A is 40 % more efficient than B and B is 20 % less efficient than C. If A takes 6 days less than C to complete a work, then in how many days will B complete this work ?

Answer

To solve the problem step by step, we will first establish the relationships between the efficiencies of A, B, and C based on the information given in the question.

### Step 1: Define the efficiencies
Let the efficiency of B be \( E_B \). According to the problem:
- A is 40% more efficient than B.
- Therefore, the efficiency of A, \( E_A \), can be expressed as:
\[ E_A = E_B + 0.4E_B = 1.4E_B \]

- B is 20% less efficient than C.
- Therefore, the efficiency of C, \( E_C \), can be expressed as:
\[ E_B = E_C - 0.2E_C = 0.8E_C \quad \Rightarrow \quad E_C = \frac{E_B}{0.8} = 1.25E_B \]

### Step 2: Establish the relationship between days taken
Let the total work be \( W \). The time taken by A and C to complete the work can be expressed as:
- Time taken by A:
\[ T_A = \frac{W}{E_A} = \frac{W}{1.4E_B} \]
- Time taken by C:
\[ T_C = \frac{W}{E_C} = \frac{W}{1.25E_B} \]

### Step 3: Use the information about the time difference
According to the problem, A takes 6 days less than C to complete the work:
\[ T_C - T_A = 6 \]
Substituting the expressions for \( T_A \) and \( T_C \):
\[ \frac{W}{1.25E_B} - \frac{W}{1.4E_B} = 6 \]

### Step 4: Solve the equation
To solve this equation, we first find a common denominator:
\[ \frac{W \cdot 1.4 - W \cdot 1.25}{1.25 \cdot 1.4 E_B} = 6 \]
\[ \frac{W(1.4 - 1.25)}{1.75E_B} = 6 \]
\[ \frac{W(0.15)}{1.75E_B} = 6 \]
\[ W = \frac{6 \cdot 1.75E_B}{0.15} \]
\[ W = 70E_B \]

### Step 5: Find the time taken by B
Now, we can find the time taken by B to complete the work:
\[ T_B = \frac{W}{E_B} = \frac{70E_B}{E_B} = 70 \text{ days} \]

### Conclusion
Thus, B will complete the work in **70 days**.

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