What a ratio claims, and why it is not a difference

0:00Two measurements, changed by one and the same factor, keep the shape they started with. That sentence is true, and it is far too long to say out loud every time you need it. So it gets a shorthand, and the shorthand is a single mark: a colon, sitting between the two measurements. Sixty to forty. Thirty to twenty. Ninety to sixty. Three pictures that kept their shape, written three times.

0:29But a colon is not a way of writing down two measurements. It is a claim about how two quantities go together, and everything that follows depends on reading it as one. Because some things you can do to those two numbers leave the claim standing, and some things quietly replace it. Sixty to forty does not say that anything is sixty across. It says: for every two units down, there are three across.

1:00Say it out loud that way once, and the notation stops being decoration and starts being a sentence you can argue with. Thirty to twenty says exactly the same thing. So does ninety to sixty. Three different pictures, at three different sizes, making one claim written three different ways. And notice what the claim does not contain: any information at all about how big the thing is. Twenty feet of wall per bag of cement could be a wall twenty feet long or two hundred.

1:36The two numbers have names of their own. They are called the terms, and they are the first term and the second term. Each term is a measurement of something, and that something needs naming just as much as the number does. Three apples for every two bananas comes to three halves. Sixty millimetres across for every forty down also comes to three halves. Those come to the same number, and they are completely different claims about completely different things.

2:09So a ratio is not finished when you have written the numbers down. Strip the units off and what you are left holding is a piece of arithmetic, not a statement about anything at all. Turn a ratio round and you have not rearranged it. You have said something else. Take seven to twelve, and set it beside twelve to seven. Read both of them out. For every twelve there are seven. For every seven there are twelve.

2:40Those are opposite claims, built out of the same two numbers. Of the five pictures, exactly one reads the same number whichever way round you take it, and that is the square one. And not even that one is the same claim reversed, because across and down are different measurements. A ratio points one way, from the first term to the second. The colon is not a comma. Two people are building a wall around a compound, and they split it between them.

3:14He builds the long side, sixty feet, and it takes three bags of cement. She builds the short side, forty feet, on two bags. He decides her wall must be the weaker one, because less cement went into it. He has done something very natural and completely wrong: he compared one column of the table instead of comparing the pairs. Sixty to three is twenty feet of wall a bag. Forty to two is twenty feet of wall a bag.

3:46Different numbers, and the same claim, which is the whole of the answer to his worry. Nothing so far said the two terms have to measure the same kind of thing. Feet of wall against bags of cement. Teachers against students. Millilitres of coffee against millilitres of milk. Five teachers and a hundred and seventy students is one to thirty-four. One teacher for every thirty-four, which is a claim you can compare with another school.

4:20Three cups of coffee at fifteen to thirty-five, twenty to thirty, and ten to forty are three different claims. No two of them agree, and you can hear that as soon as you read them out. Both units get said, every time, or the claim cannot be used by anybody but you. Now the trap, with the word finally available for it. Take a picture at sixty to forty and remove twenty from each measurement.

4:51What you land on is forty to twenty, and it looks like a reasonable thing to have done. But three halves has quietly become two, and two is not three halves by any reading. The same amount came off both, and it is a different claim. Multiplying both terms leaves the claim exactly where it was. Adding to both never does. And nothing in the arithmetic itself gives you the slightest warning that the two operations behave differently.

5:24A girl of three, whose mother is thirty. Three to thirty, which is one to ten. Nine years later she is twelve and her mother is thirty-nine. That is twelve to thirty-nine, which reduces to four to thirteen. One tenth has become four thirteenths. Nine was added to both terms, and that is the same arithmetic as the picture with the sign turned round. The claim moved towards an even one-for-one, and it will keep moving that way.

5:57Nobody did anything to it. The years did it, and the years do it to everybody. Towards an even one-for-one, every time, and never towards anywhere else. Here is the reason. Set a to b against a plus k to b plus k, and cross-multiply. The difference between the two comes out as k times b minus a, in one line. Positive whenever the second term is the bigger one, which is what lopsided means.

6:29Swept over six hundred and sixty lopsided claims and amounts added, every single one moves nearer an even claim. Swept over another six hundred and sixty lopsided the other way up, the same thing again. Taking away pushes the other way, over two hundred and twenty cases, but only while the amount stays under the smaller term. Something in that story did not move at all while the ratio was drifting, and it is worth stopping to find it.

7:03Thirty take away three is twenty-seven. Thirty-nine take away twelve is twenty-seven. The gap is the one thing adding to both terms cannot touch. The ratio is the one thing it cannot leave alone. Those two get swapped constantly, and swapping them is the whole misconception. So try running the whole thing backwards. Take a girl of one, and a brother of five. At what age of hers do the two ages stand at one to two?

7:36The gap is four years, and it is four years for the rest of their lives. So asking when he is twice her age is really asking when the gap equals her age. It happens at four. She is four, he is eight, and four to eight reduces to one to two. Swept across eighty years of her life, that happens once and only once. Before that year the claim sits under a half, and after it the claim sits over a half, and it never comes back.

8:07Because it only ever moves one way, towards an even claim, and it never reaches one. One question, one answer, and the reason the question has an answer is a difference and not a ratio. So the last question is the practical one: when is a difference the right tool, and when is a ratio? For two ages, the difference is the thing that stays and the ratio is the thing that does not.

8:38For feet and bags there is no difference at all. Feet minus bags is nothing. There the ratio is the only comparison available to you, and it happens to be exactly the right one. If you want somebody else to be able to use a ratio, three things have to be on the page. Which quantity is first, what each term is a measurement of, and nothing you have not actually shown.

9:05And a ratio does not tell you how much there is of anything. It never did, and it was never trying to. It tells you how the two go together, and once you have said that properly, it is enough.