Tricky percentages: why a 50% margin followed by a 50% discount leaves a 25% loss

0:00Every percentage quietly names a number underneath it. Fifty per cent of what? That missing half of the sentence is where almost every mistake in this topic lives. When two percentages in one story name the same number underneath, they add, and nothing goes wrong. When they name different numbers, they stop behaving like numbers you can add at all. And the trouble is that the wrong answer is always close enough to be believed.

0:32Two clubs in one school, both calling themselves the fastest growing. The first has six hundred and seventy-eight members, and it grew by eighty per cent. The second grew by a hundred per cent. It doubled. A hundred per cent sounds unbeatable. But the second club has five members, so doubling brought in five people. Eighty per cent of six hundred and seventy-eight is more than five hundred new members.

1:04The smaller club wins on proportion and loses on people, and both of those sentences are true. A percentage ranks the share. The share is not the size. The same disagreement, now in money. Two offers. Stake a hundred and get three hundred back, or stake a thousand and get one thousand five hundred back. The first hands you two hundred on a stake of a hundred. A gain of two hundred per cent.

1:35The second hands you five hundred on a stake of a thousand. A gain of fifty per cent. Two hundred per cent against fifty. Five hundred against two hundred. The two rankings point in opposite directions, and which is better depends on whether what you have is a hundred or a thousand. The percentage has thrown the amount away. That is what a percentage is for. A shop offers a choice on any basket over a hundred and fifty. Twenty per cent off, or fifty off.

2:11On a basket of a hundred and eighty, twenty per cent is thirty-six, so the flat fifty is better. On two hundred and twenty-five it is forty-five. Still short of fifty. On three hundred it is sixty, and now the proportion wins. So somewhere between those baskets the two offers cross, and they cross at two hundred and fifty. Below that a fixed amount is the better discount, and above it a share is.

2:41The crossing point is the answer. Neither offer is. Two cake shops on the same street. One advertises thirty per cent off, and then twenty per cent off. The other advertises fifty per cent off, flat. Thirty and twenty make fifty, so they ought to be the same shop. Take a cake priced at two hundred. Thirty per cent off leaves a hundred and forty. Twenty per cent off that leaves a hundred and twelve. The flat fifty leaves a hundred.

3:15Twelve apart, on a cake that started at two hundred. The whole reason sits in two words. Off that. The first discount comes off two hundred. The second comes off a hundred and forty, because that is the price by then. The second percentage is standing on a smaller number, so it removes less than you expected it to. Multiply the fractions that survive instead. Nought point seven times nought point eight is nought point five six.

3:50So the pair is a forty-four per cent discount, whatever the cake happens to cost. And the six per cent that went missing is thirty times twenty over a hundred - the second discount's bite out of the first discount's bite. The order makes no difference either. Taking the big one first changes nothing at all. Now the same fault, with a seller's own money at stake. She buys cookware from a wholesaler and marks everything up by fifty per cent.

4:23Sales are slow, so she clears the stock at fifty per cent off, expecting to come out level. Fifty on, fifty off. Surely that is where she started. Call what she paid x. She marks it up to one point five x. Half off that is nought point seven five x. She ends with three quarters of what the goods cost her. A twenty-five per cent loss, while she believed she had broken even.

4:54Look at what each of those two fifties was standing on. The mark-up added fifty per cent of x. That is half of what she paid. The discount removed fifty per cent of one point five x. That is three quarters of what she paid. The same percentage. Half as much again in money, because the second base is one and a half times the first. Put real numbers on it. Stock that went out at twelve thousand had cost her sixteen thousand.

5:27She lost four thousand. And four thousand on sixteen thousand is twenty-five per cent, the same quarter as before, which never depended on the rupees at all. So what discount would have brought her back to level? She needs one point five x pulled down to x, so the discount has to be the half she added, divided by the one and a half it is taken off. One third. Thirty-three point three three per cent.

5:58In general a mark-up of m is undone by a discount of m over one plus m, and that is always smaller than m itself. Gentler numbers behave the same way. Thirty-five per cent on, thirty per cent off. That leaves nought point nine four five, which is a loss of five and a half per cent. It reads like a five per cent gain, and it is a loss.

6:26The same trap fits inside a single sentence. I have twenty per cent more marbles than you. So you have twenty per cent fewer than me? No. If mine is a hundred and twenty per cent of yours, then yours is a hundred divided by a hundred and twenty of mine. Five sixths. Eighty-three point three three per cent. So you are short by one sixth, which is about sixteen point six seven per cent, and not by twenty.

6:57Two hundred take away a hundred and twenty is eighty, and eighty is simply a different question's answer. Reversing the comparison reverses the base with it. Three shops, identical goods, a hundred each. Buy one get one free. Buy two get one free. Buy three get one free. The first feels like a hundred per cent off, because the second item cost nothing. But you handed over a hundred and walked out with two items, so each one cost you fifty. That is fifty per cent off.

7:35Count the free items against everything you carried home, never against the ones you paid for. One in two, one in three, one in four. Fifty per cent, thirty-three point three three, and twenty-five. And if you need exactly four items, the first shop charges two hundred while both the others charge three hundred. Four is not a whole number of their deals, and the offer quietly stops helping. One last one, and it is the best of them.

8:08Ninety-nine of the hundred people in a room are left-handed. How many must walk out for that to fall to ninety-eight per cent? Almost everybody says one, or two. The answer is fifty. Stop looking at the ninety-nine. Exactly one person in that room is right-handed, and she never moves. For her to be two per cent of the room, the room has to hold fifty. So fifty left-handers leave and forty-nine stay. Half the room walks out to move the figure by one single point.

8:44The fixed quantity was the group nobody was counting. And underneath every one of these: percentages of one base do add - nine per cent and nine per cent of four hundred and fifty come to eighty-one - and the moment the base moves, they multiply instead.