Translating and Solving Basic Percent Equations

Learning Outcomes

  • Identify the amount, the base, and the percent in a percent problem
  • Translate and solve basic percent equations

We will solve percent equations by using the methods we use to solve equations with fractions or decimals. You can translate word sentences into algebraic equations, and then solve the equations.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

  • percent: has the percent symbol (%) or the word “percent”
  • amount: part of the whole
  • base: the whole amount

The Percent Equation

Percent of the Base is the Amount

[latex]\text{Percent}\cdot\text{Base}=\text{Amount}[/latex]

Percent of a Whole

We can use this equation to help us solve equations that require us to find the percent of a whole.

For example, if we know a container holds [latex]5[/latex] quarts of liquid, and we want to know how many quarts are in [latex]\frac{1}{4}[/latex] of the container, we would find [latex]\frac{1}{4}[/latex] of [latex]5[/latex] quarts by multiplying: [latex]\frac{1}{4} \cdot 5 = \frac{5}{4} = 1.25[/latex] quarts. Similarly, if we wanted to find [latex]25 \%[/latex] of [latex]5[/latex] quarts, we could find this by multiplying, but first we need to convert [latex]25 \%[/latex] to a decimal:

[latex]25 \% \ \mathrm{of} \ 5 \ \mathrm{quarts} = 0.25 \cdot 5 = 1.25 \ \mathrm{quarts}[/latex]

Finding a Percent of a Whole

To find a percent of a whole,

  • Write the percent as a decimal by moving the decimal two places to the left
  • Then multiply the percent by the whole (the base)

Example

Find [latex]30 \%[/latex] of [latex]200[/latex].

Example

What is [latex]5.3 \% \ \mathrm{of} \ 180[/latex]?

The following video contains an example that is similar to the one above.

try it

The video that follows shows how to use the percent equation to find the amount in a percent equation when the percent is greater than [latex]100 \%[/latex].

Solve for the Base

In the next examples, we are asked to find the base.

Once you have an equation, you can solve it and find the unknown value. For example, to solve  [latex]40\%\cdot{x}=50[/latex] you can divide [latex]50[/latex] by [latex]40\%[/latex] to find the unknown:

[latex]40\%\cdot{x}=50[/latex]

You can solve this by writing the percent as a decimal or fraction and then dividing.

[latex]0.40\cdot{x}=50[/latex]

[latex]x=50 \div 0.40=125[/latex]

Example

[latex]156[/latex] is [latex]65 \%[/latex] of what number?

In the following video we show another example of how to find the base or whole given percent and amount.

try it

Solve for the Percent

In the next example, we will solve for the percent.

example

What percent of [latex]36[/latex] is [latex]27?[/latex]

try it

In the next video we show another example of how to find the percent given amount and the base.