Lokesh starts a business with Rs 2,40,000. After 3 months, Vishal joins with Rs 2,10,000. At the end of the year, in what ratio should their profits be shared?

Introduction / Context:
This is a standard partnership problem from profit and loss and ratio topics. Profits in a partnership business are shared in the ratio of the products of capital and time for which the capital is invested. Here, Lokesh and Vishal invest different amounts and also invest for different durations. You must combine both factors to determine the correct profit-sharing ratio at the end of one year.

Given Data / Assumptions:

• Lokesh invests Rs 2,40,000 and stays invested for 12 months.
• Vishal invests Rs 2,10,000 and joins after 3 months, so he stays invested for 9 months.
• Total business duration considered = 12 months.
• Profits are shared in the ratio of capital multiplied by time.

Concept / Approach:
In a partnership where profit is shared according to investment and duration, each partner’s share of profit is proportional to capital * time. For example, if one partner invests more money or invests for a longer period, their profit share increases accordingly. To find the ratio of their profits, calculate the product of capital and time for each partner and express these products as a simplified ratio.

Step-by-Step Solution:
1) Lokesh’s capital = Rs 2,40,000; time invested = 12 months. 2) Vishal’s capital = Rs 2,10,000; he joins after 3 months and thus invests for 9 months. 3) Lokesh’s capital-time product = 2,40,000 * 12. 4) Vishal’s capital-time product = 2,10,000 * 9. 5) Cancel common zeros to simplify: consider 240 * 12 and 210 * 9. 6) Compute 240 * 12 = 2,880. 7) Compute 210 * 9 = 1,890. 8) So initial profit ratio = 2,880 : 1,890. 9) Divide both by 30 to simplify: (2,880 / 30) : (1,890 / 30) = 96 : 63. 10) Further divide both by 3: 96 : 63 becomes 32 : 21. 11) Therefore, Lokesh and Vishal should share profits in the ratio 32 : 21.
Verification / Alternative check:
We can express the ratio directly in terms of capital and relative time without full multiplication. Lokesh’s effective investment factor is 2,40,000 * 12. Vishal’s effective factor is 2,10,000 * 9. Divide both capital-time products by 30,000 to reduce mental arithmetic: Lokesh gets (2,40,000 * 12) / 30,000 = 8 * 12 = 96; Vishal gets (2,10,000 * 9) / 30,000 = 7 * 9 = 63. Ratio 96 : 63 simplifies to 32 : 21, confirming the same result with less computation.

Why Other Options Are Wrong:
Option B (16 : 13) and option C (8 : 5) underestimate Lokesh’s advantage in either capital or time. Option D (45 : 39) and option E (4 : 3) do not match the correct capital-time products when simplified. It is only with the ratio 32 : 21 that the relative contributions of both capital and time for Lokesh and Vishal are reflected correctly in the profit share.

Common Pitfalls:
A common mistake is to ignore the time factor and share profits simply in the ratio of the invested amounts 2,40,000 : 2,10,000. Another error is to apply the time adjustment incorrectly, such as using 3 months for Vishal instead of the 9 months he is actually invested. Always carefully read when each partner joins and multiply the capital by the exact period for which it remains in the business.

Final Answer:
Lokesh and Vishal should share the business profit in the ratio 32 : 21.