Translating Words to Equations and Solving

Learning Outcomes

Translate a phrase that contains division into an equation and solve
Use the multiplication and division property of equality to solve equations that contain fractions

Translate Sentences to Equations and Solve

Recall the four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference. We will use these to solve equations that contain fractions.

Subtraction Property of Equality:

For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$

if $a=b$ , then $a-c=b-c$ .

Addition Property of Equality:

For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$

if $a=b$ , then $a+c=b+c$ .

Division Property of Equality:

For any numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c,}}$ where $\mathit{\text{c}}\ne \mathit{0}$

if $a=b$ , then $\Large\frac{a}{c}= \Large\frac{b}{c}$

Multiplication Property of Equality:

For any real numbers $\mathit{\text{a, b,}}$ and $\mathit{\text{c}}$

if $a=b$ , then $ac=bc$

When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
In the next few examples, we’ll translate sentences that contain fractions into equations and then solve the equations. The first property of equality we will use is multiplication.

Example

Translate and solve: $n$ divided by $6$ is $-24$ .

Solution:

Translate.
Multiply both sides by $6$ .		$\color{red}{6}\cdot\Large\frac{n}{6}\normalsize=\color{red}{6}(-24)$
Simplify.		$n=-144$
Check:	Is $-144$ divided by $6$ equal to $-24$ ?
Translate.	$\Large\frac{-144}{6}\normalsize\stackrel{?}{=}-24$
Simplify. It checks.	$-24=-24\quad\checkmark$

Try It

Example

Translate and solve: The quotient of $q$ and $-5$ is $70$ .

Show Solution

Solution:

Translate.
Multiply both sides by $-5$ .		$\color{red}{5}\Large(\frac{q}{-5}) \normalsize= \color{red}{-5}(70)$
Simplify.		$q=-350$
Check:	Is the quotient of $-350$ and $-5$ equal to $70$ ?
Translate.	$\Large\frac{-350}{-5}\normalsize\stackrel{?}{=}70$
Simplify. It checks.	$70=70\quad\checkmark$

Try It

Example

Translate and solve: Two-thirds of $f$ is $18$ .

Show Solution

Solution:

Translate.
Multiply both sides by $\Large\frac{3}{2}$ .		$\color{red}{\Large\frac{3}{2}}\cdot\Large\frac{2}{3}\normalsize f=\color{red}{\Large\frac{3}{2}}\cdot \normalsize18$
Simplify.		$f=27$
Check:	Is two-thirds of $27$ equal to $18$ ?
Translate.	$\Large\frac{2}{3}\normalsize(27)\normalsize\stackrel{?}{=}18$
Simplify. It checks.	$18=18\quad\checkmark$

try It

Example

Translate and solve: The quotient of $m$ and $\Large\frac{5}{6}$ is $\Large\frac{3}{4}$ .

Show Solution

Solution:

		The quotient of $m$ and $\Large\frac{5}{6}$ is $\Large\frac{3}{4}$ .
Translate.		$\Large\frac{m}{\LARGE\frac{5}{6}}=\Large\frac{3}{4}$
Multiply both sides by $\Large\frac{5}{6}$ to isolate $m$ .		$\Large\frac{5}{6}\left(\Large\frac{m}{\LARGE\frac{5}{6}}\right)=\Large\frac{5}{6}\left(\Large\frac{3}{4}\right)$
Simplify.		$m=\Large\frac{5\cdot 3}{6\cdot 4}$
Remove common factors and multiply.		$m=\Large\frac{5}{8}$
Check:
Is the quotient of $\Large\frac{5}{8}$ and $\Large\frac{5}{6}$ equal to $\Large\frac{3}{4}$ ?	$\Large\frac{\LARGE\frac{5}{8}}{\LARGE\frac{5}{6}}\stackrel{?}{=}\Large\frac{3}{4}$
Rewrite as division.	$\Large\frac{5}{8}\div\Large\frac{5}{6}\stackrel{?}{=}\Large\frac{3}{4}$
Multiply the first fraction by the reciprocal of the second.	$\Large\frac{5}{8}\cdot\Large\frac{6}{5}\stackrel{?}{=}\Large\frac{3}{4}$
Simplify.	$\Large\frac{3}{4}=\Large\frac{3}{4}\quad\checkmark$

Our solution checks.

Try It

Example

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

Show Solution

Solution:

Translate.
Use the Subtraction Property of Equality to subtract $\Large\frac{3}{8}$ from both sides.	$\Large\frac{3}{8}+\normalsize x-\Large\frac{3}{8}=\normalsize3\Large\frac{1}{2}-\Large\frac{3}{8}$
Combine like terms on the left side.	$x=3\Large\frac{1}{2}-\Large\frac{3}{8}$
Convert mixed number to improper fraction.	$x\Large\frac{7}{2}-\Large\frac{3}{8}$
Convert to equivalent fractions with LCD of $8$ .	$x=\Large\frac{28}{8}-\Large\frac{3}{8}$
Subtract.	$x=\Large\frac{25}{8}$
Write as a mixed number.	$x=3\Large\frac{1}{8}$

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

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

	$\Large\frac{3}{8}\normalsize+3\Large\frac{1}{8}\normalsize\stackrel{?}{=}3\Large\frac{1}{2}$
Add.	$3\Large\frac{4}{8}\normalsize\stackrel{?}{=}3\Large\frac{1}{2}$
Simplify.	$3\Large\frac{1}{2}\normalsize=3\Large\frac{1}{2}\quad\checkmark$

The solution checks.

Module 4: Fractions