document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=321853>Question 321853</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A car dealer offers you a choice of 0% financing for 60 months or $2500 cash back on a new vehicle.  You have a pre-approved 60-month loan you can use from your credit union at a 4% interest rate.  If the monthly payments at 0% are $16.67 per $1000 financed, and the monthly payments at 4% are $18.41 per $1000 financed, what is the range of new car prices for which the cash back option will cost you less?  For what range of car prices should you take the 0% financing?\r<BR>\n" );
document.write( "\n" );
document.write( "This is what I have..\r<BR>\n" );
document.write( "\n" );
document.write( " $16.67 * $1000 = $16,670. <BR>\n" );
document.write( " $16,670. /60 = $277.83<BR>\n" );
document.write( " $18.41 * $1000 = $18,410. - $2500 = $15,910.<BR>\n" );
document.write( " $15,910 /60 = $265.17<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #230421 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=230421\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The problem states that if you take the 0% financing, then you don't get $2500 cash back, but if you take the 4% financing, then you do get $2500 cash back.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A basic assumption is that you use the $2500 cash back as a down payment on the car.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that if you take the 4% financing with the $2500 cash back, then the amount you have to finance is $2500 less.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let x = the price of the car.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "With 0% financing, you have to finance the full price of the car which is equal to x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "With 4% financing, you have to finance $2500 less because you are given $2500 Cash Back which you use as a down payment on the car.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means you have to finance (x - 2500).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You have 2 equations to work with.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your first equation with the 0% financing is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "total payments = 60 * (16.67 / 1000) * x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your second equation with the 4% financing is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "total payments = 60 * (18.41 / 1000) * (x - 2500)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You want to know when the 0% financing plan is better than the 4% financing plan.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "It is better when the total payments under the 0% financing plan are less than the total payments under the 4% financing plan.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You find this by setting up an equation that compares the total payments under the 0% financing plan with the total payments under the 4% financing plan.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Total Payments with 0% financing plan < Total Payments with 4% financing plan.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Replace the above statement with the equations for each of those options to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "60 * (16.67 / 1000) * x < 60 * (18.41 / 1000) * (x - 2500)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You solve this equation in the following manner:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide both sides of this equation by 60 and multiply both sides of this equation by 1000 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "16.67 * x < 18.41 * (x-2500)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Simplify this equation by multiplying out the right hand side of the equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(16.67 * x) < (18.41 * x) - (18.41 * 2500)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Simplify this further to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(16.67 * x) < (18.41 * x) - 46025.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add 46025 to both sides of this equation and subtract 16.67 * x from both sides of this equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "46025 < (18.41 * x) - (16.67 * x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Simplify this further to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "46025 < 1.75 * x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divide both sides of this equation by 1.74 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "26451.14943 < x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is the same as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x > 26451.14943\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since x is the price of the car, what this means is that you will be better off with the 0% financing option when the price of the car is greater than $26,451.14943.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In order to confirm that this value is good, we need to put in some values for the price of the car to see when the 0% financing plan is better and when the 4% financing plan is better.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the price of the car is less than $26,451.14943, the 0% financing plan should cost more (total payments will be higher).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the price of the car is greater than $26,451.14943, the 0% financing plan should cost less (total payments will be lower).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Your break even point should be when the price of the car is exactly $26,451.14943.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I will create a table assuming the price of the car is either:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "$20,000 (less than $26,451.14943)<BR>\n" );
document.write( "$26,451.14943 (EQUAL TO $26,451.14943)<BR>\n" );
document.write( "$30,000 (greater than $26,451.14943)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The table is shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "POC = Price of car.\r\n" );
document.write( "TP0 = Total Payments with the 0% financing plan.\r\n" );
document.write( "TP4 = Total Payments with the 4% financing plan.\r\n" );
document.write( "\r\n" );
document.write( "POC                     TP0                     TP4	          Result\r\n" );
document.write( "\r\n" );
document.write( "$20,000.00		$20,004.00		$19,330.50        TP0 > TP4\r\n" );
document.write( "$26,451.15		$26,456.44		$26,456.44        TP0 = TP4\r\n" );
document.write( "$30,000.00		$30,006.00		$30,376.50        TP0 < TP4\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From the above Table:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When the price of the car is less than $26,451.15, the 0% financing plan is more expensive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When the price of the car is equal to $26,451.15, the 0% financing plan and the 4% financing plan cost the same.   This is the break even point.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When the price of the car is greater than $26,451.15, the 0% financing plan is less expensive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "These numbers confirm the results of the equation analysis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The formulas used are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Total Payments with the 0% financing plan = (60 * 16.67)/1000 * x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Total Payments with the 4% financing plan = (60 * 18.41)/1000 * (x-2500)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x is the price of the car.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "With the 0% financing plan, you have to finance x.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "With the 4% financing plan, you have to finance (x - 2500).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is because you got $2500 cash back with the 4% financing plan and you used that $2500 as a down payment on the car which meant that you had to finance $2500 less. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );