Percentages come up constantly — discounts, pay rises, interest rates, exam scores, VAT. Most people guess or reach for a calculator without knowing the underlying formula. Once you know three core methods, everything else is a variation.
Method 1: Find X% of a number
Formula: Number × (Percentage ÷ 100)
Shortcut: Number × decimal equivalent
Common decimal equivalents:
- 10% = × 0.10
- 15% = × 0.15
- 20% = × 0.20
- 25% = × 0.25
- 50% = × 0.50
Examples:
- 15% tip on a £64 restaurant bill: 64 × 0.15 = £9.60
- 20% deposit on a £280,000 house: 280,000 × 0.20 = £56,000
- 3% pension contribution on a £42,000 salary: 42,000 × 0.03 = £1,260/year
Method 2: What percentage is A of B?
Formula: (A ÷ B) × 100
Examples:
- You scored 54 out of 75 on a test: (54 ÷ 75) × 100 = 72%
- You spend £380 on food out of a £1,900 monthly budget: (380 ÷ 1,900) × 100 = 20%
- A company's profit is £180,000 on revenue of £1,200,000: (180,000 ÷ 1,200,000) × 100 = 15% profit margin
Method 3: Percentage change
Formula: ((New − Old) ÷ Old) × 100
Positive result = increase. Negative result = decrease.
Examples:
- Salary rises from £35,000 to £38,500: ((38,500 − 35,000) ÷ 35,000) × 100 = +10%
- House price falls from £310,000 to £287,000: ((287,000 − 310,000) ÷ 310,000) × 100 = −7.4%
- Monthly energy bill drops from £190 to £152: ((152 − 190) ÷ 190) × 100 = −20%
The trap: A 50% increase followed by a 50% decrease does not return to the original. £100 → +50% → £150 → −50% → £75. The percentages are calculated on different bases each time.
Method 4: Reverse percentage (find the original)
You know the final price and the percentage change. You need the original.
After an increase: Original = Final ÷ (1 + decimal)
After a decrease: Original = Final ÷ (1 − decimal)
Examples:
- A jacket costs £119 after a 30% increase. Original price: £119 ÷ 1.30 = £91.54
- A TV costs £340 after a 15% discount. Original price: £340 ÷ 0.85 = £400
- Your salary after a 5% pay cut is £33,250. Original salary: £33,250 ÷ 0.95 = £35,000
Method 5: Adding and removing VAT
UK VAT at the standard rate is 20%. These two operations cover 99% of VAT calculations:
Add VAT (ex-VAT to inc-VAT): Price × 1.20
Remove VAT (inc-VAT to ex-VAT): Price ÷ 1.20
- A service costs £850 ex-VAT. VAT-inclusive price: 850 × 1.20 = £1,020
- An invoice shows £2,400 inc-VAT. Ex-VAT amount: 2,400 ÷ 1.20 = £2,000. VAT element: £400.
For the reduced 5% VAT rate (energy, children's car seats, etc.): multiply by 1.05 to add, divide by 1.05 to remove.
Mental shortcuts worth memorising
- 10% of anything: move the decimal point one place left. 10% of £847 = £84.70
- 5%: find 10% and halve it. 5% of £847 = £42.35
- 15%: find 10% + 5%. 15% of £847 = £84.70 + £42.35 = £127.05
- 1%: move the decimal two places left. 1% of £2,600 = £26
- Any %: build from 1% and 10%
Frequently asked questions
How do you calculate a percentage of a number?
Multiply the number by the percentage and divide by 100. For example: 20% of £350 = (350 × 20) ÷ 100 = £70. Or simply multiply by the decimal equivalent: 350 × 0.20 = £70. This works for any percentage — 15% of 80 = 80 × 0.15 = 12.
How do you calculate percentage change?
Percentage change = ((New Value − Old Value) ÷ Old Value) × 100. If a price rises from £40 to £52: ((52 − 40) ÷ 40) × 100 = 30% increase. If it falls from £52 to £40: ((40 − 52) ÷ 52) × 100 = −23.1% decrease. Note that a 30% rise followed by a 23.1% fall brings you back to the same number.
How do you work out what percentage one number is of another?
Divide the part by the whole and multiply by 100. If 45 students out of 180 passed: (45 ÷ 180) × 100 = 25%. You scored 68 out of 85 on a test: (68 ÷ 85) × 100 = 80%. This formula works for any 'X is what % of Y' question.
How do you reverse a percentage (find the original number)?
Divide the final number by (1 + the percentage as a decimal) for an increase, or (1 − the decimal) for a decrease. A TV costs £399 after a 20% discount — what was the original price? £399 ÷ (1 − 0.20) = £399 ÷ 0.80 = £498.75. This is called reverse percentage or working backwards from a percentage.
How do you add VAT to a price?
Multiply the price by 1.20 for standard 20% UK VAT. £85 + VAT = £85 × 1.20 = £102. To remove VAT from a VAT-inclusive price, divide by 1.20: £102 ÷ 1.20 = £85 (ex-VAT). For the 5% reduced VAT rate, multiply by 1.05 or divide by 1.05 to remove it.