In a tank `1^(st)` pipe can fill 1/2 of a tank in 1 hour, `2^(nd)` pipe can fill 1/3 of total in 1 hour and third pipe is for emptying it. 7/12 of the total tank is filled with water in 1 hour if all the three pipes are opened together. The third pipe can empty the fill tank in that no. of hours______.

Options

A. 6 hours
B. 5 hours
C. 4 hours
D. 8 hours

Answer

To solve the problem step by step, let's break down the information given and calculate the time taken by the third pipe to empty the tank.

### Step 1: Determine the filling rates of the pipes
1. **First Pipe**: It can fill \( \frac{1}{2} \) of the tank in 1 hour.
- Therefore, the time taken to fill the whole tank (M) is \( 2 \) hours.
- Filling rate of the first pipe = \( \frac{1}{2} \) tank/hour.

2. **Second Pipe**: It can fill \( \frac{1}{3} \) of the tank in 1 hour.
- Therefore, the time taken to fill the whole tank (N) is \( 3 \) hours.
- Filling rate of the second pipe = \( \frac{1}{3} \) tank/hour.

### Step 2: Set up the equation for the combined filling and emptying
Let the time taken by the third pipe to empty the tank be \( x \) hours. The emptying rate of the third pipe will be \( \frac{1}{x} \) tank/hour.

When all three pipes are opened together, the net filling rate is given as \( \frac{7}{12} \) tank/hour.

### Step 3: Write the equation
The equation for the combined effect of the three pipes can be written as:
\[ \text{Filling rate of first pipe} + \text{Filling rate of second pipe} - \text{Emptying rate of third pipe} = \text{Net filling rate} \]
Substituting the values:
\[ \frac{1}{2} + \frac{1}{3} - \frac{1}{x} = \frac{7}{12} \]

### Step 4: Solve the equation
1. Find a common denominator for the fractions on the left-hand side. The least common multiple of 2 and 3 is 6.
- Convert \( \frac{1}{2} \) to \( \frac{3}{6} \)
- Convert \( \frac{1}{3} \) to \( \frac{2}{6} \)

So, the equation becomes:
\[ \frac{3}{6} + \frac{2}{6} - \frac{1}{x} = \frac{7}{12} \]

2. Combine the fractions:
\[ \frac{5}{6} - \frac{1}{x} = \frac{7}{12} \]

3. To eliminate the fractions, find a common denominator for \( 6 \) and \( 12 \), which is \( 12 \):
- Convert \( \frac{5}{6} \) to \( \frac{10}{12} \)

Now the equation is:
\[ \frac{10}{12} - \frac{1}{x} = \frac{7}{12} \]

4. Rearranging gives:
\[ \frac{10}{12} - \frac{7}{12} = \frac{1}{x} \]
\[ \frac{3}{12} = \frac{1}{x} \]
\[ \frac{1}{4} = \frac{1}{x} \]

### Step 5: Solve for \( x \)
From \( \frac{1}{4} = \frac{1}{x} \), we find:
\[ x = 4 \]

### Conclusion
The third pipe can empty the filled tank in **4 hours**.