Percentage Calculator
Work out percentages, percentage change and what percent of one number another is — including why a rise and fall of the same percentage never cancel.
What is X% of Y?
X is what percent of Y?
Percentage change from X to Y
Up ten per cent then down ten per cent leaves you down one
A rise of p multiplies by (1 + p/100); a fall of p multiplies by (1 − p/100). Do both and you have multiplied by 1 − (p/100)², which is never 1:
| Start at 100 | End at | Net | p²/100 |
|---|---|---|---|
| +5% then −5% | 99.75 | -0.25% | 0.25% |
| +10% then −10% | 99.00 | -1.00% | 1.00% |
| +20% then −20% | 96.00 | -4.00% | 4.00% |
| +50% then −50% | 75.00 | -25.00% | 25.00% |
| +90% then −90% | 19.00 | -81.00% | 81.00% |
The shortfall is exactly p squared over 100, as a percentage — the last two columns match in every row. Because it grows as the square, small round trips barely register and large ones are brutal: 90% up and 90% down leaves you with 19.
And it never goes your way. 1 − x² is below 1 for every non-zero x, so a matched round trip is always a loss — and the order genuinely does not matter, because multiplication commutes. There is no arrangement that saves you and no percentage that breaks even except zero.
So what does it take to undo a rise?
Since the same number will not do it, the useful question is which number will. To reverse a rise of p you need a fall of p/(100 + p):
| Rise | Fall that reverses it | Check |
|---|---|---|
| +10% | −9.09% | 100.00 |
| +25% | −20.00% | 100.00 |
| +50% | −33.33% | 100.00 |
| +100% | −50.00% | 100.00 |
The last row is the one everybody already knows: a doubling is undone by a halving. That is the same rule — +100% reversed by −50% — and it is worth noticing that the familiar case and the confusing ones are the same arithmetic.
The asymmetry always points the same way: the fall you need is smaller than the rise it undoes. Which is the round-trip loss again, seen from the other side — if the matching fall were enough, there would be no loss.
Per cent against percentage points
The other thing a calculator cannot settle for you is which of the two somebody meant:
| Rate moves | In points | In per cent |
|---|---|---|
| 4% → 5% | +1 | +25.0% |
| 40% → 41% | +1 | +2.5% |
| 2% → 3% | +1 | +50.0% |
| 50% → 51% | +1 | +2.0% |
Every row above is the same one-point move, and the relative change runs from 50% down to 2% depending only on where it started. Both descriptions are correct and they are not interchangeable.
This is not pedantry, it is where the misleading headline lives. "Interest rates rose 25 per cent" and "interest rates rose a quarter of a point" can describe the same Tuesday, and only one of them sounds like news. If a figure matters, ask which one it is.
How to use
- Pick the type of calculation you need.
- Enter your two numbers.
- Read the result and the working.
- Check which number is the base for a change calculation.
Frequently asked questions
Why does a 50 per cent fall need a 100 per cent rise to recover?
Because each is measured against a different base. Falling from 100 to 50 is a 50 per cent drop measured against 100; rising from 50 back to 100 is a 100 per cent gain measured against 50. Percentage changes are not symmetrical, which is why investment losses hurt more than equivalent gains help.
What is the difference between percentage points and per cent?
A rate moving from 4 per cent to 6 per cent has risen by 2 percentage points, which is a 50 per cent increase. Both descriptions are correct and they mean very different things, which is why the distinction matters in reporting on interest rates, unemployment and polling.
How do I calculate percentage change?
Subtract the old value from the new, divide by the old value, then multiply by 100. The old value is always the base — using the new one is the standard mistake and produces a different answer in every case except no change at all.
Do successive percentage changes add up?
No. A 10 per cent rise followed by a 10 per cent fall leaves you at 99 per cent of where you started, not back at 100, because the fall applies to the larger amount. Chained percentages multiply rather than add, which is the same arithmetic that makes compounding work.
How do I add or remove a percentage from a total?
To add, multiply by one plus the rate. To remove, divide by one plus the rate — do not subtract the percentage, which uses the wrong base. Removing 20 per cent from 120 gives 100, not 96.
What does a percentage over 100 mean?
That the part exceeds the whole it is measured against, which is perfectly ordinary for growth — 150 per cent of last year's figure means one and a half times it. It becomes suspect when the base is genuinely a maximum, which is why claims of giving 110 per cent are a joke rather than a statistic.
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