Percentage Traps: Why a 50% Loss Needs a 100% Gain

Nothing here is hard arithmetic. All of it is arithmetic people get wrong in reports, in pricing, and in investment maths — usually because the base changed and nobody said so.

Losses and gains are not symmetric

Drop	Gain needed to break even
10%	11.1%
20%	25%
50%	100%
80%	400%
90%	900%

required gain = loss / (1 - loss).

This is why drawdown control dominates long-run returns: a −50% year cannot be repaired by a +50% year. Up 10% then down 10% is not zero either:

100 × 1.1 × 0.9 = 99

Volatility itself has a cost, and it compounds.

Percentage points vs percent

"Rates fell from 4% to 3%" has two readings:

• 1 percentage point (4 − 3)
• 25 percent (relative change, 1/4)

A factor of 25 between the two phrasings. Business writing mixes them constantly, so when a percentage is used as the base of another percentage, always state which.

The grammar is simple: subtracting two percentages gives percentage points; a change relative to a value gives percent.

Compounding is multiplication, not addition

Four quarters at +10% each:

1.1⁴ = 1.4641  → +46.41%, not +40%

The extra comes from growth on growth. Two years at −20% each:

0.8² = 0.64  → −36%

You cannot average growth rates with an average

Growth rates of 10%, 20%, −15%, 30% average arithmetically to 11.25%. That number describes nothing real.

The correct measure is the geometric mean (CAGR):

CAGR = (end / start)^(1/n) − 1
     = (1.1 × 1.2 × 0.85 × 1.3)^(1/4) − 1
     ≈ 10.4%

The arithmetic mean is always ≥ the geometric mean, and the gap widens with volatility. Quoting the arithmetic mean as "average annual return" systematically overstates performance.

A fund advertising "15% average annual return over three years" might mean a 45% cumulative gain divided by three. The actual CAGR is 13.2%. Not illegal, but material.

Discounts and markups do not cancel

"Mark up 20%, then take 20% off":

1.2 × 0.8 = 0.96

96% of the original — a 4% discount, not a return to the original price.

Order does not matter for the final number (multiplication commutes), but the base changes at each step, which is why "raise prices 20% then discount 20%" never restores the margin:

• Start 100 → +20% = 120 → −20% = 96
• Start 100 → −20% = 80 → +20% = 96

Both land at 96. The margin, though, is computed on the new price, so the recovered amount differs from what was lost.

Everyday examples: "spend 300, get 50 off" is 16.7% off, not 50%. "Buy one get one 50% off" is 25% off the pair.

Percentages that exceed 100%

A multiple-choice survey: WeChat 80%, Alipay 75%, cash 20% — 175% total.

Not an error. The denominator is respondents; each could pick several. The same applies to satisfaction surveys where people can skip questions: totals can fall below 100%.

The check: is the question single-choice? If not, a share above 100% is expected and says nothing about "the rest".

Rounding breaks totals

Round first, then sum, and you get a different answer than summing first:

three values of 0.4:  round → 0+0+0 = 0
                     sum → 1.2 → 1

This is why report tables show "totals may not add due to rounding". The rule: keep full precision through the calculation, round only at display; if you must round early, say so in a footnote.

The three questions to ask

Before trusting any percentage:

1. What is the base? Which number is changing.
2. Is it relative change or percentage points?
3. Is it a single period or cumulative?

Most "the numbers don't add up" conversations end after these three.

Percentage Calculator separates relative change from percentage points explicitly, which is faster than working out which one a colleague meant. When rounding decisions matter, Number Rounding shows the difference between rounding modes at the .5 boundary — a place where toFixed and half-up disagree.