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. At the end of one year, a total profit of Rs. 378 is to be divided among them. What is the share of B in this profit?

Introduction / Context:
This partnership problem involves three partners with capital in fractional ratios and a change in one partner's capital after a certain period. The business runs for one full year, and A withdraws half of his capital after 2 months. We must compute the effective investments for all three partners and determine B's share of the total profit of Rs. 378.

Given Data / Assumptions:

• Capital ratio at the start for A : B : C is 1/2 : 1/3 : 1/4.
• The business runs for 12 months.
• For the first 2 months, all partners keep their full capital invested.
• After 2 months, A withdraws half of his capital and keeps the reduced capital invested for the remaining 10 months.
• B and C keep their original capital invested for the full 12 months.
• Total profit at the end of the year is Rs. 378.
• Profit is distributed in proportion to capital * time.

Concept / Approach:
We convert the fractional capitals into a simpler ratio by choosing a convenient common multiple. Then we compute each partner's effective investment by multiplying the appropriate capital by the number of months it stays in the business. The ratios of these effective investments give the ratios of profits. Using this ratio, we can then find B's exact share from the total profit.

Step-by-Step Solution:
Step 1: Write the initial capital ratio A : B : C = 1/2 : 1/3 : 1/4. Step 2: To remove fractions, multiply each term by 12 (the least common multiple of 2, 3 and 4). This gives A : B : C = 6 : 4 : 3. Step 3: For the first 2 months, capitals are A = 6, B = 4, C = 3 (in ratio units). Effective investment for A in this period = 6 * 2 = 12, for B = 4 * 2 = 8 and for C = 3 * 2 = 6. Step 4: After 2 months, A withdraws half of his capital, so his new capital becomes 6 / 2 = 3 units. Step 5: For the remaining 10 months, effective investment of A = 3 * 10 = 30, B = 4 * 10 = 40 and C = 3 * 10 = 30. Step 6: Total effective investment for each partner: A = 12 + 30 = 42, B = 8 + 40 = 48, C = 6 + 30 = 36. Step 7: Divide these by 6 to simplify the ratio: A : B : C = 7 : 8 : 6. Step 8: Total number of parts = 7 + 8 + 6 = 21. Profit per part = 378 / 21 = 18. Step 9: B's share = 8 parts = 8 * 18 = Rs. 144.

Verification / Alternative check:
We can verify by computing the shares for all partners: A gets 7 * 18 = 126, B gets 8 * 18 = 144 and C gets 6 * 18 = 108. Adding these, 126 + 144 + 108 = 378, which matches the total profit. This confirms that the effective investment calculations and final distribution are correct.

Why Other Options Are Wrong:
The alternative options 169, 225 and 339 do not align with the ratio 7 : 8 : 6 when allocated from a total of 378. For example, if B's share were 169, the shares of A and C computed from the same ratio would not be integers and would not add up correctly to 378, so these options are inconsistent with the problem data.

Common Pitfalls:
A typical mistake is to use only the initial capital ratio 1/2 : 1/3 : 1/4 and ignore the change in A's capital after 2 months. Another error is to assume the business runs for only 10 months instead of a full year. Always pay attention to when capital changes occur and over what period profit is calculated, and always convert to capital * time before forming the final ratio.

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