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Three pipes P, Q and R can separately fi...

Three pipes P, Q and R can separately fill a cistern in 4,8 and 12 hours respectively. Another pipe S can empty the completely filled cistern in 10 hours. Which of the following arrangements will fill the empty cistern in less time than others ?

A

Q alone is open.

B

P and S are open.

C

P, R and S are open.

D

P, Q and S are open.

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to determine how long it will take to fill a cistern using different combinations of pipes P, Q, R, and S. ### Given: - Pipe P can fill the cistern in 4 hours. - Pipe Q can fill the cistern in 8 hours. - Pipe R can fill the cistern in 12 hours. - Pipe S can empty the cistern in 10 hours. ### Step 1: Calculate the filling rates of each pipe - The rate of Pipe P = 120 liters / 4 hours = 30 liters/hour - The rate of Pipe Q = 120 liters / 8 hours = 15 liters/hour - The rate of Pipe R = 120 liters / 12 hours = 10 liters/hour - The rate of Pipe S (emptying) = 120 liters / 10 hours = 12 liters/hour ### Step 2: Determine the effective rates for each combination 1. **Option 1: Q alone** - Rate of Q = 15 liters/hour - Time to fill = 120 liters / 15 liters/hour = 8 hours 2. **Option 2: P and S together** - Rate of P = 30 liters/hour - Rate of S = -12 liters/hour (since it empties) - Effective rate = 30 - 12 = 18 liters/hour - Time to fill = 120 liters / 18 liters/hour = 20/3 hours ≈ 6.67 hours 3. **Option 3: P, R, and S together** - Rate of P = 30 liters/hour - Rate of R = 10 liters/hour - Rate of S = -12 liters/hour - Effective rate = 30 + 10 - 12 = 28 liters/hour - Time to fill = 120 liters / 28 liters/hour = 15/7 hours ≈ 2.14 hours 4. **Option 4: P, Q, and S together** - Rate of P = 30 liters/hour - Rate of Q = 15 liters/hour - Rate of S = -12 liters/hour - Effective rate = 30 + 15 - 12 = 33 liters/hour - Time to fill = 120 liters / 33 liters/hour = 120/33 hours ≈ 3.64 hours ### Step 3: Compare the times - Option 1: 8 hours - Option 2: 20/3 hours ≈ 6.67 hours - Option 3: 15/7 hours ≈ 2.14 hours - Option 4: 120/33 hours ≈ 3.64 hours ### Conclusion The arrangement that fills the cistern in the least amount of time is **Option 3: P, R, and S together**, taking approximately 2.14 hours. ---
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