Two pipes A and B can fill a tank in 20 hours and 24 hours respectively. If the two pipes are opened at 5 in the morning, then at what time the pipe A should be closed to completely fill the tank exactly at 5 in the evening?

Options

A. `3` pm
B. `2` pm
C. `1` pm
D. `11` am

Answer

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

### Step 1: Determine the work done by each pipe
- Pipe A can fill the tank in 20 hours, so its work rate (efficiency) is:
\[ \text{Efficiency of A} = \frac{1 \text{ tank}}{20 \text{ hours}} = \frac{1}{20} \text{ tanks per hour} \]
- Pipe B can fill the tank in 24 hours, so its work rate (efficiency) is:
\[ \text{Efficiency of B} = \frac{1 \text{ tank}}{24 \text{ hours}} = \frac{1}{24} \text{ tanks per hour} \]

### Step 2: Find the least common multiple (LCM) of the filling times
- The LCM of 20 and 24 is 120. This means we can consider the total work in terms of units, where the total work to fill the tank is 120 units.

### Step 3: Calculate the work done by each pipe in terms of units
- The work done by Pipe A in one hour:
\[ \text{Work done by A in 1 hour} = \frac{120 \text{ units}}{20 \text{ hours}} = 6 \text{ units per hour} \]
- The work done by Pipe B in one hour:
\[ \text{Work done by B in 1 hour} = \frac{120 \text{ units}}{24 \text{ hours}} = 5 \text{ units per hour} \]

### Step 4: Calculate the combined work done by both pipes in one hour
- The combined work done by both pipes A and B in one hour is:
\[ \text{Combined work} = 6 + 5 = 11 \text{ units per hour} \]

### Step 5: Determine the total time from 5 AM to 5 PM
- The total time from 5 AM to 5 PM is 12 hours.

### Step 6: Set up the equation for the total work done
- Let \( x \) be the number of hours that Pipe A works. Then Pipe B will work for the full 12 hours.
- The work done by Pipe A in \( x \) hours is:
\[ \text{Work done by A} = 6x \text{ units} \]
- The work done by Pipe B in 12 hours is:
\[ \text{Work done by B} = 5 \times 12 = 60 \text{ units} \]

### Step 7: Write the equation for total work
- The total work done by both pipes must equal the total work required to fill the tank (120 units):
\[ 6x + 60 = 120 \]

### Step 8: Solve for \( x \)
- Rearranging the equation gives:
\[ 6x = 120 - 60 \]
\[ 6x = 60 \]
\[ x = 10 \]

### Step 9: Determine the time when Pipe A should be closed
- Since Pipe A works for 10 hours starting from 5 AM, it should be closed at:
\[ 5 \text{ AM} + 10 \text{ hours} = 3 \text{ PM} \]

### Final Answer
- Pipe A should be closed at **3 PM**.

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