6.1 Understanding Percent

The federal budget in the fiscal year 2021. The two circled flow chart shows total outlays and total revenues. The total outlay is $6.8 trillion and 30.5 percent of the G D P. Three categories are labeled mandatory, discretionary, and net interest, and the G D P percentages are 21.6, 7.3, and 1.6 respectively. The total revenue is $4.0 trillion and 18.1 percent of the G D P. Four categories are labeled individual income taxes, payroll taxes, corporate income taxes, and other, and the G D P percentages are 9.1, 5.9, 1.7, and 1.4 respectively. — **Figure 6.2** The federal budget describes how money is spent and how money is earned.The federal budget describes how money is spent and how money is earned. (credit: "Breakdown of revenues and outlays in 2021 US Federal budget" Wikimedia Commons, Public Domain)

Learning Objectives

After completing this section, you should be able to:

Define and calculate percent.
Convert between percent, decimal, and fractional values.
Calculate the total, percent, or part.
Solve application problems involving percents.

In 2020, the U.S. federal government budgeted $3.5 billion for the National Park Service, which appears to be a very large number (and is!) and a large portion of the total federal budget. However, the total outlays from the U.S. federal government in 2020 was $6.6 trillion. So, the amount budgeted for the National Park Service was less than one-tenth of 1 percent, or 1/10%, of the total outlays. This percent describes a specific number. Understanding that ratio puts the $3.5 billion budgeted to the National Park Service in perspective.

This chapter focuses on percent as a primary tool for understanding money management. The interest paid on debt, the interest earned through investments, and even taxes are entirely determined using percent. This section introduces the basics of working with this invaluable tool.

Define and Calculate Percent

The word percent comes from the Latin phrase per centum, which means “by the hundred.” So any percent is a number divided by 100. Changing a percent to a fraction is to write the percent in its fractional form. To write $n$ % in its fractional form is to write the percent as the fraction $\frac{n}{100}$ .

Convert Between Percent, Decimal, and Fractional Values

When any calculation with a percent is to be performed, the form of the percent must be changed, either to its fractional form or its decimal form. We can change a percent into decimal form by dividing the percent by 100 and representing the result as a decimal.

You should notice that, to convert from percent to decimal form, you can simply move the decimal two places to the left without performing the division.

You should notice that, to convert from decimal form to percent form, you can simply move the decimal two places to the right without performing the multiplication.

Calculate the Total, Percent, or Part

The word “of” is used to indicate multiplication using fractions, as in “one-fourth of 56.” To find “one-fourth of 56” we would multiply 56 by one-fourth. We can think of percents as fractions with a specific denominator—100. So, to calculate “25% of 52,” we multiply 52 by 25%. But, first we need to convert the percent to either fractional form (25/100) or decimal form. Using the decimal form of 25% we have 0.25 × 52, which equals 13.

In this problem, 52 is the total or base, 25 is the percentage, and 13 is the percentage of 52, or the part of 52. This is sometimes referred to as the amount.

Knowing any two of the values in our formula allows us to calculate the third value. In the following example, we know the total and the percent, and are asked to find the percentage of the total.

In the previous example, we knew the total and the percent and found the part using our formula. We may instead know the percent and the part, but not the total. We can use our formula again to solve for the total.

Example 5 Finding the Total from the Percent and the Part

What is the total if 35% of the total is 70?
What is the total if 10% of the total is 4,000?

Step 1: The percent is 35, which in decimal form is 0.35. We were given that 35% of the total is 70, so the part is 70. We are to find the total. Substituting into the formula, we have
$\begin{matrix} part & = & percent \times total \\ 70 & = & 0.35 \times total \end{matrix}$

Step 2: To find the total, we solve the equation for the total.
$\begin{matrix} 70 & = & 0.35 \times total \\ \frac{70}{0.35} & = & \frac{0.35 \times total}{0.35} \\ 200 & = & \frac{0.35 \times total}{0.35} \\ 200 & = & total \end{matrix}$

From this we see that 200 is the total, or, that 35% of 200 is 70.
Step 1: The percent is 10, which in decimal form is 0.1. We were given that 10% of the total is 4,000, so the part is 4,000. Substituting into the formula, we have
$\begin{matrix} part & = & percent \times total \\ 4,000 & = & 0.1 \times total \end{matrix}$

Step 2: To find the total, we solve the equation for the total.
$\begin{matrix} 4,000 & = & 0.1 \times total \\ \frac{4,000}{0.1} & = & \frac{0.1 \times total}{0.1} \\ 40,000 & = & \frac{0.1 \times total}{0.1} \\ 40,000 & = & total \end{matrix}$

From this we see that 40,000 is the total, or that 10% of 40,000 is 4,000.

Similarly, the percent can be found if the total and the percent of the total (the part) are known. This will result in the decimal form of the percent, so it must be converted to percent form.

Example 6 Finding the Percent from the Total and the Part

What percent of 500 is 175?
What percent of 228 is 155?

Step 1: The total is 500, the percent of the total is 175. Substituting into the formula, we have
$\begin{matrix} part & = & percent \times total \\ 175 & = & percent \times 500 \end{matrix}$

Step 2: To find the percent, we solve the equation for the percent.
$\begin{matrix} 175 & = & percent \times 500 \\ \frac{175}{500} & = & \frac{percent \times 500}{500} \\ 0.35 & = & \frac{percent \times 500}{500} \\ 0.35 & = & percent \end{matrix}$

We see the percent in decimal form is 0.35. Converting from the decimal form yields 35%. We say that 175 is 35% of 500.
Step 1: The total is 228, the percent of the total is 155. Substituting into the formula, we have
$\begin{matrix} part & = & percent \times total \\ 155 & = & percent \times 228 \end{matrix}$

Step 2: To find the percent, we solve the equation for the percent.
$\begin{matrix} 155 & = & percent \times 228 \\ \frac{155}{228} & = & \frac{percent \times 228}{228} \\ 0.6798 & = & \frac{percent \times 228}{228} \\ 0.6798 & = & percent \end{matrix}$

We see the percent is 0.6798 (rounded to four decimal places). Converting from the decimal form yields 67.98%. We say that 155 is 67.98% of 228.

Solve Application Problems Involving Percents

Percents are frequently used in finance, research, science experiments, and even casual conversation. Understanding these types of values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percents will help you become an informed consumer of this information.

In most cases, working through what is presented requires you to identify that you are indeed working with a question of percents, which two of the three values that are related through percents are known, and which of the three values you need to find.

Example 7 Retention Rate at College

Justine applies to a medium size university outside her hometown and finds out that the retention rate (percent of students who return for their sophomore year) for the 2021 academic year at the university was 84%. During a visit to the registrar’s office, she finds out that 1,350 people had enrolled in academic year 2021. How many students from the academic year 2021 are returning for the 2022 academic year?

The percent of students who will return for the 2022 academic year (the retention rate) is 84%. The total number of students who enrolled in the 2021 academic year was 1,350. This means the percent is known and the total is known. From this, we can determine the number of students who will return (percent of the total) for the 2022 academic year using the formula $part = percent \times total$ . Substituting into the formula and calculating, we find that the number of students that are returning is

$\begin{matrix} part & = & percent \times total \\ = & 0.84 \times 1,350 \\ = & 1,134 \end{matrix}$

So 1,134 students will return for the 2022 academic year.

For any answer, round to two decimal places, if necessary.

Key Terms

Percent
Fractional form
Decimal form
Total
Base
Percent of the total
Part
Amount

Key Concepts

Know what a percent is as a fraction, a decimal, and as a part of the whole.
Use the percent equation to find any of the three values that are related by the equation.
Apply the percent equation in applications.

Video

Formulas

part = percent x total

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.