Question

suppose that you decide to buy a car for $26,635, including taxes and license fees. you saved $7000 for a down payment and can get a five-year car loan at 7.64%. use $pmt = \frac{pleft( \frac{r}{n}
ight)}{left 1 - left( 1 + \frac{r}{n}
ight)^{-nt}
ight}$ to find the monthly payment and the total interest for the loan.
the monthly payment is $\square$
(round to the nearest cent as needed)

Explanation:

Step1: Calculate the loan amount

The cost of the car is \(P = 26635\), and the down - payment is \(7000\). So the loan amount \(A=26635 - 7000=19635\).

Step2: Identify the values for the formula

The annual interest rate \(r = 7.64\%=0.0764\), the number of times compounded per year \(n = 12\) (monthly compounding), and the number of years \(t = 5\). So \(mt=12\times5 = 60\) and \(\frac{r}{n}=\frac{0.0764}{12}\).

Step3: Substitute into the PMT formula

$$
LATEXBLOCK0
$$

First, calculate \(\frac{0.0764}{12}\approx0.006367\), then \(\frac{0.0764}{12}\times19635\approx0.006367\times19635\approx125.04\).

Next, calculate \((1+\frac{0.0764}{12})^{-60}=(1 + 0.006367)^{-60}\). Let \(x = 1.006367\), then \(x^{-60}=\frac{1}{x^{60}}\). Using a calculator, \(x^{60}\approx1.457\), so \(x^{-60}\approx\frac{1}{1.457}\approx0.686\).

Then \(1-(1+\frac{0.0764}{12})^{-60}\approx1 - 0.686=0.314\).

Finally, \(PMT=\frac{125.04}{0.314}\approx398.22\)

Answer:

\(398.22\)