Percentages are taught at about age eleven and then quietly get harder, because the everyday cases are not the ones on the worksheet. Four of them catch almost everybody.
The tool Percentage Calculator The four percentage sums, each with the arithmetic written out underneath.First, the four questions
Nearly every percentage problem is one of these. It helps enormously to notice which one you are actually in.
| The question | The sum | Example |
|---|---|---|
| What is 15% of 80? | 80 × 0.15 | 12 |
| 12 is what per cent of 80? | 12 ÷ 80 × 100 | 15% |
| From 80 to 92 — what change? | (92 − 80) ÷ 80 × 100 | +15% |
| 92 is 80 plus 15%. What was it before? | 92 ÷ 1.15 | 80 |
The fourth is the one that goes wrong, and it has its own section below because it is worth the space.
1. Two discounts do not add up
A shop advertises 20% off, and at the till there is a further 10% off. That is not 30% off.
The second discount applies to the already-reduced price, so the sums multiply rather than add:
0.80 × 0.90 = 0.72 — you pay 72%, which is 28% off, not 30%.
On a €200 jacket that is €144 rather than the €140 you expected — worth checking on a real basket before you decide the till is wrong. The gap grows with the numbers: 50% off then 50% off is 75% off, not free, which is the version that makes it obvious.
The same rule runs the other way with increases. Two 10% rises are a 21% rise, because 1.1 × 1.1 = 1.21. Compounding is just this, repeated.
2. Adding a percentage and taking it away does not undo it
Add 20% to 100 and you get 120. Take 20% off 120 and you get 96.
You are 4% short, and nothing went wrong. The 20% you added was 20% of 100; the 20% you removed was 20% of 120, which is a bigger number. 1.2 × 0.8 = 0.96, always.
Which leads directly to the mistake that costs real money.
Working backwards from a price that includes tax
An invoice says €121 including 21% VAT. How much is the tax?
Not 21% of 121, which is €25.41. The tax was calculated on the amount before it was added, so you have to reverse the multiplication:
121 ÷ 1.21 = 100. The net amount is €100 and the tax is €21.
The general form: to remove a percentage that has already been added, divide by 1 plus the rate. To remove 10%, divide by 1.1. To remove 21%, divide by 1.21. Multiplying by 0.79 gives the wrong answer every time, and it is the single most common percentage error on invoices — try it both ways on a figure of your own and the size of the gap is immediately obvious.
3. Per cent and percentage points are different things
An interest rate goes from 5% to 6%. Has it risen by 1% or by 20%?
Both, and that is exactly why the confusion exists. It is a rise of one percentage point, and a rise of 20 per cent — because 1 is a fifth of 5.
The convention is worth knowing because it is used to make things sound small or large on purpose. A tax going from 20% to 22% is “two percentage points” if you want it to sound modest and “a ten per cent increase” if you do not. Neither is wrong; they are answers to different questions.
When somebody says a rate rose by 2%, and it matters, ask which one they mean.
4. A fall and a rise of the same size are not the same size
Something loses 50% of its value. How much does it have to gain to get back?
Not 50%. It has halved, so it has to double — a 100% rise — to return to where it started.
| Fall of | Rise needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 50% | 100% |
| 80% | 400% |
The reason is the same as before: the fall is measured against the starting figure and the recovery against the smaller one. It is why “down 30% then up 30%” is not flat, and why averaging percentage changes over time gives an answer that is not true of anything.
The one that catches people who sell things: markup is not margin
Buy something for 100 and sell it for 150. Is that a 50% markup or a 33% margin?
Both, and they describe the same transaction from two ends.
- Markup is the profit as a percentage of what it cost you: 50 ÷ 100 = 50%.
- Margin is the profit as a percentage of what you sold it for: 50 ÷ 150 = 33.3%.
Margin is always the smaller number, and mistaking one for the other is how a business prices itself into trouble while believing it is doing fine. A 100% markup is a 50% margin; a 50% margin needs a 100% markup.
A trick that is genuinely useful
X% of Y is always the same as Y% of X.
Both are just X × Y ÷ 100, so the order does not matter. That sounds like a curiosity until you need 18% of 50 in your head, notice it is the same as 50% of 18, and answer 9 immediately.
It works every time. 4% of 75 is 75% of 4, which is 3. 8% of 25 is 25% of 8, which is 2.
Where each of these turns up
| Situation | Which trap |
|---|---|
| A sale with a second discount at the till | Discounts multiply, they do not add |
| An invoice with tax already included | Divide by 1 + the rate, do not subtract |
| A news report about a rate change | Per cent or percentage points? |
| Anything that dropped and came back | The rise needed is bigger than the fall |
| Pricing something you are selling | Markup and margin are different numbers |
All of it is arithmetic rather than advice — every example above can be checked in seconds with the percentage calculator, or with the discount calculator for the two-discount case, and both work with no connection at all.