A cistern has three pipes A, B and C. Pipes A and B are inlet pipes whereas C is an outlet pipe. Pipes A and B

Question

Q. A cistern has three pipes A, B and C. Pipes A and B are inlet pipes whereas C is an outlet pipe. Pipes A and B can fill the cistern separately in 3 hours and 4 hours respectively; while pipe C can empty the completely filled cistern in 1 hour. If the pipes A, B and C are opened in order at 5, 6 and 7 a.m. respectively, at what time will the cistern be empty ?

CBSECBSE Class XII Board 2024Subjective· 5mImportance★★★★★

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Concept understanding — Work Rate With Leakage

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:

\text{Net rate} = \frac{1}{6} - \frac{1}{12} = \frac{2}{12} - \frac{1}{12} = \frac{1}{12}

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

Watch out

A common mistake is to add the rates: $\frac{1}{6} + \frac{1}{12} = \frac{1}{4}$ , 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 = $\frac{1}{A}$ (work per unit time)
Leakage rate = $\frac{1}{B}$ (work per unit time)

When both operate simultaneously, the net work rate is:

\text{Net rate} = \frac{1}{A} - \frac{1}{B}

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

\text{Time} = \frac{1}{\frac{1}{A} - \frac{1}{B}} = \frac{AB}{B - A}

Important

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 …

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Let emptying occur $n$ hours after 5 a.m.; A runs $n$ h, B runs $n-1$ h, C runs $n-2$ h. Net fill $=0$ gives $n=\tfrac{21}{5}=4.2$ h $\Rightarrow$ 9:12 a.m.

T

Pipes: A fills in $3$ h (rate $+\tfrac13$ ), B fills in $4$ h ( $+\tfrac14$ ), C empties in $1$ h ( $-1$ ). A opens 5 a.m., B 6 a.m., C 7 a.m.

Let the cistern become empty $n$ hours after 5 a.m.
Running times: A runs $n$ h, B runs $(n-1)$ h (started 1 h later), C runs $(n-2)$ h (started 2 h later). …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2024Set 465/S/RQPS/44 marks
$Q. Case Study - 3 For providing water to the families of a colony, a large water tank with two inlet pipes A and B and an outlet pipe C, is installed. Pipes A and B can fill the tank in 10 hours and 12 hours respectively; whereas pipe C can empty the tank in 15 hours. Based on the above information, answer the following questions :$
(i) If both pipes A and B are opened together, then find the time in which the tank will be filled completely. [1]
(ii) If both pipes A and C are opened together, then find the time in which the tank will be filled completely. [1]
(iii)
(a) If all the three pipes A, B and C are opened together, then find the time in which the tank will be filled completely. [2]
(OR)
$(b) Pipes A and B are opened together for some time and then pipe B is turned off after some time. If the tank is completely filled in 6 hours, then after how many hours is pipe B turned off ? [2]$
›Reveal solutionSolution
A fills $\tfrac1{10}$ /h, B fills $\tfrac1{12}$ /h, C empties $\tfrac1{15}$ /h. Combining rates: A+B $=\tfrac{60}{11}$ h, A+C $=30$ h, A+B+C $=\tfrac{60}{7}$ h; and B is shut after $4.8$ h.

$=\dfrac{1}{\text{time to fill/empty}}$

Rates: A $=\dfrac1{10}$ per hour, B $=\dfrac1{12}$ per hour, C $=-\dfrac1{15}$ per hour.

A and B together 2. Combined rate $=\dfrac1{10}+\dfrac1{12}=\dfrac{6}{60}+\dfrac{5}{60}=\dfrac{11}{60}$ per hour. 3. Time to fill $=\dfrac{60}{11}$ hours.

A and C together 4. Combined rate $=\dfrac1{10}-\dfrac1{15}=\dfrac{3}{30}-\dfrac{2}{30}=\dfrac{1}{30}$ per hour. 5. Time to fill $=30$ hours. (iii)(a) A, B and C together 6. Combined rate $=\dfrac1{10}+\dfrac1{12}-\dfrac1{15}=\dfrac{6}{60}+\dfrac{5}{60}-\dfrac{4}{60}=\dfrac{7}{60}$ per hour. 7. Time to fill $=\dfrac{60}{7}$ hours. …
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