# 2.3: Percents

- Page ID
- 130927

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The final two sections of this part will focus on a very specific type of ratio: a *percent*. In this chapter we'll focus on the one basic equation that describes all problems involving basic percents, and practice using it in context.

- Understand and use the word "percent" in context
- Recognize and convert between various representations of percents
- Use the Basic Percent Equation to solve problems involving percents

## Definition and Representation of Percents

We start with the main definition:

A **percent** is a ratio with a denominator of 100. The percent sign \(\%\) is equivalent to the fraction \(\frac{n}{100}\).

Recall that the denominator is the bottom of the fraction; that is, the second number in the ratio. The word *percent* literally means "per \(100\)." That is, a percent expressed as a portion of a whole when that whole is divided into hundredths.

A quick note about language: some people use the words *percent* and *percentage* interchangeably, but technically they don't mean the same thing. A percentage is a relative amount (for example, "a large percentage") and a percent is a specific amount (for example, "\(20\) percent"). This isn't a point we'll belabor, but this book will specifically use the word *percent* when referring to a specific ratio.

Before we solve problems involving percents, it's important to know that a percent can be expressed in three ways: using a percent sign, as a decimal, or as a fraction. It is helpful to know how to convert between these. We'll do a couple examples in detail, and then provide a table of other examples for reference.

- Express \(96\%\) as a decimal and as a fraction.
- Express \(\frac{3}{20}\) as a decimal and as a percent.

###### Solution

- The percent denoted \(96\%\) is equivalent to the fraction \(\frac{96}{100}\) by definition of percent. The equivalent decimal is \(.96\).
- Calculating \(\frac{3}{20} = 3 \div 20\), we see that the fraction \(\frac{3}{20}\) is equivalent to the decimal \(.15\). This decimal is \(15\) hundredths, or \(\frac{15}{100}\) so the corresponding percent is \(15\%\). Note in this example we see that multiple fractions can correspond to the same percent.

In the previous example, you may have noticed a quick way to convert between decimals and percents:

To convert a decimal to a percent, move the decimal point two places *right*, and then put a percent sign at the end. For example: \[.127 = 12.7\%\]

To convert a percent to a decimal, move the decimal point two places *left*, and then drop the percent sign. For example: \[56.3\% = .563\]

Additional examples are shown below to make these rules more clear:

Percent |
Decimal |
Fraction |

\(14\%\) | \(.14\) | \(\frac{14}{100}\) |

\(8\%\) | \(.08\) | \(\frac{8}{100}\) |

\(.5\%\) or "half a percent" | \(.005\) | \(\frac{.5}{100}\) or \(\frac{5}{1000}\) |

\(25\%\) | \(.25\) | \(\frac{25}{100}\) or \(\frac{1}{4}\) |

\(123\%\) | \(1.23\) | \(\frac{123}{100}\) |

## The Basic Percent Equation

There is one equation that governs all percent relationships.

The equation that describes all relationships involving percents is

\[\text{percent } \times \text{ whole } = \text{ part}\]

We will call this the** Basic Percent Equation**. It will be used to solve all basic percent problems.

The Basic Percent Equation can be rearranged using Division Undoes Multiplication in the following way:

\[\text{percent } = \frac{\text{part}}{\text{whole}}\]

or as

\[\text{whole } = \frac{\text{part}}{\text{percent}}\]

There is one important caveat to using this equation:

**To use the Basic Percent Equation, your percent must be expressed** ** as a decimal**.

Let's see some examples of the Basic Percent Equation in action.

\(91%\) of people in the world are right-handed. In a randomly selected group of \(745\) people, how many do you expect to be right-handed?

###### Solution

We need to identify the percent, whole, and part in this equation. The percent, if given, will always have a percent sign next to it. So the percent in this case is \(91\%\), which we need to express *as the decimal,* \(.91\).

In this problem, the number \(745\) represents the whole, because that is the total number of people. We are being asked to find the *part* of that whole that is right-handed. So we will be finding the part.

We use the Basic Percent Equation: \[\text{percent} \times \text{whole} = \text{part}\] and fill in what we know, which is \(\text{percent } =.91\) and whole \(=745\). We now have \[.91 \times 745= \text{part}\] Once we compute the left side, we find that \[ 677.95 = \text{ part}\] We will round this to the nearest whole since we're talking about a number of people. Thus, \(678\) people will be right-handed.

Here's another example:

According to the WOU website, \(34\%\) of WOU undergraduates are men. Suppose there are currently \(1123\) undergraduate men at WOU. How many total undergraduates are there at WOU?

###### Solution

Once again, we start by identifying the percent, whole, and part in this equation. The percent is \(34\%\), which we express as the decimal \(.34\). In this case, we are being asked to find the total, so the "whole" is the unknown quantity. We are told that the part — the number of undergraduate men — is equal to \(1123\).

Using the Basic Percent Equation,

\[\text{percent} \times \text{ whole } = \text{part}\]

we substitute what we are given:

\[\text{percent} \times \text{whole } = \text{part}\]

\[.34 \times \text{ whole } = 1123\]

Now, since Division undoes Multiplication, we have that

\[\text{whole } = \frac{1123}{.34} \approx 3306\]

Since this is a number of people, we rounded to the nearest whole. This means that there are approximately \(3306\) WOU undergraduates.

In the next section, we'll see some particular applications of percents.

## Exercises

Remember to read carefully and answer the question that is being asked!

- Fill in the missing spots in this table. Copy the entire table in your answer.
**Percent****Decimal****Fraction**\(.24\) \(.1\%\) \(.098\) \(\frac{19}{100}\) \(238\%\) \(\frac{1}{5}\) - Twelve percent of Polk County residents speak Spanish fluently. There are \(9048\) fluent Spanish speakers in Polk County. How many total residents are there in Polk County?
- In a certain dorm on campus, \(13\) people are social science majors, \(12\) people are natural science majors, \(17\) people are education majors, and \(9\) people have other majors. What percentage of people in the dorm are natural science majors? Round to the nearest tenth of a percent.
- In a certain acre of forest, there are \(457\) deciduous trees and \(1035\) trees in total. What percentage of the trees in this acre of forest are non-deciduous? Round to the nearest tenth of a percent.
- You're trying to save up to put a \(10\%\) down payment on a house. You hope to purchase a \(\$315,000\) house. Your plan to save equal amounts of money each month for four years to reach your goal. How much will you need to save each month? (Assume the saved money earns no interest.)
- Read this article. After reading the article, answer the following questions:
- Oregon's population in \(2020\), when this article was written, was estimated to be \(4,455,920\) people. According to the statistics given in this article, what number of people in Oregon are non-Hispanic white people? Make sure to show your calculation.
- Assume there are \(135,438\) people of color under the age of \(15\) living in Oregon. According to a statistic given in this article (in the second half), calculate how many total people under the age of \(15\) live in Oregon. Make sure to show your calculation.
- Answer at least one of the following questions, writing at least \(2\) sentences.
- Did anything surprise you when reading this article? If so, what was it?
- Are you curious about other statistics relating to the population of Oregon? If so, what would you ask?