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Sagot :
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.
### 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.
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