A tank can be filled by tap A in 4 hours while tap B can fill the same in 6 hours. Tap C can empty the filled tank in 8 hours. If all the three taps are opened simultaneously. How much approximate time will be taken to fill tank completely ?

Options

A. 3 hours, 26 minutes
B. 2 hours, 26 minutes
C. 3 hours, 16 minutes
D. 3 hours, 20 minutes

Answer

To solve the problem of how long it will take to fill the tank when all three taps A, B, and C are opened simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Filling Rates of Each Tap:**
- Tap A fills the tank in 4 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{4} \text{ tanks per hour} \]
- Tap B fills the tank in 6 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{6} \text{ tanks per hour} \]
- Tap C empties the tank in 8 hours. Therefore, its rate (as a negative value) is:
\[ \text{Rate of C} = -\frac{1}{8} \text{ tanks per hour} \]

2. **Calculate the Combined Rate of All Taps:**
- The combined rate when all taps are opened is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
- Substituting the rates:
\[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{6} - \frac{1}{8} \]

3. **Find a Common Denominator:**
- The least common multiple (LCM) of 4, 6, and 8 is 24. We will convert each fraction:
\[ \frac{1}{4} = \frac{6}{24}, \quad \frac{1}{6} = \frac{4}{24}, \quad -\frac{1}{8} = -\frac{3}{24} \]
- Now, adding these fractions:
\[ \text{Combined Rate} = \frac{6}{24} + \frac{4}{24} - \frac{3}{24} = \frac{7}{24} \text{ tanks per hour} \]

4. **Calculate the Time to Fill the Tank:**
- The time taken to fill the tank can be calculated using the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} \]
- Here, the total work (filling one tank) is 1 tank:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{7}{24} \text{ tanks per hour}} = \frac{24}{7} \text{ hours} \]

5. **Convert the Time into Hours and Minutes:**
- To convert \(\frac{24}{7}\) hours into hours and minutes:
- Divide 24 by 7:
\[ 24 \div 7 = 3 \text{ hours} \quad \text{(with a remainder of 3)} \]
- The remainder (3) corresponds to:
\[ \frac{3}{7} \text{ hours} \times 60 \text{ minutes/hour} = \frac{180}{7} \approx 25.71 \text{ minutes} \approx 26 \text{ minutes} \]
- Therefore, the total time is approximately:
\[ 3 \text{ hours and } 26 \text{ minutes} \]

### Final Answer:
The approximate time taken to fill the tank completely when all three taps are opened simultaneously is **3 hours and 26 minutes**.