Explanation:

- A's investment: Rs. 4000 for 12 months.

- B's investment: Rs. 6000 for 12 months.

- C's investment: Rs. 5000 for (12-x) months.

Profit Sharing:

- A's share: Rs. 240

- C's share: Rs. 100

The profit is based on the ratio of their investments multiplied by their time in business.

1. Calculate the total profit share:

- A's profit share per unit investment per month: $240/(4000\ast 12)$

- C's profit share per unit investment per month: $100/(5000\ast (12-x))$

2. Set the ratios equal, since it’s the same total profit basis:

- $(4000\ast 12)/(5000\ast (12-x))=240/100$

3. Solve the equation for x.

- Option 1 (x=4)

- Option 2 (x=5)

- Option 3 (x=6)

- Option 4 (x=8)

- Option 5 (x=9)

Given that the equation holds when calculating x and substituting into the ratio, Option 4 - 8 is the correct solution.