A salary of 100 gets a 20% raise, taking it to 120. The next year, budget cuts bring it back down by 20%. It feels like a wash — up 20%, down 20%, net zero. The actual result is 96, not 100. Nobody made an arithmetic mistake; percentage increase and percentage decrease are not mirror-image operations, because each one is calculated on a different base.
Why Up 20% Then Down 20% Doesn't Cancel Out
100 × 1.20 = 120 ← +20%, base was 100
120 × 0.80 = 96 ← −20%, base is now 120, not 100
The 20% decrease is 20% of 120 (which is 24), not 20% of the original 100 (which would be 20). Removing a bigger chunk than was added is exactly why the final number lands below where it started — and this isn't specific to 20%. Any increase of X% followed by a decrease of the same X% always ends below the starting value, for every X other than zero. The only way to get back to exactly 100 is to decrease by a smaller percentage than the increase used — specifically, decreasing by X / (1 + X/100) percent.
The Formula
Percentage increase:
New Value = Original × (1 + X/100)
Percentage decrease:
New Value = Original × (1 − X/100)
Percentage change (given two known values, working out the percentage):
% Change = (New Value − Original Value) / Original Value × 100
A negative result from that last formula means a decrease; positive means an increase. This is the formula to reach for whenever the question is "by what percent did this change," rather than "what does X% of this look like."
Worked Example: Finding the Percentage That Actually Reverses a Change
A price rises from 100 to 120 — a 20% increase. What percentage decrease brings it back to exactly 100?
Required decrease % = (120 − 100) / 120 × 100 ≈ 16.67%
Not 20%. A 16.67% decrease from 120 gives 120 × 0.8333 = 100, exactly reversing the increase. The asymmetry gets more pronounced the larger the percentage: a 50% increase (100 → 150) needs only a 33.3% decrease to reverse, while a 100% increase (100 → 200) needs just a 50% decrease. The larger the original increase, the bigger the base the decrease has to work against, so proportionally less decrease is needed to reverse it.
Percentage vs Percentage Points — a Different Mix-Up Entirely
This one shows up constantly in news and reporting, and it's a separate error from the increase/decrease asymmetry above. If an interest rate moves from 5% to 7%:
- Percentage point change: 7 − 5 = 2 percentage points
- Percentage (relative) change: (7 − 5) / 5 × 100 = 40%
Both statements are true about the same change, but they describe wildly different-sounding magnitudes. "Rates rose 2 percentage points" and "rates rose 40%" refer to the identical move from 5% to 7% — reporting or reading one when the other was intended is a common source of "wait, that doesn't sound right" confusion, especially with anything already expressed as a percentage (interest rates, tax rates, conversion rates, survey results).
Where This Actually Trips People Up
Sequential markups and markdowns in business. A supplier cost that goes up 15% one quarter and down 15% the next does not return to the original cost — same asymmetric-base issue as the salary example, and it compounds further with each additional round of adjustment.
"Restored" prices after a sale. If an item's price was cut by 25% for a sale and then "restored" by increasing the sale price by 25%, the restored price is lower than the original — reversing a percentage decrease requires a larger percentage increase than the one that was applied, using the same asymmetric logic as the salary example above, just in the opposite direction.
Reading statistics involving rates. Any headline comparing two percentages (unemployment rate, approval rating, tax rate) as a plain difference is reporting percentage points — recognizing that distinction changes how large the actual relative shift is.
Quick FAQ
Is there a scenario where increase % and decrease % genuinely are symmetric? Only when working with a fixed absolute amount, not a percentage of a changing base — e.g., adding 20 units then subtracting 20 units always returns to the start, because both operations reference the same fixed number, not a shifting percentage base.
Why does compounding make repeated percentage changes even more confusing? Because each successive percentage applies to the result of the previous step, not the original value — the same principle behind why stacked discounts don't simply add together.
What's the fastest way to check "does X% up then X% down cancel out?"
It never does, for any nonzero X — multiplying (1 + X/100) × (1 − X/100) algebraically simplifies to 1 − (X/100)², which is always less than 1 whenever X isn't zero.
Does the order matter — increase first or decrease first? No — multiplication is commutative, so a 20% increase followed by a 20% decrease gives the same final result as a 20% decrease followed by a 20% increase (both give 96 on a base of 100).
Skip the Manual Multiplication
Rather than working out × 1.20 × 0.80 by hand or second-guessing whether a decrease will fully reverse an increase, the Percentage Calculator handles percentage increase, decrease, and percentage-change calculations directly.
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