tank can be filled by two pipes, A and B. Pipe A can fill the tank in 30 minutes, while Pipe B can fill the tank in 45 minutes. There is also a drain pipe C that can empty the tank in 36 minutes. If all three pipes are opened together, how long will it take to fill the tank

To find out how long it will take to fill the tank when all three pipes are opened together, we need to calculate the net rate at which the tank is being filled.

1. **Rate of Pipe A**:
   Pipe A can fill the tank in 30 minutes. Hence, its rate is:
   \[
   \text{Rate of Pipe A} = \frac{1}{30} \text{ tank per minute}
   \]

2. **Rate of Pipe B**:
   Pipe B can fill the tank in 45 minutes. Hence, its rate is:
   \[
   \text{Rate of Pipe B} = \frac{1}{45} \text{ tank per minute}
   \]

3. **Rate of Pipe C (drain)**:
   Pipe C can empty the tank in 36 minutes. Hence, its rate is:
   \[
   \text{Rate of Pipe C} = -\frac{1}{36} \text{ tank per minute}
   \]
   (The negative sign indicates that it's draining the tank.)

4. **Net Rate**:
   The net rate at which the tank is being filled when all three pipes are opened together is the sum of the rates of Pipe A, Pipe B, and Pipe C:
   \[
   \text{Net Rate} = \frac{1}{30} + \frac{1}{45} - \frac{1}{36}
   \]

   To add these fractions, we need a common denominator. The least common multiple (LCM) of 30, 45, and 36 is 180. Converting each fraction to have the denominator of 180:
   \[
   \frac{1}{30} = \frac{6}{180}
   \]
   \[
   \frac{1}{45} = \frac{4}{180}
   \]
   \[
   \frac{1}{36} = \frac{5}{180}
   \]

   Now, adding these fractions:
   \[
   \text{Net Rate} = \frac{6}{180} + \frac{4}{180} - \frac{5}{180} = \frac{6 + 4 - 5}{180} = \frac{5}{180} = \frac{1}{36} \text{ tank per minute}
   \]

5. **Time to Fill the Tank**:
   Since the net rate of filling the tank is \(\frac{1}{36}\) tank per minute, it will take:
   \[
   \text{Time} = \frac{1 \text{ tank}}{\frac{1}{36} \text{ tank per minute}} = 36 \text{ minutes}
   \]

Therefore, it will take 36 minutes to fill the tank when all three pipes are opened together.

Would you like any more details or have any questions?

Here are 5 questions you might ask next:
1. How do you calculate the LCM of multiple numbers?
2. Can you explain how to add and subtract fractions?
3. What happens if only two of the pipes are used?
4. How would the calculation change if Pipe C had a different drainage rate?
5. Can you solve a similar problem with different times for filling and draining?

**Tip:** Always double-check your LCM and fraction arithmetic to ensure accuracy in rate problems.

Learn how to calculate the time required to fill a tank using pipes A, B, and C. Understand the rates of filling and draining involved, and how to compute the net rate using fraction arithmetic. Suitable for students in Grades 7-9.

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