## 15. Metro in Mexico City

A worker is walking along the tracks in one of the tunnels of the capital's subway, on a section between two stations that are far apart. Every 5 minutes a train passes him coming from the opposite direction, and every 6 minutes another train overtakes him. The worker walks at a constant speed, the speeds of the subway trains (which are the same for trains going in both directions) do not change, nor do the intervals between trains (which are also the same for trains going in both directions).

When the worker stopped to fix a malfunction that had forced him to set out on his journey, a train passed by. How long will it take for the next train going in the same direction to pass by?
15. Let $x$ be the time between the passage of two consecutive trains moving in the same direction; $I$ be the distance between two consecutive trains; $V$ be the speed of each train, and finally, $v$ be the speed of the worker. Then, relative to the worker, each following train moves at a speed of $V-v$, and each oncoming train moves at a speed of $V+v$.

From this, we obtain the following system of equations:

1) $5=I / (V+v)$
2) $6=I / (V-v)$,
3) $x=I / V$.

It is easy to see that [due to (1)-(3)] $x=60 / 11=5$ min $27 \mathrm{s}$.