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Answer :
To find the correct expression to calculate the monthly payment of a 30-year loan for [tex]$190,000 at an 11.4% annual interest rate, compounded monthly, let's use the formula for calculating monthly payments on a fixed-rate mortgage, which is:
\[ P = \frac{L \cdot c \cdot (1 + c)^n}{(1 + c)^n - 1} \]
where:
- \( P \) is the monthly payment.
- \( L \) is the loan amount.
- \( c \) is the monthly interest rate (annual interest rate divided by 12).
- \( n \) is the total number of payments (loan term in years multiplied by 12).
Given:
- Loan amount (\( L \)) = \$[/tex]190,000
- Annual interest rate = 11.4%
- Monthly interest rate ([tex]\( c \)[/tex]) = 11.4% / 12 = 0.0095
- Total number of payments ([tex]\( n \)[/tex]) for 30 years = 30 \times 12 = 360
Now, plugging these into the formula will allow us to calculate the monthly payment.
Let's evaluate which of the given options matches this formula:
Option A:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1-0.0095)^{300}}{(1-0.0095)^{300}-1} \][/tex]
This uses [tex]\( (1 - c) \)[/tex] instead of [tex]\( (1 + c) \)[/tex], which is incorrect for calculating monthly payments using the standard formula.
Option B:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1+0.0095)^{360}}{(1+0.0095)^{360}+1} \][/tex]
This formula attempts to incorporate the correct base [tex]\( (1 + c) \)[/tex] but adds 1 in the denominator, which is incorrect.
Option C:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1-0.0095)^{300}}{(1-0.0095)^{300}+1} \][/tex]
This option also incorrectly uses [tex]\( (1 - c) \)[/tex].
Option D:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1+0.0095)^{360}}{(1+0.0095)^{300}-1} \][/tex]
In this option, the numerator is mostly correct, since it uses [tex]\( (1 + c) \)[/tex] and the correct power of 360, but the denominator is [tex]\( (1+c)^{300} \)[/tex] instead of [tex]\( (1+c)^{360} \)[/tex], which is incorrect.
The desired choice correctly following the standard formula should be:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1+0.0095)^{360}}{(1+0.0095)^{360}-1} \][/tex]
Thus, none of the given options exactly correspond to the correct formula. However, if we approximate based on the numerical output tested: Option B gives us a value of around 1746.92. Therefore, it closely matches the situation, considering errors or other factors, but it was structured incorrectly in the denominator, thus does not completely match the requirement.
At this point, it's essential to compare these values, and we can see that the closest match to the expected formula output is Option B when constraints are considered, revealing common calculation errors.
\[ P = \frac{L \cdot c \cdot (1 + c)^n}{(1 + c)^n - 1} \]
where:
- \( P \) is the monthly payment.
- \( L \) is the loan amount.
- \( c \) is the monthly interest rate (annual interest rate divided by 12).
- \( n \) is the total number of payments (loan term in years multiplied by 12).
Given:
- Loan amount (\( L \)) = \$[/tex]190,000
- Annual interest rate = 11.4%
- Monthly interest rate ([tex]\( c \)[/tex]) = 11.4% / 12 = 0.0095
- Total number of payments ([tex]\( n \)[/tex]) for 30 years = 30 \times 12 = 360
Now, plugging these into the formula will allow us to calculate the monthly payment.
Let's evaluate which of the given options matches this formula:
Option A:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1-0.0095)^{300}}{(1-0.0095)^{300}-1} \][/tex]
This uses [tex]\( (1 - c) \)[/tex] instead of [tex]\( (1 + c) \)[/tex], which is incorrect for calculating monthly payments using the standard formula.
Option B:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1+0.0095)^{360}}{(1+0.0095)^{360}+1} \][/tex]
This formula attempts to incorporate the correct base [tex]\( (1 + c) \)[/tex] but adds 1 in the denominator, which is incorrect.
Option C:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1-0.0095)^{300}}{(1-0.0095)^{300}+1} \][/tex]
This option also incorrectly uses [tex]\( (1 - c) \)[/tex].
Option D:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1+0.0095)^{360}}{(1+0.0095)^{300}-1} \][/tex]
In this option, the numerator is mostly correct, since it uses [tex]\( (1 + c) \)[/tex] and the correct power of 360, but the denominator is [tex]\( (1+c)^{300} \)[/tex] instead of [tex]\( (1+c)^{360} \)[/tex], which is incorrect.
The desired choice correctly following the standard formula should be:
[tex]\[ \frac{\$190,000 \cdot 0.0095 \cdot (1+0.0095)^{360}}{(1+0.0095)^{360}-1} \][/tex]
Thus, none of the given options exactly correspond to the correct formula. However, if we approximate based on the numerical output tested: Option B gives us a value of around 1746.92. Therefore, it closely matches the situation, considering errors or other factors, but it was structured incorrectly in the denominator, thus does not completely match the requirement.
At this point, it's essential to compare these values, and we can see that the closest match to the expected formula output is Option B when constraints are considered, revealing common calculation errors.
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