To find the correct expression for calculating the monthly payment for a 20-year loan of [tex]$170,000 at an annual interest rate of 12.6%, compounded monthly, we can break it down step-by-step.
1. Convert the Annual Interest Rate to a Monthly Rate:
The annual interest rate is 12.6%, which we need to convert to a monthly rate by dividing by 12:
\[
r = \frac{12.6\%}{12} = 0.126 / 12 = 0.0105
\]
2. Calculate the Number of Monthly Payments:
Since the loan term is 20 years, and there are 12 months in a year, the total number of monthly payments (n) will be:
\[
n = 20 \times 12 = 240
\]
3. Use the Monthly Payment Formula for Fixed-Rate Mortgages:
The general formula for the monthly payment (M) on a fixed-rate mortgage is:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
where \( P \) is the principal amount (loan amount), \( r \) is the monthly interest rate, and \( n \) is the number of payments.
Plugging the given values into the formula:
\[
M = \frac{\$[/tex] 170000 \cdot 0.0105 \cdot (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}
\]
Now, let's compare this with the given options:
- Option A:
[tex]\[
\frac{\$ 170 \rho 00 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}+1}
\][/tex]
The numerator here seems correct, but the denominator has an incorrect [tex]\( +1 \)[/tex] instead of [tex]\( -1 \)[/tex].
- Option B:
[tex]\[
\frac{\$ 170 p 00 \cdot 0.0105(1-0.0105)^{240}}{(1-0.0105)^{240}-1}
\][/tex]
This option incorrectly uses [tex]\( 1 - 0.0105 \)[/tex] instead of [tex]\( 1 + 0.0105 \)[/tex].
- Option C:
[tex]\[
\frac{\$ 170000 \cdot 0.0105(1-0.0105)^{240}}{(1-0.0105)^{240}+1}
\][/tex]
This option also incorrectly uses [tex]\( 1 - 0.0105 \)[/tex] instead of [tex]\( 1 + 0.0105 \)[/tex], and the denominator is incorrect.
- Option D:
[tex]\[
\frac{\$ 170000 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}-1}
\][/tex]
This option uses the correct formula.
Therefore, the correct expression to calculate the monthly payment is:
[tex]\[
\boxed{\frac{\$ 170000 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}-1}}
\][/tex]
