Answer :
Let's start by analyzing each component of Rebecca's monthly payments.
### Step-by-Step Solution:
1. House Price and Down Payment:
- House Price: \[tex]$210,000 - Down Payment Rate: 5% \[ \text{Down Payment} = \$[/tex]210,000 \times 0.05 = \[tex]$10,500 \] 2. Loan Amount: - Loan Amount = House Price - Down Payment \[ \text{Loan Amount} = \$[/tex]210,000 - \[tex]$10,500 = \$[/tex]199,500
\]
3. Mortgage Interest Rate and Term:
- Annual Mortgage Interest Rate: 4.5%
- Monthly Mortgage Interest Rate:
[tex]\[ \text{Monthly Interest Rate} = \frac{4.5\%}{12} = 0.00375 \][/tex]
4. Number of Monthly Payments:
- Mortgage Term: 15 years
[tex]\[ \text{Number of Payments} = 15 \times 12 = 180 \text{ months} \][/tex]
5. Monthly Mortgage Payment:
- Using the fixed-rate mortgage formula:
[tex]\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the loan principal (\[tex]$199,500) - \( r \) is the monthly interest rate (0.00375) - \( n \) is the number of payments (180) \[ M = 199,500 \times \frac{0.00375(1+0.00375)^{180}}{(1+0.00375)^{180} - 1} \approx \$[/tex]1526.16
\]
6. Monthly Property Tax:
- House Value Assessment: \[tex]$198,000 - Local Tax Rate: 4.5% \[ \text{Monthly Property Tax} = \frac{\$[/tex]198,000 \times 0.045}{12} = \[tex]$742.50 \] 7. Monthly Homeowners Insurance: - Yearly Homeowners Insurance: \$[/tex]840
[tex]\[ \text{Monthly Homeowners Insurance} = \frac{\$840}{12} = \$70.00 \][/tex]
8. Private Mortgage Insurance (PMI):
- PMI Rate: 0.5%
[tex]\[ \text{Monthly PMI} = \frac{\$199,500 \times 0.005}{12} = \$83.12 \][/tex]
9. Total Monthly Payment:
- Sum of Monthly Mortgage Payment, Monthly Property Tax, Monthly Homeowners Insurance, and Monthly PMI
[tex]\[ \text{Total Monthly Payment} = \$1526.16 + \$742.50 + \$70.00 + \$83.12 \approx \$2421.79 \][/tex]
### Conclusion:
Rebecca's total monthly payments would be approximately \[tex]$2421.79. Therefore, the correct answer is not explicitly provided in the given options (they might have been incorrectly calculated or a typo might be present), but based on our detailed step-by-step calculations, we determine it to be approximately \$[/tex]2421.79.
### Step-by-Step Solution:
1. House Price and Down Payment:
- House Price: \[tex]$210,000 - Down Payment Rate: 5% \[ \text{Down Payment} = \$[/tex]210,000 \times 0.05 = \[tex]$10,500 \] 2. Loan Amount: - Loan Amount = House Price - Down Payment \[ \text{Loan Amount} = \$[/tex]210,000 - \[tex]$10,500 = \$[/tex]199,500
\]
3. Mortgage Interest Rate and Term:
- Annual Mortgage Interest Rate: 4.5%
- Monthly Mortgage Interest Rate:
[tex]\[ \text{Monthly Interest Rate} = \frac{4.5\%}{12} = 0.00375 \][/tex]
4. Number of Monthly Payments:
- Mortgage Term: 15 years
[tex]\[ \text{Number of Payments} = 15 \times 12 = 180 \text{ months} \][/tex]
5. Monthly Mortgage Payment:
- Using the fixed-rate mortgage formula:
[tex]\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \][/tex]
where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the loan principal (\[tex]$199,500) - \( r \) is the monthly interest rate (0.00375) - \( n \) is the number of payments (180) \[ M = 199,500 \times \frac{0.00375(1+0.00375)^{180}}{(1+0.00375)^{180} - 1} \approx \$[/tex]1526.16
\]
6. Monthly Property Tax:
- House Value Assessment: \[tex]$198,000 - Local Tax Rate: 4.5% \[ \text{Monthly Property Tax} = \frac{\$[/tex]198,000 \times 0.045}{12} = \[tex]$742.50 \] 7. Monthly Homeowners Insurance: - Yearly Homeowners Insurance: \$[/tex]840
[tex]\[ \text{Monthly Homeowners Insurance} = \frac{\$840}{12} = \$70.00 \][/tex]
8. Private Mortgage Insurance (PMI):
- PMI Rate: 0.5%
[tex]\[ \text{Monthly PMI} = \frac{\$199,500 \times 0.005}{12} = \$83.12 \][/tex]
9. Total Monthly Payment:
- Sum of Monthly Mortgage Payment, Monthly Property Tax, Monthly Homeowners Insurance, and Monthly PMI
[tex]\[ \text{Total Monthly Payment} = \$1526.16 + \$742.50 + \$70.00 + \$83.12 \approx \$2421.79 \][/tex]
### Conclusion:
Rebecca's total monthly payments would be approximately \[tex]$2421.79. Therefore, the correct answer is not explicitly provided in the given options (they might have been incorrectly calculated or a typo might be present), but based on our detailed step-by-step calculations, we determine it to be approximately \$[/tex]2421.79.
