4.7 Solve equations with fractions  (Page 5/5)

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The quotient of $f$ and $-3$ is $-18.$

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

$\frac{f}{-4}=-20;f=80$

The quotient of $g$ and twelve is $8.$

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

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

Three-fourths of $q$ is $12.$

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

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

Seven-tenths of $p$ is $-63.$

Four-ninths of $p$ is $-28.$

$\frac{4}{9}p=-28;p=-63$

$m$ divided by $4$ equals negative $6.$

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

$\frac{h}{2}=43$

Three-fourths of $z$ is the same as $15.$

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

$\frac{\phantom{\rule{0.2em}{0ex}}a\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}=\frac{3}{4}$

The sum of five-sixths and $x$ is $\frac{1}{2}.$

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

$\frac{3}{4}+x=\frac{1}{8};x=-\frac{5}{8}$

The difference of $y$ and one-fourth is $-\frac{1}{8}.$

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

$y-\frac{1}{3}=-\frac{1}{6};y=\frac{1}{6}$

Everyday math

Shopping Teresa bought a pair of shoes on sale for $\text{48}.$ The sale price was $\frac{2}{3}$ of the regular price. Find the regular price of the shoes by solving the equation $\frac{2}{3}p=48$

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 $\frac{3}{5}h=18.$

30 inches

Writing exercises

[link] describes three methods to solve the equation $-y=15.$ Which method do you prefer? Why?

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

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Visualize Fractions

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

$\frac{5}{9}$

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

$\frac{3}{2}$

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

$\frac{58}{15}$

$\frac{63}{11}$

$5\frac{8}{11}$

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

$12\frac{1}{4}$

$9\frac{4}{5}$

$\frac{49}{5}$

Find three fractions equivalent to $\frac{2}{5}.$ Show your work, using figures or algebra.

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

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

$\frac{5}{8},\frac{4}{3},3\frac{3}{4},4$

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

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

$-1___-\frac{2}{5}$

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

>

Multiply and Divide Fractions

In the following exercises, simplify.

$-\frac{63}{84}$

$-\frac{90}{120}$

$-\frac{3}{4}$

$-\frac{14a}{14b}$

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

$-\frac{x}{y}$

In the following exercises, multiply.

$\frac{2}{5}·\frac{8}{13}$

$-\frac{1}{3}·\frac{12}{7}$

$-\frac{4}{7}$

$\frac{2}{9}·\left(-\frac{45}{32}\right)$

$6m·\frac{4}{11}$

$\frac{24}{11}m$

$-\frac{1}{4}\left(-32\right)$

$3\frac{1}{5}·1\frac{7}{8}$

6

In the following exercises, find the reciprocal.

$\frac{2}{9}$

$\frac{15}{4}$

$\frac{4}{15}$

$3$

$-\frac{1}{4}$

−4

Fill in the chart.

Opposite Absolute Value Reciprocal
$-\frac{5}{13}$
$\frac{3}{10}$
$\frac{9}{4}$
$-12$

In the following exercises, divide.

$\frac{2}{3}÷\frac{1}{6}$

4

$\left(-\frac{3x}{5}\right)÷\left(-\frac{2y}{3}\right)$

$\frac{4}{5}÷3$

$\frac{4}{15}$

$8÷2\frac{2}{3}$

$8\frac{2}{3}÷1\frac{1}{12}$

8

Multiply and Divide Mixed Numbers and Complex Fractions

In the following exercises, perform the indicated operation.

$3\frac{1}{5}·1\frac{7}{8}$

$-5\frac{7}{12}·4\frac{4}{11}$

$-\frac{268}{11}$

$8÷2\frac{2}{3}$

$8\frac{2}{3}÷1\frac{1}{12}$

8

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

the quotient of $8$ and $y$

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

$\frac{V}{h-6}$

In the following exercises, simplify the complex fraction

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{4}{5}\phantom{\rule{0.2em}{0ex}}}$

$\frac{\frac{8}{9}}{-4}$

$-\frac{2}{9}$

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{n}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\phantom{\rule{0.2em}{0ex}}}$

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

22

In the following exercises, simplify.

$\frac{5+16}{5}$

$\frac{8·4-{5}^{2}}{3·12}$

$\frac{7}{36}$

$\frac{8·7+5\left(8-10\right)}{9·3-6·4}$

Add and Subtract Fractions with Common Denominators

In the following exercises, add.

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

$\frac{5}{8}$

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

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

$\frac{3}{5}$

$\frac{15}{32}+\frac{9}{32}$

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

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

In the following exercises, subtract.

$\frac{8}{11}-\frac{6}{11}$

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

$\frac{1}{2}$

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

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

$-\frac{19}{15}$

$\frac{3}{2}-\left(\frac{3}{2}\right)$

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

$\frac{8}{15}$

Add and Subtract Fractions with Different Denominators

In the following exercises, find the least common denominator.

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{12}$

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{4}{5}$

15

$\frac{8}{15}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{11}{20}$

$\frac{3}{4},\frac{1}{6},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{10}$

60

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

$\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{1}{5},\phantom{\rule{0.2em}{0ex}}\text{LCD}=15$

$\frac{3}{8}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{6},\phantom{\rule{0.2em}{0ex}}\text{LCD}=24$

$\frac{9}{24}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{20}{24}$

$-\frac{9}{16}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{5}{12},\phantom{\rule{0.2em}{0ex}}\text{LCD}=48$

$\frac{1}{3}\text{,}\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{4}{5},\phantom{\rule{0.2em}{0ex}}\text{LCD}=60$

$\frac{20}{60},\frac{15}{60}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{48}{60}$

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

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

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

$\frac{1}{4}$

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

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

$-\frac{77}{90}$

$-\frac{22}{25}+\frac{9}{40}$

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

$\frac{3y-10}{30}$

$\frac{2}{5}+\left(-\frac{5}{9}\right)$

$\frac{4}{11}÷\frac{2}{7d}$

$\frac{14d}{11}$

$\frac{2}{5}+\left(-\frac{3n}{8}\right)\left(-\frac{2}{9n}\right)$

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

$\frac{256}{225}$

$\left(\frac{11}{12}+\frac{3}{8}\right)÷\left(\frac{5}{6}-\frac{1}{10}\right)$

In the following exercises, evaluate.

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

1. $y=-\frac{4}{5}$
2. $y=\frac{1}{4}$

1. $-\frac{8}{5}$
2. $-\frac{11}{20}$

$6m{n}^{2}$ when $m=\frac{3}{4}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n=-\frac{1}{3}$

Add and Subtract Mixed Numbers

In the following exercises, perform the indicated operation.

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

$13\frac{2}{3}$

$6\frac{2}{5}+7\frac{3}{5}$

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

$8\frac{1}{11}$

$3\frac{5}{8}+3\frac{7}{8}$

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

$5\frac{1}{10}$

$2\frac{3}{10}-1\frac{9}{10}$

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

$\frac{10}{3}$

$8\frac{6}{11}-2\frac{9}{11}$

Solve Equations with Fractions

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

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

• $x=1$
• $x=\frac{2}{3}$
• $x=-\frac{1}{3}$

1. no
2. yes
3. no

$y+\frac{3}{5}=\frac{5}{9}\text{:}$

1. $y=\frac{1}{2}$
2. $y=\frac{52}{45}$
3. $y=-\frac{2}{45}$

In the following exercises, solve the equation.

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

$n=-\frac{5}{11}$

$x-\frac{1}{6}=\frac{7}{6}$

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

$h=-\frac{51}{40}$

$\frac{x}{5}=-10$

$-z=23$

z = −23

In the following exercises, translate and solve.

The sum of two-thirds and $n$ is $-\frac{3}{5}.$

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

$q-\frac{1}{10}=\frac{1}{2};q=\frac{3}{5}$

The quotient of $p$ and $-4$ is $-8.$

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

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

Chapter practice test

Convert the improper fraction to a mixed number.

$\frac{19}{5}$

Convert the mixed number to an improper fraction.

$3\frac{2}{7}$

$\frac{23}{7}$

Locate the numbers on a number line.

$\frac{1}{2},1\frac{2}{3},-2\frac{3}{4},\text{and}\phantom{\rule{0.2em}{0ex}}\frac{9}{4}$

In the following exercises, simplify.

$\frac{5}{20}$

$\frac{1}{4}$

$\frac{18r}{27s}$

$\frac{1}{3}·\frac{3}{4}$

$\frac{1}{4}$

$\frac{3}{5}·15$

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

16 u

$-5\frac{7}{12}·4\frac{4}{11}$

$-\frac{5}{6}÷\frac{5}{12}$

−2

$\frac{7}{11}÷\left(-\frac{7}{11}\right)$

$\frac{9a}{10}÷\frac{15a}{8}$

$\frac{12}{25}$

$-6\frac{2}{5}÷4$

$\left(-15\frac{5}{6}\right)÷\left(-3\frac{1}{6}\right)$

5

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

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{p}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{q}{5}\phantom{\rule{0.2em}{0ex}}}$

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

$\frac{-\frac{4}{15}}{-2\frac{2}{3}}$

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

13

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

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

$-\frac{7}{13}$

$-\frac{22}{25}+\frac{9}{40}$

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

−1

$-\frac{3}{10}+\left(-\frac{5}{8}\right)$

$-\frac{3}{4}÷\frac{x}{3}$

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

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

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

3

Evaluate.

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

1. $x=\frac{2}{3}$
2. $x=-\frac{5}{6}$

In the following exercises, solve the equation.

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

$y=\frac{4}{5}$

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

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

$f=\frac{13}{12}$

$\frac{m}{-2}=-16$

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

c = −27

Translate and solve: The quotient of $p$ and $-4$ is $-8.$ Solve for $p.$

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