Darla purchased a new car during a special sales promotion by the manufacturer. She secured a loan from the manufacturer in the amount of $23,000 at a rate of 4.6%/year compounded monthly. Her bank is now charging 6.2%/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.)
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Why did the car go to the bank?
Because it needed a "loan!"
To calculate the amount of interest paid at the end of 3 years for each loan, 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 = Principal loan amount
r = Monthly interest rate
n = Number of months
For the loan from the manufacturer:
Principal loan amount (P) = $23,000
Monthly interest rate (r) = 4.6% / 12 = 0.046/12
Number of months (n) = 36
Substituting the values into the formula, we get:
Monthly payment (manufacturer loan) = (23000 * (0.046/12) * (1+(0.046/12))^36) / (((1+(0.046/12))^36) - 1)
Using a calculator, the monthly payment for the manufacturer loan is approximately $686.90.
Now, let's calculate the total interest paid for the manufacturer loan:
Total interest paid (manufacturer loan) = (Monthly payment * n) - P
Total interest paid (manufacturer loan) = (686.90 * 36) - 23000
Using a calculator, the total interest paid for the manufacturer loan is approximately $4,428.40.
For the loan from Darla's bank, the principal loan amount remains the same at $23,000, but the monthly interest rate is now 6.2% / 12 = 0.062/12. The number of months is still 36.
Using the same formula as before, we can calculate the new monthly payment:
Monthly payment (bank loan) = (23000 * (0.062/12) * (1+(0.062/12))^36) / (((1+(0.062/12))^36) - 1)
Using a calculator, the monthly payment for the bank loan is approximately $706.95.
Calculating the total interest paid for the bank loan:
Total interest paid (bank loan) = (Monthly payment * n) - P
Total interest paid (bank loan) = (706.95 * 36) - 23000
Using a calculator, the total interest paid for the bank loan is approximately $5,609.20.
To find the difference in interest payments between the two loans:
Difference in interest payments = Total interest paid (bank loan) - Total interest paid (manufacturer loan)
Difference in interest payments = $5,609.20 - $4,428.40
Using a calculator, the difference in interest payments is approximately $1,180.80.
Therefore, Darla would have paid approximately $1,180.80 less in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank.
To find the amount of interest paid at the end of 3 years for each loan, we need to calculate the monthly installment and the total amount paid over 36 months for each loan.
First, let's calculate the monthly installment for each loan using the formula for a loan payment:
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
Where:
M = Monthly installment
P = Loan amount
r = Monthly interest rate
n = Number of months
For the loan from the manufacturer:
P = $23,000
r = 4.6% / 100% / 12 (convert the annual interest rate to a monthly rate)
n = 36 months
Using the formula:
\[ M_{manufacturer} = \frac{23,000 \cdot 0.046/12 \cdot (1 + 0.046/12)^{36}}{(1 + 0.046/12)^{36} - 1} \]
Calculating this equation gives us a monthly payment of $683.85 (rounded to the nearest cent).
Next, let's calculate the monthly installment for the bank loan:
P = $23,000
r = 6.2% / 100% / 12 (convert the annual interest rate to a monthly rate)
n = 36 months
Using the formula again:
\[ M_{bank} = \frac{23,000 \cdot 0.062/12 \cdot (1 + 0.062/12)^{36}}{(1 + 0.062/12)^{36} - 1} \]
Calculating this equation gives us a monthly payment of $698.22 (rounded to the nearest cent).
Now we can calculate the total amount paid over 36 months for each loan by multiplying the monthly installment by the number of months:
Total amount paid with the manufacturer's loan = $683.85 * 36 = $24,617.56
Total amount paid with the bank's loan = $698.22 * 36 = $25,095.12
To find the amount of interest paid for each loan, subtract the original loan amount from the total amount paid:
Interest paid with the manufacturer's loan = $24,617.56 - $23,000 = $1,617.56
Interest paid with the bank's loan = $25,095.12 - $23,000 = $2,095.12
Therefore, after 3 years, Darla would have paid $1,617.56 in interest for the manufacturer's loan and $2,095.12 in interest for the bank's loan.
To calculate how much less she will have paid in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank, subtract the interest paid for the manufacturer's loan from the interest paid for the bank's loan:
Interest savings = $2,095.12 - $1,617.56 = $477.56
Therefore, Darla would have paid $477.56 less in interest payments over the life of the loan by borrowing from the manufacturer instead of her bank.
Use amortization equation for
monthly payment = A =P*i*(1+i/12)^(12n) / [(1+i/12)^(12n)-1]
P=amount borrowed (present value)
i=annual interest rate
n=duration of loan in years
compounding frequency = 12 times per year.
Case 1:
P=23000
i=4.6% = 0.046
n=3 (years)
Monthly payment,
A1=23000(.046/12)*(1+0.046/12)^(12*3) / [(1+0.046/12)^(12*3)-1]
Interest paid, I1 = (12*3)*A1-23000
Case 2:
P=23000
i=6.2% = 0.062
n=3 (years)
Monthly payment,
A2=23000(.062/12)*(1+0.062/12)^(12*3) / [(1+0.062/12)^(12*3)-1]
Interest paid, I2 = (12*3)*A2-23000
Difference in interest paid = I2-I1.
Will let you work with your calculator, and feel free to post your answers for a check.