Suppose that you decide to buy a car for $25,635, including taxes and license fees. You saved $6000 for a down payment and can get a five-year car loan at 6.43%. Use PMT = [tex]$\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$[/tex] to find the monthly payment and the total interest for the loan.
The monthly payment is $ [ ].
(Round to the nearest cent as needed.)
Answer (2)
Calculate the loan amount by subtracting the down payment from the car price: P = 25635 − 6000 = 19635 .
Use the PMT formula to calculate the monthly payment: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) = [ 1 − ( 1 + 12 0.0643 ) − 12 × 5 ] 19635 ( 12 0.0643 ) = 383.54 .
Calculate the total amount paid: T o t a lP ai d = PMT × n × t = 383.54 × 12 × 5 = 23012.4 .
Calculate the total interest paid: T o t a l I n t eres t = T o t a lP ai d − P = 23012.4 − 19635 = 3377.4 . The monthly payment is 383.54 .
Explanation
Understanding the Problem Let's break down this car loan problem step by step! We'll start by identifying the key information and then use the provided formula to calculate your monthly payment and the total interest you'll pay over the life of the loan.
Calculating the Loan Amount First, we need to determine the loan amount, which is the car price minus the down payment. The car costs $25 , 635 and you're putting down $6 , 000 . So, the loan amount ( P ) is: P = 25635 − 6000 = 19635
Identifying Interest Rate, Payments per Year, and Loan Term Next, we need to identify the annual interest rate ( r ), the number of payments per year ( n ), and the loan term in years ( t ). The interest rate is 6.43% , which as a decimal is 0.0643 . Since you're making monthly payments, n = 12 . The loan term is 5 years, so t = 5 .
Calculating the Monthly Payment Now we can plug these values into the PMT formula to find the monthly payment: PMT = [ 1 − ( 1 + n r ) − n t ] P ( n r ) PMT = [ 1 − ( 1 + 12 0.0643 ) − 12 × 5 ] 19635 ( 12 0.0643 ) PMT = [ 1 − ( 1 + 0.00535833 ) − 60 ] 19635 ( 0.00535833 ) PMT = [ 1 − ( 1.00535833 ) − 60 ] 105.176 PMT = [ 1 − 0.73515 ] 105.176 PMT = 0.26485 105.176 PMT = 397.10
Calculating Total Interest Paid To find the total interest paid over the loan term, we first calculate the total amount paid: T o t a lP ai d = PMT × n × t = 383.54 × 12 × 5 = 23012.4 Then, we subtract the loan amount from the total amount paid to find the total interest: T o t a l I n t eres t = T o t a lP ai d − P = 23012.4 − 19635 = 3377.4
Final Answer Therefore, the monthly payment is $383.54 and the total interest paid is $3377.4 .
Examples
Understanding car loans is essential for making informed financial decisions. For example, knowing how to calculate monthly payments and total interest helps you compare different loan offers and choose the one that best fits your budget. This knowledge can also be applied to other types of loans, such as mortgages or personal loans, enabling you to manage your finances effectively and avoid overpaying for borrowed money. By understanding the components of a loan, you can make smarter decisions about borrowing and saving money.
The monthly payment for the car loan is approximately $397.10, and the total interest paid over the loan term is about $4,189.00.
;
