Pipes and Cisterns; Time, Speed and Distance; Problems on Trains

Pipes and Cisterns

13.1 Concept

Pipes and Cisterns problems are exactly like Time and Work:

• Inlet pipe (filling) = Positive work
• Outlet pipe (emptying) = Negative work

If a pipe fills a tank in n hours → it fills 1/n of the tank in 1 hour.

13.2 Key Formulas

13.3 LCM Method for Pipes

Example: Pipe A fills in 6 hr, B fills in 8 hr, C empties in 12 hr. Time to fill:

LCM(6, 8, 12) = 24 (capacity of tank)

• A's rate = 24/6 = 4 units/hr (fills)
• B's rate = 24/8 = 3 units/hr (fills)
• C's rate = 24/12 = 2 units/hr (empties)
• Net rate = 4 + 3 - 2 = 5 units/hr
• Time = 24/5 = 4.8 hours

13.4 Alternate Opening of Pipes

If pipes work alternately for fixed periods, calculate work done in each cycle.

Example: A fills in 10 hr, B empties in 15 hr. They open alternately every hour (A first):

• In 1st hour (A): +1/10
• In 2nd hour (B): -1/15
• Net in 2 hours = 1/10 - 1/15 = 3/30 - 2/30 = 1/30
• Time for full tank = 2 × 30 = 60 hours

Time, Speed and Distance

14.1 Core Formula

Distance = Speed × Time

Speed = Distance / Time

Time = Distance / Speed

14.2 Unit Conversions

Memory Trick:

• km/hr → m/s: divide by 3.6 (or × 5/18)
• m/s → km/hr: multiply by 3.6 (or × 18/5)

Example: 72 km/hr = 72 × 5/18 = 20 m/s

14.3 Average Speed

When equal DISTANCES are covered at different speeds:

Average Speed = 2S₁S₂ / (S₁ + S₂) (Harmonic Mean)

When equal TIMES are spent at different speeds:

Average Speed = (S₁ + S₂) / 2 (Arithmetic Mean)

Important: Do NOT use arithmetic mean when distances are equal!

14.4 Relative Speed

14.5 Meeting Problems

Two persons start simultaneously from A and B moving towards each other:

• They meet after time = Total Distance / (Sum of speeds)

Two persons start simultaneously from A and B moving away:

• Distance between them after t hours = (S₁ + S₂) × t

14.6 Problems on Chasing

If A is ahead of B by distance 'd' and both move in same direction:

• B is faster (B catches A): Time = d / (S_B - S_A)
• A runs away from B: They keep diverging if A is faster

Problems on Trains

15.1 Key Rules

• Length of train is crucial — it must cross the obstacle completely
• Distance = Length of train (when crossing a pole/person/tree)
• Distance = Length of train + Length of object (when crossing bridge/platform/another train)

15.2 Formulas

15.4 Common Pattern

Example: Train 200m long passes a platform 150m long in 14 seconds. Find speed:

• Total distance = 200 + 150 = 350 m
• Speed = 350 / 14 = 25 m/s = 90 km/hr