To calculate the monthly mortgage payment, we can use the formula for a fixed-rate mortgage:
\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \(M\) is the total monthly mortgage payment.
- \(P\) is the loan principal (the amount borrowed).
- \(r\) is the monthly interest rate (annual rate divided by 12).
- \(n\) is the number of payments (loan term in months).
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Calculate the down payment and loan amount (P):
- Price of the townhome = $122,250
- Down payment = 10% of $122,250 \[ \text{Down payment} = 0.10 \times 122,250 = 12,225 \]
- Loan amount (P) = Price - Down payment \[ P = 122,250 - 12,225 = 110,025 \]
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Calculate the monthly interest rate (r):
- Annual interest rate = 5.75%
- Monthly interest rate = \(\frac{5.75}{100} \div 12\) \[ r = 0.0575 \div 12 = 0.00479167 \]
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Calculate the number of payments (n):
- Loan term = 20 years = 20 \times 12 months \[ n = 20 \times 12 = 240 \]
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Calculate the monthly mortgage payment (M):
\[ M = 110,025 \frac{0.00479167(1 + 0.00479167)^{240}}{(1 + 0.00479167)^{240} - 1} \]
First, calculate \( (1 + r)^n \): \[ (1 + 0.00479167)^{240} \approx 3.075209 \]
Next, substitute back into the formula: \[ M = 110,025 \frac{0.00479167 \times 3.075209}{3.075209 - 1} \] \[ M = 110,025 \frac{0.014743}{2.075209} \approx 110,025 \times 0.007102 \] \[ M \approx 782.64 \]
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Conclusion:
Rounding to the nearest dollar gives us a monthly payment of approximately $783.
Since $783 is not one of the options given (603, 689, 712, 772), the correct choice would be:
None of these.