A and B started a business with investments of Rs 6000 and Rs 8000 respectively. After 9 months, A withdrew half of his investment, and B increased his investment by Rs $X$. If at the end of the year the profit-sharing ratio of A to B is 21:34, find the value of $X$.

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**Solution:**

1. Let us calculate the effective investment of A and B over the 12 months.

2. A's investment:
   - For the first 9 months, A invested Rs 6000.
   - For the last 3 months, A withdrew half, so the investment was Rs 3000.

Therefore, A's total investment in terms of month-rupees:
   $$6000 \times 9 + 3000 \times 3 = 54000 + 9000 = 63000$$

3. B's investment:
   - For the first 9 months, B invested Rs 8000.
   - For the last 3 months, B added Rs $X$, so the investment was $8000 + X$.

Therefore, B's total investment in terms of month-rupees:
   $$8000 \times 9 + (8000 + X) \times 3 = 72000 + 24000 + 3X = 96000 + 3X$$

4. Given the profit-sharing ratio of A to B is 21:34, so:
   $$\frac{63000}{96000 + 3X} = \frac{21}{34}$$

5. Cross-multiplied:
   $$21(96000 + 3X) = 34 \times 63000$$

$$2016000 + 63X = 2142000$$

6. Solving for $X$:
   $$63X = 2142000 - 2016000 = 126000$$

$$X = \frac{126000}{63} = 2000$$

**Answer:** Rs $2000$

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