The three percentage problems
A percentage is a fraction of 100. The word itself means per hundred, so 25% is 25 out of 100, or 0.25 as a decimal. That single idea is behind every percentage calculation you will ever do — the only thing that changes is which number you know and which one you're looking for.
Almost every real percentage question is one of three types. Once you can recognise which type you're facing, the formula follows automatically:
- Finding a part — you know the percentage and the total, and you want the amount. What is 20% of 150?
- Finding a percentage — you know the part and the total, and you want the percentage. 30 is what percent of 120?
- Finding a change — you know two values, and you want the movement between them. Prices went from 80 to 100 — by how much?
Most of the confusion people have with percentages isn't arithmetic. It's picking the wrong one of these three, or getting the two numbers the wrong way round inside the right one.
Type 1: What is X% of Y?
This is the discount, tip, tax, and commission calculation. Convert the percentage to a decimal by dividing by 100, then multiply.
A useful shortcut: 10% of anything is that number with the decimal point moved one place left. Once you have 10%, you can build most other percentages in your head — 20% is double it, 5% is half of it, 15% is 10% plus half of 10%. This is how people work out restaurant tips without reaching for a phone.
Type 2: X is what percent of Y?
This one turns a raw pair of numbers into a comparable figure — test scores, completion rates, conversion rates, budget shares. Divide the part by the whole, then multiply by 100.
The thing that goes wrong here is the order. The number you're measuring goes on top; the total it's being measured against goes on the bottom. Flip them and 30 out of 120 becomes 400% instead of 25% — an answer that's obviously wrong, which at least makes the mistake easy to spot. A quick sanity check: if the part is smaller than the whole, your answer must be under 100%.
Type 3: Percentage increase and decrease
Percentage change measures movement relative to where you started. Subtract the old value from the new one, divide by the old value, and multiply by 100.
The original value is always the denominator. This is not an arbitrary convention — it's what makes the percentage meaningful. A £20 rise means something very different on a £40 item than on a £4,000 one, and dividing by the starting value is what captures that difference.
Worked example — a sale price
A jacket is reduced from $80 to $60. What's the discount?
Using type 3: ((60 − 80) ÷ 80) × 100 = (−20 ÷ 80) × 100 = −25%. The jacket is 25% off.
Now check it with type 1: 25% of 80 is 0.25 × 80 = 20, and 80 − 20 = 60. The two calculations agree, which is a good habit — working a percentage backwards is the fastest way to catch an error.
Why a 20% drop doesn't undo a 20% rise
This trips up almost everyone, and it matters in real situations — investments, sale prices, traffic figures. Percentage changes are not symmetrical, because each one is measured against a different starting number.
Start with 100 and add 20%: you get 120. Now take 20% off that 120 — but 20% of 120 is 24, not 20. You land on 96, not back at 100. The second percentage was calculated on a bigger number, so it removed more.
| Start | Change | Result | Net vs start |
|---|---|---|---|
| 100 | +20% | 120 | +20% |
| 120 | −20% | 96 | −4% |
| 100 | −50% | 50 | −50% |
| 50 | +50% | 75 | −25% |
Look at the bottom two rows: a 50% loss needs a 100% gain to recover, not another 50%. This is why a portfolio that halves has to double to break even, and why "we lost 30% but then gained 30%" is never actually break-even. The larger the swing, the wider the gap.
Margin vs markup — the same profit, two answers
This is the percentage confusion that costs businesses real money. Both describe the profit on a sale, but they measure it against different bases:
- Markup is profit as a percentage of what the item cost you.
- Margin is profit as a percentage of what you sold it for.
Worked example — buying at 50, selling at 75
Profit is 25 either way. But:
Markup = (25 ÷ 50) × 100 = 50% — measured against the cost.
Margin = (25 ÷ 75) × 100 = 33.3% — measured against the sale price.
Same transaction, same profit, two very different-looking percentages. Margin is always the smaller of the two.
The practical danger: someone who wants a 40% margin but applies a 40% markup will under-price every item they sell. To get a 40% margin you need roughly a 67% markup. If you're setting prices, always confirm which of the two a figure refers to before you use it.
Common mistakes with percentages
Dividing by the new value instead of the old one. Percentage change always uses the starting value as the denominator. Using the new value gives a different — and wrong — answer.
Adding percentages that have different bases. A 10% discount followed by another 10% discount is not 20% off. The second discount applies to the already-reduced price, so the total is 19%.
Confusing percent with percentage points. If a rate goes from 4% to 6%, that's a rise of 2 percentage points — but a 50% increase. Both are correct; they answer different questions, and mixing them up misleads badly.
Reversing a percentage by subtracting it. If a price after a 20% discount is $60, the original was not $72. Divide instead: 60 ÷ 0.80 = $75.