If a bike has a 52-tooth front chainring and a 9-tooth rear sprocket, how many turns of the rear wheel occur for each full turn of the crank?

What this question is really asking

Each full crank turn rotates the front chainring once, pulling the chain forward by a number of teeth equal to the chainring size. That chain motion turns the rear sprocket, and the wheel turns with it.

Using the gear ratio (teeth in, teeth out)

• Front chainring: $52$ teeth
• Rear sprocket: $9$ teeth

One crank revolution moves the chain past $52$ teeth. The rear sprocket needs $9$ teeth of chain movement to turn once, so the number of rear-sprocket (and wheel) revolutions per crank revolution is:

$$\text{wheel turns per crank turn} = \frac{52}{9}$$

Converting to a decimal

$$\frac{52}{9} = 5.777\ldots \approx 5.78$$

So the rear wheel turns about $5.78$ times per full turn of the crank.

Assumption being made

This answer assumes a standard drivetrain where the rear sprocket is fixed to the rear wheel (no internal gear hub changing the ratio).