Question

Two gears are connected and are rotating simultaneously. The smaller gear has a radius of 2 inches, and the larger gear has a radius of 8 inches.
two circles touching at one point. Larger circle has radius of 8 inches. Smaller circle has radius of 2 inches.

Part 1: What is the angle measure, in degrees and rounded to the nearest tenth, through which the larger gear has rotated when the smaller gear has made one complete rotation?

Part 2: How many rotations will the smaller gear make during one complete rotation of the larger gear?

Answer 1

R - radius of the larger circle
r - radius of the smaller circle

Circumference of the larger circle = $2R\pi=2\times 8\pi=16\pi$
Circumference of the smaller circle = $2r\pi=2\times 2\pi=4\pi$

Part 2:  During one complete rotation of the larger gear both gears pass the same distance $16\pi$.
Therefore, the smaller gear makes $\frac{16\pi}{4\pi}=4$ complete rotations.

Part 1: During one complete rotation of the smaller gear both gears pass the same distance $4\pi$.
Therefore, the larger gear makes $\frac{4\pi}{16\pi}=\frac{1}{4}$ of a complete rotation. 
Since one complete rotation corresponds to 360 degrees, 1/4 of a complete rotation corresponds to 360/4=90 degrees.