Mathematics » Introducing Percents » Solve Proportions and their Applications

Writing Percent Equations As Proportions

Writing Percent Equations As Proportions

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, \(\text{60%}=\frac{60}{100}\) and we can simplify \(\frac{60}{100}=\frac{3}{5}.\) Since the equation \(\frac{60}{100}=\frac{3}{5}\) shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

\(\cfrac{\text{amount}}{\text{base}}=\cfrac{\text{percent}}{100}\)

\(\phantom{\rule{1.2em}{0ex}}\cfrac{3}{5}=\cfrac{60}{100}\)

Definition: Percent Proportion

The amount is to the base as the percent is to \(100.\)

\(\cfrac{\text{amount}}{\text{base}}=\cfrac{\text{percent}}{100}\)

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

\(\mathit{\text{The amount is to the base as the percent is to one hundred.}}\)

We could also say:

\(\mathit{\text{The amount out of the base is the same as the percent out of one hundred.}}\)

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.Writing Percent Equations As Proportions
Restate as a proportion.Writing Percent Equations As Proportions
Set up the proportion. Let \(n=\text{number}\).\(\frac{n}{90}=\frac{75}{100}\)

Example

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

Solution

Identify the parts of the percent proportion.Writing Percent Equations As Proportions
Restate as a proportion.Writing Percent Equations As Proportions
Set up the proportion. Let \(n=\text{number}\).\(\frac{19}{n}=\frac{25}{100}\)

Example

Translate to a proportion. What percent of \(27\) is \(9?\)

Solution

Identify the parts of the percent proportion.Writing Percent Equations As Proportions
Restate as a proportion.Writing Percent Equations As Proportions
Set up the proportion. Let \(p=\text{percent}\).\(\frac{9}{27}=\frac{p}{100}\)

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