Math Problem Statement

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

Solution

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: Rate of Pipe A=130 tank per minute\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: Rate of Pipe B=145 tank per minute\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: Rate of Pipe C=136 tank per minute\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: Net Rate=130+145136\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: 130=6180\frac{1}{30} = \frac{6}{180} 145=4180\frac{1}{45} = \frac{4}{180} 136=5180\frac{1}{36} = \frac{5}{180}

    Now, adding these fractions: Net Rate=6180+41805180=6+45180=5180=136 tank per minute\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 136\frac{1}{36} tank per minute, it will take: Time=1 tank136 tank per minute=36 minutes\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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Rates
Fraction Arithmetic

Formulas

Rate calculation formula
Fraction addition and subtraction

Theorems

-

Suitable Grade Level

Grades 7-9