Why a 20% Increase and a 20% Decrease Don’t Cancel Out
A 20% increase followed by a 20% decrease does not return you to the original value. Learn why with simple examples and calculations.
If a $100 item goes up 20% and then comes back down 20%, is it back to $100? It's tempting to say yes — the percentages cancel, so the price should too. It doesn't. The item ends up at $96, not $100, and the reason comes down to a detail that's easy to miss: a percentage increase and a percentage decrease are calculated from two different starting points.
The question people get wrong
This shows up constantly: a stock drops 20% one week and rises 20% the next — is the investor back to even? A store marks a price up 20% for a sale event, then discounts it 20% — is it the original price? In both cases, the intuitive answer ("yes, they cancel") is wrong, and the gap between intuition and reality gets bigger the larger the percentage.
Walking through the numbers
Start with $100.
The item ends at $96, four dollars short of where it started. Nothing was calculated incorrectly — both steps used the correct formula. The issue is that the 20% decrease was 20% of $120, not 20% of the original $100.
Why it happens: different base values
Every percentage calculation needs a base — the number the percentage is of. The 20% increase used $100 as its base. The 20% decrease used $120 as its base, because that had become the new current value. Since 20% of a bigger number ($120) is a bigger dollar amount ($24) than 20% of a smaller number ($100) is ($20), the decrease removes more than the increase added, and you end up below where you started.
The multiplication method
There's a faster way to see this without doing each step separately. A 20% increase multiplies a value by 1.20. A 20% decrease multiplies a value by 0.80. Apply both in sequence:
1.20 × 0.80 = 0.96
Multiplying by 0.96 means the net effect is a 4% decrease — regardless of what the starting dollar amount was. Try $100 (ends at $96), $500 (ends at $480, also a 4% drop), or $37 (ends at $35.52, still a 4% drop): the percentages always net out to −4%, even though the dollar amounts differ.
The reverse question
If a value drops 20%, what increase gets it back to where it started? Not 20% — you need more than that, because you're now calculating a percentage of a smaller number.
$100 drops 20% to $80. To go from $80 back to $100, the required increase is:
(100 − 80) ÷ 80 × 100 = 25%
A 20% loss requires a 25% gain to fully recover. This asymmetry gets worse the larger the drop: a 50% loss requires a 100% gain to recover, and a 90% loss requires a 900% gain. You can check any recovery percentage with our by entering the reduced value and the original decrease percentage.
A quick reference table
| Starting value | +20% then −20% | Net result |
|---|---|---|
| $100 | $120 → $96 | −4% |
| $250 | $300 → $240 | −4% |
| $1,000 | $1,200 → $960 | −4% |
| Loss | Gain needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 50% | 100% |
The key lesson
The dollar amount a percentage represents always depends on what it's being calculated from. Once a value changes, the same percentage no longer refers to the same dollar amount — which is why equal-and-opposite percentage changes never fully cancel out, and why recovering from a loss always takes a proportionally larger gain than the loss itself. For a deeper look at how percentage change is calculated in general, see .
Frequently Asked Questions
Try it yourself
Use the Reverse Percentage Calculator to run this calculation with your own numbers.