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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 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 for $ X $, the number of regular payments, we can use the concept of the present value of an annuity and a lump sum to model the loan repayment. Here's the breakdown of the solution process:

### Step 1: Recognize the loan and payment structure
- You borrowed \$240,203, so this is the **present value** of the loan.
- You will be making regular annual payments of \$22,820, with the first payment made today (an annuity due).
- In 6 years, you will also make a special payment of \$49,700, which needs to be accounted for as a lump sum.
- The interest rate is 9.76% per year.

### Step 2: Formula for the present value of an annuity due
For an **annuity due**, where the first payment is made immediately, the present value (PV) is calculated using the formula:

\[
PV_{\text{annuity due}} = P \times \left( 1 + i \right) \times \left[ 1 - \left( 1 + i \right)^{-n} \right] / i
\]

Where:
- $ P $ is the regular payment (\$22,820),
- $ i $ is the interest rate per period (9.76% or 0.0976),
- $ n $ is the number of regular payments (which is what we are solving for).

### Step 3: Present value of the lump sum payment
The lump sum payment of \$49,700 is made in 6 years. The present value of this future payment can be found using the formula for the present value of a lump sum:

\[
PV_{\text{lump sum}} = \frac{F}{(1 + i)^t}
\]

Where:
- $ F $ is the future lump sum payment (\$49,700),
- $ i $ is the interest rate (0.0976),
- $ t $ is the time until the payment is made (6 years).

### Step 4: Total present value equation
The total present value of the loan is the sum of the present value of the annuity due and the present value of the lump sum:

\[
PV_{\text{loan}} = PV_{\text{annuity due}} + PV_{\text{lump sum}}
\]

We know that $ PV_{\text{loan}} = 240,203 $, so we can substitute and solve for $ n $.

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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 annual payments (X) for a loan of $240,203 with a 9.76% interest rate. This problem involves present value calculations for annuities and lump sum payments, and is ideal for finance students. Includes formulas and step-by-step breakdown.

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