Two pipes A and B can fill a tank in 36 min and 45 min, respectively. A pipe 'C' can empty the tank in 30 min. If all the three pipes are opened simultaneously, then in how much time the tank will be completely filled?

Options

A. 6h
B. 1 h
C. 3 h
D. 4 h

Answer

To solve the problem of how long it will take to fill the tank when all three pipes A, B, and C are opened simultaneously, we will follow these steps:

### Step 1: Determine the filling rates of pipes A and B and the emptying rate of pipe C.
- Pipe A fills the tank in 36 minutes, so its rate is \( \frac{1}{36} \) of the tank per minute.
- Pipe B fills the tank in 45 minutes, so its rate is \( \frac{1}{45} \) of the tank per minute.
- Pipe C empties the tank in 30 minutes, so its rate is \( \frac{1}{30} \) of the tank per minute (but we will consider it as negative since it is emptying).

### Step 2: Calculate the combined rate of filling when all pipes are open.
The combined rate of filling the tank when all pipes are open is given by:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
Substituting the rates we found:
\[ \text{Combined Rate} = \frac{1}{36} + \frac{1}{45} - \frac{1}{30} \]

### Step 3: Find a common denominator to simplify the equation.
The least common multiple (LCM) of 36, 45, and 30 is 180. We will convert each fraction to have a denominator of 180:
- \( \frac{1}{36} = \frac{5}{180} \)
- \( \frac{1}{45} = \frac{4}{180} \)
- \( \frac{1}{30} = \frac{6}{180} \)

Now substituting back into the combined rate:
\[ \text{Combined Rate} = \frac{5}{180} + \frac{4}{180} - \frac{6}{180} = \frac{5 + 4 - 6}{180} = \frac{3}{180} \]

### Step 4: Simplify the combined rate.
\[ \text{Combined Rate} = \frac{1}{60} \]
This means that together, the pipes fill \( \frac{1}{60} \) of the tank in one minute.

### Step 5: Calculate the time to fill the entire tank.
If the combined rate is \( \frac{1}{60} \) of the tank per minute, then the time taken to fill the entire tank is the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = 60 \text{ minutes} \]

### Final Answer:
The tank will be completely filled in **60 minutes**.

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