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Puzzle:

A farmer was getting old and so he decided to divide his paddy field between himself and his four sons in proportion to their work rates. He knew that Rahul, Anil, Sunil and Vineel together could plant a field in five hours whereas Anil, Sunil and Vineel and himself together could manage the same task in six hours. So the farmer divided his field into a two-digit square number of parts and kept just one part for himself. Now, Anil, Sunil and Vineel were identical triplets so each received the same whole number of parts. How many parts did each son get?

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Solution:

Ravi received 9 parts and the triplets each got 13 parts. Let the work rates of the triplets combined, Ravi and farmer be x, y, z, respectively. Then

5(x + y) = 6(x + z) and so y = (x + 6z)/5

Since z has one part, we can let x = nz where n is a whole number. We then have y = (6 + n)z/5. The ratio of the work rates of x, y, z is now

n : (6 + n)/5 : 1

Let the total of parts be m, a two-digit square number. We then have n + (6 + n)/5 + 1 =m so that n = (5m - 11)/6. The only solutions are n=19, m=25 or n=39, m=49. Since n is the total number of parts the triplets receive, only the value divisible by three applies (n=39).