What is a percentage?
A percentage is a number expressed as a fraction of 100. The word comes from the Latin *per centum*, 'by the hundred'. So 15% means 15 out of every 100, or 0.15 as a decimal.
Percentages make it easy to compare quantities of different sizes, which is why they are used for discounts, interest rates, pay rises, exam results, VAT and statistics. To turn a percentage into a decimal, divide by 100; to turn a decimal into a percentage, multiply by 100.
The four percentage formulas
The calculator uses exactly these formulas. For percentage change it divides by the absolute value of A, so the sign always shows whether the value went up (+) or down (−).
| Question | Formula | Example |
|---|---|---|
| What is A% of B? | A ÷ 100 × B | 15% of 200 = 30 |
| A is what % of B? | A ÷ B × 100 | 45 of 60 = 75% |
| % change from A to B | (B − A) ÷ A × 100 | 80 → 92 = +15% |
| Add or subtract A% to B | B × (1 ± A ÷ 100) | 150 + 20% = 180 |
Worked example 1: a percentage of a number
What is 15% of £200?
15 ÷ 100 = 0.15
0.15 × £200 = £30
Shortcut: 10% of £200 is £20 and 5% is half of that, £10, so 15% is £30.
Worked example 2: what percentage one number is of another
You scored 45 out of 60 in a test. What percentage is that?
45 ÷ 60 = 0.75
0.75 × 100 = 75%
Worked example 3: percentage increase and decrease
A weekly shop rose from £80 to £92; a train fare fell from £250 to £200
Shop: (92 − 80) ÷ 80 × 100 = 12 ÷ 80 × 100 = +15%
Fare: (200 − 250) ÷ 250 × 100 = −50 ÷ 250 × 100 = −20%
Always divide by the starting value (A), not the new value.
Worked example 4: adding and subtracting a percentage
Add or take 20% from £150
20% of £150 = £30
£150 increased by 20% = £150 × 1.2 = £180 (for example, adding VAT)
£150 decreased by 20% = £150 × 0.8 = £120 (for example, a 20% sale discount)
Common percentage mistakes
- Reversing an increase with the same percentage. If a price rises 20% from £100 to £120, a 20% cut takes it to £96, not £100. To undo a 20% rise you must divide by 1.2 (a cut of 16.67%).
- Adding successive percentages. A 10% rise followed by another 10% rise is a 21% increase overall (1.1 × 1.1 = 1.21), not 20%.
- Confusing percentage points with per cent. If an interest rate goes from 4% to 5%, it has risen by 1 percentage point but by 25% in relative terms.
- Dividing by the wrong base. Percentage change is always measured against the original value.
- Removing VAT by subtracting 20%. A VAT-inclusive price must be divided by 1.2; use the VAT calculator for this.
Quick mental arithmetic tips
- 10%: move the decimal point one place to the left (10% of £64 = £6.40).
- 5%: half of 10%. 1%: move the decimal point two places.
- 25%: divide by 4. 50%: divide by 2. 75%: three quarters.
- Swap trick: A% of B equals B% of A, so 8% of 50 is the same as 50% of 8, which is 4.
- Tips and service charges: 12.5% is one eighth of the bill.