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Pipe A can fill a tank in 20 hours and pi...

Pipe A can fill a tank in 20 hours and pipe B can fill the same tank in 24 hours. Pipe C can empty the same full tank in 12 hours. If all the pipes are opened together, then what portion of the tank will be filled after 12 hours?

A

`1//10`

B

`1//5`

C

`1//15`

D

`9//11`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will follow these steps: ### Step 1: Determine the filling and emptying rates of each pipe. - **Pipe A** can fill the tank in 20 hours. Therefore, in 1 hour, it fills: \[ \text{Rate of Pipe A} = \frac{1 \text{ tank}}{20 \text{ hours}} = \frac{1}{20} \text{ tanks/hour} \] - **Pipe B** can fill the tank in 24 hours. Therefore, in 1 hour, it fills: \[ \text{Rate of Pipe B} = \frac{1 \text{ tank}}{24 \text{ hours}} = \frac{1}{24} \text{ tanks/hour} \] - **Pipe C** can empty the tank in 12 hours. Therefore, in 1 hour, it empties: \[ \text{Rate of Pipe C} = \frac{1 \text{ tank}}{12 \text{ hours}} = \frac{1}{12} \text{ tanks/hour} \] ### Step 2: Calculate the combined rate of all pipes when opened together. When all three pipes are opened together, their combined rate is: \[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \] Substituting the rates: \[ \text{Combined Rate} = \frac{1}{20} + \frac{1}{24} - \frac{1}{12} \] ### Step 3: Find a common denominator to add the fractions. The least common multiple (LCM) of 20, 24, and 12 is 120. Now we convert each fraction: - For Pipe A: \[ \frac{1}{20} = \frac{6}{120} \] - For Pipe B: \[ \frac{1}{24} = \frac{5}{120} \] - For Pipe C: \[ \frac{1}{12} = \frac{10}{120} \] Now substituting these values into the combined rate: \[ \text{Combined Rate} = \frac{6}{120} + \frac{5}{120} - \frac{10}{120} = \frac{6 + 5 - 10}{120} = \frac{1}{120} \text{ tanks/hour} \] ### Step 4: Calculate the total amount filled in 12 hours. To find out how much of the tank is filled in 12 hours: \[ \text{Amount filled in 12 hours} = \text{Combined Rate} \times 12 = \frac{1}{120} \times 12 = \frac{12}{120} = \frac{1}{10} \] ### Final Answer: The portion of the tank that will be filled after 12 hours is \(\frac{1}{10}\). ---
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