Pipe A can fill a tank in 16 minutes and pipe B emptics it in 24 minutes. If both the pipes are opened simultaneously, after how many minutes should B be closed so that the tank is filled in 30 minutes?

Options

A. 21 min
B. 20 min
C. 15 min
D. 18 min

Answer

To solve the problem step by step, we will first determine the rates at which pipes A and B fill and empty the tank, respectively. Then, we will calculate how long pipe B should remain open to ensure the tank is filled in 30 minutes.

### Step 1: Determine the rates of the pipes
- Pipe A can fill the tank in 16 minutes. Therefore, the rate of pipe A is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{16 \text{ minutes}} = \frac{1}{16} \text{ tanks per minute} \]

- Pipe B can empty the tank in 24 minutes. Therefore, the rate of pipe B is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{24 \text{ minutes}} = \frac{1}{24} \text{ tanks per minute} \]

### Step 2: Calculate the combined rate when both pipes are open
When both pipes are open, the effective rate of filling the tank is:
\[ \text{Effective Rate} = \text{Rate of A} - \text{Rate of B} = \frac{1}{16} - \frac{1}{24} \]

To perform this subtraction, we need a common denominator. The least common multiple (LCM) of 16 and 24 is 48. Thus, we convert the rates:
\[ \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{24} = \frac{2}{48} \]
Now, we can subtract:
\[ \text{Effective Rate} = \frac{3}{48} - \frac{2}{48} = \frac{1}{48} \text{ tanks per minute} \]

### Step 3: Set up the equation for the total filling time
Let \( x \) be the number of minutes that pipe B is open. Therefore, pipe A will work for 30 minutes, and pipe B will work for \( x \) minutes. The tank must be filled in 30 minutes, so we can set up the equation:
\[ \text{Amount filled by A in 30 minutes} - \text{Amount emptied by B in } x \text{ minutes} = 1 \text{ tank} \]
This translates to:
\[ 30 \times \frac{1}{16} - x \times \frac{1}{24} = 1 \]

### Step 4: Solve the equation
Substituting the rates into the equation:
\[ \frac{30}{16} - \frac{x}{24} = 1 \]
To eliminate the fractions, we can multiply through by 48 (the LCM of 16 and 24):
\[ 48 \times \frac{30}{16} - 48 \times \frac{x}{24} = 48 \]
This simplifies to:
\[ 90 - 2x = 48 \]
Now, rearranging gives:
\[ 2x = 90 - 48 \]
\[ 2x = 42 \]
\[ x = \frac{42}{2} = 21 \]

### Conclusion
Pipe B should be closed after **21 minutes**.