document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167447>Question 1167447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You can afford a $200 per month car payment. You've found a 5 year loan at 2% interest. How big of a loan can you afford? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852249 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52802)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852249\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> . <BR>\n" );
document.write( "You can afford a $200 per month car payment. You've found a 5 year loan at 2% interest. <BR>\n" );
document.write( "How big of a loan can you afford?<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;This my solution goes second after the solution by @Theo.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;@Theo used an online calculator in his solution.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Since this web-site is to teach Math, the expected solution should be mathematical.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, I present here a standard mathematical solution.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Calculator can be used as a secondary tool for verification.\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Then the sequence of steps would be normally educative.\r<BR>\n" );
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document.write( "Use the standard formula for the monthly payment for a loan\r\n" );
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document.write( "    M = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=L%2A%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29 ALIGN=MIDDLE  ALT=\"L%2A%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "where L is the loan amount; r = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.02%2F12 ALIGN=MIDDLE  ALT=\"0.02%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the effective interest rate per month;\r\n" );
document.write( "n is the number of payments (same as the number of months); M is the monthly payment.\r\n" );
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document.write( "From this formula, the expression for the maximum loan is\r\n" );
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document.write( "    L = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%2F%28%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29%29 ALIGN=MIDDLE  ALT=\"M%2F%28%28r%2F%281-%281%2Br%29%5E%28-n%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "In this problem  M = $200;  r = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.02%2F12 ALIGN=MIDDLE  ALT=\"0.02%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  n = 5*12 = 60 monthly payments.\r\n" );
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document.write( "Substitute these values into the formula and get for the maximum loan amount\r\n" );
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document.write( "    L = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=200%2F%28%28%280.02%2F12%29%2F%281-%281%2B0.02%2F12%29%5E%28-60%29%29%29%29 ALIGN=MIDDLE  ALT=\"200%2F%28%28%280.02%2F12%29%2F%281-%281%2B0.02%2F12%29%5E%28-60%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = $11,410.47.\r\n" );
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document.write( "<U>ANSWER</U>.  The maximum loan amount is $11,410.47.\r\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "Compare my answer with the answer by @Theo: they coincide \"cent-to-cent\".\r<BR>\n" );
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document.write( "In such calculations, it is important to avoid intermediate rounding.<BR>\n" );
document.write( "Rounding can be done at the end, only.\r<BR>\n" );
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document.write( "Intermediate rounding is PROHIBITED, since it leads to wrong answer.\r<BR>\n" );
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document.write( "Also, a calculator should provide the necessary precision.\r<BR>\n" );
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document.write( "I use MS Excel in my computer. It works with 15 decimals in mantissa <BR>\n" );
document.write( "and provides the necessary precision.\r<BR>\n" );
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document.write( "I simply copy-paste my numerical formula into Excel spreadsheet and get the answer in one click.\r<BR>\n" );
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document.write( "Many standard online calculators provide the necessary precision.\r<BR>\n" );
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document.write( "You may use, for example, these free-of-charge online calculators\r<BR>\n" );
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document.write( "https://www.omnicalculator.com/finance/annuity-future-value\r<BR>\n" );
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document.write( "https://www.calculatorsoup.com/calculators/financial/future-value-annuity-calculator.php\r<BR>\n" );
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document.write( "https://www.calculator.net/future-value-calculator.html\r<BR>\n" );
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document.write( "They are reliable, have convenient user-friendly interface, have complete instructions and descriptions, <BR>\n" );
document.write( "provide the necessary precision and were checked million times. You may use them to check my/your calculations.\r<BR>\n" );
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document.write( "- - - - - added for your better understanding - - - - - \r<BR>\n" );
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document.write( "Notice that the full outpayment in 5 years in this problem is 5*12*200 = 12000 dollars.\r<BR>\n" );
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document.write( "It does not contradict to my answer, since the outpayment for the loan ($12000 in this case)<BR>\n" );
document.write( "is always greater than the loaned amount ($11,410.47).\r<BR>\n" );
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document.write( "The difference is what the financial company earns on this loan.\r<BR>\n" );
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document.write( "//////////////// INTERESTING and useful to know ////////////////\r<BR>\n" );
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document.write( "If you request an AI with this assignment, there is non-zero probability<BR>\n" );
document.write( "that the AI will produce incorrect answer.\r<BR>\n" );
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document.write( "For example, today, Oct.1, 2025, I submitted this problem to the Google AI called \"Overview\".<BR>\n" );
document.write( "It produced incorrect answer \"A loan of approximately $11,302.28. can be afforded. \"\r<BR>\n" );
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document.write( "Compare with my correct answer $11,410.47.\r<BR>\n" );
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document.write( "At the end of their solution, they say that their value is approximate, so this excuses them.\r<BR>\n" );
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document.write( "But in banking calculations and in solving school problems, where the answer should match precisely,<BR>\n" );
document.write( "such a discrepancy is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28UNACCEPTABLE%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28UNACCEPTABLE%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This discrepancy is a consequence of the fact that their calculation procedure is DEFECTIVE.\r<BR>\n" );
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document.write( "I observed similar discrepancy  in many their solved problems in Finance <BR>\n" );
document.write( "(more than in 10 cases and more than in 20), so it happens REGULARLY.\r<BR>\n" );
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document.write( "Many times I pointed them to this their deficiency, but it looks like <BR>\n" );
document.write( "they do not understand the problem - so they still did not fix it, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28highlight%28UNFORTUNATELY%29%29 ALIGN=MIDDLE  ALT=\"highlight%28highlight%28UNFORTUNATELY%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "X X X X X X X X X X X X X X X X X X X X X\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Come again to this forum soon to learn something new \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;or to get a precise solution in the future,\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;with complete and comprehensive explanations.\r<BR>\n" );
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