An orchestra of seventy musicians takes seventy minutes to play the Ninth Symphony by Beethoven. How long will it take an orchestra of one hundred and forty musicians to play the same symphony?
Of course, this parody of a mathematical problem is not serious, and if its statement surreptitiously tempts us to answer 140, there is no proportionality that holds in this situation. The symphony lasts seventy minutes and that’s it.
A little more subtly, other problems appear to hide a form of proportionality, but this can be reversed and more difficult to control. The faucet stories are a good example: if one faucet fills a sink in an hour, two identical faucets will fill it in half an hour, three in a third of an hour, four in a quarter of an hour, and so on. following. The multiplication of taps divides the filling time, nothing complicated. Where things get complicated is when the taps are no longer the same. Here is an example.
It takes two hours for a faucet to fill a container with water. Another faucet takes three hours to fill the same sink. How long will it take for both taps to open at the same time to fill it?
And while we add a third touch, you’ll need all your acumen to avoid drowning!
Three taps can be turned on to fill a sink. The first and second combined fill it in three hours. The second and third in four hours. The third and first together last six hours. How long will it take for the pond to fill if all three are open at the same time?
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