Let r2 = rate of the messanger.
d = distance covered by messanger.
t = time to move one mile = 1 hour.
The meeting point of the messanger and the person at the rear is a distance of r2/(1+r2) from the starting point of the messanger. This is just the ratio of the messanger's rate divided by the sum of the two rates.
The messanger will have to cover the distance to the meeting point, turn around and cover it again, and then march one more mile. So...
d = 2*r2/(1+r2).
distance = rate * time, so...
2*r2/(1+r2) = r2*1
2*r2/(1+r2) = r2 -1
2*r2 = r22 -1
r22 - 2*r2 - 1 = 0
Use the quadratic equation to solve for r2...
r2 = (2 + 81/2)/2 = 1+21/2 =~ 2.4142 miles/hr
Since he is marching for one hour is distance is 2.4142 miles.
I would like to thank Michael Whitfield for suggesting this problem.
Michael Shackleford, A.S.A.