2A Fractions and Decimals

Learning Outcomes

  • Convert a fraction to a decimal
  • Identify a fraction whose decimal form is repeating
  • Add a fraction and decimal by converting between forms

Converting Fractions to Decimals

Remember that the fraction bar indicates division. So [latex]{\Large\frac{4}{5}}[/latex] can be written [latex]4\div 5[/latex] or [latex]5\overline{)4}[/latex]. This means that we can convert a fraction to a decimal by treating it as a division problem.

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.

 

example

Write the fraction [latex]{\Large\frac{3}{4}}[/latex] as a decimal.

Solution

A fraction bar means division, so we can write the fraction [latex]\Large\frac{3}{4}[/latex] 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 [latex]{\Large\frac{3}{4}}[/latex] is equal to [latex]0.75[/latex].

 

try it

The following video contains an example of how to write a fraction as a decimal.

example

Write the fraction [latex]-{\Large\frac{7}{2}}[/latex] as a decimal.

 

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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 [latex]{\Large\frac{4}{3}}[/latex] to a decimal. First, notice that [latex]{\Large\frac{4}{3}}[/latex] is an improper fraction. Its value is greater than [latex]1[/latex]. The equivalent decimal will also be greater than [latex]1[/latex].

We divide [latex]4[/latex] by [latex]3[/latex].

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 [latex]1[/latex], and the threes in the quotient will go on forever. The number [latex]\text{1.333}\dots[/latex] is called a repeating decimal. Remember that the “…” means that the pattern repeats.

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 [latex]1.333\dots[/latex] we use a shorthand notation by placing a line over the digits that repeat. The repeating decimal [latex]1.333\dots[/latex] is written [latex]1.\overline{3}[/latex]. The line above the [latex]3[/latex] tells you that the [latex]3[/latex] repeats endlessly. So [latex]\text{1.333}\dots=1.\overline{3}[/latex]

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

[latex]\text{1.333}\ldots=1.\overline{3}[/latex] [latex]3[/latex] is the repeating digit
[latex]\text{4.1666}\ldots=4.1\overline{6}[/latex] [latex]6[/latex] is the repeating digit
[latex]\text{4.161616}\ldots=4.\overline{16}[/latex] [latex]16[/latex] is the repeating block
[latex]\text{0.271271271}\ldots =0.\overline{271}[/latex] [latex]271[/latex] is the repeating block

example

Write [latex]{\Large\frac{43}{22}}[/latex] as a decimal.

 

try it

The next video example shows an example of converting fractions to decimals when the result is repeating.

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: [latex]{\Large\frac{7}{8}}+6.4[/latex]

 

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