Simran starts a software business by investing Rs. 50,000. After six months, Nanda joins her with a capital of Rs. 80,000. After 3 years from the start, the total profit is Rs. 24,500. What is Simran's share of the profit?

Introduction / Context:
This is a partnership question where Simran starts a business and Nanda joins after six months. The total profit after 3 years is known. Since the partners invest different amounts for different periods, their shares of profit must be determined using capital time units. The question focuses on calculating Simran's share of the total profit.

Given Data / Assumptions:

• Simran invests Rs. 50,000 at the beginning.
• Nanda joins after 6 months with Rs. 80,000.
• Total duration from the start is 3 years, that is 36 months.
• Total profit at the end of 3 years is Rs. 24,500.
• Profits are shared in proportion to capital multiplied by time.
• We must find Simran's profit share.

Concept / Approach:
Simran is in the business for the full 36 months, while Nanda is present only from month 7 to month 36, that is 30 months. We compute capital time units for both: capital multiplied by months. The ratio of their profit shares will be the ratio of these capital time units. With the total profit known, we can split it according to this ratio and obtain Simran's share.

Step-by-Step Solution:
Step 1: Simran invests Rs. 50,000 for 36 months, so Simran's capital time units are 50000 * 36 = 1800000. Step 2: Nanda joins after 6 months and therefore invests for 36 - 6 = 30 months. Step 3: Nanda's capital time units are 80000 * 30 = 2400000. Step 4: The ratio of Simran's to Nanda's profit shares is 1800000 : 2400000. Step 5: Simplify this ratio by dividing both numbers by 600000 to get 3 : 4. Step 6: Total profit is Rs. 24,500, which corresponds to 3 + 4 = 7 parts in this ratio. Step 7: Value of one part is 24500 / 7 = Rs. 3500. Step 8: Simran's share is 3 parts, so 3 * 3500 = Rs. 10,500.

Verification / Alternative check:
Nanda's share should be 4 parts = 4 * 3500 = Rs. 14,000. Adding both shares, 10500 + 14000 = Rs. 24,500, which matches the total profit given. This verifies that the ratio 3 : 4 and the resulting shares are correct, confirming Simran's profit as Rs. 10,500.

Why Other Options Are Wrong:
Other amounts such as Rs. 12421, Rs. 9423, Rs. 11600 or Rs. 9800 would not divide the total profit in the ratio 3 : 4. For instance, if Simran received Rs. 11600, Nanda would receive 24500 - 11600 = Rs. 12900, and the ratio 11600 : 12900 would not simplify to 3 : 4. Only Rs. 10500 gives a complementary share of Rs. 14000 for Nanda, producing the correct ratio and matching the total profit.

Common Pitfalls:
A common error is miscalculating the time period for which Nanda invests, sometimes taking it as 6 months or 36 months instead of 30 months. Another mistake is to use the capital ratio 50,000 : 80,000 directly without adjusting for time, which would lead to an incorrect profit ratio of 5 : 8. Careful reading of the joining time and accurate computation of capital time units are crucial in partnership problems.

Final Answer:
Simran's share of the profit is Rs. 10,500, which corresponds to option D.