Question

The equation for gear ratio

Answer 1

In a gear train with two gears, the gear ratio is defined as follows
$R= \frac{\omega _A}{\omega _B}$ 

where $\omega _A$ is the angular velocity of the input gear while $\omega _B$ is the angular velocity of the output gear. 

This can be rewritten as a function of the number of teeth of the gears. In fact, the angular velocity of a gear is inversely proportional to the radius r of the gear:
$\omega = \frac{v}{r}$
But the radius is proportional to the number of teeth N of the gear. Therefore we can rewrite the gear ratio also as
$R= \frac{\omega _A}{\omega _B} = \frac{r_B}{r_A} = \frac{N_B}{N_A}$