Why forcing every fraction onto a scale of 100 makes them comparable

0:00Two kinds of biscuit come out of a test kitchen, and somebody has to say which is sweeter. In the first, sugar is nine parts out of thirty-four. In the second, thirteen parts out of forty-five. Look at those two for as long as you like and they will not settle it. The top numbers say the second one, because thirteen parts of sugar is more sugar than nine. The bottom numbers say the first one, because thirty-four is a smaller biscuit to spread nine parts through.

0:33Both readings are reasonable, and only one of them can be right. Nothing has gone wrong, though. Those are two perfectly good fractions, written against two different wholes, and a comparison needs one. First, the method most people reach for, because it works often enough to be dangerous. Compare the top numbers and ignore the bottoms. On the biscuits that gives the second variety, and the second variety is the right answer.

1:06On three quarters against one quarter it is right again, and on seven tenths against two thirds it is right a third time. But put it through nine pairs and it agrees on three, which is not a method. It is a coin that keeps landing the right way while you are watching. And it fails in a way worth seeing. Three quarters against three eighths: equal tops, so it calls them level, and one is twice the other.

1:36One half against fifty out of a hundred are level, and that is the one tie it cannot see. So here is the fix. Put both fractions over the same bottom number. One quarter and two fifths. Twenty takes both of them. One quarter is five twentieths. Two fifths is eight twentieths. Now compare the tops, and this time the comparison is honest, because both are counting the same size of piece.

2:08The top of a fraction is a count, and a count means nothing until you know what is being counted. Fix the bottom, and the top becomes the entire comparison. The only thing left to decide is which bottom number to fix. That decision was made a long time ago and has never been revisited. A hundred. Per cent means out of a hundred. That is the whole of what the word says.

2:38And the reason it is worth having a word for it is that the hundred is agreed in advance and never negotiated, so every number quoted that way is already comparable with every other. Twenty-five per cent means twenty-five in every hundred. Twenty-five people in every hundred people. Twenty-five coins in every hundred coins. Twenty-five marks in every hundred marks. The same number, three completely different things measured, and that is the point rather than a complication.

3:16Read one backwards and it turns straight back into a fraction. Fifty per cent is fifty over a hundred, which is one half. Twenty per cent is twenty over a hundred, which reduces to two tenths, and then again to one fifth. Thirty-three per cent is thirty-three over a hundred, and that one does not reduce at all. So a percentage is not a new kind of number. It is an ordinary fraction wearing a bottom number that somebody fixed in advance.

3:50Everything you know about fractions still works on it. A painter is mixing red and yellow to catch a sunset, and red is three quarters of the mixture. Three out of every four. Which is also six out of every eight. Which is thirty out of every forty. Which is seventy-five out of every hundred. Nothing changed along that chain. The same tin of paint is described four times, in four different sizes of piece.

4:23The last description has the bottom number everyone agreed on, so the red is seventy-five per cent, and the yellow is whatever is left, which is twenty-five. Seventy-five and twenty-five come to a hundred, which they had better, because between them they are the whole tin. There are two ways to walk that chain, and they feel different. The first rewrites the fraction. Multiply the top and the bottom of three quarters by twenty-five, and you have seventy-five over a hundred.

5:00The second asks a question. Three quarters of a hundred is what number? Seventy-five. They land in the same place, and not by coincidence. Rescaling three quarters by twenty-five and forcing three quarters onto a hundred are one move with two descriptions, so use whichever of them you can see. The same painter keeps two fifths of a prize for canvas. Two fifths is twenty fiftieths, and twenty fiftieths is forty hundredths. Forty per cent.

5:36Now draw the prize as a bar, nothing at one end and a hundred per cent at the other, cut into fifths. That leaves four cuts, and each one wants a number. One fifth is twenty. Two fifths is forty. Three fifths is sixty. Four fifths is eighty. The forty is the canvas, sitting exactly where it already was. The bar told us nothing the arithmetic did not, but it put every fifth on one scale at once, which a calculation cannot.

6:10Run it backwards and something slightly surprising falls out. Twenty-four per cent is twenty-four over a hundred. What else is it? It is six twenty-fifths. It is twelve fiftieths. It is eighteen seventy-fifths. Keep going past a hundred and there are four more, out to forty-eight over two hundred. Eight fractions in that stretch alone, and every one of them is the same number. So a percentage does not name one fraction. It names a whole family, and points at the member whose bottom number is a hundred.

6:48And a hundred per cent is the whole thing itself, sitting on every bottom number there is, starting with one. Six fractions, then, to put through it. Three fifths is sixty per cent. Seven fourteenths is fifty. Nine twentieths is forty-five. Seventy-two out of a hundred and fifty is forty-eight. Seven fourteenths catches people, because neither the seven nor the fourteen is the answer. It is a half, so it is fifty.

7:22Then one third, and five elevenths. One third is exactly a hundred thirds of a per cent, and five elevenths is five hundred elevenths, and neither will sit on a hundred as a whole number. That is not a failure but a fact about those two fractions, and the honest move is to say so rather than round quietly and pretend. Fifteen white marbles out of twenty-five is sixty per cent.

7:52Fifteen students walking to school out of eighty is eighteen point seven five, because a percentage is under no obligation to be a whole number. Four runners on a road, fifteen minutes in, and six numbers offered for how far along they got. Fifty-five, twenty, thirty-eight, seventy-two, eighty-four, ninety-three. Six numbers and four runners, so two belong to nobody. The first is about a third of the way, and only thirty-eight is anywhere near a third.

8:28The second is a little past halfway, and only fifty-five is. The fourth is almost at the finish, and only ninety-three is. But the third runner sits between the second and the fourth, and so do seventy-two and eighty-four. Both fit, so the picture does not decide it, and saying that beats picking one and sounding certain. Then two comparisons that need no calculation at all. Three elevenths against sixty-one per cent: one is below a half and the other above it, so a half settles it without either being worked out.

9:07And thirty per cent against one third, where both are below a half, so the half says nothing and this is the one you have to work. Back to the biscuits. Nine out of thirty-four is twenty-six point four seven per cent. Thirteen out of forty-five is twenty-eight point eight nine. Nothing about the biscuits changed. Only the bottom number did, and now that both tops count the same size of piece, reading them is all there is to do.

9:41One last question. Why a hundred, and not ten? Put those two on a scale of ten and round to a whole number, and both read three. Ten is too coarse to tell them apart. On a hundred, rounded the same way, they read twenty-six and twenty-nine. And a thousand? Of the four fractions that converted cleanly, ten took two, a hundred took all four, and a thousand took no more and wrote more digits doing it.

10:12A hundred is where the scale has stopped being too rough and has not yet started being wasteful. Every whole percentage lands exactly on two decimal places, and that is the same fact as counting in tens.