Two meshing gears strike a bargain: the big one turns slower but pushes harder, the small one faster but weaker, and the trade is exactly the ratio of their teeth. It is the oldest way to turn speed into force. Slide the driven gear (or tap) and watch the deal.
A gear pair. The ratio is r = N₂/N₁ (driven teeth over driver teeth). Speed and force trade inversely and exactly: the output turns at RPM_in/r and delivers torque_in·r — a big driven gear means slower rotation, more torque (a low gear), a small one the reverse. Crucially POWER is conserved: RPM × torque out equals RPM × torque in (ignoring friction), because the trade is exact. With the driver fixed at 12 teeth and 100 RPM in, the readout tracks the deal. A fail-loud self-check throws unless output RPM × output torque equals the input power for every ratio.
Ideal rigid gears, no friction or backlash (real drivetrains lose a few % and wear); torque is in arbitrary units. The teeth ratio, the inverse speed/force trade and power conservation are exact.