Recurring Decimals to Fractions –

Converting recurring decimals to fractions is a Higher tier GCSE Maths topic that appears regularly on exam papers. The method uses algebra to eliminate the repeating part, producing a clean fraction.

What Is a Recurring Decimal?

A recurring decimal is a decimal number in which one or more digits repeat infinitely. A single dot above a digit means that digit repeats; dots above the first and last digits of a group mean that whole group repeats.

For example, 0.333... (written 0.3 with a dot over the 3) = 1/3. The decimal 0.363636... (written with dots over the 3 and the 6) has a two-digit repeating block.

Every recurring decimal can be written as a fraction, and the algebraic method below works for any repeating pattern.

Key Formulas

Step-by-Step Method

1. Let x equal the recurring decimal.
2. Count how many digits are in the repeating block — call this n.
3. Multiply both sides by 10^n to shift the decimal point past one complete repeating block.
4. Subtract the original equation from the new equation to eliminate the repeating part.
5. Solve for x and simplify the resulting fraction.

Worked Example 1 — Foundation Level

Question: This topic is Higher only, but here is a straightforward single-digit recurrence. Convert 0.777... to a fraction.

Working:

Step 1 — Let x = 0.777...

Step 2 — One digit repeats, so multiply by 10: 10x = 7.777...

Step 3 — Subtract: 10x − x = 7.777... − 0.777... → 9x = 7.

Step 4 — Solve: x = 7/9.

Answer: 7/9

Worked Example 2 — Higher Level

Question: Convert 0.363636... to a fraction in its simplest form.

Working:

Step 1 — Let x = 0.363636...

Step 2 — Two digits repeat, so multiply by 100: 100x = 36.363636...

Step 3 — Subtract: 100x − x = 36.363636... − 0.363636... → 99x = 36.

Step 4 — Solve: x = 36/99. Simplify by dividing by HCF(36, 99) = 9: x = 4/11.

Answer: 4/11

Worked Example 3 — Exam Style

Question: Prove that 0.2181818... = 12/55.

Working:

Step 1 — Let x = 0.2181818... The repeating block is 18 (two digits), but there is a non-repeating 2 first.

Step 2 — Multiply by 10 to move past the non-repeating digit: 10x = 2.181818...

Step 3 — Multiply by 1000 to move past the non-repeating digit plus one full repeating block: 1000x = 218.181818...

Step 4 — Subtract: 1000x − 10x = 218.1818... − 2.1818... → 990x = 216.

Step 5 — Solve: x = 216/990. Simplify by dividing by HCF(216, 990) = 18: x = 12/55.

Answer: 0.2181818... = 12/55 (proven)

Common Mistakes

• Multiplying by the wrong power of 10. The power must match the number of repeating digits, not the total number of decimal places.
• Forgetting to deal with non-repeating digits before the repeating block. When there are non-repeating digits, you need two multiplications and must subtract carefully.
• Not simplifying the final fraction. Always divide numerator and denominator by their HCF.

Exam Tips

• Set out your working clearly with "Let x = ..." — examiners look for this structure.
• For "prove" or "show that" questions, you must reach the exact fraction given in the question.
• Check your answer by dividing the numerator by the denominator on your calculator to verify you get the original recurring decimal.

Practice Questions

Q1 (Higher): Convert 0.444... to a fraction.

Q2 (Higher): Convert 0.272727... to a fraction in its simplest form.

Q3 (Higher): Prove that 0.1666... = 1/6.

Practise recurring decimals to fractions questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Fractions
• Converting Fractions, Decimals and Percentages
• Algebraic Proof

Summary

• A recurring decimal has digits that repeat infinitely in a pattern.
• Use the algebraic method: let x equal the decimal, multiply by the appropriate power of 10, and subtract.
• The power of 10 matches the number of digits in the repeating block.
• If there are non-repeating digits before the block, use two different multiplications.
• Always simplify your final fraction by dividing by the HCF.