Three partners with fractional capital-time contributions: A invests 1/6 of capital for 1/6 of the total time; B invests 1/3 of capital for 1/3 of the time; C invests the remaining capital for the whole time. If the total profit is ₹ 4,600, find B’s share.

Introduction / Context:
When each partner invests a fraction of the total capital for a fraction of the total time, compute the effective contribution as (capital fraction * time fraction). The profit split is proportional to these effective contributions. Then multiply by total profit to get each share.

Given Data / Assumptions:

• A: capital 1/6 for time 1/6 ⇒ weight = 1/36.
• B: capital 1/3 for time 1/3 ⇒ weight = 1/9.
• C: remaining capital = 1 − 1/6 − 1/3 = 1/2 for full time ⇒ weight = 1/2.
• Total profit = ₹ 4,600.

Concept / Approach:
Convert fractional weights to a common denominator to read integer parts and compute B’s fraction of the total. Then multiply by the total profit for the rupee amount.

Step-by-Step Solution:

Verification / Alternative check:
A’s share = 4,600 * (1/23) = ₹ 200; C’s share = 4,600 * (18/23) = ₹ 3,600; sum = 200 + 800 + 3,600 = ₹ 4,600, confirming correctness.

Why Other Options Are Wrong:
₹ 1,000, ₹ 960, ₹ 900, ₹ 650 correspond to incorrect weights or arithmetic errors with denominators.

Common Pitfalls:
Forgetting that C holds the remaining capital (1/2) and is in for the full time; or using sums of fractions incorrectly.

Final Answer:
Rs. 800