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.

To solve this problem, we can break it down into two parts: the regular payments and the special payment. Since the first regular payment is made today, this is an annuity due problem. We will also account for the special payment in year 6. Let's compute $ X $, the remaining loan balance after considering the first regular payment and the special payment.

### Given:
- Loan amount: $ P = 240,203 $
- Annual interest rate: $ r = 9.76\% = 0.0976 $
- Regular annual payments: $ A = 22,820 $
- Special payment: $ S = 49,700 $
- Time of special payment: $ t = 6 $ years

### Steps:
1. **Determine the present value of the regular annuity payments** (since the first payment is made immediately, we use an annuity due formula).
2. **Calculate the present value of the special payment**, discounted back to today.
3. **Combine the present value of the annuity and special payment**, then solve for the total value $ X $.

We will calculate this step-by-step.The remaining balance after accounting for the present values of the regular payments and the special payment is approximately $101,919.81.

This represents the amount that is not covered by the annuity and special payment, meaning you need to pay this balance.

Would you like further details or clarification?

Here are 5 related questions you might find helpful:
1. How is an annuity due different from an ordinary annuity?
2. What would happen if the interest rate were higher or lower?
3. How do you calculate the future value of an annuity due?
4. What role does time value of money play in loan payments?
5. How does changing the frequency of payments (e.g., monthly) affect the loan calculation?

**Tip:** Always distinguish between annuity due and ordinary annuity when dealing with loans, as the timing of payments impacts the total present value.

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.

Solve a loan repayment problem with $240,203 borrowed, annual payments of $22,820, and a special payment of $49,700 in year 6 at an interest rate of 9.76%. Learn how to calculate the present value of annuity payments and special payments with detailed steps. Suitable for undergraduate finance students.

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