Which gear ratio produces the required output speed?

A reduction ratio is a kinematic statement and nothing else: the output shaft turns once for every i turns of the motor. Dividing the motor speed by the ratio gives the output speed, and multiplying the motor torque by the ratio gives the ideal output torque. Everything difficult about gear selection is what that statement leaves out.

The first thing it leaves out is load. A brushed DC motor does not hold its speed. It sits on a torque-speed line that falls from the no-load speed to zero at stall, so the motor speed you divide by the ratio is only correct when the shaft is doing no work. A mechanism that loads the motor to a quarter of its stall torque runs at roughly three quarters of the speed this page reports. The number here is the ceiling, not the operating point.

The second is that published reductions are a discrete set. A mechanism that wants 137:1 gets 120:1 or 160:1, and the choice is an engineering decision rather than a rounding error. Running fast and controlling speed electronically costs a driver and a feedback device. Running slow costs cycle time. Changing the motor winding moves the no-load speed and can bring the required ratio onto a published value, which is often the cheapest fix and the one that gets missed.

The third is that a catalog ratio is usually a rounded label. A spur or planetary train has a ratio set by tooth counts, and the exact value can differ from the printed one by a percent or more. Where a mechanism has to index a fixed number of turns, or synchronize with something else, ask for the exact tooth-count ratio and design to that instead.

The fourth is direction. Each meshing pair of external spur gears reverses rotation, so a spur train's output direction depends on the number of stages. A planetary train with a fixed ring gear keeps the input direction. Getting this wrong is cheap to fix in a drawing and expensive to fix in tooling, so the direction of the specific published reduction is worth confirming before anything is cut.

Finally, stage count is a real cost. Every stage multiplies its own efficiency into the train, adds backlash, adds length and adds noise. A four-stage 1000:1 reduction can lose more than half the input torque. That is why this tool reports the stage count next to the ratio: two reductions that both hit your speed are not equivalent if one of them takes an extra stage to do it.