A <simple gear train> transmits rotation through individual intermediate <gears>, each intermediate gear engaging its neighbors without a separate, differently sized gear rigidly sharing that intermediate rotation. For external meshes on fixed parallel axes, consecutive signed ratios multiply and intermediate tooth counts cancel:
$$
\frac{\omega_{\rm out}}{\omega_{\rm in}}=(-1)^m\frac{N_{\rm in}}{N_{\rm out}},
$$
where $m$ is the number of external meshes. An intermediate <idler gear> changes placement and potentially output direction, but not this ideal endpoint tooth-count ratio.<KHK Gear Nomenclature>[Basic Guide B, §1.1, pp.4–5, fixed-axis external mesh ratios and gear-train connections]{ref}

For example, a chain with tooth counts 18, 30, 42 and 72 has three meshes and
$$
\left(-\frac{18}{30}\right)\left(-\frac{30}{42}\right)\left(-\frac{42}{72}\right)=-\frac14.
$$
At input speed $800$ revolutions per minute, output speed is $200$ revolutions per minute in the opposite direction. Removing one intermediate gear while maintaining a compatible two-mesh layout preserves the fourfold reduction but reverses the direction conclusion.

The cancellation concerns kinematics, not every engineering consequence. Intermediate tooth counts affect physical sizes, axis positions and <angular speeds> of intermediate rotating bodies; additional meshes and supports can change <gear efficiency> and <backlash>. Moreover, if two different intermediate <gears> are rigidly coupled, their tooth counts do not cancel by the same argument: that arrangement is a <compound gear train>.


 Simple gear train (source code)