Recurring Decimals (AQA GCSE Maths): Revision Note

Recurring decimals

What are recurring decimals?

• When writing a rational number as a decimal, it will either be:

  • A decimal that stops, called a "terminating" decimal

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  • Or a decimal that repeats with a pattern, called a "recurring" decimal

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• The recurring part can be written with a dot above the digit that repeats

• If multiple digits repeat, dots are used on the first and last digits that repeat

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How do I write recurring decimals as fractions?

Write out the first few decimal places to show the recurring pattern and then:

• STEP 1
  Write the recurring decimal as

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• STEP 2
  Multiply both sides by 10 repeatedly until two lines have the same recurring decimal part

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    • Note that x and 100x have 35 repeating after the decimal point, the repeating pattern after 10x is 53 repeating

• STEP 3
  Subtract the two lines which have matching recurring decimal parts

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• STEP 4
  Divide both sides to get
  Cancel if necessary to get fraction in its lowest terms

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