Three percentage sums are wrong, and the rest are just fiddly
Most of what a percentage calculator does is arithmetic anybody could do given a minute and a piece of paper. Three of them are different: people get them wrong confidently, get a plausible-looking answer, and never find out.
One. Taking a percentage off is not undone by adding it back on
A coat is 20% off at £80. What was it before? The instinctive answer is £96, and it is wrong. It was £100.
Removing a percentage is a division. Eighty divided by 0.8 is a hundred. Adding 20% back on works with the wrong base: 20% of £80 is £16, but the discount that was taken off was 20% of £100, which is £20.
The error always runs the same way. It understates the original, never overstates it, and it gets worse the bigger the percentage: 1% out at a 10% discount, 25% out at a 50% one. It is the same arithmetic as removing VAT, which is why a lot of small businesses have quietly under-recorded their net figures for years.
Two. Per cent and percentage points are different units
A rate goes from 4% to 6%. It has risen by 2 percentage points. It has also risen by 50 per cent. Both are correct, and they are not the same statement.
Which one gets used tells you something about who is talking. If the number is a rate, points is the unit that describes what happened to it. Per cent is the one that sounds larger, so it is the one a headline will pick when the direction suits the story, and the smaller-sounding one when it does not. The tool shows both for the same pair of numbers so the gap is visible.
Three. Rises and falls of the same size do not cancel
Down 50%, then up 50%, leaves you 25% down. Not level. The rise is applied to the smaller number, so it buys back less than the fall took away.
To recover a 50% fall you need a 100% rise. A 20% fall needs 25% back. A 90% fall needs 900%. That asymmetry is why "down 20% then up 20%" investments are still down, and why recovery percentages quoted against a low point are so much larger than the fall that produced the low point.
Two changes in a row never add up
Discounts compound, so they come out smaller than their sum. Twenty per cent off, then a further 10% off, is 28% off, not 30%. The second discount is applied to the already reduced price.
Increases compound the other way and come out larger. Up 10% then up 10% is 21%, not 20%. Two years of 3% pay rises is 6.09%, not 6%, which sounds trivial until it is thirty years of it.
Why every box answers at once
Almost every percentage calculator asks you to choose a mode first, and choosing the mode is the step that goes wrong. Somebody wanting to reverse a discount lands on the percentage change box, types 80 and 20, gets a number, and leaves with it.
So there is no mode here. Five boxes, each labelled with its question in plain words, all live at the same time. If the answer you are looking at is not the one you wanted, the one you did want is already on the screen next to it.
A note on money
Nothing on this page is rounded until it is displayed, and then only to two places, so the compounding effects above stay visible rather than being smoothed away by intermediate rounding.
But if you are working with actual money, use a money tool for the final figure. A percentage of a price is exactly where fractions of a penny appear, and once they are carried into a second calculation they compound into a real discrepancy on an invoice somebody gets paid against. The VAT calculator and the markup and margin tool hold everything in whole pence for that reason.
Common questions
A coat is 20% off at £80. What was it before?
One hundred pounds, not £96. Adding 20% back onto £80 gives £96 and it is wrong, because taking a percentage off is a division rather than a subtraction. Divide by 0.8. This is the most common percentage error there is and it always errs the same way: it understates the original.
What is the difference between per cent and percentage points?
They are different units. A rate going from 4% to 6% has risen by 2 percentage points and by 50 per cent. Both statements are true and they describe the same move. If the numbers are rates, points is the honest unit; per cent is the one a headline will reach for because it sounds larger.
If something falls 50% and then rises 50%, am I back where I started?
No, you are 25% down. The rise applies to the smaller number, so it does not undo the fall. Recovering a 50% fall takes a 100% rise. This is the arithmetic behind recovery headlines that do not add up, and it works the same way on a 20% fall, which needs 25% back.
How do I remove VAT from a price?
Divide by 1.2, do not subtract 20%. A price of £120 including VAT at 20% is £100 plus £20 of VAT. Subtracting 20% gives £96, which is £4 out on every hundred pounds. Use the reverse box on this page, or the VAT calculator for anything where the pence have to reconcile.
Do two discounts add up?
No, they compound and come out smaller than the sum. Twenty per cent off followed by a further 10% off is 28% off, not 30%, because the second discount applies to the already reduced price. The same effect runs the other way on increases: up 10% then up 10% is up 21%.
Why does this show all five answers at once?
Because picking a mode is where people go wrong on every other percentage calculator. They land on the percentage change box and type a discount into it. Every box here is labelled with the question it answers in plain words and all of them are live, so there is no mode to choose wrongly.