Recurring Decimals to Fractions Questions with Answers

Free maths practice · Skills Lab

Recurring Decimals to Fractions Questions with Answers

Here are 16 recurring decimals questions with answers, split into 8 levels that start with one repeating digit and build up to digits before the repeat, numbers bigger than 1 and fractions back to decimals. For a fresh set of 10 questions at any level, open the same skill in our free Skills Lab, which also steps through the working one line at a time.

  • 8levels, easiest first
  • 16questions on this page, with answers
  • 10fresh questions every time in the Lab
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A worksheet shows the answer. The Skills Lab shows how to get there. Pick a level and the working appears one step at a time, each time you press the button, so your child can follow every move and then try ten questions of their own.

Skills Lab teach mode working through converting recurring decimals to fractions step by step
Teach: the working, one step at a timeThe repeating digits line up and cancel on screen, so the subtraction step finally makes sense.
Skills Lab practise mode with ten converting recurring decimals to fractions questions
Practise: ten fresh questionsPress New 10 for a new set at the same level. Reveal each answer to check, or switch on Type the answer and let the Lab mark it.
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Practise converting recurring decimals to fractions in the Skills Lab

How to convert a recurring decimal to a fraction

Call the decimal x. Multiply by 10, 100 or 1000 so the repeating digits line up under each other, then subtract to make the repeating part disappear. Divide to get x as a fraction and simplify it.

Write 0.4̇5̇ as a fraction

  1. x = 0.454545…
  2. Two digits repeat, so 100x = 45.454545…
  3. 100x − x = 99x = 45, so x = 45/99 = 5/11

Write 0.16̇ as a fraction

  1. x = 0.1666…, 10x = 1.666…, 100x = 16.666…
  2. 100x − 10x = 90x = 15
  3. x = 15/90 = 1/6

Recurring Decimals to Fractions Questions by level

Click a level to practise it
Level Question Question
Level 1One repeating digit 1. 0.5̇ 2. 0.4̇
Level 2Two repeating digits 3. 0.2̇9̇ 4. 0.8̇5̇
Level 3Three repeating digits 5. 0.8̇97̇ 6. 0.0̇15̇
Level 4A digit before the repeat 7. 0.37̇ 8. 0.23̇8̇
Level 5Two digits before the repeat 9. 0.592̇ 10. 0.828̇
Level 6Greater than 1 11. 5.2̇ 12. 8.5̇
Level 7Fraction to recurring decimal 13. 1/6 14. 7/15
Level 8Everything mixed 15. 0.84̇ 16. 1.779̇6̇
Show all 16 answers
Question Answer
1. 0.5̇ 5/9
2. 0.4̇ 4/9
3. 0.2̇9̇ 29/99
4. 0.8̇5̇ 85/99
5. 0.8̇97̇ 299/333
6. 0.0̇15̇ 5/333
7. 0.37̇ 17/45
8. 0.23̇8̇ 118/495
9. 0.592̇ 533/900
10. 0.828̇ 373/450
11. 5.2̇ 47/9
12. 8.5̇ 77/9
13. 1/6 0.16̇
14. 7/15 0.46̇
15. 0.84̇ 38/45
16. 1.779̇6̇ 5873/3300

Common mistakes

  • Multiplying by the wrong power of ten. Count the repeating digits. One digit repeating needs ×10, two need ×100, three need ×1000.
  • Subtracting decimals that do not line up. When a digit comes before the repeat, as in 0.16̇, use 100x − 10x so the 6s line up.
  • Not simplifying the fraction. 45/99 is right but not finished. Divide by 9 to get 5/11.

Questions parents and students ask

What does the dot over a digit mean?

It shows the digit repeats forever. 0.4̇ means 0.444…, and dots over the first and last digit, as in 0.4̇5̇, mean the whole block 45 repeats.

Is converting recurring decimals to fractions on GCSE Foundation?

No. Proving a recurring decimal is a fraction is on GCSE Higher. Foundation students should know that fractions like 1/3 give recurring decimals.

Is there a shortcut?

For a block that starts straight after the point, put the block over the same number of 9s: 0.4̇5̇ = 45/99. Exam questions usually want the full algebra method shown.

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