“Old MacDonald had a farm, E-I-E-I-O”, says the old children’s song. But Old MacDonald did have a farm! And on that farm he had some horses, cows, pigs and 69 water troughs for the animals to drink from. Only horses drank from the horse troughs, exactly two horses for each trough. Only cows drank from cow troughs, exactly three cows per trough. And only pigs drank from p*g troughs, exactly eight pigs per trough. Old MacDonald’s farm has the same number of cows, horses and pigs. How many animals does Old MacDonald have on his farm?
+129840 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
