Fractions, Decimals & Percentages
Fractions, Decimals & Percentages
Understanding how to convert between fractions, decimals and percentages — and apply them to real-world problems — is one of the most important skills in GCSE Maths.
Converting Between Forms
Every fraction, decimal and percentage is just a different way of writing the same value.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 3/10 | 0.3 | 30% |
| 1/3 | 0.333... | 33.3̄% |
Fraction → Decimal: Divide the numerator by the denominator. For example, 3/8 = 3 ÷ 8 = 0.375.
Decimal → Percentage: Multiply by 100. For example, 0.375 × 100 = 37.5%.
Percentage → Fraction: Write over 100 and simplify. For example, 37.5% = 375/1000 = 3/8.
Recurring decimals to fractions (Higher): Let x equal the recurring decimal, multiply to shift the repeating block, then subtract.
Worked Example: Convert 0.2̄7̄ to a fraction.
- Let x = 0.272727...
- 100x = 27.272727...
- 100x − x = 27
- 99x = 27
- x = 27/99 = 3/11
Percentage of an Amount
To find a percentage of an amount, convert the percentage to a decimal and multiply.
Worked Example: Find 17.5% of £240.
- 17.5% = 0.175
- 0.175 × 240 = £42
Percentage Change
Percentage change = (change ÷ original) × 100
Worked Example: A phone drops from £800 to £680. Find the percentage decrease.
- Change = 800 − 680 = 120
- Percentage decrease = (120 ÷ 800) × 100 = 15%
Reverse Percentages
When you know the value after a percentage change, work backwards to find the original.
Worked Example: A coat costs £66 after a 20% reduction. Find the original price.
- After a 20% reduction, £66 represents 80% of the original
- 80% = £66
- 1% = £66 ÷ 80 = £0.825
- 100% = £0.825 × 100 = £82.50
Compound Interest (Higher)
With compound interest, interest is added to the total each period, so you earn interest on interest.
Formula: Final amount = P × (1 + r/100)ⁿ
Where P = principal (starting amount), r = interest rate per period, n = number of periods.
Worked Example: £5000 is invested at 3% compound interest per year for 4 years.
- Final amount = 5000 × (1.03)⁴
- = 5000 × 1.12550881
- = £5627.54 (to 2 d.p.)
For depreciation (value decreasing), use (1 − r/100)ⁿ instead.
Worked Example: A car worth £12,000 depreciates by 15% each year for 3 years.
- Value = 12000 × (0.85)³ = 12000 × 0.614125 = £7369.50
Exam Tips
- In reverse percentage problems, identify what percentage the given value represents — not the percentage that was added or removed
- For compound interest, use the multiplier method (e.g. 3% increase → multiply by 1.03) rather than calculating interest separately each year
- Always check your answer makes sense — a value after a decrease should be smaller than the original