To solve this problem, let’s break down the situation and understand the elements involved in calculating the monthly mortgage payment for a loan.
### Step-by-Step Calculation:
1. Initial Amount and Interest Rate:
- The loan amount (principal, [tex]\( L \)[/tex]) is \[tex]$305,000.
- The annual interest rate is 7.8%, so the monthly interest rate (\( r \)) is:
\[
r = \frac{7.8\%}{12} = 0.0065
\]
2. Loan Term:
- The loan term is 25 years. To find the number of monthly payments (\( n \)):
\[
n = 25 \times 12 = 300 \text{ months}
\]
3. Monthly Payment Formula:
The monthly payment \( P \) for a fixed-rate mortgage can be calculated using the formula:
\[
P = \frac{L \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
4. Calculate Monthly Payment:
Using our values:
\[
L = 305000, \quad r = 0.0065, \quad n = 300
\]
The numerator of the formula would be:
\[
305000 \cdot 0.0065 \cdot (1 + 0.0065)^{300}
\]
The denominator of the formula would be:
\[
(1 + 0.0065)^{300} - 1
\]
5. Comparison with Given Expressions:
We need to compare our calculated terms with each of the given options to see which one matches the monthly payment formula.
### Analysis of the Given Expressions:
1. Expression A:
\[
\frac{305000 \cdot 0.0055 \cdot (1-0.0055)^{320}}{(1-0.0065)^{302}-1}
\]
- This expression uses incorrect values for the interest rate and the terms are in inappropriate formats.
2. Expression B:
\[
\frac{305000 \cdot 0.0055 \cdot (1+0.0065)^{300}}{(1+0.0065)^{300}-1}
\]
- This expression closely resembles our payment equation but uses `0.0055` instead of `0.0065` in the numerator.
3. Expression C:
\[
\frac{305000 \cdot 0.0055 \cdot (1+0.0065)^{300}}{(1+0.0065)^{500}+1}
\]
- The denominator here is incorrect and does not match the monthly payment formula.
4. Expression D:
\[
\frac{305000 \cdot 0.0055 \cdot (1-0.0065)^{300}}{(1-0.0065)^{301}+1}
\]
- This expression again uses a different form and does not align with the correct mortgage payment formula.
### Conclusion:
The correct expression should match the form derived from the standard mortgage payment formula.
Upon evaluating each provided option, Expression B is the one that most closely approximates the correct setup for the mortgage payment, even though the monthly interest rate in the numerator seems to be slightly off (using `0.0055` instead of `0.0065`).
Thus, the expression that best matches the monthly payment formula for a 25-year loan of \$[/tex]305,000 at 7.8% interest, compounded monthly, is:
[tex]\[
\text{B.} \frac{305000 \cdot 0.0055 (1 + 0.0065)^{300}}{(1 + 0.0065)^{300} - 1}
\][/tex]
Note that for a perfect match, we'd expect the interest part to be `0.0065` throughout, but given the choices, B is the closest.
