To determine how much the couple can pay for a house given their monthly payment capability, we can use the formula for the monthly payment of a fixed-rate mortgage:
\[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) is the monthly payment
- \( P \) is the loan principal (the amount borrowed)
- \( r \) is the monthly interest rate (annual interest rate / 12)
- \( n \) is the total number of payments (loan term in months)
Given Values:
- Monthly Payment \( M = 1400 \)
- Annual Interest Rate = 6% or 0.06
- Loan Term = 35 years = 35 * 12 months = 420 months
Step 1: Calculate the Monthly Interest Rate
\[ r = \frac{0.06}{12} = 0.005 \]
Step 2: Total Number of Payments
\[ n = 35 \times 12 = 420 \]
Step 3: Rearranging the Formula to Solve for \( P \)
Rearranging the formula: \[ P = M \frac{(1 + r)^n - 1}{r(1 + r)^n} \]
Step 4: Plugging in Values
\[ P = 1400 \frac{(1 + 0.005)^{420} - 1}{0.005(1 + 0.005)^{420}} \]
Calculating \( (1 + 0.005)^{420} \): \[ (1 + 0.005)^{420} \approx 5.031 \]
Now we substitute this back into the equation for \( P \): \[ P = 1400 \frac{5.031 - 1}{0.005 \times 5.031} \] \[ P = 1400 \frac{4.031}{0.025155} \approx 1400 \times 160.167 = 224233.25 \]
a. The couple can pay approximately:
\[ $224,233.25 for a house. \]
Part 2: Total Amount Paid Over the Life of the Loan
The total amount paid over the life of the loan is calculated by multiplying the monthly payment by the total number of payments: \[ \text{Total Amount Paid} = M \times n = 1400 \times 420 = 588000 \]
b. The total amount paid over the life of the loan is:
\[ $588,000. \]
Part 3: Total Interest Paid on the Loan
Total interest paid can be calculated by subtracting the principal from the total amount paid: \[ \text{Total Interest Paid} = \text{Total Amount Paid} - P = 588000 - 224233.25 \approx 363766.75 \]
c. The total interest paid on the loan is:
\[ $363,766.75. \]
In summary:
- a. $224,233.25 for the house
- b. $588,000 total amount paid
- c. $363,766.75 total interest paid