What a percentage greater than 100 does and does not mean

0:00Here is a belief almost everybody holds, and it is wrong about half the time. A percentage cannot be more than a hundred. It feels obviously true. You cannot get more than every mark on a test, or eat more than all of a cake. And every percentage you meet for years obeys it, which is how a habit becomes a rule. So before testing it, be exact about what a hundred per cent means.

0:29It means the thing you chose to measure against. That is all of it. And no law says a quantity must be smaller than the thing you chose to measure it against. So choose one out loud, and watch. A shop opens, and the owner sets a target. Takings of five thousand in a day. Five thousand is now what a hundred per cent means, for everything that follows. Notice that this was a decision. Nobody discovered it. The target could have been four thousand, or six.

1:04A hundred per cent is not a property of the money. It is a label somebody attached to a number. Draw it as a bar. The full bar is five thousand, and the full bar is a hundred per cent. Day one. Takings of two thousand. Two thousand out of five thousand is two fifths, which is nought point four, which is forty per cent. Day two. Three thousand five hundred. That is seven tenths, nought point seven, seventy per cent.

1:38Now say those same two facts the other way round. On day one he was sixty per cent short, on day two thirty. Forty and sixty. Seventy and thirty. Both pairs come to a hundred, and it is worth asking why they must. Because the target is one whole thing, made of the part he reached and the part he did not, and those two pieces make up the bar.

2:06Day three. Five thousand exactly. Five thousand out of five thousand is one. One whole. A hundred per cent. The bar is full. Nothing is sticking out and nothing is missing. This is where the old belief feels safest, because here a hundred really does look like a ceiling. It is not a ceiling. It is a line. And a line is something you can be short of, level with, or past. Day three is the middle one of those three.

2:39Day four. Six thousand. A thousand more than the target, and a thousand is a fifth of five thousand, which is twenty per cent. So the bar is the whole bar and then a fifth more. A hundred per cent plus twenty. A hundred and twenty per cent. The direct way lands in the same place. Six thousand over five thousand is six fifths, which is one point two, which is a hundred and twenty per cent.

3:10Nothing broke. The bar simply ran past the line. And notice what day four does not have. It has no missing part at all. Ask what percentage he fell short by and there is no answer, because the question assumes something that is no longer true. Day five, seven thousand eight hundred. Day six, nine thousand five hundred and fifty. Those no longer fit a frame that stops at the target, so stretch the scale to ten thousand and mark two hundred per cent at the far end.

3:47Day five sticks out past the line by two thousand eight hundred. Day six by four thousand five hundred and fifty. Estimate before you compute. Day six looks like nearly double, so call it roughly two hundred per cent. It is a hundred and ninety-one. The estimate was out by nine, which is a good estimate and a wrong answer. And day five is a hundred and fifty-six. Now run it backwards, because a percentage above a hundred is also an instruction.

4:23Suppose he reached a hundred and fifty per cent of target. What did he take? A hundred and fifty per cent is one point five, and that times five thousand is seven thousand five hundred. Two hundred and ten per cent is two point one, and that is ten thousand five hundred. One mistake is worth naming, the commonest in the subject. A hundred and fifty per cent, so multiply five thousand by a hundred and fifty.

4:54That gives seven hundred and fifty thousand, which is not a day's takings anywhere. It is exactly a hundred times too big, because the hundred in per cent got left out. Take a day of two thousand five hundred, and count the ways of saying so. He reached half his target. He reached fifty per cent of it. He reached nought point five of it. One number in three costumes, none more correct than the others.

5:25Now take a day of ten thousand. He reached twice his target. He doubled it. He reached two times it. He reached two hundred per cent of it. That last one says what the three before it say, and it is the only one that makes anybody uneasy. Two hundred per cent is twice. Not twice as much again on top. Twice. Here are nine percentages, from fifteen at the smallest to three hundred and fifty-eight at the largest.

5:57Six of the nine are past the hundred mark, and that is not an accident in the list. Every one has a fraction and a decimal. Two hundred and fifty per cent is five halves, two point five. Three hundred and five per cent is sixty-one twentieths, three point nought five. Now draw a line from nought per cent to four hundred, and under the ticks write nought, one, two, three, four.

6:26That second row is the same information written twice. A percentage is a multiplier with the point moved. Ninety and a hundred and ten land either side of the one, a fifth apart, and neither is doing anything strange. So far the thing underneath has been a target. Here is a previous value instead. A field yielded two hundred and sixty kilograms one year, and six hundred and fifty the next.

6:58Six hundred and fifty over two hundred and sixty is two and a half, which is two hundred and fifty per cent. This year is two hundred and fifty per cent of last year. Two and a half times it. The crop grew by three hundred and ninety kilograms, a third number different from both the others. Nothing here is a share of anything. Last year is not a container this year sits inside. It is a yardstick standing beside it.

7:31Now the other case, because sometimes the ceiling is completely real. A snack packet, sixty-five grams, and the label lists what is inside. Potato seventy per cent. Oil twenty-four. Salt three. Spices three. Those four come to exactly a hundred, and they had to. Take each share of sixty-five grams and you get forty-five and a half, fifteen point six, and one point nine five twice, adding back to sixty-five.

8:05Could one ingredient be a hundred and ten per cent? Only if another were minus ten, and there is no such thing as minus ten per cent of a packet. So here the ceiling is arithmetic. It is not disapproval. And that is the test, and it is the only thing worth taking away. Ask what a hundred per cent stands for. If it is a whole your quantity is part of, nothing can pass a hundred. If it is a yardstick beside it, anything can.

8:39Which leaves the words, where this is actually lost. Take forty. A hundred and twenty per cent of forty is forty-eight. A hundred and twenty per cent more than forty is eighty-eight, because that adds a hundred and twenty per cent on top. And twenty per cent more than forty is forty-eight, the first answer again. So of it and more than it sit exactly a hundred apart, every time, on any base.

9:08One last one. Demand for something falls by eighty-five per cent. Which of these six says the same thing? Is it eighty-five per cent of what it was? Eighty-five per cent more? Fifteen per cent of what it was? Is what it was fifteen per cent of it, or a hundred and eighty-five per cent of it? Or is it a hundred and eighty-five per cent of what it was?

9:34Exactly one of those six is true, and it is the one saying fifteen per cent of what it was. Read the other way round, what it was is about six hundred and sixty-seven per cent of what it is now, and neither eighty-five nor a hundred and eighty-five comes into it.