# 1.8: Percent

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## 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 (%)

##### Definition: Percent

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

Thus, $$1\%$$ = $$\dfrac{1}{100}$$ or $$1\%$$ = $$0.01$$

## Converting A Fraction To A Percent

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

##### Example $$\PageIndex{1}$$

\begin{aligned} \dfrac{3}{5} &=\dfrac{3}{5} \cdot \dfrac{100}{100} & \text { Multiply the fraction by } 1 . \\ &=\dfrac{3}{5} \cdot 100 \cdot \dfrac{1}{100} & \text { Since } \dfrac{100}{100}=100 \cdot \dfrac{1}{100} \\ &=\dfrac{300}{5} \cdot \dfrac{1}{100} & \text { Divide } 300 \text { by } 5 \\&=60 \cdot \dfrac{1}{100} & \text{ Multiply the fractions. } \\&=60 \% & \text{ Replace } \dfrac{1}{100} \text{ with the % symbol.} \end{aligned}

##### Fraction to Percent

To convert a fraction to a percent, multiply the fraction by $$1$$ in the form $$100 \cdot \dfrac{1}{100}$$, then replace $$\dfrac{1}{100}$$ with the $$\%$$ symbol.

## Sample Set A

Convert each fraction to a percent.

##### Example $$\PageIndex{2}$$

\begin{aligned} \dfrac{1}{4} &=\dfrac{1}{4} \cdot 100 \cdot \dfrac{1}{100} \\ &=\dfrac{100}{4} \cdot \dfrac{1}{100} \\ &=25 \cdot \dfrac{1}{100} \\ &=25 \% \end{aligned}

##### Example $$\PageIndex{2}$$

\begin{aligned} \dfrac{8}{5} &=\dfrac{8}{5} \cdot 100 \cdot \dfrac{1}{100} \\ &=\dfrac{800}{5} \cdot \dfrac{1}{100} \\ &=160 \% \end{aligned}

##### Example $$\PageIndex{2}$$

\begin{aligned} \dfrac{4}{9} &=\dfrac{4}{9} \cdot 100 \cdot \dfrac{1}{100} \\ &=\dfrac{400}{9} \cdot \dfrac{1}{100} \\ &=(44.4 \ldots) \cdot \dfrac{1}{100} \\ &=(44 . \overline{4}) \cdot \dfrac{1}{100} \\ &=44 . \overline{4} \% \end{aligned}

## 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 $$\dfrac{1}{100}$$.

##### Example $$\PageIndex{3}$$

\begin{aligned} &0.75=0.75 \cdot 100 \cdot \dfrac{1}{100} \quad \text { Multiply the decimal by } 1\\ &\begin{array}{l} =75 \cdot \dfrac{1}{100} \\ =75 \% \end{array} \quad \quad \text { Replace } \dfrac{1}{100} \text { with the } \% \text { symbol. } \end{aligned}

##### Decimal to Percent

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

## Sample Set B

Convert each decimal to a percent.

##### Example $$\PageIndex{4}$$

\begin{aligned} 0.62 &=0.62 \cdot 100 \cdot \dfrac{1}{100} \\ &=62 \cdot \dfrac{1}{100} \\ &=62 \% \end{aligned}

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

##### Example $$\PageIndex{5}$$

\begin{aligned} 8.4 &=8.4 \cdot 100 \cdot \dfrac{1}{100} \\ &=840 \cdot \dfrac{1}{100} \\ &=840 \% \end{aligned}

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

##### Example $$\PageIndex{6}$$

\begin{aligned} 0.47623 &=0.47623 \cdot 100 \cdot \dfrac{1}{100} \\ &=47.623 \cdot \dfrac{1}{100} \\ &=47.623 \% \end{aligned}

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 $$\dfrac{1}{100}$$.

##### Example $$\PageIndex{7}$$

\begin{aligned} &12 \%=12 \cdot \dfrac{1}{100} \quad \text { Replace } \% \text { with } \dfrac{1}{100}\\ &\begin{array}{ll} =\dfrac{12}{100} & \text { Multiply the fractions. } \\ =0.12 & \text { Divide } 12 \text { by } 100 . \end{array} \end{aligned}

##### Percent to Decimal

To convert a percent to a decimal, replace the % symbol with $$\dfrac{1}{100}$$, 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.

##### Example $$\PageIndex{8}$$

\begin{aligned} 48 \% &=48 \cdot \dfrac{1}{100} \\ &=\dfrac{48}{100} \\ &=0.48 \end{aligned}

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

##### Example $$\PageIndex{9}$$

\begin{aligned} 659 \% &=659 \cdot \dfrac{1}{100} \\ &=\dfrac{659}{100} \\ &=6.59 \end{aligned}

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

##### Example $$\PageIndex{10}$$

\begin{aligned} 0.4113 \% &=0.4113 \cdot \dfrac{1}{100} \\ &=\dfrac{0.4113}{100} \\ &=0.004113 \end{aligned}

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.

##### Exercise $$\PageIndex{1}$$

$$\dfrac{2}{5}$$

40%

##### Exercise $$\PageIndex{2}$$

$$\dfrac{7}{8}$$

##### Exercise $$\PageIndex{3}$$

$$\dfrac{1}{8}$$

12.5%

##### Exercise $$\PageIndex{4}$$

$$\dfrac{5}{16}$$

##### Exercise $$\PageIndex{5}$$

$$15 \div 22$$

66.18%

##### Exercise $$\PageIndex{6}$$

$$\dfrac{2}{11}$$

##### Exercise $$\PageIndex{7}$$

$$\dfrac{2}{9}$$

22.22%

##### Exercise $$\PageIndex{8}$$

$$\dfrac{16}{45}$$

##### Exercise $$\PageIndex{9}$$

$$\dfrac{27}{55}$$

49.09%

##### Exercise $$\PageIndex{10}$$

$$\dfrac{7}{27}$$

15

1500%

##### Exercise $$\PageIndex{12}$$

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%

5.785

86.98

8698%

21.26

14

1400%

##### Exercise $$\PageIndex{22}$$

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%