Cheryl and Greg agreed to pay $\$ 900000$ for a four-bedroom colonial home in Salem, Massachusetts, with a $9 \%$ down payment. They have a 25-year mortgage at a fixed rate of $6 \%$. (a) What is the amount of the downpayment? Round to the nearest cent. (b) What is the amount that Greg and Cheryl plan to borrow? Round to the nearest cent. (c) How much are the monthly payments? Round to the nearest cent. (d) Identify the type of the problem.

Step 1 ：The problem is about calculating the downpayment, the amount to borrow, and the monthly payments for a mortgage.

Step 2 ：The downpayment is calculated as a percentage of the total cost of the house. In this case, it is 9% of $900000. So, the downpayment is \(0.09 \times 900000 = \$81000.0\).

Step 3 ：The amount to borrow is the total cost of the house minus the downpayment. So, the amount to borrow is \(900000 - 81000 = \$819000.0\).

Step 4 ：The monthly payments can be calculated using the formula for the monthly payment on a fixed-rate mortgage, which is: \(P = \frac{r \times PV}{1 - (1 + r)^{-n}}\), where P is the monthly payment, r is the monthly interest rate (annual rate divided by 12), PV is the present value, or the amount borrowed, and n is the number of payments, or the term of the loan in months. Substituting the given values, we get the monthly payment as \$5276.83.

Step 5 ：Final Answer: \(\boxed{\text{(a) The amount of the downpayment is \$81000.0}}\)

Step 6 ：Final Answer: \(\boxed{\text{(b) The amount that Greg and Cheryl plan to borrow is \$819000.0}}\)

Step 7 ：Final Answer: \(\boxed{\text{(c) The monthly payments are \$5276.83}}\)