# 4.7 Solve equations with fractions  (Page 4/5)

 Page 4 / 5

Translate and solve. The quotient of $n$ and $\frac{2}{3}$ is $\frac{5}{12}.$

$\frac{\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}=\frac{5}{12};n=\frac{5}{18}$

Translate and solve The quotient of $c$ and $\frac{3}{8}$ is $\frac{4}{9}.$

$\frac{\phantom{\rule{0.2em}{0ex}}c\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{3}{8}\phantom{\rule{0.2em}{0ex}}}=\frac{4}{9};c=\frac{1}{3}$

Translate and solve: The sum of three-eighths and $x$ is three and one-half.

## Solution

 Translate. Use the Subtraction Property of Equality to subtract $\frac{3}{8}$ from both sides. Combine like terms on the left side. Convert mixed number to improper fraction. Convert to equivalent fractions with LCD of 8. Subtract. Write as a mixed number.

We write the answer as a mixed number because the original problem used a mixed number.

Check:

Is the sum of three-eighths and $3\frac{1}{8}$ equal to three and one-half?

 $\frac{3}{8}+3\frac{1}{8}\stackrel{?}{=}3\frac{1}{2}$ Add. $3\frac{4}{8}\stackrel{?}{=}3\frac{1}{2}$ Simplify. $3\frac{1}{2}=3\frac{1}{2}\phantom{\rule{0.2em}{0ex}}✓$

The solution checks.

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

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

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

$1\frac{3}{4}-x=\frac{5}{6};\phantom{\rule{0.2em}{0ex}}x=\frac{11}{12}$

## Key concepts

• Determine whether a number is a solution to an equation.
1. Substitute the number for the variable in the equation.
2. Simplify the expressions on both sides of the equation.
3. Determine whether the resulting equation is true. If it is true, the number is a solution. If it is not true, the number is not a solution.
• Addition, Subtraction, and Division Properties of Equality
• For any numbers a, b, and c,
if $a=b$ , then $a+c=b+c$ . Addition Property of Equality
• if $a=b$ , then $a-c=b-c$ . Subtraction Property of Equality
• if $a=b$ , then $\frac{a}{c}=\frac{b}{c}$ , $c\ne 0$ . Division Property of Equality
• The Multiplication Property of Equality
• For any numbers $ab$ and $c,a=b$ , then $ac=bc$ .
• If you multiply both sides of an equation by the same quantity, you still have equality.

## Practice makes perfect

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.

$x-\frac{2}{5}=\frac{1}{10}\text{:}$

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

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

1. $y=1$
2. $y=\frac{3}{4}$
3. $y=-\frac{3}{4}$

1. no
2. yes
3. no

$h+\frac{3}{4}=\frac{2}{5}\text{:}$

1. $h=1$
2. $h=\frac{7}{20}$
3. $h=-\frac{7}{20}$

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

1. $k=1$
2. $k=\frac{13}{30}$
3. $k=-\frac{13}{30}$

1. no
2. yes
3. no

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

In the following exercises, solve.

$y+\frac{1}{3}=\frac{4}{3}$

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

$m=\frac{1}{2}$

$f+\frac{9}{10}=\frac{2}{5}$

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

$h=-\frac{2}{3}$

$a-\frac{5}{8}=-\frac{7}{8}$

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

c = −1

$x-\left(-\frac{3}{20}\right)=-\frac{11}{20}$

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

z = −1

$n-\frac{1}{6}=\frac{3}{4}$

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

$p=\frac{37}{40}$

$s+\left(-\frac{1}{2}\right)=-\frac{8}{9}$

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

$k=-\frac{7}{15}$

$5j=17$

$7k=18$

$k=\frac{18}{7}$

$-4w=26$

$-9v=33$

$v=-\frac{11}{3}$

Solve Equations with Fractions Using the Multiplication Property of Equality

In the following exercises, solve.

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

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

b = −27

$\frac{y}{7}=-21$

$\frac{x}{8}=-32$

x = −256

$\frac{p}{-5}=-40$

$\frac{q}{-4}=-40$

q = 160

$\frac{r}{-12}=-6$

$\frac{s}{-15}=-3$

s = 45

$-x=23$

$-y=42$

y = −42

$-h=-\frac{5}{12}$

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

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

$\frac{4}{5}n=20$

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

p = 100

$\frac{3}{8}q=-48$

$\frac{5}{2}m=-40$

m = −16

$-\frac{2}{9}a=16$

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

b = −21

$-\frac{6}{11}u=-24$

$-\frac{5}{12}v=-15$

v = 36

Mixed Practice

In the following exercises, solve.

$3x=0$

$8y=0$

y = 0

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

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

$g=\frac{1}{9}$

$p+\frac{2}{3}=\frac{1}{12}$

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

$q=-\frac{3}{4}$

$\frac{7}{8}m=\frac{1}{10}$

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

$n=\frac{14}{5}$

$-\frac{2}{5}=x+\frac{3}{4}$

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

$y=-\frac{25}{24}$

$\frac{11}{20}=\text{−}\mathit{\text{f}}$

$\frac{8}{15}=\text{−}\mathit{\text{d}}$

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

Translate Sentences to Equations and Solve

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

$n$ divided by eight is $-16.$

$n$ divided by six is $-24.$

$\frac{n}{6}=-24;n=-144$

$m$ divided by $-9$ is $-7.$

$m$ divided by $-7$ is $-8.$

$\frac{m}{-7}=-8;m=56$

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