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

Let the length of the train be $L$ meters and its speed be $v\text{m/s}$.
1. When passing the standing person, the train covers its own length in $5$ seconds:

$$v=\frac{L}{5}⟹L=5v$$

2. When passing through the $120\text{m}$ platform, the train covers a total distance of $(L+120)$ meters in $15$ seconds:

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

3. Substitute $L=5v$ into the second equation:

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

$$15v=5v+120$$

$$10v=120⟹v=12\text{m/s}$$

4. Now, find the length of the train ($L$):

$$L=5\times 12=60\text{m}$$