https://file.aiquickdraw.com/mathbot/file/17284347606038uluxlad.jpg

### Problem Analysis:
The problem involves calculating a 30-year mortgage with a given APR and finding the remaining balance after 9 years of payments. Let's address each part.

#### Part (a):
For calculating the monthly payment, we can use the formula for a fixed-rate mortgage:

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

Where:
- $ M $ = Monthly Payment
- $ P $ = Principal amount ($307,000)
- $ r $ = Monthly interest rate (Annual rate divided by 12)
- $ n $ = Total number of payments (30 years × 12 months/year)

**Input values:**
- Principal $ P = 307,000 $
- Annual Rate = 10.4%, so Monthly Rate $ r = \frac{10.4\%}{12} = \frac{0.104}{12} \approx 0.00866667 $
- Number of Payments $ n = 30 \times 12 = 360 $

The calculated monthly payment $ M \approx 2785.32 $, which is confirmed in the screenshot.

#### Part (b):
To find the remaining balance after 108 payments (9 years), we use the formula for the remaining balance on a mortgage:

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

Where:
- $ B $ = Remaining balance
- $ P $ = Principal ($307,000)
- $ r $ = Monthly interest rate (same as above)
- $ n $ = Total number of payments (360)
- $ p $ = Number of payments already made (108)

Let's compute the remaining balance using the given values.### Results:
1. **Monthly Payment (Part a)**: The monthly payment is confirmed to be **$2785.32** (matching the given value in the screenshot).

2. **Remaining Balance after 9 Years (Part b)**: The remaining balance on the original loan after 108 payments (9 years) is **$284,856.70**.

Let me know if you need more details or have any questions.

### Related Questions:
1. What would the monthly payment be if the interest rate was reduced to 8%?
2. How much total interest would they pay over the 30-year period if they don’t refinance?
3. What would be the remaining balance after 15 years instead of 9 years?
4. How would the monthly payment change if the loan term was reduced to 20 years?
5. What would be the impact on the monthly payment if an additional $10,000 was added to the principal?

### Tip:
For calculating remaining mortgage balances, always ensure that the payment period is correctly aligned with the monthly interest rate and total number of payments.

Kosumi and Regina take out a 30-year mortgage for $307,000 at the APR of 10.4%, compounded monthly. (a) What will their monthly payments be? (b) After they had made 9 years of payments (108 payments) they decide to refinance the remaining loan balance. What is the balance remaining on the original loan of $307,000?

This problem involves calculating monthly payments and the remaining balance on a 30-year mortgage with an APR of 10.4%. Learn how to compute mortgage payments and the outstanding balance after 9 years of payments using the fixed-rate mortgage amortization formula.

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