## Converting Percents to Fractions and Decimals

Contents

Since percents are ratios, they can easily be expressed as fractions. Remember that **percent** means per \(100,\) so the denominator of the fraction is \(100.\)

### How to Convert a percent to a fraction.

- Write the percent as a ratio with the denominator \(100.\)
- Simplify the fraction if possible.

## Example

Convert each percent to a fraction:

- \(\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\text{36%}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\)
- \(\phantom{\rule{0.2em}{0ex}}\text{125%}\)

### Solution

\(36%\) | |

Write as a ratio with denominator 100. | \(\frac{36}{100}\) |

Simplify. | \(\frac{9}{25}\) |

\(125%\) | |

Write as a ratio with denominator 100. | \(\frac{125}{100}\) |

Simplify. | \(\frac{5}{4}\) |

The previous example shows that a **percent** can be greater than \(1.\) We saw that \(\text{125%}\) means \(\frac{125}{100},\) or \(\frac{5}{4}.\) These are improper fractions, and their values are greater than one.

## Example

Convert each percent to a fraction:

- \(\phantom{\rule{0.2em}{0ex}}\text{24.5%}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\)
- \(\phantom{\rule{0.2em}{0ex}}33\frac{1}{3}%\)

### Solution

\(24.5%\) | |

Write as a ratio with denominator 100. | \(\frac{24.5}{100}\) |

Clear the decimal by multiplying numerator and denominator by 10. | \(\frac{24.5\left(10\right)}{100\left(10\right)}\) |

Multiply. | \(\frac{245}{1000}\) |

Rewrite showing common factors. | \(\frac{5·49}{5·200}\) |

Simplify. | \(\frac{49}{200}\) |

\(33\frac{1}{3}%\) | |

Write as a ratio with denominator 100. | \(\frac{33\frac{1}{3}}{100}\) |

Write the numerator as an improper fraction. | \(\frac{\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\frac{100}{3}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}}{100}\) |

Rewrite as fraction division, replacing 100 with \(\frac{100}{1}\). | \(\frac{100}{3}÷\frac{100}{1}\) |

Multiply by the reciprocal. | \(\frac{100}{3}\cdot \frac{1}{100}\) |

Simplify. | \(\frac{1}{3}\) |

In Mathematics 105, we learned how to convert fractions to decimals. To convert a percent to a decimal, we first convert it to a fraction and then change the fraction to a decimal.

### How to Convert a percent to a decimal.

- Write the percent as a ratio with the denominator \(100.\)
- Convert the fraction to a decimal by dividing the numerator by the denominator.

## Example

Convert each percent to a decimal:

- \(\phantom{\rule{0.4em}{0ex}}\text{6%}\phantom{\rule{0.4em}{0ex}}\)
- \(\phantom{\rule{0.2em}{0ex}}\text{78%}\)

### Solution

Because we want to change to a decimal, we will leave the fractions with denominator \(100\) instead of removing common factors.

\(6%\) | |

Write as a ratio with denominator 100. | \(\frac{6}{100}\) |

Change the fraction to a decimal by dividing the numerator by the denominator. | \(0.06\) |

\(78%\) | |

Write as a ratio with denominator 100. | \(\frac{78}{100}\) |

Change the fraction to a decimal by dividing the numerator by the denominator. | \(0.78\) |

## Example

Convert each percent to a decimal:

- \(\phantom{\rule{0.3em}{0ex}}\text{135%}\phantom{\rule{0.3em}{0ex}}\)
- \(\phantom{\rule{0.3em}{0ex}}\text{12.5%}\)

### Solution

\(135%\) | |

Write as a ratio with denominator 100. | \(\frac{135}{100}\) |

Change the fraction to a decimal by dividing the numerator by the denominator. | \(1.35\) |

\(12.5%\) | |

Write as a ratio with denominator 100. | \(\frac{12.5}{100}\) |

Change the fraction to a decimal by dividing the numerator by the denominator. | \(0.125\) |

Let’s summarize the results from the previous examples in the table below, and look for a pattern we could use to quickly convert a percent number to a decimal number.

Percent | Decimal |
---|---|

\(\text{6%}\) | \(0.06\) |

\(\text{78%}\) | \(0.78\) |

\(\text{135%}\) | \(1.35\) |

\(\text{12.5%}\) | \(0.125\) |

Do you see the pattern?

To convert a **percent** number to a decimal number, we move the decimal point two places to the left and remove the \(%\) sign. (Sometimes the decimal point does not appear in the percent number, but just like we can think of the integer \(6\) as \(6.0,\) we can think of \(\text{6%}\) as \(\text{6.0%}.\)) Notice that we may need to add zeros in front of the number when moving the decimal to the left.

The figure below uses the percents in the table above and shows visually how to convert them to decimals by moving the decimal point two places to the left.

## Example

Among a group of business leaders, \(\text{77%}\) believe that poor math and science education in the U.S. will lead to higher unemployment rates.

Convert the percent to: a fraction a decimal

### Solution

\(77%\) | |

Write as a ratio with denominator 100. | \(\frac{77}{100}\) |

\(\frac{77}{100}\) | |

Change the fraction to a decimal by dividing the numerator by the denominator. | \(0.77\) |

## Example

There are four suits of cards in a deck of cards—hearts, diamonds, clubs, and spades. The probability of randomly choosing a heart from a shuffled deck of cards is \(\text{25%}.\) Convert the percent to:

- a fraction
- a decimal

### Solution

\(25%\) | |

Write as a ratio with denominator 100. | \(\frac{25}{100}\) |

Simplify. | \(\frac{1}{4}\) |

\(\frac{1}{4}\) | |

Change the fraction to a decimal by dividing the numerator by the denominator. | \(0.25\) |

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