Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $24,000 at a rate of 4.2%/year compounded monthly. Her bank is now charging 6.8%/year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of 3 years for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank? (Round your answers to the nearest cent.)

Question

Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $24,000 at a rate of 4.2%/year compounded monthly. Her bank is now charging 6.8%/year compounded monthly for new car loans. Assuming that each loan would be amortized by 36 equal monthly installments, determine the amount of interest she would have paid at the end of 3 years for each loan. How much less will she have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank? (Round your answers to the nearest cent.)

Answer 1

To calculate the amount of interest paid at the end of 3 years for each loan, we need to use the formula for calculating the monthly payment on a loan and then subtract the principal amount borrowed.

Here's how to calculate the monthly payment for each loan:

Loan from the manufacturer:
Principal amount borrowed: $24,000
Annual interest rate: 4.2%
Compounding period: Monthly
Loan term: 36 months

First, we need to convert the annual interest rate to a monthly interest rate by dividing it by 12 and converting it to a decimal:

Monthly interest rate = (4.2% / 12) / 100 = 0.0035

Next, we'll use the formula for the monthly payment on a loan:

Monthly payment = P * r * (1 + r)^n / ((1 + r)^n - 1)

Where:
P = Principal amount
r = Monthly interest rate
n = Number of monthly payments

Substituting the values into the formula:

Monthly payment = 24000 * 0.0035 * (1 + 0.0035)^36 / ((1 + 0.0035)^36 - 1)

Calculating this, the monthly payment for the manufacturer's loan is approximately $709.06.

To calculate the amount of interest paid, we can multiply the monthly payment by the number of payments and subtract the principal borrowed:

Total interest = (Monthly payment * Number of payments) - Principal borrowed
Total interest = (709.06 * 36) - 24000
Total interest = $4,926.16

Now let's calculate the interest paid for the loan from the bank:

Principal amount borrowed: $24,000
Annual interest rate: 6.8%
Compounding period: Monthly
Loan term: 36 months

Following the same steps:

Monthly interest rate = (6.8% / 12) / 100 = 0.00567

Monthly payment = 24000 * 0.00567 * (1 + 0.00567)^36 / ((1 + 0.00567)^36 - 1)

Calculating this, the monthly payment for the bank's loan is approximately $734.94.

Total interest = (Monthly payment * Number of payments) - Principal borrowed
Total interest = (734.94 * 36) - 24000
Total interest = $7,256.64

To determine how much less Darla will have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank, we subtract the total interest paid for the manufacturer's loan from the total interest paid for the bank's loan:

Interest savings = Total interest (bank) - Total interest (manufacturer)
Interest savings = $7,256.64 - $4,926.16
Interest savings = $2,330.48

Therefore, Darla would have paid $2,330.48 less in interest by borrowing from the manufacturer instead of her bank over the life of the loan.