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Two pipes U and V can fill a cistern in ...

Two pipes U and V can fill a cistern in 12 hours and 20 hours respectively. Pipe W can empty the tank in 8 hours. If all the three pipes are opened, then in how many hours the cistern will be full?

A

60

B

120

C

150

D

100

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine how long it will take for the cistern to be filled when all three pipes (U, V, and W) are opened together. Here’s a step-by-step breakdown of the solution: ### Step 1: Determine the filling rates of pipes U and V - Pipe U can fill the cistern in 12 hours. Therefore, in 1 hour, it fills: \[ \text{Rate of U} = \frac{1}{12} \text{ cisterns per hour} \] - Pipe V can fill the cistern in 20 hours. Therefore, in 1 hour, it fills: \[ \text{Rate of V} = \frac{1}{20} \text{ cisterns per hour} \] ### Step 2: Determine the emptying rate of pipe W - Pipe W can empty the cistern in 8 hours. Therefore, in 1 hour, it empties: \[ \text{Rate of W} = \frac{1}{8} \text{ cisterns per hour} \] ### Step 3: Calculate the net filling rate when all pipes are opened - When all three pipes are opened together, the net rate of filling the cistern is: \[ \text{Net Rate} = \text{Rate of U} + \text{Rate of V} - \text{Rate of W} \] Substituting the rates we found: \[ \text{Net Rate} = \frac{1}{12} + \frac{1}{20} - \frac{1}{8} \] ### Step 4: Find a common denominator to combine the rates - The least common multiple (LCM) of 12, 20, and 8 is 120. Now we convert each rate: \[ \frac{1}{12} = \frac{10}{120}, \quad \frac{1}{20} = \frac{6}{120}, \quad \frac{1}{8} = \frac{15}{120} \] - Now substituting these values into the net rate: \[ \text{Net Rate} = \frac{10}{120} + \frac{6}{120} - \frac{15}{120} = \frac{10 + 6 - 15}{120} = \frac{1}{120} \text{ cisterns per hour} \] ### Step 5: Calculate the time to fill the cistern - If the net rate of filling the cistern is \(\frac{1}{120}\) cisterns per hour, then the time taken to fill 1 cistern is the reciprocal of the net rate: \[ \text{Time} = \frac{1}{\text{Net Rate}} = 120 \text{ hours} \] ### Final Answer Thus, when all three pipes are opened together, the cistern will be full in **120 hours**. ---
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