Percentages, three ways
Three calculations that people reach for constantly and that go wrong in predictable ways — particularly the third one, where taking a percentage off and then putting it back on does not return you to where you started.
20% of 250
50
250 × 20 ÷ 100
The third one is the one that goes wrong
“X per cent of a number” is arithmetic nobody gets wrong. Percentage change is arithmetic people get wrong occasionally, usually by dividing by the new figure instead of the old one. Working backwards is arithmetic people get wrong systematically, and always in the same direction.
Here is the shape of it. Take a hundred, add ten per cent, and you have a hundred and ten. Take ten per cent off a hundred and ten, and you have ninety-nine. You are not back where you started, and you never will be, because the second ten per cent was a percentage of a bigger number than the first.
| Step | Result |
|---|---|
| Start | 100.00 |
| Add 10% | 110.00 |
| Take 10% off that | 99.00 |
| The correct way back: divide by 1.10 | 100.00 |
The third mode of the calculator above does the fourth row. Give it the percentage that was applied and the figure that resulted, and it returns what the figure was before.
This is the same mistake as reverse VAT
Exactly the same, and it is the most expensive place people make it. A price including 23% VAT is not the net price plus 23% that you can undo by subtracting 23% — you have to divide.
If that is what you are actually doing, the reverse VAT calculator is the right tool: it knows the Irish rates, it shows the working, and it names the source of every percentage. This page is the general-purpose version for everything else.
Percentage change, and the word “point”
If something goes from 20% to 25%, it has risen by five percentage points and by twenty-five per cent. Both statements are true and they are different numbers. The calculator’s change mode gives you the second one, because that is what “percentage change” means — but if you are comparing two figures that are themselves percentages, the first is usually what you want, and it is a subtraction.
Why this page exists at all
Honestly: as glue. It is here because people who need one of the calculations above often need one of the others on the same afternoon, and sending them off-site for it is worse than carrying it. It makes no claim to be a better percentage calculator than the hundred that already exist, and this site expects to rank for nothing on it.
What it does have that most do not is the working shown underneath the answer, and no rounding you cannot see.