# Translating and Solving Basic Percent Equations

### Learning Outcomes

- Solve percent equations for percent, amount, and base

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations.

We'll look at a common application of percent—tips to a server at a restaurant—to see how to set up a basic percent application. When Aolani and her friends ate dinner at a restaurant, the bill came to [latex]\text{\$80}[/latex]. They wanted to leave a [latex]\text{20%}[/latex] tip. What amount would the tip be? To solve this, we want to find what*amount*is [latex]\text{20%}[/latex] of [latex]\text{\$80}[/latex]. The [latex]\text{\$80}[/latex] is called the

*base*. The amount of the tip would be [latex]0.20\left(80\right)[/latex], or [latex]\text{\$16}[/latex] See the image below. To find the amount of the tip, we multiplied the percent by the base. A [latex]\text{20%}[/latex] tip for an [latex]\text{\$80}[/latex] restaurant bill comes out to [latex]\text{\$16}[/latex].

## Solve for Amount

In the next examples, we will find the amount. We must be sure to change the given percent to a decimal when we translate the words into an equation.### example

What number is [latex]\text{35%}[/latex] of [latex]90?[/latex] SolutionTranslate into algebra. Let [latex]n=[/latex] the number. | |

Multiply. | [latex]n=31.5[/latex] |

[latex]31.5[/latex] is [latex]35\text{%}[/latex] of [latex]90[/latex] |

### try it

[ohm_question]80094[/ohm_question]### example

[latex]\text{125%}[/latex] of [latex]28[/latex] is what number?Answer: Solution

Translate into algebra. Let [latex]a=[/latex] the number. | |

Multiply. | [latex]35=a[/latex] |

[latex]125\text{%}[/latex] of [latex]28[/latex] is [latex]35[/latex] |

### try it

[ohm_question]146672[/ohm_question]## Solve for the Base

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

Translate and solve: [latex]36[/latex] is [latex]\text{75%}[/latex] of what number?Answer: Solution

Translate. Let [latex]b=[/latex] the number. | |

Divide both sides by [latex]0.75[/latex]. | [latex]{\Large\frac{36}{0.75}}={\Large\frac{0.75b}{0.75}}[/latex] |

Simplify. | [latex]48=b[/latex] [latex]36[/latex] is [latex]75\%[/latex] of [latex]48[/latex]. |

### try it

[ohm_question]80098[/ohm_question]### example

[latex]\text{6.5%}[/latex] of what number is [latex]\text{\$1.17}[/latex]?Answer: Solution

Translate. Let [latex]b=[/latex] the number. | |

Divide both sides by 0.065. | [latex]{\Large\frac{0.065n}{0.065}}={\Large\frac{1.17}{0.065}}[/latex] |

Simplify. | [latex]n=18[/latex] [latex]\color{blue}{\text{6.5%}}[/latex] of [latex]\color{blue}{\text{\$18}}[/latex] is [latex]\color{blue}{\text{\$1.17}}[/latex] |

### try it

[ohm_question]146692[/ohm_question]## Solve for the Percent

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

What percent of [latex]36[/latex] is [latex]9?[/latex]Answer: Solution

Translate into algebra. Let [latex]p=[/latex] the percent. | |

Divide by [latex]36[/latex]. | [latex]{\Large\frac{36p}{36}}={\Large\frac{9}{36}}[/latex] |

Simplify. | [latex]p={\Large\frac{1}{4}}[/latex] |

Convert to decimal form. | [latex]p=0.25[/latex] |

Convert to percent. | [latex]p=\text{25%}[/latex] [latex]\color{blue}{\text{25%}}[/latex] of [latex]\color{blue}{36}[/latex] is [latex]\color{blue}{9}[/latex] |

### try it

[ohm_question]146693[/ohm_question]### example

[latex-display]144[/latex] is what percent of [latex]96?[/latex-display]Answer: Solution

Translate into algebra. Let [latex]p=[/latex] the percent. | |

Divide by [latex]96[/latex]. | [latex]{\Large\frac{144}{96}}={\Large\frac{96p}{96}}[/latex] |

Simplify. | [latex]1.5=p[/latex] |

Convert to percent. | [latex]150\%=p[/latex] [latex]\color{blue}{144}[/latex] is [latex]\color{blue}{\text{150%}}[/latex] of [latex]\color{blue}{96}[/latex] |

### try it

[ohm_question]146866[/ohm_question]## Licenses & Attributions

### CC licensed content, Original

- Question ID: 146672, 146692, 146693, 146866.
**Authored by:**Alyson Day.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC-BY + GPL.

### CC licensed content, Shared previously

- Find the Percent of a Number.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Use the Percent Equation to Find a Percent.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Use a Percent Equation to Solve for a Base or Whole Amount.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Specific attribution

- Prealgebra.
**Provided by:**OpenStax**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected].