Percentage Points vs Percent — and Five Other Percentage Traps
Percentages are the most common piece of maths in everyday life: prices, pay, interest rates, exam marks, news headlines. They are also where otherwise careful people make the same few mistakes again and again. The mistakes rarely come from bad arithmetic. They come from the question, “a percentage of what?”
Here are six traps, each with a worked example. Every number can be checked in the Percentage Calculator, which shows the formula under each answer.
1. Percentage points are not percent
A central bank raises its rate from 4% to 5%. How much has it gone up?
- By 1 percentage point — the plain difference between the two rates.
- By 25 percent — because 1 is a quarter of 4.
Both statements are true. They answer different questions, which is exactly why they get swapped. A headline saying “mortgage rates up 1%” usually means one point, and a monthly payment that goes up by far more than 1% as a result.
The same trap works the other way with small risks. If something raises a risk from 1 in 10,000 to 2 in 10,000, that is “risk doubles” (+100%) and also “risk up by one hundredth of a percentage point”. A report that picks only one of these is choosing how you feel about it.
Rule: when both numbers are already percentages, say “points” for the difference and “percent” for the relative change.
2. Up 10%, then down 10% is not break-even
A share price of 100 rises 10% to 110, then falls 10%. It does not return to 100. It ends at 99, because the fall is 10% of 110, which is 11.
It gets starker with bigger moves. A 50% fall takes 100 to 50, and a 50% rise from there only gets back to 75. To recover a 50% loss you need a 100% gain. Percentage changes multiply rather than add, and the Several changes in a row calculator shows the true net result for any sequence.
3. Taking VAT off by subtracting 20%
A price of £120 includes 20% VAT. What was it before tax? The tempting answer is £120 minus 20%, which is £96. That is wrong.
The VAT was 20% of the net price, not of £120. So £120 is 120% of the original, and the original is:
£120 ÷ 1.20 = £100
The same applies to any “before” figure. After a 25% discount a price of £75 was originally £75 ÷ 0.75 = £100, not £93.75. This is called a reverse percentage: divide by the multiplier and never subtract the percentage again.
4. Stacked discounts do not add up
“30% off, plus an extra 20% off at the till.” Is that 50% off? No. The second discount applies to the already reduced price:
0.70 × 0.80 = 0.56, so you pay 56% — a 44% discount.
On a £200 coat that is £112, not £100. Retailers know most people will add the numbers up. Multiply them instead.
5. Inflation compounds
Prices rise 4% a year for three years. That is not 12%:
1.04 × 1.04 × 1.04 = 1.124864, a rise of about 12.49%.
Half a percent sounds trivial. Over a decade at 4% it becomes 48.02% compounded against 40% added, and the gap keeps widening. The same compounding is why an early loan overpayment saves so much. That is worked through in how much interest loan overpayments save.
6. Percentage change depends on which way you look
Sales went from 80 to 100. That is a 25% increase. Now they fall back from 100 to 80. That is only a 20% decrease. Same two numbers, different percentages, because percentage change always divides by the starting value.
When neither number is a natural starting point — comparing two shops’ prices, or two classes’ results — use percentage difference instead. It divides by the average of the two, so 40 against 60 is 40% whichever way round you put them.
A quick reference
| Question | Formula |
|---|---|
| What is X% of Y? | X ÷ 100 × Y |
| X is what % of Y? | X ÷ Y × 100 |
| % change from A to B | (B − A) ÷ A × 100 |
| Increase Y by X% | Y × (1 + X ÷ 100) |
| Original before +X% | result ÷ (1 + X ÷ 100) |
| % difference | |A − B| ÷ average × 100 |
Check the working, not just the answer
Most calculator sites give you a number and nothing else. That is how the mistakes above survive: you cannot tell whether the tool answered the question you meant. The percentage calculator shows the formula filled in with your numbers under every answer, and it runs entirely in your browser. The reasons that matters, even for maths, are in private online tools for everyday tasks. For splitting a bill with a service charge on top, the Bill Splitter does the percentages and the rounding for you.