Assume the car can be purchased for 0% down for 60 months (in lieu of rebate). A car with a sticker price of $42,500 with factory and dealer rebates of $5,100 (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.)
To find the monthly payment for each scenario, we can use the formula for loan payments:
Monthly payment = P / n
Where:
P = Principal amount of the loan (sticker price minus rebates)
n = Number of months
We will go through each scenario step by step:
(a) Financing at 0% APR for 60 months:
The principal amount is the sticker price minus rebates:
P = $42,500 - $5,100 = $37,400
The number of months is given:
n = 60
Applying the formula, we get:
Monthly payment = $37,400 / 60 = $623.33 (rounded to the nearest cent)
Therefore, the monthly payment is $623.33.
(b) Financing at 2.5% add-on interest for 60 months:
To calculate the monthly payment with add-on interest, we need to add the interest amount to the principal and then divide by the number of months.
The interest amount is calculated as:
Interest = Principal * Interest rate * Number of years
In this case, the interest rate is 2.5% (or 0.025) and the number of years is 5 (60 months divided by 12 months/year).
Principal = $37,400
Interest = $37,400 * 0.025 * 5 = $4,675
Total loan amount = Principal + Interest = $37,400 + $4,675 = $42,075
Using the formula for monthly payment, we get:
Monthly payment = $42,075 / 60 = $701.25 (rounded to the nearest cent)
Therefore, the monthly payment is $701.25.
(c) Finding the APR approximation using the APR approximation formula:
The APR approximation formula is:
APR = (2 * Add-on interest rate * 12) / (Principal * (n + 1))
In this case, the add-on interest rate is 2.5% (or 0.025), the principal is $37,400, and the number of months is 60.
Applying the formula, we get:
APR = (2 * 0.025 * 12) / ($37,400 * (60 + 1))
= 0.6 / ($37,400 * 61)
= 0.6 / $2,279,400
≈ 0.00026
Converting the decimal to a percentage, we get:
APR ≈ 0.00026 * 100 ≈ 0.026%
Therefore, the APR for part (b) is approximately 0.026%.