Percentage Calculator

A percentage is a fraction with the bottom half fixed at a hundred, and almost everything that goes wrong with them comes from losing track of what the hundred is a hundred of.

The three questions that sound alike

What is 15% of 80?        12
12 is what % of 80?       15%
12 is 15% of what?        80

Three sums, three answers, one set of numbers. The word
"of" is doing all the work and it is easy to lose.

These are the reason a percentage calculator exists at all. Written out they are obviously different sums. Said out loud in a meeting they are almost indistinguishable, which is why all three sit at the top of this page as sentences instead of behind a dropdown.

Up then down does not come back

£100 plus 20%   = £120
£120 minus 20%  = £96

Not £100. Going up by a fifth and coming back down by a
fifth are different sizes of step, because the second
fifth is a fifth of a bigger number.

A 20% rise followed by a 20% fall leaves you 4% down, and the order does not matter. This is not a rounding artefact, it is what percentages are: the second step is measured against a different starting point from the first.

It is also why a share that falls 50% needs to rise 100% to get back, and why "we cut costs 30% and then they went up 30%" does not mean costs are where they started.

Taking VAT back out

A £120 price including 20% VAT

wrong   120 - 20%     = £96      and £24 of VAT
right   120 / 1.20    = £100     and £20 of VAT

The VAT is 20% of the net price, not 20% of the price
on the label.

This one costs real money. Tax is charged on the price before tax, so removing it means dividing by 1.20, not subtracting a fifth. The gap on a £120 invoice is £4, which does not sound like much until it is every invoice for a year.

The panel above does both directions from the same figure, so you can see which one you meant.

Percentage points, which is the one journalists get wrong

A rate moves from 4% to 5%

up 1 percentage point
up 25 per cent

Both true. Headlines pick whichever is more dramatic.

When the thing being measured is already a percentage, there are two honest ways to describe a move and they give wildly different numbers. Unemployment going from 4% to 5% is one percentage point, or a 25% rise, and only one of those makes a good headline.

The rule: percentage points for the gap between two percentages, per cent for the relative change. When somebody quotes a big number without saying which, they have usually picked the one that helps.

Percentage change and percentage difference are not the same

Change is measured against the starting number, so going from 80 to 100 is a 25% rise and going from 100 to 80 is a 20% fall. Same pair of numbers, different answers, because the base moved.

Percentage difference divides by the average of the two instead, which makes it symmetric: 80 and 100 differ by 22.2% whichever order you put them in. Use change when one number came first, and difference when neither did.

Zero has no percentage

There is no percentage change from zero to anything. The formula divides by the starting number and you cannot divide by nothing, so the honest answer is the raw increase and a note that a percentage does not apply. Anything reporting infinity or a very large number has substituted a small value for the zero.

Nothing here is rounded until it is shown

Every figure is calculated at full precision and rounded only for display, so chaining one answer into the next box does not accumulate error. Money shows two decimal places because that is what money has, and the other answers show as many as they need up to six.