1.8: Applications Involving Percent

SECTION 1.8 Learning Objectives

1.8: Applications Involving Percent

  • Use an algebraic equation to solve a basic percent question
  • Use an algebraic equation to solve application problems involving percent

Percent of a Whole

Percents are the ratio of a number and 100. Percents are used in many different applications. Percents are used widely to describe how something changed. For example, you may have heard that the amount of rainfall this month had decreased by 12% from last year, or that the number of jobless claims has increase by 5% this quarter over last quarter.

A graph showing the unemployment rate, with the y-axis representing percent and the x-axis representing time.

Unemployment rate as percent by year between 2004 and 2014.

We regularly use this kind of language to quickly describe how much something increased or decreased over time or between significant events.

Before we dissect the methods for finding percent change of a quantity, let’s learn the basics of finding percent of a whole.

For example, if we knew a gas tank held 14 gallons, and wanted to know how many gallons were in [latex]\frac{1}{4}[/latex] of a tank, we would find [latex]\frac{1}{4}[/latex] of 14 gallons by multiplying:

[latex] \frac{1}{4}\,\cdot \,14=\frac{1}{4}\,\cdot \,\frac{14}{1}=\frac{14}{4}=3\frac{2}{4}=3\frac{1}{2}\,\,\,\text{gallons}[/latex]

Likewise, if we wanted to find 25% of 14 gallons, we could find this by multiplying, but first we would need to convert the 25% to a decimal:

[latex]25\%\,\,\text{of}\,\,14\,\,\,\text{gallons}=0.25\,\cdot \,14=3.5\,\,\,\text{gallons}[/latex]

 

Finding a Percent of a Whole

To find a percent of a whole,

  • Write the percent as a decimal by moving the decimal two places to the left
  • Then multiply the percent by the whole amount

Example 1

What is 10% of $200?

Notice in the example above that 10% of 200 is 20. Finding 10% of a number is one of the easiest percentages to calculate. To calculate 10 percent of a number, we always simply divide it by 10 or move the decimal point one place to the left.  For example, 10% of 325 is 32.5.  Being able to recall what 10% of a value is might often be helpful when reflecting on the reasonableness of your answers to problems involving different percentages as you will see in some examples in this section.

The following video contains an example that is similar to the one above.

 

Use an algebraic equation to solve a basic percent question

From the previous examples, we can identify some important parts to finding the percent of a whole. We are going to focus on the equation method to solve problems involving percent.  To translate the following problems into equations, we need to know a few key words and what they mean.

  • “of” means multiply
  • “is” means equals
  • “what” can be represented by the variable

The following examples show how to use an algebraic equation to solve a basic percent question:

Example 2

What is 12% of 270?

 

Example 3

30 is 20% of what number?

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

Example 4

What percent of 72 is 9?

Below are some additional video examples to view:

Try some of the examples on your own before revealing the answer.

Example 5

What is 110% of 24?

Example 6

What percent of 70.52 is 13.27?  Round answer to the nearest tenth of a percent.

Note: Often times, problems like the ones in this section will ask you to round your final answer. If you need a review on rounding, refer back to Module 0 of this textbook:  Module 0: Rounding Decimals Review

Use an algebraic equation to solve application problems involving percent

Percents have a wide variety of applications to everyday life, showing up regularly in taxes, discounts, markups, and interest rates. We will look at several examples of how to use percent to calculate markups, discounts, and interest earned or owed.

Example 7

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 of the $220 original price.

Example 8

Tracey paid $185 for an item that was originally priced at $390.  What percent of the original price did Tracey pay?  Round answer to the nearest tenth of a percent.