## Converting Fractions to Decimals

Earlier, we learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar indicates division. So \(\frac{4}{5}\) can be written \(4÷5\) or \(5\overline{)4}.\) This means that we can convert a fraction to a decimal by treating it as a division problem.

### How to Convert a Fraction to a Decimal

To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.

### Optional Video: Converting a Fraction to a Decimal

## Example

Write the fraction \(\frac{3}{4}\) as a decimal.

### Solution

A fraction bar means division, so we can write the fraction \(\frac{3}{4}\) using division. | |

Divide. | |

So the fraction \(\frac{3}{4}\) is equal to \(0.75.\) |

### Optional Video: Example on Converting a Fraction to a Decimal (Repeating)

## Example

Write the fraction \(-\frac{7}{2}\) as a decimal.

### Solution

The value of this fraction is negative. After dividing, the value of the decimal will be negative. We do the division ignoring the sign, and then write the negative sign in the answer. | \(\phantom{\rule{0.5em}{0ex}}-\frac{7}{2}\) |

Divide \(7\) by \(2.\) | |

So, \(-\frac{7}{2}=-3.5.\) |

### Repeating Decimals

So far, in all the examples converting fractions to decimals the division resulted in a remainder of zero. This is not always the case. Let’s see what happens when we convert the fraction \(\frac{4}{3}\) to a decimal. First, notice that \(\frac{4}{3}\) is an improper fraction. Its value is greater than \(1.\) The equivalent decimal will also be greater than \(1.\)

We divide \(4\) by \(3.\)

No matter how many more zeros we write, there will always be a remainder of \(1,\) and the threes in the quotient will go on forever. The number \(\text{1.333…}\) is called a repeating decimal. Remember that the “…” means that the pattern repeats.

### Definition: Repeating Decimal

A **repeating decimal** is a decimal in which the last digit or group of digits repeats endlessly.

How do you know how many ‘repeats’ to write? Instead of writing \(1.333\dots \) we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal \(1.333\dots \) is written \(1.\stackrel{–}{3}.\) The line above the \(3\) tells you that the \(3\) repeats endlessly. So \(\text{1.333…}=1.\stackrel{–}{3}\)

For other decimals, two or more digits might repeat. The table below shows some more examples of repeating decimals.

\(\text{1.333…}=1.\stackrel{–}{3}\) | \(3\) is the repeating digit |

\(\text{4.1666…}=4.1\stackrel{–}{6}\) | \(6\) is the repeating digit |

\(\text{4.161616…}=4.\stackrel{\text{—}}{16}\) | \(16\) is the repeating block |

\(\text{0.271271271…}=0.\stackrel{\text{–––}}{271}\) | \(271\) is the repeating block |

## Example

Write \(\frac{43}{22}\) as a decimal.

### Solution

Divide \(43\) by \(22.\)

Notice that the differences of \(120\) and \(100\) repeat, so there is a repeat in the digits of the quotient; \(54\) will repeat endlessly. The first decimal place in the quotient, \(9,\) is not part of the pattern. So,

It is useful to convert between fractions and decimals when we need to add or subtract numbers in different forms. To add a fraction and a decimal, for example, we would need to either convert the fraction to a decimal or the decimal to a fraction.

## Example

Simplify: \(\frac{7}{8}+6.4.\)

### Solution

\(\frac{7}{8}+6.4\) | ||

Change \(\frac{7}{8}\) to a decimal. | \(0.875+6.4\) | |

Add. | \(7.275\) |

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