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This problem involves calculating mortgage payments under two different interest rates and determining the final payment after refinancing.

### Step 1: Initial Loan Information
- Principal: $ P = 270,000 $ dollars
- Initial interest rate: $ r_1 = 15.6\% $ per year, compounded monthly.
  - Monthly rate: $ r_1/12 = 0.156/12 = 0.013 $ per month
- Loan term: 26 years or 312 months.
- First 66 months are paid under this rate.

#### Monthly payment formula:
The monthly payment is calculated using the formula for a fixed-rate mortgage:

\[
M = \frac{P \cdot \left(\frac{r}{12}\right)}{1 - (1 + \frac{r}{12})^{-n}}
\]

Where:
- $ P $ is the loan amount.
- $ r $ is the annual interest rate (15.6% initially).
- $ n $ is the total number of payments.

Now let me calculate the initial monthly payment using this formula and then determine how much principal is left after 66 payments. Then, I'll apply the new interest rate and determine the final payment after refinancing. 

Let me start by calculating the initial monthly payment.Dave's initial monthly payment is approximately \$3,573.53.

### Step 2: Calculate the remaining balance after 66 payments
Next, I'll determine the remaining balance on the loan after Dave has made 66 payments.

The remaining balance on a mortgage after a certain number of payments can be calculated using the formula:

\[
B = P \times \left(1 + r_1 / 12\right)^{66} - M_1 \times \frac{\left(1 + r_1 / 12\right)^{66} - 1}{r_1 / 12}
\]

Where:
- $ P $ is the original loan amount.
- $ r_1 $ is the monthly interest rate (15.6% compounded monthly).
- $ M_1 $ is the monthly payment.
- $ 66 $ is the number of payments made so far.

Let me compute the remaining balance.After making 66 payments, the remaining balance on Dave's mortgage is approximately \$263,425.40.

### Step 3: Refinancing the Loan
- New interest rate: $ r_2 = 8.4\% $ compounded monthly
  - Monthly rate: $ r_2/12 = 0.084/12 = 0.007 $
- Dave wants to reduce his monthly payments by \$400.

His new monthly payment will be:
\[
M_2 = M_1 - 400
\]
Now, let me calculate the new monthly payment under the new interest rate.After refinancing, Dave's new monthly payment is approximately \$3,173.53, which is \$400 less than his original payment.

### Step 4: Final Loan Payment Calculation
Since the loan's remaining balance after 66 payments is \$263,425.40, I will now determine how long Dave will continue to pay the new reduced monthly amount of \$3,173.53 and calculate the final smaller payment, assuming it will be needed to cover any leftover balance.

Using the new interest rate of 8.4%, I'll calculate the number of payments and the final small payment required.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to help manually guide through the final steps of the calculation or assist with something else? Let me know if you need more clarification on the steps so far.

Here are 5 related questions to explore:
1. How would the loan change if Dave refinanced earlier or later?
2. What would happen if Dave chose a longer loan term during refinancing?
3. How does the monthly payment formula work for different loan types (e.g., fixed-rate vs. variable-rate mortgages)?
4. How does the total interest paid compare between the two interest rates Dave considered?
5. How would extra payments affect the total loan duration and interest paid?

**Tip:** Refinancing to a lower interest rate earlier in the loan term can significantly reduce the total interest paid over the life of the loan.

Dave takes out a 26-year mortgage of 270,000 dollars for his new house. Dave gets an interest rate of 15.6 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 400 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 400 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?

Explore a detailed solution to Dave's mortgage problem, where he refinances his 26-year mortgage after 66 payments. Learn how to calculate the remaining balance, new monthly payments, and final smaller payment using compound interest and loan amortization formulas.

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