Advantage
A machine cannot do more work than you put in
Gears are one of the simple machines. They can change the force you have to apply, but they do not invent energy.
Levers, gears, pulleys, wedges, and screws are some examples of machines. OpenStax College Physics 2e Energy is still conserved for these devices, because a machine cannot do more work than the energy put into it. OpenStax College Physics 2e Machines can reduce the input force that is needed to perform the job. OpenStax College Physics 2e
The ratio of output to input force magnitudes for any simple machine is called its mechanical advantage (MA). OpenStax College Physics 2e
A crank is a lever that can be rotated 360° about its pivot. OpenStax College Physics 2e The mechanical advantage for a crank is simply the ratio of the radii ri/r0. OpenStax College Physics 2e Wheels and gears have this same simple expression for their mechanical advantages. OpenStax College Physics 2e
The mechanical advantage can be greater than 1, as it is for the crank, or less than 1. OpenStax College Physics 2e
Teeth and radii
Britannica counts teeth to explain how a pair trades speed for turning force. OpenStax gives the mechanical advantage of wheels and gears as the ratio of the radii, written ri/r0. For ordinary circular gears the two pictures agree, because the number of teeth is proportional to the radius; they are two ways of writing the same trade-off.
The same idea, written by counting teeth, is on the ratios page.