Three questions, three formulas
Nearly every percentage problem is one of three questions in disguise. The hard part is never the arithmetic — it is spotting which of the three you are being asked, and which number is the base you divide by.
What is 15% of 64? — tips, tax, discounts
- Convert the percentage
- 15 ÷ 100 = 0.15
- Multiply by the number
- 0.15 × 64
15% of 64 = 9.6
42 is what percent of 60? — scores, completion rates
- Divide part by whole
- 42 ÷ 60 = 0.7
- Multiply by 100
- 0.7 × 100
42 is 70% of 60
From 80 to 100 — raises, growth, price changes
- Find the difference
- 100 − 80 = 20
- Divide by the original
- 20 ÷ 80 = 0.25
- Multiply by 100
- 0.25 × 100
A 25% increase
The traps that catch almost everyone
Percentages behave in ways that feel wrong until you have been burned once. These three account for most real-world mistakes:
A 50% rise and a 50% fall do not cancel out
Add 50% to 100 and you get 150. Take 50% off 150 and you get 75, not 100. Each step measures against a different base, so percentage changes multiply rather than add. This is why a stock that halves needs to double — a 100% gain — just to break even.
Stacked discounts are smaller than their sum
30% off followed by a further 10% off is not 40% off. The second cut applies to the already-reduced price, giving 0.70 × 0.90 = 0.63, or 37% off in total. Retailers rely on the sum sounding better than it is.
Percentage points are not percent
An interest rate moving from 2% to 3% has risen by one percentage point — but by 50% in relative terms. Both statements are true and they describe the same change. News headlines routinely pick whichever sounds more dramatic.
Reversing a percentage
The question that trips people up most is working backwards: an item costs $62.30 after 30% off, so what was the original price? The instinct is to add 30% back, which gives $80.99 — and is wrong. The 30% was taken from the original, not from the sale price.
The correct move is to divide by what remains. After 30% off, 70% of the price is left, so the original is 62.30 ÷ 0.70 = $89. The same logic strips tax out of a gross figure or recovers a pre-raise salary. Whenever you need to undo a percentage, divide by one minus the rate rather than adding the rate back on.