Compound Percentage Change Explained
A percentage change from one value to another is a single step: ((new − original) ÷ original) × 100. When several percentage changes happen in a sequence — a rise one year and another rise the next, or a markup followed by a discount — the overall change is not the sum of the individual percentages. The changes compound, because each step applies to the value left by the step before.
Two familiar surprises follow from that. A 10% increase followed by another 10% increase is a 21% increase, not 20%. A 20% increase followed by a 20% decrease does not return you to the start; it leaves you 4% down. This article shows the multiplicative rule, works those examples in full, and compares the overall change with a one-step percentage change between the first and last values. For a single step, use the percentage change calculator or the percentage increase calculator.
Percentage Increase Calculator
Measure how much a value has grown, using ((New − Original) ÷ Original) × 100.
Result
The multiplicative rule
Each percentage change P corresponds to a multiplier of 1 + P/100. A 10% increase multiplies by 1.10. A 10% decrease multiplies by 0.90. A 20% increase multiplies by 1.20. A 20% decrease multiplies by 0.80.
For a sequence of changes, multiply the multipliers, then convert the product back to a percentage:
That overall change is exactly the ordinary percentage change from the starting value to the finishing value. Compounding is not a different kind of mathematics; it is what you get when you apply several changes in order and then ask one question at the end: “from start to finish, what was the percentage change?”
Example: 10% up, then 10% up
Start with 100, to make the money easy to see.
- After the first 10%: 100 × 1.10 = 110.
- After the second 10%: 110 × 1.10 = 121.
The overall change is (121 − 100) ÷ 100 × 100 = 21%. In multipliers: 1.10 × 1.10 = 1.21, and (1.21 − 1) × 100 = 21%. The extra 1 percentage point is 10% of the first gain of 10. The second increase is applied to 110, not to 100, so it contributes 11, not 10.
The same 21% result holds for any starting value, not just 100, because both steps scale the current amount. Starting at 50: 50 → 55 → 60.50, and (60.50 − 50) ÷ 50 × 100 = 21%.
Example: 20% up, then 20% down
- After a 20% increase: 100 × 1.20 = 120.
- After a 20% decrease: 120 × 0.80 = 96.
The overall change is (96 − 100) ÷ 100 × 100 = −4%. In multipliers: 1.20 × 0.80 = 0.96, and (0.96 − 1) × 100 = −4%. You do not return to 100, because 20% of 120 is 24, and 20% of 100 is only 20. The fall is larger, in raw units, than the earlier rise.
Example: 10% down, then 10% up
- After a 10% decrease: 100 × 0.90 = 90.
- After a 10% increase: 90 × 1.10 = 99.
The overall change is −1%. Multipliers: 0.90 × 1.10 = 0.99. An equal-looking pair of percentages in opposite directions does not cancel, and the order changes the intermediate values — but 0.90 × 1.10 and 1.10 × 0.90 are the same product, so the end point is 99 either way when both steps are simple percentages of the current value.
Example: 5% a year for three years
If a quantity grows by 5% each year for three years, the overall change is not 15%.
- Multiplier: 1.05 × 1.05 × 1.05 = 1.05³ = 1.157625.
- Overall change: 15.7625%.
Starting at 100, the value goes 105, then 110.25, then 115.7625. The “15%” figure would be correct only if each 5% were taken on the *original* 100 every year, which is a different rule (sometimes called simple growth). When each year applies 5% to the new balance, the growth compounds, and the three-year figure is 15.7625%.
This is the same structure as compound interest, but the article is not a finance product review and does not quote market rates. It is only the percentage arithmetic.
Relating the overall change to a single percentage change
If you know the start and the end, you do not have to reconstruct every intermediate multiplier. Use the ordinary formula:
From 100 to 121, that is 21%. From 100 to 96, that is −4%. From 100 to 115.7625, that is 15.7625%. Compounding is the story of how you got there; the one-step formula is the summary.
If the start is 0, a later value cannot be expressed as a percentage change from that start, because the formula would divide by zero. That limit is the same one discussed for ordinary change on the percentage change calculator page and in the calculation methodology.
How this connects to successive discounts
Successive discounts are compound *decreases*. 20% off and then 10% off means multipliers 0.80 and 0.90, and an overall change of (0.72 − 1) × 100 = −28%, which is 28% off. That is the same arithmetic as How successive discounts stack, written as a percentage change instead of as a sale price.
Common mistakes
- Adding the percentages. Two 10% increases are not 20% overall.
- Assuming that +P% and −P% cancel. They do not, as the 20% up and 20% down example shows.
- Using the original value as the base for every later step when the real process applies each rate to the latest value.
- Comparing percentage points with compounded change without naming the base. If an interest *rate* goes from 4% to 5% to 6%, that is a change in percentage points, which is a different object from compounding a quantity at those rates. See Percentage increase vs. percentage points.
Frequently asked questions
Why is a 10% increase followed by a 10% increase equal to 21%, not 20%?
The second 10% is taken on the already increased value. Multipliers: 1.10 × 1.10 = 1.21, which is a 21% increase.
If a price goes up 20% and then down 20%, do I get back to the original price?
No. You end at 96% of the start, which is a 4% decrease. 1.20 × 0.80 = 0.96.
How do I combine several percentage changes?
Convert each change to a multiplier 1 + P/100, multiply the multipliers, subtract 1, and multiply by 100.
Is the overall change the same as an ordinary percentage change from start to end?
Yes. Compounding explains the path; ((end − start) ÷ start) × 100 is the result.
Does the order of the changes matter?
For simple percentage-of-current-value steps, the final multiplier is the product, so the order does not change the end point. Intermediate values do change with order, and the end point can change if a step is a flat amount rather than a percentage.