Kaushalya can complete a piece of work in 20 days, while Kaikeyi can complete the same work in 25 days. They start the work together. After some time, Sumitra also joins them and together all three complete the entire work in a total of 10 days. If they receive Rs. 700 in all, distributed in proportion to the amount of work done by each, what is Sumitra's share?

Introduction / Context:
This is a time and work question combined with a profit sharing or wage sharing concept. The workers are paid in proportion to the work they have done. Even though Sumitra joins later and we are not told exactly when, we can still find how much work must have been done by her based on the fact that the total job finishes in 10 days and the capacities of Kaushalya and Kaikeyi are known.

Given Data / Assumptions:

• Kaushalya alone can complete the work in 20 days.
• Kaikeyi alone can complete the work in 25 days.
• They start together and work for the entire 10 days.
• Sumitra joins after some unknown number of days but stays until completion on the 10th day.
• Total payment is Rs. 700, distributed in proportion to each person's work.
• We must find Sumitra's share.

Concept / Approach:
The key idea is that we do not actually need to know the exact day Sumitra joined. Over 10 days, Kaushalya and Kaikeyi will each have done a fixed fraction of the total job because their rates and days of work are fully known. The remaining fraction of the work must then belong to Sumitra. Once we know the fraction of the total work done by Sumitra, we can multiply this fraction by the total payment to get her share.

Step-by-Step Solution:
Let total work be 1 unit. Kaushalya's daily work rate = 1 / 20. Kaikeyi's daily work rate = 1 / 25. They both work for the full 10 days. Work done by Kaushalya in 10 days = 10 * (1 / 20) = 1 / 2. Work done by Kaikeyi in 10 days = 10 * (1 / 25) = 2 / 5. Total work done by Kaushalya and Kaikeyi together = 1 / 2 + 2 / 5. Compute: 1 / 2 = 5 / 10 and 2 / 5 = 4 / 10, so sum = 9 / 10. Hence, remaining work done by Sumitra = 1 − 9 / 10 = 1 / 10. Sumitra's share of the work = 1 / 10 of total job. Total payment = Rs. 700, so Sumitra's share = 1 / 10 of 700 = Rs. 70.

Verification / Alternative check:
You can quickly verify that the other two shares sum to Rs. 630. Since their combined work is 9 / 10 of the job, the ratio of their combined payment to Sumitra's payment must be 9 : 1, i.e. 630 : 70 = 9 : 1, which is correct. This consistency check confirms that our approach is valid.

Why Other Options Are Wrong:
Rs. 130 or Rs. 185 would correspond to larger fractions of the work than 1 / 10, which contradicts the computed contributions of Kaushalya and Kaikeyi over 10 days. The option “can't be determined” is tempting but incorrect because the missing information about the exact joining day of Sumitra is not actually required for determining her share of work.

Common Pitfalls:
Many students try to first find the exact time when Sumitra joined the work, which is not necessary and can lead to algebraic confusion. Focusing instead on the total work done by those whose full working period is known (here Kaushalya and Kaikeyi) simplifies the problem significantly.

Final Answer:
Sumitra's share of the Rs. 700 is Rs. 70.