To find out how much Derek owes on his mortgage after 114 payments, we need to calculate the remaining balance on the mortgage. The formula for the remaining balance \( B \) on a fixed-rate mortgage after \( n \) payments is given by:
\[ B = P \frac{(1 + r)^N - (1 + r)^n}{(1 + r)^N - 1} \]
where:
- \( P \) is the principal (initial loan amount).
- \( r \) is the monthly interest rate (annual rate divided by 12).
- \( N \) is the total number of payments.
- \( n \) is the number of payments made so far.
Given values:
- Principal \( P = 311,156.00 \)
- Annual interest rate = 5.98%, so the monthly interest rate \( r = \frac{5.98%}{12} = \frac{0.0598}{12} \approx 0.00498333 \)
- Total number of payments \( N = 30 \times 12 = 360 \)
- Payments made \( n = 114 \)
Step 1: Calculate \( (1 + r)^N \)
\[ (1 + r)^N = (1 + 0.00498333)^{360} \approx 5.029356 \]
Step 2: Calculate \( (1 + r)^n \)
\[ (1 + r)^n = (1 + 0.00498333)^{114} \approx 1.648611 \]
Step 3: Calculate the remaining balance \( B \)
\[ B = 311,156 \frac{(5.029356 - 1.648611)}{(5.029356 - 1)} \]
Calculating \( 5.029356 - 1.648611 \): \[ 5.029356 - 1.648611 \approx 3.380745 \]
Now, calculating \( 5.029356 - 1 \): \[ 5.029356 - 1 \approx 4.029356 \]
Now substituting these values into the formula for \( B \): \[ B = 311,156 \frac{3.380745}{4.029356} \]
Calculating the fraction: \[ \frac{3.380745}{4.029356} \approx 0.839803 \]
Now, calculating \( B \): \[ B \approx 311,156 \times 0.839803 \approx 261,205.07 \]
Conclusion
After making 114 payments, Derek owes approximately $261,205.07 on the mortgage.