Solving a proportion problem, and the Trairasika rule of three

0:00Here is a question that sounds like cooking and turns out to be about ratios. A regular cup of coffee is fifteen millilitres of coffee to thirty-five of milk. A strong cup is twenty to thirty. A light one is ten to forty. Nobody tastes these and reaches for a calculator, and yet everybody can rank them. Written down, fifteen to thirty-five is three parts coffee to seven parts milk.

0:30Twenty to thirty is more coffee for the milk that is there, and ten to forty is less. That is the whole of what stronger and lighter mean here: one comparison, written down instead of tasted. Look at those three cups again, because something quietly convenient is going on. Fifteen and thirty-five make fifty. So do twenty and thirty, and so do ten and forty. All three cups hold the same amount, so the coffee alone settles which is strongest.

1:05More coffee in the same size cup is more coffee in every mouthful, and you need nothing else. Now take five mixtures instead: three hundred to six hundred, a hundred and fifty to five hundred, two hundred to four hundred, twenty-four to fifty-six, and a hundred to three hundred. No two of them come to the same total, so that shortcut is gone and every one has to be reduced.

1:33Against three to seven, two come out stronger, two come out lighter, and exactly one is the regular mixture again at nothing like the same amounts. Six glasses of lemonade took ten spoons of sugar. Someone asks for eighteen more glasses, made the same way. Three of the four numbers are known and the fourth is not, and that is the entire shape of the problem. Six is to ten as eighteen is to something.

2:05Eighteen is three times six, so the factor carrying the first ratio into the second is three. Apply that same three to the ten and you get thirty spoons. Read it instead as a new total of twenty-four glasses and the answer is forty, which is a different reading of the question and not a careless one. Nothing anywhere says that factor has to be a whole number. Start from fourteen to twenty-one and fill in three blanks.

2:38Blank to forty-two: forty-two is twice twenty-one, so the missing term is twenty-eight. Six to blank: six is three sevenths of fourteen, so its partner is three sevenths of twenty-one, which is nine. Two to blank: two is a seventh of fourteen, so its partner is a seventh of twenty-one, which is three. Two of those three factors are smaller than one, and the method never noticed the difference. A factor is a factor whether it makes things bigger or smaller.

3:14A school of a hundred and twenty children needs fifteen kilograms of rice for lunch. On a wet day only eighty of them come. A hundred and twenty is to fifteen as eighty is to something. Eighty over a hundred and twenty is two thirds. Two thirds of fifteen is ten kilograms. The answer moved down, and it moved down for a reason you can point at: the factor is below one.

3:42Fewer children, less food, and the arithmetic worked that out without being told which way the answer ought to go. Every one of those was the same move, so it is worth writing down once and never again. Take a is to b as c is to d. The third term is the first times some factor: c is f times a. The fourth is the second times that same factor: d is f times b.

4:12So f is c over a, and f is also d over b, which means c over a equals d over b. Multiply both sides by a times b, and the fractions clear away: b times c equals a times d. That is cross multiplication, and it is not a trick anybody invented. It is the shared factor written without fractions. Rearrange it and d is b times c, over a.

4:44Now the fourth term comes straight out of the three you were given. On the lemonade, ten times eighteen over six is thirty. On the rice, fifteen times eighty over a hundred and twenty is ten. On the second blank, twenty-one times six over fourteen is nine. Same three answers, and no factor had to be spotted first. Two different routes to one number, and every problem in this video is put through both of them and lands in the same place each time.

5:19This method is very old. Long before anybody wrote a colon or a letter for an unknown, Aryabhata set the whole thing down as a sentence. He gives the three known quantities and the unknown one their own names. Pramana, the measure. Phala, the fruit. Ichchha, the requisition. And ichchhaphala, the yield. Those are the first, second and third terms, and the fourth one you are after. The recipe is one line: multiply the fruit by the requisition and divide by the measure.

5:55Which is exactly d equals b times c over a, said out loud. The notation is the new part; the method is not. A car covers ninety kilometres in a hundred and fifty minutes. How far does it get in four hours at the same speed? The obvious first line is: a hundred and fifty is to ninety as four is to something. That line is wrong, and the arithmetic has nothing to do with it.

6:26A hundred and fifty is minutes and four is hours, so the first and third terms are not measuring the same thing, and a proportion between them says nothing at all. Four hours is two hundred and forty minutes. Now it reads: a hundred and fifty is to ninety as two hundred and forty is to something. The factor is eight fifths, the distance is a hundred and forty-four kilometres, and both cross products come out at twenty-one thousand six hundred.

7:01Here is an old problem in which every single number is awkward. Two and a half measures of saffron cost three sevenths of a coin. How much saffron will nine coins buy? Nine divided by three sevenths is twenty-one, so the factor is twenty-one. Twenty-one times two and a half is fifty-two and a half measures. Nothing about the method changed because the numbers were fractions. It only looks harder. It is the same three lines it was for the lemonade.

7:36Now watch the whole thing break. A rider covers a road in two hours at fifty kilometres an hour. How long does that same road take at seventy-five? Set it up exactly the way everything else in this video was set up: fifty is to two as seventy-five is to something. The rule answers. It does not hesitate and it does not complain. It says three hours. Read what that claims: go half again as fast, and arrive an hour later.

8:09The answer is not merely wrong by a bit. It moved in the wrong direction, and the method gave no warning whatever. So what is different about that one? In every problem that worked, the two quantities were held together by a shared factor: triple the glasses, triple the sugar. Here they are held together by a fixed product instead. What cannot change is the road. Fifty times two is a hundred, and a hundred is what it stays.

8:42So seventy-five times the true time is also a hundred, and the true time is four thirds of an hour, which is one hour and twenty minutes. Faster, and less time, which is what anyone would have said before the arithmetic started. So ask one question before you write the ratio down: when one of them goes up, does the other go up with it, or does their product stay put?

9:10Two oxen take six hours an acre, so a hundred and twenty hours for twenty acres, and a machine four times as fast takes thirty of them. The rule of three would have told you four hundred and eighty.