Puzzle no.3 -- how much distance did the last person cover in that time.

Tuesday, December 07, 2010

Question:
There is a 50m long army platoon marching ahead. The last person in the platoon wants
to give a letter to the first person leading the platoon. So while the platoon is marching he
runs ahead, reaches the first person and hands over the letter to him and without stopping
he runs and comes back to his original position.
In the mean time the whole platoon has moved ahead by 50m.
The question is how much distance did the last person cover in that time. Assuming that
he ran the whole distance with uniform speed.

Answer:
The last person covered 120.71 meters.
It is given that the platoon and the last person moved with uniform speed. Also,
they both moved for the identical amount of time. Hence, the ratio of the distance
they covered - while person moving forward and backword - are equal.
Let's assume that when the last person reached the first person, the platoon
moved X meters forward.
Thus, while moving forward the last person moved (50+X) meters whereas the
platoon moved X meters.
Similarly, while moving back the last person moved [50-(50-X)] X meters
whereas the platoon moved (50-X) meters.
Now, as the ratios are equal,
(50+X)/X = X/(50-X)
(50+X)*(50-X) = X*X
Solving, X=35.355 meters
Thus, total distance covered by the last person
= (50+X) + X
= 2*X + 50
= 2*(35.355) + 50
= 120.71 meters
Note that at first glance, one might think that the total distance covered by the
last person is 100 meters, as he ran the total lenght of the platoon (50 meters)
twice. TRUE, but that's the relative distance covered by the last person i.e.
assuming that the platoon is stationary.