Mohan, Namit and Pranav travel from Agra to Delhi. They have a two seater bike which can be driven by only Mohan. It is known that due to very stringent traffic rules only two persons can ride at a time. Delhi is 180km from Agra. All of them can walk at 6 kmph, but reach to Delhi simultaneously also they started their journey simultaneously.

And the questions are,

Q1) If the speed of the bike is 36 kmph, then what is the total distance that the bike travels.?

Q2) If the speed of the bike is 42kmph, then what is the shortest possible time in which all three of them can complete the journey?

If I consider that Mohan travels with Pranav for some distance and leaves him there and returns to pick up Namit and meanwhile Pranav starts walking towards the destination like that, I couldn't understand how to solve it further.

Though I tried my best to solve this based on my assumption, I couldn't get the required answer.

Could any one please help me with this problem.


Assume Mohan drives the bike with one of his friends from the start. The walking person will have traversed $x = 6t$ km, where $t$ is measured in hours.

The bike takes off to Delhi, and Mohan turns around to pick up his other friend after he drops off the first friend. He will meet up with the second friend when he is $x$ km short of $360$ km (a round trip). If he's going $36$ km/h then $360 - x = 36t$.

These two equations let you solve for $x$ and $t$, the place and time where the second friend is picked up. From there, it's a single leg the rest of the way at a constant speed to get everyone to their destination.

To solve question 2 you'll need to use $42$ km/h as the bike speed, of course.

  • $\begingroup$ But here all of them reach the destination simultaneously. That's where I couldn't understand how to solve it. $\endgroup$ – Omkar Reddy Sep 2 '16 at 18:08
  • $\begingroup$ @GaneshReddy: If they don't reach the destination simultaneously, you can reduce the time for the last to arrive by changing how long each spends on the bike. You can speed up the last to arrive at the price of delaying the first to arrive. If the criterion is last to arrive, this is a good deal. $\endgroup$ – Ross Millikan Sep 3 '16 at 4:03
  • $\begingroup$ @RossMillikan Could you please elucidate? $\endgroup$ – Omkar Reddy Sep 3 '16 at 4:10
  • 1
    $\begingroup$ Say Namit arrives later than the rest. If you drop Pranav a little earlier, you can pick up Namit a little earlier and get him to the destination faster. As long as Pranav's arrival is not after the new Namit arrival, you have gained. Keep doing this until everybody arrives at once and you have the optimum. Equilibrium is a powerful concept. Once everybody arrives at once, any small perturbation makes things worse. $\endgroup$ – Ross Millikan Sep 3 '16 at 4:30

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