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.

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: [latex]n[/latex] divided by [latex]6[/latex] is [latex]-24[/latex].

Solution:

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

Example

Translate and solve: The quotient of [latex]q[/latex] and [latex]-5[/latex] is [latex]70[/latex].

Show Solution

Solution:

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

Example

Translate and solve: Two-thirds of [latex]f[/latex] is [latex]18[/latex].

Show Solution

Solution:

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

Example

Translate and solve: The sum of three-eighths and [latex]x[/latex] is three and one-half.

Show Solution

Solution:

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

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

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

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

The solution checks.

We’ve seen several examples of how to translate a given mathematical relationship from words into equation form in order to solve it. Let’s see some examples now that reverse the process. The following video shows examples of translating an equation into words as an aid to solving it. Note that this is different from the written examples on this page because these examples start with the mathematical equation then translate it into words.