Gear A has 12 teeth and 12 notches and gear B has 16 teeth and 16 notches. Suppose that before the gears begin to turn, a tooth from gear A is in a notch in gear B. As the gears turn, watch this place where the two gears are in contact. Gear A turns 40 times and, since 12  40 = 480, 480 teeth from gear A pass this contact point. Hence 480 notches from gear B pass this point. But  ⁴⁸⁰/₁₆ = 30 and hence gear B turns 30 times.

Now apply the same argument to the point of contact of gears B and C.