# A train question

A commuter train starts at rest from station A and travels toward station B at 08:45 AM. It travels at 200km/h for a third of the journey, before accelerating to 600km/h for a third of the journey, before slowing to 300km/h for a third of the journey. A second train sets off from station B to station A at the same time going the opposite way on the same route. It travels at 300km/h for the first half of the journey, then 400km/h for the second half.

My train arrives first.

What train am I on?

• Nope, but good idea. – David Apr 13 '18 at 14:02
• Does the phrase "a third of the journey" refer to a third of the total time, or a third of the total distance? – Riley Apr 13 '18 at 14:02
• I was thinking distance, but i'm not sure if it changes the outcome. – David Apr 13 '18 at 14:03
• I am indeed on one of these trains, though you are correct in assuming the answer is not as straightforwardly mathematical it appears. – David Apr 13 '18 at 14:05
• these are REALLY FAST commuter trains though – Flying_whale Apr 13 '18 at 14:20

are you on

The maglev - the fastest commuter train in the world? I mean, 600km/h, even a bullet train can't do that

• Well done :) I am indeed. – David Apr 13 '18 at 14:35
• @David Reading kraby15's answer, I can't agree with this, how can you be on that train if you stated on kraby15's answer that you were on the other? – Saeïdryl Apr 13 '18 at 14:46
• I don't think I can do spoilers in comments so suffice it to say, it doesn't matter – David Apr 13 '18 at 14:55
• @David That's too confusing, you should have checked your values to be on the right train, or be on the second to arrive... – Saeïdryl Apr 13 '18 at 15:09
• @Saeïdryl [Warning, spoilers ahead]: no, it really doesnt matter. the SCMaglev is the line type, both trains are the same. – David Apr 13 '18 at 17:24

You are on

Train 2

Given that any distance between station A and B would be acceptable, I set the distance between the two stations as 3000 km.

For the first train, it's journey is broken into three parts, each being 1000 km long. It takes 5hrs for the first part, 1hr and 40min for the second part, and 3hrs and 20min for the third part. If you add those together, the total time ends up being 10hrs.

For the second train,

it is split into two parts - each of which being 1500 km in distance. For the first half, it takes the train 5hrs. For the second half, it takes the train 3hrs and 45min. The total amounts to 8hrs and 45 min.

In conclusion

Train 1 takes 10hrs and train 2 takes 8hrs and 45min.

• Correct, and yet, not the answer I am looking for. – David Apr 13 '18 at 14:21
• What does that mean? – kraby15 Apr 13 '18 at 14:23
• that's part of the puzzle, i'll add a clue in a bit if no one spots it – David Apr 13 '18 at 14:25
• I would advise adding a clue in a few hours, after more people see it. Not only 30 min after posting the original question. – kraby15 Apr 13 '18 at 14:28

You can be

on both trains, depending on how you define 'third' and 'half'.

If you

take it to mean $1/3$ and $1/2$ of the duration of the journey, you can just take the average of the speeds. For the A->B train, this is $1100/3 = 366.66...$ km/h; for the B->A train, it's $700/2 = 350$ km/h. That means that (if the trains take the same route in opposite directions, which I assume, otherwise it's rather pointless) you travel from A to B.

On the other hand,

if $1/3$ and $1/2$ refer to the distance of the journey, the average speed of the trains is the harmonic mean of the parts, so for the A->B train it's $\frac{3}{200^{-1} + 600^{-1} + 300^{-1}} = 300$ km/h, and for the B->A train it's $\frac{2}{300^{-1} + 400^{-1}} = 342.857...$ km/h. So in that case you're on the B->A train.

Alternatively, since it's now tagged :

"at the same time" might refer to the time the train from A->B arrived at B. In that case, you're certainly on the A->B train.

• Correct, and yet, not the answer I am looking for. – David Apr 13 '18 at 14:21
• @David see my update, maybe this is what you were looking for? – Glorfindel Apr 13 '18 at 14:23
• nope. sorry! i'll give a clue in a bit if no one spots it – David Apr 13 '18 at 14:25

The train you are on is

indeterminate

because

the math (as documented by the others) shows that train 2 arrives first.  This is “your” train; you own it.  But we have no information indicating which train you were on.

Clearly this would be an unacceptable answer if it weren’t for the tag.