Ramesh started a software business by investing Rs. 50,000. After six months, Rajesh joined him with a capital of Rs. 75,000. At the end of 3 years, they earned a total profit of Rs. 27,000. Assuming the profit is shared in proportion to capital multiplied by time, what was Rajesh's share of the profit?

Introduction / Context:
This question is a classic partnership and percentage profit sharing problem. Two partners invest different amounts at different times, and we are asked to divide the total profit according to their effective investment contributions. The effective contribution is determined by multiplying the invested capital by the time period for which it was used. This concept is widely used in partnership questions in aptitude exams.

Given Data / Assumptions:

• Ramesh invested Rs. 50,000 at the start.
• Rajesh joined after six months with Rs. 75,000.
• Total duration of the business is 3 years (36 months).
• Profit at the end of 3 years is Rs. 27,000.
• Profit is distributed in the ratio of (capital * time) for each partner.

Concept / Approach:
The effective investment for each partner is capital multiplied by the number of months for which it remains invested. Once we find these products, we can form a ratio. The total profit is then split in this ratio. Specifically, we compute the ratio of Ramesh's effective investment to Rajesh's effective investment and then use this ratio to find Rajesh's share of Rs. 27,000.

Step-by-Step Solution:
Step 1: Total time for the business is 3 years = 36 months. Step 2: Ramesh invests Rs. 50,000 for all 36 months. Step 3: Rajesh joins after 6 months, so he invests Rs. 75,000 for 30 months. Step 4: Effective investment for Ramesh = 50000 * 36 = 1800000. Step 5: Effective investment for Rajesh = 75000 * 30 = 2250000. Step 6: Ratio of their effective investments is 1800000 : 2250000. Step 7: Simplify the ratio: divide both by 450000 to get 4 : 5. Step 8: Therefore, profit is shared in the ratio 4 : 5 (Ramesh : Rajesh). Step 9: Total profit = 27000, so each ratio part = 27000 / (4 + 5) = 27000 / 9 = 3000. Step 10: Rajesh's share = 5 parts = 5 * 3000 = 15000. Step 11: Hence, Rajesh receives Rs. 15,000 as his share of the profit.

Verification / Alternative check:
We can compute Ramesh's share as 4 parts = 4 * 3000 = 12000. Then, 12000 + 15000 = 27000, which matches the total profit. Also, the share division 12000 : 15000 simplifies to 4 : 5, matching the ratio of their effective investments. This confirms that the calculation for Rajesh's share is correct.

Why Other Options Are Wrong:
10,000 and 12,500 correspond to smaller shares than justified by Rajesh's larger effective investment. 20,000 and 25,000 are too large; they would upset the balance of the 4 : 5 ratio and either exceed the total profit or leave an incorrect amount for Ramesh. Only 15,000 fits the correct investment time ratio and matches the total profit distribution.

Common Pitfalls:
One common mistake is to ignore the different times for which the investments were made and compare only the raw capital amounts 50000 and 75000. Another is to treat 3 years as 3 months or miscount the months of Rajesh's investment. Students also sometimes invert the ratio or forget to add the ratio parts correctly before dividing the total profit. Careful attention to the time factor and ratio arithmetic is essential.

Final Answer:
Rajesh's share in the profit is Rs. 15,000.