171 §4.6 source ↗

Use a model to subtract:

1 - \frac{1}{4} .

Answer

$\frac{3}{4}$

$A series of three circles demonstrating fractions: a fully shaded circle, a circle divided into four shaded quadrants, and a circle with three out of four quadrants shaded.$ $A series of three circles demonstrating fractions: a fully shaded circle, a circle divided into four shaded quadrants, and a circle with three out of four quadrants shaded.$

Try It 4.172 §4.6 source ↗

Use a model to subtract:

1 - \frac{1}{5} .

Answer

$\frac{4}{5}$

Try It 4.173 §4.6 source ↗

Use a model to subtract:

2 - \frac{1}{5} .

Answer

$\frac{9}{5}$

$Six circles illustrating fractions. The first four are solid gray. The fifth circle is divided into five equal gray segments. The sixth shows four out of five segments gray and one white.$ $Six circles illustrating fractions. The first four are solid gray. The fifth circle is divided into five equal gray segments. The sixth shows four out of five segments gray and one white.$

Try It 4.174 §4.6 source ↗

Use a model to subtract:

2 - \frac{1}{3} .

Answer

$\frac{5}{3}$

$A grid of six circles illustrating fractions. The top row shows whole circles. The bottom row progresses from a whole circle to one divided into thirds, ending with two-thirds shaded.$ $A grid of six circles illustrating fractions. The top row shows whole circles. The bottom row progresses from a whole circle to one divided into thirds, ending with two-thirds shaded.$

Try It 4.175 §4.6 source ↗

Use a model to subtract:

2 - 1 \frac{1}{3} .

Answer

$\frac{2}{3}$

$A grid of six circles, illustrating various fractions including whole circles, 3/3 (a whole), and 2/3 of a circle, useful for teaching fundamental concepts of fractions and whole numbers.$ $A grid of six circles, illustrating various fractions including whole circles, 3/3 (a whole), and 2/3 of a circle, useful for teaching fundamental concepts of fractions and whole numbers.$

Try It 4.176 §4.6 source ↗

Use a model to subtract:

2 - 1 \frac{1}{4} .

Answer

$\frac{3}{4}$

The image uses circles to illustrate the subtraction problem.

Try It 4.177 §4.6 source ↗

Use a model to subtract. Draw a picture to illustrate your model.

$1 \frac{1}{3} - \frac{2}{3}$

Answer

$\frac{2}{3}$

A circle divided into three sections, two of which are shaded.

Try It 4.178 §4.6 source ↗

Use a model to subtract. Draw a picture to illustrate your model.

$1 \frac{1}{5} - \frac{4}{5}$

Answer

$\frac{2}{5}$

A circle divided into 5 sections, 2 of which are shaded.

Try It 4.179 §4.6 source ↗

Find the difference:

6 \frac{4}{9} - 3 \frac{7}{9} .

Answer

2 \frac{2}{3}

Try It 4.180 §4.6 source ↗

Find the difference:

4 \frac{4}{7} - 2 \frac{6}{7} .

Answer

1 \frac{5}{7}

Try It 4.181 §4.6 source ↗

Find the difference by converting the mixed numbers to improper fractions:

$6 \frac{4}{9} - 3 \frac{7}{9} .$

Answer

2 \frac{2}{3}

Try It 4.182 §4.6 source ↗

Find the difference by converting the mixed numbers to improper fractions:

$4 \frac{4}{7} - 2 \frac{6}{7} .$

Answer

1 \frac{5}{7}

Try It 4.183 §4.6 source ↗

Add:

1 \frac{5}{6} + 4 \frac{3}{4} .

Answer

6 \frac{7}{12}

Try It 4.184 §4.6 source ↗

Add:

3 \frac{4}{5} + 8 \frac{1}{2} .

Answer

12 \frac{3}{10}

Try It 4.185 §4.6 source ↗

Find the difference:

8 \frac{1}{2} - 3 \frac{4}{5} .

Answer

4 \frac{7}{10}

Try It 4.186 §4.6 source ↗

Find the difference:

4 \frac{3}{4} - 1 \frac{5}{6} .

Answer

2 \frac{11}{12}

Try It 4.187 §4.6 source ↗

Subtract:

1 \frac{3}{4} - 6 \frac{7}{8} .

Answer

- \frac{41}{8}

Try It 4.188 §4.6 source ↗

Subtract:

10 \frac{3}{7} - 22 \frac{4}{9} .

Answer

- \frac{757}{63}

4.6-437 §4.6 source ↗

Model Addition of Mixed Numbers

In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.

$2 \frac{1}{3} + 1 \frac{1}{3}$

Answer

Four circles are shown. The first three are shaded. The last circle is divided into 3 equal parts. 2 parts are shaded.

$3 \frac{2}{3}$

4.6-439 §4.6 source ↗

Model Addition of Mixed Numbers

In the following exercises, use a model to find the sum. Draw a picture to illustrate your model.

$1 \frac{5}{6} + 1 \frac{5}{6}$

Answer

$3 \frac{2}{3}$

4.6-441 §4.6 source ↗

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

$2 \frac{4}{9} + 5 \frac{1}{9}$

Answer

7 \frac{5}{9}

4.6-443 §4.6 source ↗

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

$7 \frac{9}{10} + 3 \frac{1}{10}$

Answer

4.6-445 §4.6 source ↗

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

$9 \frac{2}{3} + 1 \frac{2}{3}$

Answer

11 \frac{1}{3}

4.6-447 §4.6 source ↗

Add Mixed Numbers with a Common Denominator

In the following exercises, add.

$8 \frac{4}{9} + 2 \frac{8}{9}$

Answer

11 \frac{1}{3}

4.6-449 §4.6 source ↗

Model Subtraction of Mixed Numbers

In the following exercises, use a model to find the difference. Draw a picture to illustrate your model.

$1 \frac{1}{8} - \frac{5}{8}$

Answer

$\frac{1}{2}$

4.6-451 §4.6 source ↗

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

$2 \frac{7}{12} - 1 \frac{5}{12}$

Answer

1 \frac{1}{6}

4.6-453 §4.6 source ↗

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

$19 \frac{13}{15} -$ $13 \frac{7}{15}$

Answer

6 \frac{2}{5}

4.6-455 §4.6 source ↗

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

$5 \frac{2}{9} - 3 \frac{4}{9}$

Answer

1 \frac{7}{9}

4.6-457 §4.6 source ↗

Subtract Mixed Numbers with a Common Denominator

In the following exercises, find the difference.

$2 \frac{5}{12} - 1 \frac{7}{12}$

Answer

\frac{5}{6}

4.6-459 §4.6 source ↗

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

$2 \frac{1}{6} + 5 \frac{3}{4}$

Answer

7 \frac{11}{12}

4.6-461 §4.6 source ↗

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

$7 \frac{2}{3} + 8 \frac{1}{2}$

Answer

16 \frac{1}{6}

4.6-463 §4.6 source ↗

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

$6 \frac{4}{5} - 1 \frac{1}{4}$

Answer

5 \frac{11}{20}

4.6-465 §4.6 source ↗

Add and Subtract Mixed Numbers with Different Denominators

In the following exercises, write the sum or difference as a mixed number in simplified form.

$2 \frac{7}{8} - 4 \frac{1}{3}$

Answer

−1 \frac{11}{24}

4.6-467 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$1 \frac{2}{3} \cdot 4 \frac{1}{6}$

Answer

6 \frac{17}{18}

4.6-469 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$\frac{2}{9} + \frac{5}{9}$

Answer

\frac{7}{9}

4.6-471 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$2 \frac{3}{10} \div \frac{1}{10}$

Answer

4.6-473 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$15 \frac{5}{8} - 6 \frac{7}{8}$

Answer

8 \frac{3}{4}

4.6-475 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$\frac{11}{15} - \frac{7}{15}$

Answer

\frac{4}{15}

4.6-477 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$6 - \frac{2}{5}$

Answer

5 \frac{3}{5}

4.6-479 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$\frac{7}{24} \div \frac{14}{3}$

Answer

\frac{1}{16}

4.6-481 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$8 \frac{5}{13} + 4 \frac{9}{13}$

Answer

13 \frac{1}{13}

4.6-483 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$2 \frac{5}{6} + 4 \frac{1}{5}$

Answer

7 \frac{1}{30}

4.6-485 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$\frac{5}{12} \cdot \frac{8}{9}$

Answer

\frac{10}{27}

4.6-487 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$6 \frac{5}{9} - 4 \frac{2}{5}$

Answer

2 \frac{7}{45}

4.6-489 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

$4 \frac{3}{8} - 3 \frac{2}{3}$

Answer

\frac{17}{24}

4.6-491 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

Sewing Pauline has $3 \frac{1}{4}$ yards of fabric to make a jacket. The jacket uses $2 \frac{2}{3}$ yards. How much fabric will she have left after making the jacket?

Answer

\frac{7}{12}

yards

4.6-493 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

Framing a picture Tessa bought a picture frame for her son’s graduation picture. The picture is $8$ inches wide. The picture frame is $2 \frac{5}{8}$ inches wide on each side. How wide will the framed picture be?

Answer

13 \frac{1}{4}

inches

4.6-495 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

Edgar will have to pay $$ 3.75$ in tolls to drive to the city.

ⓐ Explain how he can make change from a $$ 10$ bill before he leaves so that he has the exact amount he needs.

ⓑ How is Edgar’s situation similar to how you subtract $10 - 3 \frac{3}{4} ?$

Answer

Answers will vary.

4.6-497 §4.6 source ↗

Mixed Practice

In the following exercises, perform the indicated operation and write the result as a mixed number in simplified form.

Subtract $3 \frac{7}{8} - 4 \frac{5}{12}$ twice, first by leaving them as mixed numbers and then by rewriting as improper fractions. Which method do you prefer, and why?

Answer

Answers will vary.

Be Prepared 4.17 §4.7 source ↗

Evaluate

x + 4

when

x = −3

If you missed this problem, review Example 3.23.

Example 3.23 in the book ↗

Answer

1

Be Prepared 4.18 §4.7 source ↗

Solve:

2 y - 3 = 9 .

If you missed this problem, review Example 3.61.

Example 3.61 in the book ↗

Answer

y = 6

Be Prepared 4.19 §4.7 source ↗

Solve:

y - 3 = −9

If you missed this problem, review Example 4.28.

Example 4.28 in the book ↗

Answer

- 6

Try It 4.189 §4.7 source ↗

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

$x - \frac{2}{3} = \frac{1}{6}$ :

ⓐ $x = 1$

ⓑ $x = \frac{5}{6}$

Answer

ⓐ no

ⓑ yes

Try It 4.190 §4.7 source ↗

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

$y - \frac{1}{4} = \frac{3}{8}$ :

ⓐ $y = 1$

ⓑ $y = - \frac{5}{8}$

Answer

ⓐ no

ⓑ no

Try It 4.191 §4.7 source ↗

Solve:

y + \frac{11}{12} = \frac{5}{12} .

Answer

- \frac{1}{2}

Try It 4.192 §4.7 source ↗

Solve:

y + \frac{8}{15} = \frac{4}{15} .

Answer

- \frac{4}{15}

Try It 4.193 §4.7 source ↗

Solve:

a - \frac{3}{5} = - \frac{8}{5} .

Answer

−1

Try It 4.194 §4.7 source ↗

Solve:

n - \frac{3}{7} = - \frac{9}{7} .

Answer

- \frac{6}{7}

Try It 4.195 §4.7 source ↗

Solve:

12 u = −76 .

Answer

- \frac{19}{3}

Try It 4.196 §4.7 source ↗

Solve:

8 m = 92 .

Answer

\frac{23}{2}

Try It 4.197 §4.7 source ↗

Solve:

\frac{f}{5} = −25 .

Answer

−125

Try It 4.198 §4.7 source ↗

Solve:

\frac{h}{9} = −27 .

Answer

−243

Try It 4.199 §4.7 source ↗

Solve:

\frac{c}{−7} = −35 .

Answer

245

Try It 4.200 §4.7 source ↗

Solve:

\frac{x}{−11} = −12 .

Answer

132

Try It 4.201 §4.7 source ↗

Solve:

- y = 48 .

Answer

−48

Try It 4.202 §4.7 source ↗

Solve:

- c = −23 .

Answer

Try It 4.203 §4.7 source ↗

Solve:

\frac{2}{5} n = 14 .

Answer

Try It 4.204 §4.7 source ↗

Solve:

\frac{5}{6} y = 15 .

Answer

Try It 4.205 §4.7 source ↗

Solve:

- \frac{4}{7} a = 52 .

Answer

−91

Try It 4.206 §4.7 source ↗

Solve:

- \frac{7}{9} w = 84 .

Answer

−108

Try It 4.207 §4.7 source ↗

Translate and solve:

n

divided by

7

is equal to

−21 .

Answer

\frac{n}{7} =

−21;

n =

−147

Try It 4.208 §4.7 source ↗

Translate and solve:

n

divided by

8

is equal to

−56 .

Answer

\frac{n}{8} =

−56;

n =

−448

Try It 4.209 §4.7 source ↗

Translate and solve: The quotient of

q

and

−8

72 .

Answer

\frac{q}{−8} =

72;

q =

−576

Try It 4.210 §4.7 source ↗

Translate and solve: The quotient of

p

and

−9

81 .

Answer

\frac{p}{−9} =

81;

p =

−729

Try It 4.211 §4.7 source ↗

Translate and solve: Two-fifths of

f

16 .

Answer

\frac{2}{5} f = 16; f = 40

Try It 4.212 §4.7 source ↗

Translate and solve: Three-fourths of

f

21 .

Answer

\frac{3}{4} f = 21; f = 28

Try It 4.213 §4.7 source ↗

Translate and solve. The quotient of

n

and

\frac{2}{3}

\frac{5}{12} .

Answer

\frac{n}{\frac{2}{3}} =

\frac{5}{12};

n =

\frac{5}{18}

Try It 4.214 §4.7 source ↗

Translate and solve The quotient of

c

and

\frac{3}{8}

\frac{4}{9} .

Answer

\frac{c}{\frac{3}{8}} = \frac{4}{9}; c = \frac{1}{6}

Try It 4.215 §4.7 source ↗

Translate and solve: The sum of five-eighths and

x

is one-fourth.

Answer

\frac{5}{8} +

x =

\frac{1}{4};

x =

-

\frac{3}{8}

Try It 4.216 §4.7 source ↗

Translate and solve: The difference of one-and-three-fourths and

x

is five-sixths.

Answer

1 \frac{3}{4} -

x =

\frac{5}{6};

x =

\frac{11}{12}

4.7-499 §4.7 source ↗

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

$y - \frac{1}{3} = \frac{5}{12}$ :

ⓐ $y = 1$

ⓑ $y = \frac{3}{4}$

Answer

ⓐ no

ⓑ yes

4.7-501 §4.7 source ↗

Determine Whether a Fraction is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

$k + \frac{2}{5} = \frac{5}{6}$ :

ⓐ $k = 1$

ⓑ $k = \frac{13}{30}$

Answer

ⓐ no

ⓑ yes

4.7-503 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$m + \frac{3}{8} = \frac{7}{8}$

Answer

m = \frac{1}{2}

4.7-505 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$h + \frac{5}{6} = \frac{1}{6}$

Answer

h = - \frac{2}{3}

4.7-507 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$c - \frac{1}{4} = - \frac{5}{4}$

Answer

c = −1

4.7-509 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$z -$ $(- \frac{5}{12}) =$ $-$ $\frac{7}{12}$

Answer

z = −1

4.7-511 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$p - \frac{3}{10} = \frac{5}{8}$

Answer

p = \frac{37}{40}

4.7-513 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$k + (- \frac{1}{3}) = - \frac{4}{5}$

Answer

k = - \frac{7}{15}

4.7-515 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$7 k = 18$

Answer

k = \frac{18}{7}

4.7-517 §4.7 source ↗

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In the following exercises, solve.

$−9 v = 33$

Answer

v = - \frac{11}{3}

4.7-519 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$\frac{b}{3} = −9$

Answer

b = −27

4.7-521 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$\frac{x}{8} = −32$

Answer

x = −256

4.7-523 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$\frac{q}{−4} = −40$

Answer

q = 160

4.7-525 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$\frac{s}{−15} = −3$

Answer

s = 45

4.7-527 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$- y = 42$

Answer

y = −42

4.7-529 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$- k = - \frac{17}{20}$

Answer

k = \frac{17}{20}

4.7-531 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

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

Answer

p = 100

4.7-533 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$\frac{5}{2} m = −40$

Answer

m = −16

4.7-535 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$- \frac{3}{7} b = 9$

Answer

b = −21

4.7-537 §4.7 source ↗

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

$- \frac{5}{12} v = −15$

Answer

v = 36

4.7-539 §4.7 source ↗

Mixed Practice

In the following exercises, solve.

$8 y = 0$

Answer

y = 0

4.7-541 §4.7 source ↗

Mixed Practice

In the following exercises, solve.

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

Answer

g = \frac{1}{9}

4.7-543 §4.7 source ↗

Mixed Practice

In the following exercises, solve.

$q + \frac{5}{6} = \frac{1}{12}$

Answer

q = - \frac{3}{4}

4.7-545 §4.7 source ↗

Mixed Practice

In the following exercises, solve.

$\frac{1}{4} n = \frac{7}{10}$

Answer

n = \frac{14}{5}

4.7-547 §4.7 source ↗

Mixed Practice

In the following exercises, solve.

$- \frac{2}{3} = y + \frac{3}{8}$

Answer

y = - \frac{25}{24}

4.7-549 §4.7 source ↗

Mixed Practice

In the following exercises, solve.

$\frac{8}{15} = - d$

Answer

d = - \frac{8}{15}

4.7-551 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

$n$ divided by six is $−24 .$

Answer

\frac{n}{6} =

−24;

n =

−144

4.7-553 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

$m$ divided by $−7$ is $−8 .$

Answer

\frac{m}{−7} = −8; m = 56

4.7-555 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

The quotient of $f$ and $−4$ is $−20 .$

Answer

\frac{f}{−4} =

−20;

f =

80

4.7-557 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

The quotient of $g$ and nine is $14 .$

Answer

\frac{g}{9} = 14; g = 126

4.7-559 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

Two-fifths of $q$ is $20 .$

Answer

\frac{2}{5} q = 20; q = 50

4.7-561 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

Four-ninths of $p$ is $−28 .$

Answer

\frac{4}{9} p =

−28;

p =

−63

4.7-563 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

The quotient of $h$ and $2$ is $43 .$

Answer

\frac{h}{2} = 43; h = 86

4.7-565 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

The quotient of $a$ and $\frac{2}{3}$ is $\frac{3}{4} .$

Answer

\frac{a}{\frac{2}{3}} = \frac{3}{4}; a = \frac{1}{2}

4.7-567 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

The sum of three-fourths and $x$ is $\frac{1}{8} .$

Answer

\frac{3}{4} +

x =

\frac{1}{8};

x =

-

\frac{5}{8}

4.7-569 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

The difference of $y$ and one-third is $- \frac{1}{6} .$

Answer

y -

\frac{1}{3} =

-

\frac{1}{6};

y =

\frac{1}{6}

4.7-571 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

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

Answer

30 inches

4.7-573 §4.7 source ↗

Translate Sentences to Equations and Solve

In the following exercises, translate to an algebraic equation and solve.

Richard thinks the solution to the equation $\frac{3}{4} x = 24$ is $16 .$ Explain why Richard is wrong.

Answer

Answers will vary.

4.R-575 §4.R source ↗

In the following exercises, name the fraction of each figure that is shaded.

Answer

\frac{5}{9}

4.R-577 §4.R source ↗

In the following exercises, name the improper fractions. Then write each improper fraction as a mixed number.

Answer

\frac{3}{2} = 1 \frac{1}{2}

4.R-579 §4.R source ↗

In the following exercises, convert the improper fraction to a mixed number.

$\frac{63}{11}$

Answer

5 \frac{8}{11}

4.R-581 §4.R source ↗

In the following exercises, convert the mixed number to an improper fraction.

$9 \frac{4}{5}$

Answer

\frac{49}{5}

4.R-583 §4.R source ↗

In the following exercises, convert the mixed number to an improper fraction.

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

Answer

Answers may vary.

4.R-585 §4.R source ↗

In the following exercises, locate the numbers on a number line.

$\frac{1}{4},$ $-$ $\frac{1}{4},$ $1 \frac{1}{3},$ $−1 \frac{1}{3},$ $\frac{7}{2},$ $-$ $\frac{7}{2}$

Answer

4.R-587 §4.R source ↗

In the following exercises, order each pair of numbers, using $<$ or $>.$

$−2 \frac{1}{2}___−3$

Answer

4.R-589 §4.R source ↗

In the following exercises, simplify.

$- \frac{90}{120}$

Answer

- \frac{3}{4}

4.R-591 §4.R source ↗

In the following exercises, simplify.

$- \frac{8 x}{8 y}$

Answer

- \frac{x}{y}

4.R-593 §4.R source ↗

In the following exercises, multiply.

$- \frac{1}{3} \cdot \frac{12}{7}$

Answer

- \frac{4}{7}

4.R-595 §4.R source ↗

In the following exercises, multiply.

$6 m \cdot \frac{4}{11}$

Answer

\frac{24 m}{11}

4.R-597 §4.R source ↗

In the following exercises, multiply.

$\frac{16}{5} \cdot \frac{15}{8}$

Answer

4.R-599 §4.R source ↗

In the following exercises, find the reciprocal.

$\frac{15}{4}$

Answer

\frac{4}{15}

4.R-601 §4.R source ↗

In the following exercises, find the reciprocal.

$- \frac{1}{4}$

Answer

−4

4.R-603 §4.R source ↗

In the following exercises, divide.

$\frac{2}{3} \div \frac{1}{6}$

Answer

4.R-605 §4.R source ↗

In the following exercises, divide.

$\frac{4}{5} \div 3$

Answer

\frac{4}{15}

4.R-607 §4.R source ↗

In the following exercises, divide.

$\frac{5}{18} \div (- \frac{b}{9})$

Answer

- \frac{5}{2 b}

4.R-609 §4.R source ↗

In the following exercises, perform the indicated operation.

$−5 \frac{7}{12} \cdot 4 \frac{4}{11}$

Answer

- \frac{268}{11}

4.R-611 §4.R source ↗

In the following exercises, perform the indicated operation.

$8 \frac{2}{3} \div 1 \frac{1}{12}$

Answer

4.R-613 §4.R source ↗

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

the quotient of $V$ and the difference of $h$ and $6$

Answer

\frac{V}{h - 6}

4.R-615 §4.R source ↗

In the following exercises, simplify the complex fraction

$\frac{\frac{8}{9}}{−4}$

Answer

- \frac{2}{9}

4.R-617 §4.R source ↗

In the following exercises, simplify the complex fraction

$\frac{−1 \frac{5}{6}}{- \frac{1}{12}}$

Answer

4.R-619 §4.R source ↗

In the following exercises, simplify.

$\frac{8 \cdot 4 - 5^{2}}{3 \cdot 12}$

Answer

\frac{7}{36}

4.R-621 §4.R source ↗

In the following exercises, add.

$\frac{3}{8} + \frac{2}{8}$

Answer

\frac{5}{8}

4.R-623 §4.R source ↗

In the following exercises, add.

$\frac{2}{5} + \frac{1}{5}$

Answer

\frac{3}{5}

4.R-625 §4.R source ↗

In the following exercises, add.

$\frac{x}{10} + \frac{7}{10}$

Answer

\frac{x + 7}{10}

4.R-627 §4.R source ↗

In the following exercises, subtract.

$\frac{11}{12} - \frac{5}{12}$

Answer

\frac{1}{2}

4.R-629 §4.R source ↗

In the following exercises, subtract.

$- \frac{31}{30} - \frac{7}{30}$

Answer

- \frac{19}{15}

4.R-631 §4.R source ↗

In the following exercises, subtract.

$\frac{11}{15} -$ $\frac{5}{15} -$ $(- \frac{2}{15})$

Answer

\frac{8}{15}

4.R-633 §4.R source ↗

In the following exercises, find the least common denominator.

$\frac{1}{3}$ and $\frac{4}{5}$

Answer

4.R-635 §4.R source ↗

In the following exercises, find the least common denominator.

$\frac{3}{4}, \frac{1}{6},$ and $\frac{5}{10}$

Answer

4.R-637 §4.R source ↗

In the following exercises, change to equivalent fractions using the given LCD.

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

Answer

\frac{9}{24}

and

\frac{20}{24}

4.R-639 §4.R source ↗

In the following exercises, change to equivalent fractions using the given LCD.

$\frac{1}{3}, \frac{3}{4}$ and $\frac{4}{5},$ LCD $= 60$

Answer

\frac{20}{60}, \frac{45}{60}

and

\frac{48}{60}

4.R-641 §4.R source ↗

In the following exercises, perform the indicated operations and simplify.

$\frac{11}{12} - \frac{2}{3}$

Answer

\frac{1}{4}

4.R-643 §4.R source ↗

In the following exercises, perform the indicated operations and simplify.

$- \frac{11}{36} - \frac{11}{20}$

Answer

- \frac{77}{90}

4.R-645 §4.R source ↗

In the following exercises, perform the indicated operations and simplify.

$\frac{y}{10} - \frac{1}{3}$

Answer

\frac{3 y - 10}{30}

4.R-647 §4.R source ↗

In the following exercises, perform the indicated operations and simplify.

$\frac{4}{11} \div \frac{2}{7 d}$

Answer

\frac{14 d}{11}

4.R-649 §4.R source ↗

In the following exercises, perform the indicated operations and simplify.

$\frac{{(\frac{2}{3})}^{2}}{{(\frac{5}{8})}^{2}}$

Answer

\frac{256}{225}

4.R-651 §4.R source ↗

In the following exercises, evaluate.

$y - \frac{4}{5}$ when

ⓐ $y = - \frac{4}{5}$

ⓑ $y = \frac{1}{4}$

Answer

ⓐ $- \frac{8}{5}$

ⓑ $- \frac{11}{20}$

4.R-653 §4.R source ↗

In the following exercises, perform the indicated operation.

$4 \frac{1}{3} + 9 \frac{1}{3}$

Answer

13 \frac{2}{3}

4.R-655 §4.R source ↗

In the following exercises, perform the indicated operation.

$5 \frac{8}{11} + 2 \frac{4}{11}$

Answer

8 \frac{1}{11}

4.R-657 §4.R source ↗

In the following exercises, perform the indicated operation.

$9 \frac{13}{20} - 4 \frac{11}{20}$

Answer

5 \frac{1}{10}

4.R-659 §4.R source ↗

In the following exercises, perform the indicated operation.

$2 \frac{11}{12} - 1 \frac{7}{12}$

Answer

1 \frac{1}{3}

4.R-661 §4.R source ↗

In the following exercises, determine whether the each number is a solution of the given equation.

$x - \frac{1}{2} = \frac{1}{6}$ :

ⓐ $x = 1$

ⓑ $x = \frac{2}{3}$

Answer

ⓐ no

ⓑ yes

4.R-663 §4.R source ↗

In the following exercises, solve the equation.

$n + \frac{9}{11} = \frac{4}{11}$

Answer

n = - \frac{5}{11}

4.R-665 §4.R source ↗

In the following exercises, solve the equation.

$h - (- \frac{7}{8}) = - \frac{2}{5}$

Answer

h = - \frac{51}{40}

4.R-667 §4.R source ↗

In the following exercises, solve the equation.

$- z = 23$

Answer

z = −23

4.R-669 §4.R source ↗

In the following exercises, translate and solve.

The difference of $q$ and one-tenth is $\frac{1}{2} .$

Answer

q -

\frac{1}{10} =

\frac{1}{2};

q =

\frac{3}{5}

4.R-671 §4.R source ↗

In the following exercises, translate and solve.

Three-eighths of $y$ is $24 .$

Answer

\frac{3}{8} y = 24; y = 64

4.T-673 §4.T source ↗

Convert the mixed number to an improper fraction.

$3 \frac{2}{7}$

Answer

\frac{23}{7}

4.T-675 §4.T source ↗

In the following exercises, simplify.

$\frac{5}{20}$

Answer

\frac{1}{4}

4.T-677 §4.T source ↗

In the following exercises, simplify.

$\frac{1}{3} \cdot \frac{3}{4}$

Answer

\frac{1}{4}

4.T-679 §4.T source ↗

In the following exercises, simplify.

$−36 u (- \frac{4}{9})$

Answer

16u

4.T-681 §4.T source ↗

In the following exercises, simplify.

$- \frac{5}{6} \div \frac{5}{12}$

Answer

−2

4.T-683 §4.T source ↗

In the following exercises, simplify.

$\frac{9 a}{10} \div \frac{15 a}{8}$

Answer

\frac{12}{25}

4.T-685 §4.T source ↗

In the following exercises, simplify.

$(−15 \frac{5}{6}) \div (−3 \frac{1}{6})$

Answer

4.T-687 §4.T source ↗

In the following exercises, simplify.

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

Answer

\frac{5 p}{2 q}

4.T-689 §4.T source ↗

In the following exercises, simplify.

$\frac{9^{2} - 4^{2}}{9 - 4}$

Answer

4.T-691 §4.T source ↗

In the following exercises, simplify.

$- \frac{3}{13} + (- \frac{4}{13})$

Answer

- \frac{7}{13}

4.T-693 §4.T source ↗

In the following exercises, simplify.

$\frac{2}{5} + (- \frac{7}{5})$

Answer

−1

4.T-695 §4.T source ↗

In the following exercises, simplify.

$- \frac{3}{4} \div \frac{x}{3}$

Answer

- \frac{9}{4 x}

4.T-697 §4.T source ↗

In the following exercises, simplify.

$\frac{\frac{5}{14} + \frac{1}{8}}{\frac{9}{56}}$

Answer

4.T-699 §4.T source ↗

In the following exercises, solve the equation.

$y + \frac{3}{5} = \frac{7}{5}$

Answer

y = \frac{4}{5}

4.T-701 §4.T source ↗

In the following exercises, solve the equation.

$f + (- \frac{2}{3}) = \frac{5}{12}$

Answer

f = \frac{13}{12}

4.T-703 §4.T source ↗

In the following exercises, solve the equation.

$- \frac{2}{3} c = 18$

Answer

c = −27

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Problems and answers are adapted from Prealgebra 2e by Lynn Marecek, MaryAnne Anthony-Smith and Andrea Honeycutt Mathis, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/prealgebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.