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?

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

Step-by-step 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.