Two pipes A and B can fill a tank in 20 minutes and 30 minutes respectively. If initially only pipe B was kept open for the 2/5th part of the total time and both pipes A and B were kept open for the rest time being, the tank would be filled. How many minutes should both pipes take?

Options

A. `16(5)/(23)`
B. `13(1)/(23)`
C. `13(5)/(19)`
D. `15(15)/(19)`

Answer

To solve the problem step by step, we will follow a structured approach:

### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 20 minutes, so its rate of filling is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{20 \text{ minutes}} = \frac{1}{20} \text{ tanks per minute} \]
- Pipe B can fill the tank in 30 minutes, so its rate of filling is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{30 \text{ minutes}} = \frac{1}{30} \text{ tanks per minute} \]

### Step 2: Calculate the combined rate of both pipes
- The combined rate of both pipes A and B when they are both open is:
\[ \text{Combined Rate} = \frac{1}{20} + \frac{1}{30} \]
- To add these fractions, find a common denominator (which is 60):
\[ \text{Combined Rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \text{ tanks per minute} \]

### Step 3: Set up the equation for the total time
- Let the total time taken to fill the tank be \( x \) minutes.
- Pipe B is open for \( \frac{2}{5} \) of the total time, which is:
\[ \text{Time B is open} = \frac{2}{5}x \]
- The remaining time when both pipes are open is:
\[ \text{Time A and B are open} = x - \frac{2}{5}x = \frac{3}{5}x \]

### Step 4: Calculate the amount of tank filled by each pipe
- The amount filled by pipe B in \( \frac{2}{5}x \) minutes is:
\[ \text{Amount filled by B} = \frac{1}{30} \times \frac{2}{5}x = \frac{2x}{150} = \frac{x}{75} \]
- The amount filled by both pipes A and B in \( \frac{3}{5}x \) minutes is:
\[ \text{Amount filled by A and B} = \left(\frac{1}{12}\right) \times \frac{3}{5}x = \frac{3x}{60} = \frac{x}{20} \]

### Step 5: Set up the equation for the total tank filled
- The total amount filled by both pipes should equal 1 tank:
\[ \frac{x}{75} + \frac{x}{20} = 1 \]

### Step 6: Solve the equation
- To solve the equation, find a common denominator for \( 75 \) and \( 20 \), which is \( 300 \):
\[ \frac{4x}{300} + \frac{15x}{300} = 1 \]
\[ \frac{19x}{300} = 1 \]
\[ 19x = 300 \]
\[ x = \frac{300}{19} \text{ minutes} \]

### Step 7: Convert to mixed fraction
- To express \( x \) in mixed fraction:
\[ x = 15 \frac{15}{19} \text{ minutes} \]

### Final Answer
The total time taken to fill the tank with both pipes A and B is \( 15 \frac{15}{19} \) minutes.

---