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 \( \$ 114,000 \). 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. b. Find the amount of the mortgage. \( \$ 91,200 \) c. How much must be paid for the three points at closing? \( \$ 2,736 \) (Round to the nearest dollar as needed.) d. Find the monthly payment (excluding escrowed taxes and insurance). \( \$ \square \) (Round to the nearest dollar as needed.)

To find the monthly payment amount, we can use the formula for the regular payment amount (PMT) in a fixed-rate mortgage: \[ \text{PMT} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} \] Given: - Price of the home: $114,000 - Down payment: 20% of $114,000 = $22,800 - Cost of the home financed: $114,000 - $22,800 = $91,200 - Interest rate: 10% - Loan term: 30 years First, we need to calculate the amount of the mortgage, which is the cost of the home financed. a. Amount of the mortgage: $91,200 Next, we need to find the amount to be paid for the three points at closing. b. Amount to be paid for the three points at closing: $2,736 Now, we can calculate the monthly payment amount using the formula for PMT. c. Monthly payment amount (excluding escrowed taxes and insurance): \[ \text{PMT} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} \] Substitute the values: - \( P = \$91,200 \) - \( r = 10\% = 0.10 \) - \( n = 12 \) (monthly payments) - \( t = 30 \) years \[ \text{PMT} = \frac{91,200\left(\frac{0.10}{12}\right)}{\left[1-\left(1+\frac{0.10}{12}\right)^{-12 \times 30}\right]} \] Let's calculate the monthly payment amount. Calculate the value by following steps: - step0: Calculate: \(\frac{91200\left(\frac{0.1}{12}\right)}{\left(1-\left(1+\frac{0.1}{12}\right)^{-12\times 30}\right)}\) - step1: Remove the parentheses: \(\frac{91200\left(\frac{0.1}{12}\right)}{1-\left(1+\frac{0.1}{12}\right)^{-12\times 30}}\) - step2: Divide the terms: \(\frac{91200\left(\frac{0.1}{12}\right)}{1-\left(1+\frac{1}{120}\right)^{-12\times 30}}\) - step3: Add the numbers: \(\frac{91200\left(\frac{0.1}{12}\right)}{1-\left(\frac{121}{120}\right)^{-12\times 30}}\) - step4: Divide the terms: \(\frac{91200\times \frac{1}{120}}{1-\left(\frac{121}{120}\right)^{-12\times 30}}\) - step5: Multiply the numbers: \(\frac{91200\times \frac{1}{120}}{1-\left(\frac{121}{120}\right)^{-360}}\) - step6: Multiply the numbers: \(\frac{760}{1-\left(\frac{121}{120}\right)^{-360}}\) - step7: Subtract the numbers: \(\frac{760}{\frac{121^{360}-120^{360}}{121^{360}}}\) - step8: Multiply by the reciprocal: \(760\times \frac{121^{360}}{121^{360}-120^{360}}\) - step9: Multiply: \(\frac{760\times 121^{360}}{121^{360}-120^{360}}\) The monthly payment amount (excluding escrowed taxes and insurance) is approximately $800.35.