UK Secondary (7-11)
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Recurring Decimals as/from Fractions
Lesson

Numbers that go forever!

Decimals that you have already learnt about tend to have a finite number of decimal places. For example, $\frac{1}{2}$12 is $0.5$0.5 and $\frac{3}{10}$310 is $0.3$0.3.

Recurring decimals however have an infinite number of decimal places. In other words, the number of decimal places goes on forever and you'll often see a pattern that forms.

Examples

If we wanted to write $\frac{1}{3}$13 as a decimal, we would work it out as $1\div3$1÷​3

Let's try and solve this division question.

This is an example of a recurring decimal as the threes keep going forever but because we don't want to write a million threes, we write it as $0.\overline{3}$0.3. The little line above the number means that they recur or repeat forever..

Some time, more than one number repeats in a pattern. For example, if we wanted to write $0.1777777777...$0.1777777777... in recurring decimal form, we would write $0.1\overline{7}$0.17. Notice that the line is only above the $7$7, which means only that number is repeated.

If our recurring decimal had two numbers with a line over the two of them, that means that both numbers are repeated. For example,  $0.\overline{15}$0.15 would be $0.15151515...$0.15151515...

Some recurring patterns are even longer!

Example

Evaluate: $4\div7$4÷​7

When we work this out as a decimal, our answer is $0.571428571428...$0.571428571428...

Notice that the recurring pattern is $5,7,1,4,2,8$5,7,1,4,2,8, which occurs over and over again.

To write this in recurring decimal form, we place a line over the entire recurring pattern.

$0.\overline{571428}$0.571428

Worked Examples

Question 1

Convert $\frac{1}{3}$13 to a recurring decimal.

question 2

Complete the following:

  1. Express $0.\overline{3}$0.3 as a fraction in simplest form.

  2. Express $0.\overline{6}$0.6 as a fraction in simplest form.

question 3

Use your calculator to express $6\frac{2}{7}$627 as a recurring decimal.

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