Question

One gear turns $33\frac{1}{3}$ times in a minute. Another gear turns 45 times in a minute. Initially, a mark on each gear is pointing due north. After how many seconds will the two gears next have both their marks pointing due north?

Answer 1

It will take 36 seconds for the gears to point due north again.

Explanation:

One gear turns 33 + 1/3 times in a minute while the other one turns 45 times.

The first gear takes, as a result, 60/ (33+1/3) = 1.8 seconds in turning once while the other one takes 60/45 = 4/3 seconds.

We need to find integers k and j such that 1.8 * k = 4/3 * j. In order to remove the fractions, we can multiply both sides by 15, obtaining the smplified equation

27k = 20j

The number we obtain from the equality should be a multile of both 27 and 20. Both numbers are coprime (they dont have a prime in common), thus the smallest multiple of both of them is their product 27*20 = 540.

In order to satisfy the equality for the smallest values of k and j possible we take k * 20 and j = 27. The amount of seconds passed is

1.8*20 = 4/3*27 = 36 seconds.