How many trains are there on the Paris Underground

brain teaser guess-estimates
Recent activity on Oct 01, 2017
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Anonymous A asked on Oct 01, 2017

Can someone help to answer this type of question:

That's how I would approach this problem:

1. #trains = #line multiply by #train per line

2. #train per line= frequency of train (how long I wait at a given station) multiply by average time between two station

Assumptions:

- Let's say that during peak hours I have to wait maximum 3 minutes at a given station to have a train and that the length of time between two station is 1min30

- there is 14 lines in Paris and on average 21 stations per line

________

if I must wait 3 minutes to get a train it means that there is one train every three stations at a given time (so 21/3= 7 trains per line)

As there is two ways in a line we must double it to 14 trains per line

Finally there is 14 lines so we must multiply 14 by 14 which gives me 196 trains

By doing some research I found out that this number is relatively low (approximately 500 trains during peak hours)

How would you approach this problem?

Thanks for your help

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Mark replied on Oct 01, 2017

Hey,

I am new to this too, but I think I have some ideas for your problem.

Your approach looks good, but I think you have a mistake in how you time the trains. You say that there is one train every three stations and that it takes 1m30s to travel the distance from one station to the next. But how long does the train spend at each station?

My approach would be to represent each line as a timeline (the time it would take one train to go from one side to the other). All the trains on the line in a given direction should then be distributed evenly along this timeline in increments of the "average time between trains" + "waiting time at the station" as you already did.

Assumptions:

  • 1m 30s travel time between stations
  • 1m 00s waiting time per station
  • 3m 00s between trains
  • 21 Stops per line
  • 14 lines

Based on these assumptions we take a total of 2m30s per stop. Now I would also assume that the trains changing direction at the end of the line take the equivalent of one stop to do so, so let's change the average amount of stops per line to 22.

Now the length of our timeline can be calculated as 22stops * 2m30s = 55min.

If there is a train every 3 min we can round the number of trains per leg to 14 (55min/(3min+1min)).

So the total number of trains would now be 14 trains per leg * 2 legs * 14 lines = 392 trains.

Obviously, the assumptions make a major difference, so changing the time between trains to 2min will lead to 504 trains at peak time.

Hope that helps!

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