Percentages
What is a Percentage?
A percentage (%) means per hundred — it is a way of expressing a number as a fraction of 100.
50% = 50 out of 100 = 1/2 = 0.5
Percentages appear everywhere: sales discounts, tax rates, test scores, statistics. Mastering them is one of the most practical maths skills you can develop.
Key Terms
- Percentage: a number expressed as a part of 100
- Multiplier: the decimal you multiply by to apply a percentage change (e.g. 1.15 for a 15% increase, 0.85 for a 15% decrease)
- Original value: the starting amount before any percentage change
- Percentage change: how much something has risen or fallen, expressed as a percentage
Converting Between Formats
Every percentage can also be written as a fraction and a decimal. Knowing all three forms makes calculations much faster.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333... | 33.3...% |
Fraction to Percentage: divide numerator by denominator, then multiply by 100
- 3/4 = 0.75 × 100 = 75%
Percentage to Decimal: divide by 100
- 30% = 30 ÷ 100 = 0.30
Decimal to Percentage: multiply by 100
- 0.08 × 100 = 8%
Finding a Percentage of a Quantity
Method 1 — Decimal multiplier (fastest): What is 30% of 200? → 0.30 × 200 = 60
Method 2 — Build up from 10%: 10% of 200 = 20, so 30% = 3 × 20 = 60
Method 3 — Fraction method: 30% = 30/100, so (30/100) × 200 = 6000/100 = 60
All three methods give the same answer. Choose whichever feels most natural.
Percentage Increase and Decrease
Using the multiplier method is the most efficient approach for exam questions.
Increase: Multiply by (1 + rate)
- Increase £80 by 20%: multiplier = 1.20 → 80 × 1.20 = £96
- Increase 150 by 6%: multiplier = 1.06 → 150 × 1.06 = £159
Decrease: Multiply by (1 - rate)
- Decrease £80 by 15%: multiplier = 0.85 → 80 × 0.85 = £68
- Decrease 200 by 35%: multiplier = 0.65 → 200 × 0.65 = £130
Worked Example — Percentage Increase
A shop increases the price of a laptop from £640 to £720. Find the percentage increase.
- Change = 720 - 640 = £80
- Percentage increase = (80 ÷ 640) × 100 = 12.5%
Percentage Change Formula
Percentage change = (change ÷ original) × 100
- A positive result = percentage increase
- A negative result = percentage decrease
Example: A value drops from 80 to 60. Change = 60 - 80 = -20 Percentage change = (-20 ÷ 80) × 100 = -25% (a 25% decrease)
Finding the Original Value (Reverse Percentages)
When you know the result after a percentage change and need to find the original, divide by the multiplier.
After a 20% increase, a price is £96. Find the original.
- Multiplier used = 1.20
- Original = 96 ÷ 1.20 = £80
After a 15% discount, a coat costs £68. Find the original price.
- Multiplier used = 0.85
- Original = 68 ÷ 0.85 = £80
Common Mistakes to Avoid
- Adding 20% then subtracting 20% does NOT get back to the start. If you increase 100 by 20% you get 120. Decreasing 120 by 20% gives 96, not 100.
- 1% is not 0.1 — 1% = 0.01 as a decimal (0.1 is 10%).
- Do not find the percentage of the wrong value. For percentage change, always divide by the original value, not the new one.
- Percentages over 100% are valid. 150% = 1.5 as a decimal (more than the original).
Tips and Tricks
- Memorise common multipliers: 10% → ×0.1, 25% → ×0.25, 50% → ×0.5
- To find 17.5%, find 10% + 5% + 2.5% (halve each time)
- For reverse percentage: always divide by the multiplier, never subtract the percentage from the result
- Check answers make sense — after a discount, the price must be lower than the original