How many horses on the merry-go-round?

Question

A boy on a merry-go-round says: "One third of the horses ahead of me, plus three quarters of the horses behind me, is exactly the number of horses on the ride." How many horses are on the merry-go-round?

Accepted Answer

Thirteen horses. Ahead and behind on a circle. On a merry-go-round every other horse is both ahead of the boy (going round one way) and behind him (going round the other way). So "the horses ahead of me" and "the horses behind me" are the same set: all the horses except his own, that is, n minus 1 horses if there are n in total. The equation. (1/3)(n minus 1) + (3/4)(n minus 1) = n. The left side is (1/3 + 3/4)(n minus 1) = (13/12)(n minus 1). So 13(n minus 1) = 12n, giving 13n minus 13 = 12n, so n = 13. Check. Twelve horses other than the boy's: one third is 4, three quarters is 9, and 4 + 9…

How many horses on the merry-go-round?

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