Scope and assumptions
Successive changes apply one after another to the running total; order does not change the final result for multiplication. Illustrative.
Add a 10% gain to a 10% loss and the instinct says you're back where you started. You're not — you end at 99%. Each percentage acts on whatever the last one left behind, so they stack by multiplying rather than adding, and the order you apply them in doesn't change where you land.
Stacking by multiplication
Final = Start × (1 + r₁) × (1 + r₂) × …
Each change is a multiplier: +10% → ×1.10, −10% → ×0.90. You multiply them in sequence, never add them.
| Start | Changes | Calculation | Final | Net |
|---|---|---|---|---|
| 100 | +10%, then −10% | 100 × 1.10 × 0.90 | 99 | −1% |
| 200 | +20%, then +10% | 200 × 1.20 × 1.10 | 264 | +32% |
| 500 | −25%, then +25% | 500 × 0.75 × 1.25 | 468.75 | −6.25% |
| 80 | +50%, then −33.33% | 80 × 1.50 × 0.6667 | 80 | 0% |
Why +10% then −10% ≠ 0
The −10% is taken off the new 110, not the original 100. 10% of 110 is 11, so you drop to 99. Equal-size opposite percents never cancel because the base shifts each step.
Finding the net percent
Multiply the multipliers, then subtract 1: 1.10 × 0.90 = 0.99 → net −1%; 1.20 × 1.10 = 1.32 → net +32%. The net percent is the single rate that, applied once to Start, reproduces Final.
More than two steps
Chain more factors: +5%, +5%, −3% → 1.05 × 1.05 × 0.97 = 1.0689 → +6.89% net. Each extra step multiplies onto the running product; you never add the individual percents. This is exactly the compound-interest mechanic in disguise — a +10% "year" then −10% "year" mirrors a rate that goes up then down.
Successive Changes That Don't Compound
- Adding the percents — +10% + (−10%) is not 0; multiply the factors.
- Re-basing on the original — each step uses the running total, not the start.
- Assuming order matters — ×1.10×0.90 = ×0.90×1.10, same final value.
- Trusting "big gain then equal loss" recovers — it never does; the loss base is larger.
- Forgetting the −1 for net % — the product is a factor; subtract 1 to get the percent.
Related advanced operations
For the rest of the advanced percentage operations, start at the hub: Percentage Difference Formula. Adjacent: Percentage of a Percentage and Continuous Percentage Growth.
FAQ
Does the order really never matter? For the final value, multiplication commutes, so +10% then −10% equals −10% then +10%. But the path matters for fees, taxes, or caps applied at intermediate steps, and for how volatile the journey felt. The endpoint is identical; the route isn't.
What if one change uses a different base? The model assumes each step acts on the running total. If a "10% bonus" is on the original salary while a "10% tax" is on the new total, the bases differ and you can't chain multipliers blindly — identify the base of each step.
How do I reverse a net change? If you know the final and the net factor, Original = Final ÷ (product of factors). For a +32% net (factor 1.32), 264 ÷ 1.32 = 200. Reversing successive changes is just dividing by the combined factor.
Is there a quick estimate when changes are small? For several small changes, the net is roughly the sum of the percents plus the sum of their pairwise products — a fast sanity check before reaching for a calculator.
Can I chain more than two periods with varying rates? Yes — string as many factors as you have: 1.05 × 0.97 × 1.03 … The product minus 1 is the net. This is exactly how a multi-year returns history collapses to one figure.
Sources: Khan Academy, Percent word problems; Maths is Fun, Percentage change.
Disclaimer: Illustrative. Rounding each step (e.g., to whole cents) can introduce tiny deviations from the ideal multiplication. Informational only.
Sources and references
- Khan Academy — Percent word problemshttps://www.khanacademy.org/math
- Maths is Fun — Percentage changehttps://www.mathsisfun.com/numbers/percentage-change.html
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