Three taps A, B and C can fill a tank in 20, 30 and 36 hours respectively. If all the taps are opened together, then in how many hours will the tank be filled?

Options

A. a. 6
B. b. 9
C. c. 12
D. d. 15

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 filling rates of each tap.
- Tap A fills the tank in 20 hours, so its rate is \( \frac{1}{20} \) of the tank per hour.
- Tap B fills the tank in 30 hours, so its rate is \( \frac{1}{30} \) of the tank per hour.
- Tap C fills the tank in 36 hours, so its rate is \( \frac{1}{36} \) of the tank per hour.

### Step 2: Find the total filling rate when all taps are opened together.
To find the combined rate of all three taps, we add their individual rates:
\[ \text{Total rate} = \frac{1}{20} + \frac{1}{30} + \frac{1}{36} \]

### Step 3: Calculate the least common multiple (LCM) of the denominators.
The LCM of 20, 30, and 36 is 180. We will convert each fraction to have a denominator of 180:
- For \( \frac{1}{20} \):
\[ \frac{1}{20} = \frac{9}{180} \]
- For \( \frac{1}{30} \):
\[ \frac{1}{30} = \frac{6}{180} \]
- For \( \frac{1}{36} \):
\[ \frac{1}{36} = \frac{5}{180} \]

### Step 4: Add the fractions.
Now we can add the fractions:
\[ \text{Total rate} = \frac{9}{180} + \frac{6}{180} + \frac{5}{180} = \frac{20}{180} \]

### Step 5: Simplify the total rate.
\[ \frac{20}{180} = \frac{1}{9} \]
This means that together, the three taps can fill \( \frac{1}{9} \) of the tank in one hour.

### Step 6: Calculate the total time to fill the tank.
If the combined rate is \( \frac{1}{9} \) of the tank per hour, then the time taken to fill the entire tank is the reciprocal of the rate:
\[ \text{Time} = \frac{1}{\text{Total rate}} = \frac{1}{\frac{1}{9}} = 9 \text{ hours} \]

### Final Answer:
The tank will be filled in **9 hours** when all taps A, B, and C are opened together.

---