Percentages express a quantity as parts per hundred. Nearly every practical percentage question is one of six forms, and this calculator answers all of them at once so you can see how they relate.
The six questions
| Question | Formula |
|---|---|
| What is X% of Y? | Y × X / 100 |
| X is what percent of Y? | X / Y × 100 |
| X is Y% of what? | X / (Y / 100) |
| Percent change from X to Y | (Y − X) / |X| × 100 |
| Increase X by Y% | X × (1 + Y / 100) |
| Decrease X by Y% | X × (1 − Y / 100) |
Change and difference are not the same
Percent change is directional and needs a starting point: going from 40 to 50 is a 25% increase, while going from 50 to 40 is a 20% decrease. Same gap, different base.
Percent difference is symmetric and uses the average of both values as the base, which makes it appropriate when neither value is "first".
Percent change = (new − old) / |old| × 100 Percent difference = |a − b| / ((a + b) / 2) × 100
Why successive percentages do not add
A 20% increase followed by a 20% decrease does not return you to the start. Take 100 → 120 → 96: a net 4% loss, because the decrease applies to the larger figure.
The same asymmetry means a 50% loss requires a 100% gain to recover. Multiply the factors instead of adding the percentages.
Net factor = (1 + a) × (1 + b) Example: 1.20 × 0.80 = 0.96 → 4% net decrease
Percentage points versus percent
If a rate moves from 5% to 7%, that is a rise of 2 percentage points — but a 40% increase. Conflating the two is one of the most common errors in reporting figures, and the distinction matters enormously in finance and statistics.
Worked example
Using the values pre-loaded in the calculator above:
| Input | Value |
|---|---|
| Question | What is X% of Y? |
| X | 25 |
| Y | 200 |
| Output | Value |
|---|---|
| 25% of 200 | 50 |
| X% of Y | 50 |
| X as % of Y | 12.5% |
| Percent change X → Y | 700% |
| Percent difference | 155.5556% |
| X increased by Y% | 75 |
| X decreased by Y% | -25 |
Frequently asked questions
How do I calculate a percentage increase?
Subtract the old value from the new, divide by the old value, and multiply by 100. From 80 to 100 is (100 − 80) / 80 × 100 = 25%.
How do I remove a percentage that is already included?
Divide rather than subtract. To strip 20% tax from a gross of $120, compute 120 / 1.20 = $100. Subtracting 20% would incorrectly give $96.
What is a percentage point?
The arithmetic difference between two percentages. Moving from 5% to 7% is 2 percentage points and simultaneously a 40% relative increase.
Why does a 50% loss need a 100% gain to recover?
Because the gain is calculated on the reduced base. $100 falling 50% leaves $50, and returning to $100 from $50 requires doubling — a 100% gain.