document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174878>Question 1174878</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bus travels between towns A and B. It travels half the distance between A and B at 40 mph, and the rest of the distance at 50 mph. On the return trip, the bus travels for the first 2 hours at 35 mph, and the rest of the time at 40 mph. What is the distance between A and B, if the trip from B to A is an hour longer than the trip from A to B? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #800357 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=800357\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Draw segment AB.<BR>\n" );
document.write( "Let C be the midpoint of segment AB<BR>\n" );
document.write( "This means AC = CB<BR>\n" );
document.write( "C is at the halfway point between the two endpoints A and B.\r<BR>\n" );
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document.write( "Let's say that the distance from A to C is x miles. This would mean from C to B is also the same distance x. <BR>\n" );
document.write( "Furthermore, AB = 2*AC = 2x<BR>\n" );
document.write( "x is some positive real number.\r<BR>\n" );
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document.write( "The bus travels from A to C at a speed of 40 mph going x miles. This means it travels for x/40 hours. I'm using the idea that<BR>\n" );
document.write( "distance = rate*time<BR>\n" );
document.write( "to solve for time to get<BR>\n" );
document.write( "time = distance/rate\r<BR>\n" );
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document.write( "The bus goes from C to B going 50 mph<BR>\n" );
document.write( "time = distance/rate = x/50\r<BR>\n" );
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document.write( "When going from A to B, the total time taken is: <BR>\n" );
document.write( "x/40+x/50 = 5x/200+4x/200 = 9x/200 hours\r<BR>\n" );
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document.write( "We're told that \"The trip from B to A is an hour longer than the trip from A to B\"<BR>\n" );
document.write( "So whatever 9x/200 is equal to, we add on 1 hour to get the expression<BR>\n" );
document.write( "9x/200 + 1 = 9x/200 + 200/200 = (9x+200)/200\r<BR>\n" );
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document.write( "The time it takes to go from B to A is (9x+200)/200 hours.<BR>\n" );
document.write( "We'll use this expression later.\r<BR>\n" );
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document.write( "As the bus is traveling on the return trip, it travels for 2 hours going 35 mph.<BR>\n" );
document.write( "This means the bus travels a distance of<BR>\n" );
document.write( "distance = rate*time<BR>\n" );
document.write( "distance = 35*2<BR>\n" );
document.write( "distance = 70<BR>\n" );
document.write( "After these 70 miles are up, let's say the bus arrives at point D.<BR>\n" );
document.write( "This point D may or may not be at the same location as point C.\r<BR>\n" );
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document.write( "We have defined point D such that segment BD = 70 miles.<BR>\n" );
document.write( "This makes AD = 2x-70\r<BR>\n" );
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document.write( "You start with the total distance 2x, and subtract off what you've done so far.\r<BR>\n" );
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document.write( "Because the bus travels from D to A at 40 mph, we can say,<BR>\n" );
document.write( "distance = rate*time<BR>\n" );
document.write( "2x-70 = 40*time<BR>\n" );
document.write( "time = (2x-70)/40<BR>\n" );
document.write( "This represents the amount spent on the remaining part of the trip<BR>\n" );
document.write( "It represents the number of hours that elapse when going from D to A.\r<BR>\n" );
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document.write( "Add this to the 2 hours spent, when we traveled BD = 70 miles, and we get<BR>\n" );
document.write( "(2x-70)/40 + 2 = (2x-70)/40 + 80/40 = (2x-70+80)/40 = (2x+10)/40 = 2(x+5)/40 = (x+5)/20\r<BR>\n" );
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document.write( "The bus spends (x+5)/20 hours going from B to A.\r<BR>\n" );
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document.write( "Recall earlier, at the end of the first section, we found that the time spent going from B to A was (9x+200)/200 hours.<BR>\n" );
document.write( "At the end of the second section we also found the time spent going from B to A was (x+5)/20 hours.\r<BR>\n" );
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document.write( "Equate the two time values and solve for x\r<BR>\n" );
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document.write( "(x+5)/20 = (9x+200)/200<BR>\n" );
document.write( "200(x+5) = 20(9x+200)<BR>\n" );
document.write( "200x+1000 = 180x+4000<BR>\n" );
document.write( "200x-180x = 4000-1000<BR>\n" );
document.write( "20x = 3000<BR>\n" );
document.write( "x = 3000/20<BR>\n" );
document.write( "x = 150\r<BR>\n" );
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document.write( "The distance from A to C is 150 miles. So is the distance from C to B. <BR>\n" );
document.write( "So AB = AC+CB = 150+150 = <font color=red>300 miles</font>\r<BR>\n" );
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document.write( "Let's check our answer:\r<BR>\n" );
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document.write( "If you travel AC = 150 miles at 40 mph, then you spend 150/40 = 3.75 hours doing so.<BR>\n" );
document.write( "If you travel CB = 150 miles at 50 mph, then you spend 150/50 = 3 hours doing so.<BR>\n" );
document.write( "The total time going from A to B is 3.75+3 = 6.75 hours.\r<BR>\n" );
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document.write( "If you take 1 hour longer going from B to A, then you travel for 6.75+1 = 7.75 hours<BR>\n" );
document.write( "2 of those hours is spent traveling that 70 miles mentioned earlier (from point B to point D), so we have 7.75-2 = 5.75 hours left to travel the remaining distance of 300-70 = 230 miles.\r<BR>\n" );
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document.write( "The last leg of the trip, from D to A, has us going 40 mph for 5.75 hours<BR>\n" );
document.write( "distance = rate*time<BR>\n" );
document.write( "distance = 40*5.75<BR>\n" );
document.write( "distance = 230<BR>\n" );
document.write( "This matches with the 230 we found in the previous paragraph, so this confirms our answer.\r<BR>\n" );
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document.write( "Answer: Distance from A to B = <font color=red>300 miles</font><BR>\n" );
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