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.

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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

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

Answer4/11

Example 2

Convert 0.41̄ (= 0.4111…) to a fraction.

Working

Let x = 0.4111…
Multiply by 10: 10x = 4.111…
Multiply by 100: 100x = 41.111…
100x − 10x = 41.111… − 4.111…
90x = 37
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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