Arun and Varun start a business with investments of Rs. 20,000 and Rs. 15,000 respectively. After six months, Akhil joins them with an investment of Rs. 20,000. If the total profit at the end of 2 years from the start is Rs. 25,000, what is Varun's share of the profit?

Introduction / Context:
This partnership question involves three partners with different capital contributions and different times of joining. Arun and Varun start the business together, and Akhil joins after six months. The business runs for two years. The goal is to compute Varun's share of the total profit based on capital time units for each partner.

Given Data / Assumptions:

• Arun invests Rs. 20,000 from the start.
• Varun invests Rs. 15,000 from the start.
• Akhil joins after 6 months with Rs. 20,000.
• Total business duration is 2 years = 24 months.
• Total profit at the end of 2 years is Rs. 25,000.
• Profits are shared in proportion to capital multiplied by time.

Concept / Approach:
The method is to convert each partner's investment into capital time units, which are equal to capital multiplied by the number of months for which it is invested. Arun and Varun invest for 24 months each, while Akhil invests for 18 months (joining 6 months after the start). The ratio of profits is the ratio of their capital time units. Once this ratio is known, we treat it as parts of the total profit and compute Varun's share accordingly.

Step-by-Step Solution:
Step 1: Arun invests Rs. 20,000 for 24 months, so Arun's capital time units are 20000 * 24 = 480000. Step 2: Varun invests Rs. 15,000 for 24 months, so Varun's capital time units are 15000 * 24 = 360000. Step 3: Akhil invests Rs. 20,000 but only after 6 months, so his investment period is 24 - 6 = 18 months. Step 4: Akhil's capital time units are 20000 * 18 = 360000. Step 5: The ratio of capital time units for Arun : Varun : Akhil is 480000 : 360000 : 360000. Step 6: Divide each term by 120000 to simplify the ratio to 4 : 3 : 3. Step 7: Total parts in the ratio are 4 + 3 + 3 = 10. Step 8: Total profit is Rs. 25000, so one part is 25000 / 10 = Rs. 2500. Step 9: Varun's share corresponds to 3 parts, so Varun's profit is 3 * 2500 = Rs. 7500.

Verification / Alternative check:
Arun's share should be 4 parts = 4 * 2500 = Rs. 10000. Akhil's share should be 3 parts = Rs. 7500. Adding all three shares, 10000 + 7500 + 7500 = Rs. 25000, which matches the total profit given. This confirms that the ratio and Varun's share are calculated correctly.

Why Other Options Are Wrong:
Values such as Rs. 4520, Rs. 8750, Rs. 6350 or Rs. 6800 do not fit the ratio 4 : 3 : 3 when distributed over a total of Rs. 25000. For instance, if Varun's share were Rs. 8750, then his share would be more than 3 parts out of 10, which contradicts the calculated capital time ratio. Only Rs. 7500 preserves the correct ratio among all three partners and sums to Rs. 25000.

Common Pitfalls:
A common mistake is to divide the profit simply in the ratio of capitals 20,000 : 15,000 : 20,000 without accounting for different time durations. Another error is miscounting the months for Akhil, wrongly taking his period as 6 months instead of 18. It is essential to understand that the partner who joins later still remains until the end, and the correct time difference must be used in the calculation of capital time units.

Final Answer:
Varun's share of the profit is Rs. 7500, which is option A.