Partnership profit sharing over unequal entry times: Ajay invests ₹ 25,000 at the start and Vijay joins after 3 months with ₹ 30,000. At the end of 12 months the total profit is ₹ 38,000. Compute Ajay’s share of the profit using capital × time weights.

Introduction / Context:
Partnership questions in quantitative aptitude commonly require allocating profit in proportion to each partner’s capital multiplied by the time for which that capital remained invested. Here, Vijay enters 3 months late, so his effective time is less than Ajay’s. We use capital × time (also called “money-months”) to split the profit fairly.

Given Data / Assumptions:

• Ajay’s capital = ₹ 25,000 for 12 months.
• Vijay’s capital = ₹ 30,000 for 9 months (joins after 3 months).
• Total profit at year end = ₹ 38,000.
• Profit share ∝ capital * time.

Concept / Approach:
Multiply each capital by its investment duration (in months). The resulting weights form the profit-sharing ratio. Then apply the ratio to the total profit to obtain the exact shares.

Step-by-Step Solution:

Verification / Alternative check:
Vijay’s share = 38,000 − 20,000 = ₹ 18,000. The ratio 20,000 : 18,000 reduces to 10 : 9, matching the weights, so the allocation is consistent.

Why Other Options Are Wrong:
₹ 10,000 and ₹ 15,000 imply incorrect weighting or ignoring time difference; ₹ 18,000 is Vijay’s amount, not Ajay’s; ₹ 21,000 breaks the 10 : 9 proportion with the given total profit.

Common Pitfalls:
Forgetting to multiply by time; using 12 months for both partners; or distributing profit merely by capital without accounting for staggered entry.

Final Answer:
₹ 20000