### Learning Outcomes

- Write percent equations as proportions
- Translate and solve percent proportion equations

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

Using the vocabulary we used earlier:

[latex]{\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}[/latex]

[latex]{\Large\frac{3}{5}}={\Large\frac{60}{100}}[/latex]

### Percent Proportion

The amount is to the base as the percent is to [latex]100[/latex].

[latex]{\Large\frac{\text{amount}}{\text{base}}}={\Large\frac{\text{percent}}{100}}[/latex]

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 [latex]\text{75%}[/latex] of [latex]90[/latex]?

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 [latex]90[/latex] is the same as [latex]75[/latex] out of [latex]100[/latex]? |

Set up the proportion. Let [latex]n=\text{number}[/latex] . | [latex]{\Large\frac{n}{90}}={\Large\frac{75}{100}}[/latex] |

### try it

### example

Translate to a proportion. [latex]19[/latex] is [latex]\text{25%}[/latex] of what number?

### try it

### example

Translate to a proportion. What percent of [latex]27[/latex] is [latex]9[/latex]?

### try it

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 [latex]\text{45%}[/latex] of [latex]80[/latex]?

### try it

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

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

### example

Translate and solve using proportions: [latex]\text{125%}[/latex] of [latex]25[/latex] is what number?

### try it

Percents with decimals and money are also used in proportions.

### example

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

### try it

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 [latex]72[/latex] is [latex]9?[/latex]

### try it

Watch the following video to see a similar problem.