A stock falls 50 percent on Monday and rises 50 percent on Tuesday. It is not back where it started; it is down 25 percent. Percentages are not symmetric, and almost every expensive mistake with them comes from assuming they are.
The Four Calculations You Actually Need
Nearly every percentage question reduces to one of these.
What is X percent of Y? Multiply: `Y × (X / 100)`. Twenty percent of 250 is `250 × 0.2 = 50`.
X is what percent of Y? Divide and scale: `(X / Y) × 100`. If 37 of 200 users converted, that is `37 / 200 × 100 = 18.5%`.
Percentage change from A to B. `((B - A) / A) × 100`. The denominator is always the starting value, and getting that wrong is the most common error in the whole topic. From 80 to 100 is a 25 percent increase; from 100 to 80 is a 20 percent decrease. Same gap, different bases.
Reverse percentage. If a price of 90 already includes a 25 percent discount, the original was not `90 × 1.25`. It is `90 / 0.75 = 120`. To undo a percentage you divide by the multiplier, never multiply by it.
Percent and percentage point are different units
If a conversion rate moves from 2% to 3%, that is a rise of one percentage point and a rise of 50 percent. Both are true. Reporting the larger number without saying which you mean is the most common way statistics mislead honestly. State the unit every time.
Why Losses Hurt More Than Equal Gains
The asymmetry in the opening example is arithmetic, not psychology. A decrease is measured against the original, larger value, and the subsequent increase is measured against the new, smaller one.
Drop 100 by 50 percent and you have 50. Raise 50 by 50 percent and you have 75. To get back to 100 from 50 you need a 100 percent gain.
The general rule: to recover from a loss of `L` percent, you need a gain of `L / (100 - L) × 100` percent. A 20 percent loss needs 25 percent to recover. A 50 percent loss needs 100 percent. A 90 percent loss needs 900 percent.
The same effect appears in growth reporting. Traffic that falls 30 percent one month and rises 30 percent the next is down 9 percent overall, not flat. Anyone comparing month-on-month percentages without checking the absolute numbers will read that as a full recovery.
Loss and the Gain Required to Break Even
The gap widens sharply past 50 percent, which is why deep drawdowns are so hard to recover from.
| Loss | Value remaining from 100 | Gain needed to return to 100 |
|---|---|---|
| 10% | 90 | 11.1% |
| 20% | 80 | 25% |
| 30% | 70 | 42.9% |
| 40% | 60 | 66.7% |
| 50% | 50 | 100% |
| 60% | 40 | 150% |
| 75% | 25 | 300% |
| 90% | 10 | 900% |
Discounts Do Not Add Up
"Take an extra 30 percent off sale prices, already reduced by 40 percent" is not 70 percent off. Each discount applies to the price after the previous one.
Start at 100. Forty percent off gives 60. Thirty percent off that gives 42. The total reduction is 58 percent, not 70. The shortfall grows with the size of the discounts, which is why stacked offers sound better than they are.
The general form is to multiply the remaining fractions: `0.6 × 0.7 = 0.42`, so 42 percent of the original price remains. This holds for any number of stacked changes, and it works for increases too. Three consecutive 10 percent rises give `1.1³ = 1.331`, a 33.1 percent increase rather than 30.
Tax works the same way but in a specific order that matters. A 20 percent discount followed by 20 percent tax does not return you to the original price: `100 × 0.8 × 1.2 = 96`. Whether tax is applied before or after a discount changes the total, which is why it is written into the terms of most promotions.
Run the calculation
Percentage of a number, percentage change, and reverse percentages.
Reading Percentage Claims Critically
Four questions expose most of the misleading ways percentages get reported.
Percent of what?
A 200 percent increase in incidents means nothing without the base. From 1 to 3 is a 200 percent increase, and so is a rise from 1,000 to 3,000.
Percent or percentage points?
"Support rose 5 percent" and "support rose 5 points" describe very different movements when the starting figure is 10 percent.
Is the average of percentages meaningful?
Averaging percentages computed over different-sized groups gives the wrong answer. A 50 percent success rate over 2 attempts and 10 percent over 100 do not average to 30 percent; the real rate is about 10.8 percent.
Over what period, compounded how?
Ten percent growth per month is 214 percent per year, not 120. Any repeated percentage change compounds, and stating the period without the compounding understates it substantially.
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