Pipe A can fill a tank in 20 h while pipe B alone can fill it in 10 h and pipe C can empty the full tank in 30 h . If all the pipes are open together , how much time will be needed to make the tank full ?

Answer

To solve the problem step by step, we will first determine the rates at which each pipe fills or empties the tank, then combine these rates to find the total time taken to fill the tank when all pipes are open together.

### Step 1: Determine the rates of each pipe
- **Pipe A** can fill the tank in 20 hours. Therefore, its rate of filling is:
\[ \text{Rate of A} = \frac{1}{20} \text{ tank/hour} \]

- **Pipe B** can fill the tank in 10 hours. Therefore, its rate of filling is:
\[ \text{Rate of B} = \frac{1}{10} \text{ tank/hour} \]

- **Pipe C** can empty the tank in 30 hours. Therefore, its rate of emptying is:
\[ \text{Rate of C} = -\frac{1}{30} \text{ tank/hour} \]
(Note: It's negative because it empties the tank.)

### Step 2: Combine the rates
When all pipes are open together, the combined rate of filling the tank is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
Substituting the values:
\[ \text{Combined Rate} = \frac{1}{20} + \frac{1}{10} - \frac{1}{30} \]

### Step 3: Find a common denominator
The least common multiple (LCM) of 20, 10, and 30 is 60. We will convert each fraction to have a denominator of 60:
- For \(\frac{1}{20}\):
\[ \frac{1}{20} = \frac{3}{60} \]
- For \(\frac{1}{10}\):
\[ \frac{1}{10} = \frac{6}{60} \]
- For \(-\frac{1}{30}\):
\[ -\frac{1}{30} = -\frac{2}{60} \]

### Step 4: Combine the fractions
Now we can combine the fractions:
\[ \text{Combined Rate} = \frac{3}{60} + \frac{6}{60} - \frac{2}{60} = \frac{3 + 6 - 2}{60} = \frac{7}{60} \text{ tank/hour} \]

### Step 5: Calculate the time to fill the tank
To find the time taken to fill the tank when all pipes are open, we take the reciprocal of the combined rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{7}{60} \text{ tank/hour}} = \frac{60}{7} \text{ hours} \]

### Step 6: Convert to mixed fraction
To convert \(\frac{60}{7}\) into a mixed fraction:
- Divide 60 by 7, which gives 8 with a remainder of 4.
- Thus, \(\frac{60}{7} = 8 \frac{4}{7}\) hours.

### Final Answer
The time needed to fill the tank when all pipes are open together is:
\[ \text{Time} = 8 \frac{4}{7} \text{ hours} \]

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