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Answer :
We start with the standard formula for the monthly payment on an amortized loan:
[tex]$$
\text{Payment} = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1},
$$[/tex]
where:
- [tex]$P$[/tex] is the principal (in this case, \[tex]$190,000),
- $[/tex]r[tex]$ is the monthly interest rate,
- $[/tex]n[tex]$ is the total number of monthly payments.
Since the annual interest rate is $[/tex]11.4\%[tex]$, the monthly interest rate is
$[/tex][tex]$
r = \frac{0.114}{12} \approx 0.0095.
$[/tex][tex]$
The loan term is 30 years, so the number of monthly payments is
$[/tex][tex]$
n = 30 \times 12 = 360.
$[/tex][tex]$
Substituting these values into the formula gives
$[/tex][tex]$
\text{Payment} = \frac{190000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1}.
$[/tex][tex]$
Now, let’s compare this with the given choices:
- Option A: Uses $[/tex](1 - 0.0095)[tex]$ instead of $[/tex](1 + 0.0095)[tex]$ and an exponent of 360 in the numerator with 380 in the denominator.
- Option B: Has an exponent of 380 in both numerator and denominator and has an addition in the denominator rather than a subtraction.
- Option C: Has the correct numerator, $[/tex][tex]$190000 \cdot 0.0095 \cdot (1+0.0095)^{360},$[/tex][tex]$ but its denominator is written as $[/tex][tex]$(1+0.0095)^{380} - 1.$[/tex][tex]$
- Option D: Uses $[/tex](1 - 0.0095)[tex]$ instead of $[/tex](1 + 0.0095)[tex]$.
The expression that most closely matches the structure of the standard formula is the one in Option C. Although the denominator in Option C uses the exponent $[/tex]380[tex]$ instead of the precise $[/tex]360[tex]$, it is the only option based on the structure
$[/tex][tex]$
\frac{190000 \cdot 0.0095 \cdot (1+0.0095)^{\text{(number of payments)}}}{(1+0.0095)^{\text{(number of payments)}} - 1}
$[/tex][tex]$
when comparing all the options. With the computed monthly payment of approximately \$[/tex]1867.07, the corresponding correct choice is Option C.
Thus, the answer is:
[tex]$$
\textbf{Option C.}
$$[/tex]
[tex]$$
\text{Payment} = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1},
$$[/tex]
where:
- [tex]$P$[/tex] is the principal (in this case, \[tex]$190,000),
- $[/tex]r[tex]$ is the monthly interest rate,
- $[/tex]n[tex]$ is the total number of monthly payments.
Since the annual interest rate is $[/tex]11.4\%[tex]$, the monthly interest rate is
$[/tex][tex]$
r = \frac{0.114}{12} \approx 0.0095.
$[/tex][tex]$
The loan term is 30 years, so the number of monthly payments is
$[/tex][tex]$
n = 30 \times 12 = 360.
$[/tex][tex]$
Substituting these values into the formula gives
$[/tex][tex]$
\text{Payment} = \frac{190000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1}.
$[/tex][tex]$
Now, let’s compare this with the given choices:
- Option A: Uses $[/tex](1 - 0.0095)[tex]$ instead of $[/tex](1 + 0.0095)[tex]$ and an exponent of 360 in the numerator with 380 in the denominator.
- Option B: Has an exponent of 380 in both numerator and denominator and has an addition in the denominator rather than a subtraction.
- Option C: Has the correct numerator, $[/tex][tex]$190000 \cdot 0.0095 \cdot (1+0.0095)^{360},$[/tex][tex]$ but its denominator is written as $[/tex][tex]$(1+0.0095)^{380} - 1.$[/tex][tex]$
- Option D: Uses $[/tex](1 - 0.0095)[tex]$ instead of $[/tex](1 + 0.0095)[tex]$.
The expression that most closely matches the structure of the standard formula is the one in Option C. Although the denominator in Option C uses the exponent $[/tex]380[tex]$ instead of the precise $[/tex]360[tex]$, it is the only option based on the structure
$[/tex][tex]$
\frac{190000 \cdot 0.0095 \cdot (1+0.0095)^{\text{(number of payments)}}}{(1+0.0095)^{\text{(number of payments)}} - 1}
$[/tex][tex]$
when comparing all the options. With the computed monthly payment of approximately \$[/tex]1867.07, the corresponding correct choice is Option C.
Thus, the answer is:
[tex]$$
\textbf{Option C.}
$$[/tex]
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