Pipe A and pipe B can fill up a tank in 40 min and 80 min respectively Both the pipes are opened simultaneously for some time and then pipe B is closed. If the total time to fill up the tank is 30 minutes then how long the pipe B was open for ?

Options

A. 5 min
B. 10 min
C. 15 min
D. 20 min

Answer

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

### Step 1: Determine the efficiency of each pipe
- Pipe A can fill the tank in 40 minutes, so its efficiency is \( \frac{1}{40} \) tanks per minute.
- Pipe B can fill the tank in 80 minutes, so its efficiency is \( \frac{1}{80} \) tanks per minute.

### Step 2: Calculate the combined efficiency of both pipes
- The combined efficiency of Pipe A and Pipe B when both are open is:
\[ \text{Combined Efficiency} = \frac{1}{40} + \frac{1}{80} \]
- To add these fractions, find a common denominator (which is 80):
\[ \frac{1}{40} = \frac{2}{80} \]
Thus,
\[ \text{Combined Efficiency} = \frac{2}{80} + \frac{1}{80} = \frac{3}{80} \]
- Therefore, together they can fill \( \frac{3}{80} \) of the tank per minute.

### Step 3: Define the variables
- Let \( x \) be the time in minutes that both pipes A and B are open together.
- The remaining time that only Pipe A is open is \( 30 - x \) minutes.

### Step 4: Set up the equation for total work done
- The work done by both pipes together in \( x \) minutes is:
\[ \text{Work by A and B} = \text{Combined Efficiency} \times x = \frac{3}{80}x \]
- The work done by Pipe A alone in the remaining \( 30 - x \) minutes is:
\[ \text{Work by A} = \frac{1}{40}(30 - x) \]
- The total work done to fill the tank is equal to 1 tank:
\[ \frac{3}{80}x + \frac{1}{40}(30 - x) = 1 \]

### Step 5: Solve the equation
- Convert \( \frac{1}{40} \) to have a common denominator with \( \frac{3}{80} \):
\[ \frac{1}{40} = \frac{2}{80} \]
- Substitute this into the equation:
\[ \frac{3}{80}x + \frac{2}{80}(30 - x) = 1 \]
- Multiply through by 80 to eliminate the denominators:
\[ 3x + 2(30 - x) = 80 \]
- Distribute the 2:
\[ 3x + 60 - 2x = 80 \]
- Combine like terms:
\[ x + 60 = 80 \]
- Subtract 60 from both sides:
\[ x = 20 \]

### Step 6: Conclusion
- Pipe B was open for **20 minutes**.