Unit conversion is a proportion in disguise

0:00Two packets of tea, from two different farms, and one simple question. Which is dearer? The small farm sells two hundred gram packets, and one packet costs two hundred. The large estate sells one kilogram packets, and one packet costs eight hundred. Write each as a ratio of weight to price. The first is two hundred to two hundred, which reduces to one to one. The second is one to eight hundred, and already something has quietly gone wrong.

0:34One of those ratios has grams on the left and the other kilograms, and nothing in the numbers says so. So write the second packet again, this time with its weight in grams. One kilogram is a thousand grams, so that ratio is a thousand to eight hundred. And a thousand to eight hundred reduces to five to four. The same packet of tea. The same weight, the same price, the same everything.

1:05Written one way its simplest form is one to eight hundred. Written the other way it is five to four. Not one leaf and not one coin changed. All that changed was the unit the first number was counted in. That looks like a contradiction. It is not. A simplest form is not a fact about two quantities. It is a fact about two numbers. And the numbers depend on what you decided to count in before writing them down.

1:38Which means a ratio is only half a statement until you say what its terms are measured in. So one to one and five to four really are different comparisons, and the packets are not proportional. But before the units agreed, that question could not honestly be asked at all. The step that repaired it was a conversion, and a conversion is a ratio like any other. One kilogram to a thousand grams. Beside the packet, that is two ratios and four numbers, three of them known.

2:15Which is the rule of three, arriving with most of its work already done. The only difference is that this ratio was agreed once and never changed again. The speed of a car changes. The price of tea changes. A kilogram does not. So a conversion factor is a ratio nobody measures twice, which is why converting needs no new machinery. Multiply going one way, divide going the other. Every conversion is that one step.

2:50Now, which tea is actually dearer? Dearer means more for the same amount, so choose an amount and price it in both places. A kilogram from the estate costs eight hundred, because that is what one packet is. Two hundred grams from the small farm costs two hundred, and a kilogram is five of those packets. Five two hundreds is a thousand, so a kilogram from the small farm costs a thousand.

3:19A thousand against eight hundred. The small farm's tea is the dearer of the two, by a quarter. Which is the reverse of what the packet prices suggest, purely because the packets differ in size. The same trap catches a faster-looking problem. A car covers ninety kilometres in a hundred and fifty minutes. How far does it go in four hours, at the same speed? The obvious first line puts a hundred and fifty against four, and says nothing whatsoever.

3:55Those two numbers are not two times. One of them is minutes and the other is hours. As written it reduces to seventy-five to two, a perfectly tidy fact about nothing. So convert first, before the proportion is written. Four hours is two hundred and forty minutes. Now a hundred and fifty against two hundred and forty reduces to five to eight, a factor the distance can be scaled by.

4:25Most conversions come off a list, and a list is worth reading before you trust it. A metre is three point two eight one feet. A square metre is ten point seven six four square feet. An acre is forty-three thousand five hundred and sixty square feet. A hectare is ten thousand square metres, and the same list also calls it two point four seven one acres. Some of those are definitions and some are measurements rounded off, and they are not the same kind of fact.

5:01An acre is exactly forty-three thousand five hundred and sixty square feet, because that is what an acre was defined to be. But three point two eight one is a decimal that stops, and the true number does not, so it carries a small error onwards. One entry on that list is not independent, and hunting for it teaches the commonest mistake here. If one metre is three point two eight one feet, how many square feet are in a square metre?

5:35The answer most people reach for is three point two eight one, and it is badly wrong. A square metre is one metre by one metre, and both of those lengths have to be converted. So the factor is three point two eight one multiplied by itself, which comes to ten point seven six four nine six one. And that is the area entry, agreeing to within a thousandth of a square foot.

6:03So the area factor was never a second measurement. It is the length factor squared, and it follows from what area is. The hectare entry is not independent either. Ten thousand square metres, at ten point seven six four square feet each, is a hundred and seven thousand six hundred and forty. Divide by forty-three thousand five hundred and sixty and you get two point four seven one acres, the other number on the list.

6:37So that one figure does an enormous amount of work. A plot two hundred feet by five hundred is a hundred thousand square feet, about two point three acres. At ten tonnes of manure to the acre that is a little under twenty-three tonnes, and the area had to be in square feet first, because that is the unit the acre is in. Or turn it round. If an acre costs one and a half million, what does two thousand four hundred square feet cost?

7:09That is about five and a half percent of an acre, so about eighty-two thousand six hundred. Not a round answer, though every input was. Volume is where somebody designed this step to disappear. One millilitre is one cubic centimetre. Not approximately. Exactly, on purpose. And a litre is a thousand of either. So a tap that fills a five hundred millilitre mug in fifteen seconds, asked for a ten litre bucket, is barely a problem.

7:45Ten litres is ten thousand millilitres, which is twenty mugs. Twenty fifteens is three hundred seconds, which is five minutes. The whole difficulty was the litre to millilitre step. A given ratio can do a conversion's job too. Equal volumes of gold and water have masses thirty-seven to two, and a litre of water is a kilogram, so a litre of gold is eighteen and a half. And then there is temperature, on the same list and not the same kind of thing at all.

8:22Water freezes at zero on one scale and thirty-two on the other. It boils at a hundred, and at two hundred and twelve. So take twenty-five degrees on the first scale and read it off the boiling pair as a proportion. Twenty-five times two hundred and twelve, divided by a hundred, is fifty-three. The true answer is seventy-seven. The proportion is wrong by twenty-four degrees, and gave no warning. The two scales disagree about where zero sits, and a single factor cannot carry an offset.

8:59A proportion says double the one and you double the other. Temperature does not behave like that. What is proportional is the first scale against the second measured from freezing. At twenty-five that is forty-five, and twenty-five to forty-five is five to nine. One last habit, the same one in different clothes. One city covers one thousand four hundred and eighty-four square kilometres and holds about thirty million people. Another covers five hundred and fifty square kilometres and holds about twenty million.

9:39Which is more crowded? Not which is bigger. Which is more crowded. Thirty million is the larger total, so on people the first city wins. But thirty million spread over one thousand four hundred and eighty-four is about twenty thousand two hundred people to the square kilometre. And twenty million over five hundred and fifty is about thirty-six thousand four hundred. The second city is nearly twice as crowded, and the totals ranked it the other way.

10:12Divide before you compare. It is the same instruction as convert before you write the proportion, and between them that is almost all of this.