Buy something at 20% off, then see it rise 25% and conclude you are back where you started. The arithmetic is attractive and the conclusion is wrong: you are still 10% below the original. Percent change is not a reversible operation, and the reason is structural rather than accidental — it is the single most misunderstood calculation in ordinary conversation.
The formula, and where the base matters
% change = (new − old) ÷ |old| × 100
The absolute value in the denominator keeps the sign sensible when the original is negative — a loss from −50 to −30 is a 40% decrease, not −40%. A positive result is an increase, negative a decrease.
Everything difficult about percentages follows from one fact: the base moves. Each calculation is relative to its own starting point, and reversing a move means the base has already changed.
The asymmetry, made concrete
Start at 100. A 50% loss takes you to 50. To get back to 100 from 50, what percentage increase do you need? Solve 50 × (1 + r) = 100, so r = 100% — a doubling.
The pattern generalises, and it is worth having memorised:
- Down 10% → needs +11.1% to recover
- Down 20% → needs +25%
- Down 50% → needs +100%
- Down 80% → needs +400%
- Down 90% → needs +900%
The deeper you fall, the more it costs to climb out. A 90% loss needs nine times the original value just to get back even, which is why small percentages near zero are so dangerous — and why the recovery maths in finance, and the general "never average down a losing position" instinct, is not superstition.
Worked example: price changes
A price rises from $80 to $92: (92 − 80) ÷ 80 × 100 = +15%, a $12 increase, and 92 ÷ 80 = 1.15× the original.
Now read the same two prices in reverse — 92 down to 80: (80 − 92) ÷ 92 × 100 = −13.04%. Different number, same two prices. This is why financial reports always state the base period explicitly, and why "up 15%" and "down 13%" can both be true in the same conversation about the same object.
Reversing a percentage properly
To undo a percentage change, divide rather than subtract:
original = new ÷ (1 + change)
A 15% increase is undone by dividing by 1.15, not by subtracting 15% — the base has moved, so a flat subtraction lands in the wrong place. For a 20% discount, a price of $64 came from $80 because 64 ÷ 0.8 = 80; subtracting 20% from 64 gives $51.20, which is not the original and never was. This is also the discount arithmetic trap: the "reverse" change from $64 to $80 reads as +25%, not +20%.
The percent change calculator reports the reverse change explicitly, because the most common question about a percentage is what it started from.
Three calculations people confuse
Percent of. What portion one number is of another. 50% of 200 = 100. No before-and-after, no base problem.
Percent change. The before-and-after comparison above, using the original as base.
Percent difference. Comparing two peers with no direction, using their mean as the base. Two test scores of 80 and 90: percent difference = 10 ÷ 85 ≈ 11.8%, whereas percent change from 80 to 90 would be 12.5%. Neither is wrong; they answer different questions. Use percent difference for "how different are these two values" and percent change for "how much did this change".
When small percentages are not small
A 1% rise on a large base is a large number, and compounding makes it worse. A 1% monthly increase doubles in roughly 70 months (the rule of 72: 72 ÷ 1 ≈ 72). A 0.5% monthly fee on a portfolio of $500,000 is $250 a month, $3,000 a year — which is the entire reason expense ratios matter, and why the "0.1% is negligible" argument fails in exactly the places where fees are quoted.
Compare on a consistent base and a consistent period. "4% versus 5%" and "0.4% versus 0.5%" describe different products; quoting one as an annual figure and the other as a monthly one is a comparison that flatters whoever chose the wording.
Compound percentages: the small number that becomes large
A single percentage is a one-off comparison. Repeated percentages are not additive at all, and this is where most mental arithmetic goes wrong.
Growing 5% a year for 10 years does not give +50%. It gives 1.05¹⁰ − 1 = 62.9%, because each year's gain is calculated on a larger base. Doubling time is the memorable case: at 7% a year money doubles in about 10 years (the rule of 72 says 72 ÷ 7 ≈ 10.3), which is why the compounding curve visibly bends upward while the numbers in any single year look modest.
The same asymmetry applies downward and is the reason a decaying quantity is called exponential decay. If something halves every 10 years, it is at 12.5% of its original after 30 years, not 50% — each halving is a 50% loss applied to what is left.
So when comparing an option that compounds against one that does not, the horizon matters enormously. Two paths that look level after a year can be far apart after a decade, and the gap is entirely explained by which base each percentage is applied to.
Three cross-period comparison traps
Different periods. A 0.5% monthly fee is roughly 6.2% a year, not 0.5%. Dividing or multiplying by 12 is only approximate because compounding makes it non-linear — 1.005¹² = 1.0617.
Different bases. "Rose 5% from a low base" and "fell 5% from a high base" can describe the same absolute movement with opposite-sounding results. Always ask what it is a percentage of.
Percentage points versus percent. A tax rate rising from 5% to 7% is a rise of 2 percentage points but a rise of 40% in the rate. Financial reporting distinguishes these carefully, and news headlines frequently do not.
A quick way to sanity-check any percentage
Convert to the ratio form and check the base. If someone says a price "fell 20% and rose 20% back", the two figures are measured against different bases and the claim is false by construction. Asking "20% of what?" is the cheapest and most effective check available, and it catches most published percentage errors on sight.
Frequently asked questions
How do I calculate percent change?
Subtract the original from the new value, divide by the absolute original, and multiply by 100: % change = (new − old) ÷ |old| × 100. The sign tells you increase or decrease.
Why is going up after a big loss not symmetric?
Because the base changes. Dropping 50% to 25 then rising 50% only reaches 37.5. Recovering a 50% loss requires a 100% gain, since the increase is calculated on the smaller number.
What is the difference between percent change and percent difference?
Percent change has a direction and uses the original value as the base. Percent difference compares two similar values with no direction, using their mean as the base.
How do I reverse a percent change?
Divide the new value by (1 + percent change). A 15% increase is undone by dividing by 1.15, not by subtracting 15% — the base has moved.