A person standing on a railway platform $120\text{meters}$ long notices that a train, which passed him in $5\text{seconds}$, passes completely through the station in $15\text{seconds}$. The length of the train is:

Let the length of the train be $L\text{meters}$ and its constant speed be $v\text{m/s}$.
1. When the train passes the stationary person, it covers a distance equal to its own length:

$$v=\frac{L}{5}\text{— (Equation 1)}$$

2. When the train passes completely through the platform station, it covers a total distance equal to the sum of its own length and the platform length ($L+120$):

$$v=\frac{L+120}{15}\text{— (Equation 2)}$$

Equating the two expressions for speed $v$:

$$\frac{L}{5}=\frac{L+120}{15}$$

Cross-multiply to simplify:

$$15L=5(L+120)$$

$$15L=5L+600$$

$$10L=600⟹L=60\text{m}$$