Two pipes can fill the cistern in 10hr and 12hr respectively, while the third empty it in 20hr. If all pipes are opened simultaneously then the cistem will be filled in

Options

A. 7.5 hr
B. 8 hr
C. 8.5 hr
D. 10 hr

Answer

To solve the problem of how long it will take to fill the cistern when all three pipes are opened simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Rates of Each Pipe**:
- Pipe A fills the cistern in 10 hours. Therefore, its rate of filling is:
\[ \text{Rate of A} = \frac{1}{10} \text{ cisterns per hour} \]
- Pipe B fills the cistern in 12 hours. Therefore, its rate of filling is:
\[ \text{Rate of B} = \frac{1}{12} \text{ cisterns per hour} \]
- Pipe C empties the cistern in 20 hours. Therefore, its rate of emptying is:
\[ \text{Rate of C} = -\frac{1}{20} \text{ cisterns per hour} \]

2. **Calculate the Combined Rate of All Pipes**:
- When all pipes are opened simultaneously, the combined rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
- Substituting the rates:
\[ \text{Combined Rate} = \frac{1}{10} + \frac{1}{12} - \frac{1}{20} \]

3. **Find a Common Denominator**:
- The least common multiple (LCM) of 10, 12, and 20 is 60. We will convert each rate to have a denominator of 60:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{20} = \frac{3}{60} \]

4. **Combine the Rates**:
- Now, substituting these values into the combined rate:
\[ \text{Combined Rate} = \frac{6}{60} + \frac{5}{60} - \frac{3}{60} = \frac{8}{60} \]
- Simplifying this gives:
\[ \text{Combined Rate} = \frac{2}{15} \text{ cisterns per hour} \]

5. **Calculate the Time to Fill the Cistern**:
- To find the time taken to fill one cistern, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ cistern}}{\frac{2}{15} \text{ cisterns per hour}} = \frac{15}{2} \text{ hours} = 7.5 \text{ hours} \]

### Final Answer:
The cistern will be filled in **7.5 hours** when all pipes are opened simultaneously.

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