https://file.aiquickdraw.com/mathbot/file/1726885369217ac0yzrub.jpg

I see that the image contains a few questions related to loan amortization and interest calculations. Let's break them down and solve them one by one.

### Problem 1:
**Question:**  
What monthly payment is required to amortize a loan of $35,000 over 11 years if interest at the rate of 9%/year is charged on the unpaid balance and interest calculations are made at the end of each month? (Round your answer to the nearest cent.)

To solve this, we can use the amortization formula:
\[
M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]
Where:
- $M$ = Monthly payment
- $P$ = Loan principal ($35,000)
- $r$ = Monthly interest rate (annual rate / 12 months)
- $n$ = Total number of payments (years × 12 months)

Let's compute this.

### Problem 2:
**Question:**  
The Flemings secured a bank loan of $392,000 to help finance the purchase of a house. The bank charges interest at a rate of 5%/year on the unpaid balance, and interest computations are made at the end of each month. The Flemings have agreed to repay the loan in equal monthly installments over 25 years. What should be the size of each repayment if the loan is to be amortized at the end of the term? (Round your answer to the nearest cent.)

We'll use the same formula as above, but with the new values:
- $P$ = $392,000
- $r$ = 5%/12 (monthly interest rate)
- $n$ = 25 years × 12 months

### Problem 3:
**Question:**  
A group of private investors purchased a condominium complex for $3 million. They made an initial down payment of 15% and obtained financing for the balance. If the loan is to be amortized over 13 years at an interest rate of 8.6%/year compounded quarterly, find the required quarterly payment. (Round your answer to the nearest cent.)

Here, the formula will be similar, but compounded quarterly:
- Loan principal after the down payment: $3 million × (1 - 0.15)
- Interest rate = 8.6%/4 (quarterly interest rate)
- $n$ = 13 years × 4 quarters

Let me calculate each of these step by step. I'll start with the first one.

The required monthly payment for the loan of $35,000 at a 9% annual interest rate, amortized over 11 years, is **$418.63**.

Now, let's move to the second problem and calculate the monthly payment for the $392,000 loan over 25 years at a 5% annual interest rate.The required monthly payment for the loan of $392,000 at a 5% annual interest rate, amortized over 25 years, is **$2,291.59**.

Next, let's solve the third problem, which involves quarterly payments for the loan on the condominium complex.The required quarterly payment for the loan on the condominium complex, with a 15% down payment and an 8.6% annual interest rate amortized over 13 years, is **$81,929.84**.

### Summary of Results:
1. **Loan 1:** Monthly payment for $35,000 loan at 9% interest over 11 years = **$418.63**.
2. **Loan 2:** Monthly payment for $392,000 loan at 5% interest over 25 years = **$2,291.59**.
3. **Loan 3:** Quarterly payment for the $3 million condo loan (after 15% down) at 8.6% interest over 13 years = **$81,929.84**.

If you have any questions or need more details, feel free to ask!

Here are some related questions:
1. What happens to the monthly payment if the interest rate changes in Loan 1?
2. How much total interest would be paid over the life of Loan 2?
3. What would happen to the quarterly payment if the condo loan term was extended to 15 years?
4. How does amortization work with varying interest rates throughout the loan term?
5. Can you calculate the remaining balance of Loan 2 after 10 years of payments?

**Tip:** When considering loan terms, shorter terms often mean higher monthly payments but lower overall interest paid.

Learn how to calculate monthly and quarterly payments for amortized loans with varying interest rates. This problem set covers key financial concepts, including amortization formulas and compound interest calculations, ideal for college-level students.

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