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. \( \$ \square \)

We start with a \$420,000 house and the bank requires a 10% down payment. Step (a): Down Payment 10% of \$420,000 = 0.10 × 420,000 = \$42,000. Step (b): Mortgage Amount The mortgage is the cost of the house minus the down payment. Mortgage amount = \$420,000 − \$42,000 = \$378,000. Step (c): Monthly Payment The mortgage is a fixed‐rate loan over 30 years at 5% annual interest with monthly payments. That is, there are n = 30 × 12 = 360 monthly payments, and the monthly interest rate is r = 5%/12 = 0.05/12. The formula for the monthly payment (M) on a loan of amount L is:   M = L × [r / (1 − (1 + r)^(–n))]. Substitute the values:   L = \$378,000, r = 0.05/12, and n = 360. First, compute the numerator:   L × r = 378,000 × (0.05/12). Since 0.05/12 = 0.00416666667, then   378,000 × 0.00416666667 = 1,575. Now, compute the denominator:   1 − (1 + r)^(–n) = 1 − (1.00416666667)^(–360). We can compute (1.00416666667)^360. (Using a calculator, one finds roughly:)   (1.00416666667)^360 ≈ 4.46774431. Thus,   (1.00416666667)^(–360) = 1 / 4.46774431 ≈ 0.22313016, and   Denom = 1 − 0.22313016 ≈ 0.77686984. Finally, calculate the monthly payment:   M = 1,575 / 0.77686984 ≈ 2,027.27. Rounded to the nearest cent, the monthly payment is about \$2,027.27. Summary of Answers: (a) Down Payment: \$42,000.00 (b) Mortgage Amount: \$378,000.00 (c) Monthly Payment: \$2,027.27