Key takeaway
Net percentage change after several moves is NOT their sum. Multiply the growth factors: net = (1 + r1)(1 + r2)… − 1, where each r is the decimal rate. A +10% then −10% on 100 → 110 → 99, a net −1%, not 0%. Only equal-percent moves on the same base net to zero; sequential moves compound on changing bases, so order and base both matter.
Scope and assumptions
General arithmetic for successive multiplicative changes. Additive rebases (e.g. a fee on top) need their own setup. This site is informational and not official.
Short answer: Net percentage change after several moves is NOT their sum. Multiply the growth factors: net = (1 + r1)(1 + r2)… − 1, where each r is the decimal rate. A +10% then −10% on 100 → 110 → 99, a net −1%, not 0%. Only equal-percent moves on the same base net to zero; sequential moves compound on changing bases, so order and base both matter.
Quick navigation
- Increase & decrease → percentage-increase-and-decrease
- Percent change → percent-change-between-two-numbers
- Reverse a drop → reverse-a-percentage-decrease
- The calculator → Percentage Calculator
1. Why percents don't add
A percent change rescales a value. After the first move, the base is different, so the second percent acts on a new number. Adding them ignores that the second percent is "of a different thing".
2. The factor method
Turn each percent into a factor: +10% → 1.10, −10% → 0.90.
net factor = (1 + r1)(1 + r2) … net % = net factor − 1
3. Worked: +10% then −10%
Start 100.
- After +10%: 100 × 1.10 = 110
- After −10%: 110 × 0.90 = 99
Net = (1.10 × 0.90) − 1 = 0.99 − 1 = −1%. Not zero.
4. Worked: +20% then −20%
- 1.20 × 0.80 = 0.96 → −4% net
The bigger the percent, the bigger the gap from "zero".
5. Worked: +10% then +10%
- 1.10 × 1.10 = 1.21 → +21% net
Two +10% moves beat +20% because the second acts on the inflated base.
6. Comparison table
| Moves | Factor product | Net % |
|---|---|---|
| +10%, −10% | 1.10 × 0.90 = 0.99 | −1% |
| +20%, −20% | 1.20 × 0.80 = 0.96 | −4% |
| +10%, +10% | 1.10 × 1.10 = 1.21 | +21% |
| +50%, +50% | 1.50 × 1.50 = 2.25 | +125% |
| −25%, +33.3% | 0.75 × 1.333 = 1.00 | 0% (recovery) |
7. Three or more moves
A value changes +10%, −5%, +20%:
1.10 × 0.95 × 1.20 = 1.254 → +25.4% net
Order among a rise and a fall of fixed size does not matter for two moves, but with three mixed moves the product is still order-independent (multiplication commutes) — the base at each step differs, yet the final product is the same regardless of sequence. What matters is that you multiply, never add.
8. A price example
A stock: +15% Monday, −10% Tuesday, +5% Wednesday.
1.15 × 0.90 × 1.05 = 1.08675 → +8.68% net
From $100 you end at $108.68, not $100 + (15−10+5) = $110.
9. Common mistakes
- Adding the percents — the #1 error; always multiply factors.
- Assuming +x then −x = 0 — it is a net loss because the drop is on a larger base.
- Re-adding from the final number — reversing needs division by the factor product.
- Confusing net with average — net is total effect; per-period average is a geometric mean.
- Using the wrong sign — a decrease is a factor below 1 (0.90), not −0.10 added loosely.
10. A final checklist
- Convert each percent to a factor (1 + r).
- Multiply the factors; subtract 1 for the net.
- Never add percents for sequential moves.
- A +x then −x never returns to start.
- Use division by the factor product to reverse.
Run this checklist on any "what was the total change" question and the common traps disappear. The single rule to remember is that every later percent rescales an already-changed value, so multiplication — not addition — is the only honest operation.
FAQ
Q: How do I find the net percentage change? Multiply the factors (1+r1)(1+r2)… then subtract 1. For +10% then −10%: 1.10×0.90 − 1 = −0.01 = −1%.
Q: Why don't percentages just add? Each later percent acts on an already-changed base, so they compound, not add.
Q: Is +20% then −20% equal to zero? No — 1.20×0.80 = 0.96, a net −4%.
Q: Does order matter? For a single rise and a single fall of the same size, no (1.10×0.90 = 0.90×1.10). Order never matters for the product — multiplication commutes, so even three or more mixed moves give the same final result regardless of sequence; what changes is each step's base, not the total.
Q: What if all moves are increases? They still compound; +10% then +10% = +21%, not +20%.
Q: How do I reverse a net change? Divide by the product of factors; see reverse-a-percentage-decrease for a single drop.
Q: Can net change exceed 100%? Yes — several large increases compound past doubling (e.g. +100% twice = +300%).
Q: Is this the same as average percent change? No — net is the total effect; average per period is a different (geometric-mean) figure.
Sources: Investopedia — Percentage Change; Khan Academy percentages.
Disclaimer: General arithmetic for successive multiplicative changes. Additive rebases (e.g. a fee on top) need their own setup. This site is informational and not official.
Sources and references
- Investopedia — Percentage Changehttps://www.investopedia.com/terms/p/percentage-change.asp
- Khan Academy percentageshttps://www.khanacademy.org/math/pre-algebra/x31947de81e3f6453:percentages