Work done by A in 3 days is equal to the work done by B in 4 days. What is the ratio of time taken by A and B to complete a work?

Answer

To solve the problem, we need to find the ratio of the time taken by A and B to complete a work based on the information given.

### Step-by-Step Solution:

1. **Understanding Work Done**:
The work done by a person can be expressed as:
\[ \text{Work Done} = \text{Efficiency} \times \text{Time} \]

2. **Setting Up the Equation**:
According to the problem, the work done by A in 3 days is equal to the work done by B in 4 days. We can express this mathematically as:
\[ \text{Efficiency of A} \times 3 = \text{Efficiency of B} \times 4 \]

3. **Expressing Efficiency**:
Let's denote the efficiency of A as \( E_A \) and the efficiency of B as \( E_B \). Thus, we can rewrite the equation as:
\[ E_A \times 3 = E_B \times 4 \]

4. **Finding the Ratio of Efficiencies**:
Rearranging the equation gives us:
\[ \frac{E_A}{E_B} = \frac{4}{3} \]
This means that A is more efficient than B.

5. **Finding the Ratio of Time Taken**:
The time taken to complete the work is inversely proportional to efficiency. Therefore, we can express the ratio of time taken by A and B as:
\[ \frac{\text{Time taken by A}}{\text{Time taken by B}} = \frac{E_B}{E_A} \]
Substituting the values we found:
\[ \frac{\text{Time taken by A}}{\text{Time taken by B}} = \frac{3}{4} \]

6. **Final Ratio**:
Thus, the ratio of time taken by A and B to complete the work is:
\[ \text{Time taken by A : Time taken by B} = 3 : 4 \]

### Final Answer:
The ratio of time taken by A and B to complete the work is **3 : 4**.

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