The mathematics you actually use

Stand in front of a market stall for a minute and count the maths on offer. Prices per kilo, prices per punnet, a handwritten sign saying £1.50 each, 2 for £2, a chalked claim about being cheaper than the supermarket. All of it is arithmetic you were taught by the age of twelve, and most people work through it by feel and hope for the best.

That's not stupidity. It's what happens when a subject gets organised around techniques rather than around decisions. You leave school with a set of procedures you'll forget inside five years and a settled conviction that you're bad at maths, and the handful of ideas that would genuinely help with money, risk and news reporting are barely covered at all.

Here's that handful. There's no notation and nothing here needs a calculator, though a couple of the examples reward doing slowly rather than quickly.

Percentages only ever move in one direction at a time

A percentage is a fraction with the bottom fixed at a hundred, which is what makes different fractions comparable. The arithmetic is never the problem. The problem is that a percentage is always of something, and the something keeps changing while nobody says so.

A £100 coat is cut by 20 per cent, so it's £80. The sale ends and the price goes back up by 20 per cent. You'd expect £100. It's £96, because the second 20 per cent was a fifth of £80, which is £16, not a fifth of £100. Percentage changes don't cancel out and they never have.

To actually get back from £80 to £100 you'd need a rise of 25 per cent. The general shape of this is that recovering from a fall always takes a bigger percentage than the fall itself, and the gap widens fast — a drop of 50 per cent needs a rise of 100 per cent to undo, and a portfolio down 90 per cent has to increase tenfold just to return to where it started. Anyone who's watched an investment fall a long way already knows this in their stomach without necessarily having the arithmetic to hand.

The same trick runs in reverse in shops. "Half price, plus an extra 20 per cent off" isn't 70 per cent off. It's £100 down to £50 down to £40, which is 60 per cent off, and it's a perfectly honest way of writing a smaller discount so it reads as a larger one.

Then there's the distinction between percentage points and per cent, which is where reporting gets slippery. An interest rate moving from 2 per cent to 3 per cent has risen by one percentage point, and it has also risen by fifty per cent. Both are true. They provoke entirely different reactions, and which one appears in the headline isn't often an accident.

One genuinely useful piece of arithmetic while you're here. VAT in the UK is 20 per cent, and to strip it out of a price you divide by six, not by five. A £60 bill contains £10 of VAT, because £50 plus a fifth of £50 is £60. Subtracting 20 per cent from £60 gives £48 and is simply the wrong operation, though it appears on invoices constantly.

The only fair comparison is per unit

Back to the stall, and the sign reading £1.50 each, 2 for £2.

Two for £2 is a pound each, so the offer takes a third off — not the "50p off" that the numbers suggest at a glance. Working in price-per-item rather than in the sizes on the label is the single most reliable defence against packaging, and it's why unit pricing on supermarket shelf labels was worth arguing for.

It doesn't always favour the big pack. 750 grams at £2.40 works out at £3.20 a kilo. The 1.2 kilogram box at £3.60 comes to £3.00 a kilo, so the big one wins here by about six per cent — but the same sum done on a different pair of products will happily tell you the small one is cheaper, and the large-size-is-better-value assumption is wrong often enough to be worth ten seconds of checking.

Every comparison in the rest of this article is a version of the same move. Get both things onto the same denominator before you decide anything.

Compounding is the one that ambushes everybody

Growth applied to a total that already contains the previous growth doesn't add up. It accelerates, and human intuition extrapolates in straight lines, so the answer always arrives as a surprise.

The shortcut worth memorising is the rule of 72. Divide 72 by a growth rate to get the approximate doubling time. At 6 per cent a year something doubles in about twelve years — the exact answer is 11.9, which is close enough for a pub. At 3 per cent it's 24 years, and the true figure is 23.4. Halving the rate doesn't slow things down a bit, it doubles the wait.

The rule is at its most accurate around 8 per cent, where it's very nearly perfect, and it drifts at the extremes. At 20 per cent it suggests three and a half years when the honest answer is closer to four. Fine for mental arithmetic, not for a spreadsheet.

Run it forwards and the numbers get silly. A thousand pounds growing at 5 per cent a year becomes about £4,320 in thirty years — more than four times the original, with nothing added. At 1 per cent per day, which sounds trivially small, you'd multiply your money by roughly 38 in a year. Losing 1 per cent a day for a year leaves you with about two and a half per cent of what you started with.

The old chessboard story is the cleanest illustration: one grain of rice on the first square, doubling each time. The last square alone carries more grains than every previous square added together, which is exactly true of any doubling sequence and is the whole reason the story keeps being told.

The mechanism doesn't care which way it's pointing. It makes long-term saving work and it makes high-interest debt genuinely dangerous, and it's the reason a difference of two percentage points in a fee, which looks like a rounding error, quietly eats an enormous share of a pension over forty years.

Averages conceal at least as much as they report

"Average" almost always means the mean: add everything up, divide by how many. The mean is fragile. One extreme value drags it a long way, which is how average income in a place can climb while nobody living there is any better off.

The median is the middle value once everything's sorted. It ignores how extreme the extremes are, so it's far more representative whenever the data is lopsided — and income, wealth, house prices, waiting times and company sizes are all strongly lopsided. Median is nearly always the more honest number, and mean is nearly always the more flattering one. Notice which gets quoted.

The mode, the most common value, is the third and gets forgotten, though it's the only one that makes sense for things that come in whole units. The mean number of children per household is never a whole number, and no household has ever contained a fraction of a child.

There's a subtler trap, which is averaging things that can't be averaged. Drive 60 kilometres out at 30 km/h and 60 kilometres back at 60 km/h, and your average speed isn't 45. The first leg takes two hours, the second takes one, so you've covered 120 kilometres in three hours, which is 40 km/h. Averaging rates directly gives the wrong answer whenever the underlying amounts differ, and the same mistake shows up in blended interest rates, in fuel consumption figures and in combined test scores.

Ask how common the thing is before anything else

The most consequential mistake in everyday probability is forgetting to ask how rare something was to begin with.

Take a test for a condition that affects one person in a thousand. The test is 99 per cent accurate in both directions. You test positive. It feels like near-certainty. It isn't close.

Do it with ten thousand people. Ten of them have the condition, and the test finds essentially all ten. Of the 9,990 who don't have it, one per cent get a positive anyway — that's about a hundred people. So roughly a hundred and ten positive results, ten of which are real. Under one in ten.

Nothing there is advanced. It's just that when a condition is rare, most of your positives come from the enormous healthy group rather than the tiny affected one, and no amount of test accuracy changes that unless the accuracy is extraordinary.

The follow-up is the cheering bit. Test the hundred and ten again, assuming the second test's errors are independent of the first, and the ten real cases stay positive while only about one of the hundred false positives repeats. Now you've got roughly eleven positives and ten of them are genuine. This is why doctors retest rather than diagnose on a single result, and why screening an entire population for something rare produces a mountain of frightening letters.

The same structure sits under security alerts, fraud flags, and every system that hunts for something uncommon in a very large pile.

"Doubles your risk" is not information

A short one, and it follows directly.

Relative risk without absolute risk is unreadable. If something raises your chance of a particular outcome from 1 in 10,000 to 2 in 10,000, that's a doubling, a hundred per cent increase, and an entirely accurate way of describing an extra one case in ten thousand. If it takes a risk from 1 in 4 to 1 in 2, that's also a doubling.

They're the same headline and they aren't remotely the same news. Whenever a percentage change in risk turns up without the underlying numbers, the underlying numbers are the ones you want.

Expected value, and when to ignore it

Expected value is what an uncertain thing is worth on average: multiply each outcome by its probability, add them up.

Invent a lottery to see it work. A ticket costs £2 and gives you a one-in-ten-million chance of winning £5 million. The expected return is five million divided by ten million, which is 50p, against £2 spent. Every real lottery has this shape, since the prizes have to come out of the ticket money with a good deal held back, and no strategy touches it. The ticket buys about ninety seconds of imagining something, which may well be worth two pounds — it just isn't an investment.

The tool works the other way too. A bet that pays £100 on a 60 per cent chance and loses £100 on a 40 per cent chance has an expected value of plus £20 per go, which is excellent.

Now the part people skip. Excellent per go is not the same as excellent once. Play that bet a thousand times with small stakes and you'll finish comfortably ahead. Play it once for everything you own and you've got a 40 per cent chance of ruin, and ruin doesn't average out with anything, because you're not there for the next round.

That gap is precisely why insurance exists and why buying it is sensible. Insurance has a negative expected value by construction — the company needs the difference to survive — and you buy it anyway, because a loss you can't absorb isn't simply a larger version of a loss you can.

Enormous numbers need something to lean on

Intuition copes with numbers up to a few hundred and then gives up quietly without telling you. A million, a billion and a trillion arrive feeling like a sequence of similar large things. They're nothing like each other.

Convert them to time and the gap becomes physical. A million seconds is about eleven and a half days. A billion seconds is nearly thirty-two years. A trillion seconds is roughly thirty-one thousand seven hundred years, which puts you well before any recorded history. Same three words, wildly different objects.

The habit that helps is turning any big figure into something per person, or per day, or as a share of something you already have a feel for. A national total divided by the population is almost always more informative than the total, and it's noticeable how rarely that division gets done for you.

Correlation, causation, and the third thing

Everyone can recite that correlation doesn't imply causation, and the recitation does very little work. The useful move is to have the alternatives ready, because there are only a few and running through them takes seconds.

• It's backwards. B is causing A rather than the reverse.
• Something else causes both. Ice cream sales and drownings rise together because both follow hot weather.
• Selection. The pattern lives in how the data got collected, not in the world.
• Chance. Compare enough pairs of unrelated things and some of them will match impressively for no reason whatsoever.

The common-cause case is the one that gets people, because a story connecting A directly to B is usually easy to construct and sounds sensible when told. The discipline isn't doubting the correlation. It's asking what would produce this exact pattern if the obvious explanation happened to be false.

Who got asked matters more than how many

A badly drawn sample can't be rescued by enlarging it, and the definitive demonstration is nearly a century old.

In 1936 an American magazine ran a straw poll on the presidential election, mailed out something like ten million ballots and got well over two million back — a sample size no serious pollster has attempted since. It predicted a comfortable win for Alf Landon. Franklin Roosevelt took 46 of the 48 states. George Gallup, working with a sample smaller by a factor of roughly forty, called it correctly.

The magazine had drawn its names from telephone directories and car registration lists in the middle of the Depression, which selected quietly for the better-off half of the country. Every additional reply made the same bias heavier and lent the final number a credibility it hadn't earned.

Modern versions are everywhere and mostly unlabelled. Online polls answered by whoever felt strongly enough to click. Reviews written by the delighted and the furious while the indifferent middle says nothing. Customer satisfaction surveys that by definition cannot reach anyone who already left. When a figure surprises you, the first question isn't how many were asked. It's who couldn't have been counted.

Things don't scale the way they look

Double the width of something and the area goes up four times, not two. Double it in three dimensions and the volume goes up eight times. That single fact settles more everyday questions than it has any right to.

Pizza is the classic. A 12-inch pizza has about 113 square inches of surface; a 16-inch has about 201. A third more diameter, but nearly 80 per cent more pizza — which means one 16-inch comfortably beats two 10-inch pizzas, and a large costing 40 per cent more than a medium is usually the better deal even though it doesn't look it.

The same relationship explains why a doubled tin of paint covers four times the wall only in the advertisement, why small animals lose heat so fast relative to their size, and why halving the linear dimensions of anything cuts its material cost by a factor of eight.

Estimating roughly beats calculating precisely

The most transferable skill in this whole area isn't calculation. It's approximating well enough, fast enough, to notice when a number can't possibly be right.

Physicists call these Fermi problems, after Enrico Fermi, who had a habit of producing decent estimates from almost no information by breaking a question into parts he could each guess within a factor of two or three. Errors in opposite directions tend to cancel, and the answer usually lands within an order of magnitude of the truth. At the Trinity nuclear test in 1945 he dropped scraps of paper as the blast wave passed and estimated the yield from how far they blew — landing within a factor of two of the figure the instruments produced later.

You don't need his arithmetic. You need the reflex. When a claim turns up — a cost, a saving, a statistic, a projection — break it into two or three pieces you can each roughly guess and see whether the result is in the same postcode as the claim. It catches an enormous amount of nonsense, including a fair bit of your own.

The related habit is distrusting precision that hasn't been earned. A figure quoted to two decimal places has usually been through an estimate somewhere upstream, and the decimals are decoration. Rounding aggressively while you think is not sloppiness. It's what keeps the arithmetic honest enough to check.