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Two pipes A and B can fill a tank in 12 ...

Two pipes A and B can fill a tank in 12 minutes and 15 minutes, respectively. When an outlet pipe C is also opened, then the three pipes together can fill the tank in 10 minutes. In how many minutes can C alone empty the full tank?

A

18

B

20

C

16

D

24

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we will analyze the filling rates of pipes A, B, and C, and then determine how long it takes for pipe C to empty the tank alone. ### Step 1: Determine the filling rates of pipes A and B - Pipe A can fill the tank in 12 minutes. Therefore, its rate of filling is: \[ \text{Rate of A} = \frac{1}{12} \text{ tanks per minute} \] - Pipe B can fill the tank in 15 minutes. Therefore, its rate of filling is: \[ \text{Rate of B} = \frac{1}{15} \text{ tanks per minute} \] ### Step 2: Determine the combined filling rate of pipes A and B - The combined rate of A and B is: \[ \text{Rate of A + B} = \frac{1}{12} + \frac{1}{15} \] - To add these fractions, we need a common denominator. The least common multiple (LCM) of 12 and 15 is 60. Thus: \[ \text{Rate of A + B} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \text{ tanks per minute} \] ### Step 3: Determine the combined filling rate of pipes A, B, and C - When all three pipes (A, B, and C) are opened, they can fill the tank in 10 minutes. Therefore, their combined rate is: \[ \text{Rate of A + B + C} = \frac{1}{10} \text{ tanks per minute} \] ### Step 4: Set up the equation to find the rate of pipe C - We know that: \[ \text{Rate of A + B + C} = \text{Rate of A + B} + \text{Rate of C} \] - Substituting the known rates: \[ \frac{1}{10} = \frac{3}{20} + \text{Rate of C} \] - To isolate the rate of C, we first convert \(\frac{1}{10}\) to a fraction with a denominator of 20: \[ \frac{1}{10} = \frac{2}{20} \] - Now, we can rewrite the equation: \[ \frac{2}{20} = \frac{3}{20} + \text{Rate of C} \] - Rearranging gives: \[ \text{Rate of C} = \frac{2}{20} - \frac{3}{20} = -\frac{1}{20} \text{ tanks per minute} \] ### Step 5: Determine the time taken by pipe C to empty the tank - The negative sign indicates that pipe C is emptying the tank. The rate of C is \(-\frac{1}{20}\) tanks per minute, which means it takes: \[ X = 20 \text{ minutes to empty the full tank} \] ### Final Answer Pipe C can empty the full tank in **20 minutes**. ---
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