# Trains travelling at different speeds towards a station C

Here's the question:

Consider two stations A and B located 100 kilometer apart. There is a station C, located between A and B. Now trains from station A and B start moving towards station C at different speeds. The train starting from A moves at 40 km/h while the train starting from B moves at 60 km/h. These trains have to stop at station C. A bird also starts from station B and flies at a speed of 80 km/h towards station A. When it reaches the train coming from station A, it switches direction – flying back towards station B and when it reaches the train coming from station B, it turns around again, heading towards the direction of station A. This goes on for some time till both the trains reach station C. What is the total distance covered by the bird in the time it takes for the two trains to reach station C?

This is what I thought:

The distance travelled by the bird should depend on location of station C.

Reason: If Ta and Tb is the time taken by the trains to reach station C respectively, and the bird will have to keep on flying until both trains reach station C, the time for which the bird will have to fly is max(Ta, Tb).

So the answer should be max(Ta, Tb) x 80.

The fact that the answer will depend on location of C can also be verified by taking 2 cases. In the first case, let C be 1km from station A, and in the other one, let C be 1km from station B. You will get different answers.

Can someone please explain where I went wrong?

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I can't fault your reasoning. I would guess station C is supposed to be at the point where the trains meet i.e. it's position is determined by the relative speeds of the trains.

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Actually, I just verified it. The question isn't worded properly. What you said is correct! –  Rushil Apr 20 '12 at 11:19

You are right here, that's the correct solution.

The question is ambiguous regarding the location of station C. It looks like they meant "midpoint" when they said "between"--which gives an answer of $100 \:\rm km$.

Or, they may have meant that its in a position such that both trains reach C at the same time--in that case, the answer is $80 \:\rm{km}$

Were any of these the given answer?

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By the way, this is a nice problem that I ask others sometimes. Half of them create a recursive relation and whatnot :) –  Manishearth Apr 20 '12 at 10:56
The answer is actually 80, the question isn't worded properly. In the question, they're talking about a special case : If the trains move until they meet, the point where they meet is station C. –  Rushil Apr 20 '12 at 11:21
@Rushil: Yep, so they wanted you to interpret it that way. I hate it when questions are of the form "YOU tell me what I mean". :/ –  Manishearth Apr 20 '12 at 15:29
For bonus marks: How many times does the bird reverse its direction of flight? –  DJohnM Jul 22 '13 at 17:54