Pipe A and B can fill a water tank in 30 and 45 minutes respectively while pipe C can take out all the water is 60 minutes. If the three pipes are opened simultaneously . How long it fill the empty tank

Options

A. `(25 +5/7)` min
B. 60 min
C. `(34+1/2)` min
D. `(18+5/7)` min

Answer

To solve the problem step by step, we will calculate the rates at which each pipe fills or empties the tank and then combine these rates to find out how long it will take to fill the tank when all pipes are opened simultaneously.

### Step 1: Determine the rate of each pipe
- **Pipe A** can fill the tank in 30 minutes. Therefore, its rate is:
\[ \text{Rate of Pipe A} = \frac{1}{30} \text{ tank per minute} \]

- **Pipe B** can fill the tank in 45 minutes. Therefore, its rate is:
\[ \text{Rate of Pipe B} = \frac{1}{45} \text{ tank per minute} \]

- **Pipe C** can empty the tank in 60 minutes. Therefore, its rate (as it is emptying) is:
\[ \text{Rate of Pipe C} = -\frac{1}{60} \text{ tank per minute} \]

### Step 2: Combine the rates of the pipes
When all three pipes are opened simultaneously, the combined rate of filling the tank is:
\[ \text{Combined Rate} = \text{Rate of Pipe A} + \text{Rate of Pipe B} + \text{Rate of Pipe C} \]
Substituting the rates we found:
\[ \text{Combined Rate} = \frac{1}{30} + \frac{1}{45} - \frac{1}{60} \]

### Step 3: Find a common denominator
To add these fractions, we need to find the least common multiple (LCM) of the denominators 30, 45, and 60. The LCM is 180.

Now we convert each fraction:
- \(\frac{1}{30} = \frac{6}{180}\)
- \(\frac{1}{45} = \frac{4}{180}\)
- \(-\frac{1}{60} = -\frac{3}{180}\)

### Step 4: Add the fractions
Now we can add the fractions:
\[ \text{Combined Rate} = \frac{6}{180} + \frac{4}{180} - \frac{3}{180} = \frac{6 + 4 - 3}{180} = \frac{7}{180} \]

### Step 5: Calculate the time to fill the tank
To find the time taken to fill the tank, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{7}{180}} = \frac{180}{7} \text{ minutes} \]

### Step 6: Simplify the time
\[ \frac{180}{7} \approx 25.71 \text{ minutes} \quad \text{(or } 25 \frac{5}{7} \text{ minutes)} \]

### Final Answer
The time taken to fill the tank when all three pipes are opened simultaneously is \(\frac{180}{7}\) minutes.

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