Dave takes out a 23-year mortgage of 220000 dollars for his new house. Dave gets an interest rate of 13.2 percent compounded monthly. He agrees to make equal monthly payments, the first coming in one month. After making the 66th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 700 dollars, and to get a better interest rate. In particular, he negotiates a new rate of 8.4 percent compounded monthly, and agrees to make equal monthly payments (each 700 dollars less than his original payments) for as long as necessary, followed by a single smaller payment. How large will Dave's final loan payment be?

To solve this problem, we need to first understand the situation and break it down into stages:

1. **Initial Loan Information:**
   - Loan Amount ($ P $): \$220,000
   - Mortgage Duration: 23 years (which means $ 23 \times 12 = 276 $ months)
   - Interest Rate: 13.2% compounded monthly, i.e., monthly interest rate $ r = \frac{13.2\%}{12} = 0.011 $.
   - Monthly payment amount: unknown for now, denoted as $ M $.

2. **Step 1: Calculate Dave’s original monthly payment $ M $:**
   We use the formula for monthly payments on a fixed-rate mortgage:
   \[
   M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
   \]
   Where:
   - $ P = 220,000 $ is the principal
   - $ r = 0.011 $ is the monthly interest rate
   - $ n = 276 $ months is the total number of payments

3. **Step 2: Determine the loan balance after 66 months:**
   After 66 months, Dave wants to refinance. To find the remaining balance after 66 months, we use the balance formula:
   \[
   B = P(1 + r)^n - \frac{M[(1 + r)^n - (1 + r)^k]}{r}
   \]
   Where:
   - $ B $ is the remaining balance after $ k = 66 $ payments
   - $ M $ is the monthly payment we will calculate in Step 1
   - $ r = 0.011 $
   - $ n = 276 $, the original loan term

4. **Step 3: Refinance details:**
   After refinancing, Dave’s new interest rate is 8.4% compounded monthly, i.e., monthly interest rate $ r_{\text{new}} = \frac{8.4\%}{12} = 0.007 $.

   The new monthly payment will be $ M - 700 $, where $ M $ is the original monthly payment.

   Let the number of payments for the new mortgage be $ N $, followed by a single smaller payment. The goal is to calculate the final payment after these equal payments.

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Let's proceed with the calculations. I'll start by finding Dave's original monthly payment.Dave's original monthly payment is approximately \$2,544.23.

Next, we will calculate the loan balance after Dave makes 66 payments and wants to refinance his mortgage. Let me do that now.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can still guide you through the steps to calculate the remaining balance and final payment manually. Let me know if you'd like further details or have any questions.

Here are some related questions to expand on:
1. How does compound interest impact mortgage payments?
2. What is the amortization formula and how is it used in loan calculations?
3. How would Dave's monthly payment change if the interest rate were lower from the start?
4. How is the remaining loan balance calculated after making a certain number of payments?
5. What are the benefits of refinancing a mortgage?

**Tip**: Refinancing a loan at a lower interest rate can significantly reduce your long-term costs, even if it means extending the loan duration.

Learn how to calculate Dave's final loan payment after refinancing his mortgage. This real-world problem covers key concepts like compound interest, loan amortization, and refinancing. Discover the steps involved in determining the remaining loan balance and new payment terms, ideal for college-level math students.

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