Rohith starts a business by investing Rs. 3600. After some months, Puneeth joins him by investing Rs. 2400. At the end of one year, the profit is divided in the ratio 2 : 1 between Rohith and Puneeth. After how many months from the start did Puneeth join the business?

Introduction / Context:
This is a classic partnership question where one partner starts the business and another joins later. The profit ratio at the end of the year is known, and we must use this to determine the time for which the second partner was involved in the business.

Given Data / Assumptions:

• Rohith invests Rs. 3600 at the start.
• Puneeth invests Rs. 2400 after some months and keeps it invested until the end of the year.
• Total business period is 12 months.
• The final profit ratio Rohith : Puneeth is 2 : 1.
• We must find after how many months Puneeth joined, that is, how long after Rohith started.

Concept / Approach:
Profit sharing in partnership is proportional to the products of capital and time. If one partner invests longer, he can receive a greater share even with similar capital. By letting the second partner's time be a variable, we can form a ratio equation equating capital time products to the known profit ratio and solve for the unknown number of months.

Step-by-Step Solution:
Step 1: Rohith invests Rs. 3600 for the full 12 months. Step 2: His capital time product = 3600 * 12 = 43,200. Step 3: Let Puneeth invest Rs. 2400 for x months. Step 4: Puneeth's capital time product = 2400 * x. Step 5: The profit ratio Rohith : Puneeth is 2 : 1, so capital time ratio must be 2 : 1. Step 6: Set up the proportion: 43,200 : 2400x = 2 : 1. Step 7: Cross multiply: 43,200 * 1 = 2 * 2400x, so 43,200 = 4800x. Step 8: Solve for x: x = 43,200 / 4800 = 9 months. Step 9: Puneeth invests for 9 months, so he must have joined after 12 - 9 = 3 months from the start.

Verification / Alternative check:
If Puneeth joins after 3 months, he invests for 9 months. Then his capital time product is 2400 * 9 = 21,600. Rohith's product is 43,200 as computed earlier. The ratio 43,200 : 21,600 simplifies by dividing both terms by 21,600 to 2 : 1, which matches the given profit ratio. This confirms that the joining time of 3 months is consistent with the profit division.

Why Other Options Are Wrong:
If Puneeth joined after 2, 2.5 or 3.5 months, his investment duration would be 10, 9.5 or 8.5 months respectively. These would give capital time products that do not produce the exact ratio 2 : 1 when compared with Rohith's 43,200. Therefore those joining times do not satisfy the required profit ratio.

Common Pitfalls:
One common error is to use the joining time directly as the variable in the proportion instead of using the actual duration of investment. Another mistake is to swap the ratio order, using 1 : 2 instead of 2 : 1 and ending up with the reciprocal of the true duration. Always define the variable clearly as the number of months of investment for the later partner, then convert to joining time at the end.

Final Answer:
Puneeth joined the business after 3 months from the start.