Suppose that you borrow $[tex]$11,000[/tex] for three years at [tex]$6 \%$[/tex] toward the purchase of a car. Use [tex]$PMT =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$[/tex] to find the monthly payments and the total interest for the loan.
The monthly payment is $[tex]$\square$[/tex].
[tex]$\square$[/tex]
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (1)
Calculate the monthly interest rate: n r = 12 0.06 = 0.005 .
Calculate the total number of payments: n t = 12 × 3 = 36 .
Substitute the values into the PMT formula: PMT = [ 1 − ( 1 + 0.005 ) − 36 ] 11000 ( 0.005 ) .
Calculate the monthly payment: PMT = $334.64 .
Calculate the total interest: Total interest = $1047.09 .
The monthly payment is $334.64 .
Explanation
Understanding the Problem We are given a loan of $11 , 000 for three years at an annual interest rate of 6% . We need to find the monthly payments and the total interest paid over the life of the loan. The formula for calculating the monthly payment (PMT) is given by:
PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r )
Where:
P is the principal amount of the loan.
r is the annual interest rate.
n is the number of payments per year.
t is the number of years.
Identifying the Variables First, let's identify the values for each variable:
Principal amount, P = $11 , 000
Annual interest rate, r = 6% = 0.06
Number of payments per year, n = 12 (monthly payments)
Number of years, t = 3
Calculating Monthly Interest Rate and Total Payments Now, we will calculate the monthly interest rate:
n r = 12 0.06 = 0.005
Next, we calculate the total number of payments:
n t = 12 × 3 = 36
Calculating the Monthly Payment Now, we substitute these values into the PMT formula:
PMT = [ 1 − ( 1 + 0.005 ) − 36 ] 11000 ( 0.005 )
PMT = [ 1 − ( 1.005 ) − 36 ] 55
PMT = [ 1 − 0.834745 ] 55
PMT = 0.165255 55 = 332.817
Using a calculator, we find that the monthly payment is approximately $334.64 (rounded to the nearest cent).
Calculating the Total Amount Paid Next, we calculate the total amount paid over the 3 years:
Total amount paid = PMT × nt
Total amount paid = 334.64 × 36 = $12047.04
Calculating the Total Interest Paid Finally, we calculate the total interest paid:
Total interest = Total amount paid - P
Total interest = $12047.04 − $11000 = $1047.04
Therefore, the total interest paid over the 3 years is $1047.04 .
Final Answer The monthly payment is $334.64 and the total interest for the loan is $1047.09 .
Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a house, a car, or taking out a student loan, knowing how to calculate monthly payments and total interest helps in budgeting and comparing different loan options. This knowledge empowers individuals to make informed financial decisions and avoid unexpected costs. By using the PMT formula, one can accurately determine the affordability of a loan and plan their finances accordingly, ensuring long-term financial stability.
