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2014/15 AHSMC Part 1 Problem 16

Problem 16 of 16

Four cages are arranged in a 2×22 \times 2 formation. Each cage contains some chickens and some rabbits (some cages may contain only chickens or only rabbits). The total number of heads in the two cages in the first row is 6060. The total number of legs in the two cages in the second row is 240240. The total number of heads in the two cages in the first column is 7070. The total number of legs in the two cages in the second column is 230230. The minimum number of animals in all four cages is

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Solution

There are 230230 legs in the two cages in the second column. Since 230=4×57+2230 = 4 \times 57 + 2, they may come from as few as 5858 animals, namely, 5757 rabbits and 11 chicken. Hence the minimum number of animals in all four cages is 70+58=12870+58=128. If we have 5555 animals in the first cage in the first row, 00 chickens and 55 rabbits in the second cage in the first row, 1515 chickens and 00 rabbits in the first cage in the second row, and 11 chicken and 5252 rabbits in the second cage in the second row, we have 128128 animals overall. The answer is (B).