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1.8: Percent

  • Page ID
    49344
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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 \(\PageIndex{1}\)

    \(\dfrac{2}{5}\)

    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%


    This page titled 1.8: Percent is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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