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Roads in town looks like this:

enter image description here

Where $t=45$ means the path takes 45 minutes, and $t=\frac{N}{100}$ means the path takes as many minutes as the number of drivers who chose that path divided by $100$. For example, if $200$ drivers go from home to B, that path takes $\frac{200}{100}=2$ minutes.

Every day, $4000$ drivers leave home, heading for work. Whenever there is a choice between two paths, every driver will choose the shortest path.

What is the average commute time, in minutes, from home to work?

Solution: Initially both the path costs same time (N=0, hence 45 minutes). Now since A is taken by one driver it costs (45+1/100) So next driver will take path B which has still t=45+0/100, therefore he will take that path. So on so forth,

$$ T(A) = 45+45+1/100+45+2/100+...+45+1999/100$$

T(B) will be same as T(A). Therefore solving this summation we get Avg commute time as 54.995 minutes .

What's wrong in this analysis? P.S. this question is form Expii-Travel

Edit: So correct answer is 65, 45+ N/100 i.e. $$T(A)=T(B)=45+2000/100=65$$ The problem is poorly stated and assumed reader to assume somethings. But if asked to do analysis in the above stated manner then would it be correct?

Parth Gor
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    What is $T(A)$? Also, what is the value that you are trying to calculate? – 5xum Mar 16 '23 at 12:28
  • Looks good, what makes you think this is incorrect? – QC_QAOA Mar 16 '23 at 12:40
  • @5xum T(A) is sum of time of every driver passed through point A. We can also take average there like did in edited. We want to calculate the average time taken by drivers. – Parth Gor Mar 16 '23 at 13:37
  • @QC_QAOA the site is giving incorrect answer. Updated thier assumption in edit. – Parth Gor Mar 16 '23 at 13:38
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    I think you and the website are making different assumptions leading to different solutions.

    You are defining $N$ as the number of drivers on the given road at the specific time a new driver is about to go onto the road and you assume that drivers go onto the road one by one.

    The website is either defining $N$ as the eventual (total) number of drivers on the road i.e. $2000$, or is assuming that all the drivers go onto the road at the same time.

     – joseville Mar 16 '23 at 13:54
  • Yes exactly, Assuming drivers go one by one is my solution correct then? – Parth Gor Mar 16 '23 at 14:16
  • Yes, assuming drivers go one by one your way looks good. – QC_QAOA Mar 16 '23 at 15:01

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