Work Rate With Leakage – A First Look
Imagine you are filling a bucket with a hose. The hose pours water in at a certain speed. But there is a small hole at the bottom of the bucket — water is leaking out at the same time. How long will it actually take to fill the bucket?
That is the core idea of work rate with leakage: one agent is doing positive work (filling, building, adding), while another is doing negative work (emptying, destroying, removing). The net result is what matters.
The Intuition
If a pipe fills a tank in 6 hours, its filling rate is 61 of the tank per hour. If a leak empties the same tank in 12 hours, its emptying rate is 121 of the tank per hour.
When both are open together, the tank fills at a net rate:
Net rate=61−121=122−121=121
So together, they fill 121 of the tank per hour — meaning it takes 12 hours to fill the tank.
A common mistake is to add the rates: 61+121=41, giving 4 hours. That would be correct if both were filling. But one is emptying — you must subtract the leakage rate.
The Precise Statement
Let:
- A = time taken by the filling agent alone to complete the work (or fill the tank)
- B = time taken by the leakage alone to empty the work (or empty the tank)
Then:
- Filling rate = A1 (work per unit time)
- Leakage rate = B1 (work per unit time)
When both operate simultaneously, the net work rate is:
Net rate=A1−B1
The time taken to complete the work (fill the tank) when both are active is:
Time=A1−B11=B−AAB
This formula only works when B>A — the leakage must be slower than the filling. If B=A, the net rate is zero and the tank never fills. If B<A, the tank never fills at all (it empties faster than it fills).
A Worked Example
Problem: A tap fills a cistern in 8 hours. A leak at the bottom empties it in 12 hours. If both are opened together, how long will it take to fill the cistern?
Step 1 – Find the rates
- Filling rate: 81 cistern per hour
- Leakage rate: 121 cistern per hour
Step 2 – Net rate
81−121=243−242=241
Step 3 – Time
Time=2411=24 hours
If the problem gives you the time with leakage and asks for the leakage time alone, set up the equation: A1−B1=T1, where T is the time with leakage. Solve for B.
Why This Matters
Work rate with leakage appears in:
- Pipes and cisterns (most common)
- Workers with a helper who undoes work (e.g., a machine that occasionally breaks down)
- Projects where one team builds and another demolishes
The key is always the same: identify which agent adds and which subtracts, then work with net rate. Everything else follows from the basic idea that rate × time = work done.