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 hr
B. 4 hr
C. 3 hr
D. 5hr

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 each tap
- Tap A can fill the tank in 4 hours. Therefore, the rate of tap A is:
\[ \text{Rate of A} = \frac{1}{4} \text{ tank per hour} \]

- Tap B can fill the tank in 6 hours. Therefore, the rate of tap B is:
\[ \text{Rate of B} = \frac{1}{6} \text{ tank per hour} \]

- Tap C can fill the tank in 12 hours. Therefore, the rate of tap C is:
\[ \text{Rate of C} = \frac{1}{12} \text{ tank per hour} \]

### Step 2: Calculate the combined rate of all taps
To find the combined rate 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 rates we calculated:
\[ \text{Combined Rate} = \frac{1}{4} + \frac{1}{6} + \frac{1}{12} \]

### 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:
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}, \quad \frac{1}{12} = \frac{1}{12} \]

### Step 4: Add the fractions
Now we can add the fractions:
\[ \text{Combined Rate} = \frac{3}{12} + \frac{2}{12} + \frac{1}{12} = \frac{6}{12} = \frac{1}{2} \]

### Step 5: Calculate the total time to fill the tank
The combined rate of \(\frac{1}{2}\) means that together, the taps can fill \(\frac{1}{2}\) of the tank in one hour. To find the total time \(T\) to fill the entire tank, we take the reciprocal of the combined rate:
\[ T = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{2}} = 2 \text{ hours} \]

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

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