https://file.aiquickdraw.com/mathbot/file/1725936485647d2pfil5c.jpg

Let's analyze and solve the mortgage-related questions shown in the image. Here's a breakdown of the information provided:

### Question 8:

Diane took a loan of $320,000 five years ago to finance a property purchase with the following details:
- **Original loan amount:** $320,000
- **Loan term:** 30 years
- **Interest rate:** 8% per year, compounded monthly

The questions are:

(a) What is Diane's current monthly mortgage payment?  
(b) What is Diane's current outstanding principal?  
(c) If Diane decides to refinance at a rate of 6.5% per year compounded monthly, what will her new monthly mortgage payment be?  
(d) How much less will Diane's monthly mortgage payment be if she refinances?

### Question 9:

George took an adjustable-rate mortgage (ARM) loan 5 years ago with the following details:
- **Original loan amount:** $300,000
- **Loan term:** 30 years
- **Interest rate for the first 5 years:** 9% per year, compounded monthly
- **New interest rate after 5 years:** 5.5% per year, compounded monthly

The question is:
- What will be the new monthly payment for George?

### Solution Approach:

To solve these problems, we'll use the formula for the monthly mortgage payment:

\[
M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]

where:
- $ M $ is the monthly mortgage payment,
- $ P $ is the loan principal,
- $ r $ is the monthly interest rate (annual rate divided by 12),
- $ n $ is the total number of payments (loan term in years multiplied by 12).

#### Let's calculate:

1. **Diane's Current Monthly Payment (a)**
2. **Diane's Current Outstanding Principal (b)**
3. **New Monthly Payment if Diane Refinances (c)**
4. **Difference in Payment after Refinancing (d)**
5. **George's New Monthly Payment (Question 9)**

I'll calculate these step-by-step now.Here are the calculations for the questions based on the given dataHere are the calculations for the questions based on the given 

### Question 8: Diane's Mortgage Calculations

(a) **Diane's current monthly mortgage payment:**  
\[
\text{Monthly Payment} = \$2,348.05
\]

(b) **Diane's current outstanding principal after 5 years:**  
\[
\text{Outstanding Principal} = \$304,223.54
\]

(c) **New monthly mortgage payment if Diane refinances at 6.5% per year:**  
\[
\text{New Monthly Payment} = \$1,922.90
\]

(d) **How much less will Diane's monthly mortgage payment be if she refinances?**  
\[
\text{Payment Difference} = \$425.15
\]

### Question 9: George's Mortgage Calculation

**George's new monthly mortgage payment after the rate is adjusted to 5.5% per year:**  
\[
\text{New Monthly Payment} = \$1,703.37
\]

Would you like more details on any of these calculations or have any specific questions?

### Follow-Up Questions:

1. How is the formula for calculating mortgage payments derived?
2. What happens to the mortgage payment if the loan term is reduced?
3. How does refinancing affect the total interest paid over the loan's duration?
4. What are the pros and cons of adjustable-rate mortgages (ARMs)?
5. How can you calculate the remaining principal for a different time period (e.g., 10 years)?

### Tip:  
Always consider refinancing options when interest rates drop significantly. It can lead to substantial savings on monthly payments and overall interest paid.

Diane took a $320,000 loan 5 years ago with an interest rate of 8% compounded monthly. George took a $300,000 adjustable-rate mortgage (ARM) loan with an initial rate of 9%, now adjusted to 5.5%.

Calculate the monthly mortgage payments for Diane's $320,000 loan at 8% interest and George's adjustable-rate mortgage at 9% now adjusted to 5.5%. Learn about loan amortization, refinancing, and how interest rate changes affect payments.

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