Successive Percentage Changes: Why +20% Then -20% Is Not Back to Zero
October 29, 2026
Percentage changes do not simply add or subtract when applied one after another, because each change applies to a different base. To combine them, convert each to a multiplier (1 + change) and multiply. A 20% increase is × 1.20, and a 20% decrease is × 0.80. Together: 1.20 × 0.80 = 0.96, a net decrease of 4%.
Starting with 100: after +20% you have 120, and 20% of 120 is 24, so −20% gives 96. The order does not matter: reversing gives 100 × 0.8 × 1.2 = 96 too.
Skip the math and get your answer instantly:
Open the Percentage Calculator →Recovering From a Loss
The gain needed to recover from a loss of L is L ÷ (1 − L). A 10% loss needs 11.1%, because 0.90 × 1.111 = 1. A 20% loss requires 25%. A 33.3% loss requires 50%. A 50% loss requires 100%, and a 75% loss requires 300%. This asymmetry is why avoiding large losses matters in investing. A portfolio that loses 50% and then gains 50% ends at 75% of its starting value.
| Loss | Gain needed to break even | Check |
|---|---|---|
| 10% | 11.1% | 0.90 x 1.111 = 1.00 |
| 20% | 25.0% | 0.80 x 1.25 = 1.00 |
| 33.3% | 50.0% | 0.667 x 1.5 = 1.00 |
| 50% | 100.0% | 0.50 x 2.00 = 1.00 |
| 75% | 300.0% | 0.25 x 4.00 = 1.00 |
Stacked Discounts
A store offers 20% off, then an additional 10% off the reduced price: the combined discount is not 30% but 1 − 0.80 × 0.90 = 28%. Add a 5% coupon: 0.80 × 0.90 × 0.95 = 0.684, a total discount of 31.6%. On a $200 item, the price is $136.80. When comparing offers, compute the multiplier for each and compare the final price. A single 30% off is better than 20% plus 10%, which is only 28%.
Multi-Year Returns
An investment gaining 10% and 10% and then losing 15% over three years is 1.10 × 1.10 × 0.85 = 1.0285, a total gain of 2.85%, not 5%. The average annual return computed geometrically is 1.0285^(1/3) − 1 = 0.94% per year. The arithmetic average of the three numbers, (10 + 10 − 15) ÷ 3 = 1.67%, overstates actual growth. Use the compound growth rate to describe performance over time.
Reverse Percentages
To undo a percentage increase, divide by the multiplier instead of subtracting the same percentage: a price of $120 after a 20% increase was $120 ÷ 1.20 = $100, not $120 × 0.80 = $96. The same applies for taxes: a $108 total including 8% sales tax had a pre-tax price of $108 ÷ 1.08 = $100. Mistakes here are common in markup and margin calculations.
Frequently Asked Questions
How do I combine two percentage changes?
Multiply their multipliers (1 + change) and subtract 1.
What gain do I need to recover from a 50% loss?
100%.
Is 20% off plus 10% off the same as 30% off?
No. It is 28% off.
Does the order of percentage changes matter?
Not for the final result, since multiplication is commutative.
How do I undo a 20% increase?
Divide by 1.20.