Question 1027166
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A battalion 20 miles long advances 20 miles. During this time, a messenger on a horse travels from the rear of the battalion 
to the front and immediately turns around, ending up precisely at the rear of the battalion upon the completion of the 20-mile journey. 
How far has the messenger traveled?
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<pre>
Let "b" be the batalion' speed (in mph), and "m" be the messenger speed (relative the ground, of course).

Let "t" be the time the messenger spent for the journey from the rear of the batalion to its front.

During this time, the front moved forward to the distance b*t, so the messenger covered the distance L + bt, 
where L is the batalion length, which is 20 miles according the condition.

Thus the equation for this part of the journey is 

L + b*t = m*t.   (1)


Now let us consider the second part of the journey: the messenger is moving back from the instant front position 
of the batalion to its rear, while the rear is moving forward towards the messenger.

Let "s" be the time when they "met" each other and the messenger get the current (updated) rear position.

The equation for this part of the journey is 

m*s + b*s = L.   (2)


We are almost at the finish line. Now re-write the equations (1) and (2) in the equivalent form:

m*t - b*t = L,    (1')
m*s + b*s = L.    (2')

Add equations (1') and (2')  (both sides). You will get

m*(t+s) = 2L.     (3)

t+s is the total time of the messenger journey forward and back.
Hence, m*(t+s) is the total distance traveled by the messenger.

The equation (3) says that this distance is 2L = 2*20 = 40 miles.

<U>Answer</U>. Messenger traveled 40 miles.
</pre>

<U>Notice</U>. &nbsp;Interesting that the answer does not depend on how far the batalion had advanced.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In other words, the answer does not depend on the speed of the batalion.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;As well as does not depend on the speed of the messenger (assuming it is higher then the speed of the batalion).