A couple plans to purchase a house. The bank requires a \( 10 \% \) down payment on the \( \$ 310,000 \) house. The couple will finance the rest of the cost with a fixed-rate mortgage at \( 4.5 \% \) annual interest with monthly payments over 30 years. Complete the parts below. Do not round any intermediate computations. Round your final answers to the nearest cent if necessary. If necessary, refer to the list of financial formulas. (a) Find the required down payment. \( s \square] \) (b) Find the amount of the mortgage. s \( \square \) (c) Find the monthly payment. \( \$ \square \)

To solve this step-by-step, let's start with the required down payment. First, let's calculate the down payment: (a) The required down payment is \( 10\% \) of the house price, so we can calculate it as follows: \[ \text{Down Payment} = 0.10 \times 310,000 = 31,000 \] The down payment is **$31,000**. Next, we find the amount of the mortgage. (b) The mortgage amount will be the total house price minus the down payment: \[ \text{Mortgage Amount} = 310,000 - 31,000 = 279,000 \] The mortgage amount is **$279,000**. Finally, let's compute the monthly payment using the formula for a fixed-rate mortgage payment: - The formula for monthly payment is: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \] where: - \( M \) is the total monthly mortgage payment. - \( P \) is the loan principal (mortgage amount). - \( r \) is the monthly interest rate (annual rate / 12). - \( n \) is the number of payments (loan term in months). Given: - \( P = 279,000 \) - Annual interest rate = \( 4.5\% \), so \( r = \frac{4.5\%}{12} = \frac{0.045}{12} = 0.00375 \) - Loan term = \( 30 \, \text{years} = 30 \times 12 = 360 \, \text{months} \) Now we can substitute into the formula: \[ M = 279,000 \frac{0.00375(1 + 0.00375)^{360}}{(1 + 0.00375)^{360} - 1} \] Calculating, \[ (1 + 0.00375)^{360} \approx 3.8495 \] Thus, \[ M \approx 279,000 \frac{0.00375 \times 3.8495}{3.8495 - 1} \approx 279,000 \frac{0.01443625}{2.8495} \approx 279,000 \times 0.005065 \] So, \[ M \approx 1,413.43 \] This rounds to **$1,413.43**. Here are your final answers: (a) Down payment: **$31,000** (b) Mortgage amount: **$279,000** (c) Monthly payment: **$1,413.43**