A recurring decimal repeats forever. Converting it to a fraction makes its exact value visible. Set x equal to the whole decimal, then multiply by powers of 10 that line up the recurring tails. Subtraction removes those tails because they are identical. When some digits do not repeat, both equations must start their repeating sections in the same decimal position. Finish by dividing and simplifying. A rounded calculator display is an approximation and cannot replace an exact recurring value.
Learning objectives
- Recognise the repeating block in a decimal.
- Convert recurring decimals to simplified exact fractions.
Key ideas
An overbar identifies the digits that repeat forever.
Multiply by powers of ten so the recurring tails match.
Subtract the aligned equations to remove the infinite recurring tails.
Divide to find the fraction, then simplify using a common factor.
Worked examples
Example 1
Convert the decimal to an exact fraction; only the digit six repeats.
Answer
Example 2
Convert the repeating decimal to a fraction.
Answer
Question styles to prepare for
Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.
AO1 · Methods
Convert a recurring decimal to an exact fraction.
AO2 · Reasoning
Justify the powers of ten used to eliminate the recurring part.
AO3 · Problem solving
Combine exact recurring values in a calculation rather than using rounded decimals.
When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.
Avoid this mistake
Match the recurring tails before subtracting. Do not treat a non-repeating digit as part of the repeating block.
Try it yourself
Attempt each question before opening its answer.
Check 1: Convert 0.454545... to a fraction.
Answer: 5/11
100x - x = 45, so x = 45/99 = 5/11.
Check 2: Convert 0.277777... to a fraction. Only 7 repeats.
Answer: 5/18
100x - 10x = 27 - 2 = 25, so x = 25/90 = 5/18.
Explain it in your own words
- Explain why multiplying by 100 helps with a two-digit repeating block.
- Write two equations that align the tails of 0.122222... .
- Explain the difference between an exact fraction and a rounded decimal.
Keep practising
Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes
Specification source
AQA Number specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.