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] |
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[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]\frac{36}{0.75}=\frac{0.75b}{0.75}[/latex] |
Simplify. |
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[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]\frac{0.065n}{0.065}=\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] |
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[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]\frac{36p}{36}=\frac{9}{36}[/latex] |
Simplify. | [latex]p=\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] |
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[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]\frac{144}{96}=\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] |
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[ohm_question]146866[/ohm_question]Licenses & Attributions
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].