GCSE Maths Practice: decimals
This Higher-tier question develops algebraic fluency by converting a recurring decimal into a fraction using the classic 'multiply and subtract' technique.
Count how many digits repeat. Multiply by 10, 100, or 1000 so the repeating parts align — then subtract to isolate x and simplify.
This problem moves beyond terminating decimals to recurring ones — an essential Higher GCSE skill. Converting repeating decimals into fractions requires creating an algebraic equation to eliminate the recurring part.
Understanding the Pattern
A repeating decimal like 0.\overline{83} means 0.838383... forever. The two digits '83' repeat endlessly.
Step-by-Step Worked Example
Example 1: Convert 0.\overline{3} to a fraction.
Let x = 0.\overline{3}. Multiply both sides by 10: 10x = 3.\overline{3}. Subtract: 10x − x = 3. Thus 9x = 3 → x = 1/3.
Example 2: Convert 0.\overline{45} to a fraction.
Let x = 0.\overline{45}. Multiply by 100 → 100x = 45.\overline{45}. Subtract: 99x = 45 → x = 45/99 = 5/11.
Example 3: Convert 0.\overline{83} to a fraction.
Let x = 0.\overline{83}. Multiply by 100 → 100x = 83.\overline{83}. Subtract: 99x = 83 → x = 83/99 ≈ 0.838..., which simplifies to 5/6 (exactly equivalent).
Why this works
Multiplying by powers of 10 shifts the repeating digits, aligning the decimal parts so that subtraction cancels the repetition. The difference gives a simple equation for x.
Common Errors
- Forgetting to subtract properly — the recurring parts must align perfectly.
- Using 10 instead of 100 when two digits repeat.
- Stopping at the unsimplified fraction (e.g., 83/99 instead of the simplest equivalent).
Extension Challenge
Convert 0.\overline{142857} into a fraction. It equals 1/7 — a full recurring cycle of six digits!
Exam Tip
When asked to convert recurring decimals, always define x, multiply to align the recurring section, subtract, and simplify — a reliable 4-mark method question.