A and B invested Rs. $X$ and Rs. $(X + 900)$ respectively in a business. After 8 months, B left the business. At the end of 1 year, the total profit is Rs. 5800, and B's share of the profit is Rs. 1000 more than A's share. Find the value of $X$ (in Rs.).

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**Solution:**

1. Let A's investment be Rs. $X$ for 12 months.
2. B invests Rs. $(X + 900)$ but leaves after 8 months.

3. Profit share is proportional to the product of investment and time period:
   - A's share = $X 	imes 12 = 12X$
   - B's share = $(X + 900) 	imes 8 = 8(X + 900) = 8X + 7200$

4. Total profit = Rs. 5800.

5. Let A's profit share be $P_A$ and B's profit share be $P_B$.

6. Given, $P_B = P_A + 1000$ and $P_A + P_B = 5800$.

7. From the above, $P_A + (P_A + 1000) = 5800 \Rightarrow 2P_A = 4800 \Rightarrow P_A = 2400$ and $P_B = 3400$.

8. Profit shares are in the ratio of investments multiplied by time:
   $$\frac{P_A}{P_B} = \frac{12X}{8X + 7200} = \frac{2400}{3400} = \frac{12}{17}$$

9. Cross-multiplied:
   $$17 	imes 12X = 12 	imes (8X + 7200)$$
   $$204X = 96X + 86400$$

10. Simplify:
    $$204X - 96X = 86400$$
    $$108X = 86400$$
    $$X = \frac{86400}{108} = 800$$

**Answer:** Rs. 800

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