1.9: Solving Problems Using Percents

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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 $\text{\$80}$. They wanted to leave a $20\%$ tip. What amount would the tip be?

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

A $20\%$ tip for an $\$80$ restaurant bill comes out to $\$16$.

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

$\text{percent}\cdot{\text{base}}=\text{amount}$

Example

Find $50\%$ of $20$

Solution:

First identify each piece of the problem:

percent: $50\%$ or $.5$

base: $20$

amount: unknown

Now plug them into your equation $\text{percent}\cdot{\text{base}}=\text{amount}$

$.5\cdot{20}= ?$

$.5\cdot{20}= 10$

Therefore, $10$ is the amount or part that is $50\%$ of $20$.

Example

What is $25\%$ of $80$?

[reveal-answer q=”813233″]Show Answer[/reveal-answer]
[hidden-answer a=”813233″]

The base is $80$ and the percent is $25\%$, so amount $= 80(0.25) = 20$

[/hidden-answer]

Try It

[ohm_question]80094[/ohm_question]

Thumbnail for the embedded element "Find the Percent of a Number"

A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

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.

$\Large{\frac{\text{amount}}{\text{base}}}\normalsize=\text{percent}$

Example

What percent of $320$ is $80$?

Solution:

First identify each piece of the problem:

percent: unknown

base: $320$

amount: $80$

Now plug the values into your equation $\Large{\frac{\text{amount}}{\text{base}}}\normalsize=\text{percent}$

$\large\frac{80}{320}\normalsize=?$

$\large\frac{80}{320}\normalsize=.25$

Therefore, $80$ is $25\%$ of $320$.

TRY IT

[ohm_question]80097[/ohm_question]

Thumbnail for the embedded element "Use the Percent Equation to Find a Percent"

A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

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.

$\Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}$

EXample

$60$ is $40\%$ of what number?

Solution:

First identify each piece of the problem:

percent:$40\%$ or $.4$

base: unknown

amount: $60$

Now plug the values into your equation $\Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}$

$(60)\div(.4)=?$

$(60)\div(.4)=150$

Therefore, $60$ is $40\%$ of $150$.

Example

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

[reveal-answer q=”731314″]Show Answer[/reveal-answer]
[hidden-answer a=”731314″]

The percent is $15\%$, and $$30,000$ is the amount (or part of the whole). We are looking for the base.

base = $30000\div(.15)=$200000$

The non-profit receives $$200000$ a year in donations

[/hidden-answer]

TRY IT

[ohm_question]157022[/ohm_question]

Thumbnail for the embedded element "Use a Percent Equation to Solve for a Base or Whole Amount"

A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

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

try it

[ohm_question]146672[/ohm_question]

try it

[ohm_question]146692[/ohm_question]

try it

[ohm_question]146693[/ohm_question]

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.

example

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

Solution

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

try it

[ohm_question]146694[/ohm_question]

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

Thumbnail for the embedded element "Percent Application - Tipping"

A YouTube element has been excluded from this version of the text. You can view it online here: http://pb.libretexts.org/afm-2/?p=78

example

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

The figures shows the nutrition facts for cereal.
[reveal-answer q=”744443″]Show Answer[/reveal-answer]
[hidden-answer a=”744443″]

Solution

What are you asked to find?	the total daily amount of potassium recommended (whole)
What formula/equation should you use?	$\Large{\frac{\text{amount}}{\text{percent}}}\normalsize=\text{base}$
Substitute in the correct values.	$\Large{\frac{85}{.02}}$
Solve.	$\Large{\frac{85}{.02}}\normalsize=4250$
Write a complete sentence that answers the question.	The amount of potassium that is recommended is $4,250$ mg.

[/hidden-answer]

try it

[ohm_question]146697[/ohm_question]

[ohm_question]146702[/ohm_question]

try it

[ohm_question]146703[/ohm_question]

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