A and B started a business with investments of Rs 2400 and Rs $X$, respectively. After 9 months, A added Rs 600 to his investment, and B withdrew Rs 600 from his investment. If at the end of one year, the profit share of A to B is 17:23, find the value of $X$.

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**Solution:**

1. Let the initial investments of A and B be Rs $2400$ and Rs $X$ respectively.

2. A invests Rs $2400$ for 9 months, then adds Rs $600$ for the remaining 3 months.

3. B invests Rs $X$ for 9 months, then withdraws Rs $600$ for the last 3 months.

4. Calculate A's total investment in terms of month-rupees:
   - For first 9 months: $2400 \times 9 = 21600$
   - For last 3 months: $(2400 + 600) \times 3 = 3000 \times 3 = 9000$
   - Total for A = $21600 + 9000 = 30600$

5. Calculate B's total investment in terms of month-rupees:
   - For first 9 months: $X \times 9 = 9X$
   - For last 3 months: $(X - 600) \times 3 = 3X - 1800$
   - Total for B = $9X + 3X - 1800 = 12X - 1800$

6. Given the profit ratio of A to B is $17:23$, so their investment ratio is also $17:23$.

7. Set up the equation:
   $$\frac{30600}{12X - 1800} = \frac{17}{23}$$

8. Cross-multiply:
   $$30600 \times 23 = 17 \times (12X - 1800)$$

9. Calculate left side:
   $$30600 \times 23 = 703800$$

10. Expand right side:
    $$17 \times 12X - 17 \times 1800 = 204X - 30600$$

11. Equation becomes:
    $$703800 = 204X - 30600$$

12. Add 30600 to both sides:
    $$703800 + 30600 = 204X$$
    $$734400 = 204X$$

13. Solve for $X$:
    $$X = \frac{734400}{204} = 3600$$

**Answer:** Rs $3600$

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