GCSE MATHS • HIGHER TIER

Recurring Decimals Made Simple

Recognise recurring decimals and convert accurately between recurring decimals and fractions.

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Recurring Decimals in Four Moves

1

Recognise

Identify the digit or block of digits that repeats forever.

2

Set Up

Let x equal the recurring decimal and multiply by a suitable power of 10.

3

Subtract

Subtract the original equation so the recurring digits cancel.

4

Solve

Solve for x and simplify the resulting fraction.

What You’ll Learn

Recognise Recurring Decimals

Identify decimals with repeating digits or repeating blocks.

Use Recurring Notation

Understand dots or bars used to show recurring digits.

Convert Fractions to Decimals

Recognise fractions that produce recurring decimals.

Convert Decimals to Fractions

Use algebra to convert recurring decimals exactly.

Work with Repeating Blocks

Handle recurring decimals with one or two repeating digits.

Key Rules

Recurring Digits Repeat Forever

A recurring decimal contains a digit or block of digits that repeats indefinitely.

Use the Correct Power of 10

Multiply by 10, 100, 1000 and so on to move one complete recurring block.

Align the Recurring Digits

The repeating digits must line up before subtracting the equations.

Subtract to Cancel

Subtract the original equation from the multiplied equation to remove the recurring part.

Worked Proof: Recurring Decimals

Step-by-Step Method

  • 1. Identify the digit or block of digits that repeats.
  • 2. Let x equal the recurring decimal.
  • 3. Multiply both sides by 10, 100 or another suitable power of 10.
  • 4. Subtract the original equation to cancel the recurring digits.
  • 5. Solve for x and simplify the fraction fully.

More Worked Examples

1

1 ÷ 3 = 0.333333...

Therefore :

1/3 = 0.3̇

2

2 ÷ 3 = 0.666666...

Therefore :

2/3 = 0.6̇

3

Convert 0.333... into a fraction .

Let x = 0.333...

10x = 3.333...
10x − x = 3
9x = 3
x = 1/3

4

Convert 0.666... into a fraction .

Let x = 0.666...

10x = 6.666...
10x − x = 6
9x = 6
x = 6/9 = 2/3
  • Common GCSE Mistakes
  • Algebra: Do not forget to introduce x when converting a recurring decimal to a fraction.
  • Power of 10: Choose the power that moves one full recurring block.
  • Simplifying: Always simplify the final fraction where possible.
  • Decimal type: Do not confuse terminating decimals with recurring decimals.
  • Exam Tips
  • Identify exactly which digit or block repeats before starting.
  • For one recurring digit, multiplying by 10 is often appropriate.
  • For two recurring digits, multiplying by 100 is often appropriate.
  • Write both equations clearly before subtracting.
  • Check that the recurring parts line up and cancel.
  • Simplify the final fraction fully.

Skills Used in Recurring Decimals

Fractions

Decimals

Recurring Notation

Algebra

Powers of 10

Subtraction

Simplifying Fractions

Higher Tier Number Skills

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Frequently Asked Questions

A recurring decimal is a decimal in which one or more digits repeat forever.

The source explains that a dot or bar can be placed above the recurring digit or digits.

Yes. The source states that every recurring decimal can be written exactly as a fraction.

It shifts one complete recurring block so subtraction can remove the repeating decimal part.

Use a power of 10 that moves the full two-digit block, as shown in the source examples using 100.

Yes. Simplifying the resulting fraction is part of the method and is highlighted in the revision checklist.