# 1.7 Percent

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

## Overview

• The Meaning of Percent
• Converting A Fraction To A Percent
• Converting A Decimal To A Percent
• Converting A Percent To A Decimal

## The meaning of percent

The word percent comes from the Latin word “per centum,” “per” meaning “for each,” and “centum” meaning “hundred.”

## Percent (%)

Percent means “for each hundred” or “for every hundred.” The symbol % is used to represent the word percent.

Thus, $\begin{array}{rrrrr}\hfill 1%=\frac{1}{100}& \hfill & \hfill \text{or}& \hfill & \hfill 1%=0.01.\end{array}$

## Converting a fraction to a percent

We can see how a fraction can be converted to a percent by analyzing the method that $\frac{3}{5}$ is converted to a percent. In order to convert $\frac{3}{5}$ to a percent, we need to introduce $\frac{1}{100}$ (since percent means for each hundred).

$\begin{array}{rrrrr}\hfill \frac{3}{5}& \hfill =& \frac{3}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{100}{100}\hfill & \hfill & \text{Multiply\hspace{0.17em}the\hspace{0.17em}fraction\hspace{0.17em}by\hspace{0.17em}1}.\hfill \\ \hfill & \hfill =& \frac{3}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill & \hfill & \text{Since \hspace{0.17em}}\frac{100}{100}=100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}.\hfill \\ \hfill & \hfill =& \frac{300}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill & \hfill & \text{Divide\hspace{0.17em}}300\text{\hspace{0.17em}by\hspace{0.17em}5}.\hfill \\ \hfill & \hfill =& 60\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill & \hfill & \text{Multiply\hspace{0.17em}the\hspace{0.17em}fractions}.\hfill \\ \hfill & \hfill =& 60%\hfill & \hfill & \text{Replace\hspace{0.17em}}\frac{1}{100}\text{\hspace{0.17em}}\text{with\hspace{0.17em}the\hspace{0.17em}}%\text{\hspace{0.17em}symbol}.\hfill \end{array}$

## Fraction to percent

To convert a fraction to a percent, multiply the fraction by 1 in the form $100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}$ , then replace $\frac{1}{100}$ with the % symbol.

## Sample set a

Convert each fraction to a percent.

$\begin{array}{lll}\frac{1}{4}\hfill & =\hfill & \frac{1}{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \frac{100}{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 25\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 25%\hfill \end{array}$

$\begin{array}{lll}\frac{8}{5}\hfill & =\hfill & \frac{8}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \frac{800}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 160%\hfill \end{array}$

$\begin{array}{lll}\frac{4}{9}\hfill & =\hfill & \frac{4}{9}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \frac{400}{9}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \left(44.4...\right)\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \left(44.\overline{4}\right)\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 44.\overline{4}%\hfill \end{array}$

## Converting a decimal to a percent

We can see how a decimal is converted to a percent by analyzing the method that $0.75$ is converted to a percent. We need to introduce $\frac{1}{100}.$

$\begin{array}{lllll}0.75\hfill & =\hfill & 0.75\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill & \hfill & \text{Multiply\hspace{0.17em}the\hspace{0.17em}decimal\hspace{0.17em}by\hspace{0.17em}1}\text{.}\hfill \\ \hfill & =\hfill & 75\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 75%\hfill & \hfill & \text{Replace\hspace{0.17em}}\frac{1}{100}\text{\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}%\hspace{0.17em}symbol}.\hfill \end{array}$

## Decimal to percent

To convert a fraction to a percent, multiply the decimal by 1 in the form $100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}$ , then replace $\frac{1}{100}$ with the % symbol. This amounts to moving the decimal point 2 places to the right.

## Sample set b

Convert each decimal to a percent.

$\begin{array}{lll}0.62\hfill & =\hfill & 0.62\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 62\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 62%\hfill \end{array}$

Notice that the decimal point in the original number has been moved to the right 2 places.

$\begin{array}{lll}8.4\hfill & =\hfill & 8.4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 840\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 840%\hfill \end{array}$

Notice that the decimal point in the original number has been moved to the right 2 places.

$\begin{array}{lll}0.47623\hfill & =\hfill & 0.47623\text{\hspace{0.17em}}·\text{\hspace{0.17em}}100\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 0.47623\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & 47.623%\hfill \end{array}$

Notice that the decimal point in the original number has been moved to the right 2 places.

## Converting a percent to a decimal

We can see how a percent is converted to a decimal by analyzing the method that 12% is converted to a decimal. We need to introduce $\frac{1}{100}.$

$\begin{array}{lllll}12%\hfill & =\hfill & 12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill & \hfill & \text{Replace}\text{\hspace{0.17em}}%\text{\hspace{0.17em}}\text{with}\text{\hspace{0.17em}}\frac{1}{100}.\hfill \\ \hfill & =\hfill & \frac{12}{100}\hfill & \hfill & \text{Multiply\hspace{0.17em}the\hspace{0.17em}fractions}.\hfill \\ \hfill & =\hfill & 0.12\hfill & \hfill & \text{Divide\hspace{0.17em}12\hspace{0.17em}by\hspace{0.17em}1}00.\hfill \end{array}$

## Percent to decimal

To convert a percent to a decimal, replace the % symbol with $\frac{1}{100},$ then divide the number by 100. This amounts to moving the decimal point 2 places to the left.

## Sample set c

Convert each percent to a decimal.

$\begin{array}{lll}48%\hfill & =\hfill & 48\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \frac{48}{100}\hfill \\ \hfill & =\hfill & 0.48\hfill \end{array}$

Notice that the decimal point in the original number has been moved to the left 2 places.

$\begin{array}{lll}659%\hfill & =\hfill & 659\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \frac{659}{100}\hfill \\ \hfill & =\hfill & 6.59\hfill \end{array}$

Notice that the decimal point in the original number has been moved to the left 2 places.

$\begin{array}{lll}0.4113%\hfill & =\hfill & 0.4113\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{100}\hfill \\ \hfill & =\hfill & \frac{0.4113}{100}\hfill \\ \hfill & =\hfill & 0.004113\hfill \end{array}$

Notice that the decimal point in the original number has been moved to the left 2 places.

## Exercises

For the following problems, convert each fraction to a percent.

$\frac{2}{5}$

$40%$

$\frac{7}{8}$

$\frac{1}{8}$

$12.5%$

$\frac{5}{16}$

$15÷22$

$68.18%$

$\frac{2}{11}$

$\frac{2}{9}$

$22.22%$

$\frac{16}{45}$

$\frac{27}{55}$

$49.09%$

$\frac{7}{27}$

15

$1500%$

8

For the following problems, convert each decimal to a percent.

$0.36$

$36%$

$0.42$

$0.446$

$44.6%$

$0.1298$

$4.25$

$425%$

$5.875$

$86.98$

$8698%$

$21.26$

14

$1400%$

12

For the following problems, convert each percent to a decimal.

$35%$

$0.35$

$76%$

$18.6%$

$0.186$

$67.2%$

$9.0145%$

$0.090145$

$3.00156%$

$0.00005%$

$0.0000005$

$0.00034%$

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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