Use PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndEndFraction
to determine the regular payment​ amount, rounded to the nearest cent. The cost of a home is financed with a ​$130,000. 30-year ​fixed-rate mortgage at 4.5%.
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.

To solve this problem, we'll use the given formula for the monthly payment (PMT) on a mortgage, which is a variant of the standard loan payment formula:

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

Where:
- $ P $ is the loan amount (principal), which is $130,000.
- $ r $ is the annual interest rate (in decimal), which is 4.5% or 0.045.
- $ n $ is the number of payments per year (monthly payments means $ n = 12 $).
- $ t $ is the loan term in years, which is 30 years.

### Part (a): Find the Monthly Payment and Total Interest

#### Step 1: Plug the values into the formula:
- $ P = 130,000 $
- $ r = 0.045 $
- $ n = 12 $
- $ t = 30 $

Now, let's calculate the monthly payment (PMT). I'll compute it step by step.

#### Step 2: Calculate the total interest.
The total interest paid over the loan's life is calculated by subtracting the principal from the total payments made over the life of the loan:
\[
\text{Total Interest} = \left( \text{Monthly Payment} \times 12 \times 30 \right) - P
\]

I'll perform the calculations for both the monthly payment and total interest.

#### Part (b): Loan Amortization Schedule (First Three Months)

For the loan amortization schedule, we need to break down the monthly payment into the amount going toward interest and the amount going toward principal. The formula for the interest and principal portion of each payment is:

- **Interest Payment for the month**:
\[
\text{Interest}_{\text{month}} = \text{remaining principal} \times \frac{r}{n}
\]

- **Principal Payment for the month**:
\[
\text{Principal}_{\text{month}} = \text{PMT} - \text{Interest}_{\text{month}}
\]

The remaining principal after each payment is updated as:
\[
\text{New Principal} = \text{Old Principal} - \text{Principal}_{\text{month}}
\]

I'll calculate the first three months of this schedule.### Part (a): Monthly Payment and Total Interest

1. **Monthly Payment (PMT)**:  
   The monthly payment is approximately **$658.69**.

2. **Total Interest**:  
   Over the life of the loan (30 years), the total interest paid will be approximately **$107,128.72**.

### Part (b): Loan Amortization Schedule for the First Three Months

| Month | Payment (PMT) | Interest Payment | Principal Payment | Remaining Principal |
|-------|---------------|------------------|-------------------|---------------------|
| 1     | $658.69       | $487.50          | $171.19           | $129,828.81         |
| 2     | $658.69       | $486.86          | $171.83           | $129,656.98         |
| 3     | $658.69       | $486.21          | $172.48           | $129,484.50         |

As you can see, with each payment, a slightly larger portion goes towards paying off the principal, while the interest portion decreases.

Would you like any more details or calculations?

### Related Questions:
1. How is the total loan balance affected over the first year?
2. What is the principal remaining after 10 years of payments?
3. How would a change in the interest rate to 5% affect the monthly payment?
4. What would be the effect of making an additional monthly payment of $100 towards the principal?
5. How does the amortization schedule change in a 15-year mortgage instead of a 30-year one?

### Tip:
Making extra payments towards the principal can significantly reduce the total interest paid over the life of the loan.

Use PMT equals StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndEndFraction to determine the regular payment​ amount, rounded to the nearest cent. The cost of a home is financed with a ​$130,000. 30-year ​fixed-rate mortgage at 4.5%. 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.

Learn how to calculate the monthly mortgage payment for a $130,000 loan with a 30-year fixed rate of 4.5%. This guide covers step-by-step calculations, including finding the total interest paid and preparing a detailed loan amortization schedule for the first three months. Suitable for advanced high school and college-level students.

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