Finding the joining time from a profit ratio: A begins with ₹ 2,250. B later joins with ₹ 2,700. At year end, profits are divided in the ratio 2 : 1 (A : B). After how many months did B join?

Introduction / Context: When profits are divided in a known ratio, and the capitals are known, we can find the missing time for a late entrant by equating the capital × time weights to the profit ratio. This is a straightforward application of proportional reasoning in partnerships.

Given Data / Assumptions:

• A’s capital = ₹ 2,250 for 12 months.
• B’s capital = ₹ 2,700 for t months (unknown).
• Profit ratio A : B = 2 : 1.

Concept / Approach: Profit ratio equals the ratio of capital × time. Set up 2,250*12 : 2,700*t = 2 : 1 and solve for t. Ensure proper simplification to avoid arithmetic slips.

Step-by-Step Solution:

Verification / Alternative check: B weight = 2,700*5 = 13,500. A : B weights = 27,000 : 13,500 = 2 : 1, matching the profit ratio.

Why Other Options Are Wrong: 3, 4, 6, 7 months do not yield the stated 2 : 1 profit split when converted to weights with given capitals.

Common Pitfalls: Using 12 months for both partners or failing to divide by the common factor correctly when simplifying the proportion.

Final Answer: 5 months