EST. 2024 · LONDON·MMXXVI SPECIFICATION
AQA·Edexcel·OCR|Foundation + Higher
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Number · Higher

Recurring decimals to fractions

Recurring decimals are decimals where one or more digits repeat infinitely. Every recurring decimal can be written as an exact fraction using an algebraic method.

Revising recurring decimals to fractions the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Recurring decimals to fractions GCSE” on the site for the complete guide.

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Key facts to remember

  • 1A recurring decimal is shown with a dot above the repeating digit(s): 0.3̄ = 0.333…
  • 2Two dots show the start and end of a repeating block: 0.1̄2̄ = 0.121212…
  • 3Method: let x = the decimal, multiply by 10ⁿ (where n = length of recurring block) to shift it, then subtract.
  • 4The difference eliminates the recurring part, leaving a simple equation to solve.
  • 5All recurring decimals are rational numbers (can be written as fractions).
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Worked examples

Example 1

Convert 0.363636… to a fraction.

Working

  1. Let x = 0.363636…
  2. The recurring block is "36" (length 2), so multiply by 100: 100x = 36.363636…
  3. Subtract: 100x − x = 36.363636… − 0.363636…
  4. 99x = 36
  5. x = 36/99
  6. Simplify by dividing by HCF of 9: x = 4/11
Answer4/11
Example 2

Convert 0.41̄ (= 0.4111…) to a fraction.

Working

  1. Let x = 0.4111…
  2. Multiply by 10: 10x = 4.111…
  3. Multiply by 100: 100x = 41.111…
  4. 100x − 10x = 41.111… − 4.111…
  5. 90x = 37
  6. x = 37/90
Answer37/90
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Common mistakes

✗Multiplying by the wrong power of 10 — count the digits in the repeating block carefully.
✗Subtracting incorrectly and losing the non-recurring part of the decimal.
✗Forgetting to simplify the resulting fraction.
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Exam tips

✓Write out the decimal carefully before multiplying — identify exactly which digits recur.
✓After subtracting, check that the recurring part cancels completely before solving.

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