How it works
The four percentage calculations you'll ever need
Almost every real-world percentage question is one of four shapes. Once you spot which shape a problem is, the calculation is the same each time.
1. Find X% of Y
Turn the percentage into a decimal (÷ 100) and multiply.
20% of £65 → 0.20 × 65 = £13.
7.5% of £1,200 → 0.075 × 1,200 = £90.
2. What percentage is A of B?
Divide A by B, multiply by 100.
£34 out of £200 → 34 / 200 × 100 = 17%.
45 out of 60 questions → 45 / 60 × 100 = 75%.
3. Percentage increase or decrease
((new − old) / old) × 100. Positive = rise, negative = fall.
Price goes from £80 to £92: (92 − 80) / 80 × 100 = +15%.
Price goes from £80 to £68: (68 − 80) / 80 × 100 = −15%.
4. Increase or decrease a number by X%
Multiply by (1 ± X/100).
Increase £200 by 12% → 200 × 1.12 = £224.
Decrease £200 by 12% → 200 × 0.88 = £176.
A common shortcut: to add VAT (20%) to a net price, multiply by 1.20.
Reverse percentage: the one that trips everyone up
If something has already been increased or decreased, you can't just subtract the percentage. Example: the sale price is £68 after a 15% discount — what was the original?
Sale price = original × (1 − 0.15) = original × 0.85. So original = 68 / 0.85 = £80.
Same idea with VAT: £240 gross at 20% → net = 240 / 1.20 = £200. People often wrongly subtract 20% of £240 (= £48) to get £192 — which is incorrect by £8.
Percentage vs percentage points
A small but important distinction. If an interest rate rises from 4% to 5%, that's a 1 percentage point rise — but a 25% relative increase (because 1 is 25% of 4). Financial commentators use both, and mixing them up inflates or deflates stories by orders of magnitude.
Everyday British uses
- Sales and discounts — "30% off" means new price = old × 0.70.
- Tips in restaurants — discretionary service is usually 10–12.5%. Some add it automatically to the bill.
- Mortgage rate movements — expressed in basis points (1 bp = 0.01%); a 25 bp cut is a 0.25 percentage point cut.
- NHS waiting-list statistics — often expressed as percentage of patients seen within 18 weeks.
- Wage rises — compare annual rises against CPI inflation to see real-terms change.
Quick mental-maths shortcuts
- 10% — shift the decimal one place left. 10% of £47.50 = £4.75.
- 5% — take 10% and halve it. 5% of £47.50 = £2.375.
- 1% — shift the decimal two places left. 1% of £47.50 = £0.475.
- 15% — 10% + 5%. 15% of £47.50 = £4.75 + £2.375 = £7.125.
- 20% — double 10%. 20% of £47.50 = £9.50.
- Percentages reverse: X% of Y equals Y% of X. 18% of 50 is the same as 50% of 18 = 9.
Compounding percentages — why two discounts rarely give you the sum
A frequent shop-floor myth is that a 20 % discount followed by another 10 % discount equals 30 % off. It does not. The two reductions compound multiplicatively, not additively. The general rule is: apply each factor in turn.
Worked example: a £200 jacket with 20 % off and a staff discount of 10 % on top. After the first cut: 200 × 0.80 = £160. After the second: 160 × 0.90 = £144. Total discount = (200 − 144) / 200 = 28 %, not 30 %. The same logic applies to price rises: a 10 % rise followed by a 10 % rise is 21 %, not 20 %.
Knowing this saves money in sales and stops over-claiming in reports. Marketers exploit the ambiguity; careful shoppers do the maths.
Margin versus mark-up — two answers to one product question
Retail, hospitality and freelance invoicing collide on this almost weekly. The two words describe different ratios and knowing which is which changes your pricing.
Mark-up
Mark-up is the profit expressed as a percentage of the cost. A product that costs £40 and sells for £50 has £10 profit on £40 of cost = 25 % mark-up.
Margin
Margin is the same profit expressed as a percentage of the selling price. The same £40 cost / £50 sale has £10 profit on £50 of revenue = 20 % margin.
Converting between them
If you know the margin m (as a decimal), mark-up = m / (1 − m). 20 % margin → 0.20 / 0.80 = 25 % mark-up. If you know the mark-up k, margin = k / (1 + k). 50 % mark-up → 0.50 / 1.50 = 33 % margin.
Five everyday UK scenarios worked from end to end
Drop these into your day-to-day and watch how often percentages are quietly doing the heavy lifting.
1. The VAT receipt check
A taxi gives you a £42 receipt marked "inc. VAT". To pull out the VAT for your expense claim, divide the gross by 1.20 to get the net (£35) and take the difference: VAT = £7. Do not take 20 % of £42 — that would be £8.40, and you'll be out of pocket.
2. The pay-rise and inflation comparison
Your salary rises from £32,000 to £33,200 — a 3.75 % increase. CPI inflation for the same period is 2.3 %. Your real rise is not 3.75 − 2.3 = 1.45 %; it is (1.0375 / 1.023) − 1 = 1.42 %. The approximation is close enough for small numbers but diverges at higher inflation.
3. The mortgage rate swing
The Bank of England raises base rate from 4.25 % to 5.00 %. That is a 75 basis-point rise (0.75 percentage points) but an 18 % relative increase. For a tracker borrower on base + 0.85 %, the monthly rate moved from 5.10 % to 5.85 % — a 14.7 % jump in the interest slice of every payment.
4. The sale-and-size-up
A supermarket doubles the pack size of a 240 g cereal to 480 g and adds 40 % to the price. Is that better value? Unit price went from £3.60 / 240 g = £1.50 per 100 g to £5.04 / 480 g = £1.05 per 100 g — about 30 % cheaper per gram. Percentage discounts on unit cost, not the sticker price, is the value test.
5. The service charge at dinner
A £78 restaurant bill includes 12.5 % service. To find the pre-service price: 78 / 1.125 = £69.33. If the service charge is discretionary and the food was indifferent, asking to remove it leaves you owing the smaller figure.
Common percentage mistakes — and how to spot them fast
Most percentage errors fall into one of four pits. Recognising the pit is usually enough to step around it.
- Averaging percentages of different totals. Two branches: 20 % of 200 sales = 40 complaints; 10 % of 400 sales = 40 complaints. The "average complaint rate" is not 15 % — it is 80 / 600 = 13.3 %, the weighted average. Always average the underlying counts.
- Subtracting after an increase to "undo" it. A 20 % rise on £100 becomes £120. A 20 % cut from £120 goes to £96, not £100. To reverse an X % rise, you need to cut by X / (100 + X).
- Using the wrong base. "Profit grew 50 %" needs a 2024 figure and a 2025 figure. Without the base, the number is meaningless.
- Forgetting percentage of a negative. If a company's loss shrinks from −£4 m to −£1 m, that is a 75 % improvement, not a 75 % loss. Sign matters as much as magnitude.
A reference table for the most common percentages
Pin this to the side of your monitor. The rows you use most often become second nature after a week.
| Percentage | Decimal | Fraction | Mental shortcut |
|---|---|---|---|
| 1% | 0.01 | 1/100 | Shift decimal two places left |
| 5% | 0.05 | 1/20 | Take 10% and halve |
| 10% | 0.10 | 1/10 | Shift decimal one place left |
| 12.5% | 0.125 | 1/8 | Divide by 8 |
| 20% | 0.20 | 1/5 | Divide by 5 |
| 25% | 0.25 | 1/4 | Divide by 4 |
| 33.3% | 0.333 | 1/3 | Divide by 3 |
| 50% | 0.50 | 1/2 | Halve it |
| 66.6% | 0.666 | 2/3 | Divide by 3 then double |
| 75% | 0.75 | 3/4 | Divide by 4 then triple |
Percentage points, basis points and percentiles — who uses what
Different fields use different units for the same core idea, and they are not interchangeable. Knowing who uses what stops a lot of crossed wires.
- Percentage point (pp) — the absolute difference between two percentages. Unemployment moving from 5.0 % to 5.5 % is a 0.5 pp rise, not 0.5 %. Economic commentary, polling, public-health reporting and government statistics all use pp for changes between two rates.
- Basis point (bp) — one hundredth of a percentage point. 100 bp = 1 pp. A 25 bp cut by the Bank of England is a 0.25 percentage-point cut. Used in fixed income, mortgages and FX because the moves are often too small to express cleanly in per-cent.
- Percentile — the place of a single observation in a distribution. Your child's height at the 80th centile means they are taller than 80 % of peers. Used in health, exams and pay benchmarking, and commonly mis-called "per-cent" in news stories.
- Per mille (‰) — parts per thousand. Used in birth and death rates, very old maritime insurance tables and some dental-fluoride concentrations.
- Parts per million (ppm) — used in environmental, water and air-quality rules. 400 ppm of CO₂ is 0.04 %.
A percentage-literacy checklist before any big decision
Use this list the next time a salesperson, an advert or a colleague quotes a percentage. It takes thirty seconds and will catch most mistakes.
- What is the base — is the percentage of a price, a total, a change or a rate?
- Over what period? A "20 % rise" over five years is very different from a 20 % rise in a month.
- Are you comparing to a percentage point or a relative percentage? Write both if in doubt.
- Has anything compounded? Two successive percentage changes never simply add.
- Does the sign make sense? A 110 % reduction is impossible; anything above 100 % must be a rise.
- Sanity-check against a round number. 18 % of 50 should be a little under 10 — if you get 90 you misplaced a decimal.