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

To determine when two gears will align again with marks pointing due north, one rotating at 33 1/3 rpm and the other at 45 rpm, we find the Least Common Multiple of their periods in seconds, resulting in 36 seconds.

Step-by-step explanation:

The question asks after how many seconds two gears, one turning at 33 1/3 revolutions per minute (rpm) and the other at 45 rpm, will both have their marks pointing due north again. To solve this, we need to find the Least Common Multiple (LCM) of the two gear's rotations in seconds.

First, we convert the revolutions per minute to revolutions per second by dividing by 60 (since there are 60 seconds in a minute). For the first gear:
33 1/3 rpm = (33 + 1/3)/60 = (100/3)/60 = 5/9 revolutions per second.
For the second gear:
45 rpm = 45/60 = 3/4 revolutions per second.
To find when both gears point north again, we need to find the LCM of their rotation periods in seconds. The first gear takes 9/5 seconds for one revolution (since 1/(5/9) = 9/5), and the second gear takes 4/3 seconds for one revolution.
We now need to determine the LCM of 9/5 and 4/3. The LCM of two fractions is the LCM of their numerators divided by the Greatest Common Divisor (GCD) of their denominators. The LCM of 9 and 4 is 36, and the GCD of 5 and 3 is 1. Thus, the LCM of the periods is 36/1 = 36 seconds.

Both gears will have their marks pointing due north again every 36 seconds.