How to Use This Tool
Enter two numbers and a percentage. All three of the questions people mean by "percentage" are answered at once, from the same inputs.
Three questions, one word
The reason percentage calculators are confusing is that the word covers three different operations, and with round numbers they can all produce the same answer:
what is 15% of 200 200 x 0.15 = 30 30 is what percent of 200 30 / 200 = 15% 200 to 230, what change (230-200) / 200 = 15% down 10% -> up 11.1% to recover down 20% -> up 25.0% down 50% -> up 100.0% down 80% -> up 400.0% +50% then -50% 100 -> 150 -> 75 -50% then +50% 100 -> 50 -> 75 same either way
Why a fall needs a bigger rise
Because the rise is measured against a smaller base. Fall 50 per cent and you are at half; to get back you need to add the whole of what remains, which is 100 per cent of it. Fall 80 per cent and the recovery is 400 per cent, which is why deep losses are so much worse than they look.
The same asymmetry means successive changes never add up. Plus 50 followed by minus 50 leaves you 25 per cent down, and the order makes no difference because multiplication commutes.
Percentage points are a different unit
If an interest rate moves from 4 per cent to 6 per cent, it has risen 2 percentage points and 50 per cent. Both are true and they describe the same event.
Which one gets quoted usually depends on which sounds better. A fee rising from 1 per cent to 1.5 per cent is "half a percentage point" or "a 50 per cent increase" — identical fact, very different impression. When a headline gives a percentage change of something that is itself a percentage, it is worth checking which one it means.
Discounts do not stack the way they look
Twenty per cent off, then a further 10 per cent off, is not 30 per cent off. It is 0.8 x 0.9 = 0.72, which is 28 per cent off. The second discount applies to the already-reduced price, and the gap widens with larger numbers: 50 and 50 is 75 per cent off, not 100.
Reverse percentages need division, not subtraction
If a price includes 20 per cent tax and is $120, the pre-tax price is not $120 minus 20 per cent, which would be $96. It is 120 / 1.2 = $100. Subtracting the percentage takes it off the wrong base, and this is one of the most common arithmetic mistakes in everyday life.
