Dave takes out a 24-year mortgage of 260000 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 68th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 600 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 600 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 break it down into two stages: (1) the original mortgage plan and (2) the refinanced mortgage plan.

### Stage 1: Original Mortgage Plan
We are given the following information:
- Loan amount: $ 260,000 $ dollars
- Annual interest rate: 15.6%, compounded monthly
- Term: 24 years
- Payments: monthly

#### Step 1: Determine the monthly payment in the original mortgage plan

The monthly interest rate $ i_1 $ is calculated as:

\[
i_1 = \frac{15.6\%}{12} = \frac{0.156}{12} = 0.013
\]

The total number of payments $ n_1 $ for a 24-year mortgage with monthly payments is:

\[
n_1 = 24 \times 12 = 288
\]

The formula for the monthly mortgage payment $ P $ is:

\[
P = \frac{i_1 \cdot L}{1 - (1 + i_1)^{-n_1}}
\]

Where:
- $ L = 260,000 $ (the loan amount)
- $ i_1 = 0.013 $ (monthly interest rate)
- $ n_1 = 288 $ (number of payments)

Let's calculate the monthly payment $ P $.

#### Step 2: Calculate the remaining balance after 68 payments

After making 68 payments, Dave has reduced the loan balance. The remaining balance $ B_{68} $ can be calculated using the following formula:

\[
B_{68} = L \cdot \left( \frac{(1 + i_1)^{68} - 1}{(1 + i_1)^{288} - 1} \right)
\]

Let's calculate this remaining balance after 68 months.

---

### Stage 2: Refinanced Mortgage Plan

Dave refinances his mortgage after 68 months. In the new plan, the information is:
- New interest rate: 8.4%, compounded monthly
- The new monthly payment is $ 600 $ dollars less than the original payment $ P $.

#### Step 3: Determine the new monthly payment

The new monthly interest rate $ i_2 $ is:

\[
i_2 = \frac{8.4\%}{12} = \frac{0.084}{12} = 0.007
\]

The new monthly payment is $ P - 600 $. The number of new payments $ n_2 $ is unknown.

#### Step 4: Calculate the final loan payment

Dave makes equal monthly payments in the new plan for as long as necessary, followed by a final smaller payment. The remaining balance formula will be used to find out the final payment after paying the reduced monthly payments for the remaining period.

Let's calculate all of these values step by step. I'll begin by computing the original monthly payment $ P $.The original monthly payment Dave was making is approximately $ 3463.96 $ dollars.

Next, let's compute the remaining balance after making 68 payments.It seems that the calculated remaining balance after 68 payments is too low, indicating an issue with the formula used for the remaining balance. Let me correct the approach and recompute the balance using the correct mortgage balance formula:

\[
B_{68} = P \cdot \frac{1 - (1 + i_1)^{-(n_1 - 68)}}{i_1}
\]

I'll recalculate this balance for you.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can guide you through the steps manually if you'd like to continue with the problem, or you can upload more details for a different calculation. Let me know how you’d like to proceed!

Would you like to:
1. Understand the remaining balance formula after 68 payments?
2. Calculate the reduced monthly payment with the new interest rate?
3. Determine how the refinancing impacts the loan term?
4. Discuss how interest rates affect mortgage costs?
5. Explore how Dave's final payment is affected by monthly reductions?

**Tip:** Refinancing a mortgage typically reduces monthly payments, but can increase the total interest paid over time unless carefully planned!

Dave takes out a 24-year mortgage of 260000 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 68th payment, Dave wants to buy a boat, so he wants to refinance his house to reduce his monthly payment by 600 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 600 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?

Solve a mortgage refinancing problem where Dave initially takes out a 24-year mortgage of $260,000 at 15.6% interest. After 68 payments, he refinances with an 8.4% interest rate, reducing his monthly payment by $600. This example covers key concepts such as amortization, compound interest, and loan refinancing.

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