A tap can fill a tank in 6 hours. After half the tank is filled, three more similar taps are opened. What is the total time taken to fill the tank completely?

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Accepted Answer

1. **Determine the total work (capacity of the tank):**  
   - Let the total work = 6 units (LCM of time taken by one tap).

2. **Efficiency of one tap:**  
   - Efficiency of one tap = $$\frac{6}{6} = 1$$ unit/hour.

3. **Time taken to fill half the tank with one tap:**  
   - Half work = $$\frac{6}{2} = 3$$ units.  
   - Time taken = $$\frac{3}{1} = 3$$ hours.

4. **Remaining work after half the tank is filled:**  
   - Remaining work = $$6 - 3 = 3$$ units.

5. **Efficiency of all four taps working together:**  
   - Total efficiency = $$1 + 1 + 1 + 1 = 4$$ units/hour.

6. **Time taken to complete the remaining work:**  
   - Time = $$\frac{	ext{Remaining Work}}{	ext{Efficiency}} = \frac{3}{4}$$ hours = $$45$$ minutes.

7. **Total time taken to fill the tank completely:**  
   - Total time = Time for first half + Time for second half:  
     $$
     3 \, 	ext{hours} + 45 \, 	ext{minutes} = 3 \, 	ext{hours and }45 \, 	ext{minutes}.
     $$

Thus, the total time taken to fill the tank is **3 hours and 45 minutes**, making option **(4)** correct.

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