Percentage calculator

How to use it

1. Pick the question that matches yours — the three are separate on purpose.
2. Type the two numbers; the answer updates as you type.
3. Percentage change is signed: a fall shows as a negative number.

Successive percentage changes multiply, they do not add

A rise of 10% followed by a fall of 10% does not return you to the start. Multiply by 1.10, then by 0.90, and you land on 0.99 of the original figure. Percentage changes compose as a product of factors, never as a sum, which also means the order is irrelevant: the same two changes in either sequence give the same answer.

That compounding is why the recovery asymmetry generalises beyond halving. Undoing a fall of d percent takes a rise of d/(100 - d): down 20 needs up 25, down 50 needs up 100, down 80 needs up 400. It is also why averaging a run of annual growth rates arithmetically overstates the outcome, and why the geometric mean is the figure that actually reproduces the end value.

Basis points exist because percent and percentage points get confused

Money markets invented a third unit to escape the ambiguity entirely. A basis point is one hundredth of a percentage point, so a rate moving from 4.00% to 4.25% has risen 25 basis points and nobody has to work out whether an increase of 25% meant a move to 5%. Outside finance the confusion survives untreated: a party polling 20% one month and 25% the next has gained five points, which is a quarter of its support.

The other half of the problem is a base that goes unstated. A risk that doubles from 1 in 10,000 to 2 in 10,000 has risen by 100% and by one hundredth of a percentage point, both true and only one of them alarming. The same trap sits in trade pricing: a 25% mark-up on cost is a 20% margin on the selling price, because the two percentages are taken of different numbers.

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