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Two pipes X and Y can fill a cistern in ...

Two pipes X and Y can fill a cistern in 6 hours and 10 hours respectively. Pipe Z can empty the tank in 4 hours. If all the three pipes are open then in how many hours the cistern will be full?

A

30

B

45

C

50

D

60

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will first determine the rates at which each pipe fills or empties the cistern and then combine these rates to find out how long it will take to fill the cistern when all three pipes are open. ### Step-by-Step Solution: 1. **Determine the filling rates of pipes X and Y:** - Pipe X can fill the cistern in 6 hours. Therefore, its rate is: \[ \text{Rate of X} = \frac{1 \text{ cistern}}{6 \text{ hours}} = \frac{1}{6} \text{ cistern per hour} \] - Pipe Y can fill the cistern in 10 hours. Therefore, its rate is: \[ \text{Rate of Y} = \frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{1}{10} \text{ cistern per hour} \] 2. **Determine the emptying rate of pipe Z:** - Pipe Z can empty the cistern in 4 hours. Therefore, its rate is: \[ \text{Rate of Z} = -\frac{1 \text{ cistern}}{4 \text{ hours}} = -\frac{1}{4} \text{ cistern per hour} \] (The negative sign indicates that it is emptying the tank.) 3. **Combine the rates of all three pipes:** - The combined rate when all three pipes are open is: \[ \text{Combined Rate} = \text{Rate of X} + \text{Rate of Y} + \text{Rate of Z} \] Substituting the values: \[ \text{Combined Rate} = \frac{1}{6} + \frac{1}{10} - \frac{1}{4} \] 4. **Find a common denominator to add the fractions:** - The least common multiple (LCM) of 6, 10, and 4 is 60. We convert each fraction: \[ \frac{1}{6} = \frac{10}{60}, \quad \frac{1}{10} = \frac{6}{60}, \quad -\frac{1}{4} = -\frac{15}{60} \] - Now, we can add them: \[ \text{Combined Rate} = \frac{10}{60} + \frac{6}{60} - \frac{15}{60} = \frac{10 + 6 - 15}{60} = \frac{1}{60} \text{ cistern per hour} \] 5. **Calculate the time to fill the cistern:** - If the combined rate is \(\frac{1}{60}\) cistern per hour, then the time taken to fill 1 cistern is: \[ \text{Time} = \frac{1 \text{ cistern}}{\frac{1}{60} \text{ cistern per hour}} = 60 \text{ hours} \] ### Final Answer: The cistern will be full in **60 hours** when all three pipes are open.
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