◐ ai-teacher

🧮 Aptitude & Reasoning · Chapter 4

Work, rate and pipes

A fills the tank in 4 hours and B in 6 — together it is not 5 hours and not 10, it is 2.4, because you add speeds of work, never the days.

Study this lesson free in the app →

The basics

Every "time and work" question is a speed question in disguise. If Asha paints a wall in 4 days, her rate is 1/4 of the wall per day. Rates add; days do not.

The problem it solves: working out how long a job takes when several people, machines or pipes — some fast, some slow, some working against you — share it.

The mental model: give the job a size. Pick the total work as the LCM of everyone's days — say a 12-litre tank for pipes that take 4 and 6 hours. Then A fills 3 litres an hour, B fills 2, together 5, so the tank takes 12/5 = 2.4 hours. No fractions, and leaks are just negative litres.

For crews of people, the same idea becomes man-days: 10 men for 18 days is 180 man-days of work, however you re-split it.

Words you must know 5

Rate
work per unit time; the reciprocal of time alone (4 days → 1/4 per day)
Total work (LCM units)
a convenient job size that makes every rate a whole number
Efficiency
relative rate; twice as efficient means half the time
Man-days
men × days (× hours) for a fixed job
Outlet pipe
a pipe or leak with a negative rate

Explained

Pipe A fills a tank in 4 hours. Pipe B fills it in 6. Both open together.

The two wrong answers everyone reaches for: 5 hours (the average) and 10 hours (the sum). Both are longer than A alone — impossible, since B is helping.

The real answer is 2.4 hours. Make the tank 12 litres: A pours 3 L/h, B pours 2 L/h, together 5 L/h. 12 ÷ 5 = 2.4. Add the speeds, never the times.

The overhead tank in a Pune society

Every morning the watchman of a Kothrud housing society fills the overhead tank. The motor alone takes 4 hours. On days the municipal line has pressure, it adds water too, and the tank fills well before 4.

One summer the tank also starts leaking. The watchman notices the motor now takes 6 hours instead of 4.

He doesn't think in hours — he thinks in how much per hour. Motor: a quarter of the tank an hour. With the leak: a sixth. So the leak is draining 1/4 − 1/6 = 1/12 of a tank every hour, and on its own would empty a full tank in 12 hours.

Helpers add their rates, leaks subtract theirs. That's the entire chapter.

The method, the shortcut, and where it breaks

Standard method — rates. A does a job in 12 days, B in 24.

A's rate = 1/12   B's rate = 1/24
together = 1/12 + 1/24 = 3/24 = 1/8  → 8 days

LCM method — same maths, no fractions. Job = LCM(12, 24) = 24 units. A does 2/day, B 1/day, together 3/day → 24/3 = 8 days.

Two-person shortcut: ab/(a + b) = 12 × 24 / 36 = 8.

When the shortcut fails:

The LCM method handles all of these; the shortcut handles one.

Picture it: the job as a bar of units

A alone: 12 days. B alone: 24 days. Make the job 24 units.

job  |□□□□□□□□□□□□□□□□□□□□□□□□|  24 units

A    |■■|                         2 units/day  (24 ÷ 12)
B    |■|                          1 unit/day   (24 ÷ 24)
A+B  |■■■|                        3 units/day

day: 1   2   3   4   5   6   7   8
     ■■■ ■■■ ■■■ ■■■ ■■■ ■■■ ■■■ ■■■  = 24 ✓

A tank with a leak works the same way, with one bar pointing backwards:

fill   +3 units/h  →→→
leak   −1 unit/h   ←
net    +2 units/h  →→

Solving any work or pipes question

If your combined time is longer than the fastest worker alone, you averaged or added the days.

How bad is the leak?

A pipe fills a tank in 6 hours. Because of a crack at the bottom, it actually takes 8 hours to fill.

AMCAT quant section, 50 seconds left

Pipes A and B can fill a tank in 12 and 15 hours. Pipe C can empty it in 20 hours. All three are opened together. How long until the tank is full?

Step 1 — job size: LCM(12, 15, 20) = 60 units.

Step 2 — rates with signs:

A: 60/12 = +5 per hour
B: 60/15 = +4 per hour
C: 60/20 = −3 per hour
net        +6 per hour

Step 3 — divide: 60 ÷ 6 = 10 hours.

Step 4 — check: A and B alone would take 60/9 ≈ 6.7 h. The outlet slows that down, so 10 h is longer than 6.7 h — direction makes sense.

The trap option was 5 hours: 60/(5 + 4 + 3), treating C as a filler. One sign error, and it's always in the options.

The traps examiners set

Words to know

RateWork done per unit time; a job done in n days alone means a rate of 1/n per day.
Total work (LCM units)A chosen job size — the LCM of all the times — so every rate is a whole number.
EfficiencyRelative rate. Twice as efficient means twice the rate and half the time.
Man-daysMen × days (× hours) needed for a fixed job; constant for that job.
Outlet pipeA pipe or leak that removes water — its rate is subtracted.
Net rateThe sum of all rates with signs — inlets minus outlets.

In two minutes

Test yourself

1. A can finish a job in 12 days and B in 24 days. Working together, how long do they take?

  1. 8 days
  2. 18 days
  3. 36 days
  4. 6 days
Show the answer

8 days — 1/12 + 1/24 = 3/24 = 1/8 → 8 days. 18 days averages the times and 36 adds them — both slower than A alone, which cannot be right when B is helping.

2. Why do you add rates, and not days, when two people work together?

  1. Because days are always whole numbers
  2. Because adding days gives the time for the faster worker
  3. Because work done per day is what accumulates; the days each takes alone are reciprocals of that
  4. Because the slower worker sets the pace
Show the answer

Because work done per day is what accumulates; the days each takes alone are reciprocals of that — Each day, A completes 1/12 and B 1/24 of the job, and those pieces stack up. 'Days alone' is 1/rate — adding reciprocals has no physical meaning.

3. How many days will 15 men take to finish a job that 10 men finish in 18 days?

  1. 27 days
  2. 12 days
  3. 13 days
  4. 10 days
Show the answer

12 days — Work = 10 × 18 = 180 man-days. 180/15 = 12 days. 27 days treats men and days as directly proportional — more men cannot take longer.

4. A vs B: A is twice as efficient as B and finishes the job 30 days sooner. How long do they take together?

  1. 20 days
  2. 15 days
  3. 45 days
  4. 10 days
Show the answer

20 days — If A takes t, B takes 2t; 2t − t = 30 → A 30 days, B 60 days. Together 30 × 60/90 = 20 days. 45 averages the two times.

5. What happens to the filling time if a leak that alone would empty the tank in 12 hours is open while a pipe that fills it in 4 hours runs?

  1. It becomes 3 hours
  2. It becomes 8 hours
  3. It becomes 6 hours
  4. It becomes 16 hours
Show the answer

It becomes 6 hours — Net rate 1/4 − 1/12 = 2/12 = 1/6 → 6 hours. 3 hours adds the leak as if it were a filler; 8 hours subtracts the times (12 − 4).

6. Pipes A and B fill a tank in 10 and 15 hours; pipe C empties it in 30 hours. If all three are opened, how long does the tank take to fill?

  1. 5 hours
  2. 6 hours
  3. 10 hours
  4. 7.5 hours
Show the answer

7.5 hours — LCM 30: A +3, B +2, C −1 → net 4 per hour → 30/4 = 7.5 hours. 5 hours treats C as a filler; 6 hours ignores C entirely.

There are 10 questions, 3 puzzles and 5 recall cards for this lesson in the app — the recall cards come back on later days, just before you would forget.

Learn it so it stays. Read it, test yourself, then recall it on two different days. The app schedules that for you, free.

Start this lesson →