Fractional relations between two quantities

0:00A quarter of a litre of milk, poured equally into five cups. How much is in one cup? That is a sharing, and every sharing is a division. A quarter, divided by five. Which means: five times what makes a quarter? Five turns into one fifth, and a quarter of a fifth is one twentieth. One twentieth of a litre in each cup. Check it. Twenty of those cups would be a whole litre, so five of them is a quarter. It fits.

0:32Nothing new there. But notice the shape of that, because the shape is what this whole video is about. Second one. You have a floor of seven and a half square units to cover. The bricks are square, one fifth of a unit along each side. First question. How much area does one brick cover? One fifth by one fifth. One twenty-fifth of a square unit. Now, how many of those fit into seven and a half?

1:03Seven and a half is fifteen over two. Fifteen over two, divided by one twenty-fifth. And dividing by a twenty-fifth is multiplying by twenty-five. Three hundred and seventy-five over two. Stop on that answer for a moment, because it is worth looking at. Three hundred and seventy-five over two is a hundred and eighty-seven and a half. A hundred and eighty-seven and a half bricks. You cannot lay half a brick, and nobody is pretending you can.

1:39But the arithmetic is not wrong. Whether an answer is a usable instruction is a different question from whether it is correct. A hundred and eighty-seven whole bricks leaves a gap. A hundred and eighty-eight overshoots. So the honest answer is the one with the half in it, and the practical answer is that you cut one brick. Do not round a result just because it looks untidy. Say what it means instead.

2:10Third one, and this is the one people get wrong. Four fountains fill the same tank. On its own, the first takes a day, the second half a day, the third a quarter of a day, the fourth a fifth of a day. All four run together. How long? The tempting move is to add the times. A day, plus a half, plus a quarter, plus a fifth. That comes to nearly two days. Which is longer than the slowest fountain takes by itself, with four taps running. Obviously wrong.

2:45Here is what actually adds. How many tanks each one fills in a day. One. Two. Four. Five. Together, twelve tanks a day. So one tank takes a twelfth of a day. Two hours. Faster than any of them alone, which is what you would expect. Now the real subject of this video. Here is a square. I am going to draw some lines inside it. Both diagonals. Both mid-lines. And then the top right quarter gets cut up further still.

3:19Somewhere inside all of that there is a small shaded sliver. The question is: what fraction of the whole square is that sliver? And here is the difficulty. Nobody is going to measure it against the whole square. It is far too small and far too awkwardly placed. So we will not. We will compare it with the thing immediately around it, then compare that with the thing around it, and keep going outward.

3:48A chain of relations. Then one multiplication at the end. First link. The top right quarter. The two mid-lines cut the square into four equal squares. So the top right one is a quarter of the whole square. Call the whole square one square unit. Then that corner square is a quarter of a square unit. Now forget the big square for a moment. Genuinely forget it. From here on, the corner square is our whole.

4:20Everything in the next step is measured against that, and not against the original. This is the most important habit in the whole topic. Always know what your current whole is. Second link. Enlarge the corner square so we can see inside it. It has a diagonal drawn across it. And a diagonal cuts any square into two identical triangles. So the triangle is one half of the corner square.

4:51Not one half of the big square. One half of the corner square. The whole has changed. Although we can already say what the triangle is against the original, if we want to. Half of a quarter. One eighth of the big square. And now the whole changes again, because the shaded part sits inside that triangle. Third link. Enlarge the triangle. The shaded region is three quarters of it. Let me show you why, rather than just telling you.

5:25The triangle is itself cut into four equal pieces, by lines that are already in the figure. Three of those four are shaded. One is not. Three quarters of the triangle. And that is the last link, because there is nothing smaller sitting inside it. So we have three statements, and every one of them uses a different whole. A quarter of the square. A half of that. Three quarters of that.

5:55Now put them together, and this is the step that does all the work. Three quarters, of one half, of one quarter. The word of is multiplication. So it is three quarters, times one half, times one quarter. Multiply the tops. One times one times three is three. Multiply the bottoms. Four times two times four is thirty-two. Three over thirty-two. Already in lowest form. And now it is measured against the big square again, because the chain reached all the way back out to it.

6:32Notice what you must not do. Adding those three fractions gives three halves, which is bigger than the whole square. The picture refuses that instantly. And notice this as well. I could have multiplied them in any order. All six orders give three over thirty-two. A product does not care which end you start from. Here is a problem from a mathematician called Bhaskara, written around eleven fifty. A traveller asks a miser for money.

7:07The miser gives him one fifth, of one sixteenth, of one quarter, of one half, of two thirds, of three quarters of a coin. That is deliberately exhausting, and it is meant to be. It sounds enormous. But it is exactly what we have just done. Six links instead of three. Multiply the tops. One, one, one, one, two, three. Six. Multiply the bottoms. Five, sixteen, four, two, three, four. Seven thousand six hundred and eighty.

7:41Six over seven thousand six hundred and eighty, which reduces to one over one thousand two hundred and eighty. One one-thousand-two-hundred-and-eightieth of a coin. That is the gift. And that sounds like almost nothing, which is where most people stop reading. But there is one more step, and it is the punchline. The coin in the story is a silver dramma. And in that story, one dramma is worth one thousand two hundred and eighty cowrie shells.

8:16A cowrie shell was a real unit of currency, and the smallest one there was. So one twelve-hundred-and-eightieth of a dramma, times one thousand two hundred and eighty shells to the dramma, is exactly one shell. He did not give a fraction of anything. He gave one cowrie shell. The most elaborate-sounding gift in the problem turns out to be the smallest coin in circulation. That is the joke, and it needed a chain of six fractions to land it.

8:50One last thing, about those coins, and it comes with a warning attached. There were gold coins, silver coins, copper coins and cowrie shells, and you can lay them out as a ladder. But the rates between them were never fixed. They changed with the region, with the century, with the economy, and with the weight and purity of the actual metal. So take what follows as an assumption, not as a fact.

9:20Assume one gold dinar is twelve silver drammas, and one dramma is four copper panas. Then one pana, as a fraction of a gold dinar, is one twelfth of one quarter. One forty-eighth. The same chaining, on money instead of shapes. And here is the honest part. That ladder gives a hundred and twenty cowries to a dramma, but the story used one thousand two hundred and eighty. They do not agree. Which is exactly why you hedge. Chain the relations, multiply along them, and be clear about which links you assumed.