document.write( "Question 1027166: 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? \n" ); document.write( "

Algebra.Com's Answer #642440 by ikleyn(52814) $\"\"$ $\"About$

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\n" ); document.write( "A battalion 20 miles long advances 20 miles. During this time, a messenger on a horse travels from the rear of the battalion
\n" ); document.write( "to the front and immediately turns around, ending up precisely at the rear of the battalion upon the completion of the 20-mile journey.
\n" ); document.write( "How far has the messenger traveled?
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document.write( "Let \"b\" be the batalion' speed (in mph), and \"m\" be the messenger speed (relative the ground, of course).\r\n" );
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document.write( "Let \"t\" be the time the messenger spent for the journey from the rear of the batalion to its front.\r\n" );
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document.write( "During this time, the front moved forward to the distance b*t, so the messenger covered the distance L + bt, \r\n" );
document.write( "where L is the batalion length, which is 20 miles according the condition.\r\n" );
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document.write( "Thus the equation for this part of the journey is \r\n" );
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document.write( "L + b*t = m*t.   (1)\r\n" );
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document.write( "Now let us consider the second part of the journey: the messenger is moving back from the instant front position \r\n" );
document.write( "of the batalion to its rear, while the rear is moving forward towards the messenger.\r\n" );
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document.write( "Let \"s\" be the time when they \"met\" each other and the messenger get the current (updated) rear position.\r\n" );
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document.write( "The equation for this part of the journey is \r\n" );
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document.write( "m*s + b*s = L.   (2)\r\n" );
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document.write( "We are almost at the finish line. Now re-write the equations (1) and (2) in the equivalent form:\r\n" );
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document.write( "m*t - b*t = L,    (1')\r\n" );
document.write( "m*s + b*s = L.    (2')\r\n" );
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document.write( "Add equations (1') and (2')  (both sides). You will get\r\n" );
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document.write( "m*(t+s) = 2L.     (3)\r\n" );
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document.write( "t+s is the total time of the messenger journey forward and back.\r\n" );
document.write( "Hence, m*(t+s) is the total distance traveled by the messenger.\r\n" );
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document.write( "The equation (3) says that this distance is 2L = 2*20 = 40 miles.\r\n" );
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document.write( "Answer. Messenger traveled 40 miles.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Notice. Interesting that the answer does not depend on how far the batalion had advanced.\r
\n" ); document.write( "\n" ); document.write( " In other words, the answer does not depend on the speed of the batalion.\r
\n" ); document.write( "\n" ); document.write( " As well as does not depend on the speed of the messenger (assuming it is higher then the speed of the batalion).\r
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