Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example, $\text{60%}={\Large\frac{60}{100}}$ and we can simplify ${\Large\frac{60}{100}}={\Large\frac{3}{5}}$. Since the equation ${\Large\frac{60}{100}}={\Large\frac{3}{5}}$ shows a percent equal to an equivalent ratio, we call it a percent proportion.

Using the vocabulary we used earlier:

${\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}$
${\Large\frac{3}{5}}={\Large\frac{60}{100}}$

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

The amount is to the base as the percent is to one hundred.

We could also say:

The amount out of the base is the same as the percent out of one hundred.

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

example

Translate to a proportion. What number is $\text{75%}$ of $90$?

Solution
If you look for the word “of”, it may help you identify the base.

Identify the parts of the percent proportion.
Restate as a proportion.	What number out of $90$ is the same as $75$ out of $100$?
Set up the proportion. Let $n=\text{number}$ .	${\Large\frac{n}{90}}={\Large\frac{75}{100}}$

example

Translate to a proportion. $19$ is $\text{25%}$ of what number?

Show Solution

Solution

Identify the parts of the percent proportion.
Restate as a proportion.	$19$ out of what number is the same as $25$ out of $100$?
Set up the proportion. Let $n=\text{number}$ .	${\Large\frac{19}{n}}={\Large\frac{25}{100}}$

example

Translate to a proportion. What percent of $27$ is $9$?

Show Solution

Solution

Identify the parts of the percent proportion.
Restate as a proportion.	$9$ out of $27$ is the same as what number out of $100$?
Set up the proportion. Let $p=\text{percent}$ .	${\Large\frac{9}{27}}={\Large\frac{p}{100}}$

Now that we have written percent equations as proportions, we are ready to solve the equations.

example

Translate and solve using proportions: What number is $\text{45%}$ of $80$?

Show Solution

Solution

Identify the parts of the percent proportion.
Restate as a proportion.	What number out of $80$ is the same as $45$ out of $100$?
Set up the proportion. Let $n=$ number.	${\Large\frac{n}{80}}={\Large\frac{45}{100}}$
Find the cross products and set them equal.	$100\cdot{n}=80\cdot{45}$
Simplify.	$100n=3,600$
Divide both sides by $100$.	${\Large\frac{100n}{100}}={\Large\frac{3,600}{100}}$
Simplify.	$n=36$
Check if the answer is reasonable.
Yes. $45$ is a little less than half of $100$ and $36$ is a little less than half $80$.
Write a complete sentence that answers the question.	$36$ is $45\text{%}$ of $80$.

The following video shows a similar example of how to solve a percent proportion.

In the next example, the percent is more than $100$, which is more than one whole. So the unknown number will be more than the base.

example

Translate and solve using proportions: $\text{125%}$ of $25$ is what number?

Show Solution

Solution

Identify the parts of the percent proportion.
Restate as a proportion.	What number out of $25$ is the same as $125$ out of $100$?
Set up the proportion. Let $n=$ number.	${\Large\frac{n}{25}}={\Large\frac{125}{100}}$
Find the cross products and set them equal.	$100\cdot{n}=25\cdot{125}$
Simplify.	$100n=3,125$
Divide both sides by $100$.	${\Large\frac{100n}{100}}={\Large\frac{3,125}{100}}$
Simplify.	$n=31.25$
Check if the answer is reasonable.
Yes. $125$ is more than $100$ and $31.25$ is more than $25$.
Write a complete sentence that answers the question.	$125\text{%}$ of $25$ is $31.25$.

Percents with decimals and money are also used in proportions.

example

Translate and solve: $\text{6.5%}$ of what number is $\text{\$1.56}$?

Show Solution

Solution

Identify the parts of the percent proportion.
Restate as a proportion.	$\text{\$1.56}$ out of what number is the same as $6.5$ out of $100$?
Set up the proportion. Let $n=$ number.	${\Large\frac{1.56}{n}}={\Large\frac{6.5}{100}}$
Find the cross products and set them equal.	$100\cdot{1.56}=n\cdot{6.5}$
Simplify.	$156=6.5n$
Divide both sides by $6.5$ to isolate the variable.	${\Large\frac{156}{6.5}}={\Large\frac{6.5n}{6.5}}$
Simplify.	$24=n$
Check if the answer is reasonable.
Yes. $6.5\text{%}$ is a small amount and $\text{\$1.56}$ is much less than $\text{\$24}$.
Write a complete sentence that answers the question.	$6.5\text{%}$ of $\text{\$24}$ is $\text{\$1.56}$.

In the following video we show a similar problem, note the different wording that results in the same equation.

example

Translate and solve using proportions: What percent of $72$ is $9?$

Show Solution

Solution

Identify the parts of the percent proportion.
Restate as a proportion.	$9$ out of $72$ is the same as what number out of $100$?
Set up the proportion. Let $n=$ number.	${\Large\frac{9}{72}}={\Large\frac{n}{100}}$
Find the cross products and set them equal.	$72\cdot{n}=100\cdot{9}$
Simplify.	$72n=900$
Divide both sides by $72$.	${\Large\frac{72n}{72}}={\Large\frac{900}{72}}$
Simplify.	$n=12.5$
Check if the answer is reasonable.
Yes. $9$ is $\Large\frac{1}{8}$ of $72$, and $\Large\frac{1}{8}$ is $12.5\text{%}$.
Write a complete sentence that answers the question.	$12.5\text{%}$ of $72$ is $9$.

Watch the following video to see a similar problem.