A percentage is a fraction with 100 fixed on the bottom. Every percentage question reduces to one of three underlying shapes — finding a part from a whole, finding what fraction one number is of another, or finding how much something changed — and knowing which shape you are looking at is most of the actual work. The arithmetic itself is trivial once the shape is identified.
The three shapes, worked through
Finding a part. “What is 15% of 80?” Multiply: 80 × 0.15 = 12. This is the most mechanical of the three and rarely trips anyone up.
Finding a percentage. “80 is what percent of 120?” Divide the part by the whole and multiply by 100: (80/120) × 100 = 66.7%. The common error here is dividing the wrong number by the other — always divide the smaller reference quantity into the total it is being compared against, not the other way round.
Finding a change. “A price moved from 80 to 120 — what percentage change is that?” Subtract, then divide by the originalvalue: (120 − 80) / 80 × 100 = 50%. Dividing by the new value instead of the original is the single most common percentage-change mistake, and it silently produces a smaller, wrong answer — dividing by 120 here would incorrectly give 33.3%.
Increase then decrease does not return you home
Add 20% to 100 and you get 120. Take 20% off 120 and you get 96, not 100. The two percentages are calculated on different bases — the increase is 20% of 100, but the decrease is 20% of 120, a larger number, so it removes more than was added. To exactly reverse a 20% increase you must divide by 1.2 rather than subtract 20%, which works out to a 16.7% decrease, not 20%. This asymmetry is behind a surprising amount of misleading retail pricing — a store advertising “40% off, then an extra 20% off” is not offering 60% off; it is offering 1 − (0.6 × 0.8) = 52% off, calculated by successively discounting what remains rather than adding the two percentages together.
Percentage points versus percent — a distinction worth keeping straight
If an interest rate moves from 4% to 6%, that is a rise of two percentage points, but it is a rise of 50 percent relative to the original rate (a change of 2 divided by the original 4, times 100). Both descriptions are correct and they describe very different magnitudes. News coverage and marketing material frequently blur the two — sometimes carelessly, sometimes because the percent figure sounds more dramatic than the percentage-point one — and the gap between them widens as the original percentage gets smaller. A rate moving from 1% to 2% is only a single percentage point, but it is a 100% increase.
Compounding percentages multiply, they do not add
Successive percentage changes multiply rather than sum. Three consecutive 10% increases give a factor of 1.1³ = 1.331, a 33.1% overall rise — not the 30% that naive addition would suggest, because each 10% is taken on an already-larger base. The same logic runs in reverse for losses: recovering from a 50% loss requires a 100% gain, not a 50% gain, since the 50% gain would only be calculated on the already-diminished amount. This is one of the more counterintuitive but consequential facts in everyday percentage arithmetic, and it is the entire reason a portfolio that drops 50% and then rises 50% ends up smaller than where it started, not back to even.