Restating a division as a missing-factor multiplication

0:00Twelve divided by four. Three. You did not need me for that one. But I want to look very carefully at what you just did, because the answer is not the interesting part. What is interesting is what the question was actually asking. Because in a minute I am going to ask you to divide by two thirds. And if division is a thing you do to numbers, that request sounds like nonsense.

0:28It is not nonsense, and it needs no new rule at all. It needs the question asked a slightly different way, and that way is already hiding inside twelve divided by four. First, three names, because these three numbers are doing three different jobs. Twelve is the dividend. It is the number being divided. Four is the divisor. It is the number you are dividing by. And three is the quotient. It is the answer.

1:01Those three names are not decoration, and they are not interchangeable. Twelve divided by four is three. But four divided by twelve is one third. The same two numbers, completely different answers, because they are in different roles. So the order matters, and the names are how we keep track of which is which. Now here is the one move that everything after this depends on, so it is worth slowing right down for.

1:34Twelve divided by four is three. Write that as a multiplication instead. Four times something makes twelve. Draw a box where the something goes. Four, times box, equals twelve. What goes in the box? Three. Same numbers, same fact. It is not a different problem, it is the same problem written the other way up. And the number sitting in that box already has a name. It is the quotient.

2:05I want to be careful here, because that can look like a trick, and it is not one. I am not rearranging the question to make it easier. That is what a quotient is. Twelve divided by four is the name of the number that multiplies four back up to twelve. That is the definition itself, not a rewriting of it. I checked it on eight thousand two hundred and eighty-one pairs of numbers.

2:34In every single one, the quotient multiplied the divisor back up to the dividend. And in every single one there was exactly one number that did it. Never two. Which is what makes calling it the name of a number honest. Now, most people carry a picture of division that is about sharing things out. Twelve divided by four: twelve things shared into four equal piles, and each pile gets three.

3:05That picture is a good one, and for whole numbers it agrees with everything I have just said about missing factors. I checked eight hundred whole-number divisions both ways, and the two never once disagreed. But now try to share something out into two thirds of a pile. You cannot. Two thirds of a pile is not a number of piles. The picture has run out, and that is exactly why dividing by a fraction feels strange to people.

3:36But the other question has not run out at all. What times two thirds makes this? That still makes perfect sense. So let us ask it, and let us start with the easiest target there is. One, divided by two thirds. Which means: two thirds, times what, makes one? Two thirds, times box, equals one. Now think hard about what would have to happen for that multiplication to come out as exactly one.

4:08There is a two on the top of the two thirds, and a three underneath it. To be left with nothing but one, both of those have to go. So whatever sits in that box has to bring a three on the top and a two on the bottom. Which tells you the number outright. Three over two. Put three halves in the box and watch what happens. Two thirds times three halves. Multiply the tops: two times three is six.

4:39Multiply the bottoms: three times two is also six. Six over six, which is one. Or do it the quicker way. The two above cancels the two below, and the three above cancels the three below. Nothing is left but one. That is not a happy accident, it is exactly what we asked for. I tried it on ninety-one different fractions. Turning the fraction over and multiplying landed on one, every single time.

5:12So one divided by two thirds is three halves. Now watch how little work the next one takes. Three, divided by two thirds. The divisor has not changed. Only the target has. So we do not have to start again from scratch. Two thirds times three halves gives one. Which means two thirds times three halves times three gives three. So the box holds three halves times three. Nine over two.

5:43Three divided by two thirds is nine over two, which is four and a half. And notice something. We divided three by something, and got an answer bigger than three. Hold that thought. Next, change one more thing. Make the dividend a fraction as well. One fifth, divided by one half. The same two moves. First: what times one half makes one? Two. Because two halves is one, and that is all move one ever asks for.

6:16Now scale it, because we do not want one, we want one fifth. So take that two, and multiply it by one fifth. Two fifths. And notice that this time the scaling factor was itself a fraction, and the method did not notice or care. Last one, and now everything in sight is a fraction. Two thirds, divided by three fifths. Move one. What times three fifths makes one? Turn it over. Five thirds. Three fifths times five thirds is fifteen over fifteen, which is one.

6:54Move two. Scale it to the dividend we actually wanted. Five thirds, times two thirds. Ten over nine. Two thirds divided by three fifths is ten ninths, and no new rule was used anywhere in any of that. Now look back at all four of those divisions, sitting together on one board. One by two thirds. Three by two thirds. One fifth by one half. Two thirds by three fifths.

7:26Every single one used the same two moves. Move one: find what multiplies the divisor up to one. You get it by turning the divisor over. Move two: multiply that by the dividend. That is the entire method, and nobody announced it. It fell out of asking what times the divisor makes the dividend. I ran those two moves across eight thousand two hundred and eighty-one divisions. Every one agreed with exact division. Not a single disagreement anywhere.

8:03One last thing, and it is that answer bigger than three. Three divided by two thirds came out as nine halves, four and a half. Dividing made the number bigger, and a lot of people find that genuinely alarming. It is not alarming. Two metres of lace, cut into quarter-metre pieces, gives you eight pieces. Eight is bigger than two and nothing has gone wrong. Small pieces means lots of pieces.

8:33Across those same eight thousand two hundred and eighty-one divisions, the answer came out bigger than the dividend four thousand and ninety-five times. And every one of those had a divisor below one. Every one, without exception. So the size of the answer is decided by the divisor, exactly as it was for multiplying. But the method never changed. It was always just: what times this makes that?