Pipe A can fill a tank in 10 hours and pipe B can fill the same tank in 12 hours. Pipe C can empty the same full tank in 6 hours. If all the pipes are opened together, then what portion of the tank will be filled after 12 hours?

Options

A. `3//4`
B. `1//5`
C. `1//4`
D. `4//5`

Answer

To solve the problem step by step, let's break down the information given and calculate the portion of the tank filled after 12 hours when all pipes are opened together.

### Step 1: Determine the filling and emptying rates of the pipes
- **Pipe A** can fill the tank in 10 hours.
- Rate of Pipe A = \( \frac{1}{10} \) tank/hour
- **Pipe B** can fill the tank in 12 hours.
- Rate of Pipe B = \( \frac{1}{12} \) tank/hour
- **Pipe C** can empty the tank in 6 hours.
- Rate of Pipe C = \( -\frac{1}{6} \) tank/hour (negative because it empties)

### Step 2: Find the combined rate of all pipes
To find the combined rate when all pipes are opened together, we add the rates of Pipe A and Pipe B and subtract the rate of Pipe C:
\[ \text{Combined rate} = \left(\frac{1}{10} + \frac{1}{12} - \frac{1}{6}\right) \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 10, 12, and 6 is 60. We will convert each rate to have a denominator of 60:
- Rate of Pipe A: \( \frac{1}{10} = \frac{6}{60} \)
- Rate of Pipe B: \( \frac{1}{12} = \frac{5}{60} \)
- Rate of Pipe C: \( -\frac{1}{6} = -\frac{10}{60} \)

### Step 4: Calculate the combined rate
Now we can add these fractions:
\[ \text{Combined rate} = \frac{6}{60} + \frac{5}{60} - \frac{10}{60} = \frac{6 + 5 - 10}{60} = \frac{1}{60} \text{ tank/hour} \]

### Step 5: Calculate the total amount filled in 12 hours
Now, we multiply the combined rate by the number of hours (12 hours):
\[ \text{Amount filled in 12 hours} = \text{Combined rate} \times 12 = \frac{1}{60} \times 12 = \frac{12}{60} = \frac{1}{5} \text{ of the tank} \]

### Conclusion
The portion of the tank that will be filled after 12 hours is \( \frac{1}{5} \).

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