Question
Derek borrows $284,857.00 to buy a house. He has a 30-year mortgage with a rate of 5.33%. After making 113.00 payments, how much does he owe on the mortgage?
Answers
GPT-4o mini
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. \( (1 + 0.00444167)^{360} \)
\[
(1 + 0.00444167)^{360} \approx 5.250690017
\]
2. \( (1 + 0.00444167)^{113} \)
\[
(1 + 0.00444167)^{113} \approx 1.647095048
\]
### 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**.
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. \( (1 + 0.00444167)^{360} \)
\[
(1 + 0.00444167)^{360} \approx 5.250690017
\]
2. \( (1 + 0.00444167)^{113} \)
\[
(1 + 0.00444167)^{113} \approx 1.647095048
\]
### 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**.
123
What values do I enter for N, PMT, i/y, FV and PV on a finance calculator to solve
123
So, what values do I enter for N, PMT, i/y, FV and PV on a finance calculator to solve for this problem