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Convert Between Fractions and Percentages
> ... Math > Division and Percentages > Convert Between Fractions and Percentages

Introduction

In this lesson you will learn how to convert a fraction to a percentage, and a percentage to a fraction. You may want to review the following lessons to prepare you for this lesson:

  • Place Values After the Decimal Point
  • Division
  • Multiplication and Division by Powers of 10
  • Converting Between Decimals and Percentages
  • Simplifying Fractions

This video illustrates the lesson material below. Watching the video is optional.

Converting from a Fraction to Percentage

Fractions can be represented as percentages and percentages can be represented as fractions. A fraction is a portion of something. A percentage is a portion out of 100.

To convert a fraction to a percentage, follow these steps:

  • Convert the fraction to a decimal:
    • Divide the numerator (top half of the fraction) by the denominator (bottom half of the fraction).
  • Convert the decimal to a percentage:
    • Multiply the decimal by 100 or move the decimal point two digits to the right. Don’t forget to add the percentage symbol (%).

Example 1
Convert \(\frac{1}{2}\) to a percentage.

First, convert \(\frac{1}{2}\) to a decimal by dividing the numerator by the denominator. The decimal is 0.5.

\begin{align*} \frac{1}{2} = 1\div2 = 0.5 \end{align*}

Next convert 0.5 into a percentage by multiplying it by 100:
\begin{align*} 0.5\times100=50\% \end{align*}

This makes sense because \(\frac{1}{2}\) is the same as 50%.

Example 2
Convert \(\frac{3}{8}\) into a fraction.

First, convert the fraction into a decimal, then convert the decimal into a percentage.

\begin{align*}
& \frac{3}{8} = 0.375 &\color{red}\small\text{Convert fraction to decimal by dividing}\\\\ &0.375\times100=37.5\% &\color{red}\small\text{Convert decimal to percentage by multiplying by \(100\)}\\\\
\end{align*}

Another way to convert decimal values to a percentage is by moving the decimal point 2 places to the right:

\begin{align*} 0\underrightarrow{.37}5 = 37.5\%\end{align*}

Example 3
\(\frac {3}{4} \rightarrow \%\)

  1. Divide the numerator by the denominator to get a decimal number.
    \begin{align*} \frac {3}{4} = 3\: \div4 = 0.75\end{align*}
  2. Multiply by 100. This moves the decimal place to the right two places.
    \begin{align*} 0.75 \times 100 = 75\end{align*}
  3. Put a % symbol at the end.
    \begin{align*} 75 → 75\%\end{align*}

Final answer: \(\frac {3}{4} = 75\%\)

Converting from a Percentage to Fraction

To convert a percentage to a fraction, follow these steps:

  • Use the definition of percentage—which literally means out of 100:
    • Put the number of the percentage as the numerator and 100 as the denominator.
  • Simplify the fraction:
    • The way to simplify is by dividing both the numerator and denominator to their simplest form. To do this, we need to find the Greatest Common Factor (GCF) from both the numerator and denominator and then simplify.

Example 4
Convert 45% to a simplified fraction.

When simplifying fractions, remember to find the greatest common factor (GCF) between the numerator and denominator.

\begin{align*}&\frac{45}{100} & \color{red}\small\text {Write \(45\%\) as a fraction}\\\\&\frac{9\cdot 5}{20\cdot 5} & \color{red}\small\text {Find greatest common factor} \\\\&\frac{9\cdot\color{red}\cancel5}{20\cdot\color{red}\cancel5} & \color{red}\small\text {Cancel greatest common factor (5)}\\\\&\frac{9}{20} & \color{red}\small\text {Simplified answer}\\\end{align*}

Final answer: 45% is the same as \(\frac{9}{20}\).

Example 5
\(40\% \rightarrow\text{Fraction}\)

  1. Remove the % symbol from the number and divide by 100.
    \begin{align*} 40\% → \dfrac {40}{100}\end{align*}
  2. Simplify the fraction.
    \begin{align*} \frac {40}{100}=\frac {\color{red}\cancel{20} \color{black}⋅2} {\color{red}\cancel{20} \color{black}⋅5}= \frac {2}{5}\end{align*}

Final answer: \(40\% = \dfrac {2}{5}\)

Things to Remember


  • Fraction to percentage:
    1. Convert the fraction to a decimal,
    2. Convert the decimal to a percentage.
  • Percentage to fraction:
    1. Use the definition of a percentage,
    2. Simplify the fraction.
  • The definition of percentage is a number out of 100.

Practice Problems

  1. Convert \(22\%\) to a fraction. Simplify the fraction as much as possible. (
    Solution
    x
    Solution:
    \(\displaystyle \frac{22}{100}=\frac{11}{50}\)
    )
  2. Convert \(5\%\) to a fraction. Simplify the fraction as much as possible. (
    Solution
    x
    Solution:
    \(\dfrac{1}{20}\)
    Details:
    \(\dfrac{5}{100}\)

    Simplify the fraction as much as possible.

    \(\displaystyle \frac{5}{100}=\frac{5\cdot1}{5\cdot20}=\frac{1}{20}\)
    )
  3. Convert \(250\%\) to a fraction. Simplify as much as possible. Convert to a mixed fraction if needed. (
    Solution
    x
    Solution:
    \(\displaystyle \frac{250}{100}=\frac{5\cdot50}{2\cdot50}=\frac{5}{2}=2\frac{1}{2}\)
    )
  4. Convert \(\dfrac{3}{5}\) to a percentage. (
    Solution
    x
    Solution:
    \(60\%\)
    Details:
    \(\displaystyle \frac{3}{5} = 0.6\)
    \(0.6 \times 100\) with a % symbol = \(60\%\)
    )
  5. Convert \(\dfrac{5}{8}\) to a percentage. (
    Solution
    x
    Solution:
    \(62.5\%\)
    Details:
    \(\displaystyle \frac{5}{8} = 0.625\)
    \(0.625 \times 100\) with a % symbol = \(62.5\%\)
    )
  6. Convert \(\dfrac{15}{8}\) to a percentage. (
    Solution
    x
    Solution:
    \(187.5\%\)
    Details:
    \(\displaystyle \frac{15}{8} = 1.875\)
    \(1.875 \times 100\) with a % symbol = \(187.5%\)
    )

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