Inferring investment duration from profit ratio: A and B invest in the ratio 12 : 11. Their annual profits are in the ratio 4 : 1. If A invested for 11 months, for how many months did B invest?

Introduction / Context:
Profit in partnerships is proportional to (capital × time). Knowing the capital ratio and profit ratio, we can infer one partner’s investment duration from the other’s time period.

Given Data / Assumptions:

• Capital ratio A : B = 12 : 11.
• Profit ratio A : B = 4 : 1.
• A’s time = 11 months; B’s time = t months (unknown).

Concept / Approach:
Set up the equality: (12 × 11) : (11 × t) = 4 : 1. Solve for t by cross-multiplying. This directly uses the money-time product model of profit sharing.

Step-by-Step Solution:
(12 × 11) / (11 × t) = 4 / 1.132 / (11t) = 4 ⇒ 132 = 44t ⇒ t = 3 months.

Verification / Alternative check:
Check weights: A weight = 12 × 11 = 132; B weight = 11 × 3 = 33; ratio 132 : 33 = 4 : 1, matching the profit ratio.

Why Other Options Are Wrong:

• 6 months gives ratio 132 : 66 = 2 : 1, not 4 : 1.
• 32/3 and 4 months also fail to match the required ratio.

Common Pitfalls:

• Cancelling the 11s prematurely without maintaining the ratio relationship properly.

Final Answer:
3 months