GCSE Recurring Decimals to Fractions | The Maths and Science Tutor

Converting recurring decimals to fractions is a Higher tier topic that almost everybody finds slightly mystical the first time. Once you've seen the algebraic trick that drives it, it's mechanical. This guide walks through the routine for the three flavours that come up — single recurring digit, multiple recurring digits, and recurring after some non-recurring digits.

The notation

A dot above a digit means it recurs. A dot above the first and last of a group means everything between them recurs.

0.3̇ means 0.3333…
0.3̇6̇ means 0.363636…
0.27̇ means 0.27777… (only the 7 recurs)

The standard routine

Three steps:

Let x equal the recurring decimal.
Multiply x by a power of 10 to shift the decimal point past the recurring part. Choose the power that makes the recurring digits line up with x.
Subtract — the recurring part disappears. Solve for x.

Type 1 — Single recurring digit

Worked example 1

Convert 0.4̇ (= 0.4444…) to a fraction.

Let x = 0.4444…

10x = 4.4444…

10x − x = 4.4444… − 0.4444…

9x = 4

x = 4/9

Type 2 — Two or more recurring digits

Worked example 2

Convert 0.2̇7̇ (= 0.272727…) to a fraction.

Two recurring digits, so multiply by 100.

Let x = 0.2727…

100x = 27.2727…

100x − x = 27

99x = 27

x = 27/99 = 3/11

Examiner's note: Always simplify the final fraction. The mark scheme awards a mark for the correct fraction and a mark for it being fully simplified. Use the highest common factor.

Type 3 — Mixed (recurring after some non-recurring digits)

Worked example 3

Convert 0.16̇ (= 0.1666…) to a fraction.

Multiply once by 10 (to move the non-recurring 1 past the decimal), and once by 100 (to get the recurring part lined up).

Let x = 0.1666…

10x = 1.6666…

100x = 16.6666…

100x − 10x = 16.6666… − 1.6666… = 15

90x = 15

x = 15/90 = 1/6

The mistakes that cost the most marks

Mistake 1 — Wrong power of 10. The number of zeros must equal the number of recurring digits. One recurring digit → ×10. Two → ×100. Three → ×1000.

Mistake 2 — Subtracting without lining up the decimals. The whole point of choosing the power of 10 is so the recurring tail cancels. If it doesn't cancel, you've used the wrong power.

Mistake 3 — Not simplifying the final fraction. 27/99 is correct but 3/11 is fully simplified. The unsimplified version often loses one mark.

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Recurring Decimals to Fractions