Divide `40608 rs` into two parts such that if one part is invested in `8% 100rs` shares at `8%` discount and the other part is invested in `9% Rs 100` shares at `8%` premium , the annual incomes from both the investments are equal.

Answer

To solve the problem of dividing ₹40,608 into two parts such that the annual incomes from both investments are equal, we can follow these steps:

### Step-by-Step Solution:

1. **Let the two parts be defined**:
- Let the first part be \( x \).
- Therefore, the second part will be \( 40,608 - x \).

2. **Calculate the effective investment for the first part**:
- The first part \( x \) is invested in 8% shares at an 8% discount.
- The market price of the shares is \( 100 - 8 = 92 \) (since there is an 8% discount).
- The annual income from this investment can be calculated as:
\[ \text{Income from first part} = \left( \frac{x}{92} \right) \times 8 = \frac{8x}{92} = \frac{2x}{23} \]

3. **Calculate the effective investment for the second part**:
- The second part \( 40,608 - x \) is invested in 9% shares at an 8% premium.
- The market price of the shares is \( 100 + 8 = 108 \) (since there is an 8% premium).
- The annual income from this investment can be calculated as:
\[ \text{Income from second part} = \left( \frac{40,608 - x}{108} \right) \times 9 = \frac{9(40,608 - x)}{108} = \frac{3(40,608 - x)}{36} = \frac{3(40,608 - x)}{36} \]

4. **Set the incomes equal**:
- According to the problem, the incomes from both investments are equal:
\[ \frac{2x}{23} = \frac{3(40,608 - x)}{36} \]

5. **Cross-multiply to eliminate the fractions**:
- Cross-multiplying gives:
\[ 2x \times 36 = 3(40,608 - x) \times 23 \]
- Simplifying this:
\[ 72x = 3(40,608 \times 23 - 23x) \]

6. **Distribute and simplify**:
- Expanding the right side:
\[ 72x = 121824 - 69x \]
- Combine like terms:
\[ 72x + 69x = 121824 \]
\[ 141x = 121824 \]

7. **Solve for \( x \)**:
- Dividing both sides by 141:
\[ x = \frac{121824}{141} = 864 \]

8. **Calculate the second part**:
- The second part is:
\[ 40,608 - x = 40,608 - 864 = 39,744 \]

### Final Answer:
- The first part is ₹8,864.
- The second part is ₹39,744.