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# Solution

Okay, let's suppose Anil had 'C' number of cats and 'D' number of dogs initially. The total number of cats and dogs is 55, therefore C + D = 55.

Exactly one-fifth of the cats and one-fourth of the dogs were adopted. Therefore the number of cats adopted is C/5 and the number of dogs adopted is D/4. The total number of cats and dogs adopted is C/5 + D/4 = (4C + 5D)/20.

It is given that exactly one-fifth of the cats were adopted. Hence the total number of cats has to be 5 or multiples of 5, and not exceeding 55. For example, if there were 23 cats, then one-fifth that number would be 4.6; you cannot adopt 4.6 cats! The number of cats cannot exceed 55 because the total number of cats and dogs is 55 and it is quite possible that there were no dogs initially; all were cats.

Therefore, the number of cats can be 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 or 55.

Therefore, if we consider there were 5 cats then the number of dogs is 50 (because C + D = 55). So, the (cat, dog) numbers can be (5, 50), (10, 45), (15, 40), (20, 35), (25, 30), (30, 25), (35, 20), (40, 15), (45, 10), (50,5), (55, 0). But, the number of dogs is greater than the number of cats; so, only the first five pairs are possible: (5, 50), (10, 45), (15, 40), (20, 35), (25, 30).

Similarly, exactly one-fourth of the dogs were adopted. Therefore the total number of dogs has to be 4 or multiples of 4, and not exceeding 55. Therefore, the number of dogs could be 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48 or 52. So, in this case, the (cat, dog) numbers can be: (51, 4), (47, 8), (43, 12), (39, 16), (35, 20), (31, 24), (27, 28), (23, 32), (19, 36), (15, 40), (11, 44), (7, 48), (3, 52). The number of dogs is greater than the number of cats, so only the last seven pairs are feasible: (27, 28), (23, 32), (19, 36), (15, 40), (11, 44), (7, 48), (3, 52).

Only the (cat, dog) pair of (15, 40) is common between the two sequence of pairs. Thus, only (cat, dog) pair of (15, 40) satisfies the condition.

There were, thus, 15 cats initially and one-fifth, or 3, of them were adopted. There were 40 dogs initially and one-fourth, or 10, of them were adopted.

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- Union Public Service Commission - www.upsc.gov.in
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