Madhu starts a business by investing Rs. 76,000. After some months, Amar joins the business with an investment of Rs. 57,000. At the end of the year, the profit is divided between them in the ratio 2 : 1 (Madhu : Amar). After how many months from the start does Amar join the business?

Introduction / Context:
In this partnership problem, one partner, Madhu, starts the business alone, and another partner, Amar, joins later. The profit is shared in the ratio 2 : 1 in favour of Madhu at the end of the year. Since their capitals are different and Amar joins late, we must use the relation between profits and capital time units to determine when Amar joins.

Given Data / Assumptions:

• Madhu invests Rs. 76,000 at the start of the year.
• Amar invests Rs. 57,000 but joins after some months.
• The total business period is 1 year = 12 months.
• At the end of the year, the profit is divided in the ratio 2 : 1 (Madhu : Amar).
• Profit shares are proportional to capital multiplied by time.
• We must find how many months after the start Amar joins the business.

Concept / Approach:
Let Amar be in the business for t months. Then the capital time units for Madhu are 76000 * 12, and for Amar they are 57000 * t. The ratio of these products must equal the profit ratio 2 : 1. From this equation, we first find t, which is the duration of Amar's investment. Since the whole business period is 12 months, the time delay after which Amar joins is 12 - t months.

Step-by-Step Solution:
Step 1: Madhu invests Rs. 76,000 for the full 12 months, so Madhu's capital time units are 76000 * 12. Step 2: Let Amar invest Rs. 57,000 for t months, so Amar's capital time units are 57000 * t. Step 3: The profit ratio of Madhu to Amar is 2 : 1. Step 4: Therefore, 76000 * 12 : 57000 * t = 2 : 1. Step 5: Simplify by writing the equation (76000 * 12) / (57000 * t) = 2 / 1. Step 6: Compute 76000 * 12 = 912000. So 912000 / (57000 * t) = 2. Step 7: Cross multiply to get 912000 = 2 * 57000 * t. Step 8: Compute 2 * 57000 = 114000, so 912000 = 114000 * t. Step 9: Solve for t: t = 912000 / 114000 = 8 months. Step 10: Since the total year is 12 months and Amar is present for the last 8 months, he must have joined after 12 - 8 = 4 months from the start.

Verification / Alternative check:
Check the capital time products with t = 8 and a joining delay of 4 months. Madhu has 76000 * 12 = 912000 units. Amar has 57000 * 8 = 456000 units. The ratio 912000 : 456000 simplifies by dividing both by 456000 to 2 : 1, which matches the given profit ratio. Therefore, our interpretation that Amar joins after 4 months is correct.

Why Other Options Are Wrong:
If Amar had joined after 3, 5, 7 or 8 months, his period of investment would be 9, 7, 5 or 4 months respectively, leading to different capital time products and different profit ratios. For example, if Amar joined after 3 months, t would be 9, giving Madhu : Amar capital time units as 76000 * 12 : 57000 * 9 = 912000 : 513000, which does not simplify to 2 : 1. Hence these options do not match the required 2 : 1 profit ratio.

Common Pitfalls:
Some learners mistake t for the joining time instead of the period of investment and directly mark t = 8 months as the answer. It is important to distinguish between the total duration of the business and the time during which a partner is active. Another common error is incorrectly forming the ratio equation or forgetting to multiply by 12 for Madhu's full year of investment. Writing the steps clearly helps avoid confusion about what t represents.

Final Answer:
Amar joins the business after 4 months from the start, which corresponds to option D.