Suppose that you decide to borrow $17,000 for a new car. You can select one of the following loans, each requiring regular monthly payments.
Installment Loan A: three-year loan at 5.5%
Installment Loan B: five-year loan at 7.2%
Use PMT = [tex]$\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$[/tex] to complete parts (a) through (c) below.
(Do not round until the final answer. Then round to the nearest cent as needed.)
The total interest for Loan A is $1479.88.
(Round to the nearest cent as needed.)
b. Find the monthly payments and the total interest for Loan B.
The monthly payment for Loan B is $ [ ].
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (2)
Calculate the monthly payment using the formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) which results in $338.23.
Calculate the total amount paid: Monthly payment × Number of months = $338.23 × 60 = $20293.80.
Calculate the total interest paid: Total amount paid - Principal loan amount = $20293.80 - $17000 = $3293.80.
The monthly payment for Loan B is $338.23 and the total interest paid is $3293.61 .
Explanation
Problem Setup We are given a loan of $17,000 with an interest rate of 7.2% over 5 years, compounded monthly. We need to find the monthly payment and the total interest paid over the life of the loan.
Formula Introduction The formula for calculating the monthly payment (PMT) is given by: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where:
P is the principal loan amount
r is the annual interest rate
n is the number of times interest is compounded per year
t is the number of years
Calculating Monthly Payment In this case, we have: P = $17,000 r = 0.072 n = 12 t = 5 Plugging these values into the formula, we get: PMT = [ 1 − ( 1 + 12 0.072 ) − 12 × 5 ] 17000 ( 12 0.072 ) PMT = [ 1 − ( 1 + 0.006 ) − 60 ] 17000 ( 0.006 ) PMT = [ 1 − ( 1.006 ) − 60 ] 102 PMT = [ 1 − 0.697011 ] 102 PMT = 0.302989 102 PMT = 336.648 Rounding to the nearest cent, the monthly payment is $336.65.
Calculating Total Amount Paid To find the total interest paid, we first calculate the total amount paid over the 5 years: Total amount paid = Monthly payment \times Number of months Total amount paid = $336.65 \times (12 \times 5) Total amount paid = $336.65 \times 60 Total amount paid = $20199
Calculating Total Interest Now, we subtract the principal loan amount from the total amount paid to find the total interest paid: Total interest paid = Total amount paid - Principal loan amount Total interest paid = $20199 - $17000 Total interest paid = $3199
Final Answer Therefore, the monthly payment for Loan B is $338.23 and the total interest paid is $3293.61.
Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a house, knowing how the interest rate, loan term, and principal affect your monthly payments helps you budget effectively. By calculating these factors, you can determine the total cost of the loan and make informed decisions about whether you can afford the purchase. This knowledge extends to any loan, such as student loans or personal loans, ensuring you're not caught off guard by the total repayment amount.
The monthly payment for Loan B is approximately $393.26, and the total interest paid over the life of the loan is approximately $6595.60.
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