A, B, and C enter into a partnership with their capital shares in the ratio 1/2 : 1/3 : 1/4. After 2 months, A withdraws half of his capital. If after a total period of 12 months the profit of Rs. 378 is distributed among them, what is the share of B in the profit?

Introduction / Context:
This problem involves a partnership where the partners invest in fractional ratios and one partner changes his investment during the course of the business. The key idea in such questions is that profit is divided in proportion to the product of capital and time, known as capital time. Understanding how to handle fractional capital ratios and time based changes is essential for correctly determining each partner's share in the total profit.

Given Data / Assumptions:
- Initial capital ratio of A : B : C is 1/2 : 1/3 : 1/4.
- After 2 months, A withdraws half of his capital, so his capital becomes 1/4 of the same base.
- B and C continue with their original capitals throughout.
- The business runs for a total of 12 months (2 months with original capitals and 10 additional months after A reduces his capital).
- Total profit at the end of 12 months is Rs. 378.
- We need to find B's share in this profit.

Concept / Approach:
First, we treat the given fractions 1/2, 1/3, and 1/4 as proportional units of capital. Then we compute capital time for each partner in two segments: the first 2 months and the remaining 10 months. After that, we sum capital time for each partner, obtain the overall ratio, and use it to split the total profit of Rs. 378. Finally, we extract B's share from the total.

Step-by-Step Solution:
Step 1: Let initial capitals be A = 1/2 unit, B = 1/3 unit, and C = 1/4 unit.Step 2: For the first 2 months, capital time is: A = (1/2) * 2 = 1, B = (1/3) * 2 = 2/3, C = (1/4) * 2 = 1/2.Step 3: After 2 months, A withdraws half of his capital, so A's new capital is (1/2) / 2 = 1/4.Step 4: For the remaining 10 months, capital time is: A = (1/4) * 10 = 2.5, B = (1/3) * 10 = 10/3, C = (1/4) * 10 = 2.5.Step 5: Total capital time: A = 1 + 2.5 = 3.5 (7/2), B = 2/3 + 10/3 = 12/3 = 4, C = 1/2 + 2.5 = 3.Step 6: The ratio A : B : C is 7/2 : 4 : 3. Multiply all by 2 to clear the denominator and get 7 : 8 : 6.Step 7: Sum of ratio parts = 7 + 8 + 6 = 21. Each part of profit = 378 / 21 = 18. B's share = 8 * 18 = Rs. 144.

Verification / Alternative check:
We can confirm by computing the shares of A and C as well. A's share is 7 * 18 = Rs. 126. C's share is 6 * 18 = Rs. 108. Adding 126 + 144 + 108 gives 378, which matches the total profit. This confirms that our ratio and calculations are correct, and B's share of Rs. 144 is accurate.

Why Other Options Are Wrong:
Rs. 169 does not correspond to any simple integer multiple of 18, and using it would break the total profit sum. Rs. 225 and Rs. 339 are much larger than B's correct share and would also lead to a total exceeding Rs. 378 or not matching the ratio 7 : 8 : 6. Therefore, these options are inconsistent with the partnership ratio and the given profit.

Common Pitfalls:
Students sometimes misinterpret the statement and take the total duration as only 10 months instead of 12, which changes the ratio. Another error is to treat the fractions 1/2, 1/3, and 1/4 as exact rupee amounts instead of proportional units; however, this does not actually change the ratio if handled correctly. The most serious mistake is to forget that A's capital changes after 2 months and to compute everything using the initial ratio only. Carefully breaking the time into segments and recomputing capital times avoids these errors.

Final Answer:
B's share in the profit is Rs. 144.