To solve this problem, we can use the concept of profit sharing based on the ratio of investments and the time for which the investments were made.

Step 1: Understanding the problem

• Raju's investment: ₹900 for the entire year.
• Kamal's investment: ₹600, but only for a certain number of months (let's denote the number of months Kamal invested as $x$).
• The profit-sharing ratio is 2:1 in favor of Raju.

Step 2: Setting up the equations

The profit ratio is directly proportional to the product of the investment and the time the investment was active. So:

• Raju’s share: $900 \times 12$ (since he invested for the whole year)
• Kamal’s share: $600 \times x$ (where $x$ is the number of months Kamal invested)

Given the ratio of their profits is $\frac{2}{1}$:

$\frac{900 \times 12}{600 \times x} = \frac{2}{1}$

Step 3: Simplifying the equation

Now, solve for $x$:

$\frac{900 \times 12}{600 \times x} = 2$

$\frac{10800}{600 \times x} = 2$

$10800 = 1200 \times x$

$x = \frac{10800}{1200} = 9$

Step 4: Conclusion

Kamal joined the business after 3 months (since Kamal invested for $12 - 9 = 3$ months of the year).

Final Answer:

Kamal joined the business after 3 months.

---

Would you like any further details on this solution? Here are some related questions you might find interesting:

1. How would the result change if Kamal had invested ₹900 instead?
2. What if the profit-sharing ratio was different, say 3:2? How would that affect the result?
3. How would the result change if Raju invested a different amount, say ₹1200?
4. What if Kamal had joined 2 months earlier? How would that affect the profit ratio?
5. How does the timing of the investment affect the overall profit-sharing in partnership problems?

Tip: In partnership problems, always remember that the share of profit is directly proportional to both the amount invested and the duration of the investment.