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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 $80\text{\$80}. They wanted to leave a \text{20%} tip. What amount would the tip be? To solve this, we want to find what amount is \text{20%} of $80\text{\$80}. The $80\text{\$80} is called the base. The amount of the tip would be 0.20(80)0.20\left(80\right), or $16\text{\$16} See the image below. To find the amount of the tip, we multiplied the percent by the base. A \text{20%} tip for an $80\text{\$80} restaurant bill comes out to $16\text{\$16}.

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 \text{35%} of 90?90? Solution
Translate into algebra. Let n=n= the number. .
Multiply. n=31.5n=31.5
31.531.5 is 35\text{%} of 9090
 

try it

[ohm_question]80094[/ohm_question]
 

example

\text{125%} of 2828 is what number?

Answer: Solution

Translate into algebra. Let a=a= the number. .
Multiply. 35=a35=a
125\text{%} of 2828 is 3535
Remember that a percent over 100100 is a number greater than 11. We found that \text{125%} of 2828 is 3535, which is greater than 2828.

 

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: 3636 is \text{75%} of what number?

Answer: Solution

Translate. Let b=b= the number. .
Divide both sides by 0.750.75. 360.75=0.75b0.75{\Large\frac{36}{0.75}}={\Large\frac{0.75b}{0.75}}
Simplify. 48=b48=b 3636 is 75%75\% of 4848.

 

try it

[ohm_question]80098[/ohm_question]
 

example

\text{6.5%} of what number is $1.17\text{\$1.17}?

Answer: Solution

Translate. Let b=b= the number. .
Divide both sides by 0.065. 0.065n0.065=1.170.065{\Large\frac{0.065n}{0.065}}={\Large\frac{1.17}{0.065}}
Simplify. n=18n=18 \color{blue}{\text{6.5%}} of $18\color{blue}{\text{\$18}} is $1.17\color{blue}{\text{\$1.17}}

 

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 3636 is 9?9?

Answer: Solution

Translate into algebra. Let p=p= the percent. .
Divide by 3636. 36p36=936{\Large\frac{36p}{36}}={\Large\frac{9}{36}}
Simplify. p=14p={\Large\frac{1}{4}}
Convert to decimal form. p=0.25p=0.25
Convert to percent. p=\text{25%} \color{blue}{\text{25%}} of 36\color{blue}{36} is 9\color{blue}{9}

 

try it

[ohm_question]146693[/ohm_question]
 

example

144[/latex]iswhatpercentof[latex]96?144[/latex] is what percent of [latex]96?

Answer: Solution

Translate into algebra. Let p=p= the percent. .
Divide by 9696. 14496=96p96{\Large\frac{144}{96}}={\Large\frac{96p}{96}}
Simplify. 1.5=p1.5=p
Convert to percent. 150%=p150\%=p 144\color{blue}{144} is \color{blue}{\text{150%}} of 96\color{blue}{96}

 

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

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