2 Gears are in their starting position.
Gear 1 has 6 teeth.
Gear 2 has 8 teeth.
As gear 1 turns, it causes gear 2 to turn at a different rate.
Gear 1 is rotated until the two gears are back at starting position.
What is the minimum number of rotations Gear 1 requires to return to it's starting position?
I know the answer is 4 rotations but I don't understand the proper reason why?
least common multiple of 6 and 8 is 24 teeth clashing
that is 4 turns of the 6 and 3 turns of the 8
oooohh ok that makes sense! thanks!!
You are welcome :)
4 because l.c.m. of 6 & 8
impossible boi
To understand why Gear 1 requires a minimum of 4 rotations to return to its starting position, let's analyze the relationship between the number of teeth on each gear and their rotations.
Gear 1 has 6 teeth, which means that for every rotation, it moves one-sixth of a full turn.
Similarly, Gear 2 has 8 teeth, so for every rotation of Gear 1, Gear 2 moves one-eighth of a full turn.
In order for both gears to return to their starting positions, Gear 1 needs to complete a full turn, while Gear 2 needs to complete a full turn as well.
Now, let's determine when Gear 1 and Gear 2 will align back to their starting positions.
To find the least common multiple (LCM) of the number of teeth on both gears, we need to find the smallest number that is divisible by both 6 and 8.
By listing the multiples of 6 and 8, we can determine that the LCM is 24:
Multiples of 6: 6, 12, 18, 24, ...
Multiples of 8: 8, 16, 24, ...
Therefore, the gears will align back to their starting positions after 24 rotations of Gear 1.
However, we are interested in finding the minimum number of rotations for Gear 1 to return to its starting position. Since we know that a full rotation of Gear 1 is 24 steps, we can divide it by the number of teeth on Gear 1, which is 6:
24 (full rotations) ÷ 6 (teeth on Gear 1) = 4 rotations.
Hence, Gear 1 requires a minimum of 4 rotations to return to its starting position.