Percent calculations go wrong in the same few places
The hard part of a percentage is not the multiplying and dividing. It is deciding what to measure against.
Get the base right and there is only one formula: divide the part by the whole and multiply by 100.
Working out what percent something is
A $25 lunch with $3 off. What percent is the discount?
3 ÷ 25 × 100 = 12%If you can never remember which way to divide, look for "out of". The number after it goes on the bottom. It's $3 out of $25, so divide by 25.
The other direction is a multiply. 12% of $25 is 25 × 0.12 = 3. The two operations check each other: if one gives you 12%, the other should hand back the 3.
A change is measured against the old value
A subscription goes from $120 a month to $150. How big a rise is that?
(150 - 120) ÷ 120 × 100 = 25%Divide the change by the original value, not the new one.
And here something odd happens. Going back down from $150 to $120 is the same $30, but it is a 20% cut, because the base has moved to 150. A rise and the matching fall do not come out as the same percentage. That is not a mistake; that is how it works.
Down and then up does not land where you started
For the same reason, take 20% off a price and then put 20% back on and you do not return to the original.
100 → 20% off → 80
80 → 20% on → 96Going down you subtracted 20% of 100. Going up you added 20% of 80. Different bases, and $4 has gone missing. The feeling that "it fell 20%, so a 20% rise makes it even" simply does not hold for percentages.
A percentage point is a different unit
An interest rate goes from 2% to 3%. Calling that "a 1% rise" is wrong.
- The gap between 2 and 3 is 1 percentage point (1 pp)
- As a ratio, 2 becoming 3 is a 50% increase
The difference between two percentages is a subtraction and its unit is pp. The change as a ratio is a division and its unit is %. This is why careful reporting says "points": a headline reading "interest rates up 50%" would be technically true and read completely differently from 2% becoming 3%.
Polling numbers behave the same way. Approval going from 30% to 33% is a 3 pp rise, and also a 10% rise. One sounds much bigger than the other, and plenty of writing quietly picks whichever number looks more dramatic.
The four shapes, side by side
| Question | Formula | Example |
|---|---|---|
| What percent is 3 out of 25 | 3 ÷ 25 × 100 | 12% |
| What is 12% of 25 | 25 × 0.12 | 3 |
| 120 becomes 150, what's the rise | 30 ÷ 120 × 100 | 25% |
| 2% becomes 3% | a 1 pp gap, a 50% rise as a ratio |
What this does not cover
- Compound interest, where interest earns interest of its own
- Adding and removing sales tax: removing it has a trap of its own (dividing, not subtracting)
- Averaging percentages across several items: that needs a weighted average, so you need all
the underlying numbers, not just the percentages
All four shapes above live in one calculator. Choose "what percent", "ratio" or "change", give it two numbers, and the answer lands underneath; discounts and tax sit in the same panel. Any result copies with a tap.