Questions: Use PMT = P(r/n) / [1 - (1 + r/n)^(-nt)] to determine the regular payment amount, rounded to the nearest cent when a home is financed with a 140,000 30-year fixed-rate mortgage at 3.5%. a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage. a. The monthly payment is . (Do not round until the final answer. Then round to the nearest cent as needed.) The total interest for the loan is . (Use the answer from part a to find this answer. Round to the nearest cent as needed.) b. Fill out the loan amortization schedule for the first three months of the mortgage below.

Use PMT = P(r/n) / [1 - (1 + r/n)^(-nt)] to determine the regular payment amount, rounded to the nearest cent when a home is financed with a 140,000 30-year fixed-rate mortgage at 3.5%. a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage. a. The monthly payment is . (Do not round until the final answer. Then round to the nearest cent as needed.) The total interest for the loan is . (Use the answer from part a to find this answer. Round to the nearest cent as needed.) b. Fill out the loan amortization schedule for the first three months of the mortgage below.
Transcript text: Use $P M T=\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}$ to determine the regular payment amount, rounded to the nearest cent when a home is financed with a $\$ 140,000$ 30-year fixed-rate mortgage at $3.5 \%$. a. Find the monthly payments and the total interest for the loan. b. Prepare a loan amortization schedule for the first three months of the mortgage. a. The monthly payment is $\$$ $\square$ . (Do not round until the final answer. Then round to the nearest cent as needed.) The total interest for the loan is \$ $\square$ . (Use the answer from part a to find this answer. Round to the nearest cent as needed.) b. Fill out the loan amortization schedule for the first three months of the mortgage below.
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Solution

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Solution Steps

To solve this problem, we will use the given formula for calculating the monthly payment of a fixed-rate mortgage. First, we will identify the principal amount, interest rate, and loan term. Then, we will substitute these values into the formula to calculate the monthly payment. For the total interest, we will multiply the monthly payment by the total number of payments and subtract the principal. Finally, we will create a simple loan amortization schedule for the first three months by calculating the interest and principal portions of each payment.

Step 1: Calculate Monthly Payment

Using the formula for the monthly payment \( PMT \):

\[ PMT = \frac{P \left( \frac{r}{n} \right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \]

Substituting the values \( P = 140000 \), \( r = 0.035 \), \( n = 12 \), and \( t = 30 \):

\[ PMT = \frac{140000 \left( \frac{0.035}{12} \right)}{1 - \left(1 + \frac{0.035}{12}\right)^{-360}} \approx 628.6626 \]

Rounding to the nearest cent, the monthly payment is:

\[ \boxed{PMT = 628.66} \]

Step 2: Calculate Total Interest

The total payment over the life of the loan is given by:

\[ \text{Total Payment} = PMT \times n \times t = 628.6626 \times 12 \times 30 \approx 226318.5227 \]

The total interest paid is:

\[ \text{Total Interest} = \text{Total Payment} - P = 226318.5227 - 140000 \approx 86318.5227 \]

Rounding to the nearest cent, the total interest is:

\[ \boxed{\text{Total Interest} = 86318.52} \]

Step 3: Prepare Amortization Schedule for the First Three Months

For the first three months, we calculate the interest payment, principal payment, and remaining balance as follows:

  1. Month 1:

    • Interest Payment: \( 140000 \times \frac{0.035}{12} \approx 408.3333 \)
    • Principal Payment: \( 628.6626 - 408.3333 \approx 220.3292 \)
    • Remaining Balance: \( 140000 - 220.3292 \approx 139779.6708 \)

  2. Month 2:

    • Interest Payment: \( 139779.6708 \times \frac{0.035}{12} \approx 407.6907 \)
    • Principal Payment: \( 628.6626 - 407.6907 \approx 220.9719 \)
    • Remaining Balance: \( 139779.6708 - 220.9719 \approx 139558.6989 \)

  3. Month 3:

    • Interest Payment: \( 139558.6989 \times \frac{0.035}{12} \approx 407.0462 \)
    • Principal Payment: \( 628.6626 - 407.0462 \approx 221.6164 \)
    • Remaining Balance: \( 139558.6989 - 221.6164 \approx 139337.0826 \)

The amortization schedule for the first three months is summarized as follows:

  • Month 1: Interest Payment = \( 408.3333 \), Principal Payment = \( 220.3292 \), Remaining Balance = \( 139779.6708 \)
  • Month 2: Interest Payment = \( 407.6907 \), Principal Payment = \( 220.9719 \), Remaining Balance = \( 139558.6989 \)
  • Month 3: Interest Payment = \( 407.0462 \), Principal Payment = \( 221.6164 \), Remaining Balance = \( 139337.0826 \)

Final Answer

  • Monthly Payment: \( \boxed{628.66} \)
  • Total Interest: \( \boxed{86318.52} \)
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