read carefully: 

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 need to calculate the number of regular annual payments $ X $ that will fully repay the loan. We know the following:

- **Loan amount**: \$240,203.
- **Regular payment**: \$22,820 annually.
- **Special payment**: \$49,700 at year 6.
- **Interest rate**: 9.76% per year.
- **The first payment is made today** (this indicates that payments are made at the beginning of each period, i.e., it's an **annuity due**).

### Step-by-Step Plan:

1. **Break the loan into two parts**:
   - One part of the loan is covered by the regular annual payments.
   - The other part is covered by the special payment of \$49,700 in 6 years.

2. **Step 1: Discount the special payment**:
   Since the special payment of \$49,700 is made in 6 years, we need to discount it to the present value using the formula for present value of a single sum:

   \[
   PV_{\text{special}} = \frac{49,700}{(1 + 0.0976)^6}
   \]

3. **Step 2: Subtract the discounted value of the special payment**:
   After calculating the present value of the special payment, subtract it from the total loan amount to find the amount that will be repaid with regular annual payments.

4. **Step 3: Solve for the number of payments $ X $**:
   With the remaining loan amount, calculate how many annual payments of \$22,820 are needed using the present value of an annuity due formula:

   \[
   PV_{\text{annuity due}} = P \times \left( \frac{1 - (1 + r)^{-X}}{r} \right) \times (1 + r)
   \]
   
   Where:
   - $ P = 22,820 $ (regular payment),
   - $ r = 0.0976 $ (annual interest rate),
   - $ PV_{\text{annuity due}} $ is the remaining loan after subtracting the present value of the special payment.

Now let's calculate it step by step.The number of regular annual payments $ X $ required to fully repay the loan is approximately **18.73**.

Would you like more details on any part of the calculation, or do you have any questions?

Here are 5 related questions for further exploration:
1. How does changing the interest rate affect the number of payments required?
2. What would the impact be if the special payment were made in a different year?
3. How would the loan change if the payments were made at the end of each period instead of the beginning?
4. What happens to the loan term if the regular payments are increased or decreased?
5. How can one calculate the total interest paid over the life of the loan?

**Tip:** When working with loans, be sure to distinguish between annuities due (payments at the beginning of periods) and ordinary annuities (payments at the end), as this significantly impacts the calculation.

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 the final answer.

Solve a loan repayment problem involving $240,203 with regular annual payments of $22,820 and a special payment of $49,700 in 6 years. Learn how to apply the present value formula for an annuity due and calculate the number of payments needed at an interest rate of 9.76%.

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