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.

find  the REGULAR ANNUAL PAYMENTS

To solve this problem, we need to find the number of regular annual payments (denoted as $ X $) you'll make, where you are repaying a loan of $ 240,203 $ with regular payments of $ 22,820 $, a special payment of $ 49,700 $ in 6 years, and the interest rate is $ 9.76\% $ per year.

Here are the steps we'll follow:

### Step 1: Understand the Problem
The loan is being repaid in the following manner:
- You make a regular payment of $ 22,820 $ at the beginning of each year, including today.
- After 6 years, you'll make an additional special payment of $ 49,700 $.
- The loan has an annual interest rate of $ 9.76\% $.

We need to find how many regular payments $ X $ will pay off the entire loan.

### Step 2: Use Present Value (PV) Formulas
We need to split this problem into two parts:
1. **Present Value of Regular Payments (Annuity):**
   This is the present value of the regular payments, considering that the first payment is made immediately (today). This requires adjusting for payments made at the beginning of the year (an annuity due).

2. **Present Value of the Special Payment:**
   This is the present value of the special payment of $ 49,700 $, which occurs in 6 years, discounted back to today.

### Step 3: Calculate the Present Value of the Annuity
Since payments are made at the beginning of each year (annuity due), the present value of an annuity due can be calculated using the formula:

\[
PV_{\text{annuity}} = P \times \left( 1 + r \right) \times \left[ 1 - \frac{1}{(1 + r)^X} \right] \div r
\]

Where:
- $ P = 22,820 $ is the regular payment
- $ r = 0.0976 $ is the interest rate (9.76% as a decimal)
- $ X $ is the number of regular payments
- The term $ (1 + r) $ accounts for the fact that this is an annuity due.

### Step 4: Calculate the Present Value of the Special Payment
The present value of the special payment, $ 49,700 $, discounted back 6 years is:

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

### Step 5: Set up the Equation
The total present value of all payments (regular and special) must equal the initial loan amount. This gives the equation:

\[
PV_{\text{annuity}} + PV_{\text{special}} = 240,203
\]

Substitute the formulas from Step 3 and Step 4 into this equation, then solve for $ X $.

Let me calculate this for you.The number of regular annual payments, $ X $, required to repay the loan is approximately $ 18.73 $ years. 

This means you will need to make approximately 19 regular annual payments of $22,820, with the special payment of $49,700 occurring in year 6.

Would you like more details or have any questions? Here are 5 related questions that might interest you:

1. How does an annuity due differ from an ordinary annuity in terms of present value?
2. What effect does increasing the interest rate have on the number of payments?
3. How would the number of payments change if the loan amount increased?
4. What if the special payment occurred at a different year—how would that affect the results?
5. Can we calculate the total interest paid over the entire loan period?

**Tip**: Understanding the concept of present value helps in evaluating both loans and investments, as it reflects the current worth of future cash flows considering a specific interest rate.

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 regular loan payments when borrowing $240,203 with an interest rate of 9.76%, including a special payment of $49,700 in year 6. This problem involves the present value of annuities due and lump-sum payments, suitable for finance and business students.

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