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 [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.
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.
The buyer will save approximately $ $\square$ in interest.
(Round to the nearest cent as needed.)
Answer (2)
Calculate the monthly payment and total interest for Loan A (3-year loan at 5.5%) using the PMT formula: PM T A = [ 1 − ( 1 + 12 0.055 ) − 12 × 3 ] 17000 ( 12 0.055 ) ≈ 513.33 , Total Interest A = $1,479.88.
Calculate the monthly payment and total interest for Loan B (5-year loan at 7.2%) using the PMT formula: PM T B = [ 1 − ( 1 + 12 0.072 ) − 12 × 5 ] 17000 ( 12 0.072 ) ≈ 338.23 , Total Interest B = $3,293.80.
Compare the total interest paid for both loans and determine that Loan A is more economical since it has lower total interest.
Calculate the savings by subtracting the total interest of Loan A from Loan B: Savings = $3,293.80 - $1,479.88 = 1 , 813.92. T h e f ina l an s w er i s \boxed{B} . T h e b u yer w i ll s a v e a pp ro x ima t e l y \boxed{ 1813.92} in interest.
Explanation
Problem Analysis We are given two loan options for borrowing $17,000 for a new car. Our goal is to determine which loan is more economical, meaning which loan results in paying less interest overall. We will use the provided PMT formula to calculate the monthly payments for each loan and then determine the total interest paid for each.
Loan A Calculations First, let's calculate the monthly payment and total interest paid for Installment Loan A, which is a three-year loan at 5.5%. We have:
Principal (P) = $17,000 Annual interest rate (r) = 5.5% = 0.055 Loan term (t) = 3 years Number of payments per year (n) = 12 (monthly payments)
Using the PMT formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) PM T A = [ 1 − ( 1 + 12 0.055 ) − 12 × 3 ] 17000 ( 12 0.055 )
The monthly payment for Loan A is approximately $513.33. The total amount paid for Loan A is:
Total Paid_A = PMT_A × n × t = $513.33 × 12 × 3 = $18,479.88
The total interest paid for Loan A is:
Interest_A = Total Paid_A - P = $18,479.88 - $17,000 = $1,479.88
Loan B Calculations Next, let's calculate the monthly payment and total interest paid for Installment Loan B, which is a five-year loan at 7.2%. We have:
Principal (P) = $17,000 Annual interest rate (r) = 7.2% = 0.072 Loan term (t) = 5 years Number of payments per year (n) = 12 (monthly payments)
Using the PMT formula: PM T B = [ 1 − ( 1 + 12 0.072 ) − 12 × 5 ] 17000 ( 12 0.072 )
The monthly payment for Loan B is approximately $338.23. The total amount paid for Loan B is:
Total Paid_B = PMT_B × n × t = $338.23 × 12 × 5 = $20,293.80
The total interest paid for Loan B is:
Interest_B = Total Paid_B - P = $20,293.80 - $17,000 = $3,293.80
Comparison and Savings Comparing the total interest paid for both loans:
Interest_A = $1,479.88 Interest_B = $3,293.80
Since Interest_A < Interest_B, Loan A (the three-year loan at 5.5%) is more economical. The difference in total interest paid is:
Savings = Interest_B - Interest_A = $3,293.80 - $1,479.88 = $1,813.92
Conclusion Therefore, the three-year loan at 5.5% is more economical, and the buyer will save approximately $1,813.92 in interest.
Examples
Understanding loan options and calculating the total cost, including interest, is crucial when making financial decisions such as buying a car or a house. By using the PMT formula, you can determine the monthly payments and total interest paid for different loan terms and interest rates. This allows you to compare loan options and choose the one that best fits your budget and financial goals. For example, 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 paid can help you make an informed decision and save money in the long run. This skill is also useful in understanding mortgages, student loans, and other types of financing.
The more economical loan is the three-year loan at 5.5%, resulting in lower total interest paid. The buyer will save approximately $1,813.92 in interest compared to the five-year loan at 7.2%.
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