sana.noor wrote:A circular gear with a diameter of 24 centimeters is mounted directly on another circular gear with a diameter of 96 centimeters. Both gears turn on the same axle at their exact centers and each gear has a single notch, at the 12 o'clock position. At the same moment, the gears begin to turn at the same rate, with the larger gear moving clockwise and the smaller gear counterclockwise. How far, in centimeters, will the notch on the larger gear have traveled the second time the notches pass each other?

(A) 32.2 pi
(B)35.6 pi
(C) 38.4 pi
(D) 39.2 pi
(E) 40.8 pi

C

Circumference of the larger gear = 96�.
Circumference of the smaller gear = 24�.

Let L = the notch on the LARGER gear and S = the notch on the SMALLER gear.

We can plug in the answers, which represent the distance traveled by L when the notches meet the SECOND TIME.
Since L and S travel at the SAME RATE, the answer choices also represent the distance traveled by S when the notches meet the second time.

Answer choice C: 38.4�
Since 38.4�cm are traveled the SECOND time L and S meet, the distance traveled by L and S the FIRST TIME they meet = (38.4�)/2 = 19.2�cm.

Of the larger gear's circumference, the portion traveled by L upon completing 19.2�cm = 19.2� / 96� = 1/5.
Thus, L travels 1/5 of the CLOCKWISE distance.

Of the smaller gear's circumference, the portion traveled by S upon completing 19.2�cm = 19.2� / 24� = 4/5.
Thus, S travels 4/5 of the COUNTERCLOCKWISE distance.

Success!
Since L travels 1/5 of the clockwise distance, and S travels 4/5 of the counterclockwise distance, the two notches MEET when each travels 19.2�cm.

The correct answer is C.