Use PMT \( =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) to determine the regular payment amount, rounded to the nearest dollar. The price of a home is \( \$ 183,000 \). The bank requires a \( 20 \% \) down payment and three points at the time of closing. The cost of the home is financed with a 30-year fixed-rate mortgage at \( 10 \% \). Complete parts (a) through (e) below. a. Find the required down payment. \( \$ 36,600 \) b. Find the amount of the mortgage. \( \$ \square \)

To solve the problem step by step, we will first extract the known conditions and then proceed to calculate the required values. ### Known Conditions: - Price of the home \( P = 183,000 \) - Down payment percentage = \( 20\% \) - Interest rate \( r = 10\% = 0.10 \) - Loan term \( t = 30 \) years - Number of payments per year \( n = 12 \) (monthly payments) ### Step a: Find the required down payment The down payment can be calculated as follows: \[ \text{Down Payment} = P \times \text{Down Payment Percentage} \] Substituting the values: \[ \text{Down Payment} = 183,000 \times 0.20 \] Calculating this gives: \[ \text{Down Payment} = 36,600 \] ### Step b: Find the amount of the mortgage The amount of the mortgage is the price of the home minus the down payment: \[ \text{Mortgage Amount} = P - \text{Down Payment} \] Substituting the values: \[ \text{Mortgage Amount} = 183,000 - 36,600 \] Calculating this gives: \[ \text{Mortgage Amount} = 146,400 \] Now, we have the required down payment and the amount of the mortgage: - a. Required down payment: \( \$ 36,600 \) - b. Amount of the mortgage: \( \$ 146,400 \)