Time & Work, Pipes & Cisterns

Time & Work, Pipes & Cisterns

Concept

Time and work is fundamentally about rates. If someone can complete a job in n days, their daily work rate is 1/n of the job. When multiple workers combine, add their rates. The total work is always considered as 1 unit. This “rate approach” is the most reliable method — avoid trying to find LCM of days unnecessarily.

Pipes and cisterns use the same logic: a filling pipe has positive rate, an emptying pipe has negative rate. The net rate determines how fast the tank fills or empties. If a pipe can fill in x hours, its rate = 1/x per hour.

Work with Men and Machines: Sometimes the question introduces machines (like pumps or tractors) alongside men. The approach remains the same — convert each entity’s rate and add/subtract.

Negative Work (Outlet Pipes): When an outlet pipe is open while inlet pipes fill, the net rate = (sum of inlet rates) − (sum of outlet rates). If the net rate is negative, the tank empties instead of filling.

Key Points

• If A can do work in x days, B in y days, together in xy/(x+y) days.
• If A is twice as efficient as B: A’s time = B’s time/2, or A’s 1-day work = 2 × B’s 1-day work.
• If A and B work together for n days, then A completes remaining in m days: Total work = n(1/x + 1/y) + m(1/x) = 1.
• For filling + emptying pipes: Net filling time = (product of individual times) / (difference of filling and emptying rates).
• When workers join or leave mid-work, calculate work done in each segment separately.

Worked Example

Q: A pipe can fill a tank in 10 hours. Due to a leak, it takes 25 hours to fill. How long will the leak take to empty the full tank? Approach: Pipe’s rate = 1/10 per hour. With leak, effective rate = 1/25 per hour. Let leak’s emptying rate = 1/L. Then 1/10 − 1/L = 1/25 → 1/L = 1/10 − 1/25 = (5−2)/50 = 3/50 → L = 50/3 = 16⅔ hours. Answer: 16⅔ hours

SSC Pattern / Tips

• When multiple pipes fill/empty simultaneously, always subtract outlet rates from inlet rates for net rate.
• For leak problems, the leak rate = pipe rate − effective rate (positive value, as leak empties).
• If question asks “how long to fill if pipe A is open and pipe B closed for first x hours”, solve in two segments.
• Use man-days concept: M₁ × D₁ = M₂ × D₂ when same work is done.