Two pipes can fill a tank in 10 hours and 12 hours respectively. While a third pipe empted the full tank in 20 hours. If all the three pipes operate simultaneously, in how much time the tank will be filled ?

Options

A. 7 hours 30 min
B. 6 hours 40 min
C. 8 hours 30 min
D. 9 hours 30 min

Answer

To solve the problem step by step, we will first determine the rates at which the pipes fill or empty the tank and then combine these rates to find out how long it takes to fill the tank when all pipes are working together.

### Step 1: Determine the rates of the pipes
- **Pipe A** fills the tank in 10 hours. Therefore, its rate is:
\[ \text{Rate of Pipe A} = \frac{1 \text{ tank}}{10 \text{ hours}} = 0.1 \text{ tanks per hour} \]

- **Pipe B** fills the tank in 12 hours. Therefore, its rate is:
\[ \text{Rate of Pipe B} = \frac{1 \text{ tank}}{12 \text{ hours}} = \frac{1}{12} \text{ tanks per hour} \approx 0.0833 \text{ tanks per hour} \]

- **Pipe C** empties the tank in 20 hours. Therefore, its rate is:
\[ \text{Rate of Pipe C} = \frac{1 \text{ tank}}{20 \text{ hours}} = 0.05 \text{ tanks per hour} \]

### Step 2: Combine the rates of the pipes
When all three pipes are working together, the effective rate of filling the tank is the sum of the filling rates of Pipes A and B minus the emptying rate of Pipe C:
\[ \text{Effective Rate} = \text{Rate of Pipe A} + \text{Rate of Pipe B} - \text{Rate of Pipe C} \]
Substituting the rates we calculated:
\[ \text{Effective Rate} = 0.1 + \frac{1}{12} - 0.05 \]

### Step 3: Convert the rates to a common denominator
To combine these fractions, we can convert them to a common denominator (which is 60):
- \(0.1 = \frac{6}{60}\)
- \(\frac{1}{12} = \frac{5}{60}\)
- \(0.05 = \frac{3}{60}\)

Now substituting back:
\[ \text{Effective Rate} = \frac{6}{60} + \frac{5}{60} - \frac{3}{60} = \frac{8}{60} = \frac{2}{15} \text{ tanks per hour} \]

### Step 4: Calculate the time to fill the tank
To find the time taken to fill 1 tank at the effective rate of \(\frac{2}{15}\) tanks per hour, we use the formula:
\[ \text{Time} = \frac{\text{Total work}}{\text{Effective Rate}} = \frac{1 \text{ tank}}{\frac{2}{15} \text{ tanks per hour}} = \frac{15}{2} \text{ hours} = 7.5 \text{ hours} \]

### Final Answer
The tank will be filled in **7.5 hours**.