A couple plans to purchase a house. The bank requires a \( 10 \% \) down payment on the \( \$ 420,000 \) house. The couple will finance the rest of the cost with a fixed- rate mortgage at \( 5 \% \) 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. \( \$ \square \) (b) Find the amount of the mortgage. \( \$ \square \) (c) Find the monthly payment. \( \$ \square \)

To solve the couple's home financing questions, we can start with the calculations step-by-step: (a) To find the down payment, we calculate \(10\%\) of the house price: \[ \text{Down Payment} = 0.10 \times 420,000 = 42,000 \] So, the required down payment is \( \mathbf{\$42,000} \). (b) Next, we determine the amount of the mortgage, which is the total house cost minus the down payment: \[ \text{Mortgage Amount} = 420,000 - 42,000 = 378,000 \] Thus, the amount of the mortgage is \( \mathbf{\$378,000} \). (c) To find the monthly payment on a fixed-rate mortgage, we can use the mortgage payment formula: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where: - \(P = 378,000\) (the mortgage amount) - \(r = \frac{5\%}{12} = \frac{0.05}{12} = 0.00416667\) (monthly interest rate) - \(n = 30 \times 12 = 360\) (total number of payments) Substituting in the values: \[ M = 378,000 \frac{0.00416667(1 + 0.00416667)^{360}}{(1 + 0.00416667)^{360} - 1} \] Calculating inside the parentheses: \[ (1 + 0.00416667)^{360} \approx 4.46774 \] Now we can substitute and complete: \[ M = 378,000 \frac{0.00416667 \times 4.46774}{4.46774 - 1} \approx 378,000 \frac{0.018569}{3.46774} \approx 378,000 \times 0.005352 \approx 2,020.13 \] Thus, the monthly payment is approximately \( \mathbf{\$2,020.13} \).