Let the total capacity of the cistern be $1$ unit (i.e., the whole cistern is $1$).

Step 1: Determine the rates of filling and emptying

• Pipe A: It can fill the cistern in 20 minutes. So, its rate of filling is: $\text{Rate of A} = \frac{1}{20} \, \text{units per minute}.$

• Pipe B: It can fill the cistern in 30 minutes. So, its rate of filling is: $\text{Rate of B} = \frac{1}{30} \, \text{units per minute}.$

• Pipe C: It can empty the cistern in 40 minutes. So, its rate of emptying is: $\text{Rate of C} = \frac{1}{40} \, \text{units per minute}.$

Step 2: Net rate when all pipes are opened simultaneously

When all three pipes are opened, the net rate of filling is the sum of the rates of pipes A and B, minus the rate of pipe C (since it's emptying): $\text{Net rate} = \frac{1}{20} + \frac{1}{30} - \frac{1}{40}.$

Step 3: Simplify the net rate

To simplify, first find the least common denominator (LCD) of 20, 30, and 40, which is 120.

Convert each fraction to have a denominator of 120: $\frac{1}{20} = \frac{6}{120}, \quad \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120}.$

Now, substitute these values into the equation for the net rate: $\text{Net rate} = \frac{6}{120} + \frac{4}{120} - \frac{3}{120} = \frac{6 + 4 - 3}{120} = \frac{7}{120}.$

Step 4: Calculate the time to fill the cistern

The net rate of filling the cistern is $\frac{7}{120} \, \text{units per minute}$. The time taken to fill the cistern is the reciprocal of this rate: $\text{Time} = \frac{1}{\frac{7}{120}} = \frac{120}{7} \, \text{minutes}.$

Step 5: Simplify the result

$\frac{120}{7} \approx 17.14 \, \text{minutes}.$

Thus, it would take approximately 17.14 minutes to fill the cistern when all three pipes are released simultaneously.

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Would you like further details or have any questions?

Here are 5 related questions to explore:

1. How would the result change if the third pipe emptied the cistern in 30 minutes instead of 40?
2. What would happen if only pipe A and pipe C were opened?
3. How long would it take to fill the cistern if only pipes A and B were open?
4. Can you determine the percentage of the cistern filled after 10 minutes when all three pipes are open?
5. How would the result change if the cistern could hold twice as much water?

Tip: When solving problems with rates, always combine the rates by adding or subtracting based on the direction of the flow (filling vs. emptying).