How this is calculated. The ratio is simply the tooth counts,
i = Ndriven / Ndriver — diameters do not enter into it,
because for gears to mesh they must share a module, so diameter is proportional to tooth count
anyway. Speed divides by the ratio and torque multiplies by it, less whatever the mesh loses:
a well-made spur mesh is around 97–99 % efficient per stage, so a three-stage box is doing well
to reach 92 %.
Power is what you do not get for free. Pitch diameter is
d = m·N and centre distance is the average of the two, which is why changing a
ratio usually means changing the housing. One thing worth checking: if the tooth counts share a
common factor, the same teeth meet each other on every revolution and wear concentrates on
those pairs. Coprime counts — a "hunting" pair — spread it evenly.
These tooth counts share a common factor, so the same pairs of teeth mesh repeatedly. Adding or removing one tooth to make the counts coprime spreads wear across every tooth.