Profit, loss and taxes as percentages of a stated amount

0:00Three prices are hiding inside one ordinary sale, and mixing them up is most of the difficulty in this subject. There is what the seller paid to get the thing, the price written on the tag, and what the customer actually handed over. Usually three different numbers, and only the last one is money that changed hands today. The tag is a proposal. The seller can cut it, and often does.

0:29What the seller paid is not on the tag anywhere, and the customer never sees it. A profit is a fact about a sale. A profit percentage is a fact about a sale and a choice of what to measure it against. Follow one sweater from the workshop to the person who wears it. The maker spends two hundred and thirty, tags it at two hundred and fifty-five, and lets it go for two hundred and fifty-three.

0:59The wholesaler pays two hundred and fifty-three, tags it at three hundred and ten, and lets it go for three hundred. The shop pays three hundred, tags it at four hundred and eighty, and after bargaining takes four hundred and thirty. Look at what just happened to the two hundred and fifty-three. It is what the maker took in and what the wholesaler paid out. The same number. Only the label changed.

1:28So cost price and selling price are not properties of a number. They are positions in somebody's story. So the shop put in three hundred and took out four hundred and thirty. The profit is a hundred and thirty. Nobody argues with that. Now measure it. A hundred and thirty out of what? Out of the three hundred, because that is what the shop risked, and a profit percentage answers what you earned on what you put in.

2:00A hundred and thirty over three hundred is forty-three point three three per cent, and the threes go on for ever. Upstream, the maker turned two hundred and thirty into two hundred and fifty-three, which is exactly ten per cent. And the wholesaler turned his into three hundred, which is eighteen point five eight. Now the same shape with the sign turned round. A seller bought rice at thirty-five a kilo and clears ten stale kilos for three hundred.

2:34That stock cost three hundred and fifty, three hundred came back, so fifty is gone. Fifty out of three hundred and fifty. Same fraction, same base, opposite direction. It comes to fourteen point two eight five seven, and it never stops. Cut short at two places that is fourteen point two eight. Rounded it is fourteen point two nine. Two operations, two numbers, and it is worth knowing which one you just did.

3:06Here is a question worth stopping for. Could that loss have been worked out per kilo instead? Five lost on thirty-five, rather than fifty lost on three hundred and fifty. Same percentage. Exactly the same, not nearly. Both halves of the fraction were multiplied by ten, and a fraction does not notice. Tried on every deal here at five different sizes, fifty cases, the percentage did not move once. The amount moved in all fifty.

3:41That is the trade a percentage makes. It forgets how much, and remembers how big. Now run a margin the other way, because that is where people fall. Forwards is easy. Thirty-six at a twenty per cent margin, times one point two, forty-three point two. Backwards. Something sells for fifty at a twenty-five per cent margin. What did it cost? Not fifty less twenty-five per cent. The margin was never a percentage of fifty.

4:15It is a division. Fifty over one point two five, which is forty. Same with a loss. Strawberries sold for eighty at a twelve per cent loss cost eighty over nought point eight eight, which is ninety point nine one. Add twelve per cent to the eighty instead and you get eighty-nine point six, close enough to be believed. But eighty-nine point six at a twelve per cent loss comes to seventy-eight point eight four eight, short of the eighty it actually sold for by one point one five two.

4:54A discount is a percentage taken off a price nobody was going to pay. Thirty-five per cent off a cooker tagged eighteen hundred leaves eleven hundred and seventy. If the shop paid nine hundred for it, the profit is two hundred and seventy, which is thirty per cent. Two percentages in one situation, thirty-five and thirty, and they never once shared a base. The thirty-five was of eighteen hundred. The thirty was of nine hundred.

5:27Not every cost is the cost of the goods. A shop takes eighty thousand in a month and spent forty-eight thousand buying what it sold, so thirty-two thousand is left. That is the gross profit. Sales, less the goods themselves. Then transport, wages and electricity come to eight thousand, none of which ever touched the goods, and twenty-four thousand is left. That is the net profit, and it is thirty per cent of what came in.

6:00A bigger month runs identically. A hundred and fifty thousand in, a hundred thousand of goods, fifty thousand gross, five thousand of everything else, forty-five thousand net, thirty per cent again. Someone buys fertiliser at five hundred a bag and sells at seven hundred and fifty. The profit is two hundred and fifty. Against the five hundred she paid, that is fifty per cent. Against the seven hundred and fifty she took in, it is thirty-three point three three.

6:37Both correct. The two hundred and fifty never moved. The thing underneath it did. The first answers what she earned on what she invested. The second answers what share of her takings was profit. And the second is always the smaller, on every one of the ten deals measured here, equal only where there was neither profit nor loss. So a business with revenue of twenty-five million and a twenty-five per cent margin spent twenty million if that margin is on cost, and eighteen point seven five million if it is on revenue.

7:16One and a quarter million apart, and nobody said which. Now two percentages that do add, which is the other half of this. A receipt. Three bulbs at a hundred and fifty, four hundred and fifty. Then two tax lines, nine per cent each, forty point five zero each. Nine and nine of the same four hundred and fifty is eighteen per cent of it. Eighty-one. Four hundred and fifty plus eighty-one is five hundred and thirty-one, and every line on that bill is right.

7:53Two rates on one base always add. Tried twenty different ways here, it held all twenty times. So a phone at eight thousand two hundred and fifty with eighteen per cent added is that price times one point one eight, or that price plus eighteen per cent of itself. Both give nine thousand seven hundred and thirty-five. Of seven expressions offered, exactly those two work. And now two percentages that do not add, in a situation that looks almost identical.

8:30Someone sells two animals for eighty thousand each. On one he makes five per cent. On the other he loses ten. Five minus ten is minus five, so a five per cent loss overall. That is what almost everybody says, and it is wrong. Work out what each one cost. The first came from eighty thousand over one point nought five, about seventy-six thousand one hundred and ninety. The second came from eighty thousand over nought point nine, about eighty-eight thousand eight hundred and eighty-nine.

9:08Not the same number, and that is the entire problem. The five and the ten are sitting on different bases. He put in about a hundred and sixty-five thousand and seventy-nine and took a hundred and sixty thousand back, so he is down about five thousand and seventy-nine. Which is three point nought eight per cent. Not five, and not the two and a half you get by averaging. The sign came out right by luck. The size never had a chance.

9:43One last one, with no money in it anywhere. A shop prices pencils so that selling three brings in exactly what five of them cost. No price is given. None is needed. Each pencil sells for five thirds of what it cost, so the profit is two thirds of the cost. Sixty-six point six seven per cent. Cut short rather than rounded it reads sixty-six point six six, a hundredth apart.

10:13A profit is a fact. A profit percentage is a fact and a choice. And if nobody tells you what the number underneath was, you have not actually been told a percentage.