Method · Basic Math
How to Calculate Percentages: All Three Questions
Three questions get called the same thing. Knowing which one you are being asked is most of the difficulty.
The three questions
Learning how to calculate percentages means separating three different questions that all get described as 'doing percentages'.
| Question | Method | Example |
|---|---|---|
| What is A% of B? | A ÷ 100 × B | 15% of 80 = 12 |
| A is what percent of B? | A ÷ B × 100 | 15 of 80 = 18.75% |
| Change from A to B? | (B − A) ÷ A × 100 | 50 to 75 = 50% increase |
Identifying which one is being asked is most of the work. The arithmetic afterwards is one multiplication or one division.
Finding a percentage of a number
Convert the percentage to a decimal and multiply.
- 15% of 80the question
- 15 ÷ 100 = 0.15convert to a decimal
- 0.15 × 80 = 12multiply
For mental arithmetic, break the percentage into easy pieces: 10% of 80 is 8, half of that is 4 for 5%, so 15% is 12. Finding 10% by moving the decimal point one place is the single most useful percentage trick there is.
Finding what percentage one number is of another
Divide the part by the whole, then multiply by 100.
- 15 ÷ 80 = 0.1875part over whole
- 0.1875 × 100 = 18.75%scale to a percentage
The order matters. Dividing 80 by 15 answers a different question — how many times bigger the whole is than the part — and gives 533%.
Percent increase and decrease
Compare the change with where you started. The original value is always the denominator.
- 75 − 50 = 25the change
- 25 ÷ 50 = 0.5compare with the ORIGINAL, not the new value
- 0.5 × 100 = 50%a 50% increase
This also explains why a rise and a fall of the same percentage do not cancel. 100 raised by 50% is 150; 150 lowered by 50% is 75, because the second 50% is taken of a larger base.
Discounts and the multiplier shortcut
A 20% discount leaves 80% of the price, so multiplying by 0.8 does the whole calculation in one step rather than finding the discount and subtracting it.
| Change | Multiplier | £40 becomes |
|---|---|---|
| 20% off | × 0.8 | £32 |
| 10% off | × 0.9 | £36 |
| 20% VAT added | × 1.2 | £48 |
| 5% increase | × 1.05 | £42 |
Multipliers also chain. Two successive 10% discounts are × 0.9 × 0.9 = × 0.81, which is a 19% discount overall — not 20%, for the same reason a rise and fall do not cancel.
Percentages of percentages
Taking a percentage of a percentage is one multiplication, but the answer surprises people because the base changes.
- 20% repliedof the whole group
- 0.5 × 0.2 = 0.1half of that fifth
- 10% of the whole groupnot 10 percentage points off 20
Compound growth works the same way. Three years at 5% is not 15% — it is 1.05 × 1.05 × 1.05, about 15.8%, because each year is calculated on the total the previous year left behind.
Reverse percentages
The hardest common case: a price after a change is known, and the original is wanted. Divide rather than multiply.
- original × 1.2 = 48the multiplier that was applied
- 48 ÷ 1.2 = 40divide by the same multiplier
- £40the price before VAT
The tempting mistake is to take 20% off £48, giving £38.40. That is wrong because the 20% was calculated on £40, not on £48 — the base changed, and only dividing by the multiplier undoes it correctly.
Questions about how to calculate percentages
How do I find a percentage of a number?
Divide the percentage by 100 and multiply. 15% of 80 is 0.15 × 80 = 12.
Which number goes on the bottom in percent change?
The original value, always. Comparing the change with the new value answers a different question and gives a different number.
How do I take 20% off a price?
Multiply by 0.8. A 20% discount leaves 80% of the price, so one multiplication does it.
How do I find the original price before a percentage was added?
Divide by the multiplier. If 20% was added, divide by 1.2 — do not take 20% off the new price, because that percentage was calculated on the smaller original.