The Turners have purchased a house for \( \$ 150,000 \). They made an initial down payment of \( 20 \% \) and secured a mortgage with interest charged at a rate of \( 3.5 \% / \) year on the unpaid balance. Assume that the loan is amortized over 30 years. (Round all answers to the nearest cent.) a) How much is the down payment? b) how much is borrowed from the bank? c) What monthly payment will the Turners be required to make? d) What will be their total price of the house? e) How much of the total amount is interest?

To solve the problem, we will go through each part step by step. ### a) How much is the down payment? The down payment is \( 20\% \) of the purchase price of the house. \[ \text{Down Payment} = 0.20 \times 150,000 = 30,000 \] ### b) How much is borrowed from the bank? The amount borrowed is the purchase price minus the down payment. \[ \text{Amount Borrowed} = 150,000 - 30,000 = 120,000 \] ### c) What monthly payment will the Turners be required to make? To calculate the monthly payment for a mortgage, we can use the formula for a fixed-rate mortgage: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( M \) = monthly payment - \( P \) = principal (amount borrowed) - \( r \) = monthly interest rate (annual rate / 12) - \( n \) = number of payments (loan term in months) Given: - \( P = 120,000 \) - Annual interest rate = \( 3.5\% \) or \( 0.035 \) - Monthly interest rate \( r = \frac{0.035}{12} \approx 0.00291667 \) - Loan term = \( 30 \) years = \( 30 \times 12 = 360 \) months Now we can plug in the values: \[ M = 120,000 \frac{0.00291667(1 + 0.00291667)^{360}}{(1 + 0.00291667)^{360} - 1} \] Calculating \( (1 + 0.00291667)^{360} \): \[ (1 + 0.00291667)^{360} \approx 2.89828 \] Now substituting back into the formula: \[ M = 120,000 \frac{0.00291667 \times 2.89828}{2.89828 - 1} \] Calculating the numerator: \[ 0.00291667 \times 2.89828 \approx 0.008466 \] Calculating the denominator: \[ 2.89828 - 1 \approx 1.89828 \] Now substituting these values: \[ M = 120,000 \frac{0.008466}{1.89828} \approx 120,000 \times 0.00446 \approx 535.20 \] Thus, the monthly payment is approximately: \[ \text{Monthly Payment} \approx 535.20 \] ### d) What will be their total price of the house? The total price of the house is the purchase price, which is: \[ \text{Total Price of the House} = 150,000 \] ### e) How much of the total amount is interest? To find the total amount paid over the life of the loan, we multiply the monthly payment by the total number of payments: \[ \text{Total Amount Paid} = M \times n = 535.20 \times 360 \approx 192,672 \] Now, we can find the total interest paid by subtracting the amount borrowed from the total amount paid: \[ \text{Total Interest} = \text{Total Amount Paid} - \text{Amount Borrowed} = 192,672 - 120,000 \approx 72,672 \] ### Summary of Answers: a) Down payment: \( \$30,000 \) b) Amount borrowed: \( \$120,000 \) c) Monthly payment: \( \$535.20 \) d) Total price of the house: \( \$150,000 \) e) Total interest: \( \$72,672 \)