Aptitude · Pipes and Cisterns

An outlet is a worker with a minus sign

Q: Two pipes fill in 6 and 12 hours. What is the shortcut for two pipes together?

ab/(a+b) , so 6×12/18 = 4 hours. It only works for exactly two pipes, both filling. With three pipes, or with one of them emptying, go back to the unit method — there is no tidy version and the units are faster anyway.

This chapter is Time and Work with one new idea: some pipes undo the work. Once an emptying pipe is written as a negative rate, every question in the chapter becomes the same subtraction you already know how to do.

Set the pipes and walk the net rate →

Take the tank as the LCM of the times. Add the inlets, subtract the outlets, divide the tank by what is left.

01 The idea

The same chapter, with one sign added

A pipe that fills a tank is called an inlet. A pipe that drains it is an outlet, or a leak. The tank itself is the job, and its capacity is the total work. Line those three words up against Time and Work — worker, job, total work — and you have already translated the chapter.

The one genuinely new thing is that an outlet does negative work. A worker in Time and Work only ever moves you towards finishing. An outlet moves you away from it, so its rate carries a minus and you subtract instead of adding. That is the whole extension.

Rate is still work divided by time. A pipe that fills the tank in 8 hours moves one eighth of it an hour; an outlet that empties it in 24 hours removes one twenty-fourth an hour. Those fractions are correct and slow, so use the habit that makes this chapter quick: the answer cannot depend on how big the tank is, so choose its size. Take it as the LCM of the given times and every rate becomes a whole number.

With 8 and 24 hours the LCM is 24 units. The inlet brings in 24/8 = 3 units an hour and the outlet removes 24/24 = 1, so the tank gains 2 units an hour and fills in 24/2 = 12 hours. Not 8 hours, because the leak steals a third of everything the inlet delivers — and not 16 either, which is what averaging the two times would have suggested.

Inlet rates are positive, outlet rates are negative, and they all go into one sum. Time to fill is always the tank divided by that net rate.

Inlet pipeA pipe that fills. Its one-hour work is +1/a for a pipe that fills the tank alone in a hours, or +tank/a units in the LCM method.

Outlet pipeA pipe that empties, also called a leak. Same division, opposite sign: −1/c, or −tank/c units. It is the only reason this chapter is not simply Time and Work.

Net rateThe sum of every open pipe’s signed rate — what the tank actually gains in an hour. Positive means it fills, zero means the level holds, negative means it drains.

02 Worked example

One inlet at 8 hours against one outlet at 24

This pair runs the whole lesson. Pipe P can fill a tank in 8 hours and pipe Q can empty the full tank in 24 hours. Both are opened on an empty tank. How long until it is full?

Choose the size of the tankAny size gives the same answer, so pick the one that makes the arithmetic whole. That is the LCM of the two times.tank = LCM(8, 24) = 24 units

The inlet, as a positive rateP clears 24 units in 8 hours, so divide.P = 24 / 8 = +3 units per hour

The outlet, as a negative rateQ clears 24 units in 24 hours too — but outwards, so the sign flips. This is the only line that differs from a Time and Work question.Q = −24 / 24 = −1 unit per hour

Add them with their signsBoth are open, so the tank gains whatever the two of them come to together.net = +3 − 1 = 2 units per hour

Tank divided by net rate24 units of tank, gained at 2 units an hour. Had Q been shut, P alone would have finished in 8 hours.time = 24 / 2 = 12 hours (the leak costs 4 hours)

The leak did not simply add a bit of delay — it removed a third of P’s output and stretched 8 hours into 12. Check the fraction route agrees: 1/8 − 1/24 = 3/24 − 1/24 = 2/24 = 1/12, so 12 hours. Same answer, more writing. Every model in this module is this calculation with one extra complication bolted on.

03 The method

The method, and the two ways it has no answer

Four lines of method, then the check that stops you from confidently reporting a negative filling time.

tank = LCM of all the given times, then rate = tank / time with a plus for every inlet and a minus for every outlet, then time = tank / net rate. In fraction form the same thing reads 1/a + 1/b − 1/c, and the time is its reciprocal.

Look at the sign of the net rate before you divide by it. If the outlets remove at least as much as the inlets bring in, the tank never fills — and dividing a positive tank by a zero or negative net rate hands you infinity or a negative number of hours. Exam sets deliberately include one of these, and it is marked wrong far more often than it is marked “never”.

Inlets vs outlets	Net rate	What actually happens
Inlets faster	`positive`	Fills, in tank / net hours
Exactly equal	`zero`	Level never moves — never fills
Outlets faster	`negative`	Empties; a full tank drains instead
Only inlets	`positive`	Plain Time and Work, nothing to subtract
Outlet on a full tank	`negative`	Time to empty = tank / net drain
Averaging the two times	`never valid`	8 and 24 give 12, not 16

05 Cheat sheet

Pipes and Cisterns on one page

The conversions, and the three checks that catch nearly every wrong answer in the chapter.

Case	Rule	On P 8 h vs Q 24 h
Tank size	`LCM of all times`	LCM(8,24) = 24 units
Inlet rate	`+ tank / time`	+24/8 = +3 /hr
Outlet rate	`− tank / time`	−24/24 = −1 /hr
Net rate	`sum of signed rates`	+3 − 1 = +2 /hr
Time to fill	`tank / net rate`	24/2 = 12 hours
Two inlets only	`ab/(a+b)`	6 and 12 give 4 hours
Net rate zero or less	`never fills`	12 in vs 6 out = −1

The sign is the whole chapterInlets plus, outlets minus, one sum. If a solution has you adding an emptying pipe, the answer will come out too small and look plausible, which is why this error survives.

Units never matter, but be consistentMinutes, hours, days — the method does not care. It only breaks if one pipe is given in minutes and another in hours and you do not convert before taking the LCM.

The answer has a windowWith only inlets, the joint time is less than the fastest pipe alone and more than that time divided by the number of pipes. Add an outlet and the joint time must exceed every inlet’s own time. Anything outside that is arithmetic gone wrong.

06 Where & why

Where this shows up

Pipes and Cisterns is set in almost every aptitude paper, and it is set because it is Time and Work wearing a disguise — examiners get two topics tested for the price of one.

TCS NQT · Infosys

One inlet, one leak, find the time

The numbers are chosen so the LCM is small. Spot that it is a signed sum and the question is over in fifteen seconds.

SSC CGL · Bank PO

Find the missing outlet

Two inlets and a stated joint time, with the outlet unknown. Work out the inlets’ total, subtract the net rate the joint time implies, and the difference is the outlet.

CAT · XAT

The never-fills case

Stated as a normal question with numbers that make the net rate zero or negative. The mark is for noticing, not for calculating.

Real plumbing and reservoirs

Inflow, evaporation and draw-off

Any tank with a supply and a demand is this arithmetic. The negative rate is the demand, and a net rate at or below zero is the reason a reservoir falls in a dry season.

Every model in this module — a pipe closed part-way, pipes opened alternately, faster-and-slower comparisons — is this same signed sum with one extra complication. Get the sign habit right here and the rest of the module is bookkeeping.

07 Interview questions

What gets asked

Ten, from the definition through to the case that has no numerical answer at all.

What is the difference between Pipes and Cisterns and Time and Work?

Only the sign. The tank is the job and each pipe is a worker, but an emptying pipe does negative work, so its rate is subtracted instead of added. Everything else — rates add, times never do, take the total work as the LCM — carries over unchanged.

How do you write the rate of an emptying pipe?

As a negative number. A pipe that empties the tank in c hours has a one-hour work of −1/c, or −tank/c units once you have fixed the tank as the LCM. The minus is the entire adjustment.

Why take the tank as the LCM of the times?

Because the answer cannot depend on the size of the tank, so you are free to choose a convenient one. The LCM makes every pipe a whole number of units per hour, which removes all the fractions from the working. It is a choice, not a formula.

P fills a tank in 8 hours and Q empties it in 24. Both open — how long?

12 hours. Take the tank as 24 units, so P is +3 an hour and Q is −1, giving a net of +2 and 24/2 = 12 hours. In fractions, 1/8 − 1/24 = 1/12.

Two pipes fill in 6 and 12 hours. What is the shortcut for two pipes together?

ab/(a+b), so 6×12/18 = 4 hours. It only works for exactly two pipes, both filling. With three pipes, or with one of them emptying, go back to the unit method — there is no tidy version and the units are faster anyway.

When does a tank never fill?

When the outlets remove at least as much as the inlets bring in, so the net rate is zero or negative. At exactly zero the level holds where it is forever; below zero it drains. Check the sign of the net rate before dividing — a negative time is the arithmetic telling you this has happened.

Two inlets fill in 24 and 30 minutes. With an outlet also open all three fill it in 20 minutes. How long does the outlet take to empty it?

40 minutes. Take the tank as 120 units: the inlets are 5 and 4, so 9 an hour together, while the stated 20 minutes means a net of 120/20 = 6. The outlet must therefore be 9 − 6 = 3 units, and 120/3 = 40 minutes.

A full tank has only a leak open. How is that different?

It is the same arithmetic with the tank starting full instead of empty. Time to empty is the tank divided by the net drain rate. If an inlet is also open, the net drain is the outlet’s rate minus the inlet’s, and the tank only empties if that is positive.

Does it matter whether the times are in minutes or hours?

Hard

A cistern has a leak that would empty it in 20 hours. A tap admitting 6 litres a minute is turned on, and now the full cistern empties in 30 hours instead. Find the capacity of the cistern in litres.

Follow-up

Here the tank size is not yours to choose — the tap is given in real litres, so the LCM trick is unavailable and you have to carry the capacity as an unknown. It is the one problem type where the units are the question.

Show the hint

Write the leak as C/20 litres an hour and the tap as 360 litres an hour, then set the net drain equal to C/30.