Divide 3000 taka into two parts such that the ratio of the interest of the first part at a rate of 6% for 5 years and the interest of the second part at a rate of 4% for 3 years is 5:8.

To solve this problem, let the two parts of the 3000 taka be \( x \) and \( 3000 - x \).

We are given:
- The interest on the first part \( x \) is calculated using a rate of 6% for 5 years.
- The interest on the second part \( 3000 - x \) is calculated using a rate of 4% for 3 years.
- The ratio of the interest on the first part to the second part is 5:8.

### Step-by-Step Solution

1. **Simple Interest Formula**: 
   \[
   \text{Simple Interest} = \frac{P \times R \times T}{100}
   \]
   where:
   - \( P \) is the principal amount,
   - \( R \) is the rate of interest, and
   - \( T \) is the time period.

2. **Interest on the First Part (x)**:
   Using the simple interest formula for \( x \) at 6% for 5 years:
   \[
   \text{Interest}_1 = \frac{x \times 6 \times 5}{100} = \frac{30x}{100} = 0.3x
   \]

3. **Interest on the Second Part (3000 - x)**:
   Using the simple interest formula for \( 3000 - x \) at 4% for 3 years:
   \[
   \text{Interest}_2 = \frac{(3000 - x) \times 4 \times 3}{100} = \frac{12(3000 - x)}{100} = 0.12(3000 - x)
   \]

4. **Given Ratio**:
   We are given that the ratio of the interest on the first part to the interest on the second part is 5:8. Therefore:
   \[
   \frac{0.3x}{0.12(3000 - x)} = \frac{5}{8}
   \]

5. **Cross-multiply and Solve**:
   Cross-multiplying:
   \[
   8 \times 0.3x = 5 \times 0.12(3000 - x)
   \]
   Simplifying:
   \[
   2.4x = 0.6(3000 - x)
   \]
   Expanding:
   \[
   2.4x = 1800 - 0.6x
   \]
   Collecting like terms:
   \[
   2.4x + 0.6x = 1800
   \]
   \[
   3x = 1800
   \]
   Solving for \( x \):
   \[
   x = \frac{1800}{3} = 600
   \]

So, the first part is \( x = 600 \) taka, and the second part is \( 3000 - 600 = 2400 \) taka.

### Final Answer:
- The first part is 600 taka.
- The second part is 2400 taka.

Would you like further explanation or details on any steps? 

Here are 5 related questions for further practice:
1. How would the solution change if the ratio were 7:9?
2. What would happen if the interest rates were swapped?
3. How would the answer change if the total amount were 5000 taka?
4. Can you derive a general formula for splitting any amount based on different interest rates and time periods?
5. What if the time period for the second part was extended to 5 years?

**Tip**: When dealing with ratios in financial problems, simplifying the ratio using the underlying interest formulas makes the problem more manageable.

Learn how to divide 3000 taka into two parts where the simple interest for one part at 6% for 5 years and the other at 4% for 3 years forms a ratio of 5:8. Includes a step-by-step solution, covering algebraic concepts and the simple interest formula.

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