Ratios, and the gear that does not change one

The ratio of a train is the product of its stages, so the middle gear of a simple three-gear train cancels: 30/20 × 40/30 = 40/20, exactly as if it were not there. What it does change is the direction, since every external mesh reverses — which is the entire reason idlers exist. A compound train, where two gears share a shaft, does not cancel, and that is how large reductions are built without absurdly large wheels.