The percentage change formula, and the mistake everyone makes
A percentage change compares a new value with the value it replaced:
percentage change = (new − original) ÷ original × 100
Two things about that formula decide whether you get the right answer. The order of the subtraction sets the sign, so going from 250 to 300 is positive and going from 300 to 250 is negative. And the denominator is the original value — the one you started from. Dividing by the new value instead is the classic error: 50 ÷ 300 = 16.67%, which is the change measured backwards, not the change you experienced.
So 250 → 300 is 50 ÷ 250 = 0.2 = +20%. A drop from 300 to 250 is 50 ÷ 300 = 16.67%. Same 50 units, different percentages, because the base moved. This asymmetry is a property of the arithmetic rather than a trick: a rise and a fall of the same size are not the same percentage of anything.
Why “up 50% then down 50%” is not break-even
Start at 100. A 50% rise gives 150. A 50% fall from 150 gives 75 — not 100. The second percentage is taken from a larger base, so it removes more. To undo a rise of r percent you need a fall of r ÷ (1 + r/100) percent: to undo +50% you need −33.33%, to undo +25% you need −20%, and to undo +10% you need −9.09%.
| Move | Result from 100 | Move needed to return to 100 |
|---|---|---|
| +10% | 110 | −9.09% |
| +25% | 125 | −20.00% |
| +50% | 150 | −33.33% |
| +100% | 200 | −50.00% |
| −20% | 80 | +25.00% |
| −50% | 50 | +100.00% |
The table is also the fastest way to sanity-check a discount: a 20% discount does not undo a 20% markup. A shop that marks a price up 20% and then offers 20% off is charging 96% of the original price, which is why retailers describe such offers as “still cheaper” rather than “back to normal”.
Reverse mode: finding the original from a percentage
Restocking your mental toolkit takes one step: instead of subtracting the percentage from the final value, divide by the multiplier. If a price is 300 after rising 20%, the original is 300 ÷ 1.2 = 250. Subtracting 20% from 300 gives 240, which would be the original only if the rise had been 25%.
Written out: original = final ÷ (1 + p ÷ 100), and the other direction is final = original × (1 + p ÷ 100). Use a negative percentage for a fall: a price at 80 after a 20% fall came from 80 ÷ 0.8 = 100. This is the calculation people need for “what was the price before the sale”, “what was the original weight”, and “what was the pre-tax figure” — and it is also the calculation almost everyone gets wrong by subtracting the percentage instead of dividing by the multiplier.
Percentage change versus percent difference
When neither number is a natural baseline, percentage change is the wrong tool because the answer depends on which value you call first. Percentage difference solves that by dividing by the mean of the two values:
percent difference = |A − B| ÷ ((A + B) ÷ 2) × 100
For 250 and 300 the difference is 50 and the mean is 275, giving 18.18% — a single symmetric number, whereas the change is +20% one way and −16.67% the other. Use it to compare two measurements of the same thing, two supplier quotes, or two classes’ averages.
| Measure | Denominator | 250 & 300 | Use when |
|---|---|---|---|
| Percentage change | The original value | +20% (250→300) or −16.67% (300→250) | There is a before and an after |
| Percent difference | The mean of the two | 18.18% | Neither value is the baseline |
| Percent error | The true / accepted value | Depends on which value is the reference | One value is known to be correct |
Percentage points are not percent
One more trap worth knowing, because it appears in every news cycle. If an interest rate moves from 4% to 6%, that is a rise of 2 percentage points and a rise of 50%. Both statements are true and they describe the same event; they are not interchangeable. Rates, probabilities and survey shares are normally described in percentage points so the denominator problem disappears — and percentage change is reserved for quantities such as prices, weights and counts, where the original value is a meaningful base.
Where the number goes wrong in practice
- Dividing by the new value. Gives a smaller number than the truth whenever the value increased.
- Using the absolute change as a percentage. “Up 50” and “up 50%” are different statements; the tool reports both, side by side.
- Averaging percentages. A country whose two regions grow 10% and 20% has not grown 15% unless the regions are the same size; average the underlying values, then compute the percentage once.
- Chaining percentages by adding them. Three consecutive 10% rises are 1.1 × 1.1 × 1.1 = 33.1%, not 30%.
- Negative and zero bases. A change from zero has no percentage answer, and changes involving negative values (a loss shrinking from −50 to −10) are better described in absolute terms.
Frequently asked questions
How do I calculate the percentage change from 250 to 300?
Subtract the original from the new value, then divide by the original: (300 − 250) ÷ 250 = 0.2, so the change is +20%, an absolute increase of 50. Dividing by the new value instead gives 16.67%, which measures the change backwards.
How do I find the original value if I only know the final value and the percentage?
Divide by the multiplier rather than subtracting the percentage: original = final ÷ (1 + p ÷ 100). A price of 300 after a 20% rise came from 300 ÷ 1.2 = 250, not 300 − 20% = 240. For a fall, use a negative percentage: 80 after a 20% drop came from 80 ÷ 0.8 = 100.
Why is the percentage change different when I reverse the two numbers?
Because the denominator changes. From 250 to 300 the base is 250 and the change is +20%; from 300 to 250 the base is 300 and the change is −16.67%. If neither value is a baseline, use percent difference, which divides by the mean and gives the same answer either way.
What is the difference between percentage change and percent difference?
Percentage change divides by the original value and answers "how much did it move?". Percent difference divides by the mean of the two values and answers "how far apart are these two numbers?" — for 250 and 300 that is 20% versus 18.18%.
What is 250 to 300 as a percent difference?
18.18%. The absolute difference is 50 and the mean is 275, so 50 ÷ 275 × 100 = 18.18%. It is lower than the 20% percentage change because the mean is larger than the original value.
Does a 20% increase followed by a 20% decrease return to the original?
No. From 100, up 20% gives 120, and 20% off 120 leaves 96 — 4% below where you started. To undo a 20% rise you need a 16.67% fall (20 ÷ 1.2), and to undo a 50% rise you need a 33.33% fall.
What is the difference between percentage points and percent?
If a rate goes from 4% to 6%, it has risen by 2 percentage points and by 50%. Percentage points describe the arithmetic difference between two percentages; percent describes the relative change. Rates, probabilities and survey shares are usually quoted in percentage points for exactly this reason.