Percentage Calculator

Percent means per hundred. Every percentage question is therefore a division by a hundred somewhere, and the only difficulty is which way round.

The seven questions, with the operation each one needs:

What is 15% of 200? Convert and multiply: 0.15 x 200 = 30.

15 is what percent of 200? Divide and convert: 15 / 200 = 0.075, so 7.5%.

Increase 200 by 15%: multiply by 1.15, giving 230. Decrease it by 15%: multiply by 0.85, giving 170. Doing it as "find 15% then add" gives the same answer and takes two steps, so the multiplier form is worth learning.

What is the percentage change from 200 to 230? The difference over the ORIGINAL value: 30 / 200 = 15%.

30 is 15% of what number? Divide by the decimal: 30 / 0.15 = 200.

What is the percentage difference between 200 and 230? The gap over the AVERAGE of the two: 30 / 215 = 13.95%. It has no starting point, so it gives the same answer either way round, which percentage change does not: from 230 down to 200 is a fall of 13.04%.

Three things about percentages are counterintuitive and account for most mistakes made with them.

The first is that increases and decreases do not cancel, because each is taken from a different base. To undo a 50% fall you need a 100% rise, and an investment that loses 50% then gains 50% is still 25% down.

The second is percentage points. Unemployment going from 4% to 5% is up one percentage point and up 25%, and gets reported as "up a point" by the government and "up 25%" by the opposition. The convention is to use points when comparing two percentages and percent when comparing two quantities.

The third is that percentages do not average. Scoring 50% on a 10-question test and 90% on a 90-question test gives 86% overall, not 70%, because the second test carries nine times the weight. Averaging percentages is only valid when the bases are equal, which they usually are not.

One more, for compounding: a 10% rise three times running is not 30%. It is 1.1 cubed, so 33.1%. Over longer runs the gap grows quickly: 10% a year for ten years is 159%, not 100%.