In a gear train with two gears, the gear ratio is defined as follows
[tex]R= \frac{\omega _A}{\omega _B} [/tex]
where [tex] \omega _A [/tex] is the angular velocity of the input gear while [tex] \omega _B [/tex] 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:
[tex]\omega = \frac{v}{r}[/tex]
But the radius is proportional to the number of teeth N of the gear. Therefore we can rewrite the gear ratio also as
[tex] R= \frac{\omega _A}{\omega _B} = \frac{r_B}{r_A} = \frac{N_B}{N_A} [/tex]
