Three taps A, B and C can fill a tank in 27, 36 and 54 minutes respectively. If all the three taps are opened, then how much time (in minutes) it will take to completely fill the tank?

Options

A. 16
B. 12
C. 10
D. 15

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, we can follow these steps:

### Step 1: Determine the filling rates of each tap.
- Tap A can fill the tank in 27 minutes. Therefore, in 1 minute, it fills \( \frac{1}{27} \) of the tank.
- Tap B can fill the tank in 36 minutes. Therefore, in 1 minute, it fills \( \frac{1}{36} \) of the tank.
- Tap C can fill the tank in 54 minutes. Therefore, in 1 minute, it fills \( \frac{1}{54} \) of the tank.

### Step 2: Find a common denominator to add the rates.
To add the filling rates, we need to find the least common multiple (LCM) of the denominators 27, 36, and 54.

- The LCM of 27, 36, and 54 is 108.

### Step 3: Convert the rates to a common denominator.
Now we convert each rate to have a denominator of 108:
- For Tap A:
\[ \frac{1}{27} = \frac{4}{108} \]
- For Tap B:
\[ \frac{1}{36} = \frac{3}{108} \]
- For Tap C:
\[ \frac{1}{54} = \frac{2}{108} \]

### Step 4: Add the filling rates.
Now we can add the rates:
\[ \text{Total rate} = \frac{4}{108} + \frac{3}{108} + \frac{2}{108} = \frac{4 + 3 + 2}{108} = \frac{9}{108} \]

### Step 5: Simplify the total rate.
The total rate simplifies to:
\[ \frac{9}{108} = \frac{1}{12} \]
This means that together, the three taps can fill \( \frac{1}{12} \) of the tank in 1 minute.

### Step 6: Calculate the total time to fill the tank.
To find the total time to fill the tank, we take the reciprocal of the total rate:
\[ \text{Time} = \frac{1}{\frac{1}{12}} = 12 \text{ minutes} \]

Thus, the time taken to completely fill the tank when all three taps are opened is **12 minutes**.