How to calculate percentages

Percentages cause more everyday arithmetic anxiety than anything else, and almost all of it comes from not noticing that four different questions are being asked with the same word. Sort the questions out and the sums become trivial.

Percentages are one idea in four disguises

Per cent means "per hundred". 25% is 25 per hundred, which is 0.25. That is the entire concept — and the single most useful habit is to convert the percentage to a decimal immediately and then just multiply.

Everything below is that one move, applied to four questions.

Case 1: X% of a number

Multiply by the decimal. 25% of 80 = 0.25 × 80 = 20.

Two shortcuts worth knowing. First, percentages are commutative: X% of Y always equals Y% of X. 4% of 75 is awkward; 75% of 4 is obviously 3. Same answer. This one trick handles a surprising amount of mental arithmetic.

Second, build from 10%. Ten per cent is just moving the decimal point one place, and everything else is assembled from it: 5% is half of 10%, 20% is double, 15% is 10% plus half of 10%. For 15% of 60: 10% is 6, half of that is 3, total 9.

The percentage calculator does all three modes if you would rather not.

Case 2: what percentage is X of Y?

Divide, then multiply by 100. 18 out of 24 = 18 ÷ 24 = 0.75 = 75%.

The thing to get right is which number goes on the bottom. The denominator is the whole — the total, the original, the thing you are measuring against. Marks out of a total, a discount against the original price, a part against the sum.

Getting this backwards is the most common percentage error there is, and it is easy to catch: if your answer is over 100% and it should not be, you divided the wrong way round.

Case 3: percentage increase and decrease

The formula is (new − old) ÷ old × 100. The old value is always the denominator, because change is measured against where you started.

From 40 to 50: (50 − 40) ÷ 40 = 10 ÷ 40 = 0.25, so a 25% increase.

Now the part that catches people. From 50 back to 40: (40 − 50) ÷ 50 = −10 ÷ 50 = −0.2, so a 20% decrease. Up 25%, down 20%, same two numbers. The percentages are not symmetrical because the base changed.

This is why "a 50% fall needs a 100% rise to recover" is true rather than paradoxical. Halve 100 and you have 50; to get back to 100 you must double it. Anyone who has watched an investment fall knows this arithmetic personally.

To apply a change, multiply by (1 + the decimal). A 15% increase is × 1.15. A 15% decrease is × 0.85. The discount calculator does this both ways.

Case 4: reverse percentages

This is the one people get wrong most expensively, because it looks like subtraction and is actually division.

A jacket costs £60 in a 25% off sale. What was the original price? The instinct is to add 25% back: 60 × 1.25 = £75. Wrong.

£60 is not the original — it is 75% of the original. So the original is 60 ÷ 0.75 = £80. Check it: 25% of 80 is 20, and 80 − 20 = 60. Correct.

The rule: to go forwards you multiply; to go backwards you divide.

The same trap appears with tax. Adding 20% VAT to £100 gives £120. Taking 20% off £120 gives £96, not £100. To strip the tax you divide by 1.20. Do this the wrong way on an invoice and you undercount the tax every single time — the VAT and sales tax calculator handles both directions and shows the working.

The traps that catch everyone

Stacked discounts multiply, they do not add. "50% off, then an extra 20% off" is 60% off, not 70%. The second reduction applies to the already-reduced price: 0.5 × 0.8 = 0.4, so you pay 40%. The order makes no difference.

A markup then an equal discount does not return you to the start. Add 30% then take 30% off: 1.3 × 0.7 = 0.91. You end up 9% below where you began.

Percentage points are not percentages. If interest rises from 2% to 3%, that is a rise of one percentage point — but a 50% increase. Both statements are true and they describe the same event. Newspaper headlines exploit the gap constantly.

Averaging percentages usually gives nonsense. 50% of 10 and 50% of 1,000 do not average to "50% of 505" in any meaningful sense. Percentages must be weighted by the size of what they refer to.

Percentages over 100 are fine. A tripling is a 200% increase, not 300%. The increase is the part added, not the final total.

For the everyday cases there are dedicated tools: the tip calculator splits a bill, the discount calculator handles stacked sale prices, the VAT calculator adds or strips tax, and the loan calculator deals with the compounding kind. Pikkit has the rest.