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Answer :
To determine the monthly payment for a 20‑year loan of \[tex]$170,000 at an annual interest rate of 12.6% compounded monthly, we start with the standard formula for the monthly payment on a loan with compound interest:
$[/tex][tex]$
\text{Payment} = \frac{P \cdot r (1 + r)^n}{(1 + r)^n - 1}
$[/tex][tex]$
where
- $[/tex]P[tex]$ is the principal, here \$[/tex]170,000,
- [tex]$r$[/tex] is the monthly interest rate, and
- [tex]$n$[/tex] is the total number of monthly payments.
Step 1. Convert the annual interest rate to a monthly interest rate.
Given an annual rate of 12.6%, the monthly interest rate is:
[tex]$$
r = \frac{0.126}{12} = 0.0105.
$$[/tex]
Step 2. Calculate the total number of monthly payments.
For a 20‑year loan:
[tex]$$
n = 20 \times 12 = 240.
$$[/tex]
Step 3. Substitute the values into the formula.
Substituting [tex]$P=170,000$[/tex], [tex]$r=0.0105$[/tex], and [tex]$n=240$[/tex], the formula becomes:
[tex]$$
\text{Payment} = \frac{170000 \cdot 0.0105 \, (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}.
$$[/tex]
Step 4. Identify the matching expression.
When comparing with the given options:
- Option A: Uses [tex]$(1 - 0.0105)$[/tex] instead of [tex]$(1 + 0.0105)$[/tex].
- Option B: Uses [tex]$(1 + 0.0105)$[/tex] in both the numerator and denominator exactly as in our formula.
- Option C: Uses addition in the denominator instead of subtraction.
- Option D: Uses [tex]$(1+0.0105)$[/tex] in the numerator but uses addition in the denominator, which is not correct.
Thus, the correct expression is:
[tex]$$
\frac{170000 \cdot 0.0105 (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}.
$$[/tex]
This matches Option B.
Conclusion: The expression that can be used to calculate the monthly payment is Option B.
$[/tex][tex]$
\text{Payment} = \frac{P \cdot r (1 + r)^n}{(1 + r)^n - 1}
$[/tex][tex]$
where
- $[/tex]P[tex]$ is the principal, here \$[/tex]170,000,
- [tex]$r$[/tex] is the monthly interest rate, and
- [tex]$n$[/tex] is the total number of monthly payments.
Step 1. Convert the annual interest rate to a monthly interest rate.
Given an annual rate of 12.6%, the monthly interest rate is:
[tex]$$
r = \frac{0.126}{12} = 0.0105.
$$[/tex]
Step 2. Calculate the total number of monthly payments.
For a 20‑year loan:
[tex]$$
n = 20 \times 12 = 240.
$$[/tex]
Step 3. Substitute the values into the formula.
Substituting [tex]$P=170,000$[/tex], [tex]$r=0.0105$[/tex], and [tex]$n=240$[/tex], the formula becomes:
[tex]$$
\text{Payment} = \frac{170000 \cdot 0.0105 \, (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}.
$$[/tex]
Step 4. Identify the matching expression.
When comparing with the given options:
- Option A: Uses [tex]$(1 - 0.0105)$[/tex] instead of [tex]$(1 + 0.0105)$[/tex].
- Option B: Uses [tex]$(1 + 0.0105)$[/tex] in both the numerator and denominator exactly as in our formula.
- Option C: Uses addition in the denominator instead of subtraction.
- Option D: Uses [tex]$(1+0.0105)$[/tex] in the numerator but uses addition in the denominator, which is not correct.
Thus, the correct expression is:
[tex]$$
\frac{170000 \cdot 0.0105 (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}.
$$[/tex]
This matches Option B.
Conclusion: The expression that can be used to calculate the monthly payment is Option B.
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