Answer :
To determine the correct expression for calculating the monthly payment for a 30-year loan of [tex]$190,000 at an 11.4% annual interest rate compounded monthly, we can follow these steps:
1. Understand the Formula:
The formula for calculating the monthly payment of a fixed-rate loan is commonly represented as:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
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.
2. Identify the Variables:
- The principal (\( P \)) is $[/tex]190,000.
- The annual interest rate is 11.4%. To convert this to a monthly rate, divide by 12:
[tex]\[
r = \frac{11.4\%}{12} = \frac{0.114}{12} \approx 0.0095
\][/tex]
- The total number of payments ([tex]\( n \)[/tex]) for a 30-year loan is:
[tex]\[
n = 30 \times 12 = 360
\][/tex]
3. Select the Correct Expression:
Now, substitute these values into the standard formula:
[tex]\[
M = \frac{\$190,000 \times 0.0095 \times (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1}
\][/tex]
Compare this with the provided options:
- Option B matches our derived formula:
[tex]\[
\frac{\$190,000 \cdot 0.0095(1+0.0095)^{300}}{(1+0.0095)^{300}-1}
\][/tex]
After verifying, Option B is the expression that can be used to calculate the monthly payment. It aligns with the formula used to compute the monthly payments for a fixed-rate mortgage with the given parameters. The calculated monthly payment using this formula is approximately $1867.07.
1. Understand the Formula:
The formula for calculating the monthly payment of a fixed-rate loan is commonly represented as:
\[
M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1}
\]
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.
2. Identify the Variables:
- The principal (\( P \)) is $[/tex]190,000.
- The annual interest rate is 11.4%. To convert this to a monthly rate, divide by 12:
[tex]\[
r = \frac{11.4\%}{12} = \frac{0.114}{12} \approx 0.0095
\][/tex]
- The total number of payments ([tex]\( n \)[/tex]) for a 30-year loan is:
[tex]\[
n = 30 \times 12 = 360
\][/tex]
3. Select the Correct Expression:
Now, substitute these values into the standard formula:
[tex]\[
M = \frac{\$190,000 \times 0.0095 \times (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1}
\][/tex]
Compare this with the provided options:
- Option B matches our derived formula:
[tex]\[
\frac{\$190,000 \cdot 0.0095(1+0.0095)^{300}}{(1+0.0095)^{300}-1}
\][/tex]
After verifying, Option B is the expression that can be used to calculate the monthly payment. It aligns with the formula used to compute the monthly payments for a fixed-rate mortgage with the given parameters. The calculated monthly payment using this formula is approximately $1867.07.