Answer :
To solve the problem of finding the correct expression to calculate the monthly payment for a 20-year loan of [tex]$215,000 at an interest rate of 5.4% compounded monthly, we use the formula for calculating monthly payments on an installment loan. The formula is given by:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( M \) is the monthly payment,
- \( P \) is the principal loan amount,
- \( r \) is the monthly interest rate, and
- \( n \) is the total number of payments.
Step-by-step explanation:
1. Principal Loan Amount (\( P \)): This is given as $[/tex]215,000.
2. Annual Interest Rate: The given annual interest rate is 5.4%. Convert this to a monthly interest rate since we are dealing with monthly compounding:
[tex]\[
r = \frac{5.4\%}{12} = \frac{0.054}{12} \approx 0.0045
\][/tex]
Here, 0.0045 is the monthly interest rate expressed as a decimal.
3. Total Number of Payments ([tex]\( n \)[/tex]): Since it's a 20-year loan and payments are made monthly:
[tex]\[
n = 20 \times 12 = 240
\][/tex]
4. Substitute the values into the monthly payment formula:
[tex]\[
M = \frac{215000 \cdot 0.0045 \cdot (1 + 0.0045)^{240}}{(1 + 0.0045)^{240} - 1}
\][/tex]
5. Identify the correct expression based on the formula, which matches:
- [tex]\( \frac{\$ 215000 \cdot 0.0045 \cdot (1+0.0045)^{240}}{(1+0.0045)^{240}-1} \)[/tex]
Therefore, the correct answer is:
Option D: [tex]\(\frac{\$ 215000 \cdot 0.0045 \cdot (1+0.0045)^{240}}{(1+0.0045)^{240}-1}\)[/tex]
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( M \) is the monthly payment,
- \( P \) is the principal loan amount,
- \( r \) is the monthly interest rate, and
- \( n \) is the total number of payments.
Step-by-step explanation:
1. Principal Loan Amount (\( P \)): This is given as $[/tex]215,000.
2. Annual Interest Rate: The given annual interest rate is 5.4%. Convert this to a monthly interest rate since we are dealing with monthly compounding:
[tex]\[
r = \frac{5.4\%}{12} = \frac{0.054}{12} \approx 0.0045
\][/tex]
Here, 0.0045 is the monthly interest rate expressed as a decimal.
3. Total Number of Payments ([tex]\( n \)[/tex]): Since it's a 20-year loan and payments are made monthly:
[tex]\[
n = 20 \times 12 = 240
\][/tex]
4. Substitute the values into the monthly payment formula:
[tex]\[
M = \frac{215000 \cdot 0.0045 \cdot (1 + 0.0045)^{240}}{(1 + 0.0045)^{240} - 1}
\][/tex]
5. Identify the correct expression based on the formula, which matches:
- [tex]\( \frac{\$ 215000 \cdot 0.0045 \cdot (1+0.0045)^{240}}{(1+0.0045)^{240}-1} \)[/tex]
Therefore, the correct answer is:
Option D: [tex]\(\frac{\$ 215000 \cdot 0.0045 \cdot (1+0.0045)^{240}}{(1+0.0045)^{240}-1}\)[/tex]
