A and B undertake a piece of work for Rs. 250. A alone can do the work in 5 days and B alone can do it in 15 days. With the help of C, they finish the work in 3 days. If everyone is paid in proportion to the amount of work done, how much money should C receive?

Introduction / Context:
This question combines time and work concepts with proportional division of money. Three people, A, B, and C, share a contract payment based on how much work each completes. We know how long A and B take individually, and how long the three take together. From this we infer C’s contribution and then compute C’s fair share of the total payment.

Given Data / Assumptions:
- Total contract amount = Rs. 250.
- A alone can complete the work in 5 days.
- B alone can complete the work in 15 days.
- A, B, and C together complete the work in 3 days.
- Payment is proportional to work done by each individual.

Concept / Approach:
First, we compute the daily work rates of A and B. Next, we determine the combined rate of all three from the given completion time of 3 days. C’s daily rate is then found by subtracting the rates of A and B from the combined rate. Knowing C’s rate, we calculate the fraction of work done by C in 3 days and then multiply this fraction by the total payment to get C’s share.

Step-by-Step Solution:
Step 1: Assume total work = 1 unit. Step 2: Rate of A = 1 / 5 work per day. Step 3: Rate of B = 1 / 15 work per day. Step 4: Together, A, B, and C finish in 3 days, so combined rate = 1 / 3 work per day. Step 5: Rate of A + B = 1 / 5 + 1 / 15. Step 6: LCM of 5 and 15 is 15, so 1 / 5 = 3 / 15 and 1 / 15 = 1 / 15. Step 7: Rate of A + B = 3 / 15 + 1 / 15 = 4 / 15 work per day. Step 8: Rate of C = combined rate - rate of A + B = 1 / 3 - 4 / 15. Step 9: Convert 1 / 3 to denominator 15: 1 / 3 = 5 / 15. Step 10: Rate of C = 5 / 15 - 4 / 15 = 1 / 15 work per day. Step 11: Work done by C in 3 days = 3 * 1 / 15 = 3 / 15 = 1 / 5 of the work. Step 12: C’s share of money = (1 / 5) * 250 = Rs. 50.

Verification / Alternative check:
We can also compute A’s and B’s shares to verify. In 3 days, A does 3 * 1 / 5 = 3 / 5 of the work, and B does 3 * 1 / 15 = 1 / 5 of the work. Thus, the distribution of work among A, B, and C is 3 / 5, 1 / 5, and 1 / 5 respectively, corresponding to a ratio of 3 : 1 : 1. So A gets 3 / 5 * 250 = 150, B gets 1 / 5 * 250 = 50, and C also gets 50. The totals add up to 250, confirming correctness.

Why Other Options Are Wrong:
- Rs.100, Rs.150, Rs.200: These amounts would imply that C did a larger fraction of the work than 1 / 5, which contradicts the calculated work share based on rates and time.
- Rs.75: This does not correspond to any simple fraction of 250 and does not align with the ratio of work contributions.

Common Pitfalls:
Many learners forget to compute C’s rate explicitly and instead try to guess the share. Another error is to divide the total amount equally among the three people, ignoring the actual work distribution. It is important to always base payment on proportional work done, especially in time and work problems involving contracts and money.

Final Answer:
C should receive Rs. 50 from the total payment.