Percentages made simple

Most percentage confusion comes from not noticing that "percent" is used for three different questions. Separate them and the maths becomes almost trivial.

Percent means "out of a hundred"

A percentage is just a fraction with 100 on the bottom. 25% is 25/100 is 0.25. Once you can flip between the percentage and the decimal, every percentage sum turns into ordinary multiplication or division.

The three questions

Almost every real problem is one of three: what is X% of a number (multiply by the decimal), what percentage is one number of another (divide and multiply by 100), or what is the percentage change between two numbers (difference divided by the original). Spot which one you have and the method is fixed.

The reversibility trick

A neat shortcut: X% of Y always equals Y% of X. Working out 4% of 75 looks awkward, but 75% of 4 is obviously 3 — same answer, far easier. Swap the numbers whenever one order is friendlier.

Percentage points vs percent

A rate rising from 5% to 6% is a one percentage point rise, but a 20% increase. Mixing these up is behind a surprising amount of misleading reporting. Say which you mean.

Practise with the tool

The calculator below handles all three questions plus percentage change — enter your numbers and check your working.

The three questions percentages answer

Almost every percentage problem is one of three shapes, and recognising which one you are looking at is most of the work. What is X% of Y? — multiply: 15% of 80 is 0.15 × 80 = 12. X is what percent of Y? — divide and multiply by 100: 12 out of 80 is 12 ÷ 80 × 100 = 15%. X is Y% of what? — divide: if 12 is 15% of something, that something is 12 ÷ 0.15 = 80.

The third form is the one people stumble over, and it is the one that matters most in practice — working backwards from a discounted price to the original, or from a tax-inclusive total to the amount before tax. If £96 is 80% of the original price (a 20% discount), the original was 96 ÷ 0.8 = £120. Subtracting 20% from £96 gives £76.80, which is wrong and is the most common error in the whole topic.

Why percentage changes do not reverse

A value that falls 20% does not return to where it started when it rises 20%. Fall 20% from 100 and you have 80; rise 20% from 80 and you have 96. The percentages are taken from different bases, so they simply do not cancel. To recover a 20% fall you need a 25% rise; to recover a 50% fall you need 100%; to recover a 90% fall you need 900%.

This is not a curiosity. It is why an investment that halves needs to double just to break even, and why a shop that cuts prices 30% cannot restore its old price with a 30% increase. Whenever someone quotes a fall and a recovery as though they offset, the arithmetic is against them.

Percentage points versus percentages

These are different units and mixing them produces statements that are technically true and completely misleading. If an interest rate moves from 2% to 3%, it has risen by one percentage point — but by fifty per cent. Both descriptions are accurate. One sounds trivial and the other sounds alarming, and which one gets quoted usually depends on what the speaker wants you to conclude.

The convention is simple: use percentage points when comparing two percentages, and percentages when describing relative change. If you see a headline about a rate rising "50%", check whether that means fifty percentage points or a relative increase, because the difference can be enormous.

Stacking percentages

Percentages applied one after another multiply rather than add. A 10% discount followed by a further 10% is not 20% off — it is 0.9 × 0.9 = 0.81, so 19% off. A 20% rise followed by a 20% fall leaves you at 0.96 of where you started, down 4% despite the two figures looking symmetrical.

The same principle explains why compound interest outruns simple interest, and why several small annual price rises add up to more than their sum. Three consecutive 5% increases give 1.05³ = 15.8%, not 15%. Over longer periods and larger rates the gap becomes the dominant part of the answer.

Sanity checks worth keeping

Before trusting any percentage calculation, run two quick tests. First, is the answer on the right side of the original number? An increase must be larger and a decrease smaller — this alone catches most misplaced decimal points. Second, is 10% of the number obviously right? Ten per cent is just the number with the decimal shifted one place, and once you know 10% you can build 5%, 20% and 15% in your head and check the tool agrees.