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To solve the problem of finding the value of \( X \) that makes the two cash flow diagrams equivalent, we will use the concept of **equivalent present value** and apply the **time value of money**. The problem states that the interest rate is 10%, compounded annually.

### Steps to approach:

1. **Calculate the Present Value (PV) of the first cash flow diagram:**
   - Each cash flow in the left diagram has a future value at its corresponding time.
   - We'll discount each cash flow back to time 0 using the formula:
   \[
   PV = \frac{FV}{(1 + i)^t}
   \]
   where \( FV \) is the future value, \( i \) is the interest rate (0.10 in this case), and \( t \) is the time in years.

2. **Set the Present Value of the second cash flow equal to the Present Value of the first cash flow diagram:**
   - The second diagram has two unknown cash flows (both are \( X \)) occurring at time periods 3 and 5. We'll also discount them back to the present (time 0).

3. **Solve for \( X \).**

Let me first calculate the Present Values for both diagrams.The present value of the first cash flow diagram is approximately \( 2016.16 \).

Next, let's calculate the present value of the second cash flow diagram, which has two payments of \( X \) at time periods 3 and 5. We'll set their present values equal to the present value of the first diagram and solve for \( X \).The value of \( X \) that makes the two cash flow diagrams equivalent is approximately \( 1469.25 \).

Do you need more details or further clarification?

Here are five related questions for further practice:
1. What would happen to the value of \( X \) if the interest rate increased to 12%?
2. How does changing the time periods in the second cash flow diagram affect the value of \( X \)?
3. How would you calculate the future value of the first cash flow diagram at year 8?
4. If one of the cash flows in the first diagram is removed, how would you adjust the calculations?
5. How would monthly compounding interest change the equivalent value of \( X \)?

**Tip:** Always discount future cash flows to their present value when comparing different cash flow streams.

Find the value of X that makes two cash flow diagrams equivalent given a 10% annual interest rate.

Find the value of X that makes two cash flow diagrams equivalent using the concept of Present Value and a 10% annual interest rate. This problem demonstrates how to compare cash flows occurring at different times through discounting future values.

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