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 =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} to complete parts (a) through (c) below.
a. Find the monthly payments and the total interest for Loan A.
The monthly payment for Loan A is $ [ ].
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (1)
Calculate the monthly payment (PMT) using the formula: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) , where P = 17000 , r = 0.055 , n = 12 , and t = 3 .
The monthly payment for Loan A is approximately $513.33 .
Calculate the total amount paid: $513.33 × 12 × 3 = $18479.88 .
Calculate the total interest paid: $18479.88 − $17000 = $1479.88 . The monthly payment for Loan A is $513.33 .
Explanation
Understanding the Formula and Given Values We are given the formula for calculating the monthly payment (PMT) of a loan: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) where:
P is the principal loan amount, which is $17 , 000 .
r is the annual interest rate, which is 5.5% = 0.055 for Loan A.
n is the number of payments per year, which is 12 (monthly payments).
t is the number of years, which is 3 for Loan A.
Calculating Monthly Payment Now, we plug in the values for Loan A into the formula: PMT = [ 1 − ( 1 + 12 0.055 ) − 12 × 3 ] 17000 ( 12 0.055 ) PMT = [ 1 − ( 1 + 0.00458333 ) − 36 ] 17000 ( 0.00458333 ) PMT = [ 1 − ( 1.00458333 ) − 36 ] 77.91666667 PMT = [ 1 − 0.89223475 ] 77.91666667 PMT = 0.10776525 77.91666667 PMT = 513.33033067 Rounding to the nearest cent, the monthly payment for Loan A is $513.33 .
Calculating Total Amount Paid Next, we calculate the total amount paid over the 3 years: Total Paid = PMT × n × t Total Paid = $513.33 × 12 × 3 Total Paid = $$18479.88
Calculating Total Interest Now, we calculate the total interest paid for Loan A: Total Interest = Total Paid - P Total Interest = $18479.88 − $17000 Total Interest = $$1479.88
Final Answer Therefore, the monthly payment for Loan A is $513.33 and the total interest paid is $1479.88 .
Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a house, knowing how the loan term and interest rate affect your monthly payments and total interest paid helps you make informed decisions. A shorter loan term means higher monthly payments but less interest paid overall, while a longer loan term results in lower monthly payments but more interest paid. This understanding allows you to budget effectively and choose a loan that fits your financial goals.
