document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=91914>Question 91914</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Car rental agency A will rent a compact car for $35 per day and an additional charge of $0.24 per mile. Car rental agency B will charge only $0.16 per mile but charges $41 per day. If Adam wanted to rent a car for three days, how many miles would Adam have to drive to make car rental agency B a better bargain?  \r<BR>\n" );
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document.write( "By trial and error I have found that the answer is 226 but can't figure out an equation, please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #66860 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=66860\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To keep this in notation that you are probably most familiar with, let's represent the cost<BR>\n" );
document.write( "of renting the car with the letter y. And let's represent the number of miles driven with<BR>\n" );
document.write( "the letter x.<BR>\n" );
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document.write( "Adam is going to rent a car for three days.  Since agency A charges $35 per day in fixed costs,<BR>\n" );
document.write( "Adam will owe that company $35 times 3 or $105 dollars ... not including the mileage <BR>\n" );
document.write( "charges.  The mileage charges are $0.24 for every mile driven, and since x represents the<BR>\n" );
document.write( "number of miles he drives, he will owe agency A $0.24 times x or $0.24x in mileage charges.<BR>\n" );
document.write( "The total amount y that he will owe to agency A for the three day rental isis the sum of <BR>\n" );
document.write( "the fixed costs of $105 plus the mileage charge of $0.24x. So we can write the equation for <BR>\n" );
document.write( "the total dollars y that he will pay to agency A as:<BR>\n" );
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document.write( "y = 0.24x + 105<BR>\n" );
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document.write( "Now let's do a similar analysis for agency B. At $41 per day, the 3 day rental will cost<BR>\n" );
document.write( "Adam $41 times 3 which is $123 in fixed costs.  And at $0.16 per mile for the x miles that<BR>\n" );
document.write( "Adam drives he will owe agency B $0.16 times x or 0.16x in mileage charges. The total cost<BR>\n" );
document.write( "y charged by agency B is the sum of these two or:<BR>\n" );
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document.write( "y = 0.16x + 123<BR>\n" );
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document.write( "So you now have two equations to work with as follows:<BR>\n" );
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document.write( "y = 0.24x + 105<BR>\n" );
document.write( "y = 0.16x + 123<BR>\n" );
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document.write( "If you solve these equations simultaneously you find the solution that is the \"break even\"<BR>\n" );
document.write( "point ... that is the solution where the total cost to both agencies is the same.<BR>\n" );
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document.write( "Let's do that ...<BR>\n" );
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document.write( "There are a couple of ways this could be done ... variable elimination or substitution<BR>\n" );
document.write( "among them.  Let's use substitution. In the first equation we have that y is equal to<BR>\n" );
document.write( "0.24x + 105. Suppose we substitute that value for y in the second equation. In that case<BR>\n" );
document.write( "the second equation becomes:<BR>\n" );
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document.write( "0.24x + 105 = 0.16x + 123<BR>\n" );
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document.write( "Get rid of the 0.16x on the right side by subtracting 0.16x from both sides:<BR>\n" );
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document.write( "0.08x + 105 = 123<BR>\n" );
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document.write( "Then get rid of the 105 on the left side by subtracting 105 from both sides:<BR>\n" );
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document.write( "0.08x = 18<BR>\n" );
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document.write( "Solve for x by dividing both sides by 0.08 to get:<BR>\n" );
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document.write( "x = 18/0.08 = 225<BR>\n" );
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document.write( "So if Adam drives 225 miles in the three days he can use either agency and the cost should<BR>\n" );
document.write( "be the same.<BR>\n" );
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document.write( "Let's check that out.  At agency A he will pay $35 dollars a day for 3 days plus 225 miles <BR>\n" );
document.write( "times 0.24 per mile, a total of $105 plus $54 = $159<BR>\n" );
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document.write( "At agency b he will pay $41 per day for 3 days plus 225 miles times 0.16 cents per mile,<BR>\n" );
document.write( "a total of $123 + $36 = $159.<BR>\n" );
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document.write( "Just as we suspected. The costs at 225 miles are identical for either of the two agencies.<BR>\n" );
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document.write( "Since agency B charges just $0.16 per mile, Adam will save money by going with agency B<BR>\n" );
document.write( "if he drives more than 225 miles.<BR>\n" );
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document.write( "Hope this helps you to understand a mathematical way of coming up with the answer that<BR>\n" );
document.write( "you managed to work out by just using your thought power.  Good exercise for you!<BR>\n" );
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