An idler changes direction but not the end ratio

A fictional ideal external gear train has a twenty-tooth input gear, a thirty-tooth idler and a forty-tooth output gear. The input rotates at 600 revolutions per minute, and the supplied model neglects slip and loss. A learner multiplies all three tooth counts as though the idler created a separate compound shaft ratio. Another follows each mesh and cancels the intermediate tooth count when calculating the end-to-end speed ratio. This written exercise drives no machinery and specifies no allowable load. It asks for output speed and relative direction, while preserving the distinction between a simple idler and two different gears rigidly attached to a common compound shaft.