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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:

$0.333333...$

$0.425425425...$

Is this a recurring decimal?

$0.75445737$

Recurring decimals can always be written as fractions.

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

1

Let 0.2222... equal $x$

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

2

Take away $x$ from $10x$

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

3

Therefore, $9x=2$

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

4

Make $x$ the subject

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

5

Awesome! 🙌

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

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

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

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

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

1

Let $x=0.357357357...$

2

Multiply $x$ by 1000

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

3

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

4

Express this as a fraction

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

5

Simplify the fraction

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

6

Great work! 👍

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

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

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

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

$100x=21.1111...$

$10x=2.1111...$

$100x-10x=\dfrac{19}{90}$

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

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