Pipe A and B can empty a full tank in 18 hours and 24 hours,respectively.Pipe C alone can fill the tank in 36 hours.If the tank is 5/6 full and all the three pipes are opened together , then in how many hours the tank will be emptied ?

Options

A. `10(1)/(2)`
B. `12(1)/(2)`
C. 10
D. 12

Answer

To solve the problem step by step, we will analyze the rates at which each pipe works and then calculate the time taken to empty the tank when all pipes are opened together.

### Step 1: Determine the rates of each pipe
- **Pipe A** can empty the tank in 18 hours. Therefore, its rate is:
\[ \text{Rate of Pipe A} = -\frac{1}{18} \text{ tanks per hour} \]
- **Pipe B** can empty the tank in 24 hours. Therefore, its rate is:
\[ \text{Rate of Pipe B} = -\frac{1}{24} \text{ tanks per hour} \]
- **Pipe C** can fill the tank in 36 hours. Therefore, its rate is:
\[ \text{Rate of Pipe C} = \frac{1}{36} \text{ tanks per hour} \]

### Step 2: Calculate the combined rate of all pipes
To find the combined rate when all three pipes are opened together, we sum their rates:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the rates:
\[ \text{Combined Rate} = -\frac{1}{18} - \frac{1}{24} + \frac{1}{36} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 18, 24, and 36 is 72. We will convert each rate to have a denominator of 72:
- For Pipe A:
\[ -\frac{1}{18} = -\frac{4}{72} \]
- For Pipe B:
\[ -\frac{1}{24} = -\frac{3}{72} \]
- For Pipe C:
\[ \frac{1}{36} = \frac{2}{72} \]

### Step 4: Combine the rates
Now, we can combine the rates:
\[ \text{Combined Rate} = -\frac{4}{72} - \frac{3}{72} + \frac{2}{72} = -\frac{5}{72} \text{ tanks per hour} \]

### Step 5: Determine the amount of work to be done
The tank is 5/6 full, which means:
\[ \text{Work to be done} = \frac{5}{6} \text{ of a full tank} \]
In terms of units, since the total work (full tank) is 72 units:
\[ \text{Work to be done} = \frac{5}{6} \times 72 = 60 \text{ units} \]

### Step 6: Calculate the time taken to empty the tank
Using the formula:
\[ \text{Time} = \frac{\text{Work}}{\text{Rate}} \]
Substituting the values:
\[ \text{Time} = \frac{60 \text{ units}}{-\frac{5}{72} \text{ tanks per hour}} = 60 \times -\frac{72}{5} = -720 \div 5 = 12 \text{ hours} \]

### Final Answer
The time taken to empty the tank when all three pipes are opened together is **12 hours**.

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