Two gears are connected and are rotating simultaneously. the smaller gear has a radius of 4 inches, and the larger gear has a radius of 7 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?

Let r represent the radius of the smaller circle and R the radius of the larger circle.
Apply ratios: the radius of smaller circle to radius of larger circle, i.e.

r: R = 3 : 7.

I complete rotation = 360 degrees.

Part 1:

For one complete rotation of the smaller circle, the larger circle is rotated through: (3/7)*(360) = 154.3 degrees

Part 2:
For one complete rotation of the larger circle, the larger circle is rotated through: (7/3)*(360) = 840.0 degrees

This is equivalent to (840/360) = 2.3 rotations

Alternatively, use the ratios: 
The number of rotations = R/r = 7/3 = 2.3 rotations