document.write( "Question 1144575: I need help with this problem - Two cyclists ride at 8:00 am: one from A to B, the other from B to A. Each of them rode along a constant speed along the same road and when each arrived at the terminal point, immediately turned back. They met each other for the first time at 11:00 am and each of them turned exactly once before they met for the second time. Find the time of the second meeting. \n" ); document.write( "
Algebra.Com's Answer #765729 by ikleyn(52908)\"\" \"About 
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document.write( "The Figure below illustrates the situation:\r\n" );
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document.write( "    A  --->        C             <----  B\r\n" );
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document.write( "You see the starting points A and B  and the meeting point C.\r\n" );
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document.write( "They started simultaneously at 8:00 am and move towards each other.\r\n" );
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document.write( "They met at the point C at 11:00 am spending 3 hours each.\r\n" );
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document.write( "OK and very good.\r\n" );
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document.write( "Now, there is one EXCLUSIVE case, when their moving rates are the same.  In this case, they will get the ending points A and B \r\n" );
document.write( "simultaneously after 3 hours (i.e. at 2:00 pm), each will turn back, and, OBVIOUSLY, will meet next time \r\n" );
document.write( "in 3 hours after 2:00 pm (i.e. at 5:00 pm) exactly at the midpoint between A and B.\r\n" );
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document.write( "    Memorize this case and this answer:  6 hours after 11:00 am, at 5:00 pm.\r\n" );
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document.write( "Now we consider the general case, when their moving rates are different.\r\n" );
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document.write( "    It is a miracle, but we will get THE SAME ANSWER (!)\r\n" );
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document.write( "So, now I consider the general case, when they have different moving rates . . . \r\n" );
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document.write( "Again, I consider their movements from A to B and their meeting at the point C.\r\n" );
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document.write( "Now warp the segment AB into a (or \"the\") circle, by uniting points A and B.\r\n" );
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document.write( "Imagine that it is your ORIGINAL configuration and that cyclists started at the same point A=B at this circle, \r\n" );
document.write( "moving in opposite directions.\r\n" );
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document.write( "You will get an EQUIVALENT model, but now the points move not along the segment AB, but along the circle circumference.\r\n" );
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document.write( "So, they started from one single common point A=B at 8:00 am, move in opposite directions along the circle \r\n" );
document.write( "and met at 11:00 somewhere at the point C on the circle.\r\n" );
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document.write( "Fantastic !  Let's move forward . . . (and further . . . )\r\n" );
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document.write( "They covered the entire distance AB, which we treat now as the circumference of the circle.\r\n" );
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document.write( "So, next they started at the point C on the circle and move in opposite directions.  When they will meet again ?\r\n" );
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document.write( "But of course, the situation after 11:00 am is exactly as it was at 8:00 am: they started from the common point on the circle \r\n" );
document.write( "(now it is point C instead of A=B) and move in opposite directions.\r\n" );
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document.write( "When they will meet each other next time ?\r\n" );
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document.write( "-- But of source, in 3 hours, i.e. at 2:00 pm, when they cover the length of the circumference again.    \r\n" );
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document.write( "Because the situation is REPEATED.\r\n" );
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document.write( "But stop for a minute and be careful (!)\r\n" );
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document.write( "In my consideration, they met each other at 2:00 pm, but it is the meeting on the circle - not in the rectilinear segment AB.\r\n" );
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document.write( "They need to cover the total circumference ONE MORE time in 3 hours - then their next meeting at 5:00 pm will be REAL MEETING\r\n" );
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document.write( "on the segment AB after reflections at the ends (!) (!) (!)\r\n" );
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document.write( "So, the answer is: they will meet each other on the segment AB next time 6 hours after 11:00 am, i.e. at 5:00 pm.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answered and solved.\r
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\n" ); document.write( "\n" ); document.write( "And explained.\r
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\n" ); document.write( "\n" ); document.write( "And completed.\r
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document.write( "    On the circle, they meet each other every 3 hours, by covering (together) the circumference of the circle.\r\n" );
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document.write( "    But on the segment AB, they meet each other (accounting for reflections) every 6 hours.\r\n" );
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