‘If 6 cats kill 6 rats in 6 minutes, how many will be needed to kill 100 rats in 50 minutes?’
This is a good example of a phenomenon that often occurs in working problems in double proportion; the answer looks all right at first, but, when we come to test it, we find that, owing to peculiar circumstances in the case, the solution is either impossible or else indefinite, and needing further data. The ‘peculiar circumstance’ here is that fractional cats or rats are excluded from consideration, and in consequence of this the solution is, as we shall see, indefinite.
The solution, by the ordinary rules of Double Proportion, is as follows:—
But when we come to trace the history of this sanguinary scene through all its horrid details, we find that at the end of 48 minutes 96 rats are dead, and that there remain 4 live rats and 2 minutes to kill them in: the question is, can this be done?
Now there are at least four different ways in which the original feat, of 6 cats killing 6 rats in 6 minutes, may be achieved. For the sake of clearness let us tabulate them:—
a. All 6 cats are needed to kill a rat; and this they do in one minute, the other rats standing meekly by, waiting for their turn.
b. 3 cats are needed to kill a rat, and they do it in 2 minutes.
c. 2 cats are needed, and do it in 3 minutes.
d. Each cat kills a rat all by itself, and takes 6 minutes to do it.
In cases a and b it is clear that the 12 cats (who are assumed to come quite fresh from their 48 minutes of slaughter) can finish the affair in the required time; but, in case c, it can only be done by supposing that 2 cats could kill two-thirds of a rat in 2 minutes; and in case d, by supposing that a cat could kill one-third of a rat in 2 minutes. Neither supposition is warranted by the data; nor could the fractional rats (even if endowed with equal vitality) be fairly assigned to the different cats. For my part, if I were a cat in case d, and did not find my claws in good working order, I should certainly prefer to have my one-third-rat cut off from the tail end.
In cases c and d, then, it is clear that we must provide extra cat-power. In case c less than 2 extra cats would be of no use. If 2 were supplied, and if they began killing their 4 rats at the beginning of the time, they would finish them in 12 minutes, and have 36 minutes to spare, during which they might weep, like Alexander, because there were not 12 more rats to kill. In case d, one extra cat would suffice; it would kill its 4 rats in 24 minutes, and have 24 minutes to spare, during which it could have killed another 4. But in neither case could any use be made of the last 2 minutes, except to half-kill rats—a barbarity we need not take into consideration.
To sum up our results. If the 6 cats kill the 6 rats by method a or b, the answer is ‘12;’ if by method c, ‘14;’ if by method d, ‘13.’
This, then, is an instance of a solution made ‘indefinite’ by the circumstances of the case. If an instance of the ‘impossible’ be desired, take the following:—‘If a cat can kill a rat in a minute, how many would be needed to kill it in the thousandth part of a second?’ The mathematical answer, of course, is ‘60,000,’ and no doubt less than this would not suffice; but would 60,000 suffice? I doubt it very much. I fancy that at least 50,000 of the cats would never even see the rat, or have any idea of what was going on.
Or take this:—‘If a cat can kill a rat in a minute, how long would it be killing 60,000 rats?’ Ah, how long, indeed! My privat opinion is, that the rats would kill the cat.
Lewis Carroll.