When one quantity rises and the other falls by the inverse factor

0:00A motorcycle covers a stretch of road in three hours, travelling at thirty kilometres an hour. The same road is going to be driven at sixty. How long will that take? There is a rule for questions like this. Thirty is to sixty as three is to what? Cross-multiply, and the answer is six hours. Six hours. Twice as fast, and twice as long. That is four times the truth, and in the wrong direction.

0:31The honest answer is an hour and a half. The rule did not misfire - it was the wrong rule, and something has to tell you so before you reach for it. Here is the same road, four different ways. Walking, at five kilometres an hour, takes eighteen hours. A bicycle at fifteen takes six. A motorcycle at thirty takes three. A car at sixty takes an hour and a half.

1:00Look down the speeds: five, fifteen, thirty, sixty. Up they go. And the times: eighteen, six, three, one and a half. Down they come. Something is being kept, because it is the same road every time. The question is what, exactly. Start with the walk and the bicycle. Fifteen is three times five - the bicycle is three times the speed. And the time? Eighteen hours becomes six. That is a third.

1:33Three above, a third below. Now the motorcycle against the bicycle: thirty is twice fifteen, and three hours is half of six. Two above, a half below. And the car against the bicycle: four above, a quarter below. Every multiplier on the speeds sits directly over its own reciprocal on the times. Not a similar number - the exact reciprocal, three times over. That pairing is worth remembering. But it is a symptom, and the cause is one line further down.

2:11Multiply each column. Five times eighteen is ninety. Fifteen times six is ninety. Thirty times three is ninety. Sixty times one and a half is ninety. Four different journeys, and one number underneath all of them. And that number is not a coincidence, because look at its units. Kilometres an hour, times hours. The hours cancel, and what is left is kilometres. Ninety is not a number. It is the road - the distance between the two places, which does not change however you travel.

2:50And a factor times its own reciprocal is one, so of course the product stayed put. Those paired multipliers were this, seen sideways. So write it down. One quantity times the other equals a constant. Call the constant k. That is the definition. Everything else in this topic is that sentence in a different coat. Take any two columns and both products equal k, so they equal each other. That is the form you actually solve with.

3:25Rearrange it and a third form falls out: the ratio of two speeds equals the reverse ratio of their times. Try the walk against the car. Five over sixty; and the other way round, one and a half over eighteen. Both are exactly one twelfth. Not two decimals that nearly agree - one twelfth, exactly, and the decimal is that fraction cut short. Now the trap, and it is worth walking into deliberately.

3:58People remember inverse proportion as: one goes up, the other goes down. That is necessary. It is nowhere near sufficient. Here is a table. Two, four, eight along the top. Twelve, six, five underneath. The top rises every step. The bottom falls every step. It looks exactly like what you were told to look for. Multiply. Two times twelve is twenty-four. Four times six is twenty-four. Eight times five is forty.

4:33Forty is not twenty-four. There is no constant, so there is no k, so there is nothing being conserved - and this is not inverse proportion, however it moves. The second trap is subtler, and it is a trap about checking rather than about meaning. Two tables. The top row is identical in both: forty, eighty, twenty-five, sixteen. The first two columns underneath are identical too: twenty, then ten. Then they part.

5:07One has thirty-two and fifty. The other has twelve and a half, and eight. Products for the first: eight hundred, eight hundred, eight hundred, eight hundred. Inverse. Products for the second: eight hundred, eight hundred - and then three hundred and twelve and a half, and a hundred and twenty-eight. Test one column and you cannot tell them apart. Test two and you still cannot. And a single column, on its own, is both kinds of proportion at once.

5:41Once you look for the conserved product, three problems stop looking different. Twenty workers lay a road in four days. Twenty times four is eighty - eighty worker-days of road. So ten workers take eight days. Two pumps fill a tank in eighteen hours. Two times eighteen is thirty-six - thirty-six pump-hours of tankful. So four pumps take nine. Food for eighty students lasts fifteen days. Eighty times fifteen is twelve hundred - twelve hundred student-days of food. So a hundred students eat for twelve.

6:20Eighty, thirty-six, twelve hundred. Three numbers in three units, doing one job. Name the constant and its unit, and you have understood the problem before solving it. Here is one that looks the same and behaves quite differently. One person chops a pile of vegetables in an hour. Another takes an hour and a half. Together? Not two and a half hours. Two people on one job do not take longer than one of them alone.

6:53Times do not add. Rates do - because what somebody does in an hour is an amount, and amounts pile up. The first does one whole job an hour. The second does two thirds of one. Together, five thirds of a job an hour. That is more than one whole job, which already tells you the answer is under an hour. Invert it: three fifths of an hour. Thirty-six minutes.

7:21And the check is free: it has to beat both of them, not just the slower. An hour and a quarter beats the slower and loses to the faster, so it is no answer at all. With the constant in hand, most of these questions are one line. A tank supplies twenty families for six days - a hundred and twenty family-days - so thirty families get four days. Three people paint a fence in four days - twelve worker-days - so four people take three.

7:55Twenty-five rows of twelve chairs is three hundred chairs, so at twenty to a row you get fifteen rows. Eight periods of forty-five minutes is three hundred and sixty minutes; squeeze that into nine periods and each runs forty. Forty-two machines for sixty-three days is two thousand six hundred and forty-six machine-days, so to finish in fifty-four days you need forty-nine machines. And a car taking two hours at sixty is covering a hundred and twenty kilometres, so at eighty it takes an hour and a half. Which is where we came in.

8:37But not every problem about more and less is inverse, and the look-alikes are the real test. Twenty-four pencils cost a hundred and twenty. What do twenty cost? Fewer pencils, less money. Not inverse - direct, because what is held fixed is the price of one pencil. A hundred. One pump fills two tanks in six hours. Five tanks? More tanks, more hours. Direct again, and fifteen hours. Take six situations: taps against filling time, painters against days on one wall, petrol against distance, cycling speed against time on one route, cloth against its cost, pages against reading time.

9:24Three of the six are inverse - and they are exactly the three holding a product fixed: one tankful, one wall, one route. So here is the whole topic as one question. What is this situation holding fixed? If it holds a quotient fixed - a price per pencil, a speed of reading, litres per kilometre - the quantities are directly proportional. If it holds a product fixed - one road, one wall, one tankful, one store of food - they are inversely proportional.

10:01And if it holds nothing fixed, then it is neither, and no rule of three of any shape is going to help you. One last honesty. These answers rest on assumptions nobody wrote down: that ten workers are as good as twenty, that nobody gets in anybody's way, that the job divides freely. Halving the workforce doubles the days only if all of that holds. The mathematics is exact; what it is a model of is not, and saying which is which is part of the answer.