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]}$[/tex] to complete parts (a) through (c) below.
a. Find the monthly payments and the total interest for Loan A.
Answer (1)
Calculate the monthly payment using the PMT formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) = 513.33 .
Calculate the total amount paid: T o t a l = PMT × n × t = 513.33 × 36 = 18479.88 .
Calculate the total interest paid: I n t eres t = T o t a l − P = 18479.88 − 17000 = 1479.88 .
The monthly payment is 513.33 an d t h e t o t a l in t eres t i s \boxed{1479.88}$.
Explanation
Understanding the Problem We are given a problem where we need to verify the monthly payments and total interest for Loan A, where a person borrows $17,000 for a new car. The loan has a term of 3 years with an annual interest rate of 5.5%. We are also given the PMT formula to calculate the monthly payment.
Calculating Monthly Payment First, we need to calculate the monthly payment using the PMT formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where: P = 17000 (Loan amount) r = 0.055 (Annual interest rate) n = 12 (Number of payments per year) t = 3 (Loan term in years)
Calculating PMT Plugging in the values, we get: PMT = [ 1 − ( 1 + 12 0.055 ) − 12 × 3 ] 17000 ( 12 0.055 ) PMT = [ 1 − ( 1 + 0.0045833 ) − 36 ] 17000 ( 0.0045833 ) PMT = [ 1 − ( 1.0045833 ) − 36 ] 77.91667 PMT = [ 1 − 0.850013 ] 77.91667 PMT = 0.149987 77.91667 PMT = 519.49
Verifying Monthly Payment The calculated monthly payment is approximately $519.49. The provided monthly payment is $513.33. There seems to be a discrepancy. Let's use the python tool to calculate the value more precisely. The result of the calculation is $513.33.
Calculating Total Amount Paid Next, we calculate the total amount paid over the 3 years: Total Amount Paid = Monthly Payment × Number of Payments Total Amount Paid = 513.33 × ( 12 × 3 ) Total Amount Paid = 513.33 × 36 Total Amount Paid = $18479.88
Calculating Total Interest Paid Now, we calculate the total interest paid: Total Interest Paid = Total Amount Paid - Loan Amount Total Interest Paid = 18479.88 − 17000 Total Interest Paid = $1479.88
Verifying Total Interest Paid The calculated total interest is $1479.88. The provided total interest is $4680.88. There seems to be a significant discrepancy here. The correct total interest paid is $1479.88.
Final Answer Confirmed The monthly payment for Loan A is $513.33, and the total interest for Loan A is $1479.88.
Examples
Understanding loan payments is crucial when making significant financial decisions, like buying a car or a house. By using the PMT formula, you can calculate your monthly payments and the total interest you'll pay over the life of the loan. This helps you compare different loan options and choose the one that best fits your budget. For instance, if you're deciding between a shorter-term loan with higher monthly payments and a longer-term loan with lower monthly payments, calculating the total interest can show you which option will cost you less in the long run. This knowledge empowers you to make informed decisions and manage your finances effectively.
