To find out how much Derek owes on his mortgage after making 113 payments, we can use the remaining balance formula for a fixed-rate mortgage.
The formula for the remaining balance \( B \) after \( n \) payments is:
\[ B = P \times \frac{(1 + r)^N - (1 + r)^n}{(1 + r)^N - 1} \]
Where:
- \( B \) is the remaining balance on the mortgage.
- \( P \) is the original loan amount (the principal), which is $284,857.00.
- \( r \) is the monthly interest rate (annual interest rate divided by 12).
- \( N \) is the total number of payments (loan term in months).
- \( n \) is the number of payments made.
Step 1: Calculate the monthly interest rate
The annual interest rate is 5.33%, so the monthly interest rate \( r \) is:
\[ r = \frac{5.33%}{100} \div 12 = \frac{0.0533}{12} \approx 0.00444167 \]
Step 2: Determine the total number of payments
For a 30-year mortgage, the total number of payments \( N \) is:
\[ N = 30 \times 12 = 360 \]
Step 3: Plug the values into the formula
Where \( n = 113 \):
\[ B = 284,857 \times \frac{(1 + 0.00444167)^{360} - (1 + 0.00444167)^{113}}{(1 + 0.00444167)^{360} - 1} \]
Step 4: Calculate the parts of the formula
Calculate \( (1 + r)^{360} \) and \( (1 + r)^{113} \)
- \( (1 + 0.00444167)^{360} \)
\[ (1 + 0.00444167)^{360} \approx 5.250690017 \]
- \( (1 + 0.00444167)^{113} \)
\[ (1 + 0.00444167)^{113} \approx 1.647095048 \]
Step 5: Substitute back into the formula
\[ B = 284,857 \times \frac{5.250690017 - 1.647095048}{5.250690017 - 1} \]
Calculating the numerator and denominator:
- Numerator: \( 5.250690017 - 1.647095048 \approx 3.603594969 \)
- Denominator: \( 5.250690017 - 1 \approx 4.250690017 \)
So the formula now looks like this:
\[ B = 284,857 \times \frac{3.603594969}{4.250690017} \approx 284,857 \times 0.847291229 \]
Step 6: Calculate the remaining balance
\[ B \approx 241,144.6584 \]
Hence, after making 113 payments, the amount Derek owes on the mortgage is approximately $241,144.66.
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