5.2.1: Solving Percent Problems

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The NROC Project

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Learning Objectives

Identify the amount, the base, and the percent in a percent problem.
Find the unknown in a percent problem.

Introduction

Percents are a ratio of a number and 100, so they are easier to compare than fractions, as they always have the same denominator, 100. A store may have a 10% off sale. The amount saved is always the same portion or fraction of the price, but a higher price means more money is taken off. Interest rates on a saving account work in the same way. The more money you put in your account, the more money you get in interest. It’s helpful to understand how these percents are calculated.

Parts of a Percent Problem

Jeff has a coupon at the Guitar Store 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 have any 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. In the problem above, the whole price of the guitar is $220, which is the base.
The amount is the number that relates to the percent. It is always part of the whole. In the problem above, the amount is unknown. Since the percent is the percent off, the amount will be the amount off of the price.

You will return to this problem a bit later. The following examples show how to identify the three parts: the percent, the base, and the amount.

Example

Identify the percent, amount, and base in this problem.

30 is 20% of what number?

Solution

Percent: The percent is the number with the % symbol: 20%.

Base: The base is the whole amount, which in this case is unknown.

Amount: The amount based on the percent is 30.

Answer:

Percent=20%

Amount=30

Base=unknown

The previous problem states that 30 is a portion of another number. That means 30 is the amount. Note that this problem could be rewritten: 20% of what number is 30?

Exercise

Identify the percent, base, and amount in this problem:

What percent of 30 is 3?

Answer: The percent is unknown, because the problem states "What percent?" The base is the whole in the situation, so the base is 30. The amount is the portion of the whole, which is 3 in this case.

Solving with Equations

Percent problems can be solved by writing equations. An equation uses an equal sign (=) to show that two mathematical expressions have the same value.

Percents are fractions, and just like fractions, when finding a percent (or fraction, or portion) of another amount, you multiply.

The percent of the base is the amount.

Percent of the Base is the Amount.

\[\ \text { Percent } {\color{red}\cdot}\text { Base }{\color{blue}=}\text { Amount } \nonumber \]

In the examples below, the unknown is represented by the letter $\ n$. The unknown can be represented by any letter or a box $\ \square$ or even a question mark.

Example

Write an equation that represents the following problem.

30 is 20% of what number?

Solution

20% of what number is 30?

Rewrite the problem in the form “percent of base is amount.”

Percent is: 20%

Base is: unknown

Amount is: 30

Identify the percent, the base, and the amount.

$\ \text { Percent } \cdot \text { Base }=\text { Amount }$

$\ 20 \% \cdot n=30$

Write the percent equation. using $\ n$ for the base, which is the unknown value.

$\ 20 \% \cdot n=30$

Once you have an equation, you can solve it and find the unknown value. To do this, think about the relationship between multiplication and division. Look at the pairs of multiplication and division facts below, and look for a pattern in each row.

Multiplication	Division
$\ 2 \cdot 3=6$	$\ 6 \div 2=3$
$\ 8 \cdot 5=40$	$\ 40 \div 8=5$
$\ 7 \cdot 4=28$	$\ 28 \div 7=4$
$\ 6 \cdot 9=54$	$\ 54 \div 6=9$

Multiplication and division are inverse operations. What one does to a number, the other “undoes.”

When you have an equation such as $\ 20 \% \cdot n=30$, you can divide 30 by 20% to find the unknown: $\ n=30 \div 20 \%$.

You can solve this by writing the percent as a decimal or fraction and then dividing.

$\ n=30 \div 20 \%=30 \div 0.20=150$

Example

What percent of 72 is 9?

Solution

Percent: unknown Base: 72 Amount: 9	Identify the percent, base, and amount.
$\ n \cdot 72=9$	Write the percent equation: $\ \text { Percent } \cdot \text { Base }=\text { Amount }$. Use $\ n$ for the unknown (percent).
$\ n=9 \div 72$	Divide to undo the multiplication of $\ n$ times 72.
$\ \begin{array}{r} 0.125\ \\ 72\longdiv{9.000} \end{array}$	Divide 9 by 72 to find the value for $\ n$, the unknown.
$\ n=0.125$ $\ n=12.5 \%$	Move the decimal point two places to the right to write the decimal as a percent.

$\ 12.5 \% \text { of } 72 \text { is } 9$.

You can estimate to see if the answer is reasonable. Use 10% and 20%, numbers close to 12.5%, to see if they get you close to the answer.

$\ 10 \% \text { of } 72=0.1 \cdot 72=7.2$

$\ 20 \% \text { of } 72=0.2 \cdot 72=14.4$

Notice that 9 is between 7.2 and 14.4, so 12.5% is reasonable since it is between 10% and 20%.

Example

What is 110% of 24?

Solution

Percent: 110% Base: 24 Amount: unknown	Identify the percent, the base, and the amount.
$\ 110 \% \cdot 24=n$	Write the percent equation. $\ \text { Percent } \cdot \text { Base }=\text { Amount }$. The amount is unknown, so use $\ n$.
$\ 1.10 \cdot 24=n$	Write the percent as a decimal by moving the decimal point two places to the left.
$\ 1.10 \cdot 24=26.4=n$	Multiply 24 by 1.10 or 1.1.

$\ 26.4 \text { is } 110 \% \text { of } 24$.

This problem is a little easier to estimate. 100% of 24 is 24. And 110% is a little bit more than 24. So, 26.4 is a reasonable answer.

Exercise

18 is what percent of 48?

$\ 0.375 \%$
$\ 8.64 \%$
$\ 37.5 \%$
$\ 864 \%$

Answer

$\ 0.375 \%$
Incorrect. You may have calculated properly, but you forgot to move the decimal point when you rewrote your answer as a percent. The equation for this problem is $\ n \cdot 48=18$. The corresponding division is $\ 18 \div 48$, so $\ n=0.375$. Rewriting this decimal as a percent gives the correct answer, $\ 37.5 \%$.
$\ 8.64 \%$
Incorrect. You may have used $\ 18$ or $\ 48$ as the percent, rather than the amount or base. The equation for this problem is $\ n \cdot 48=18$. The corresponding division is $\ 18 \div 48$, so $\ n=0.375$. Rewriting this decimal as a percent gives the correct answer, $\ 37.5 \%$.
$\ 37.5 \%$
Correct. The equation for this problem is $\ n \cdot 48=18$. The corresponding division is $\ 18 \div 48$, so $\ n=0.375$. Rewriting this decimal as a percent gives $\ 37.5 \%$.
$\ 864 \%$
Incorrect. You probably used 18 or 48 as the percent, rather than the amount or base, and also forgot to rewrite the percent as a decimal before multiplying. The equation for this problem is $\ n \cdot 48=18$. The corresponding division is $\ 18 \div 48$, so $\ n=0.375$. Rewriting this decimal as a percent gives the correct answer, $\ 37.5 \%$.

Using Proportions to Solve Percent Problems

Percent problems can also be solved by writing a proportion. A proportion is an equation that sets two ratios or fractions equal to each other. With percent problems, one of the ratios is the percent, written as $\ \frac{n}{100}$. The other ratio is the amount to the base.

$\ \text { Percent }=\frac{\text { amount }}{\text { base }}$

Example

Write a proportion to find the answer to the following question.

30 is 20% of what number?

Solution

$\ \frac{20}{100}=\frac{\text { amount }}{\text { base }}$	The percent in this problem is 20%. Write this percent in fractional form, with 100 as the denominator.
$\ \frac{20}{100}=\frac{30}{n}$	The percent is written as the ratio $\ \frac{20}{100}$, the amount is 30, and the base is unknown.
$\ \begin{array}{r} 20 \cdot n=30 \cdot 100 \\ 20 \cdot n=3,000 \\ n=3,000 \div 20 \\ n=150 \end{array}$	Cross multiply and solve for the unknown, $\ n$, by dividing 3,000 by 20.

30 is 20% of 150.

Example

What percent of 72 is 9?

Solution

$\ \begin{array}{r} \text { Percent }=\frac{\text { amount }}{\text { base }} \\ \frac{n}{100}=\frac{9}{72} \end{array}$	The percent is the ratio of $\ n$ to 100. The amount is 9, and the base is 72.
$\ \begin{array}{r} n \cdot 72=9 \cdot 100 \\ n \cdot 72=900 \\ n=900 \div 72 \\ n=12.5 \end{array}$	Cross multiply and solve for $\ n$ by dividing 900 by 72.
$\ 12.5 \% \text { of } 72 \text { is } 9$	The percent is $\ \frac{12.5}{100}=12.5 \%$.

Example

What is 110% of 24?

Solution

$\ \begin{array}{l} \text { Percent }=\frac{\text { amount }}{\text { base }} \\ \frac{110}{100}=\frac{n}{24} \end{array}$	The percent is the ratio $\ \frac{110}{100}$. The amount is unknown, and the base is 24.
$\ \begin{array}{r} 24 \cdot 110=100 \cdot n \\ 2,640 \div 100=n \\ 26.4=n \end{array}$	Cross multiply and solve for $\ n$ by dividing 2,640 by 100.
$\ 26.4 \text { is } 110 \% \text { of } 24$

Exercise

18 is 125% of what number?

$\ 0.144$
$\ 14.4$
$\ 22.5$
$\ 694 \frac{4}{9}$ (or about $\ 694.4$)

Answer

$\ 0.144$
Incorrect. You probably didn’t write a proportion and just divided 18 by 125. Or, you incorrectly set up one fraction as $\ \frac{18}{125}$ and set this equal to the base, $\ n$. The percent in this case is 125%, so one fraction in the proportion should be $\ \frac{125}{100}$. The base is unknown and the amount is 18, so the other fraction is $\ \frac{18}{n}$. Solving the proportion $\ \frac{125}{100}=\frac{18}{n}$ gives $\ n=14.4$.
$\ 14.4$
Correct. The percent in this case is 125%, so one fraction in the proportion should be $\ \frac{125}{100}$. The base is unknown and the amount is 18, so the other fraction is $\ \frac{18}{n}$. Solving the proportion $\ \frac{125}{100}=\frac{18}{n}$ gives $\ n=14.4$.
$\ 22.5$
Incorrect. You probably put the amount (18) over 100 in the proportion, rather than the percent (125). Perhaps you thought 18 was the percent and 125 was the base. The correct percent fraction for the proportion is $\ \frac{125}{100}$. The base is unknown and the amount is 18, so the other fraction is $\ \frac{18}{n}$. Solving the proportion $\ \frac{125}{100}=\frac{18}{n}$ gives $\ n=14.4$.
$\ 694 \frac{4}{9}$ (or about $\ 694.4$)
Incorrect. You probably confused the amount (18) with the percent (125) when you set up the proportion. The correct percent fraction for the proportion is $\ \frac{125}{100}$. The base is unknown and the amount is 18, so the other fraction is $\ \frac{18}{n}$. Solving the proportion $\ \frac{125}{100}=\frac{18}{n}$ gives $\ n=14.4$.

Let’s go back to the problem that was posed at the beginning. You can now solve this problem as shown in the following example.

Example

Solution

How much is 15% of $220?	Simplify the problems by eliminating extra words.
Percent: 15% Base: 220 Amount: $\ n$	Identify the percent, the base, and the amount.
$\ 15 \% \cdot 220=n$	Write the percent equation. $\ \text { Percent } \cdot \text { Base }=\text { Amount }$
$\ 0.15 \cdot 220=33$	Convert 15% to 0.15, then multiply by 220. 15% of $220 is $33.

The coupon will take $33 off the original price.

You can estimate to see if the answer is reasonable. Since 15% is half way between 10% and 20%, find these numbers.

$\ \begin{array}{l}
10 \% \text { of } 220=0.1 \cdot 220=22 \\
20 \% \text { of } 220=0.2 \cdot 220=44
\end{array}$

The answer, 33, is between 22 and 44. So $33 seems reasonable.

There are many other situations that involve percents. Below are just a few.

Example

Evelyn bought some books at the local bookstore. Her total bill was $31.50, which included 5% tax. How much did the books cost before tax?

Solution

What number +5% of that number is $31.50?	In this problem, you know that the tax of 5% is added onto the cost of the books. So if the cost of the books is 100%, the cost plus tax is 105%.
105% of what number = 31.50? Percent: 105% Base: $\ n$ Amount: 31.50	Identify the percent, the base, and the amount.
$\ 105 \% \cdot n=31.50$	Write the percent equation. $\ \text { Percent } \cdot \text { Base }=\text { Amount }$.
$\ 1.05 \cdot n=31.50$	Convert 105% to a decimal.
$\ n=31.50 \div 1.05=30$	Divide to undo the multiplication of $\ n$ times 1.05.

The books cost $30 before tax.

Example

Susana worked 20 hours at her job last week. This week, she worked 35 hours. In terms of a percent, how much more did she work this week than last week?

Solution

35 is what percent of 20?	Simplify the problem by eliminating extra words.
Percent: $\ n$ Base: 20 Amount: 35	Identify the percent, the base, and the amount.
$\ n \cdot 20=35$	Write the percent equation. $\ \text { Percent } \cdot \text { Base }=\text { Amount }$.
$\ n=35 \div 20$	Divide to undo the multiplication of $\ n$ times 20.
$\ n=1.75=175 \%$	Convert 1.75 to a percent.

Since 35 is 175% of 20, Susana worked 75% more this week than she did last week. (You can think of this as, “Susana worked 100% of the hours she worked last week, as well as 75% more.”)

Summary

Percent problems have three parts: the percent, the base (or whole), and the amount. Any of those parts may be the unknown value to be found. To solve percent problems, you can use the equation, $\ \text { Percent } \cdot \text { Base }=\text { Amount }$, and solve for the unknown numbers. Or, you can set up the proportion, $\ \text { Percent }=\frac{\text { amount }}{\text { base }}$, where the percent is a ratio of a number to 100. You can then use cross multiplication to solve the proportion.

Multiplication	Division
\(\ 2 \cdot 3=6\)	\(\ 6 \div 2=3\)
\(\ 8 \cdot 5=40\)	\(\ 40 \div 8=5\)
\(\ 7 \cdot 4=28\)	\(\ 28 \div 7=4\)
\(\ 6 \cdot 9=54\)	\(\ 54 \div 6=9\)

Percent: unknown Base: 72 Amount: 9	Identify the percent, base, and amount.
\(\ n \cdot 72=9\)	Write the percent equation: \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\). Use \(\ n\) for the unknown (percent).
\(\ n=9 \div 72\)	Divide to undo the multiplication of \(\ n\) times 72.
\(\ \begin{array}{r} 0.125\ \\ 72\longdiv{9.000} \end{array}\)	Divide 9 by 72 to find the value for \(\ n\), the unknown.
\(\ n=0.125\) \(\ n=12.5 \%\)	Move the decimal point two places to the right to write the decimal as a percent.

\(\ \frac{20}{100}=\frac{\text { amount }}{\text { base }}\)	The percent in this problem is 20%. Write this percent in fractional form, with 100 as the denominator.
\(\ \frac{20}{100}=\frac{30}{n}\)	The percent is written as the ratio \(\ \frac{20}{100}\), the amount is 30, and the base is unknown.
\(\ \begin{array}{r} 20 \cdot n=30 \cdot 100 \\ 20 \cdot n=3,000 \\ n=3,000 \div 20 \\ n=150 \end{array}\)	Cross multiply and solve for the unknown, \(\ n\), by dividing 3,000 by 20.

\(\ \begin{array}{r} \text { Percent }=\frac{\text { amount }}{\text { base }} \\ \frac{n}{100}=\frac{9}{72} \end{array}\)	The percent is the ratio of \(\ n\) to 100. The amount is 9, and the base is 72.
\(\ \begin{array}{r} n \cdot 72=9 \cdot 100 \\ n \cdot 72=900 \\ n=900 \div 72 \\ n=12.5 \end{array}\)	Cross multiply and solve for \(\ n\) by dividing 900 by 72.
\(\ 12.5 \% \text { of } 72 \text { is } 9\)	The percent is \(\ \frac{12.5}{100}=12.5 \%\).

\(\ \begin{array}{l} \text { Percent }=\frac{\text { amount }}{\text { base }} \\ \frac{110}{100}=\frac{n}{24} \end{array}\)	The percent is the ratio \(\ \frac{110}{100}\). The amount is unknown, and the base is 24.
\(\ \begin{array}{r} 24 \cdot 110=100 \cdot n \\ 2,640 \div 100=n \\ 26.4=n \end{array}\)	Cross multiply and solve for \(\ n\) by dividing 2,640 by 100.
\(\ 26.4 \text { is } 110 \% \text { of } 24\)

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Percent: 110% Base: 24 Amount: unknown	Identify the percent, the base, and the amount.
\(\ 110 \% \cdot 24=n\)	Write the percent equation. \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\). The amount is unknown, so use \(\ n\).
\(\ 1.10 \cdot 24=n\)	Write the percent as a decimal by moving the decimal point two places to the left.
\(\ 1.10 \cdot 24=26.4=n\)	Multiply 24 by 1.10 or 1.1.

How much is 15% of $220?	Simplify the problems by eliminating extra words.
Percent: 15% Base: 220 Amount: \(\ n\)	Identify the percent, the base, and the amount.
\(\ 15 \% \cdot 220=n\)	Write the percent equation. \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\)
\(\ 0.15 \cdot 220=33\)	Convert 15% to 0.15, then multiply by 220. 15% of $220 is $33.

35 is what percent of 20?	Simplify the problem by eliminating extra words.
Percent: \(\ n\) Base: 20 Amount: 35	Identify the percent, the base, and the amount.
\(\ n \cdot 20=35\)	Write the percent equation. \(\ \text { Percent } \cdot \text { Base }=\text { Amount }\).
\(\ n=35 \div 20\)	Divide to undo the multiplication of \(\ n\) times 20.
\(\ n=1.75=175 \%\)	Convert 1.75 to a percent.