Let's buy x cocks, y hens and (100-X-Y) chickens.

5X+3Y+ 1/3( 100-X-Y)= 100

Y=25-7/4X

Because X>0, y>0

So X>0 25-7/4x > 0

And because both X and Y are positive integers, X with Y=25-7/4X must be a multiple of 4, so X can only be 4,8, 12.

When x=4, y= 18, 100-x-y=78.

When x=8, y = 1 1, 100-x-y = 8 1.

When x= 12, y=4, 100-x-y=84.

There are three solutions, including the right.