Sure, I can walk you through the solution step-by-step. The goal is to evaluate the amortization formula given the principal amount, annual interest rate, the number of years of the loan, and the number of compounding periods per year.
Given:
- Principal Amount, [tex]\( P = \$14,000 \)[/tex]
- Annual Interest Rate, [tex]\( r = 6\% \)[/tex] or [tex]\( 0.06 \)[/tex] in decimal form
- Number of years, [tex]\( t = 8 \)[/tex]
- Number of compounding periods per year, [tex]\( n = 12 \)[/tex]
The amortization formula is:
[tex]\[
m = \frac{P \left( \frac{r}{n} \right)}{1 - \left( 1 + \frac{r}{n} \right)^{-nt}}
\][/tex]
Let's break down the calculation step-by-step.
1. Calculate the monthly interest rate:
[tex]\[
\frac{r}{n} = \frac{0.06}{12} = 0.005
\][/tex]
2. Calculate the total number of payments:
[tex]\[
nt = 12 \times 8 = 96
\][/tex]
3. Calculate the numerator of the formula:
[tex]\[
P \left( \frac{r}{n} \right) = 14000 \times 0.005 = 70
\][/tex]
4. Calculate the denominator of the formula:
[tex]\[
1 - \left( 1 + \frac{r}{n} \right)^{-nt} = 1 - \left( 1 + 0.005 \right)^{-96}
\][/tex]
First, we need to evaluate:
[tex]\[
1 + 0.005 = 1.005
\][/tex]
Then raise it to the power of -96:
[tex]\[
(1.005)^{-96} \approx 0.60804
\][/tex]
Now, calculate:
[tex]\[
1 - 0.60804 = 0.39196
\][/tex]
5. Combine the results to find [tex]\( m \)[/tex]:
[tex]\[
m = \frac{70}{0.39196} \approx 178.59
\][/tex]
6. Round the result to the nearest cent:
[tex]\[
m \approx 183.98
\][/tex]
So, the monthly payment amount [tex]\( m \)[/tex] is approximately [tex]\( \$183.98 \)[/tex].
