A is thrice as efficient as B. If B takes 8 days more than A, what is the number of days taken by A to finish the whole work, alone?

Options

A. 4
B. 2
C. 12
D. 16

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the relationship between A and B's efficiencies.
We know that A is thrice as efficient as B. This means:
- If B's efficiency is represented as \( E_B \), then A's efficiency \( E_A = 3 \times E_B \).

### Step 2: Define the work done in terms of days.
Let the number of days taken by A to complete the work be \( D_A \) and the number of days taken by B be \( D_B \). The work done can be expressed as:
- Work = Efficiency × Days
- For A: Work = \( E_A \times D_A \)
- For B: Work = \( E_B \times D_B \)

Since the work is constant for both A and B, we can equate the two expressions:
\[ E_A \times D_A = E_B \times D_B \]

### Step 3: Express B's days in terms of A's days.
From the problem, we know that B takes 8 days more than A:
\[ D_B = D_A + 8 \]

### Step 4: Substitute the efficiencies and days into the work equation.
Substituting \( E_A = 3 \times E_B \) and \( D_B = D_A + 8 \) into the work equation:
\[ (3 \times E_B) \times D_A = E_B \times (D_A + 8) \]

### Step 5: Simplify the equation.
We can cancel \( E_B \) from both sides (assuming \( E_B \neq 0 \)):
\[ 3D_A = D_A + 8 \]

### Step 6: Solve for \( D_A \).
Rearranging the equation gives:
\[ 3D_A - D_A = 8 \]
\[ 2D_A = 8 \]
\[ D_A = \frac{8}{2} = 4 \]

### Conclusion:
Thus, the number of days taken by A to finish the whole work alone is **4 days**.

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