P starts a business with Rs. 3,00,000. Q joins after 3 months with Rs. 12,00,000. What is the ratio of their profits at the end of one year?

Introduction / Context:
For unequal joining times, profits are proportional to capital * time. Q joins later but with a larger capital; we compute time-weighted contributions for both partners to find the final ratio.

Given Data / Assumptions:

• P: Rs. 3,00,000 for 12 months.
• Q: Rs. 12,00,000 for 9 months (joined after 3 months).
• Profits ∝ capital * time.

Concept / Approach:
Multiply each capital by months invested to get weights, then reduce to simplest ratio P : Q.

Step-by-Step Solution:
P weight = 3,00,000 * 12 = 36,00,000. Q weight = 12,00,000 * 9 = 1,08,00,000. Ratio P : Q = 36 : 108 = 1 : 3.

Verification / Alternative check:
Dividing by 36 confirms 1 : 3. Any total profit will split in the proportion 1 part to P and 3 parts to Q.

Why Other Options Are Wrong:
2 : 5, 3 : 5, and 5 : 1 do not match the calculated 1 : 3 weight ratio.

Common Pitfalls:
Ignoring the late entry of Q and using only capital ratio; time must be included in the weighting.

Final Answer:
1 : 3