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Study Guides > Mathematics for the Liberal Arts Corequisite

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].

The figure shows a customer copy of a restaurant receipt with the amount of the bill, $80, and the amount of the tip, $16. There is a group of bills totaling $16.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] Solution
Translate 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]
Remember that a percent over [latex]100[/latex] is a number greater than [latex]1[/latex]. We found that [latex]\text{125%}[/latex] of [latex]28[/latex] is [latex]35[/latex], which is greater than [latex]28[/latex].

 

try it

[ohm_question]146672[/ohm_question]
In the next video we show another example of finding the base given a percent and the amount. https://youtu.be/jTM7ZMvAzsc

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]
In the following video we show another example of how to find the base or whole given percent and amount. https://youtu.be/3etjmUw8K3A

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]
In the next video we show another example of how to find the percent given amount and the base. https://youtu.be/p2KHHFMhJRs

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