Three taps A, B, C can fill an overhead tank in 4, 6 and 12 hours respectively. How long would the three taps take to fill the tank if all of them are opened together ?

Options

A. 2 hrs.
B. 4 hrs
C. 3 hrs
D. 5 hrs

Answer

To solve the problem of how long it would take for taps A, B, and C to fill the tank together, we can follow these steps:

### Step 1: Determine the rate of work for each tap
- Tap A can fill the tank in 4 hours. Therefore, its rate of work is:
\[ \text{Rate of A} = \frac{1}{4} \text{ tank/hour} \]
- Tap B can fill the tank in 6 hours. Therefore, its rate of work is:
\[ \text{Rate of B} = \frac{1}{6} \text{ tank/hour} \]
- Tap C can fill the tank in 12 hours. Therefore, its rate of work is:
\[ \text{Rate of C} = \frac{1}{12} \text{ tank/hour} \]

### Step 2: Calculate the combined rate of work
To find the combined rate of work when all three taps are opened together, we add their individual rates:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the values:
\[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{6} + \frac{1}{12} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 4, 6, and 12 is 12. We can convert each fraction to have a denominator of 12:
- \(\frac{1}{4} = \frac{3}{12}\)
- \(\frac{1}{6} = \frac{2}{12}\)
- \(\frac{1}{12} = \frac{1}{12}\)

Now we can add them:
\[ \text{Combined Rate} = \frac{3}{12} + \frac{2}{12} + \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \text{ tank/hour} \]

### Step 4: Calculate the time taken to fill the tank
To find the time taken to fill the tank when all taps are working together, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} \]
Since the total work to fill the tank is 1 tank, we have:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{2} \text{ tank/hour}} = 2 \text{ hours} \]

### Conclusion
Thus, the three taps A, B, and C together will take **2 hours** to fill the tank.

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