To determine 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 can break down the process using the standard formula for computing monthly mortgage payments.
The formula to calculate the monthly mortgage payment \( M \) is:
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( P \) is the principal loan amount, which is $[/tex]215,000.
- [tex]\( r \)[/tex] is the monthly interest rate, which is the annual interest rate divided by 12.
- [tex]\( n \)[/tex] is the total number of payments, which is the number of years multiplied by 12 (since payments are made monthly).
Given:
- Principal amount [tex]\( P = \$ 215,000 \)[/tex]
- Annual interest rate = 5.4%, so the monthly interest rate [tex]\( r = \frac{5.4\%}{12} = \frac{0.054}{12} = 0.0045 \)[/tex]
- Total number of monthly payments [tex]\( n = 20 \years \times 12 = 240 \)[/tex]
Let's consider each option provided and determine which fits the formula:
1. Option A:
[tex]\[ \frac{\$ 215,000 \cdot 0.0045 (1 - 0.0045)^{240}}{(1 - 0.0045)^{240} + 1} \][/tex]
2. Option B:
[tex]\[ \frac{\$ 215,000 \cdot 0.0045 (1 - 0.0045)^{240}}{(1 - 0.0045)^{240} - 1} \][/tex]
3. Option C:
[tex]\[ \frac{\$ 215,000 \cdot 0.0045 (1 + 0.0045)^{240}}{(1 + 0.0045)^{240} - 1} \][/tex]
4. Option D:
[tex]\[ \frac{\$ 215,000 \cdot 0.0045 (1 + 0.0045)^{240}}{(1 + 0.0045)^{240} + 1} \][/tex]
Since we follow the standard mortgage calculation formula, we need to use the one with [tex]\((1 + r)\)[/tex] in both the numerator and the denominator. Hence, comparing the options:
- Option A and B use [tex]\((1 - 0.0045)\)[/tex], which is incorrect.
- Option C and D use [tex]\((1 + 0.0045)\)[/tex].
Furthermore:
- Option C correctly subtracts 1 in the denominator, which aligns exactly with the standard mortgage formula.
- Option D adds 1 in the denominator, which is incorrect.
Therefore, the correct expression is:
[tex]\[ \frac{\$ 215,000 \cdot 0.0045(1 + 0.0045)^{240}}{(1 + 0.0045)^{240} - 1} \][/tex]
Thus, the correct option is C.
