The question deals with the concept of profit division in a partnership that is based on the amount of money invested and the duration of investment. Amit started the business, investing Rs. 30,000 for the entire year, and Rahul joined later, investing Rs. 20,000. The end-of-year profit was divided in the ratio of 2:1, favoring Amit.

Given the information, we can establish a proportional relationship:

Amit`s Investment/Time = 2; therefore, Rs. 30,000 * 12 months = 2.

Rahul`s Investment/Time = 1; therefore, Rs. 20,000 * X months = 1.

Dividing both sides of the equation by 20000, we get X = 3.

Option 2 is the correct answer, as if we solve the equations for X, we find that Rahul joined the business 3 months after Amit. The other options do not fit the scenario given by the problem. For example, option 1 (2 months) would have resulted in a different profit ratio, and options 3 and 4 (4 and 5 months) would result in Rahul gaining a larger share of the profit.