Aptitude · Pipes and Cisterns · Model 3

The last minute belongs to whoever is next

When pipes take turns, only one is ever open, so their rates are used one at a time. Find the work in one cycle, divide to get the whole cycles, then walk the final stretch slot by slot — because the pattern, not the arithmetic, decides who finishes the tank.

Step through a cycle at a time →

Work per cycle, then whole cycles by division, then the tail one minute at a time. Never let one pipe clear the whole remainder on its own.

01 The idea

Taking turns is a cycle, not a partnership

“Opened alternately for one minute each” means exactly one pipe is running at any moment. That single sentence rules out the move your hand wants to make, which is to add the two rates together. Added rates describe pipes open at the same time; here they are open in turns, so each rate is used on its own, for its own minute.

What repeats is the pair of minutes, and that pair is called a cycle. Work out how many units a full cycle delivers and you have a new, larger unit of progress: instead of counting minutes you count cycles, each worth a fixed amount. This is the same machinery as alternate-day work in Time and Work, and if you have met that, you have met this.

The cycle total does not depend on who starts — 3 + 4 and 4 + 3 are both 7. What the order changes is the tail: the last stretch, after the last complete cycle, when the tank is so close to full that the next slot alone can finish it. The tail is where every mark in this model is won or lost, because it is the only part that knows whose turn it is.

The wrong answer is always the same shape. You run some whole cycles, see a small remainder, and hand the remainder to the pipe that clears it fastest. That silently converts an alternating question into a “both pipes then one pipe” question. Take A at 16 minutes and B at 12, starting with A: the honest answer is 13¾ minutes, and letting B mop up the last 6 units on its own gives 13½ — close enough to look right, and wrong.

A cycle is the repeating block of slots. Whole cycles come from a division; the tail comes from following the pattern, one slot at a time, until a slot can finish the tank.

CycleThe shortest block of slots that repeats — two minutes for A / B alternating, three hours for a three-pipe rotation. Its work is the sum of its slots’ signed rates.

SlotOne pipe’s single turn, and the reason the answer can be a fraction. A slot lasts a fixed length — one minute, one hour — and a pipe may only need part of it.

TailWhat is left after the last complete cycle. You walk it slot by slot in pattern order; the first slot whose pipe can finish the remaining units ends the question.

02 Worked example

Sixteen and twelve, starting with A

One pair runs the whole lesson. Pipe A fills a tank in 16 minutes and pipe B fills it in 12 minutes. Starting with A, they are opened in alternate minutes. How long until the tank is full?

Size the tank so both pipes are wholeThe answer cannot depend on how big the tank is, so choose the LCM of the two times.tank = LCM(16, 12) = 48 units · A = 48/16 = +3/min · B = 48/12 = +4/min

One cycle is two minutes and 7 unitsMinute 1 is A’s, minute 2 is B’s, then it repeats. Add the two slots.3 + 4 = 7 units every 2 minutes

Divide to get the whole cycles48 units at 7 units a cycle. Six cycles fit; a seventh would overshoot, so six is where the counting stops and the walking starts.6 × 7 = 42 units in 12 minutes ⇒ 6 units left

Minute 13 is A’s, and A is not enoughThe pattern says A, so A runs the whole minute. It brings in 3 of the 6 units needed.42 + 3 = 45 of 48 ⇒ 3 units still short

Minute 14 is B’s, and B needs only part of itB moves 4 units a minute and only 3 are left, so B is done before the minute is.3 / 4 = ¾ minute ⇒ total 13¾ minutes

Compare the two endings. Following the pattern gives 13¾ minutes. Skipping minute 13 and letting B clear all 6 remaining units at 4 a minute gives 12 + 6/4 = 13½ — a plausible-looking number that answers a question nobody asked, because B was never entitled to two minutes in a row. Both pipes open together would have finished in 48/7 = 6 and 6/7 minutes, so alternating costs you a little over twice as long as sharing.

03 The method

Cycle, divide, walk

Three moves, in this order, and one check that costs nothing. The order matters: the division tells you where to start walking, and the walk tells you when to stop.

Cycle work = the sum of the signed rates in one full round of slots. Whole cycles = the largest count whose next slot can still finish the tank — in practice, divide the tank by the cycle work and step back if the quotient overshoots. The tail: take the slots in pattern order; a slot that cannot finish contributes its whole rate, and the first slot that can finish contributes only remaining / rate of its length.

The pipe that finishes the tank is chosen by the pattern, not by you. If your last line divides a remainder by whichever rate makes it come out neatly, you have stopped solving an alternating question. Sanity-check the answer against the both-open time: alternating must always be slower, and roughly twice as slow for two similar pipes.

What you are doing	Right	Wrong
The two rates	used one at a time, per slot	added into one joint rate
Whole cycles	48 / 7 = 6 cycles, 12 minutes	a round guess like 5 cycles
Who gets minute 13	A — the pattern says so	B, because it is faster
The last 3 units	B needs ¾ of its minute	B takes a second minute
Final answer	13¾ minutes	13½ minutes
A part-minute ending	keep the fraction	round up to a whole minute
Sanity check	slower than 6 6/7 min both-open	—

05 Cheat sheet

Alternate opening on one page

Every row is the same three moves in a different costume. The example column is the lesson pair — A at 16 minutes, B at 12, A first, 48-unit tank.

Situation	The cycle	On A 16 / B 12, A first
Two pipes, 1 minute each	`2 slots: rA + rB`	3 + 4 = 7 units / 2 min
Whole cycles	`divide the tank by the cycle`	48/7 → 6 cycles, 42 units
The tail	`slots in pattern order`	min 13 A (+3), min 14 B (¾ min)
Answer	`2 × cycles + tail`	12 + 1 + ¾ = 13¾ min
Three pipes rotating	`3 slots: rA + rB + rC`	30/24/20 h → 15 units, 8 cycles, 24 h
One always on, two alternating	`slots are (A+B) and (A+C)`	12/20/15 h → 8 and 9 units, 7⅑ h
A drain in the cycle	`one slot is negative`	peaks mid-cycle, so do not just divide

The cycle total ignores the order, the tail does notStart with B instead of A and the cycle is still 7 units, so the whole-cycle count is identical. Only the tail changes — and when the tank happens to fill exactly on a cycle boundary, even the tail agrees and the two answers are the same.

A negative slot breaks the divisionWith A filling 3 a minute and a drain removing 2, the cycle nets 1 but the level peaks at the end of A’s minute. A 30-unit tank is full during minute 55, not at the end of cycle 30 — so find the cycle whose peak reaches the top, not the one whose end does.

Clock-time questions are stages, not cycles“P at 7 pm, Q at 8 pm, R at 9 pm” is not an alternating pattern; it is three successive net rates. Work each stage to the moment the next pipe opens, then divide the remainder by the final net rate and only then convert back to a clock reading.

06 Where & why

Where this shows up

Alternate opening is set constantly, and almost always with numbers that tempt you into the shortcut that breaks the pattern. Examiners know the tail is where students give up.

TCS NQT · Cognizant

Two pipes, one minute each

The numbers are chosen so the LCM is small and the cycle count is under ten. Marks go to whoever writes the thirteenth minute down instead of assuming it away.

SSC CGL · RRB NTPC

“On alternate hours, B being first”

The same pair with the order flipped, set as a separate question. If the tank fills on a cycle boundary both orders give the same answer, and noticing that saves you the second solve.

Bank PO · SBI Clerk

One pipe always open, two rotating

The slots become A+B and A+C. It looks much harder and is the identical method with fatter slots.

Shift rosters and batch jobs

Alternating capacity

Two machines that run on alternate shifts, or a nightly job that only gets the extra worker every third night, are this arithmetic exactly — including the part-shift ending.

The reason this model is worth real attention is that it teaches you to respect a stated pattern over a convenient calculation. That habit is worth more than the formula: most wrong answers in aptitude come from quietly solving a neater question than the one on the page.

07 Interview questions

What gets asked

Ten, from the definition of a cycle through to the two source-book answers that do not survive checking.

Why can you not just add the two rates when pipes alternate?

Because they are never open together. Adding rates describes simultaneous work; alternating pipes work in turns, so each rate is applied on its own for its own slot. The sum only appears as the cycle total — the work of two separate minutes, not of one shared minute.

What is a cycle, and how do you find its work?

A cycle is the shortest block of slots that repeats. Its work is the sum of the signed rates of the slots in it. For A and B alternating a minute each with a 48-unit tank, the cycle is 2 minutes and 3 + 4 = 7 units.

A fills in 16 minutes and B in 12, alternately from A. How long?

13¾ minutes. The tank is 48 units, A is 3 a minute and B is 4, so a cycle is 7 units. Six cycles give 42 units in 12 minutes; minute 13 is A’s and adds 3 to reach 45; minute 14 is B’s and B needs only ¾ of it to bring in the last 3 units.

A fills in 20 minutes and B in 30, alternately from A. How long?

24 minutes. The tank is 60 units, the cycle is 3 + 2 = 5 units, and 60/5 = 12 cycles exactly, so 12 × 2 = 24 minutes. Some printed keys give 27 by running five cycles and then letting B clear the leftover 32 units alone — which abandons the alternation. Divide first; if the tank lands on a cycle boundary there is no tail at all.

Does it matter which pipe goes first?

It changes the tail, not the cycle. The cycle total is a sum, so the order cannot affect it, and therefore cannot affect the number of whole cycles. For A 20 / B 30 the answer is 24 minutes either way because there is no tail. For A 16 / B 12 there is one, and starting with B gives 13⅔ instead of 13¾.

What is the most common wrong answer in this model?

Letting one pipe clear the entire remainder. You run some cycles, see a few units left, and divide them by whichever rate is convenient. That answers a different question — no pipe is entitled to two consecutive slots. Always give the next slot to the pipe the pattern names, for its full length, and only then let the following pipe finish.

Pipe A is open all the time while B and C alternate, B first. A 12 h, B 20 h, C 15 h. How long?

7⅑ hours. On a 60-unit tank A is 5 an hour, B is 3 and C is 4, so the slots are A+B = 8 and A+C = 9, a cycle of 17 every 2 hours. Three cycles give 51 units in 6 hours; hour 7 is A+B and brings 8 more to 59; hour 8 is A+C at 9 an hour and needs 1/9 of an hour for the last unit. Some keys print 7⅛ by letting A+B run 9/8 of an hour, but a slot is exactly one hour long.

How do you handle “every third hour B and C are also opened”?

As a three-slot cycle: A alone, A alone, then A+B+C. With A 48 h, B 12 h and C 24 h the tank is 48 units, so the slots are 1, 1 and 7 — a cycle of 9 every 3 hours. Five cycles give 45 units in 15 hours, then hours 16 and 17 give one unit each, and the final unit takes 1/7 of hour 18, for 17⅐ hours.

What changes if one of the alternating pipes is an outlet?

Hard

Pipe P fills a tank in 6 hours, pipe Q in 5 hours, and pipe R empties the full tank in 3 hours. P is opened at 7 pm, Q at 8 pm and R at 9 pm, and all three then stay open. At what time does the tank become full?

Follow-up

This is not a cycle at all — it is three successive net rates, and the final one is very nearly zero, which is what stretches the answer past midnight. Getting the arithmetic right and the clock reading wrong loses the mark just as completely.

Show the hint

Tank 30 units: +5, +6 and −10 an hour. Work each stage to the hour the next pipe opens, then divide the remainder by the three-pipe net rate.