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