Ajit is twice as good a workman as Badrinath and therefore finishes a certain job in 30 days less than Badrinath. Working together, in how many days will Ajit and Badrinath complete the job?

Introduction / Context:
Here we have a comparative efficiency problem where Ajit is described as “twice as good” a workman as Badrinath. This means Ajit's rate of working is double that of Badrinath. Additionally, Ajit finishes a job in 30 days less than Badrinath. We must use this relationship to find their individual times and then compute how long they would take to finish the job together. This involves setting up equations for time and rate and then combining rates.

Given Data / Assumptions:
- Ajit is twice as efficient as Badrinath. - Because of this, Ajit takes 30 days less than Badrinath to complete the job alone. - Both work at constant rates when they work. - Total work is one complete job. - We want the time taken when Ajit and Badrinath work together.

Concept / Approach:
If one person is twice as efficient as another, then their times for the same work are in the inverse ratio 1 : 2. Let Badrinath's time be T days, then Ajit's time is T / 2 days. We also know that Ajit's time is 30 days less than Badrinath's. Using these two relationships, we can solve for T. Once we know each person's time, we can find their daily work rates, add these rates to get the combined rate, and then take the reciprocal to get the total time when working together.

Step-by-Step Solution:
Step 1: Let time taken by Badrinath alone = T days. Step 2: Ajit is twice as efficient, so Ajit's time = T / 2 days. Step 3: We are told that Ajit takes 30 days less than Badrinath, so T - T / 2 = 30. Step 4: Simplify: T / 2 = 30, hence T = 60 days. Step 5: Therefore, Badrinath's time = 60 days and Ajit's time = 60 / 2 = 30 days. Step 6: Ajit's daily work rate = 1 / 30 job per day. Step 7: Badrinath's daily work rate = 1 / 60 job per day. Step 8: Combined rate when working together = 1 / 30 + 1 / 60 = 2 / 60 + 1 / 60 = 3 / 60 = 1 / 20 job per day. Step 9: Time taken together = 1 / (1 / 20) = 20 days.

Verification / Alternative check:
We can verify that Ajit's time is indeed 30 days less than Badrinath's: 60 - 30 = 30 days, which matches Ajit's time. Also, working together at a rate of 1 / 20 job per day, they would finish in 20 days. Since Ajit alone takes 30 days, finishing in 20 days with help from Badrinath is reasonable and consistent.

Why Other Options Are Wrong:
10 or 15 days: These are too short and would require a higher combined rate than the actual 1 / 20 job per day.
30 days: This is just Ajit's individual time, not the combined time with Badrinath helping.
25 days: This is higher than the calculated 20 days and does not match the combined rate when both work together.

Common Pitfalls:
Students may misinterpret “twice as good” and incorrectly double the time instead of halving it. Another mistake is ignoring the 30 day difference in times, using only the efficiency relation and ending up with incomplete information. Correctly translating verbal efficiency comparisons into algebraic equations for time and then solving those equations is crucial for such problems.

Final Answer:
Ajit and Badrinath together will complete the job in 20 days.