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Answer :
To calculate the monthly payment for a loan, we use the standard formula for an annuity payment:
[tex]$$
\text{Payment} = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1},
$$[/tex]
where
- [tex]$P$[/tex] is the principal (loan amount),
- [tex]$r$[/tex] is the monthly interest rate, and
- [tex]$n$[/tex] is the total number of monthly payments.
In this problem:
1. The principal is
[tex]$$
P = \$170{,}000.
$$[/tex]
2. The annual interest rate is [tex]$12.6\%$[/tex], so the monthly interest rate is
[tex]$$
r = \frac{0.126}{12} = 0.0105.
$$[/tex]
3. The loan term is 20 years, so the total number of monthly payments is
[tex]$$
n = 20 \times 12 = 240.
$$[/tex]
Next, we substitute these values into the formula:
[tex]$$
\text{Payment} = \frac{170000 \times 0.0105 \times (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}.
$$[/tex]
This expression matches the one given in option B. In addition, when you evaluate the factors, you would find that
- the factor [tex]$(1+0.0105)^{240}$[/tex] is approximately [tex]$12.266376321322747$[/tex], and
- the denominator [tex]$(1+0.0105)^{240} - 1$[/tex] is approximately [tex]$11.266376321322747$[/tex].
Thus, the monthly payment comes out to be approximately \$1943.44.
Therefore, the correct expression for the monthly payment is given in:
Option B.
[tex]$$
\text{Payment} = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1},
$$[/tex]
where
- [tex]$P$[/tex] is the principal (loan amount),
- [tex]$r$[/tex] is the monthly interest rate, and
- [tex]$n$[/tex] is the total number of monthly payments.
In this problem:
1. The principal is
[tex]$$
P = \$170{,}000.
$$[/tex]
2. The annual interest rate is [tex]$12.6\%$[/tex], so the monthly interest rate is
[tex]$$
r = \frac{0.126}{12} = 0.0105.
$$[/tex]
3. The loan term is 20 years, so the total number of monthly payments is
[tex]$$
n = 20 \times 12 = 240.
$$[/tex]
Next, we substitute these values into the formula:
[tex]$$
\text{Payment} = \frac{170000 \times 0.0105 \times (1+0.0105)^{240}}{(1+0.0105)^{240} - 1}.
$$[/tex]
This expression matches the one given in option B. In addition, when you evaluate the factors, you would find that
- the factor [tex]$(1+0.0105)^{240}$[/tex] is approximately [tex]$12.266376321322747$[/tex], and
- the denominator [tex]$(1+0.0105)^{240} - 1$[/tex] is approximately [tex]$11.266376321322747$[/tex].
Thus, the monthly payment comes out to be approximately \$1943.44.
Therefore, the correct expression for the monthly payment is given in:
Option B.
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