The correct answer is option (C):
6months
Let's break down this profit-sharing problem step by step.
The core principle here is that the profit share is directly proportional to the investment and the time the investment is active. Since A and C receive equal profit shares, we can use this information to determine how long C's investment was active.
1. **Understand the Ratio:** A and B's investments are for the entire year (12 months). We need to figure out how many months C's investment was active. The key is understanding that since A and C receive equal profits, their effective investment contributions (investment amount multiplied by time) must be equal.
2. **Calculate A's Effective Investment:** A invested Rs. 1200 for 12 months. Therefore, A's effective investment is 1200 * 12 = Rs. 14400.
3. **Calculate C's Effective Investment:** C invested Rs. 2400. Since A and C receive equal profits, C's effective investment must also equal Rs. 14400. To find out the number of months C's investment was active, divide his effective investment by his investment amount: 14400 / 2400 = 6 months.
4. **Find the Time C joined the firm**: This calculation tells us that C's investment was active for 6 months. The problem asks *after how many months* did C join. Since the year has 12 months, C joined the firm after 12 - 6 = 6 months.
Therefore, the correct answer is 6 months.