```Question 96220
An army of soldiers is marching down a road at 5 mph. a messenger on horseback rides from the front to the rear and returns immediately, the total time taken being 10 minutes. assuming that the messenger rides at the rate of 10 mph, determine the distance from the front to the rear.
:
To convert mph to ft/min:  5280/60 = 88
:
Speed in ft/min, relative to the troops:
Riding against the troop direction: 88(10+5) = 1320 ft/min
Riding with the troop direction: 88(10-5) = 440 ft/min
:
Let x = length of the troops, from front to rear, in ft:

Write a time equation (in min): Time = distance/speed (in ft/min)
{{{x/1320}}} + {{{x/440}}} = 10
:
Get rid of the denominators, mult equation by 1320
x + 3x = 1320(10)
4x = 13200
x = 13200/4
x =  3300 ft is the length of the troop
:
In miles: 3300/5280 = 5/8 of a mile
:
:
Check our solution, find the time for the trip back, and return trip
3300/1320 = 2.5 min to the rear
3300/440  = 7.5 min return
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total time = 10 min```