The Three Questions a Percentage Answers
Almost every percentage problem is one of three: what is X per cent of Y, X is what per cent of Y, and what is the change from X to Y. They look similar and are easy to conflate, which is where most percentage errors start.
This calculator handles all three separately, so you pick the question rather than rearranging a formula in your head.
How to Use the Calculator
Identify which of the three questions you actually have before entering anything.
- For "what is 15% of 200", use the first form. Multiply the number by the percentage as a decimal, the answer is 30.
- For "30 is what per cent of 200", use the second. Divide and multiply by a hundred, the answer is 15%.
- For "the price went from 200 to 230", use the change form. That is a 15% increase, calculated from the original value.
- Check which value is the base when calculating change. Percentage change is always measured from the starting figure, and using the end figure instead is the most common error.
- For a reverse calculation, a price after a 20% rise was 240, what was it before, divide by 1.2 rather than subtracting 20%.
Why a Rise and a Fall Do Not Cancel
A number that rises 50% and then falls 50% does not return to where it started. Going from 100 up to 150 and then down by half of 150 leaves 75. The percentages are taken from different bases, so they are not opposites even though they look like them.
The same asymmetry explains why recovering from a loss takes more than the loss took away. A fall of 50% needs a rise of 100% to get back; a fall of 20% needs a rise of 25%. Anyone reading investment returns or comparing year-on-year figures runs into this, and the intuition that the two should cancel is simply wrong.
Recovering From a Percentage Fall
The rise required to return to the original value after a given decline.
| Fall | Leaves you at | Rise needed to recover |
|---|---|---|
| 10% | 90 | 11.1% |
| 20% | 80 | 25% |
| 25% | 75 | 33.3% |
| 33% | 67 | 50% |
| 50% | 50 | 100% |
| 75% | 25 | 300% |
| 90% | 10 | 900% |
The asymmetry grows sharply at the deep end, which is why a large drawdown is so much worse than it first appears. Losing 90% requires a tenfold gain to undo, not another 90%.
Percentage Points Are Not Per Cent
If an interest rate moves from 2% to 3%, that is a rise of one percentage point and also a rise of fifty per cent. Both statements are true and they describe very different things, which is why the distinction is worth keeping. Reporting that describes it as a "50% increase" is technically correct and often misleading; "one percentage point" is the clearer phrasing.
The same care applies to percentages of percentages. A vote share moving from 40% to 44% has gained four percentage points, which is a ten per cent increase in the share. Whichever you use, saying which one you mean removes an ambiguity that otherwise sits quietly in the number.
Averaging percentages is the other quiet trap. If one shop sells 10 items at a 50% margin and another sells 1,000 at 10%, the combined margin is not 30 per cent, it is barely above 10, because the average has to be weighted by volume. Percentages taken from different bases cannot be averaged directly, and doing so produces a figure that looks reasonable and describes nothing.