To solve this step by step, let's use the provided formula and data:

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

Step-by-step Solution:

a. Find the required down payment.

The bank requires a 20% down payment. The down payment is: $\text{Down Payment} = \text{Price of the Home} \times 20\%$ $\text{Down Payment} = 117,000 \times 0.2 = 23,400$

b. Find the amount of the mortgage.

The mortgage amount is the price of the home minus the down payment: $\text{Mortgage Amount} = \text{Price of the Home} - \text{Down Payment}$ $\text{Mortgage Amount} = 117,000 - 23,400 = 93,600$

c. How much must be paid for the three points at closing?

Three points represent 3% of the mortgage amount: $\text{Points} = \text{Mortgage Amount} \times 3\%$ $\text{Points} = 93,600 \times 0.03 = 2,808$

d. Find the monthly payment (excluding escrowed taxes and insurance).

Given:

• $P = 93,600$ (loan amount after down payment),
• $r = 0.07$ (annual interest rate),
• $n = 12$ (monthly payments per year),
• $t = 30$ (years of the mortgage).

Substitute these values into the formula: $PMT = \frac{P \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}$

First, calculate:

• Monthly interest rate $\frac{r}{n} = \frac{0.07}{12} = 0.005833$,
• Total number of payments $nt = 12 \times 30 = 360$.

The formula becomes: $PMT = \frac{93,600 \times 0.005833}{1 - (1 + 0.005833)^{-360}}$

Calculate step-by-step:

1. $1 + 0.005833 = 1.005833$,
2. $(1.005833)^{-360} = 0.20276$,
3. $1 - 0.20276 = 0.79724$,
4. $93,600 \times 0.005833 = 546$,
5. $\frac{546}{0.79724} = 684.86$.

The monthly payment is approximately: $PMT \approx 685$

e. Find the total cost of interest over 30 years.

The total cost of the loan is the monthly payment multiplied by the total number of payments: $\text{Total Payment} = PMT \times nt$ $\text{Total Payment} = 685 \times 360 = 246,600$

The total interest paid is: $\text{Total Interest} = \text{Total Payment} - \text{Mortgage Amount}$ $\text{Total Interest} = 246,600 - 93,600 = 153,000$

Final Answers:

1. Down Payment: $\$23,400$
2. Mortgage Amount: $\$93,600$
3. Three Points at Closing: $\$2,808$
4. Monthly Payment: $\$685$
5. Total Cost of Interest: $\$153,000$

Let me know if you want the calculations broken down further or have additional questions! Here are five relative questions to deepen understanding:

1. How does the interest rate impact the total cost of a loan over its term?
2. Why is a larger down payment beneficial in the long term?
3. What is the effect of choosing a shorter loan term on monthly payments and total interest?
4. How do "points" at closing affect the initial costs of a mortgage?
5. How would a slight change in the annual interest rate (e.g., from 7% to 6.5%) impact the monthly payment and total interest?

Tip: When planning to buy a home, consider prepaying a portion of the loan or increasing monthly payments to reduce the overall interest paid over time.