A water tank is `2/5`th full. Pipe A can fill the tank in 10 minutes and pipe B can empty it in 6 minutes. If both the pipes are open, how long will it take to empty or fill the tank completely ?

Options

A. 6 minutes to empty
B. 6 minutes to fill
C. 9 minutes to empty
D. 9 minutes to fill

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Determine the rate of work for Pipe A and Pipe B.
- Pipe A can fill the tank in 10 minutes. Therefore, the rate of Pipe A is:
\[ \text{Rate of Pipe A} = \frac{1 \text{ tank}}{10 \text{ minutes}} = \frac{1}{10} \text{ tank per minute} \]

- Pipe B can empty the tank in 6 minutes. Therefore, the rate of Pipe B is:
\[ \text{Rate of Pipe B} = \frac{1 \text{ tank}}{6 \text{ minutes}} = \frac{1}{6} \text{ tank per minute} \]

### Step 2: Calculate the combined rate of both pipes when they are open.
When both pipes are open, the net effect is the filling rate of Pipe A minus the emptying rate of Pipe B:
\[ \text{Combined Rate} = \text{Rate of Pipe A} - \text{Rate of Pipe B} = \frac{1}{10} - \frac{1}{6} \]

### Step 3: Find a common denominator to subtract the rates.
The least common multiple (LCM) of 10 and 6 is 30. We will convert both fractions:
\[ \frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{1}{6} = \frac{5}{30} \]
Now, we can subtract:
\[ \text{Combined Rate} = \frac{3}{30} - \frac{5}{30} = -\frac{2}{30} = -\frac{1}{15} \text{ tank per minute} \]

### Step 4: Interpret the combined rate.
The negative sign indicates that the tank is being emptied. The rate of emptying is:
\[ \frac{1}{15} \text{ tank per minute} \]

### Step 5: Calculate the time to empty the entire tank.
Since the tank is currently \( \frac{2}{5} \) full, we need to find out how long it will take to empty this amount:
\[ \text{Time to empty } \frac{2}{5} \text{ of the tank} = \frac{\text{Amount of tank}}{\text{Rate}} = \frac{\frac{2}{5}}{\frac{1}{15}} \]

### Step 6: Simplify the calculation.
To divide by a fraction, we multiply by its reciprocal:
\[ \text{Time} = \frac{2}{5} \times 15 = \frac{2 \times 15}{5} = \frac{30}{5} = 6 \text{ minutes} \]

### Final Answer:
The time taken to empty the tank completely is **6 minutes**.

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