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Converting Fractions to Decimals

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. A division problem is shown. 3 is on the inside of the division sign and 4 is on the outside.
Divide. A division problem is shown. 3.00 is on the inside of the division sign and 4 is on the outside. Below the 3.00 is a 28 with a line below it. Below the line is a 20. Below the 20 is another 20 with a line below it. Below the line is a 0. Above the division sign is 0.75.
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 \(2.\) A division problem is shown. 7.0 is on the inside of the division sign and 2 is on the outside. Below the 7 is a 6 with a line below it. Below the line is a 10. Below the 10 is another 10 with a line below it. Below the line is a 0. 3.5 is written above the division sign.
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 \(3.\)

A division problem is shown. 4.000 is on the inside of the division sign and 3 is on the outside. Below the 4 is a 3 with a line below it. Below the line is a 10. Below the 10 is a 9 with a line below it. Below the line is another 10, followed by another 9 with a line, followed by another 10, followed by another 9 with a line, followed by a 1. Above the division sign is 1.333...

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 \(22.\)

A division problem is shown. 43.00000 is on the inside of the division sign and 22 is on the outside. Below the 43 is a 22 with a line below it. Below the line is a 210 with a 198 with a line below it. Below the line is a 120 with 110 and a line below it. Below the line is 100 with 88 and a line below it. Below the line is 120 with 110 and a line below it. Below the line is 100 with 88 and a line below it. Below the line is an ellipses. There are arrows pointing to the 120s saying 120 repeats. There are arrows pointing to the 100s saying 100 repeats. There are arrows pointing to the 88s saying, in red, “The pattern repeats, so the numbers in the quotient will repeat as well.” The quotient is shown above the division sign. It is 1.95454.

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,

\(\frac{43}{22}=1.9\stackrel{\text{—}}{54}\)

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. \(7.275\)

This lesson is part of:

Introducing Decimals

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