A gear ratio is easy to calculate and easy to invert by mistake. GearTrain follows the convention used in the KHK reference: i equals driven teeth divided by driver teeth, which also equals input rotational speed divided by output speed. The names of the shafts matter more than which gear looks larger in a drawing.
Write the fraction in the named direction
Label the input gear as the driver with a teeth count a and the output gear as the driven gear with count b. Then i = b/a. A 12-tooth driver meshing with a 36-tooth driven gear gives 36/12 = 3. Because input speed divided by output speed is also 3, an input of 1,800 RPM gives an ideal output of 600 RPM.
Read reduction and speed increase consistently
When i is greater than one, the output turns more slowly and the pair is a speed reduction. When i is exactly one, the speeds match. When i is below one, the output turns faster. For 40 driver teeth and 20 driven teeth, i = 20/40 = 1/2, so the reciprocal speed factor is 2 and 600 RPM becomes an ideal 1,200 RPM.
One external mesh reverses direction
A simple pair of external gears rotates in opposite directions. This direction result is separate from the size of the ratio: both a reduction pair and a speed-increase pair reverse once. Internal gears, worms, bevel arrangements, crossed axes, chains, and belts have different geometry and are deliberately outside the model. Confirm that the real mechanism matches the assumed external mesh.