Two pipes can fill a tank in 15 and 12 hours respectively and third pipe can empty the tank in 4 hours. If the pipes are opened at 8, 9 and 11 o'clock respectively. The tank will be emp tied at that o'clock ____.

Options

A. 45 minutes past 1
B. 40 minutes past 1
C. 55 minutes past 1
D. 40 minutes past 2

Answer

To solve the problem, we need to analyze the filling and emptying rates of the pipes and calculate when the tank will be empty. Let's break it down step by step.

### Step 1: Determine the rates of the pipes
- **Pipe A** can fill the tank in 15 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1}{15} \text{ tank/hour} \]

- **Pipe B** can fill the tank in 12 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1}{12} \text{ tank/hour} \]

- **Pipe C** can empty the tank in 4 hours. Therefore, its rate is:
\[ \text{Rate of C} = -\frac{1}{4} \text{ tank/hour} \quad (\text{negative because it empties the tank}) \]

### Step 2: Find a common capacity for the tank
To make calculations easier, we can consider the least common multiple (LCM) of the times taken by the pipes to fill or empty the tank. The LCM of 15, 12, and 4 is 60. This means we can consider the tank's capacity as 60 units.

- **Rate of A**:
\[ \text{Rate of A} = \frac{60}{15} = 4 \text{ units/hour} \]

- **Rate of B**:
\[ \text{Rate of B} = \frac{60}{12} = 5 \text{ units/hour} \]

- **Rate of C**:
\[ \text{Rate of C} = \frac{60}{4} = 15 \text{ units/hour} \]

### Step 3: Calculate the total contribution of each pipe before 11 o'clock
- **From 8 AM to 11 AM (3 hours)**, only Pipe A is working:
\[ \text{Contribution of A} = 4 \text{ units/hour} \times 3 \text{ hours} = 12 \text{ units} \]

- **From 9 AM to 11 AM (2 hours)**, both Pipe A and Pipe B are working:
\[ \text{Contribution of B} = 5 \text{ units/hour} \times 2 \text{ hours} = 10 \text{ units} \]

- **Total contribution by 11 AM**:
\[ \text{Total contribution} = 12 + 10 = 22 \text{ units} \]

### Step 4: Calculate the net rate after 11 o'clock
At 11 AM, all three pipes are open:
\[ \text{Net rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} = 4 + 5 - 15 = -6 \text{ units/hour} \]
This means the tank is emptying at a rate of 6 units per hour.

### Step 5: Calculate the time to empty the tank
The tank has 22 units of water at 11 AM, and it is emptying at a rate of 6 units/hour. The time taken to empty 22 units is:
\[ \text{Time} = \frac{22 \text{ units}}{6 \text{ units/hour}} = \frac{11}{3} \text{ hours} = 3 \text{ hours and } 40 \text{ minutes} \]

### Step 6: Determine the time when the tank will be empty
Starting from 11 AM, we add 3 hours and 40 minutes:
- 11 AM + 3 hours = 2 PM
- 2 PM + 40 minutes = 2:40 PM

Thus, the tank will be empty at **2:40 PM**.

### Final Answer
The tank will be empty at **2:40 PM**.

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