Deepa starts a business in 2009 by investing Rs. 50,000. In 2010 she adds Rs. 20,000 more to her capital, and her friend Sushma joins at that time with an investment of Rs. 70,000. In 2011 Deepa adds another Rs. 20,000, making her capital larger again, and Avanthi joins with Rs. 70,000. At the end of these 3 years, the total profit is Rs. 3,00,000. What is Sushma's share of this profit?

Introduction / Context:
This question involves a changing capital partnership over multiple years. Different partners join at different times and capital amounts change during the three year period from 2009 to 2011. To find Sushma's share of the profit, we need to carefully compute capital time contributions for each partner across each year and then distribute the total profit in that ratio.

Given Data / Assumptions:

• Deepa invests Rs. 50,000 in 2009.
• Deepa increases her capital by Rs. 20,000 in 2010 (so it becomes Rs. 70,000 in 2010).
• Sushma joins in 2010 with Rs. 70,000.
• In 2011, Deepa adds another Rs. 20,000, making her investment Rs. 90,000 that year.
• Avanthi joins in 2011 with Rs. 70,000.
• The business runs from the start of 2009 to the end of 2011 (3 years).
• Total profit at the end of 3 years is Rs. 3,00,000.

Concept / Approach:
When capital changes or new partners join at different times, the profit sharing is based on the product of capital and the time for which that capital remains invested. We treat each year as one equal time unit and compute capital time for each partner in each year. Summing these gives relative weights for profit distribution. Then we compute Sushma's share by applying her share of total weight to the total profit.

Step-by-Step Solution:
Step 1: For 2009, only Deepa invests Rs. 50,000 for 1 year, so Deepa's capital time = 50,000 * 1 = 50,000.Step 2: For 2010, Deepa invests Rs. 70,000 for 1 year and Sushma invests Rs. 70,000 for 1 year, so Deepa contributes 70,000 and Sushma contributes 70,000 for that year.Step 3: For 2011, Deepa invests Rs. 90,000 for 1 year, Sushma invests Rs. 70,000 for 1 year and Avanthi invests Rs. 70,000 for 1 year.Step 4: Total capital time for Deepa = 50,000 + 70,000 + 90,000 = 2,10,000.Step 5: Total capital time for Sushma = 70,000 (2010) + 70,000 (2011) = 1,40,000.Step 6: Total capital time for Avanthi = 70,000 (2011) = 70,000.Step 7: The ratio Deepa : Sushma : Avanthi is 2,10,000 : 1,40,000 : 70,000 = 3 : 2 : 1.Step 8: Total ratio parts are 3 + 2 + 1 = 6. Each part of profit = 3,00,000 / 6 = Rs. 50,000.Step 9: Sushma gets 2 parts, so Sushma's share = 2 * 50,000 = Rs. 1,00,000.

Verification / Alternative check:
Check the shares: Deepa gets 3 * 50,000 = Rs. 1,50,000, Sushma gets 1,00,000 and Avanthi gets 50,000. Summing these gives 1,50,000 + 1,00,000 + 50,000 = Rs. 3,00,000, which matches the total profit. This confirms that the ratio and calculations are correct.

Why Other Options Are Wrong:
Rs. 75,000 and Rs. 85,000 do not correspond to any integer number of parts in the 3 : 2 : 1 ratio when the total profit is Rs. 3,00,000. Rs. 98,000 is close to 1,00,000 but does not match the exact computation. 'None of these' is incorrect because Rs. 1,00,000 is a valid option that satisfies all conditions precisely.

Common Pitfalls:
Many students treat the problem as if the capitals remain constant over the whole 3 years, ignoring the changes. Another error is to ignore Avanthi's late entry or to assume each year's profit is split separately. The correct method is to use capital time for the entire duration and then form a single ratio that accounts for every change in capital and membership.

Final Answer:
Sushma's correct share of the profit is Rs. 1,00,000.