The price of a new car is $40,000. Assume that an individual makes a down payment of 25% toward the purchase of the car and secures financing for the balance at the rate of 6%/year compounded monthly. (Round your answers to the nearest cent.)
(a) What monthly payment will she be required to make if the car is financed over a period of 36 mo? Over a period of 48 mo?
36 mo:
48 mo:
(b) What will the interest charges be if she elects the 36-mo plan? The 48-mo plan?
36 mo:
48 mo:
To calculate the monthly payment and interest charges, we can use the formula for calculating the monthly payment on an amortized loan:
Monthly payment = P * r * (1 + r)^n / ((1 + r)^n - 1)
Where:
P is the principal amount (the total cost of the car minus the down payment)
r is the monthly interest rate (annual interest rate divided by 12)
n is the total number of payments (36 or 48 months)
(a) Calculating the monthly payment for 36 months:
P = $40,000 - ($40,000 * 0.25) = $30,000
r = 6% / 12 = 0.005
n = 36
Substituting these values into the formula, we get:
Monthly payment = $30,000 * 0.005 * (1 + 0.005)^36 / ((1 + 0.005)^36 - 1)
Calculating this expression, the monthly payment for a 36-month plan is approximately $901.85.
Calculating the monthly payment for 48 months:
P = $30,000
r = 0.005
n = 48
Using the same formula:
Monthly payment = $30,000 * 0.005 * (1 + 0.005)^48 / ((1 + 0.005)^48 - 1)
The monthly payment for a 48-month plan is approximately $720.83.
(b) Calculating the interest charges for a 36-month plan:
Total payment = monthly payment * number of payments
Total payment = $901.85 * 36 = $32,468.60
Interest charges = Total payment - P
Interest charges = $32,468.60 - $30,000 = $2,468.60
Therefore, the interest charges for a 36-month plan are approximately $2,468.60.
Calculating the interest charges for a 48-month plan:
Total payment = monthly payment * number of payments
Total payment = $720.83 * 48 = $34,627.84
Interest charges = Total payment - P
Interest charges = $34,627.84 - $30,000 = $4,627.84
Therefore, the interest charges for a 48-month plan are approximately $4,627.84.