Two pipes X and Y can fill a tank in 6 hours and 8 hours respectively while another pipe Z can empty the tank in 4.8 hours. If all the three pipes are opened at the same time, then the time in which the tank can be filled.

Answer

To solve the problem, we need to determine how long it will take to fill the tank when all three pipes (X, Y, and Z) are opened simultaneously. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the filling rates of pipes X and Y
- Pipe X can fill the tank in 6 hours. Therefore, the rate of Pipe X is:
\[ \text{Rate of X} = \frac{1}{6} \text{ tank/hour} \]
- Pipe Y can fill the tank in 8 hours. Therefore, the rate of Pipe Y is:
\[ \text{Rate of Y} = \frac{1}{8} \text{ tank/hour} \]

### Step 2: Determine the emptying rate of pipe Z
- Pipe Z can empty the tank in 4.8 hours. Therefore, the rate of Pipe Z is:
\[ \text{Rate of Z} = -\frac{1}{4.8} \text{ tank/hour} \]
(The negative sign indicates that it is emptying the tank.)

### Step 3: Calculate the combined rate of all three pipes
To find the combined rate when all three pipes are opened together, we add the rates of pipes X and Y and subtract the rate of pipe Z:
\[ \text{Combined Rate} = \text{Rate of X} + \text{Rate of Y} + \text{Rate of Z \]
Substituting the values we found:
\[ \text{Combined Rate} = \frac{1}{6} + \frac{1}{8} - \frac{1}{4.8} \]

### Step 4: Find a common denominator
The least common multiple (LCM) of 6, 8, and 4.8 can be calculated. The LCM of 6 and 8 is 24. To include 4.8, we can convert it to a fraction:
\[ 4.8 = \frac{24}{5} \]
Thus, the LCM of 6, 8, and 4.8 is 24.

### Step 5: Convert each rate to the common denominator
Now, we convert each rate to have a denominator of 24:
- Rate of X:
\[ \frac{1}{6} = \frac{4}{24} \]
- Rate of Y:
\[ \frac{1}{8} = \frac{3}{24} \]
- Rate of Z:
\[ -\frac{1}{4.8} = -\frac{5}{24} \]

### Step 6: Combine the rates
Now we can combine the rates:
\[ \text{Combined Rate} = \frac{4}{24} + \frac{3}{24} - \frac{5}{24} = \frac{4 + 3 - 5}{24} = \frac{2}{24} = \frac{1}{12} \text{ tank/hour} \]

### Step 7: Calculate the time to fill the tank
If the combined rate is \(\frac{1}{12}\) tank/hour, then the time taken to fill the entire tank is the reciprocal of the combined rate:
\[ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{1}{12}} = 12 \text{ hours} \]

### Final Answer
The time taken to fill the tank when all three pipes are opened together is **12 hours**.

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