Amit and Brijesh invest in the ratio 12 : 11. Their annual profits are in the ratio 4 : 1. If Amit invested for 11 months, for how many months did Brijesh invest?

Introduction / Context:
Profit is proportional to capital × time. Given the capital ratio and the profit ratio, we can determine the missing time of investment for Brijesh using a simple proportional equation.

Given Data / Assumptions:

• Capitals: Amit : Brijesh = 12 : 11.
• Profits: Amit : Brijesh = 4 : 1.
• Amit's time = 11 months; Brijesh's time = t months (unknown).

Concept / Approach:
Since profit ∝ capital × time, set (12 * 11) : (11 * t) = 4 : 1 and solve for t. This uses the fact that the constant of proportionality cancels in the ratio.

Step-by-Step Solution:
Amit's weight = 12 * 11 = 132. Brijesh's weight = 11 * t. 132 : 11t = 4 : 1 ⇒ 132/(11t) = 4/1 ⇒ 132 = 44t ⇒ t = 3 months.

Verification / Alternative check:
With t = 3, Brijesh's weight = 33; Amit's weight = 132; ratio = 132 : 33 = 4 : 1 as required.

Why Other Options Are Wrong:
9, 5, 10, 7 months do not satisfy 132 : 11t = 4 : 1.

Common Pitfalls:
Reversing the ratios or forgetting that profits scale with both capital and time, not just capital.

Final Answer:
3 months