Alternating workers by day: Ram finishes a job in 7 days, Rahim in 8 days, and Robert in 6 days. They work one after another in a three-day cycle: Day 1 Ram, Day 2 Rahim, Day 3 Robert, then repeat. Who is working on the last day when the work finishes?

Introduction / Context:
In cyclic work schedules, the total work done per full cycle helps estimate how many cycles are required. Then compare the remaining work with the contribution of the next worker in sequence to determine who completes the job on the final day.

Given Data / Assumptions:

• Ram’s rate = 1/7 per day.
• Rahim’s rate = 1/8 per day.
• Robert’s rate = 1/6 per day.
• Cycle order: Ram → Rahim → Robert → repeat.

Concept / Approach:
Compute the three-day cycle contribution S = 1/7 + 1/8 + 1/6. Multiply by the number of full cycles and then evaluate the remainder relative to the next person’s daily rate. The person whose day reaches or exceeds the remaining work is the finisher.

Step-by-Step Solution:

Verification / Alternative check:
If Ram’s final-day effort slightly exceeds remaining work, it simply means he completes it that day. There is no need for the others to work further.

Why Other Options Are Wrong:
Neither Rahim nor Robert is needed after Ram’s final contribution covers the remaining 22/168 of the work.

Common Pitfalls:
Misordering the cycle or assuming equal daily contributions. Always compare the exact remaining fraction with the next worker’s full-day contribution.

Final Answer:
Ram