To solve the problem step by step, we can follow these instructions:
### Step 1: Define the efficiencies of A, B, and C
Let the efficiency of workman B be represented as \( K \). According to the problem:
- A is twice as good as B, so \( A = 2K \).
- C is thrice as good as B, so \( C = 3K \).
### Step 2: Calculate the combined efficiency of A, B, and C
Now, we can find the total efficiency when A, B, and C work together:
\[
\text{Total Efficiency} = A + B + C = 2K + K + 3K = 6K
\]
### Step 3: Determine the amount of work done in a day
According to the problem, A, B, and C together can complete the work in 2 days. This means that in one day, they complete half of the work. Therefore, the work done in one day is:
\[
\text{Work done in 1 day} = \frac{1}{2} \text{ (of the total work)}
\]
### Step 4: Set up the equation for total work
Since the total work done in one day is equal to their combined efficiency, we can set up the equation:
\[
6K = \frac{1}{2}
\]
### Step 5: Solve for K
To find \( K \), we multiply both sides by 2:
\[
12K = 1 \implies K = \frac{1}{12}
\]
### Step 6: Find A's efficiency
Now that we have \( K \), we can find A's efficiency:
\[
A = 2K = 2 \times \frac{1}{12} = \frac{1}{6}
\]
### Step 7: Calculate the time taken by A to complete the work alone
If A's efficiency is \( \frac{1}{6} \), it means A can complete \( \frac{1}{6} \) of the work in one day. Therefore, the time taken by A to complete the entire work is:
\[
\text{Time taken by A} = \frac{1}{\frac{1}{6}} = 6 \text{ days}
\]
### Final Answer
A alone can complete the work in **6 days**.
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