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Answer :
To find the monthly payment for the 20-year loan of [tex]$170,000 at an annual interest rate of 12.6%, compounded monthly, we can use the formula for calculating monthly payments on an annuity or loan, which is:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( M \) is the monthly payment,
- \( P \) is the principal loan amount,
- \( r \) is the monthly interest rate,
- \( n \) is the total number of payments (months).
Let's break down the steps to match this with one of the given expressions:
1. Identify the principal (\( P \)):
The principal amount is $[/tex]170,000.
2. Determine the monthly interest rate ([tex]\( r \)[/tex]):
The annual interest rate is 12.6%. To find the monthly rate, divide by 12:
[tex]\[ r = \frac{0.126}{12} = 0.0105 \][/tex]
3. Calculate the total number of payments ([tex]\( n \)[/tex]):
Since the loan is for 20 years and payments are monthly:
[tex]\[ n = 20 \times 12 = 240 \][/tex]
4. Plug these values into the formula:
[tex]\[
M = \frac{170000 \cdot 0.0105 \cdot (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}
\][/tex]
Now, compare this setup with the given options:
- Option A: [tex]\(\frac{\$ 170000 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}-1}\)[/tex]
This matches exactly with our formula set-up.
Therefore, the correct choice is Option A.
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( M \) is the monthly payment,
- \( P \) is the principal loan amount,
- \( r \) is the monthly interest rate,
- \( n \) is the total number of payments (months).
Let's break down the steps to match this with one of the given expressions:
1. Identify the principal (\( P \)):
The principal amount is $[/tex]170,000.
2. Determine the monthly interest rate ([tex]\( r \)[/tex]):
The annual interest rate is 12.6%. To find the monthly rate, divide by 12:
[tex]\[ r = \frac{0.126}{12} = 0.0105 \][/tex]
3. Calculate the total number of payments ([tex]\( n \)[/tex]):
Since the loan is for 20 years and payments are monthly:
[tex]\[ n = 20 \times 12 = 240 \][/tex]
4. Plug these values into the formula:
[tex]\[
M = \frac{170000 \cdot 0.0105 \cdot (1 + 0.0105)^{240}}{(1 + 0.0105)^{240} - 1}
\][/tex]
Now, compare this setup with the given options:
- Option A: [tex]\(\frac{\$ 170000 \cdot 0.0105(1+0.0105)^{240}}{(1+0.0105)^{240}-1}\)[/tex]
This matches exactly with our formula set-up.
Therefore, the correct choice is Option A.
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