The percentage change is the new value minus the original, divided by the original, times 100. A positive result is an increase and a negative one a decrease; to work backwards, divide the final value by 1 plus (or minus) the percentage.
The formula
Percentage change
change % = (new − original) ÷ |original| × 100Original value, after an increase
original = final ÷ (1 + percent ÷ 100)Original value, after a decrease
original = final ÷ (1 − percent ÷ 100)Worked example
Now go backwards. A price is 100 after a 25% increase: 100 ÷ 1.25 = 80. And a price is 75 after a 25% decrease: 75 ÷ 0.75 = 100.
When to use it, and the mistakes to avoid
Use it for prices, salaries, grades, share prices, traffic, population and any figure you want to compare over time. The reverse mode is for when you know a new price and a rise or discount and need the price before it.
- Using the wrong base. The percentage is always of the original value, not the new one.
- Adding or subtracting percentages in a row. +10% then −10% is −1%, not 0%.
- Confusing percentage points with percent. A rate going from 4% to 5% is up one percentage point but 25% in relative terms.
- Subtracting the percentage from the final value. That is not the way back: divide instead.
- Mixing it up with percentage difference. That compares two values with neither as the start; see the percentage calculator.
FAQ
How do I calculate a percentage increase?
Subtract the original value from the new value, divide by the original, and multiply by 100. From 80 to 100 that is (100 − 80) ÷ 80 × 100 = 25%. A negative answer means the value fell, so the same steps give a percentage decrease.
Why is the change from 50 to 100 +100% but from 100 to 50 only −50%?
Because the change is always measured against the starting value. Going up, the starting value is 50, and the rise of 50 is 100% of it. Coming down, the starting value is 100, and the fall of 50 is only half of it. That is why a 100% rise followed by a 50% fall gets you back to where you started.
How do I find the original value from the final value and a percentage?
Divide the final value by 1 plus the percentage if it was an increase, or by 1 minus the percentage if it was a decrease. If a price is 100 after a 25% rise, the original was 100 ÷ 1.25 = 80. If it is 75 after a 25% fall, the original was 75 ÷ 0.75 = 100.
Why is it wrong to take 25% off the final value to undo a 25% increase?
Because the 25% was worked out on the original, not on the final value. 25% of 100 is 25 and 100 − 25 = 75, not the 80 you started with. The reverse calculation divides by 1.25 instead.
What if the starting value is zero or negative?
From zero there is no percentage change, because you cannot divide by zero, and the calculator says so. From a negative value, the change is measured against its size without the minus sign, so −50 to −25 is +50%. Read percentages from negative values with care, since they can mislead.