The four questions and how each is solved
Almost every percentage problem is one of these four:
- What is 20 % of 150? Multiply, then divide by a hundred: 150 × 20 / 100 = 30. This is the sale price, the tip, the VAT portion, the deposit.
- 30 is what % of 120? Divide the part by the whole: 30 / 120 × 100 = 25 %. Exam marks, a share of a bill, how much of a budget is spent.
- From 80 to 100 — what is the change? Divide the difference by the starting value: (100 − 80) / 80 × 100 = +25 %. Price rises, salary changes, year-on-year growth.
- 200 plus 15 %? Multiply by 1.15: 230. Minus 15 % is × 0.85 = 170. Adding tax, applying a discount, a rent increase.
The third one trips people up most: the change is always measured against where you started. Going from 80 to 100 is +25 %, but going back from 100 to 80 is −20 %, because the base is now 100.
Percent and percentage points are not the same
If a VAT rate goes from 20 % to 23 %, it has risen by 3 percentage points — and by 15 percent (3 / 20). Both are true, and headlines pick whichever sounds better. When two percentages are compared, ask which of the two you are being told.
Undoing a percentage is not the same as subtracting it
A price was raised by 25 % to 100. The original was not 75: it was 100 / 1.25 = 80. The same goes for VAT — a gross price of 120 at 20 % contains 20 of tax, not 24. To strip a percentage out, divide by (1 + rate); to add it, multiply. The VAT calculator does exactly that for any EU rate, and the discount calculator does it for sale prices.
Two percentages never add up
Twenty per cent off, then another ten, is not thirty off. The second cut applies to what is left after the first: 100 → 80 → 72. That is 28 % off, and the two extra points went to the shop.
The rule holds in both directions and everywhere the same number is applied twice. A price raised 10 % and then cut 10 % ends at 99, not 100. A tax of 21 % on top of a duty of 12 % is not 33 % of the goods — it is 21 % of a base that already includes the duty. Percentages multiply; they do not stack.
In one line: chain them as factors. 0.8 × 0.9 = 0.72, so 28 % off. 1.1 × 0.9 = 0.99, so 1 % down.
Where a percentage costs real money
Taking VAT off a gross price. The commonest arithmetic error in retail, and it always errs the same way. At 21 %, a gross price of 121 contains 21 of tax — divide by 1.21. Multiplying the gross by 0.21 gives 25.41, and 4.41 of it does not exist. The percentage was applied to the wrong base: VAT is a percentage of the net, not of the gross. The VAT calculator does the division for the rate of each member state.
Margin and markup. The same two numbers, two different bases. Buy at 60, sell at 100: the markup is 67 % of cost, the margin is 40 % of price. Quote one where the other is meant and the price is wrong by a quarter — see margin and markup.
Points, not per cent. When a member state moves its standard rate from 19 % to 21 %, that is two percentage points. As a percentage it is a rise of 10.5 %, and on a repricing sheet the two produce very different numbers.
Quick percentages
| Percent | of 50 | of 100 | of 250 | of 1000 |
|---|---|---|---|---|
| 5% | 2.5 | 5 | 12.5 | 50 |
| 10% | 5 | 10 | 25 | 100 |
| 12.5% | 6.25 | 12.5 | 31.25 | 125 |
| 15% | 7.5 | 15 | 37.5 | 150 |
| 20% | 10 | 20 | 50 | 200 |
| 25% | 12.5 | 25 | 62.5 | 250 |
| 30% | 15 | 30 | 75 | 300 |
| 33.3% | 16.65 | 33.3 | 83.25 | 333 |
| 50% | 25 | 50 | 125 | 500 |
| 75% | 37.5 | 75 | 187.5 | 750 |
Frequently asked questions
Find 10 % first by moving the decimal point one place left, then build from there. 10 % of 240 is 24, so 5 % is 12, 20 % is 48 and 15 % is 24 + 12 = 36.
Because the second percentage is taken from a bigger number. 100 + 50 % = 150; 150 − 50 % = 75. The same asymmetry is why a stock that drops 50 % needs to gain 100 % to recover.
Results are shown to two decimal places. Internally nothing is rounded until display, so chaining calculations by copying the result will not drift.
Divide, do not multiply. At a rate of 21 %, divide the gross by 1.21 to get the net; the difference is the tax. Multiplying the gross by 0.21 overstates the tax every time, because VAT is a percentage of the net price and the gross already contains it.
No — it is 28 % off. The second discount applies to the already reduced price: 100 → 80 → 72. Multiply the factors instead of adding the percentages: 0.8 × 0.9 = 0.72.
A move from 19 % to 21 % is two percentage points and, at the same time, a rise of 10.5 %. Points measure the gap between two percentages; per cent measures the change relative to the starting one. Rate changes are announced in points and repriced in per cent.