Pipes and Cisterns

“Pipes and Cisterns problems are similar to Time and Work, but involve filling and emptying tanks using inlets and outlets.”

1. Basic Concepts and Formulas

  • If a pipe fills a tank in x hours, part filled in 1 hour = 1/x
  • If a pipe empties a tank in y hours, part emptied in 1 hour = 1/y
  • If one pipe fills and another empties, net work per hour = (1/x - 1/y)
  • If both are inlet pipes, net work = (1/x + 1/y)
  • Time taken to fill/empty the tank = 1 ÷ Net work per hour

2. Type-Wise Questions

  • Type 1: Single Inlet Pipe
    Example: A pipe fills a tank in 5 hours. How much part is filled in 1 hour?
    Solution: Part filled in 1 hour = 1/5
  • Type 2: Single Outlet Pipe
    Example: A pipe can empty a full tank in 6 hours. What part of the tank does it empty in 1 hour?
    Solution: Part emptied = 1/6
  • Type 3: Inlet and Outlet Working Together
    Example: Inlet fills in 4 hours, outlet empties in 6 hours. Time to fill the tank?
    Solution: Net rate = 1/4 - 1/6 = 1/12 ⇒ Time = 12 hours
  • Type 4: Two Inlet Pipes
    Example: Pipe A fills in 3 hours, Pipe B in 6 hours. Time to fill tank together?
    Solution: 1/3 + 1/6 = 1/2 ⇒ Time = 2 hours
  • Type 5: Inlet Opened, Outlet After Delay
    Example: Inlet fills tank in 4 hrs. Outlet empties in 8 hrs, opened after 1 hr. Time to fill?
    Solution: Work in 1 hr = 1/4. Net work after = 1/4 - 1/8 = 1/8. Remaining 3/4 takes (3/4) ÷ (1/8) = 6 hrs. Total = 1 + 6 = 7 hrs.
  • Type 6: Tank Capacity Questions
    Example: Pipe fills in 5 hrs. If it fills 60 liters/hour, what is capacity?
    Solution: Capacity = 60 × 5 = 300 liters
  • Type 7: Pipe A alone, then B joins
    Example: A fills in 10 hrs. After 4 hrs, B (fills in 5 hrs) joins. Time to fill?
    Solution: A’s work in 4 hrs = 4/10 = 0.4. Remaining = 0.6. Combined rate = 1/10 + 1/5 = 3/10. Time = 0.6 ÷ (3/10) = 2 hrs. Total = 4 + 2 = 6 hrs.
  • Type 8: Alternate Pipe Opening
    Example: A fills in 2 hrs, B empties in 4 hrs. A & B open alternately starting with A. Time to fill?
    Solution: In 2 hrs: A → 1/2, B → -1/4 ⇒ Net in 2 hrs = 1/4. Total = 1 ÷ 1/4 × 2 = 8 hrs.
  • Type 9: Inlet Faster than Outlet
    Example: Inlet fills in 3 hrs, outlet empties in 9 hrs. Time to fill tank?
    Solution: Net rate = 1/3 - 1/9 = 2/9 ⇒ Time = 9/2 = 4.5 hrs
  • Type 10: Inlet and Outlet Same Time
    Example: Inlet fills in 4 hrs, outlet empties in 4 hrs. What happens?
    Solution: Net rate = 1/4 - 1/4 = 0 ⇒ No water fills the tank (stays same)

3. Tips to Remember

  • Treat filling like positive work and emptying like negative work.
  • Always find net rate of work for combined inlet and outlet problems.
  • Convert hours to minutes when units mismatch. Use LCM where needed.
  • If net rate = 0, the tank’s level remains constant.
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