document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167948>Question 1167948</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Assume the car can be purchased for 0% down for 60 months (in lieu of rebate).<BR>\n" );
document.write( "A BMW that has a sticker price of $62,590 with factory and dealer rebates of $6,000<BR>\n" );
document.write( "(a) Find the monthly payment if financed for 60 months at 0% APR. (Round your answer to the nearest cent.)<BR>\n" );
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document.write( "(b) Find the monthly payment if financed at 2.5% add-on interest for 60 months. (Round your answer to the nearest cent.)<BR>\n" );
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document.write( "(c) Use the APR approximation formula to find the APR for part (b). (Round your answer to one decimal place.)<BR>\n" );
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document.write( "(d) State whether the 0% APR or the 2.5% add-on rate should be preferred.<BR>\n" );
document.write( "0% APR<BR>\n" );
document.write( "2.5% add-on rate </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852076 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852076\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "# Part (a): Monthly payment with 0% APR<BR>\n" );
document.write( "sticker_price = 62590<BR>\n" );
document.write( "rebates = 6000<BR>\n" );
document.write( "loan_amount = sticker_price - rebates<BR>\n" );
document.write( "loan_term_months = 60\r<BR>\n" );
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document.write( "monthly_payment_0_apr = loan_amount / loan_term_months<BR>\n" );
document.write( "print(f\"Monthly payment (0% APR): {monthly_payment_0_apr:.2f}\")\r<BR>\n" );
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document.write( "# Part (b): Monthly payment with 2.5% add-on interest<BR>\n" );
document.write( "add_on_rate_annual = 0.025<BR>\n" );
document.write( "loan_term_years = loan_term_months / 12\r<BR>\n" );
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document.write( "total_add_on_interest = loan_amount * add_on_rate_annual * loan_term_years<BR>\n" );
document.write( "total_amount_to_repay = loan_amount + total_add_on_interest<BR>\n" );
document.write( "monthly_payment_add_on = total_amount_to_repay / loan_term_months\r<BR>\n" );
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document.write( "print(f\"Total add-on interest: {total_add_on_interest:.2f}\")<BR>\n" );
document.write( "print(f\"Total amount to repay: {total_amount_to_repay:.2f}\")<BR>\n" );
document.write( "print(f\"Monthly payment (2.5% add-on interest): {monthly_payment_add_on:.2f}\")\r<BR>\n" );
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document.write( "# Part (c): APR for part (b) using the APR approximation formula<BR>\n" );
document.write( "n = 12  # number of payment periods in one year<BR>\n" );
document.write( "I = total_add_on_interest<BR>\n" );
document.write( "P = loan_amount<BR>\n" );
document.write( "N = loan_term_months\r<BR>\n" );
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document.write( "apr_approximation = (2 * n * I) / (P * (N + 1))<BR>\n" );
document.write( "apr_percentage = apr_approximation * 100\r<BR>\n" );
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document.write( "print(f\"APR approximation: {apr_percentage:.1f}%\")<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Monthly payment (0% APR): 943.17<BR>\n" );
document.write( "Total add-on interest: 7073.75<BR>\n" );
document.write( "Total amount to repay: 63663.75<BR>\n" );
document.write( "Monthly payment (2.5% add-on interest): 1061.06<BR>\n" );
document.write( "APR approximation: 4.9%\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "Here are the calculations for the car purchase:\r<BR>\n" );
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document.write( "**Loan Amount:**<BR>\n" );
document.write( "Sticker price: $$62,590$<BR>\n" );
document.write( "Factory and dealer rebates: $$6,000$<BR>\n" );
document.write( "Loan Amount = $$62,590 - $6,000 = $56,590$\r<BR>\n" );
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document.write( "**(a) Find the monthly payment if financed for 60 months at 0% APR.**<BR>\n" );
document.write( "Since the APR is 0%, there is no interest charged.<BR>\n" );
document.write( "Monthly Payment = Loan Amount / Number of Months<BR>\n" );
document.write( "Monthly Payment = $$56,590 / 60 = $943.1666...$<BR>\n" );
document.write( "Rounding to the nearest cent, the monthly payment is $\\boxed{$943.17}$.\r<BR>\n" );
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document.write( "**(b) Find the monthly payment if financed at 2.5% add-on interest for 60 months.**<BR>\n" );
document.write( "Loan Amount = $$56,590$<BR>\n" );
document.write( "Add-on Interest Rate = 2.5% per year<BR>\n" );
document.write( "Loan Term = 60 months = 5 years\r<BR>\n" );
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document.write( "Total Add-on Interest = Loan Amount $\\times$ Add-on Interest Rate $\\times$ Number of Years<BR>\n" );
document.write( "Total Add-on Interest = $$56,590 \\times 0.025 \\times 5 = $7,073.75$\r<BR>\n" );
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document.write( "Total Amount to Repay = Loan Amount + Total Add-on Interest<BR>\n" );
document.write( "Total Amount to Repay = $$56,590 + $7,073.75 = $63,663.75$\r<BR>\n" );
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document.write( "Monthly Payment = Total Amount to Repay / Number of Months<BR>\n" );
document.write( "Monthly Payment = $$63,663.75 / 60 = $1,061.0625$<BR>\n" );
document.write( "Rounding to the nearest cent, the monthly payment is $\\boxed{$1,061.06}$.\r<BR>\n" );
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document.write( "**(c) Use the APR approximation formula to find the APR for part (b).**<BR>\n" );
document.write( "APR Approximation Formula: $APR \\approx \\frac{2 \\times n \\times I}{P \\times (N+1)}$<BR>\n" );
document.write( "Where:<BR>\n" );
document.write( "$n$ = number of payment periods in one year = 12 (for monthly payments)<BR>\n" );
document.write( "$I$ = total interest paid = $$7,073.75$ (from part b)<BR>\n" );
document.write( "$P$ = principal (loan amount) = $$56,590$<BR>\n" );
document.write( "$N$ = total number of payments = 60\r<BR>\n" );
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document.write( "$APR \\approx \\frac{2 \\times 12 \\times $7,073.75}{$56,590 \\times (60+1)}$<BR>\n" );
document.write( "$APR \\approx \\frac{24 \\times $7,073.75}{$56,590 \\times 61}$<BR>\n" );
document.write( "$APR \\approx \\frac{$169,770}{$3,452,990}$<BR>\n" );
document.write( "$APR \\approx 0.04916$\r<BR>\n" );
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document.write( "Convert to percentage and round to one decimal place:<BR>\n" );
document.write( "$APR \\approx 0.04916 \\times 100% \\approx 4.916%$<BR>\n" );
document.write( "Rounding to one decimal place, the APR is $\\boxed{4.9%}$.\r<BR>\n" );
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document.write( "**(d) State whether the 0% APR or the 2.5% add-on rate should be preferred.**<BR>\n" );
document.write( "Comparing the two options:\r<BR>\n" );
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document.write( "  * 0% APR: Monthly payment of $$943.17$ with no interest.<BR>\n" );
document.write( "  * 2.5% add-on rate: Monthly payment of $$1,061.06$ with an effective APR of approximately 4.9%.\r<BR>\n" );
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document.write( "The 0% APR option results in a lower monthly payment and incurs no interest, making it clearly the better financial choice.\r<BR>\n" );
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document.write( "The 0% APR should be preferred. </SPAN>\n" );
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