1.8: Percent

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 $\PageIndex{1}$

$\dfrac{2}{5}$

Answer

40%

Exercise $\PageIndex{2}$

$\dfrac{7}{8}$

Exercise $\PageIndex{3}$

$\dfrac{1}{8}$

Answer

12.5%

Exercise $\PageIndex{4}$

$\dfrac{5}{16}$

Exercise $\PageIndex{5}$

$15 \div 22$

Answer

66.18%

Exercise $\PageIndex{6}$

$\dfrac{2}{11}$

Exercise $\PageIndex{7}$

$\dfrac{2}{9}$

Answer

22.22%

Exercise $\PageIndex{8}$

$\dfrac{16}{45}$

Exercise $\PageIndex{9}$

$\dfrac{27}{55}$

Answer

49.09%

Exercise $\PageIndex{10}$

$\dfrac{7}{27}$

Exercise $\PageIndex{11}$

15

Answer

1500%

Exercise $\PageIndex{12}$

8

For the following problems, convert each decimal to a percent

Exercise $\PageIndex{13}$

0.36

Answer

36%

Exercise $\PageIndex{14}$

0.42

Exercise $\PageIndex{15}$

0.446

Answer

44.6%

Exercise $\PageIndex{16}$

0.1298

Exercise $\PageIndex{17}$

4.25

Answer

425%

Exercise $\PageIndex{18}$

5.785

Exercise $\PageIndex{19}$

86.98

Answer

8698%

Exercise $\PageIndex{20}$

21.26

Exercise $\PageIndex{21}$

14

Answer

1400%

Exercise $\PageIndex{22}$

12

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

Exercise $\PageIndex{23}$

35%

Answer

0.35

Exercise $\PageIndex{24}$

76%

Exercise $\PageIndex{25}$

18.6%

Answer

0.186

Exercise $\PageIndex{26}$

67.2%

Exercise $\PageIndex{27}$

9.0145%

Answer

0.090145

Exercise $\PageIndex{28}$

3.00156%

Exercise $\PageIndex{29}$

0.00005%

Answer

0.0000005

Exercise $\PageIndex{30}$

0.00034%