Q.Pipe $A$ can fill a tank in 30 hours and pipe $B$ in 45 hours. If both the pipes are opened in an empty tank, how much time will it take to fill the tank?

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 $\frac{1}{6}$ of the tank per hour. If a leak empties the same tank in 12 hours, its emptying rate is $\frac{1}{12}$ of the tank per hour.

When both are open together, the tank fills at a net rate:

So together, they fill $\frac{1}{12}$ of the tank per hour — meaning it takes 12 hours to fill the tank.

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 = $\frac{1}{A}$ (work per unit time)
• Leakage rate = $\frac{1}{B}$ (work per unit time)

When both operate simultaneously, the net work rate is:

The time taken to complete the work (fill the tank) when both are active is:

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: $\frac{1}{8}$ cistern per hour
• Leakage rate: $\frac{1}{12}$ cistern per hour

Step 2 – Net rate

Step 3 – Time

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.