Simplest form, and using it to test whether two ratios are proportional

0:00You can get from sixty to forty across to ninety to sixty by hunting for the factor that carries one to the other. It is three halves, and once you have found it there is nothing left to do. That works. What it does not do is scale. You have to find the factor first, and there is no rule that says it will be a whole number. And when no single factor carries one to the other, a failed hunt tells you only that you failed.

0:30There is a move that goes the other way and throws the factor away instead of chasing it. Divide, rather than multiply, and the comparing takes care of itself. Take sixty to forty and divide both terms by twenty. Three to two. Take ninety to sixty and divide both terms by thirty. Three to two. Neither of those divisions was a guess. Twenty is the largest whole number that goes into both sixty and forty, and thirty is the largest that goes into both ninety and sixty.

1:09Nothing about the claim changed. Sixty over forty is three halves, and three over two is three halves. The numbers changed, and the numbers were never the claim. The pair you are left with is called the simplest form, and the two pictures landed on the same one without ever being compared to each other. Why the highest, and not whichever common factor happens to catch your eye? The whole numbers that go into both sixty and forty are one, two, four, five, ten and twenty.

1:46Divide by the smallest of them and sixty to forty becomes thirty to twenty. Which still has something to cancel, so you are not finished. Do the same to ninety to sixty and you get forty-five to thirty, which is not thirty to twenty. Two pictures making one claim, and two different answers, entirely because of which factor was noticed first. Only the highest common factor finishes the job in a single step, and of the six numbers on that list, it is the only one that does.

2:23What you are left with at the end has a property worth saying out loud. Nothing goes into both of its terms except one. That is the whole of what simplest means here. Not smallest-looking, not tidiest, not the version that fits the page. Nothing left to cancel, which is a thing you can check rather than a thing you feel. Of the five pictures, not one of them arrived in that state.

2:51Every single one had something to cancel, and every single one had exactly one place to stop. Now the reason any of this is worth doing. Write the two terms as their highest common factor times whatever is left over: g times p, and g times q. By construction, p and q share nothing at all. Any other ratio making the same claim is that same pair scaled up: f times g times p, to f times g times q.

3:27Its highest common factor is f times g, so dividing by it lands you on p to q as well. Swept over eleven hundred combinations of a coprime pair, a common factor and a scaling, every single one comes back to the same pair. So a simplest form is not a tidier way of writing one ratio. It is a name for the entire family of ratios that make that claim.

3:58And a name gives you a test. Two ratios are in proportion when their simplest forms are equal. Not when they look similar, and not when one first term happens to divide the other. There is even a symbol for saying it: two dots against two dots, set between the two ratios. The important word in that test is a word people skip. Equal names are not evidence of proportion. They are the whole of it.

4:28Two ratios with the same name cannot fail to be in proportion, because each of them is that name scaled up, so there is nothing left to check afterwards. So sort the five pictures by name and see what happens. Sixty to forty is three to two. Thirty to twenty is three to two. Ninety to sixty is three to two. Forty to twenty is two to one, and sixty to sixty is one to one, and each of those is on its own.

5:01Three of the five share one name. There are twenty ordered pairs you can make from five pictures, and exactly six of them are in proportion. And those six are precisely the ordered pairs you can build out of the three that share a name. No hunting, no judgement by eye, no factor to find. Five reductions and a comparison. Now some numbers that do not help you at all.

5:31Is three to four in proportion with seventy-two to ninety-six? Three to four has nothing to cancel, so it is already a name and there is nothing to do to it. What is the highest common factor of seventy-two and ninety-six? It is twenty-four, and twenty-four is not the first thing anybody sees. Eight numbers go into both of them, and stopping at the smallest leaves you at thirty-six to forty-eight, which looks nothing like three to four.

6:04Divide by twenty-four instead and out comes three to four, and the answer is yes, with no hunting and no doubt about it. Six claims, and the job is to say which are true. Four to seven against twelve to twenty-one. Eight to three against twenty-four to six. Seven to twelve against twelve to seven. Twenty-one to six against thirty-five to ten. Twelve to eighteen against twenty-eight to twelve. Twenty-four to eight against nine to three.

6:38Reduce every one of them and exactly three of the six come out true. But one of the false ones is false for a completely different reason from the others. Its two names are one another reversed, so the two numbers are the same numbers and the claims are opposite, and no amount of cancelling will ever fix that. The machine runs backwards just as well. Start from four to nine, which is already a name, and multiply both terms by anything you like.

7:12Eight to eighteen. Twelve to twenty-seven. Twenty to forty-five. Forty to ninety. Every one of them reduces straight back to four to nine, which is why that list has no end. Now fill in blanks against eighteen to twenty-four, which reduces to three to four. Three gives you four. Twelve gives you sixteen. Twenty-seven gives you thirty-six. And twenty gives you twenty-six and two thirds, because twenty is not a multiple of three, and that is not a misprint. It is what the machine says.

7:50One more, and this one is about looking rather than arithmetic. Here is a wall of bricks with coloured ones set into it in a pattern. The obvious move is to count the coloured bricks and the plain ones and write down the ratio. Do not. The wall carries on past the edge of the picture, and the picture stops wherever it happens to stop. Find the block that repeats first: five columns across by three courses down, fifteen bricks, nine plain and six coloured, which is three to two.

8:27Count everything on the page instead and you get thirty-three plain to eighteen coloured, which reduces to eleven to six. Those are different claims, and only one of them is the wall's. The other answers a question nobody asked. The second wall says the same thing in a way that is easier to miss. Its pattern repeats every four columns over seven courses: twenty-eight bricks, twelve coloured and sixteen plain, which is four to three.

9:01Count all seventeen columns on the page and you get seventy-one to forty-eight. That has nothing at all left to cancel, so it is a perfectly good name, and it is a name for the wrong thing. Six hundred millilitres of orange juice with nine hundred of apple is two to three, in the order the question asked for. And that is exactly where a name stops being useful. These two carry the same name and one of them is three times the size of the other, so a simplest form tells you nothing whatever about how much there is.

9:39It tells you which family you are in. That is a different question, and it is the one worth settling first.