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Answer :
To find which expression calculates the monthly payment for a 30-year loan of [tex]$190,000 at an 11.4% annual interest rate, compounded monthly, we need to determine the correct formula for a fixed-rate mortgage, which is an annuity payment formula.
### Step-by-step Explanation:
1. Identify the given values:
- Loan Amount (Principal): $[/tex]190,000
- Annual Interest Rate: 11.4% or 0.114 as a decimal
- Loan Term: 30 years
- Compounding Frequency: Monthly
2. Calculate the monthly interest rate:
- Convert the annual rate to a monthly rate by dividing by 12:
[tex]\[
\text{Monthly Interest Rate} = \frac{0.114}{12} = 0.0095
\][/tex]
3. Calculate the number of payments:
- Since the loan is for 30 years and payments are monthly:
[tex]\[
\text{Number of Payments} = 30 \times 12 = 360
\][/tex]
4. Monthly payment formula for fixed-rate loans:
- Use the annuity formula for calculating monthly payments:
[tex]\[
M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1}
\][/tex]
- Where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the principal ([tex]$190,000)
- \( r \) is the monthly interest rate (0.0095)
- \( n \) is the total number of payments (360)
5. Evaluate the given options:
- Option A:
\[
\frac{\$[/tex] 190,000 \cdot 0.0095(1-0.0095)^{350}}{(1-0.0095)^{360}-1}
\]
- Option B:
[tex]\[
\frac{\$ 190,000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{350}+1}
\][/tex]
- Option C:
[tex]\[
\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{260}}{(1+0.0055)^{364}+1}
\][/tex]
- Option D:
[tex]\[
\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{30}}{(1+0.0095)^{36}-1}
\][/tex]
6. Compare the correct formula with options:
- Among these options, match each structure with the correct annuity formula.
- We see that the valid formula involves all addition in exponents rather than subtraction.
Based on the given calculations, option C is incorrect due to mismatched values, and options A and D both feature incorrect operations in the exponents similar to option B.
None match the structure of the correct annuity formula, but a correct computation reveals that the required monthly payment should be approximately $1867.07.
7. Conclusion:
Of the expressions given, none perfectly match the correct one for calculating the monthly payment of this specific loan scenario. A standard formula similar but accurately computed is key for real applications.
### Step-by-step Explanation:
1. Identify the given values:
- Loan Amount (Principal): $[/tex]190,000
- Annual Interest Rate: 11.4% or 0.114 as a decimal
- Loan Term: 30 years
- Compounding Frequency: Monthly
2. Calculate the monthly interest rate:
- Convert the annual rate to a monthly rate by dividing by 12:
[tex]\[
\text{Monthly Interest Rate} = \frac{0.114}{12} = 0.0095
\][/tex]
3. Calculate the number of payments:
- Since the loan is for 30 years and payments are monthly:
[tex]\[
\text{Number of Payments} = 30 \times 12 = 360
\][/tex]
4. Monthly payment formula for fixed-rate loans:
- Use the annuity formula for calculating monthly payments:
[tex]\[
M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1}
\][/tex]
- Where:
- [tex]\( M \)[/tex] is the monthly payment
- [tex]\( P \)[/tex] is the principal ([tex]$190,000)
- \( r \) is the monthly interest rate (0.0095)
- \( n \) is the total number of payments (360)
5. Evaluate the given options:
- Option A:
\[
\frac{\$[/tex] 190,000 \cdot 0.0095(1-0.0095)^{350}}{(1-0.0095)^{360}-1}
\]
- Option B:
[tex]\[
\frac{\$ 190,000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{350}+1}
\][/tex]
- Option C:
[tex]\[
\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{260}}{(1+0.0055)^{364}+1}
\][/tex]
- Option D:
[tex]\[
\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{30}}{(1+0.0095)^{36}-1}
\][/tex]
6. Compare the correct formula with options:
- Among these options, match each structure with the correct annuity formula.
- We see that the valid formula involves all addition in exponents rather than subtraction.
Based on the given calculations, option C is incorrect due to mismatched values, and options A and D both feature incorrect operations in the exponents similar to option B.
None match the structure of the correct annuity formula, but a correct computation reveals that the required monthly payment should be approximately $1867.07.
7. Conclusion:
Of the expressions given, none perfectly match the correct one for calculating the monthly payment of this specific loan scenario. A standard formula similar but accurately computed is key for real applications.
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