Suppose that you decide to borrow $[tex]17,000[/tex] 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 [tex]PMT =\frac{P(\frac{r}{n})}{[1-(1+\frac{r}{n})^{-nt}]}[/tex] to complete parts (a) through (c) below. (Round to the nearest cent as needed.)
c. Compare the monthly payments and the total interest for the two loans.
Determine which loan is more economical. Choose the correct answer below.
A. The five-year loan at 7.2% is more economical.
B. The three-year loan at 5.5% is more economical.
Answer (2)
After calculating, Loan A (3-year at 5.5%) has a lower total interest of $1479.88 compared to Loan B's $3293.80. Therefore, Loan A is more economical. The correct option is B.
;
Calculate the monthly payment and total interest for Loan A (3-year loan at 5.5%) and find the total interest to be $1479.88 .
Calculate the monthly payment and total interest for Loan B (5-year loan at 7.2%) and find the total interest to be $3293.80 .
Compare the total interest paid for both loans.
Since Loan A has a lower total interest paid, it is the more economical option. The three-year loan at 5.5% is more economical. B
Explanation
Problem Analysis We are given two loan options and need to determine which one is more economical. To do this, we will calculate the monthly payments and total interest paid for each loan.
Loan A: Setup For Loan A, we have a principal of $17,000, an interest rate of 5.5% (0.055), and a loan term of 3 years. Using the PMT formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where P = 17000, r = 0.055, n = 12, and t = 3.
Loan A: Monthly Payment Plugging in the values for Loan A: PMT A = [ 1 − ( 1 + 12 0.055 ) − 12 × 3 ] 17000 ( 12 0.055 ) Calculating this gives a monthly payment of approximately $513.33.
Loan A: Total Interest The total amount paid for Loan A is the monthly payment multiplied by the number of payments: Total Paid A = 513.33 × 12 × 3 = $18479.88 The total interest paid for Loan A is the total amount paid minus the principal: Total Interest A = 18479.88 − 17000 = $1479.88
Loan B: Setup For Loan B, we have a principal of $17,000, an interest rate of 7.2% (0.072), and a loan term of 5 years. Using the PMT formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where P = 17000, r = 0.072, n = 12, and t = 5.
Loan B: Monthly Payment Plugging in the values for Loan B: PMT B = [ 1 − ( 1 + 12 0.072 ) − 12 × 5 ] 17000 ( 12 0.072 ) Calculating this gives a monthly payment of approximately $338.23.
Loan B: Total Interest The total amount paid for Loan B is the monthly payment multiplied by the number of payments: Total Paid B = 338.23 × 12 × 5 = $20293.80 The total interest paid for Loan B is the total amount paid minus the principal: Total Interest B = 20293.80 − 17000 = $3293.80
Comparison and Conclusion Comparing the total interest paid for both loans: Loan A: $1479.88 Loan B: $3293.80 Since Loan A has a lower total interest paid, it is the more economical option.
Examples
Understanding loan options is crucial when making significant purchases like a car or a house. By calculating monthly payments and total interest, you can determine the most cost-effective choice. For instance, if you're buying a house, comparing a 15-year mortgage to a 30-year mortgage involves similar calculations. The shorter loan term usually has higher monthly payments but significantly less total interest paid over the life of the loan, saving you money in the long run. This analysis helps you make informed financial decisions.
