Question 1

Two pipes A and B can fill a tank in 24h and 30 h respectively. If both the pipes are opened simultaneously in the empty tank, how much time will be taken by them to fill it?

Accepted Answer

Concept & Formula: When two pipes work together, their combined filling rate is the sum of their individual filling rates. Combined Rate = (1/A) + (1/B) Total Time = (A × B/A + B) Step-by-Step Solution: 1. Pipe A fills the tank in 24 hours. Its rate is (1/24) per hour. 2. Pipe B fills the tank in 30 hours. Its rate is (1/30) per hour. 3. Combined rate when both are open: (1/24) + (1/30) 4. Find a common denominator (LCM of 24 and 30 is 120): Combined Rate = (5/120) + (4/120) = (9/120) = (3/40) tank per hour 5. Total time taken is the reciprocal of the combined rate: Time = (40/3) hours = 13 (1/3) hours 6. Convert the fractional hour to minutes: (1/3) hour = (1/3) × 60 minutes = 20 minutes Total time = 13h 20 min. Exam Strategy & Shortcut: Use the direct formula T = (ab/a+b). T = (24 × 30/24 + 30) = (720/54) Simplify by dividing by 18: T = (40/3) = 13.33 hours, which is 13 hours and 20 minutes. Common Pitfall: A frequent error is adding the hours directly (24 + 30 = 54) instead of adding their rates (inverses). Always add rates, never times. Final Answer: Therefore, the correct answer is 13h 20 min.

Question 2

A tank is 7 metre long and 4 metre wide. At what speed should water run through a pipe 5 cm broad and 4 cm deep so that in 6 hours and 18 minutes water level in the tank rise by 4.5 metre?

Accepted Answer

Concept & Formula: The volume of water that flows through the pipe in a given time must equal the volume of water accumulated in the tank. The flow volume is the cross-sectional area of the pipe multiplied by the water speed and the time elapsed. Volume of Tank = L × W × H Flow Volume = Area of Pipe × Speed × Time Step-by-Step Solution: 1. Calculate Target Volume: Volume = 7 m (length) × 4 m (width) × 4.5 m (rise in level) Volume = 126 m^3 2. Convert Pipe Dimensions: Width = 5 cm = 0.05 m Depth = 4 cm = 0.04 m Area of Pipe = 0.05 × 0.04 = 0.002 m^2 3. Convert Time: 6 hours and 18 minutes = 6 + (18/60) hours = 6 + 0.3 = 6.3 hours 4. Set up Equation: Let the speed be v km/hr. Since dimensions are in meters, convert speed to meters/hour (1000v m/hr). Area × Speed × Time = Volume 0.002 × (1000v) × 6.3 = 126 2v × 6.3 = 126 12.6v = 126 v = 10 km/hr Exam Strategy & Shortcut: Instead of converting v to meters initially, calculate the speed required in meters per hour and convert to km/hr at the end. Speed (m/hr) = Volume / (Area × Time) = 126 / (0.002 × 6.3) = 126 / 0.0126 = 10000 m/hr. 10000 m/hr = 10 km/hr. Common Pitfall: Failing to maintain consistent units across all measurements. Mixing cm for pipe dimensions and meters for the tank without conversion will yield a massive calculation error. Final Answer: Therefore, the correct answer is 10 km/hr.

Question 3

Two pieces A and B can fill a tank in 18 hrs and 6 hrs respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

Accepted Answer

Concept & Formula: When two pipes work together to fill a tank, their combined filling rate is the sum of their individual rates. If pipe A takes x hours and pipe B takes y hours, their individual rates are (1/x) and (1/y) of the tank per hour. Combined Rate = (1/x) + (1/y) Time Taken = (x × y/x + y) Step-by-Step Solution: 1. Identify Individual Rates: - Pipe A takes 18 hours. Its rate is (1/18) tank/hour. - Pipe B takes 6 hours. Its rate is (1/6) tank/hour. 2. Calculate Combined Rate: - Combined Rate = (1/18) + (1/6) - Find a common denominator, which is 18: - Combined Rate = (1/18) + (3/18) = (4/18) = (2/9) tank/hour. 3. Determine Total Time: - Time taken is the reciprocal of the combined rate. - Time = (9/2) hours = 4 (1/2) hours. Exam Strategy & Shortcut: For two pipes filling together, use the direct formula T = (a × b/a + b). T = (18 × 6/18 + 6) = (108/24). Dividing numerator and denominator by 12 gives (9/2), which is 4.5 or 4 (1/2) hours. Common Pitfall: A common mistake is adding the times directly (18 + 6 = 24 hours) instead of adding their rates (inverses). Always add rates when combining work. Final Answer: Therefore, the correct answer is 4 (1/2) hrs.

Question 4

A tyre has two punctures. The first puncture alone would have made the tyre flat in 9 minutes and the second alone would have done it in 6 minutes. If air leaks out at a constant rate, how long does it take both the punctures together to make it fla…

Accepted Answer

Concept & Combined Rates (Pipes and Cisterns Logic): This problem is identical in logic to "Pipes and Cisterns" or "Time and Work". A puncture acts like a pipe emptying a tank. If one empties in A minutes and another in B minutes, together they empty (1/A) + (1/B) of the capacity per minute. Step-by-Step Solution: • Given: First puncture time = 9 minutes. Second puncture time = 6 minutes. • Calculation: Rate of deflation by first puncture = (1/9) of the tyre per minute. Rate of deflation by second puncture = (1/6) of the tyre per minute. • Combined rate of deflation: = (1/9) + (1/6) LCM of 9 and 6 is 18. = (2/18) + (3/18) = (5/18) of the tyre per minute. • Total time taken to make the tyre flat: = Reciprocal of combined rate = (18/5) minutes. • Converting to a mixed fraction: 18 \div 5 = 3 with a remainder of 3. Time = 3(3/5) minutes. Exam Strategy & Shortcut: For two entities working together, you can directly use the formula: Total Time = (xy/x + y) where x and y are the individual times. Here, x = 9 and y = 6. Time = (9 × 6/9 + 6) = (54/15) = (18/5) = 3(3/5) minutes. Common Pitfall: A common error is to subtract the rates if treating one as an inlet and one as an outlet. Remember both punctures are "emptying" the tyre, so their rates must be added. Final Answer: Therefore, the correct answer is 3(3/5) minutes.

Pipes and Cistern Questions

Questions

Practice smarter