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The problem presented in the image involves determining the initial payment \( K_0 \) on February 15, 2025, that would make the two repayment options equivalent in terms of value on the comparison date of July 15, 2025. Here's a step-by-step breakdown of how to solve this:

### Problem Summary:
- **Total Bill:** €10,000
- **Interest Rate:** 4% nominal per annum with linear interest.
- **Option 1:**
  - Initial payment \( K_0 \) on February 15, 2025.
  - The remainder to be paid on July 15, 2025, with interest calculated at a nominal rate of 4% per annum.
  
- **Option 2:**
  - Four equal payments of €2,575 on February 15, April 15, June 15, and July 15, 2025.
  - Total repayment sum: €10,300

### Steps to Solve:

1. **Calculate the Future Value of Option 1:**
   The future value (on July 15, 2025) of the payment \( K_0 \) made on February 15, 2025, considering linear interest:
   \[
   \text{Future Value of } K_0 = K_0 \times \left(1 + \frac{4\% \times \text{Number of Days}}{360}\right)
   \]
   Here, "Number of Days" is the time from February 15, 2025, to July 15, 2025.

2. **Determine the Equivalent Payment \( K_0 \):**
   For the two options to be equivalent on July 15, 2025, the total payments of Option 1 (initial payment and the future value of the remaining amount) must equal the present value of the total payments in Option 2.

### Calculation:
Let's compute this step by step.

1. **Days from February 15, 2025, to July 15, 2025:**  
   The number of days between these dates is approximately 150 days.

2. **Interest accrued on the remainder in Option 1:**
   Interest on the remaining amount \( (10,000 - K_0) \):
   \[
   \text{Interest} = (10,000 - K_0) \times \frac{0.04 \times 150}{360} = (10,000 - K_0) \times \frac{0.01667}{1} 
   \]

3. **The future value of the total payment on July 15, 2025, for Option 1:**
   \[
   K_0 \times 1 + (10,000 - K_0) \times 1.01667
   \]

4. **Compare the total of Option 2:**
   The total payment in Option 2 is €10,300, so you would equate this value to the future value of Option 1 to solve for \( K_0 \).

By equating the future value of Option 1 and the total of Option 2:

\[
K_0 + (10,000 - K_0) \times 1.01667 = 10,300
\]

\[
K_0 + 10,000 \times 1.01667 - K_0 \times 1.01667 = 10,300
\]

Simplifying to find \( K_0 \):

\[
K_0 \times (1 - 1.01667) + 10,000 \times 1.01667 = 10,300
\]

\[
K_0 \times (-0.01667) = 10,300 - 10,166.7
\]

\[
K_0 = \frac{133.3}{0.01667} \approx 8000 \text{ Euros}
\]

So, \( K_0 \) would be approximately €8,000.

### Final Answer:
The payment \( K_0 \) that makes the two options equivalent on July 15, 2025, is approximately **€8,000**.

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Would you like more details on any part of the process? Or have any questions?

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**Relative Questions:**

1. How does the linear interest method affect the future value compared to compound interest?
2. What would the repayment schedule look like if the interest rate was 5% instead of 4%?
3. How does the choice of the comparison date (July 15, 2025) influence the initial payment \( K_0 \)?
4. How would this calculation change if there were compounding interest?
5. What other factors should be considered when choosing between these two payment options?

**Tip:** When calculating future values, always pay attention to whether interest is compounded or linear, as it significantly affects the results.

Huber must pay a bill of €10,000 on February 15, 2025. He has two payment options: 1) An initial payment of K_0 on February 15, 2025, with the remainder paid on July 15, 2025, at a nominal interest rate of 4% p.a.; 2) Four equal payments of €2,575 on February 15, April 15, June 15, and July 15, 2025. The total repayment sum in Option 2 is €10,300. What initial payment K_0 on February 15, 2025, would make both payment options equivalent on July 15, 2025?

Learn how to determine the initial payment amount K_0 that makes two repayment options equivalent on July 15, 2025, for a €10,000 bill with a 4% nominal interest rate. The problem involves calculating the future value using simple interest and comparing it with a series of equal payments.

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