### Brain Teasers

# Long March

The Madadian army is out on manoeuvre. At the head of the platoon is Major Dumsaw and at the rear of the platoon is Private Dartes. The platoon is 50 metres long.

Private Dartes (the last person in the platoon) wants to give a letter to Major Dumsaw (the first person) leading the platoon. So while the platoon is marching Private Dartes runs ahead, reaches the major and hands over the letter to him and without stopping he runs and comes back to his original position.

In the meantime the whole platoon has moved ahead by 50m.

The question is how much distance did Private Dartes cover in that time, assuming that he ran the whole distance with uniform speed?

Private Dartes (the last person in the platoon) wants to give a letter to Major Dumsaw (the first person) leading the platoon. So while the platoon is marching Private Dartes runs ahead, reaches the major and hands over the letter to him and without stopping he runs and comes back to his original position.

In the meantime the whole platoon has moved ahead by 50m.

The question is how much distance did Private Dartes cover in that time, assuming that he ran the whole distance with uniform speed?

### Answer

Private Dartes covered 120.71 metres.It is given that the platoon and the Private 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 backwards - are equal.

Let's assume that when Dartes reached the major, the platoon moved X metres forward.

Thus, while moving forward Dartes moved (50+X) metres whereas the platoon moved X metres.

Similarly, while moving back Dartes moved [50-(50-X)] X metres whereas the platoon moved (50-X) metres.

Now, as the ratios are equal,

(50+X)/X = X/(50-X)

(50+X)*(50-X) = X*X

Solving, X=35.355 metres

Thus, total distance covered by Private Dartes

= (50+X) + X

= 2*X + 50

= 2*(35.355) + 50

= 120.71 metres

Note that at first glance, one might think that the total distance covered by Private Dartes is 100 metres, as he ran the total length of the platoon (50 metres) twice. TRUE, but that's the relative distance covered by the last person i.e. assuming that the platoon is stationary.

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