Suppose 2 boys are walking in the woods and they decide to take a shortcut through a railroad tunnel. They had walked 2/3 of the way through the tunnel, but then something horrible happened: a train was coming in the opposite direction towards the 2 boys, and it was coming close to the other entrance of the tunnel. Each boy ran in a different direction to get out of the tunnel and avoid the incoming train. Each boy ran at the same exact speed of 10 miles per hour, and each boy managed to escape the train at the exact instant in which the train would have hit and killed him. If the train was moving at a constant speed and each boy was capable of instantaneous acceleration, then how fast was the train going?

Take a look at the diagram below to visualize the problem – note that the diagram below is very rough and approximate. Let A and B represent the 2 boys, and you can see that they are both running in opposite directions. You can see that the boys had walked 2/3 of the way through the tunnel, and you can also see the train approaching the other end of the tunnel.

Rough Diagram of the problem

<--- 2/3--->|<-1/3->|   

        <- B A ->               <----TRAIN

----------------------------

At first this may seem like an algebra problem, but as soon as you start using your variables (the x and y's) you will notice that it is not really an algebra problem that deals with rates and the boys' speeds - because you are missing important information like distances and times. Clearly, this problem requires something other than algebra.




Review the relevant information

So, what other information is important here? The fact that the boys barely escape sounds pretty important and relevant to solving the problem. It's also important to note that each boy would have been hit by the train at a different time - since they were running in different directions. This means that B would have been hit by the train at a later time than A since A was running towards the train whereas B was running away from the train.

This means that A barely escapes by the time the train comes right into the tunnel. But, where is B when the train is right about to enter the tunnel? Because A had to pass 1/3 of the tunnel to get out, and because B runs at the same exact rate as A, it must mean that B has also passed another 1/3 of the tunnel by the time the train is right outside the other end of the tunnel. And because B started out 2/3 of the way into the tunnel, this means that B has to pass another 2/3 - 1/3, which equals 1/3 of the tunnel in order to get out. This is what a drawing of this would look like:

<--1/3-->-------------  

         <- B                 <-TRAIN

------------------------- A


Remember that B will get out of the tunnel at the exact moment when the train was about to hit him. This means that the train will travel the entire length of the tunnel, and B will travel just 1/3 of the tunnel - but they both get to the end of the tunnel at the same time. This also means that the train must travel 3 times as fast as B (since B only travels 1/3 of the tunnel to reach the end, but the train travels the entire length of the tunnel to reach the end, but they both reach the end at the same exact time). And, since B travels at a speed of 10 miles per hour, the train moves at 30 miles per hour (3 * 10). So, we have our answer!

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