Three pipes A , B and C can fill a tank separately in 8h , 10 h and 20h , respectively. Find the time taken by all the three pipes to fill the tank when the pipes are opened together.

Options

A. `5 (7)/(11)`
B. `5 (7)/(11)`
C. `8 (7)/(11)`
D. `3 (7)/(11)`

Answer

To solve the problem of how long it takes for three pipes A, B, and C to fill a tank when opened together, we can follow these steps:

### Step 1: Determine the filling rates of each pipe.
- Pipe A fills the tank in 8 hours, so its rate is:
\[ \text{Rate of A} = \frac{1}{8} \text{ tank/hour} \]
- Pipe B fills the tank in 10 hours, so its rate is:
\[ \text{Rate of B} = \frac{1}{10} \text{ tank/hour} \]
- Pipe C fills the tank in 20 hours, so its rate is:
\[ \text{Rate of C} = \frac{1}{20} \text{ tank/hour} \]

### Step 2: Add the rates together to find the combined rate.
When all three pipes are opened together, their combined rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the rates we found:
\[ \text{Combined Rate} = \frac{1}{8} + \frac{1}{10} + \frac{1}{20} \]

### Step 3: Find the least common multiple (LCM) of the denominators.
The denominators are 8, 10, and 20. The LCM of these numbers is 40.

### Step 4: Convert each fraction to have the same denominator.
- For \(\frac{1}{8}\):
\[ \frac{1}{8} = \frac{5}{40} \]
- For \(\frac{1}{10}\):
\[ \frac{1}{10} = \frac{4}{40} \]
- For \(\frac{1}{20}\):
\[ \frac{1}{20} = \frac{2}{40} \]

### Step 5: Add the fractions.
Now we can add the fractions:
\[ \text{Combined Rate} = \frac{5}{40} + \frac{4}{40} + \frac{2}{40} = \frac{11}{40} \]

### Step 6: Calculate the time taken to fill the tank.
If the combined rate is \(\frac{11}{40}\) of the tank per hour, then the time \(T\) taken to fill the entire tank is the reciprocal of the combined rate:
\[ T = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{11}{40}} = \frac{40}{11} \text{ hours} \]

### Step 7: Convert to a mixed number.
To convert \(\frac{40}{11}\) into a mixed number:
- Divide 40 by 11, which gives 3 with a remainder of 7.
- Therefore, \(\frac{40}{11} = 3 \frac{7}{11}\) hours.

### Final Answer:
The time taken by all three pipes to fill the tank when opened together is:
\[ 3 \frac{7}{11} \text{ hours} \]