Solving Problems Using Percents

Learning Outcome

  • Evaluate expressions and word problems involving percents

In this section we will solve percent questions by identifying the parts of the problem. 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]20\%[/latex] tip. What amount would the tip be?

To solve this, we want to find what amount is [latex]20\%[/latex] of [latex]\$80[/latex]. The [latex]\$80[/latex] is called the base. The percent is the given [latex]20\%[/latex]. The amount of the tip would be [latex]0.20(80)[/latex], or [latex]\$16[/latex] — see the image below. To find the amount of the tip, we multiplied the percent by the base.

A [latex]20\%[/latex] tip for an [latex]\$80[/latex] restaurant bill comes out to [latex]\$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.


Pieces of a Percent Problem

Percent problems involve three quantities:  the base amount (the whole), the percent, and the amount (a part of the whole or partial amount).

The amount is a percent of the base.

Let’s look at another example:

Jeff has a Guitar Strings coupon for [latex]15\%[/latex] off any purchase of [latex]$100[/latex] or more. He wants to buy a used guitar that has a price tag of [latex]$220[/latex] on it. Jeff wonders how much money the coupon will take off the original [latex]$220[/latex] price. Problems involving percents will have some combination of these three quantities to work with: the percent, the amount, and the base. The percent has the percent symbol (%) or the word percent. In the problem above, [latex]15\%[/latex] is the percent off the purchase price. The base is the whole amount or original amount. In the problem above, the “whole” price of the guitar is [latex]$220[/latex], which is the base. The amount is the unknown and what we will need to calculate.

There are thee cases: a missing amount, a missing percent or a missing base. Let’s take a look at each possibility.

Solving for the Amount

When solving for the amount in a percent problem, you will multiply the percent (as a decimal or fraction) by the base. Typically we choose the decimal value for percent.



Find [latex]50\%[/latex] of [latex]20[/latex]


First identify each piece of the problem:

percent: [latex]50\%[/latex] or [latex].5[/latex]

base: [latex]20[/latex]

amount: unknown

Now plug them into your equation [latex]\text{percent}\cdot{\text{base}}=\text{amount}[/latex]

[latex].5\cdot{20}= ?[/latex]

[latex].5\cdot{20}= 10[/latex]

Therefore, [latex]10[/latex] is the amount or part that is [latex]50\%[/latex] of [latex]20[/latex].


What is [latex]25\%[/latex] of [latex]80[/latex]?

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Solving for the Percent

When solving for the percent in a percent problem, you will divide the amount by the base. The equation above is rearranged and the percent will come back as a decimal of fraction you can report in the form asked of you.



What percent of [latex]320[/latex] is [latex]80[/latex]?


First identify each piece of the problem:

percent: unknown

base: [latex]320[/latex]

amount: [latex]80[/latex]

Now plug the values into your equation [latex]\Large{\frac{\text{amount}}{\text{base}}}\normalsize=\text{percent}[/latex]



Therefore, [latex]80[/latex] is [latex]25\%[/latex] of [latex]320[/latex].


Solving for the Base

When solving for the base in a percent problem, you will divide the amount by the percent (as a decimal or fraction). The equation above is rearranged and you will find the base after plugging in the values.



[latex]60[/latex] is [latex]40\%[/latex] of what number?


First identify each piece of the problem:

percent:[latex]40\%[/latex] or [latex].4[/latex]

base: unknown

amount: [latex]60[/latex]

Now plug the values into your equation [latex]\Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}[/latex]



Therefore, [latex]60[/latex] is [latex]40\%[/latex] of [latex]150[/latex].


An article says that [latex]15\%[/latex] of a non-profit’s donations, about [latex]$30,000[/latex] a year, comes from individual donors.  What is the total amount of donations the non-profit receives?


Here are a few more percent problems for you to try.

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Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we’ll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.


Dezohn and his girlfriend enjoyed a dinner at a restaurant, and the bill was [latex]\text{\$68.50}[/latex]. They want to leave an [latex]\text{18%}[/latex] tip. If the tip will be [latex]\text{18%}[/latex] of the total bill, how much should the tip be?


What are you asked to find? the amount of the tip
What formula/equation should you use? [latex]\text{percent}\cdot{\text{base}}=\text{amount}[/latex]
Substitute in the correct values. [latex](.18)\cdot{68.50}[/latex]
Solve.  [latex](.18)\cdot{68.50}=12.33[/latex]
Write a complete sentence that answers the question. The couple should leave a tip of [latex]\text{\$12.33}[/latex].

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In the next video we show another example of finding how much tip to give based on percent.


The label on Masao’s breakfast cereal said that one serving of cereal provides [latex]85[/latex] milligrams (mg) of potassium, which is [latex]\text{2%}[/latex] of the recommended daily amount. What is the total recommended daily amount of potassium?

The figures shows the nutrition facts for cereal.

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