Prabhas is twice as efficient a workman as Rana and can finish a piece of work in 3 hours when working alone. If Prabhas and Rana work together on the same job, in how many hours will they complete the work?

Introduction / Context:
This is a straightforward time and work question that involves comparing efficiencies of two workers. Knowing that one worker is twice as efficient as the other and that the faster worker can complete the work alone in a fixed time, we can easily find the combined time when both work together.

Given Data / Assumptions:
• Prabhas is twice as efficient as Rana.
• Prabhas alone finishes the work in 3 hours.
• Both have constant and independent work rates.
• We must find the time taken when Prabhas and Rana work together.

Concept / Approach:
Efficiency in this context means rate of work per unit time. If one worker is twice as efficient, his work rate is double. From the time taken by Prabhas alone, we find his work rate, then deduce Rana work rate. Adding the two gives the combined rate. The reciprocal of the combined rate gives the total time to finish the work together.

Step-by-Step Solution:
Let the total work be 1 job. Let Rana rate be r jobs per hour. Prabhas is twice as efficient, so his rate = 2r jobs per hour. Prabhas alone finishes the job in 3 hours, so 2r × 3 = 1. This gives 6r = 1, therefore r = 1 / 6 job per hour. So Rana rate = 1 / 6 job per hour and Prabhas rate = 2 / 6 = 1 / 3 job per hour. Combined rate when they work together = 1 / 3 + 1 / 6 = 1 / 2 job per hour. Time to complete 1 job at rate 1 / 2 = 1 ÷ (1 / 2) = 2 hours.

Verification / Alternative check:
In 2 hours, Prabhas would do 2 × (1 / 3) = 2 / 3 of the work. Rana would do 2 × (1 / 6) = 1 / 3 of the work. Together, 2 / 3 + 1 / 3 = 1 complete job. This confirms that 2 hours is exactly correct for both of them working together.

Why Other Options Are Wrong:
Any time greater than 3 hours, such as 3 or 3.5 hours, is impossible because Prabhas alone finishes in 3 hours, so adding Rana must reduce the total time. Times like 2.5 hours or 3 hours do not match the combined rate computed from their given efficiency relationship.

Common Pitfalls:
A typical mistake is to average the times directly, for example taking the mean of 3 hours and some assumed time for Rana, which is not valid. Another error is to misinterpret “twice as good a workman” as meaning that Prabhas takes half the time Rana takes, without translating this into consistent rates and rechecking the conditions.

Final Answer:
Prabhas and Rana together will complete the work in 2 hours.