GCSE — mathsHigher2 min read

Recurring Decimals to Fractions

Convert recurring decimals to fractions algebraically for GCSE Maths. Use the multiply-and-subtract method.

Tutor AI Team2026-02-25T01:43:30.341Z

A recurring (repeating) decimal has digits that repeat forever. Converting them to fractions is a Higher-tier GCSE skill that uses algebra to eliminate the repeating part.

Core Concepts

Notation

$0.\dot{3} = 0.333...$
$0.\dot{1}\dot{2} = 0.121212...$
$0.1\dot{6} = 0.1666...$

The Algebraic Method

Step 1: Let $x =$ the recurring decimal. Step 2: Multiply by 10, 100, etc. to shift the repeating block. Step 3: Subtract to eliminate the repeating part. Step 4: Solve for $x$ .

Worked Example: $0.\dot{3}$

Problem

Let $x = 0.333...$ $10x = 3.333...$ $10x - x = 3$ $9x = 3$ → $x = \frac{3}{9} = \frac{1}{3}$

Solution

Worked Example: $0.\dot{1}\dot{2}$

Problem

Let $x = 0.121212...$ $100x = 12.1212...$ $100x - x = 12$ $99x = 12$ → $x = \frac{12}{99} = \frac{4}{33}$

Solution

Worked Example: $0.1\dot{6}$

Problem

Let $x = 0.1666...$ $10x = 1.666...$ $100x = 16.666...$ $100x - 10x = 15$ $90x = 15$ → $x = \frac{15}{90} = \frac{1}{6}$

Solution

Practice Problems

1. Convert $0.\dot{7}$ to a fraction.
1. Convert $0.\dot{4}\dot{5}$ to a fraction.
1. Convert $0.2\dot{3}$ to a fraction.
1. Show that $0.\dot{9} = 1$ .

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Common Mistakes

Multiplying by the wrong power of 10. Match the number of repeating digits.
Not subtracting to eliminate the repeating part. This is the key step.
Forgetting to simplify the fraction.

Key Takeaways

✓
Use the algebraic method: let $x$ = decimal, multiply, subtract, solve.
✓
Multiply by $10^n$ where $n$ = number of repeating digits.
✓
For delayed repeats (e.g., $0.1\dot{6}$ ), use two multiplications.
✓
Always simplify the resulting fraction.

Tags:

#gcse#maths#number#recurring-decimals#fractions

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Core Concepts

Notation

The Algebraic Method

Worked Example: $0.\dot{3}$

Worked Example: $0.\dot{1}\dot{2}$

Worked Example: $0.1\dot{6}$

Practice Problems

Common Mistakes

Key Takeaways

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