Q.A tank is fitted with 3 taps $A$, $B$ and $C$. All the three taps, if opened together, can drain the full tank in $1\frac{1}{2}$ minutes. Taps $B$ and $C$ together take 2 minutes to drain the tank while $A$ and $C$ together take $2\frac{4}{13}$ minutes to drain it. How long will taps $A$ and $B$ together take to drain 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 …