The percentage increase formula
A percentage increase compares how much something grew with where it started. Take the difference, then express it as a share of the starting value:
A positive result is an increase and a negative result is a decrease. The starting value is always the one you divide by. That's why going from 40 to 50 is a 25% increase, while going from 50 back to 40 is only a 20% decrease: the same gap of 10 is measured against different starting points.
Adding or removing a percentage
To increase a number by a percentage, multiply it by one plus the percentage as a decimal. To decrease it, multiply by one minus the percentage.
A 3.5% raise on a $52,000 salary is 52,000 × 1.035 = $53,820. A jacket at $120 with 30% off costs 120 × 0.70 = $84.
Working back to the original value
A common trap is undoing a percentage by applying the same percentage in the other direction. If a price is $54 after 8% sales tax, taking 8% off gives $49.68, which is wrong. The tax was 8% of the original price, not of $54. Divide instead:
So 54 ÷ 1.08 = $50. For a decrease, divide by (1 − p ÷ 100): a laptop that costs $680 after a 15% discount was originally 680 ÷ 0.85 = $800.
Several changes in a row
Percentage changes multiply; they don't add up. A shop that raises a price by 20% and then puts it on sale at 20% off ends up 4% below where it began, because 1.20 × 0.80 = 0.96. To combine any number of changes, multiply their factors together:
| Changes | Combined factor | Overall change |
|---|---|---|
| +10%, then −10% | 1.10 × 0.90 = 0.99 | −1% |
| −50%, then +50% | 0.50 × 1.50 = 0.75 | −25% |
| +5% three years running | 1.05³ = 1.157625 | +15.76% |
| −20%, then −10% | 0.80 × 0.90 = 0.72 | −28% |
Losses hurt more than equal-sized gains help. After a 50% fall you need a 100% rise to get back to even, and after a 20% fall you need a 25% rise.
Increase, change or difference?
Use this page when there is a clear before and after, such as last year's rent and this year's. If you're comparing two numbers where neither one comes first, like the prices at two different stores, a percentage difference measures the gap against their average instead.
Questions people ask
How do I calculate a percentage increase?
Subtract the old value from the new value, divide by the old value, then multiply by 100. Going from 40 to 50 is (50 − 40) ÷ 40 × 100 = 25%.
How do I add a percentage to a number?
Multiply the number by 1 plus the percentage as a decimal. Adding 8% to $250 is 250 × 1.08 = $270.
Why does a 50% drop need a 100% rise to recover?
Because the rise is measured from the smaller number. $100 falling 50% leaves $50, and getting from $50 back to $100 is a rise of $50, which is 100% of $50.
Is a 10% increase then a 10% decrease back to where I started?
No. 100 × 1.10 = 110, then 110 × 0.90 = 99. You end 1% lower, because the decrease is taken from the bigger number.
How do I find the price before a percentage was added?
Divide by 1 plus the percentage. If $54 includes 8% tax, the pre-tax price is 54 ÷ 1.08 = $50. Taking 8% off $54 would give the wrong answer of $49.68.
Can a percentage decrease be more than 100%?
Not for something that can’t go below zero, like a price. For values that can turn negative, such as profit, a change from +$20 to −$10 is a 150% decrease.