Anil is twice as efficient a workman as Bimal and together they finish a piece of work in 9 days. In how many days will Anil alone finish the work?

Introduction / Context:
This is a typical work and time question involving relative efficiency. One worker is twice as efficient as another, and together they complete a job in a known number of days. The problem is to find the time required by the more efficient worker alone to complete the entire job.

Given Data / Assumptions:

• Anil is twice as efficient as Bimal.
• Anil and Bimal together complete the work in 9 days.
• Both have constant work rates during the entire period.
• We are required to find the number of days taken by Anil alone to do the work.

Concept / Approach:
If Anil is twice as efficient as Bimal, then Anil's rate is 2 times Bimal's rate. Let us denote Bimal's rate as x units of work per day and Anil's rate as 2x units of work per day. Together their rate is 3x units per day. Since they complete the entire work in 9 days, we can determine the total work in terms of x and then find the time Anil would need alone using his rate.

Step-by-Step Solution:
Step 1: Let Bimal's work rate be x units per day.Step 2: Since Anil is twice as efficient, his rate = 2x units per day.Step 3: Combined rate of Anil and Bimal = x + 2x = 3x units per day.Step 4: Time taken together to finish the work = 9 days. Therefore total work W = 3x * 9 = 27x units.Step 5: Time taken by Anil alone = total work / Anil's rate = 27x / (2x) days.Step 6: Simplify 27x / (2x) = 27 / 2 = 13.5 days.

Verification / Alternative check:
We can check by computing daily contributions. In 13.5 days, Anil alone working at 2x units per day would complete 13.5 * 2x = 27x units of work, which matches the total work computed earlier. Also, working together at 3x units per day for 9 days gives 9 * 3x = 27x units, again matching. This consistency confirms that Anil needs 13.5 days alone to finish the same work.

Why Other Options Are Wrong:

• 12.5 days: This would mean total work is 25x, but our combined work in 9 days would then not match 25x, contradicting the given condition.
• 11.5 days: This gives total work 23x, inconsistent with the requirement that together they finish in 9 days at rate 3x per day.
• 10.5 days: This yields 21x total work for Anil, again inconsistent with the 27x units required when both work for 9 days.

Common Pitfalls:
Some students invert the efficiency relationship and assume that if Anil is twice as efficient, he will take twice as long, which is incorrect. Efficiency and time for the same job are inversely related. Others attempt to average times directly without considering rates. Carefully define rates, use them to find total work, and then derive the required time.

Final Answer:
Anil alone will finish the work in 13.5 days.