. Two pipes A and B can separately fill in 15 minutes and 10 minutes respectively and a waste pipe C can carry off 7 liters per minute. If all the pipes are opened together a full cistern is emptied in 2 hours. How many liters does the cistern hold?

Let the total capacity of the cistern be \( x \) liters.

### Step 1: Rate of filling by pipes A and B
- Pipe A can fill the cistern in 15 minutes, so its rate of filling is \( \frac{x}{15} \) liters per minute.
- Pipe B can fill the cistern in 10 minutes, so its rate of filling is \( \frac{x}{10} \) liters per minute.

### Step 2: Rate of emptying by pipe C
- Pipe C removes water at a constant rate of 7 liters per minute.

### Step 3: Net rate of filling when all pipes are open
When all three pipes are open, the net rate of filling is the sum of the filling rates of pipes A and B minus the emptying rate of pipe C. This is:

\[
\text{Net rate} = \left( \frac{x}{15} + \frac{x}{10} - 7 \right) \text{ liters per minute}.
\]

### Step 4: Given information
When all pipes are open, the cistern is emptied in 2 hours, i.e., 120 minutes. This means that the net rate of water flow is negative, and in 120 minutes, the entire cistern is emptied. Therefore, the total volume \( x \) is emptied in 120 minutes, implying the following equation:

\[
\text{Net rate} \times 120 = -x.
\]

### Step 5: Solve for \( x \)

The net rate can be simplified by finding a common denominator for the fractions:

\[
\frac{x}{15} + \frac{x}{10} = \frac{2x}{30} + \frac{3x}{30} = \frac{5x}{30} = \frac{x}{6}.
\]

Thus, the net rate of water flow is:

\[
\frac{x}{6} - 7 \text{ liters per minute}.
\]

Now, using the fact that the cistern is emptied in 120 minutes:

\[
\left( \frac{x}{6} - 7 \right) \times 120 = -x.
\]

Expanding the equation:

\[
\frac{120x}{6} - 7 \times 120 = -x,
\]
\[
20x - 840 = -x.
\]

Now, solve for \( x \):

\[
20x + x = 840,
\]
\[
21x = 840,
\]
\[
x = \frac{840}{21} = 40 \text{ liters}.
\]

### Final Answer:
The cistern holds **40 liters**.

Would you like more details or have any questions?

Here are 5 relative questions for further practice:
1. What would be the net rate of filling if the waste pipe C was not opened?
2. How long would it take to fill the cistern if pipe B was closed?
3. If the waste pipe C increased its rate to 10 liters per minute, how would that affect the time to empty the cistern?
4. How much time would it take for pipes A and C to fill the cistern together?
5. What is the efficiency of each pipe in terms of liters filled per minute?

**Tip**: When solving problems with multiple rates, always ensure to carefully consider the net effect of all rates combined (filling vs. emptying).

Two pipes A and B can separately fill in 15 minutes and 10 minutes respectively and a waste pipe C can carry off 7 liters per minute. If all the pipes are opened together a full cistern is emptied in 2 hours. How many liters does the cistern hold?

Learn how to solve a rate of work problem involving two pipes filling and one waste pipe emptying a cistern. This step-by-step solution shows how to calculate the volume of the cistern, involving key algebraic concepts such as rate-time relationships and solving for unknowns. Suitable for students in Grades 9-11.

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