A percentage point is an absolute difference between two percentages; a per cent is a relative change. When a rate moves from 4 per cent to 5 per cent it has risen by one percentage point and by 25 per cent, because one is four per cent of the original figure and 1 ÷ 4 = 0.25.
Both statements describe the same move. Mixing them up is the most common numerical error in reporting, and the two readings can differ by an order of magnitude.
Why does the confusion matter?
Because the two numbers support opposite arguments. A campaign that lifts a conversion rate from 2 per cent to 2.5 per cent has gained half a percentage point, which sounds negligible, and 25 per cent, which sounds substantial. Neither figure is wrong and only one of them will appear in the slide.
The reliable habit is to state both: "up 0.5 points, from 2.0 to 2.5 per cent". It takes four extra words and it is unambiguous, which no single figure is.
Why does a 25 per cent fall not undo a 25 per cent rise?
Because each change applies to a different base. A hundred rising by 25 per cent gives 125; 125 falling by 25 per cent removes 31.25 and gives 93.75. You end 6.25 per cent below where you started.
The asymmetry grows with the size of the move. A 50 per cent fall needs a 100 per cent rise to recover; an 80 per cent fall needs 400 per cent. This is arithmetic rather than misfortune, and it is why investment returns are compared as compound rates rather than as averaged percentages.
| Fall | Rise needed to recover |
|---|---|
| 10% | 11.1% |
| 25% | 33.3% |
| 50% | 100% |
| 75% | 300% |
| 90% | 900% |
How do you take VAT off a gross price?
By dividing, not by subtracting. A price of £120 including 20 per cent VAT has a net of £100, because £100 plus 20 per cent is £120. Taking 20 per cent off £120 gives £96, which is wrong by four pounds and by the whole logic of the operation.
The rule: to remove a percentage that was added, divide by one plus the rate. To remove 20 per cent, divide by 1.2. To remove 7.5 per cent sales tax, divide by 1.075.
The size of the error scales with the rate. At 5 per cent, subtracting instead of dividing is out by 0.24 per cent of the gross; at 25 per cent it is out by 5 per cent — which on a five-figure invoice is a real amount of money.
Do stacked discounts add up?
No, they multiply. Twenty per cent off followed by a further 10 per cent off is 0.8 × 0.9 = 0.72, so 28 per cent off the original price rather than 30. On a £200 item that is £144 rather than the £140 the addition suggests.
The order does not matter — multiplication commutes, so 10 per cent then 20 per cent gives the same £144. What does matter is whether the second discount applies to the reduced price or to the original, and that is a policy question rather than a mathematical one.
The same structure appears in compound growth, in inflation and in interest. Anything applied successively multiplies, and anything applied to the same base adds.
Interest works the same way, which is why an account paying 1 per cent a month does not pay 12 per cent a year. Twelve successive multiplications by 1.01 give 1.1268, so the annual figure is 12.68 per cent — and the gap widens with the rate and the number of periods.
What about percentage of versus percentage off?
Three questions hide behind one word, and they have three different formulas.
- What is 15% of 80? Multiply and divide by a hundred: 80 × 15 ÷ 100 = 12.
- What percentage is 12 of 80? Divide and multiply by a hundred: 12 ÷ 80 × 100 = 15%.
- What is the change from 80 to 92? Divide the difference by the starting value: 12 ÷ 80 × 100 = 15% increase.
The second and third look identical and are not: the second compares a part to a whole, the third compares a difference to a starting point. Using the finishing value as the denominator in a change calculation is the other common error, and it always understates a rise.
Questions people ask
Is a percentage point ever written as pp? Yes, particularly in economics and polling — "up 1.2pp" is unambiguous and shorter than the alternatives.
How do I express a change in something already measured in per cent? In points, unless you say otherwise. A market share moving from 12 to 15 per cent is normally reported as three points, and "up 25 per cent" would be technically true and misleading.
What about basis points? A hundredth of a percentage point, used where precision matters — 25 basis points is 0.25 points. Interest rates and fees are quoted this way precisely to avoid the ambiguity above.
Can a percentage exceed 100? Of a base, yes — growth of 150 per cent is ordinary. Of a whole, no: a share of something cannot be more than all of it, and a figure claiming otherwise usually has a different denominator than you think.
Say points when you mean the difference and per cent when you mean the ratio, and most of the ambiguity disappears. The percentage calculator covers all three questions separately, the fraction calculator handles the same maths in a different notation, and the inflation calculator is where compound rates matter most.