The word hides its own meaning. Per cent is two Latin words — per centum, “for each hundred”. So when a label says 40%, it is telling you: for every hundred, take forty. That is the whole idea. A percentage is a fraction that has been forced to share the same bottom number — always a hundred — so that any two of them can be compared at a glance.
This is the quiet power of the symbol. A test marked 18 out of 25 and another marked 31 out of 40 are hard to weigh against each other. Turn them both into “out of a hundred” — 72% and about 78% — and the comparison is instant. Percentages are a common currency for parts of a whole, and once three things — a percentage, a fraction, a decimal — name the same amount, you can swap between them freely.
To go from a percentage to a decimal, divide by 100 (slide the dot two places left): 40% becomes 0.4. To go to a fraction, write it over 100 and cancel: 40/100 becomes 2/5. Three costumes, one amount — and you choose the one that makes the sum in front of you the kindest.
The % symbol grew out of scribes' shorthand. Italian clerks in the 1400s kept writing “per 100” so often that the words wore down — first to “p 100”, then the 100 collapsed into a little slash with two circles. By 1650 the modern % had arrived, a fossil of the words it once was.
“What is 15% of £80?” sounds like a hurdle until you see that there is always a friendly route in. You almost never need the panic method. Three ideas cover nearly everything you will meet.
There is a tidy quirk: x% of y always equals y% of x. So 18% of 50 is the same as 50% of 18 — which is just 9, found in a breath. When a percentage looks awkward, try flipping it; the flipped version is sometimes the friendly one.
Most real percentages aren't “find the part” — they're “change the whole”. A coat in a 30% sale. Savings that grow 4% in a year. A bill with 20% VAT added. There is a slow way and a fast way, and the fast way is worth learning because shops use it on you constantly.
A jacket goes up 10%, then is cut 10% in a sale. You might expect to be back where you started — but you aren't, quite. Up 10% is ×1.1; down 10% is ×0.9; together that's ×0.99. The price ends a touch below the start. Percentages of different wholes don't cancel cleanly, which is exactly how some “sales” are built.
A shop that marks something up 50%, then later takes 50% off, does not return to the first price. £100 up 50% is £150; then 50% off £150 is £75 — a quarter less than you began with. The two halves are slices of different cakes.
Every percentage you'll meet as an adult is one of the moves you've just learned, dressed for a particular occasion. Learn to recognise the costume and the maths underneath stays calm and familiar.
| Where you meet it | What it does | The move |
|---|---|---|
| VAT on most things you buy | adds 20% to the price | × 1.2 |
| A tip in a restaurant | adds ~12.5% to the bill | × 1.125 |
| Savings interest over a year | grows money by, say, 4% | × 1.04 |
| A sale on the rail | cuts the price by, say, 25% | × 0.75 |
Read the table as a single lesson: adding a percentage gives a multiplier just above 1; taking one off gives a multiplier just below 1. The further from 1, the bigger the change. A 4% savings rate (×1.04) barely lifts your money in a year — but leave it for years and that gentle lift compounds into something real.
Interest is paid not just on what you saved, but on the interest already earned. £1,000 at 4% becomes £1,040 after one year. The next year's 4% is taken on £1,040, not £1,000 — so you earn £41.60, a little more than before. Each year the snowball is bigger, so each year it gathers more. That is ×1.04 applied again and again.
VAT in the UK was 8% when it began in 1973. It has crept up over fifty years to 20% today. Some things stay at 0% on purpose — most food, children's clothes, and books — a quiet decision that the state makes about what ought to be cheaper to buy.
Shops love to dress the same discount in different clothes, hoping one looks bigger than the rest. Percentages are the tool that strips the costume off. Take one shampoo, sold three ways:
Notice the lesson hiding inside: a percentage off only means something next to a price. “50% off” a thing priced at double is no saving at all. The way to see through any deal is to carry it back to a single, plain number — pounds per bottle, pence per 100ml — and compare those.
Sometimes you know the after price and want the before. A coat costs £28 in a 30% sale; what was it first? You know £28 is 70% of the old price. So divide, don't multiply: £28 ÷ 0.7 = £40. Reversing a percentage means dividing by the multiplier you'd have used going forward.
“Buy one get one free” is a 50% discount wearing a louder coat — two items for the price of one is half price each. Stores prefer the free-sounding version because research shows the word free lights up the brain in a way that “50% off” somehow doesn't, even though the maths is identical.
Here is the 30%-off bar from earlier, with four pins. Pick a label below, then drop it on the matching pin. Two of the chips don't belong on a percentage bar at all — read each one before you place it.
Fresh one. A coat is £80 with 15% off. How much do you save?
Fresh one. A hoodie is £30 with 20% off. What do you pay?
The phrase “110 per cent”, beloved of football managers, is mathematically impossible if 100% is everything you have. It survives because we feel percentages as effort, not as a fixed whole — a reminder that the everyday and the exact don't always shake hands.
A coat is £80 in both shops. Shop A offers “30% off”. Shop B offers “£20 off, then 10% off the new price”. Work out the final price in each, say which is cheaper, and explain the trap in Shop B's wording. Show your steps — the reasoning is the answer.
Shop A: one multiply. Shop B: two steps, in order — the 10% is taken off the already reduced price, not the original. Then compare the two finals.
strong You kept the two shops on separate lines and did Shop B in the right order — the £20 first, then the 10% on what was left. That order is the whole game here, and you held it.
try this Put the two final prices side by side in one last line — £56 against £54 — so the comparison is impossible to miss. A marker reading quickly wants the answer stated, not just calculated.
to add One sentence on why B's 10% is smaller than it sounds: it's taken on £60, not £80, so it only saves £6. Naming that is the difference between getting the number and understanding the trick.
Three short films to watch alongside today's lesson — each shows you something the words and pictures can't.
You learned that “per cent” just means “out of a hundred”, and that a percentage, a fraction and a decimal can be the same amount in three coats. You can build any percentage from 10% and 1%, turn a discount into one multiply, and reverse it to find the price before. Next time you pass a sale rail, Florence, you'll know exactly what the sign is really offering.
If everyone in a room of 100 people gave just 1% of their height to the shortest person, that person would tower over everyone — the same 1% means a wildly different thing depending on the whole it's taken from. A percentage is never just a number; it's always a number of something.