A and B invest Rs. 8000 and Rs. 9000 respectively in a business. After 4 months, A withdraws half of his capital, and 2 months later B withdraws one-third of his capital. If the business runs for a total of 12 months, in what ratio should A and B share the profit?

Introduction / Context:
This is a partnership question where partners change their capital contributions at different times. Profit sharing in such cases is based on capital multiplied by time, i.e., capital-time units. We must track how long each investment level is maintained and then form the ratio of these capital-time products to determine the profit-sharing ratio between A and B.

Given Data / Assumptions:
• A initially invests Rs. 8000.
• B initially invests Rs. 9000.
• After 4 months, A withdraws half his capital, so his capital becomes Rs. 4000 for the remainder.
• 2 months after that (i.e., after 6 months from the start), B withdraws one-third of his capital, so his capital becomes Rs. 6000 for the remainder.
• Total duration of the business = 12 months.
• Profit is shared in proportion to the capital-time contributions of A and B.

Concept / Approach:
To compute profit sharing, we divide the 12-month period into segments where each partner’s capital is constant. For A, there are two segments: first with 8000 for 4 months and second with 4000 for 8 months. For B, there are also two segments: first with 9000 for 6 months and second with 6000 for 6 months. We calculate the product capital * time for each segment, sum for each partner and then take the ratio A : B.

Step-by-Step Solution:
Step 1: For A, first period: 8000 for 4 months ⇒ capital-time = 8000 * 4 = 32000.Step 2: For A, second period: 4000 for 8 months ⇒ capital-time = 4000 * 8 = 32000.Step 3: Total capital-time for A = 32000 + 32000 = 64000.Step 4: For B, first period: 9000 for 6 months ⇒ capital-time = 9000 * 6 = 54000.Step 5: For B, second period: 6000 for 6 months ⇒ capital-time = 6000 * 6 = 36000.Step 6: Total capital-time for B = 54000 + 36000 = 90000.Step 7: Profit-sharing ratio A : B = 64000 : 90000. Divide by 1000 ⇒ 64 : 90, then divide by 2 ⇒ 32 : 45.

Verification / Alternative check:
We can confirm the simplification: 64 : 90 dividing by 2 gives 32 : 45, and no further common factor exists between 32 and 45. Therefore, 32 : 45 is the simplest form of the ratio. This ratio correctly reflects that B has a larger effective investment over time than A because his capital remains higher for a longer portion of the year.

Why Other Options Are Wrong:
Ratios 21 : 29, 41 : 54 and 37 : 51 do not correspond to the computed capital-time products. If you scale them to match 64000 or 90000, they will not produce consistent capital-time values for both partners based on the given investment schedule.

Common Pitfalls:
Students sometimes forget to split the time correctly and treat investments as if they were constant over the entire year. Another mistake is to use simple capital ratios instead of capital-time ratios. Carefully accounting for when capital changes occur and multiplying by the duration of each segment is essential in partnership problems with varying investments.

Final Answer:
A and B should share the profit in the ratio 32 : 45.