Puzzle on Time Speed and DIstance - Mr. Wagle goes to work by a bus

Question:  Mr. Wagle goes to work by a bus. One day he falls asleep when the bus still has twice as far to go as it has already gone.

Halfway through the trip he wakes up as the bus bounces over some bad potholes. When he finally falls asleep again, the bus still has half the distance to go that it has already traveled. Fortunately, Mr. Wagle wakes up at the end of his trip.

What portion of the total trip did Mr. Wagle sleep?


Answer:
Mr. Wagle slept through half his trip.

Solution:
Let's draw a timeline. Picture the bus route on a line showen below:


    ---------------- ________ -------- ________________



  Start            1/3      1/2      2/3             End



--------- shows time for which Mr. Wagle was not sleeping

_____ shows time for which Mr. Wagle was sleeping



When Mr. Wagle fell asleep the first time, the bus sill had twice as far to go as it had already gone, that marks the first third of his trip.

He wake up halfway through the trip i.e slept from 1/3 mark to the 1/2 mark. He fell sleep again when the bus still had half the distance to go that it had already traveled i.e 2/3 mark.

Adding up, all sleeping times,
= (1/2 - 1/3) + (1 - 2/3)
= 1/6 + 1/3
= 1/2

Hence, Mr. Wagle slept through half his trip.