If I can do 0,81% interest rate in a period of 9h investment, how many days I will need to achieve 10% of interest rate knowing that I will be investing/working during only 8h in a 24h day?

To solve this problem, we need to calculate the number of days required to achieve a 10% interest rate, given a 0.81% interest rate for every 9-hour investment period and the fact that you only invest for 8 hours each day.

### Step-by-Step Solution

1. **Calculate the Interest Rate Per Hour:**

   Given:
   - Interest rate for 9 hours = 0.81%
   
   We can find the hourly interest rate by dividing the 9-hour interest rate by 9:
   \[
   r_{\text{hour}} = \frac{0.81\%}{9} = \frac{0.81}{9} \approx 0.09\% \text{ per hour}
   \]

2. **Calculate the Interest Rate Per 8-Hour Investment Period:**

   Since you're investing for 8 hours each day, the interest rate for 8 hours can be calculated as:
   \[
   r_{\text{8-hour}} = 0.09\% \times 8 = 0.72\% \text{ per 8 hours}
   \]

3. **Understand Compounding Over Multiple Investment Periods:**

   The total interest over time with compounding can be described by the formula:
   \[
   (1 + r)^n = (1 + R)
   \]
   Where:
   - \(r = \frac{r_{\text{8-hour}}}{100} = \frac{0.72}{100} = 0.0072\)
   - \(n\) is the number of 8-hour periods (investment periods),
   - \(R = 10\% = 0.10\) is the total desired interest.

4. **Determine the Number of 8-Hour Periods Needed to Achieve 10% Interest:**

   To find \(n\), we solve the equation:
   \[
   (1 + 0.0072)^n = 1.10
   \]

   Taking the natural logarithm of both sides:
   \[
   n \cdot \ln(1.0072) = \ln(1.10)
   \]

   Solving for \(n\):
   \[
   n = \frac{\ln(1.10)}{\ln(1.0072)}
   \]

   Let’s compute \(n\):

\[
n \approx \frac{\ln(1.10)}{\ln(1.0072)}
\]

Let me calculate this value.The number of 8-hour investment periods needed to achieve a 10% interest rate is approximately **13.29**.

### Convert Periods into Days

Since you're investing for only 8 hours each day, we need to determine how many days this corresponds to:

\[
\text{Days} = \frac{n_{\text{periods}}}{1 \, \text{period/day}} = 13.29 \, \text{days}
\]

Therefore, it will take about **13.29 days** to achieve a 10% interest rate.

### Summary

- You need approximately **13.29 days** of investing 8 hours per day to achieve a 10% interest rate, given a 0.81% interest rate for every 9-hour investment period.

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Do you have any questions or would you like more details? 

Here are 5 related questions to expand on this topic:

1. How would the result change if you could invest for 12 hours a day instead of 8?
2. What is the impact of compounding frequency on the time needed to achieve the target interest rate?
3. How would the required time change if the target interest rate was 15% instead of 10%?
4. What if the interest rate per 9-hour period was higher, say 1%? How many days would be needed then?
5. How do different compounding interest formulas affect the calculation for different scenarios?

**Tip:** Understanding how compounding works is crucial for maximizing investment growth over time.

Learn how to calculate the time required to achieve a 10% interest rate with a 0.81% interest rate per 9-hour investment period, while investing only 8 hours a day. Understand the compound interest formula and its application to investment periods. Suitable for financial planning and investment analysis.

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