Cancelling common factors before multiplying, not after

0:00Twelve sevenths, times five twenty-fourths. There are two ways to do this, and both of them work. The first way: multiply everything out, then tidy up at the end. Twelve fives are sixty. Seven twenty-fours are a hundred and sixty-eight. Sixty over a hundred and sixty-eight, which is not a form anybody wants to leave an answer in. So you go hunting for what those two share. It is twelve. Divide it out, and you get five fourteenths.

0:33The second way arrives at five fourteenths without ever writing a hundred and sixty-eight down. Same number. Far fewer numbers. That is the whole of today. Here is the move that makes the second way possible, and it is very easy to skip past. Write the product as one fraction first. Before multiplying anything out. Twelve times five, over seven times twenty-four. Look hard at what that is. It is one fraction. One bar.

1:07And above the bar and below it, two multiplications that have not been carried out yet. That is not a half-finished answer. It is the answer, in a form you can still work on. Everything that follows depends on the top and the bottom each being a product. Because a product is made of factors, and factors are the things that can be cancelled. Now, cancelling is not some special permission that multiplication hands you.

1:39It is a rule you have had for a long time, and it is about one fraction on its own. Divide the top and the bottom by the same number, and the fraction has not changed. Six eighths and three quarters are the same number. Halve both parts of one and you get the other. I checked that rule on two thousand five hundred and twenty-six cases. It never once changed the value.

2:06And notice what that rule does not mention. It says nothing whatsoever about multiplying. It is about a single fraction — and a single fraction is exactly what we have now got. So the licence was already in your hand. What changed is that it became usable a step earlier. Back to twelve times five, over seven times twenty-four. Ring the twelve on the top. Ring the twenty-four on the bottom.

2:38They share a factor. Twelve divides both of them. So divide both by twelve. The twelve becomes one. And the twenty-four becomes two. Strike each of them through and write the small number beside it. That is your working, and it is worth showing. What is left is one times five, over seven times two. Five fourteenths. And nothing needed tidying up afterwards. Now look at where those two numbers came from.

3:10The twelve was the top of the first fraction. The twenty-four was the bottom of the second. They started in different fractions, on opposite sides of a multiplication sign. And that was fine — because by the time we cancelled them, they were not in different fractions any more. They were both in the same one. One above the bar, one below it. So a factor can travel right across that multiplication sign, as long as it lands on the other side of the bar.

3:44I tried seven thousand six hundred and thirty-two of those crossings. Not a single one of them changed the value of the answer. Which raises the obvious question. What is not allowed? Two numerators, for a start. You cannot cancel a top against a top. Dividing only the top of a fraction by something gives you a different fraction. That is not a rule to learn, it is just what division does.

4:15I tried seventy-five of those. Every last one of them changed the number. And here is one that catches people much further on. You cannot do it across a plus sign. Three plus one, over three plus two, is four fifths. It is not one half. I tried one thousand seven hundred and twenty-eight of those. It came out right on a hundred and forty-four, and every one of those was a fraction equal to one, where nothing could have gone wrong anyway.

4:49A harder one. Fourteen fifteenths, times twenty-five forty-seconds. One fraction first. Fourteen times twenty-five, over fifteen times forty-two. Now go looking for a pair. Fourteen on the top, forty-two on the bottom. Forty-two is three fourteens. So divide both of them by fourteen. That leaves one, and three. One stroke done. But there is another one sitting right there. Twenty-five on the top, fifteen on the bottom. Both of those divide by five.

5:26Five, and three. One times five, over three times three. Five ninths. Two strokes, two different factors, one product. Something about that is worth saying out loud. The two strokes used different numbers. Fourteen for the first, five for the second. You are not required to find the biggest factor that fits, and you are not required to do it all in one go. The long way round would have given you three hundred and fifty over six hundred and thirty.

6:01You could divide that by two. Then by five. Then by seven. Or you could spot that they share seventy, and do the whole thing in a single stroke. Both routes land on five ninths. I checked that on one thousand five hundred and sixty fractions. Small repeated strokes always finish exactly where one big stroke finishes. So what is actually gained here? Look at the numbers each route made you write down.

6:32The long way: sixty, and a hundred and sixty-eight. The short way: five, and fourteen. On the second example it is starker still. Six hundred and thirty on one route. Nine on the other. Seventy times smaller. I ran twenty thousand seven hundred and thirty-six products down both routes. Cancelling first never once made the numbers bigger. And on twelve thousand four hundred and fifty-five of them, it made them smaller.

7:06There is a second thing gained, and it is the one that matters when somebody is marking your work. Answers are almost always wanted in lowest form. On the long route you have to go hunting for that at the end, when the numbers are at their very biggest. On the short route it tends to arrive already done. So I checked the case that matters. Both fractions given in lowest form to begin with.

7:34Eight thousand two hundred and eighty-one products. Cancelling first landed in lowest form every single time. Not one exception in the whole lot. It cannot fix everything, mind. If a fraction turns up already untidy, like four sixths, cancelling across the sign will not touch that — the two numbers are inside the same fraction. But that is an older job, and you already know how to do it. This is old, and it has a name of its own.

8:10In Sanskrit, apavartana. Reducing a fraction to its lowest terms. And the interesting part is not really the word. It is where the word turns up. A scholar writing around the year one hundred and fifty reached for it as an image — outside mathematics altogether, in a work of philosophy. That is roughly one thousand eight hundred and seventy-six years ago. And you do not borrow an idea to make a picture out of it unless your reader already knows the idea.

8:45Which tells you that reducing a fraction was ordinary enough to be a figure of speech. Not a technique somebody had just thought of. A habit. One last thing, and it is where all of this actually gets used. A fraction of a quantity is this same multiplication. Nothing new is happening. Two weights. The first is twelve fifteenths of five hundred grams. The second is three twentieths of four kilograms. Which one is heavier?

9:19Look at the fractions and the answer seems obvious. Twelve fifteenths is four fifths, which is most of it. Three twentieths is a sliver. But that is answering a completely different question, because those are fractions of different things. Twelve fifteenths of five hundred grams is four hundred grams. Three twentieths of four kilograms — and four kilograms is four thousand grams — is six hundred. The smaller fraction is the heavier weight, by two hundred grams. Match the units first. Then, and only then, the fractions.