Weedy Willie was getting too old to work the land alone so he decided to divide his cornfield between himself and his four sons in proportion to their five work rates.
He knew that Rastus, Wig, Twig and Swig together could plant a field of corn in five hours whereas Wig, Twig and Swig and himself together could manage the same task in six hours.
So Weedy divided his field into a two-digit square number of parts and kept just one part for himself. Now, Wig, Twig and Swig were identical triplets so each received the same whole number of parts.
How many parts did each son get?
__________
Rastus received 9 parts and the triplets each got 13 parts. Let the workrates of the triplets combined, Rastus and Willie 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).