You just borrowed $240,203. You plan to repay this loan by making (X) regular annual payments of $22,820 and a special payment of special payment of $49,700 in 6 years. The interest rate on the loan is 9.76 percent per year and your first regular payment will be made today. What is X? Round to 2 decimal places for final answer.

We can solve this problem by calculating the number of regular payments (X) using the present value of annuities and lump-sum formulas.

### Key information:
- Loan amount (Present value, $ PV $) = $240,203
- Regular payment = $22,820
- Special payment in 6 years = $49,700
- Interest rate $ i = 9.76\% $ or 0.0976 per year
- First regular payment is made today, which implies an **annuity due**.

### Step-by-Step Process:

1. **Adjust the present value (PV) for the special payment**:  
   The special payment will be made in 6 years, so we need to find its present value, discounted at the interest rate.

   The formula for present value of a lump sum is:
   \[
   PV_{\text{lump sum}} = \frac{\text{Special Payment}}{(1 + i)^n}
   \]
   where $ n = 6 $ years, $ i = 0.0976 $, and the special payment is $49,700.

2. **Calculate the present value of the regular annuity payments (annuity due)**:  
   The formula for present value of an annuity due is:
   \[
   PV_{\text{annuity due}} = P \times \left[ \frac{1 - (1 + i)^{-X}}{i} \right] \times (1 + i)
   \]
   where $ P = 22,820 $ (regular payment), $ i = 0.0976 $, and $ X $ is the number of regular payments to be determined.

3. **Set up the equation**:  
   The total present value (loan amount) is the sum of the present values of the special payment and the annuity payments:
   \[
   PV = PV_{\text{annuity due}} + PV_{\text{lump sum}}
   \]
   Plug in the known values and solve for $ X $.

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You just borrowed $240,203. You plan to repay this loan by making (X) regular annual payments of $22,820 and a special payment of $49,700 in 6 years. The interest rate on the loan is 9.76 percent per year and your first regular payment will be made today. What is X? Round to 2 decimal places for final answer.

Learn how to calculate the number of regular payments required for a loan of $240,203 with an interest rate of 9.76% per year, incorporating a special payment of $49,700 in 6 years. The solution involves using the present value of annuities and lump-sum formulas, suitable for finance and business students.

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