5.2.1: Solving Percent Problems

• The NROC Project

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 amountoff 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.

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.

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

Base is: unknown

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 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