Alex and Bert have the same walking speed and the same running speed. They both decide to take a lap around the same track.
Alex walks to a point and then runs such that one half of the distance is spent walking and the other half is spent running.
Bert walks to a point and then runs such that one half of his time is spent walking and the other half is spent running.
Who finishes first?
(In reply to
Puzzle Answer by K Sengupta)
The distance from the point to the endpoint of the given track =D(say)
Alex and Bert's common walking speed = w, say
Alex and Bert's common running speed = r, say
The time taken by Alex to cover the distance D = t(1), say
The time taken by Bert to cover the distance D = t(2), say
Then, we must have:
D/(2w) + D/(2r) = t(1)
=> D{1/(2w) + 1/(2r)} = t(1)
Also,
D = (1/2)*t(2)*w + (1/2)*t(2)*r = t(2)*{(w+r)/2}
=> t(2) = (2D)/(w+r)
So, we have:
t(1) - t(2)
= {D(r+w)/(2wr)} - (2D)/(w+r)
= D{(w+r)^2 -4rw)/(2wr(w+r)}
= D{(w-r)^2)/(2wr(w+r)} .....(*)
The numerator D(w-r)^2 is a multiple of a perfect square so it must be positive. The denominator is obviously positive.
Accordingly, t(1) > t(2), that is: Alex takes more time than Bert to finish the race.
Consequently, Bert will win the race.
NOTE: In the unlikeliest event that the common walking speed and the common running speed is equal, Alex and Bert will finish the race in the same instant.
Explanation to Puzzle Answer
