Three taps A, B and C can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively. How long would the three taps take to fill the empty tank, if all of them are opened together?

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 6 hours.
- Tap B can fill the tank in 8 hours.
- Tap C can fill the tank in 12 hours.

The rate of each tap can be expressed as:
- Rate of A = 1/6 tank per hour
- Rate of B = 1/8 tank per hour
- Rate of C = 1/12 tank per hour

### Step 2: Add the rates of the taps
To find the combined rate when all taps are opened together, we add their individual rates:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
\[ \text{Combined Rate} = \frac{1}{6} + \frac{1}{8} + \frac{1}{12} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 6, 8, and 12 is 24. We will convert each rate to have a denominator of 24:
- \(\frac{1}{6} = \frac{4}{24}\)
- \(\frac{1}{8} = \frac{3}{24}\)
- \(\frac{1}{12} = \frac{2}{24}\)

### Step 4: Add the fractions
Now we can add the fractions:
\[ \text{Combined Rate} = \frac{4}{24} + \frac{3}{24} + \frac{2}{24} = \frac{4 + 3 + 2}{24} = \frac{9}{24} \]

### Step 5: Simplify the combined rate
The combined rate can be simplified:
\[ \frac{9}{24} = \frac{3}{8} \text{ tank per hour} \]

### Step 6: Calculate the time to fill the tank
To find the time taken to fill the entire tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{3}{8} \text{ tank per hour}} = \frac{8}{3} \text{ hours} \]

### Step 7: Convert the time into hours and minutes
\(\frac{8}{3}\) hours can be expressed as:
- 2 hours and \(\frac{2}{3}\) of an hour.
- \(\frac{2}{3}\) of an hour is 40 minutes (since \(\frac{2}{3} \times 60 = 40\)).

Thus, the total time taken to fill the tank with all three taps open is:
\[ 2 \text{ hours and } 40 \text{ minutes} \]

### Final Answer
The three taps A, B, and C together will take **2 hours and 40 minutes** to fill the tank.

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