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

​$180 comma 000

20

​-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.

Let's break down this problem into two parts: calculating the monthly payment and then preparing the loan amortization schedule for the first three months.

### Part a: Finding the Monthly Payment and Total Interest

The formula for the monthly payment (PMT) is given by:

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

where:
- $ P $ = Principal amount (the loan amount, which is \$180,000)
- $ r $ = Annual interest rate (3.5% = 0.035)
- $ n $ = Number of payments per year (for monthly payments, $ n = 12 $)
- $ t $ = Number of years (in this case, 20 years)

#### Step 1: Plug in the values
- $ P = 180,000 $
- $ r = 0.035 $
- $ n = 12 $
- $ t = 20 $

Now, we'll calculate the monthly payment.

I'll calculate this now.The monthly payment (PMT) for the mortgage is **$1,043.93**.

#### Step 2: Calculate the total interest

To find the total interest paid over the life of the loan, we need to calculate the total amount paid and subtract the principal from it.

The total amount paid is the monthly payment multiplied by the total number of payments ($ n \times t $):

\[
\text{Total Paid} = PMT \times n \times t
\]

Then, subtract the principal to find the total interest:

\[
\text{Total Interest} = \text{Total Paid} - P
\]

I'll calculate this now.The total amount paid over the life of the loan is **$250,543.20**, and the total interest paid is **$70,543.20**.

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

For an amortization schedule, each monthly payment consists of two parts:
1. **Interest payment**: Calculated as $ \text{Interest} = \frac{r}{n} \times \text{remaining principal} $
2. **Principal payment**: The remaining part of the payment after the interest is paid.

The remaining balance decreases with each payment, and the interest is recalculated accordingly.

I'll calculate the breakdown for the first three months.Here is the loan amortization schedule for the first three months:

| Month | Payment | Interest Payment | Principal Payment | Remaining Balance |
|-------|---------|------------------|-------------------|-------------------|
| 1     | $1,043.93 | $525.00         | $518.93           | $179,481.07        |
| 2     | $1,043.93 | $523.49         | $520.44           | $178,960.63        |
| 3     | $1,043.93 | $521.97         | $521.96           | $178,438.66        |

### Summary:
- The monthly payment is **$1,043.93**.
- Total interest paid over the life of the loan will be **$70,543.20**.
- The principal and interest portions of the payments increase and decrease respectively as the loan progresses.

Do you have any questions or would you like further details on any part?

Here are 5 related questions to explore further:
1. How does increasing the interest rate affect the monthly payment and total interest?
2. What happens to the total interest paid if the loan term is reduced to 15 years?
3. How would bi-weekly payments impact the amortization schedule?
4. What is the formula for calculating the remaining balance at any point in the loan term?
5. How would the schedule look if extra payments were made monthly?

**Tip:** Making extra payments towards the principal early in the loan can significantly reduce the total interest paid over time.

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 $180,000 20-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.

Learn how to calculate the monthly payments and total interest for a $180,000 20-year fixed-rate mortgage at 3.5%. This problem demonstrates how to use the loan amortization formula and provides a loan amortization schedule for the first three months. Suitable for students at the college level or advanced high school (Grades 11-12).

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