Questions: Question 10 (8 points) For parts a-c, round your answer to the nearest cent. Do not include any symbols or commas. Example: 56789.12 A house costs 189,900. You pay 20% down and amortize the rest with equal monthly payments over a 30 year period. If you must pay 4.25% compounded monthly, a) What is the amount of the mortgage after the 20% down payment? b) what is your monthly payment? c) How much interest will you pay?

Solution Steps

The down payment is calculated as: \[ \text{Down Payment} = \text{House Cost} \times \text{Down Payment Percentage} = 189900 \times 0.20 = 37980 \]

The mortgage amount after the down payment is: \[ \text{Mortgage Amount} = \text{House Cost} - \text{Down Payment} = 189900 - 37980 = 151920 \]

The monthly payment can be calculated using the formula: \[ M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where:

• \( M \) is the monthly payment,
• \( P \) is the mortgage amount (\( 151920 \)),
• \( r \) is the monthly interest rate (\( \frac{0.0425}{12} = 0.00354166667 \)),
• \( n \) is the total number of payments (\( 30 \times 12 = 360 \)).

Substituting the values: \[ M = 151920 \times \frac{0.00354166667(1 + 0.00354166667)^{360}}{(1 + 0.00354166667)^{360} - 1} \approx 747.36 \]

The total amount paid over the life of the loan is: \[ \text{Total Paid} = M \times n = 747.36 \times 360 \approx 269047.83 \] The total interest paid is then: \[ \text{Total Interest Paid} = \text{Total Paid} - \text{Mortgage Amount} = 269047.83 - 151920 \approx 117127.83 \]

Final Answer