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Answer :
To calculate the monthly payment on a fixed-rate loan with compound interest, we use the standard amortization formula
[tex]$$
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1},
$$[/tex]
where:
• [tex]$P$[/tex] is the principal amount (the loan amount),
• [tex]$r$[/tex] is the monthly interest rate, and
• [tex]$n$[/tex] is the total number of monthly payments.
For this loan:
1. The principal is [tex]$P = \$[/tex]170,\!000[tex]$.
2. The annual interest rate is $[/tex]12.6\%[tex]$. Since interest is compounded monthly, the monthly rate is
$[/tex][tex]$
r = \frac{12.6\%}{12} = \frac{0.126}{12} = 0.0105.
$[/tex][tex]$
3. The term of the loan is 20 years. With 12 months per year, the total number of payments is
$[/tex][tex]$
n = 20 \times 12 = 240.
$[/tex][tex]$
Substitute these values into the formula:
$[/tex][tex]$
M = \frac{170000 \times 0.0105 \times (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}.
$[/tex][tex]$
This expression exactly matches option D:
$[/tex][tex]$
\frac{\$[/tex]170000 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}-1}.
[tex]$$[/tex]
Thus, the correct answer is option D.
[tex]$$
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1},
$$[/tex]
where:
• [tex]$P$[/tex] is the principal amount (the loan amount),
• [tex]$r$[/tex] is the monthly interest rate, and
• [tex]$n$[/tex] is the total number of monthly payments.
For this loan:
1. The principal is [tex]$P = \$[/tex]170,\!000[tex]$.
2. The annual interest rate is $[/tex]12.6\%[tex]$. Since interest is compounded monthly, the monthly rate is
$[/tex][tex]$
r = \frac{12.6\%}{12} = \frac{0.126}{12} = 0.0105.
$[/tex][tex]$
3. The term of the loan is 20 years. With 12 months per year, the total number of payments is
$[/tex][tex]$
n = 20 \times 12 = 240.
$[/tex][tex]$
Substitute these values into the formula:
$[/tex][tex]$
M = \frac{170000 \times 0.0105 \times (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}.
$[/tex][tex]$
This expression exactly matches option D:
$[/tex][tex]$
\frac{\$[/tex]170000 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}-1}.
[tex]$$[/tex]
Thus, the correct answer is option D.
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