Two inlet pipes A and E can fill an empty cistern in 12 minutes and 36 minutes respectively. Both the pipes A and E are opened together and after some time pipe A is closed. If the cistern gets filled in 20 minutes, then for how many minutes pipe A was open?

Options

A. `5(1)/(3)` minutes
B. 6 minutes
C. 7 minutes
D. 3 minutes

Answer

To solve the problem, we need to determine how long pipe A was open while both pipes A and E were filling the cistern together. Let's break down the solution step by step.

### Step 1: Determine the filling rates of pipes A and E
- Pipe A can fill the cistern in 12 minutes. Therefore, in 1 minute, it fills:
\[ \text{Rate of A} = \frac{1}{12} \text{ cisterns per minute} \]

- Pipe E can fill the cistern in 36 minutes. Therefore, in 1 minute, it fills:
\[ \text{Rate of E} = \frac{1}{36} \text{ cisterns per minute} \]

### Step 2: Combine the rates of A and E
When both pipes A and E are opened together, their combined filling rate per minute is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of E} = \frac{1}{12} + \frac{1}{36} \]

To add these fractions, we need a common denominator. The least common multiple of 12 and 36 is 36:
\[ \text{Combined Rate} = \frac{3}{36} + \frac{1}{36} = \frac{4}{36} = \frac{1}{9} \text{ cisterns per minute} \]

### Step 3: Determine the total filling time
We know that the cistern is filled in a total of 20 minutes. Let \( t \) be the time in minutes that pipe A was open. Therefore, pipe E was open for the entire 20 minutes, while pipe A was open for \( t \) minutes.

### Step 4: Calculate the amount filled by each pipe
- The amount filled by pipe A in \( t \) minutes is:
\[ \text{Amount filled by A} = t \cdot \frac{1}{12} \]

- The amount filled by pipe E in 20 minutes is:
\[ \text{Amount filled by E} = 20 \cdot \frac{1}{36} = \frac{20}{36} = \frac{5}{9} \]

### Step 5: Set up the equation for total filling
The total amount filled by both pipes A and E together must equal 1 (the whole cistern):
\[ \text{Amount filled by A} + \text{Amount filled by E} = 1 \]
Substituting the amounts:
\[ t \cdot \frac{1}{12} + \frac{5}{9} = 1 \]

### Step 6: Solve for \( t \)
To solve for \( t \), first isolate \( t \):
\[ t \cdot \frac{1}{12} = 1 - \frac{5}{9} \]
Finding a common denominator (which is 9):
\[ 1 = \frac{9}{9} \quad \Rightarrow \quad 1 - \frac{5}{9} = \frac{9}{9} - \frac{5}{9} = \frac{4}{9} \]
Now substituting back:
\[ t \cdot \frac{1}{12} = \frac{4}{9} \]
Multiply both sides by 12:
\[ t = \frac{4}{9} \cdot 12 = \frac{48}{9} = \frac{16}{3} \text{ minutes} \]

### Step 7: Convert to mixed number
\[ \frac{16}{3} = 5 \frac{1}{3} \text{ minutes} \]

Thus, pipe A was open for **5 minutes and 20 seconds** (or \( 5 \frac{1}{3} \) minutes).

### Final Answer
Pipe A was open for **5 minutes and 20 seconds**.

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