Question

Many bicycles have 26 inch diameter wheels. The low gear attached to your pedals has a radius of 2.2 inches. The low rear gear, driven by the chain attached to the front gear, has a diameter of 3 inches. How far does the bike travel in one rotation of the pedals, i.e. the front gear revolves once.

Answer 1

The circumference is 'Pi' times diameter.

So one rotation of the pedals will cause the front chain drive to complete one full turn, so it will move the chain the same distance as it's circumference.

So it will move the chain  (2.2 Pi )inches. Then the back drive has a circumference of  3pi inches, but it only turns 2.2pi inches. So it turns 2.2pi/3pi = 11/15 of a full rotation. Since the wheel turns at the same rate as the back chain drive, the wheel will turn  11/15 of a full turn as well.

And since the back wheel is 26 inches in diameter, it's circumference is 26pi inches. If it turned a full rotation, it would go this distance along the ground, but it is only going  11/15 of this distance, so it is going (11/15)x 26pi =(286/15)pi inches. Or approximately 59.8997 inches.