A fraction of a fraction, and why the numerators and denominators multiply

0:00A tortoise covers a quarter of a kilometre in an hour. How far does it get in half an hour? You can probably answer that in your head. But look at what the question actually is. A half, times a quarter. Both of the numbers are fractions now. And neither of the two procedures we have will reach it. You cannot lay a quarter down half a time. And cutting a quarter into two equal pieces is a move nobody has made yet.

0:31So we need a picture. And the picture turns out to do a great deal more than answer this. Here is a square. One unit along each side. It is going to stand for one whole. One hour, one kilometre, one anything. Everything from here on happens inside it, and — this is the part people lose — it never changes size. A quarter of a kilometre is a quarter of this square.

1:00Cut it into four strips of the same height, and shade one of them. That shading is the quarter. Nothing clever has happened yet. Now the question. Half of that shading. Not half of the square. Half of the part that is already shaded. So cut the shaded strip in half, the other way, with a line straight down. Half of the shading is this left-hand piece. But how big is that piece, compared with the whole square?

1:31To find out, carry the line all the way down, through every strip. And now look at what has happened to the square. Four rows, crossed by two columns. Eight pieces, every one of them the same size. One of them is shaded. One piece in eight. A half times a quarter is one eighth. And nobody multiplied anything. We counted. Two things about that are worth slowing right down for.

2:01The first: the square never changed. It was one whole before that second cut, and it is one whole after it. So the answer is read against eight pieces, not against the four we started with. The second thing is the little word: of. Half of a quarter. The second cut landed inside the first shading. It did not go next to it. Nothing was added on. A fraction of a fraction is a multiplication, and that picture is the reason why.

2:35Now a faster tortoise. Two fifths of a kilometre in an hour. And a longer walk. Three quarters of an hour. Same square, same job. Cut it into fifths this time. Five rows. Two of those rows shaded. That is the two fifths. Now for the three quarters, cut it the other way. Four columns. And count what the square is in now. Five rows, crossed by four columns. Twenty pieces.

3:05Twenty. And nobody worked that out either. It is simply what five rows crossed with four columns gives you. Of those twenty pieces, the two shaded rows account for eight. Which is still the two fifths. Eight out of twenty. Now, a quarter of an hour is one of the four columns. So take the leftmost column of the shaded part. Two pieces. Two out of twenty. That is what a quarter of two fifths comes to.

3:36But we want three quarters, so take three of the four columns instead. Six pieces out of twenty. Six twentieths. Or three tenths, which is the same number written smaller — and choosing between those two is a job of its own. Look again at where that twenty came from. It was not calculated. It was counted, off a picture that made itself. Cut a square into five rows. Then cut it into four columns.

4:08You now have twenty pieces, because that is what crossing cuts do. I tried it for every pair of row and column counts up to twenty by twenty. Four hundred squares. Every one came out with rows times columns pieces. And every one added back to exactly one whole. So the number underneath a product is not a convention, and not something anybody decided. It is a count of pieces the picture has already made for you.

4:42Now a case that looks like it ought to break all of this. A tortoise doing three halves of a kilometre in an hour, which is more than a whole kilometre. For five quarters of an hour, which is more than a whole hour. Draw the whole square, two rows crossed by four columns. Eight pieces. Three halves needs another half a square, so add four more pieces underneath. Twelve pieces in all, and the figure is three rows tall now.

5:13A quarter of it is the leftmost column. Three pieces. Against a whole of eight, that is three eighths. Five of those columns is fifteen eighths. One whole, and seven eighths over. And nothing about the method had to change. Go back to the eight-piece square for a moment, and look at one piece on its own. It is a little rectangle. How long are its two sides? Half a unit one way. A quarter of a unit the other.

5:45And its area, which is length times breadth, is one eighth of a square unit. Which is exactly the answer we counted off the picture. Eight copies of that rectangle fill the square, and eight eighths make one. So a product of two fractions is the area of a rectangle built on them. Which means a multiplication can always be drawn. A rectangle three and three quarters by nine and three fifths has an area of exactly thirty-six. Two sides that are not whole numbers, and an area that is.

6:24So the picture always works. Now let me show you where it stops helping. One twelfth, times one eighteenth. The recipe has not changed. Eighteen rows, twelve columns. That is two hundred and sixteen pieces, and twenty-eight separate cuts to draw them all. The picture is not wrong — it is perfectly correct. It is just useless. Nobody is going to count two hundred and sixteen boxes. But look at what it would tell you if you did draw it.

6:56One corner piece, out of twelve times eighteen of them. And there is the rule for fractions with a one on top. The bottoms multiply, because that is how many pieces there are. Now stop taking one piece, and take a block of them. Five twelfths, times seven eighteenths. Same square. Eighteen rows, twelve columns. Cut the seven eighteenths into twelve equal parts, and take five of the results. On the picture, that is a block five pieces wide and seven pieces tall.

7:34Five sevens. Thirty-five pieces, out of two hundred and sixteen. And there is the general rule, sitting in the picture. The tops multiply because the block you kept is a rectangle of pieces. The bottoms multiply because that is how many pieces the square was cut into. Brahmagupta set that rule down in general form in the year six hundred and twenty-eight. Very nearly fourteen hundred years ago. One last tidying-up, and it retires something.

8:10What about three times three quarters, where one of the two is a whole number? Three is a fraction. It is three over one. Three ones, if you like. And it sits in the same rule without complaint. Three over one, times three over four. Tops multiply: nine. Bottoms multiply: four. Nine quarters. Two whole ones and a quarter. And three fifths times four is three fifths times four over one. Twelve fifths.

8:43So there is no separate rule for a whole number any more. There is one rule, and underneath it a square that shows you why.