Recurring decimals to fractions – GCSE Maths

Higher · Grade 6 · Non-calculator

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A recurring decimal such as $0.1\dot{7}$ or $0.\overline{12}$ is a fraction in disguise. The method is algebra: let $x$ equal the decimal, multiply by $10$, $100$ or $1000$ until the repeating block lines up, subtract, and divide. One repeating digit: $\times 10$. Two repeating digits: $\times 100$. A delay before the repeat ($0.5\dot{3}$) needs two different multiplications so the non-repeating part cancels.

Simplify the fraction. $0.\dot{6}=\frac{2}{3}$, not $\frac{6}{9}$ left unsimplified. The reverse (fraction to recurring decimal) is long division of the numerator by the denominator. A terminating decimal is a fraction whose denominator's primes are only $2$ and $5$.

Call the decimal $x$, multiply to shift one full repeating block, subtract so the tail cancels, then simplify. One repeating digit uses $\times 10$; two digits $\times 100$. A non-repeating starter such as $0.3\dot{6}$ needs both $10x$ and $100x$.

Objectives

• Write a one-digit repeat as a fraction over 9.
• Write a two-digit repeat as a fraction over 99.
• Turn a simple fraction back into a recurring decimal.

Key points

• Read dot notation: $0.\dot{7}$ means 0.777… and $0.\dot{4}\dot{5}$ means 0.454545…
• Call the decimal $x$, multiply by 10, 100 or 1000 to shift one full repeating block, then subtract.
• Handle decimals that start with a non-repeating digit, such as $0.2\dot{3}$.
• Simplify the final fraction fully.

Worked examples

Questions and answers

GCSE Maths recurring decimals to fractions questions and answers. Try each one, then open the answer. Easy questions come first.

Common mistakes

Most lost marks on this topic are one slip.

1. Using 10 or 100 as the denominator for a repeating block

   Use 9 for one digit and 99 for two

2. Leaving 45/99 when it still cancels

   Divide top and bottom by 9

3. Rounding a recurring decimal that was asked for as a fraction

   Keep the cancelled fraction

Exam tips

• Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
• If the question says “show that”, the result is given. The marks are for the steps that reach it.
• The method is the same on AQA-style and Edexcel-style papers.
• Write 'Let $x=\dots$' and show both multiplied lines — the method marks come from these lines even if you slip at the end.
• This is non-calculator. Check your fraction by dividing mentally: $7÷9$ starts 0.77…, so $\frac{7}{9}$ looks right.

Printable worksheet

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More on Recurring decimals to fractions

• Recurring decimals to fractions worked examples and method
  The method in order, key facts and the full lesson explanation
• Recurring decimals to fractions practice questions
  A second set of questions with answers, plus quick checks

AQA-style and Edexcel-style questions. Yumio is not affiliated with, endorsed by or connected to AQA, Pearson Edexcel or any other exam board.

Last updated 9 November 2026.