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 (%)
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).
\(
\begin{aligned} \dfrac{3}{5} &=\dfrac{3}{5} \cdot \dfrac{100}{100} & \text { Multiply the fraction by } 1 . \\ &=\dfrac{3}{5} \cdot 100 \cdot \dfrac{1}{100} & \text { Since } \dfrac{100}{100}=100 \cdot \dfrac{1}{100} \\ &=\dfrac{300}{5} \cdot \dfrac{1}{100} & \text { Divide } 300 \text { by } 5
\\&=60 \cdot \dfrac{1}{100} & \text{ Multiply the fractions. }
\\&=60 \% & \text{ Replace } \dfrac{1}{100} \text{ with the % symbol.}
\end{aligned}
\)
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.
\(
\begin{aligned}
\dfrac{1}{4} &=\dfrac{1}{4} \cdot 100 \cdot \dfrac{1}{100} \\
&=\dfrac{100}{4} \cdot \dfrac{1}{100} \\
&=25 \cdot \dfrac{1}{100} \\
&=25 \%
\end{aligned}
\)
\(
\begin{aligned}
\dfrac{8}{5} &=\dfrac{8}{5} \cdot 100 \cdot \dfrac{1}{100} \\
&=\dfrac{800}{5} \cdot \dfrac{1}{100} \\
&=160 \%
\end{aligned}
\)
\(
\begin{aligned}
\dfrac{4}{9} &=\dfrac{4}{9} \cdot 100 \cdot \dfrac{1}{100} \\
&=\dfrac{400}{9} \cdot \dfrac{1}{100} \\
&=(44.4 \ldots) \cdot \dfrac{1}{100} \\
&=(44 . \overline{4}) \cdot \dfrac{1}{100} \\
&=44 . \overline{4} \%
\end{aligned}
\)
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 \(\dfrac{1}{100}\).
\(
\begin{aligned}
&0.75=0.75 \cdot 100 \cdot \dfrac{1}{100} \quad \text { Multiply the decimal by } 1\\
&\begin{array}{l}
=75 \cdot \dfrac{1}{100} \\
=75 \%
\end{array} \quad \quad \text { Replace } \dfrac{1}{100} \text { with the } \% \text { symbol. }
\end{aligned}
\)
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.
\(
\begin{aligned}
0.62 &=0.62 \cdot 100 \cdot \dfrac{1}{100} \\
&=62 \cdot \dfrac{1}{100} \\
&=62 \%
\end{aligned}
\)
Notice that the decimal point in the original number has been moved to the right 2 places.
\(
\begin{aligned}
8.4 &=8.4 \cdot 100 \cdot \dfrac{1}{100} \\
&=840 \cdot \dfrac{1}{100} \\
&=840 \%
\end{aligned}
\)
Notice that the decimal point in the original number has been moved to the right 2 places.
\(
\begin{aligned}
0.47623 &=0.47623 \cdot 100 \cdot \dfrac{1}{100} \\
&=47.623 \cdot \dfrac{1}{100} \\
&=47.623 \%
\end{aligned}
\)
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 \(\dfrac{1}{100}\).
\(
\begin{aligned}
&12 \%=12 \cdot \dfrac{1}{100} \quad \text { Replace } \% \text { with } \dfrac{1}{100}\\
&\begin{array}{ll}
=\dfrac{12}{100} & \text { Multiply the fractions. } \\
=0.12 & \text { Divide } 12 \text { by } 100 .
\end{array}
\end{aligned}
\)
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.
\(
\begin{aligned}
48 \% &=48 \cdot \dfrac{1}{100} \\
&=\dfrac{48}{100} \\
&=0.48
\end{aligned}
\)
Notice that the decimal point in the original number has been moved to the left 2 places.
\(
\begin{aligned}
659 \% &=659 \cdot \dfrac{1}{100} \\
&=\dfrac{659}{100} \\
&=6.59
\end{aligned}
\)
Notice that the decimal point in the original number has been moved to the left 2 places.
\(
\begin{aligned}
0.4113 \% &=0.4113 \cdot \dfrac{1}{100} \\
&=\dfrac{0.4113}{100} \\
&=0.004113
\end{aligned}
\)
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.
\(\dfrac{2}{5}\)
- Answer
-
40%
\(\dfrac{7}{8}\)
\(\dfrac{1}{8}\)
- Answer
-
12.5%
\(\dfrac{5}{16}\)
\(15 \div 22\)
- Answer
-
66.18%
\(\dfrac{2}{11}\)
\(\dfrac{2}{9}\)
- Answer
-
22.22%
\(\dfrac{16}{45}\)
\(\dfrac{27}{55}\)
- Answer
-
49.09%
\(\dfrac{7}{27}\)
15
- Answer
-
1500%
8
For the following problems, convert each decimal to a percent
0.36
- Answer
-
36%
0.42
0.446
- Answer
-
44.6%
0.1298
4.25
- Answer
-
425%
5.785
86.98
- Answer
-
8698%
21.26
14
- Answer
-
1400%
12
For the following problems, convert each percent to a decimal.
35%
- Answer
-
0.35
76%
18.6%
- Answer
-
0.186
67.2%
9.0145%
- Answer
-
0.090145
3.00156%
0.00005%
- Answer
-
0.0000005
0.00034%