document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=52337>Question 52337</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi There! I am studying rate questions for the GMAT, and feel pretty comfortable on the basic forms such questions can take. I am completely stuck, however, on rate questions that pose hypothetical questions, like \"If the train went 4 mph faster, it would have arrived in 1/2 the time.\" The question below is taken from an online course, and the answer is also provided. I just don't see how they figured it out.\r<BR>\n" );
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document.write( "The Question:\r<BR>\n" );
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document.write( "A train traveled 960 miles from Town A to Town B. If the train had traveled from Town A to Town B at an average speed of 12 miles per hour less, it would have taken 4 more hours.\r<BR>\n" );
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document.write( "What was the train's average speed during that journey?\r<BR>\n" );
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document.write( "The Author's Answer:\r<BR>\n" );
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document.write( "We need an average speed to have sufficiency. Let's call the train's average speed s miles per hour and let's call the time that it took the train to travel the 960 miles t hours. The distance formula says that Distance = Rate * Time. So here, st = 960.\r<BR>\n" );
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document.write( "We learn that (s – 12)(t + 4) = 960. Including the equation st = 960 from the question, we now have two equations for two variables, and we could solve. In the solution, we can substitute (960/s) for t in the equation (s – 12)(t + 4) = 960. Then (s – 12)((960/s) + 4) = 960. When this equation is solved for s, there will be two solutions, one positive and one negative. (The solution s = 60 corresponds to t = 16; the other case s = 48 which would correspond to t = 20 but would be discarded because its negative.) So there is just one answer to the question, which is that the average speed was 60 miles per hour.\r<BR>\n" );
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document.write( "Whoa! \"When this equation is solved for s..?\" How about showing how to solve for it? I follow everything up until this point. After that, I'm lost. Two different solutions? What, is this a quadratic equation? Help!\r<BR>\n" );
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document.write( "Thank you so much for your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #34869 by </B><A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><B>AnlytcPhil(1807)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=AnlytcPhil><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=AnlytcPhil><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=34869\">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 = 5><b> \r\n" );
document.write( "The Question: \r\n" );
document.write( "A train traveled 960 miles from Town A to Town B.\r\n" );
document.write( "If the train had traveled from Town A to Town B \r\n" );
document.write( "at an average speed of 12 miles per hour less, \r\n" );
document.write( "it would have taken 4 more hours. \r\n" );
document.write( "What was the train's average speed during that \r\n" );
document.write( "journey? \r\n" );
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document.write( "The explanations in some of those online courses are \r\n" );
document.write( "not so good. Here's the way I would solve it:\r\n" );
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document.write( "Make this chart:\r\n" );
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document.write( "                    DISTANCE   RATE       TIME\r\n" );
document.write( "  ACTUAL TRIP                                   \r\n" );
document.write( "HYPOTHETICAL TRIP                                  \r\n" );
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document.write( "Let the speed on the actual trip be x.  Fill that in \r\n" );
document.write( "for the RATE of the ACTUAL TRIP.\r\n" );
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document.write( "                    DISTANCE   RATE       TIME\r\n" );
document.write( "  ACTUAL TRIP                   x              \r\n" );
document.write( "HYPOTHETICAL TRIP                               \r\n" );
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document.write( ">>...If the train had traveled from Town A to Town B \r\n" );
document.write( "at an average speed of 12 miles per hour less...<<\r\n" );
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document.write( "So we subtract 12 from x, getting x-12 and fill that \r\n" );
document.write( "in for the RATE for the HYPOTHETICAL TRIP.\r\n" );
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document.write( "                    DISTANCE   RATE       TIME\r\n" );
document.write( "  ACTUAL TRIP                   x              \r\n" );
document.write( "HYPOTHETICAL TRIP              x-12                \r\n" );
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document.write( "We are told that the DISTANCE for both the ACTUAL \r\n" );
document.write( "and the HYPOTHETICAL trip is 960 miles, so we fill \r\n" );
document.write( "that in:\r\n" );
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document.write( "                    DISTANCE   RATE       TIME\r\n" );
document.write( "  ACTUAL TRIP          960      x              \r\n" );
document.write( "HYPOTHETICAL TRIP      960     x-12              \r\n" );
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document.write( "Now we fill in the TIMEs using TIME = DISTANCE/RATE\r\n" );
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document.write( "                    DISTANCE   RATE       TIME\r\n" );
document.write( "  ACTUAL TRIP          960      x        960/x \r\n" );
document.write( "HYPOTHETICAL TRIP      960     x-12    960/(x-12)\r\n" );
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document.write( "Now we look for what we haven't used:\r\n" );
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document.write( ">>...it [THE HYPOTHETICAL TRIP] would have taken 4 \r\n" );
document.write( "more hours...<<\r\n" );
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document.write( "So  \r\n" );
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document.write( "HYPOTHETICAL TRIP TIME = ACTUAL TRIP TIME + 4 HOURS\r\n" );
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document.write( "or\r\n" );
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document.write( "960/(x-12) = 960/x + 4\r\n" );
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document.write( "Divide thru by 4\r\n" );
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document.write( "240/(x-12) = 240/x + 1\r\n" );
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document.write( "Multiply through by LCD = x(x-12)\r\n" );
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document.write( "240x = 240(x-12) + x(x-12)\r\n" );
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document.write( "240x = 240x - 2880 + xē - 12x\r\n" );
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document.write( "   0 = xē - 12x - 2880\r\n" );
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document.write( "Solve this by the quadratic formula\r\n" );
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document.write( "        xē - 12x - 2880 = 0\r\n" );
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document.write( "The quadratic formula is:\r\n" );
document.write( "                  ______ \r\n" );
document.write( "            -b ą <font face = \"symbol\">Ö</font>bē-4ac\r\n" );
document.write( "        x = —————————————\r\n" );
document.write( "                2a \r\n" );
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document.write( "where a = 1; b = -12; c = -2880\r\n" );
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document.write( "                       __________________ \r\n" );
document.write( "             -(-12) ą <font face = \"symbol\">Ö</font>(-12)ē-4(1)(-2880)\r\n" );
document.write( "        x = ——————————————————————————\r\n" );
document.write( "                     2(1) \r\n" );
document.write( "                   _________ \r\n" );
document.write( "             12 ą <font face = \"symbol\">Ö</font>144+11520\r\n" );
document.write( "        x = ——————————————————\r\n" );
document.write( "                   2\r\n" );
document.write( "                   _____ \r\n" );
document.write( "             12 ą <font face = \"symbol\">Ö</font>11664\r\n" );
document.write( "        x = ————————————————\r\n" );
document.write( "                  2\r\n" );
document.write( "                    \r\n" );
document.write( "             12 ą 108\r\n" );
document.write( "        x = ——————————\r\n" );
document.write( "                2  \r\n" );
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document.write( "Using the +\r\n" );
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document.write( "             12 + 108\r\n" );
document.write( "        x = ——————————\r\n" );
document.write( "                2\r\n" );
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document.write( "             120\r\n" );
document.write( "        x = ————— = 60 mph\r\n" );
document.write( "              2\r\n" );
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document.write( "Using the -\r\n" );
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document.write( "             12 - 108\r\n" );
document.write( "        x = ——————————\r\n" );
document.write( "                2\r\n" );
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document.write( "             -96\r\n" );
document.write( "        x = ————— = -48 mph\r\n" );
document.write( "              2\r\n" );
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document.write( "so we discard that and the only answer is 60 mph.\r\n" );
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document.write( "Edwin</PRE><BR>\n" );
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