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.

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]

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

From the previous example, we can identify three important parts to finding the percent of a whole:

• the percent,  has the percent symbol (%) or the word “percent”
• the amount, the amount is part of the whole
• and the base, the base is the whole amount

The following examples show how to identify the three parts: the percent, the base, and the amount.

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?

The following video provides more examples that describe how to identify the percent, amount, and base in a percent problem.

In the next section, you will use the parts of a percent problem to find the percent increase or decrease of a quantity by writing and solving equations.

Percent 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.

In the previous section, we identified three important parts to finding the percent of a whole:

• the percent, has the percent symbol (%) or the word “percent”
• the amount, the amount is part of the whole
• and the base, the base is the whole amount

Using these parts, we can define equations that will help us answer percent problems.

In the examples below, the unknown is represented by the letter n. The unknown can be represented by any letter or a box □, question mark, or even a smiley face :)

The following example shows how to use the percent equation to find the base in a percent equation.

Once you have an equation, you can solve it and find the unknown value. For example, to solve
[latex]20%\cdot{n}=30[/latex]

you can divide 30 by 20% to find the unknown:

[latex]20\%\cdot{n}=30[/latex]

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

[latex]20\%\cdot{n}=30[/latex]

[latex]n=30\div20\%=30\div0.20=150[/latex]

In the following video example, you are shown how to use the percent equation to find the base in a percent problem.

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% of 72 = 0.1 · 72 = 7.2

20% of 72 = 0.2 · 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%.

The video that follows shows how top use the percent equation to find the amount in a percent equation.

Percent Change

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.

The example video that follows shows how to use the percent equation to find the amount of a discount from the price of a phone.

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

[latex]\begin{array}{c}10\%\,\,\text{of}\,\,220=0.1\cdot220=22\\20\%\,\,\text{of}\,\,220=0.2\cdot220=44\end{array}[/latex]

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.

In the following video example, you will be shown how to find a price before tax using the percent equation.

The video that follows explains how to use the percent equation to determine the percent increase of a given amount.

Interest

When a person takes out a loan, most lenders charge interest on the loan. Interest is a fee or change for borrowing money, typically a percent rate charged per year. We can compute simple interest by finding the interest rate percentage of the amount borrowed, then multiply by the number of years interest is earned.

In the following video, you are shown how to find how much interest is earned on a specified investment amount.

The example video that follows shows how to determine the annual simple interest rate.

Pie Charts

Circle graphs, or pie charts, represent data as sections of the circle (or “pieces of the pie”), corresponding to their percentage of the whole. Circle graphs are often used to show how a whole set of data is broken down into individual components.

Here’s an example. At the beginning of a semester, a teacher talks about how she will determine student grades. She says, “Half your grade will be based on the final exam and 20% will be determined by quizzes. A class project will also be worth 20% and class participation will count for 10%.” In addition to telling the class this information, she could also create a circle graph.

This graph is useful because it relates each part—the final exam, the quizzes, the class project, and the class participation—to the whole.

In the following video, an example of using a pie chart to determine a percent of a whole is shown.

Summary

When solving application problems with percents, it is important to be extremely careful in identifying the percent, whole, and amount in the problem. Once those are identified, use the percent equation to solve the problem. Write your final answer back in terms of the original scenario.
Put it Together

While the forgotten food in your fridge may not be what you spend your work day dreaming about devouring when you get home, those jumbled bits and pieces can be used to create something tasty enough to satisfy your hunger. Hopefully this section offered you enough tidbits to keep you full long enough to get to the next section, where you will learn how to put these forgotten skills to work as you continue to grow your mathematical prowess. Remember, you can come back to this review any time throughout the course for practice dividing fractions, simplifying with square roots, or whatever you might need to help you learn the concepts in this course more easily.

Oh, and now you know that [latex]3+5•\times2[/latex] is 13!