Three taps A, B and C can fill a tank in 10, 12 and 15 hours respectively. If all the taps are opened together, then in how many hours will the tank be filled?

Options

A. 4
B. 6
C. 8
D. 2

Answer

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

### Step 1: Determine the rate of each tap
- Tap A fills the tank in 10 hours, so its rate is \( \frac{1}{10} \) of the tank per hour.
- Tap B fills the tank in 12 hours, so its rate is \( \frac{1}{12} \) of the tank per hour.
- Tap C fills the tank in 15 hours, so its rate is \( \frac{1}{15} \) of the tank per hour.

### Step 2: Add the rates together
To find the combined rate when all taps are opened together, we add their individual rates:
\[ \text{Combined rate} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 10, 12, and 15 is 60. We will convert each rate to have this common denominator:
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)

### Step 4: Add the fractions
Now we can add the fractions:
\[ \text{Combined rate} = \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{15}{60} \]
This simplifies to:
\[ \frac{15}{60} = \frac{1}{4} \]

### Step 5: Calculate the time to fill the tank
The combined rate of \( \frac{1}{4} \) means that together, the taps can fill \( \frac{1}{4} \) of the tank in one hour. Therefore, to fill the entire tank:
\[ \text{Time} = \frac{1}{\text{Combined rate}} = \frac{1}{\frac{1}{4}} = 4 \text{ hours} \]

### Conclusion
Thus, if all the taps are opened together, the tank will be filled in **4 hours**.

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