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Recurring Decimals

Recurring Decimals

Recurring Decimals

A recurring decimal follows a pattern after the decimal point, and it is important to be able to convert recurring decimals into fractions.

A recurring decimal is a decimal which follows some kind of pattern.

For example:

0.333333...0.333333...

0.425425425...0.425425425...

Is this a recurring decimal?

0.754457370.75445737

Recurring decimals can always be written as fractions.

Let's try converting 0.2222...0.2222... into a fraction.

1

Let 0.2222... equal xx

Therefore,10×x=2.2222...10\times x=2.2222...

2

Take away xx from 10x10x

2.222..0.222...=22.222..-0.222...=2. This means we now have a whole number!

3

Therefore, 9x=29x=2

We started with 10x=2.222...10x=2.222... and subtracted x=0.222...x=0.222...

4

Make xx the subject

9x=2x=299x=2\rightarrow x=\dfrac{2}{9}

5

Awesome! 🙌

0.2222...=290.2222...=\dfrac{2}{9}

What is 0.33333...0.33333... as a fraction in its simplest form?

What is 0.55555..0.55555... as a fraction in its simplest form?

There are also recurring decimals where the pattern repeats every 3 digits such as 0.357357357...0.357357357....

Write 0.357357357...0.357357357... as a fraction in it's simplest form.

1

Let x=0.357357357...x=0.357357357...

2

Multiply xx by 1000

This means that the repeating unit is ahead of the decimal point, and leaves us with 357.357357...357.357357...

3

Subtract x=0.357...x=0.357... from 1000x=357.357...1000x=357.357...

4

Express this as a fraction

After subtracting, we have 357999\dfrac{357}{999}

5

Simplify the fraction

There is a common factor of 3! 357999=119333\dfrac{\cancel{357}}{\cancel{999}}=\dfrac{119}{333}

6

Great work! 👍

The final answer is 119333\dfrac{119}{333}

What is 0.363636...0.363636... as a fraction in its simplest form?

What is 0.757575...0.757575... as a fraction in its simplest form?

Another type of recurring decimal you might come across is 0.211111..0.211111... In this case, we perform the same process, but this time subtract 10x10xfrom 100x100x to remove the repeating unit.

100x=21.1111...100x=21.1111...

10x=2.1111...10x=2.1111...

100x10x=1990100x-10x=\dfrac{19}{90}

What is 0.17777..0.17777... as a fraction?

What is 0.288888...0.288888... as a fraction?