document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207776>Question 207776</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Loren drove 200 miles at a certain rate, and his wife lois drove 100 miles at a rate 10 mph slower. If Loren had driven for the entire trip, they would have arived 30 minutes sooner. What was Loren's rate? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157196 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157196\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Loren drove 200 miles at a certain rate, and his wife lois drove 100 miles at a rate 10 mph slower. If Loren had driven for the entire trip, they would have arived 30 minutes sooner. What was Loren's rate?<BR>\n" );
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document.write( "Basic formula is R * T = D where R is the rate of travel, T is the time it takes and D is the distance traveled.<BR>\n" );
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document.write( "first formula is:<BR>\n" );
document.write( "x * t1 = 200<BR>\n" );
document.write( "where x is loren's rate of travel and t1 is the time it took loren to go 200 miles.<BR>\n" );
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document.write( "second formula is:<BR>\n" );
document.write( "(x-10) * t2 = 100<BR>\n" );
document.write( "where (x-10) is lois' rate of travel and t2 is the time it took lois to go 100 miles.<BR>\n" );
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document.write( "third formula is:<BR>\n" );
document.write( "x * t3 = 300<BR>\n" );
document.write( "where x is loren's rate of travel and t3 is the time it took loren to go 300 miles.<BR>\n" );
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document.write( "fourth formula is:<BR>\n" );
document.write( "t3 = t1 + t2 - .5<BR>\n" );
document.write( "where t3 is the total time it would have taken loren to go 300 if he had driven the whole way.<BR>\n" );
document.write( "note that t3 is 1/2 hour less than t1 + t2<BR>\n" );
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document.write( "when they were both driving the total time it took them to go the 300 miles was x * t1 + (x-10) * t2<BR>\n" );
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document.write( "if only loren drove, the total time it would have taken him was x * t3.<BR>\n" );
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document.write( "you have 2 equations that are equal to 300.<BR>\n" );
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document.write( "they are:<BR>\n" );
document.write( "x * t3 = 300<BR>\n" );
document.write( "and<BR>\n" );
document.write( "(x * t1) + ((x-10) * t2) = 300<BR>\n" );
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document.write( "since both these equations are equal to 300 then they are both equal to each other so you have:<BR>\n" );
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document.write( "x * t3 = (x * t1) + ((x-10) * t2)<BR>\n" );
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document.write( "simplifying this equation you get:<BR>\n" );
document.write( "(x * t3) = (x * t1) + (x * t2) - (10 * t2)<BR>\n" );
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document.write( "since t3 = t1 + t2 - .5, you can substitute in this equation to get:<BR>\n" );
document.write( "(x * (t1 + t2 - .5) = (x * t1) + (x * t2) - (10 * t2)<BR>\n" );
document.write( "you can simplify this equation to become:<BR>\n" );
document.write( "(x * t1) + (x * t2) - (.5 * x) = (x * t1) + (x * t2) - (10 * t2)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the (x * t1) and the (x * t2) on each side of this equation cancel out and you are left with:<BR>\n" );
document.write( "- (.5 * x) = - (10 * t2)<BR>\n" );
document.write( "if you multiply both sides of this equation by (-1) you get:<BR>\n" );
document.write( ".5 * x = 10 * t2<BR>\n" );
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document.write( "if you divide both sides of this equation by .5 you get:<BR>\n" );
document.write( "x = 20 * t2<BR>\n" );
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document.write( "if your remember in the beginning, we had a formula of:<BR>\n" );
document.write( "(x - 10) * t2 = 100<BR>\n" );
document.write( "this was lois traveling 100 miles at (x-10) miles per hour for t2 hours.<BR>\n" );
document.write( "we can substitute 20 * t2 for x in this equation to get:<BR>\n" );
document.write( "(20 * t2 - 10)  * t2 = 100<BR>\n" );
document.write( "simplifying this equation gets:<BR>\n" );
document.write( "20 * t2^2 - 10 * t2 = 100<BR>\n" );
document.write( "subtracting 100 from both sides of this equation gets:<BR>\n" );
document.write( "20*(t2)^2 - 10*(t2) - 100 = 0<BR>\n" );
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document.write( "dividing both sides of this equation by 10 gets:<BR>\n" );
document.write( "2*(t2)^2 - (t2) - 10 = 0<BR>\n" );
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document.write( "this factors out to be:<BR>\n" );
document.write( "(2*(t2) - 5) * ((t2) + 2) = 0<BR>\n" );
document.write( "which results in:<BR>\n" );
document.write( "2 * (t2) = 5<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "(t2) = -2<BR>\n" );
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document.write( "since (t2) can't be negative, the only possible solution is:<BR>\n" );
document.write( "2 * (t2) = 5<BR>\n" );
document.write( "which results in:<BR>\n" );
document.write( "t2 = 5/2<BR>\n" );
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document.write( "since (x - 10) * t2 = 100  becomes (x - 10) * (5/2) = 100 we can solve for x to get:<BR>\n" );
document.write( "x = 50 miles per hour<BR>\n" );
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document.write( "loren's rate of travel is 50 miles per hour.<BR>\n" );
document.write( "lois' rate of travel is 40 miles per hour.<BR>\n" );
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document.write( "we have t2 = 5/2 hours<BR>\n" );
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document.write( "since the formula of x * t1 = 200 miles becomes 50 * t1 = 200 miles, we can use this formula to solve for t1 to get:<BR>\n" );
document.write( "t1 = 4 hours.<BR>\n" );
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document.write( "since we know loren's rate of travel, we can use the formula for x * t3 = 300 to find t3.<BR>\n" );
document.write( "this formula becomes 50 * t3 = 300 which becomes:<BR>\n" );
document.write( "t3 = 300 / 50 = 6 hours.<BR>\n" );
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document.write( "driving together it took them 4 hours plus 5/2 hours.<BR>\n" );
document.write( "if loren drove alone, it would have taken him 6 hours.<BR>\n" );
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document.write( "4 hours plus 5/2 hours = 13/2 hours = 6.5 hours<BR>\n" );
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document.write( "the answers have been verified as correct.<BR>\n" );
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document.write( "the answers are:<BR>\n" );
document.write( "loren's rate of speed is 50 miles per hour.<BR>\n" );
document.write( "lois' rate of speed is 40 miles per hours.<BR>\n" );
document.write( "traveling together it took them 6.5 hours.<BR>\n" );
document.write( "if loren had driven all the way it would have taken them 6 hours.<BR>\n" );
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document.write( "since they only asked for loren's rate, the real answer is:<BR>\n" );
document.write( "loren's rate of travel is 50 miles per hour.<BR>\n" );
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