Inferring a partner’s time from profit share and capital fractions: Aayush provides 1/4 of the capital for 15 months. Babloo receives 2/3 of the profit. Assuming only two partners, for how many months was Babloo’s money used?

Introduction / Context:
Profit shares in partnerships are proportional to capital × time. With only two partners, if one partner’s capital fraction and time are known, and the profit split is given, we can solve for the other partner’s time from a single proportion equation.

Given Data / Assumptions:

• Aayush capital fraction = 1/4, time = 15 months.
• Babloo capital fraction = 3/4 (rest of the capital), time = t months.
• Babloo’s profit share = 2/3; hence Aayush’s share = 1/3.

Concept / Approach:
Set the ratio of profit shares equal to the ratio of capital × time contributions: (Babloo : Aayush) = ((3/4)*t) : ((1/4)*15) = 2 : 1. Solve for t directly.

Step-by-Step Solution:

Verification / Alternative check:
Weights: Aayush = (1/4)*15 = 3.75; Babloo = (3/4)*10 = 7.5 ⇒ ratio 7.5 : 3.75 = 2 : 1, consistent with the given profit split.

Why Other Options Are Wrong:
9, 11, 7, 12 months do not produce a 2 : 1 profit ratio when combined with the stated capital fractions.

Common Pitfalls:
Using 1/4 and 3/4 as months rather than capital fractions or mixing up which partner gets 2/3 of the profit.

Final Answer:
10 months