Let the number of horse troughs = x, the number of cow troughs = y and the number of p*g troughs = z
And we know that x + y + z = 69
Further, we know that there are an equal number of animals of each type which implies that 2x = 3y = 8z
So......2x = 3y implies that y = (2/3)x and 2x = 8z implies that z = (1/4)x
So....we can make the following substitution
x + (2/3)x + (1/4)x = 69
x + (11/12)x = 69
(23/12)x = 69 multiply both sides by 12/23 and we have that
x = 69 (12/23) = (69/23) * 12 = 36 this is the number of horse troughs ...so 2x = the number of horses = 72
And 2x = 3y → 72 = 3y and y = the number of cow troughs = 24...so 3y = 72 cows
And 2x = 8z → 72 = 8z and z = 9 = the number of p*g troughs ......so 8*z = 72 pigs, as well
So.......the total number of animals = 3(72) = 216
