two-fifths of the capital was invested for 8 months with an interest rate of 4.5% and 200 euros were earned on that part of the capital in the name of interest.  the rest of the capital was invested for 9 months with 3.6% interest.  determine the amount of capital invested and the total interest earned.

Let's break down the problem and solve it step by step.

### Given:
1. Two-fifths of the capital was invested for 8 months at an interest rate of 4.5%.
2. The interest earned on this part was 200 euros.
3. The remaining capital was invested for 9 months at an interest rate of 3.6%.

### Step 1: Find the total capital

Let the total capital be \( C \) euros.

The capital invested for 8 months at 4.5% interest is:
\[
\text{Capital invested} = \frac{2}{5}C
\]

### Step 2: Calculate the interest formula

The interest earned on an investment can be calculated using the simple interest formula:
\[
I = \frac{P \times R \times T}{100}
\]
where:
- \( P \) is the principal amount (initial investment)
- \( R \) is the annual interest rate
- \( T \) is the time in years

For the first part of the capital:
\[
200 = \frac{\frac{2}{5}C \times 4.5 \times \frac{8}{12}}{100}
\]

### Step 3: Solve for \( C \)

Simplifying the equation:
\[
200 = \frac{\frac{2}{5}C \times 4.5 \times \frac{2}{3}}{100}
\]
\[
200 = \frac{\frac{9}{25}C}{100}
\]
\[
200 = \frac{9C}{2500}
\]
\[
C = \frac{200 \times 2500}{9} = \frac{500000}{9} \approx 55555.56 \text{ euros}
\]

So, the total capital \( C \) is approximately 55555.56 euros.

### Step 4: Calculate the interest earned on the remaining capital

The remaining capital \( \frac{3}{5}C \) was invested for 9 months at 3.6% interest:
\[
\text{Remaining capital} = \frac{3}{5} \times 55555.56 \approx 33333.33 \text{ euros}
\]

The interest on this part is:
\[
I = \frac{33333.33 \times 3.6 \times \frac{9}{12}}{100}
\]
\[
I = \frac{33333.33 \times 3.6 \times 0.75}{100} = \frac{90000}{100} = 900 \text{ euros}
\]

### Step 5: Calculate the total interest earned

The total interest earned is:
\[
\text{Total interest} = 200 + 900 = 1100 \text{ euros}
\]

### Final Answer:
- **Total capital invested:** Approximately 55555.56 euros.
- **Total interest earned:** 1100 euros.

Would you like more details on any part of this solution or do you have any further questions?

Here are five questions that are related:
1. How would the problem change if the interest rates were swapped?
2. What would be the total interest if both investments had the same duration?
3. How would you approach solving this if the time periods were given in days instead of months?
4. What if the interest was compounded annually instead of being simple interest?
5. How would the problem be affected if only 100 euros were earned as interest from the first investment?

**Tip:** Always double-check units, especially when converting time periods from months to years in interest calculations.

Two-fifths of the capital was invested for 8 months with an interest rate of 4.5% and 200 euros were earned on that part of the capital in the name of interest. The rest of the capital was invested for 9 months with 3.6% interest. Determine the amount of capital invested and the total interest earned.

Learn how to determine the total capital invested and the total interest earned with a detailed step-by-step solution. This problem involves concepts of simple interest, algebra, and proportion, making it suitable for students in Grades 9-10.

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