document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=184982>Question 184982</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bus with 12 soccer players broke down in a town 20 miles from its destination. The coach’s car was available but could carry only 4 players at a time. Also, it could travel only 20 miles per hour because of the traffic. The players said they could walk at 4 miles per hour when they were not riding. Suppose the coach took 4 of them part way, came back for 4 more and took them part way, and then came back for the last 4. How could they all get to their scheduled soccer game at the same time?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #138851 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=138851\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font size = 4 color = \"indigo\"><b>\r\n" );
document.write( "We make this diagram, where the broken-down bus is at A and the\r\n" );
document.write( "destination is at F.  B,C,D, and E are points along the way\r\n" );
document.write( "with unknown distances v,w,x,y,z miles between them. \r\n" );
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document.write( "|  v miles  |  w miles  | x miles  |  y miles  |  z miles  |      \r\n" );
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document.write( "A           B           C          D           E           F\r\n" );
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document.write( "We of course have the first equation as\r\n" );
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document.write( "             v + w + x + y + z = 20\r\n" );
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document.write( "Here are the 5 trips the car makes.\r\n" );
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document.write( "1. Car drives from A to D. (1st 4 get out and walk from D to F.)\r\n" );
document.write( "2. Car drives from D to B. (picks up 2nd 4, who have walked from A to B.) \r\n" );
document.write( "3. Car drives from B to E. (2nd 4 get out and walk from E to F.) \r\n" );
document.write( "4. Car drives from E to C. (picks up 3rd 4, who have walked from A to C.) \r\n" );
document.write( "5. Car drives from C to F.  \r\n" );
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document.write( "Now we'll fill in the details:\r\n" );
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document.write( "1. Car takes 1st 4 from A to D, a distance of v+w+x miles.\r\n" );
document.write( "   When car gets to D, it has traveled v+w+x miles and therefore\r\n" );
document.write( "   (v+w+x)/20 hours have passed, using TIME = DISTANCE/RATE. \r\n" );
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document.write( "   [Then the 1st 4 have y+z miles yet to walk to F, which will take them\r\n" );
document.write( "    (y+z)/4 hours.  So total time = (v+w+x)/20 + (y+z)/4 hours, or\r\n" );
document.write( "    simplifying, total time = (v+w+x+5y+5z)/20 hours.]  \r\n" );
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document.write( "2. Car drives from D to B, a distance of x+w miles, which takes \r\n" );
document.write( "   (x+w)/20 hours more. When car gets to B, (v+w+x)/20 + (x+w)/20,\r\n" );
document.write( "   or (v+2w+2x)/20 hours have passed. During this time the 2nd and\r\n" );
document.write( "   3rd 4 have walked from A to B and using Distance=Rate·Time, this \r\n" );
document.write( "   is 4(v+2w+2x)/20 or (v+2w+2x)/5 miles.  Since A to B is also v, \r\n" );
document.write( "   we have the equation v = (v+2w+2x)/5 which when simplified becomes\r\n" );
document.write( "   the equation 5v = v+2w+2x or 4v = 2w+2x or 2v = w+x \r\n" );
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document.write( "3. Car drives from B to E, a distance of w+x+y miles, which takes\r\n" );
document.write( "   (w+x+y)/20 miles.  When car gets to E, (v+2w+2x)/20 + (w+x+y)/20,\r\n" );
document.write( "   or (v+3w+3x+y)/20 hours have passed. \r\n" );
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document.write( "   [Then the 2nd 4 have z miles yet to walk from E to F, which will \r\n" );
document.write( "    take them z/4 hours.  So total time = (v+3w+3x+y)/20 + z/4 hours,\r\n" );
document.write( "    or simplifying, total time = (v+3w+3x+y+5z)/20 hours.] \r\n" );
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document.write( "    Now we have two expressions for the total time, so we can\r\n" );
document.write( "    set them equal\r\n" );
document.write( "    (v+w+x+5y+5z)/20 = (v+3w+3x+y+5z)/20\r\n" );
document.write( "         v+w+x+5y+5z = v+3w+3x+y+5z\r\n" );
document.write( "                  4y = 2w+2x\r\n" );
document.write( "                  2y = w+x\r\n" );
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document.write( "4. Car drives from E to C, a distance of y+x miles, which takes \r\n" );
document.write( "   (y+x)/20 hours more. When car gets to B, (v+3w+3x+y)/20 + (y+x)/20,\r\n" );
document.write( "   or (v+3w+4x+2y)/20 hours have passed. During this time the 3rd 4 \r\n" );
document.write( "   have walked from A to C and using DISTANCE=RATExTIME, this \r\n" );
document.write( "   is 4(v+3w+4x+2y)/20 or (v+3w+4x+2y)/5 miles.  Since A to C is also v+w, \r\n" );
document.write( "   we have the equation v+w = (v+3w+4x+2y)/5 which when simplified becomes\r\n" );
document.write( "   the equation 5v+5w = v+3w+4x+2y or 4v+2w=4x+2y or 2v+w = 2x+y.\r\n" );
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document.write( "5. Car drives from C to F, the destination, a distance of x+y+z miles, \r\n" );
document.write( "   which takes (x+y+z)/20 hours more. So the total time that has passed\r\n" );
document.write( "   is (v+3w+4x+2y)/20 + (x+y+z)/20 or (v+3w+5x+3y+z)/20.  So this is a\r\n" );
document.write( "   third expression for the total time, and we set it equal to one of the\r\n" );
document.write( "   above expressions for the total time, and get\r\n" );
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document.write( "   (v+3w+5x+3y+z)/20 = (v+w+x+5y+5z)/20\r\n" );
document.write( "        v+3w+5x+3y+z = v+w+x+5y+5z   \r\n" );
document.write( "               2w+4x = 2y+4z\r\n" );
document.write( "                 w+x = y+z\r\n" );
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document.write( "So the equations we have are\r\n" );
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document.write( "v+w+x+y+z=20    \r\n" );
document.write( "2v = w+x\r\n" );
document.write( "2y = w+x\r\n" );
document.write( "2v+w = 2x+y \r\n" );
document.write( "w+x = y+z\r\n" );
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document.write( "To solve it by matrices, we write it:\r\n" );
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document.write( " v + w +  x +  y + z = 20\r\n" );
document.write( "2v - w -  x          =  0\r\n" );
document.write( "   - w -  x + 2y     =  0\r\n" );
document.write( "2v + w - 2x -  y     =  0\r\n" );
document.write( "     w +  x -  y - z =  0\r\n" );
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document.write( "So we get v=4, w=4, x=4, y=4, z=4\r\n" );
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document.write( "So    \r\n" );
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document.write( "|  4 miles  |  4 miles  | 4 miles  |  4 miles  |  4 miles  |      \r\n" );
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document.write( "A           B           C          D           E           F\r\n" );
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document.write( "1. Car drives from A to D. That's 12 miles. \r\n" );
document.write( "2. Car drives from D to B. That's 8 miles.  \r\n" );
document.write( "3. Car drives from B to E. That's 12 miles \r\n" );
document.write( "4. Car drives from E to C. That's 8 miles. \r\n" );
document.write( "5. Car drives from C to F. That's 12 miles.\r\n" );
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document.write( "So the car went a total of 12+8+12+8+12 or 52 miles.\r\n" );
document.write( "At 20 miles per hour, that took the car 52/20 or 13/5 hours or 2 3/5 hours\r\n" );
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document.write( "The 1st 4:\r\n" );
document.write( "Rode 12 miles from A to D, which took them 12/20 or 3/5 hours,\r\n" );
document.write( "Then walked 8 miles from D to F, which took them 8/4 or 2 hours.\r\n" );
document.write( "So it took the 1st 4 also 2 3/5 hours to get from A to F     \r\n" );
document.write( "    \r\n" );
document.write( "The 2nd 4:\r\n" );
document.write( "Walked 4 miles from A to B, which took them 4/4 or 1 hour\r\n" );
document.write( "Rode 12 miles from B to E, which took them 12/20 or 3/5 hour.\r\n" );
document.write( "Walked 4 miles from E to F, which took them 4/4 or 1 hour.\r\n" );
document.write( "So it took the 2nd 4 also 2 3/5 hours to get from A to F\r\n" );
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document.write( "The 3rd 4:\r\n" );
document.write( "Walked 8 miles from A to C, which took them 8/4 or 2 hours\r\n" );
document.write( "Rode 12 miles from C to F, which took them 12/20 or 3/5 hours\r\n" );
document.write( "So it took the 3rd 4 also 2 3/5 hours to get from A to F\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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