Imagine I'm walking in a tunnel and I stop, seeing that I am three eights of the way in. The tunnel is divided into eight equal length sections.
All of a sudden, I hear the engine noise of a train behind me. Now, the speed of the train is unknown. To survive, I have to run out of the tunnel as quickly as possible.
I have two choices, running forwards or backwards. If I run backwards in the direction of the train, I will be able to get out of the tunnel right at the moment the train comes in the tunnel. If I run forwards in the way the train is coming, the train will exit the tunnel at the same time as I do.
What is my speed?
Best Answer
All I can figure out is that the train is $4$ times faster than you.
Let's say that you can run with a speed $v$, and the train, $u$. Let's also assume that the tunnel has length $8L$, and that the train is originally a distance $x$ away from the entrance of the tunnel. Then we have $$\frac{3L}{v} = \frac{x}{u}$$ and $$\frac{5L}{v} = \frac{x+8L}{u}$$
We therefore have $$\frac{5}{3} \cdot \frac{x}{u} = \frac{x+8L}{u}$$
Solving this, we get $$\frac{2}{3} x = 8L \Rightarrow x = 12L$$
Plugging it back in, we get $$\frac{3L}{v} = \frac{12L}{u} \Rightarrow \frac{u}{v} = 4$$