The five percentage calculations, and when to use each
Almost every percentage question students meet is one of five operations. Most confusion comes from applying the wrong one rather than from arithmetic error, so it is worth being able to tell them apart.
| Question | Formula | Example |
|---|---|---|
| What is X% of N? | N × X ÷ 100 | 25% of 80 = 20 |
| X is what % of N? | X ÷ N × 100 | 18 of 24 = 75% |
| % change from A to B | (B − A) ÷ A × 100 | 40 → 50 = +25% |
| Increase N by X% | N × (1 + X ÷ 100) | 80 + 15% = 92 |
| Decrease N by X% | N × (1 − X ÷ 100) | 80 − 15% = 68 |
The second row is the one that matters most for marks: converting a score into a percentage. 18 out of 24 is 75%, and the calculation is always your mark divided by the total, times 100. Getting the division the wrong way round is the single most common error here, and it produces an answer above 100% — which is a useful check.
Two traps worth understanding properly
Percentage change is not symmetric
An increase of 50% followed by a decrease of 50% does not return you to where you started. Take 100: up 50% gives 150; down 50% from 150 gives 75. You are 25% below your starting point.
The reason is that each percentage is taken of a different base. The increase was
50% of 100; the decrease was 50% of 150. To reverse a 50% increase you need a decrease of
33.3%, because 1 ÷ 1.5 = 0.667.
This matters in any context involving successive changes — compound interest, price
changes, population growth, and any exam question that applies two changes in sequence. If a
question applies two percentage changes, you multiply the factors
(1.5 × 0.5 = 0.75); you never add or subtract the percentages.
Percentage points are not percentages
If a pass rate rises from 40% to 45%, that is an increase of 5 percentage
points — but a 12.5% increase in the rate itself
(5 ÷ 40 × 100).
Both statements are true and they describe the same change. Confusing them is extremely common in reporting statistics, and in exam questions on data interpretation the distinction is frequently the point of the question. When you see "percentage points", the numbers are being subtracted directly. When you see "percent increase", a division is involved.
Weighted grades: the calculation students most often get wrong
Course grades are usually weighted, not averaged. If coursework is worth 30% and the exam 70%, and you scored 80% on coursework and 60% on the exam, your grade is not 70%. It is:
(80 × 0.30) + (60 × 0.70) = 24 + 42 = 66%
The same weighting logic drives GPA, which weights by credits rather than by percentage — our GPA calculator handles that case, and our guide to grading systems explains why percentages don't convert between countries.
Working backwards is often what you actually want: to find the exam mark needed for a
target grade, subtract the coursework contribution from your target and divide by the
exam weight. For 70% overall: (70 − 24) ÷ 0.70 = 66% needed on the
exam.
Unit conversion, and why it goes wrong
Unit conversion errors in exams are rarely arithmetic. They come from two places: converting in the wrong direction, and mishandling squared or cubed units.
Direction. The reliable method is to write the conversion as a fraction
that cancels the unit you are removing. To convert 5 kilometres to metres, multiply by
1000 m / 1 km — the kilometres cancel and you are left with metres. If your
units do not cancel to leave what you wanted, you have the fraction upside down. This is
slower than guessing and it is never wrong.
Squared and cubed units. This is the error that costs the most marks.
1 metre is 100 centimetres, but 1 square metre is 10,000 square centimetres,
not 100 — because the conversion factor is squared too (100² = 10,000). For
cubic units it is cubed: 1 m³ = 1,000,000 cm³. Any conversion involving area or volume needs
the factor raised to the same power as the unit.
A related trap: 1 litre = 1000 cm³ exactly, and 1 m³ = 1000 litres. These two facts together catch a lot of people out in chemistry and physics calculations involving concentration.
Sanity checks that catch most errors
- Estimate first. Round to convenient numbers and get an approximate answer before calculating. 23% of 412 is roughly a quarter of 400, so about 100. If your calculator says 9.5 or 950, you have a factor-of-ten error.
- Check the direction. Converting to a smaller unit gives a bigger number. Metres to millimetres should grow; grams to kilograms should shrink.
- Watch for impossible results. A score above 100% of a total, or a decrease of more than 100% of a positive quantity, means the operation was wrong.
- Keep units written down throughout. Most of the marks lost in multi-step physics and chemistry questions are lost by dropping units mid-calculation and losing track of what the number represents.
- Round at the end, not during. Rounding intermediate values propagates error, and in multi-step problems it can move your answer outside the accepted tolerance.
Common questions
Is anything I enter sent anywhere?
No. All calculations run in your browser. There is no server component and nothing you type is transmitted or stored.
How do I reverse a percentage increase?
Divide rather than subtract. If a price of 120 includes a 20% increase, the
original was 120 ÷ 1.20 = 100, not 120 − 20% = 96. This is
the same asymmetry described above, and it is a standard exam question in its own
right.
How do I remove tax from a total?
The same way. If a total of 220 includes 10% tax, the pre-tax figure is
220 ÷ 1.10 = 200 and the tax was 20. Subtracting 10% of 220 gives 198,
which is wrong.
Why does converting square units give such large numbers?
Because the conversion factor is squared along with the unit. Going from metres to centimetres multiplies by 100; going from square metres to square centimetres multiplies by 100², which is 10,000. Cubic units use the cube.
Which unit system should I use in exams?
Whatever your specification requires, and state units in your answer. Most science curricula worldwide use SI units, but some questions supply imperial values and expect a conversion. Marks are commonly available for correct units alone, so never omit them.