Question

A gear 50 inches in diameter turns a smaller gear 30 inches in diameter. If the larger gear makes 15 revolutions, how many revolutions does the smaller gear make in that time?
(A) 9
(B) 12
(C) 20
(D) 25
(E) 30

Answer 1

All right, I think it may work to consider that the large wheel's diameter is ~66% longer than the smaller gear, 50 = 30 + (2/3)*30

Answer 2

it takes longer for the the larger gear to finish a revolution because circumference is \[\pi*d=circumference\] and if you were imagine a point on the edge of the circle, in order to complete a revolution means that the gear has made a complete turn, and therefore that the point we imagined moved all the way around the circle, therefore moving a total distance equal to the circumference

Answer 3

so you are basically comparing the sum of lengths traversed by revolutions, or still imagine the point on the large gear. if it were to have completed 15 revolutions, that means it traveled a distane of \[15 * \pi*d\]
we know d is 50in.

\[15*50*\pi\] is total distance traveled by the large gear. now we can determine the number of revolutions the smaller gear completes by \[15*50*pi=(NumOfRevolutions)*pi*30\]

Now, solve for the (NumOfRevolutions)