Suppose that you decide to borrow $[tex]$15,000[/tex] for a new car. You can select one of the following loans, each requiring regular monthly payments.
Installment Loan A: three-year loan at 6.3%
Installment Loan B: five-year loan at 4.8%
Use [tex]$PMT =\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.
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 4.8% is more economical.
B. The three-year loan at 6.3% is more economical.
The buyer will save approximately $[tex]$ \square[/tex] in interest.
(Round to the nearest cent as needed.)
Answer (1)
Calculate the monthly payment for Loan A: PM T A ≈ 458.37 .
Calculate the monthly payment for Loan B: PM T B ≈ 281.70 .
Calculate the total interest for Loan A and Loan B and compare them: I n t eres t A = 1501.35 and I n t eres t B = 1901.77 .
Determine that Loan A is more economical and calculate the difference in interest: D i ff ere n ce = 400.42 . The final answer is $400.42 .
Explanation
Problem Analysis We are given two loan options and need to determine which one is more economical by comparing their total interest payments. We will use the PMT formula to calculate the monthly payments for each loan and then find the total interest paid.
Calculating Monthly Payment for Loan A First, let's calculate the monthly payment for Loan A using the formula: PM T A = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where P = 15000 , r = 0.063 , n = 12 , and t = 3 .
PM T A = [ 1 − ( 1 + 12 0.063 ) − 12 ⋅ 3 ] 15000 ( 12 0.063 ) = [ 1 − ( 1.00525 ) − 36 ] 15000 ( 0.00525 ) ≈ 458.37 So, the monthly payment for Loan A is approximately $$458.37.
Calculating Monthly Payment for Loan B Next, we calculate the monthly payment for Loan B using the same formula: PM T B = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where P = 15000 , r = 0.048 , n = 12 , and t = 5 .
PM T B = [ 1 − ( 1 + 12 0.048 ) − 12 ⋅ 5 ] 15000 ( 12 0.048 ) = [ 1 − ( 1.004 ) − 60 ] 15000 ( 0.004 ) ≈ 281.70 So, the monthly payment for Loan B is approximately $$281.70.
Calculating Total Amount Paid Now, let's calculate the total amount paid for Loan A: T o t a l A = PM T A ⋅ ( 12 ⋅ 3 ) = 458.37 ⋅ 36 ≈ 16501.35 And the total amount paid for Loan B: T o t a l B = PM T B ⋅ ( 12 ⋅ 5 ) = 281.70 ⋅ 60 ≈ 16901.77
Calculating Total Interest Paid Next, we calculate the total interest paid for Loan A: I n t eres t A = T o t a l A − 15000 = 16501.35 − 15000 = 1501.35 And the total interest paid for Loan B: I n t eres t B = T o t a l B − 15000 = 16901.77 − 15000 = 1901.77
Comparing Total Interest Paid Comparing the total interest paid, we see that Loan A has a total interest of $1501.35 , w hi l e L o an B ha s a t o t a l in t eres t o f $1901.77. Since Loan A has less total interest, it is more economical.
Calculating the Difference in Interest The difference in total interest paid is: D i ff ere n ce = ∣ I n t eres t A − I n t eres t B ∣ = ∣1501.35 − 1901.77∣ = 400.42 Therefore, the buyer will save approximately $$400.42 in interest by choosing Loan A.
Final Answer The three-year loan at 6.3% is more economical, and the buyer will save approximately $$400.42 in interest.
Examples
When buying a car or a house, understanding loan options is crucial. This problem demonstrates how to calculate and compare the total cost of different loans, helping you make an informed financial decision. By calculating monthly payments and total interest paid, you can determine which loan is more economical and save money in the long run. This type of analysis is essential for responsible borrowing and financial planning.
