How do you calculate percentage change?
Short answer. Subtract the old value from the new one, divide by the old value, then multiply by 100. Going from 40 to 50 is (50 − 40) ÷ 40 × 100 = a 25% increase. A negative answer is a decrease: 50 to 40 is −20%. The old value always goes on the bottom.
One formula covers increases and decreases, and the sign of the answer tells you which one you got. Almost every mistake people make with it is the same mistake: dividing by the wrong number.
The formula
percentage change = (new − old) / old × 100
The old value is the denominator, always. Percentage change is a comparison to where you started, so the starting point sets the scale. Swap the two and you are answering a different question, as the next section shows rather bluntly.
An increase, worked
A salary goes from 52,000 to 55,900.
change 55,900 − 52,000 = 3,900 divide 3,900 / 52,000 = 0.075 × 100 7.5% increase
Positive, so it went up. If you prefer one pass, divide the new value by the old and subtract one: 55,900 ÷ 52,000 = 1.075, and 1.075 − 1 = 0.075. Same answer, fewer keystrokes.
A decrease, worked
Now go the other way, from 50 to 40.
change 40 − 50 = −10 divide −10 / 50 = −0.2 × 100 20% decrease
Here is the asymmetry that surprises people. Going from 40 to 50 is a 25% increase, but coming back from 50 to 40 is only a 20% decrease. Same ten units of difference, two different answers, because the denominator changed from 40 to 50. Neither is wrong. They are answers to different questions.
The multiplier shortcut
For repeated work, skip the subtraction and multiply by a single number. An increase of p percent means multiplying by 1 + p/100; a decrease means multiplying by 1 − p/100.
| Change | Multiplier | Applied to 200 |
|---|---|---|
| Up 5% | × 1.05 | 210 |
| Up 15% | × 1.15 | 230 |
| Up 25% | × 1.25 | 250 |
| Up 100% | × 2 | 400 |
| Down 10% | × 0.9 | 180 |
| Down 20% | × 0.8 | 160 |
| Down 50% | × 0.5 | 100 |
Multipliers also chain, which is what makes them worth learning. Three years of 3% growth is not 9%, it is 1.03 × 1.03 × 1.03 = 1.092727, a total increase of 9.2727%. The extra 0.2727 is growth on the growth.
Down 50%, then up 50%, is not where you started
This is the trap, and it costs real money in sale prices and portfolio statements.
100 down 50% → 100 × 0.5 = 50 50 up 50% → 50 × 1.5 = 75 net change (75 − 100) / 100 × 100 = 25% decrease
The second 50% is taken from 50, not from 100, so it is only worth half as much. Gentler numbers behave the same way: 200 down 20% is 160, and 160 back up 20% is 192, a net loss of 4%. The multipliers say it in one line: 0.8 × 1.2 = 0.96.
Percentage changes cancel only when the multipliers multiply to exactly 1. Up 25% then down 20% does it, because 1.25 × 0.8 = 1. There is no shortcut for spotting these; you just have to multiply. It is the same arithmetic behind stacked discounts, where 20% off then 10% off comes to 28% off rather than 30%.
Percent versus percentage points
When the thing changing is itself a percentage, the language forks. A rate rising from 4% to 6% has gone up by 2 percentage points. As a percentage change it has gone up 50%, because (6 − 4) ÷ 4 × 100 = 50.
Both numbers are true and they describe the same event. Which one gets printed usually depends on which one sounds better, so it is worth checking which is meant before reacting to either.
Working backwards to the original
If you know the new value and the percentage change, divide by the multiplier. Do not subtract the percentage.
after a 5% rise, the value is 63 original = 63 / 1.05 = 60 after 15% off, a jacket costs 68 original = 68 / 0.85 = 80 wrong: 63 − 5% of 63 = 59.85
The 5% belonged to the original number, not to the new one, so taking 5% off the new value takes off too much. Reversing a percentage is a division, every time.
When the division does not come out clean
Real inputs rarely divide nicely. A price moving from 18.50 to 21.30 is a change of 2.80, and 2.80 ÷ 18.50 = 0.151351351..., repeating forever. Rounded, a 15.14% increase.
That rounding is fine as a final answer and dangerous as an intermediate one. Binary floating point, which is what most calculators and spreadsheets run on, cannot even hold 1.1 exactly, so a percentage change from 1.0 to 1.1 comes out as 10.000000000000009% instead of a flat 10%. Harmless once, visible after you sum a column of them. Prism's keypad works in exact decimal arithmetic rather than binary floating point, which is why 0.1 + 0.2 returns exactly 0.3 there.
Quick reference
| You want | Do this | Example |
|---|---|---|
| Percentage change | (new − old) / old × 100 | 40 to 50 is +25% |
| Apply an increase | old × (1 + p/100) | 200 up 15% is 230 |
| Apply a decrease | old × (1 − p/100) | 200 down 20% is 160 |
| Undo a change | new / multiplier | 63 after +5% came from 60 |
| Two changes in a row | multiply the multipliers | 0.5 × 1.5 = 0.75, so −25% |
If the numbers you are comparing are a measurement against a known correct value rather than a before and after, you want percent error instead. The arithmetic looks nearly identical, and the denominator means something different.
Or let the percentages tool hold the denominator for you
Prism has a percentages calculator with From and To fields: type 40 and 50, read 25% off the Change row. Percentages are free forever, with no ads and no subscription. See the app.