SSC CGL Maths · Time and Work

Pipes and Cisterns

A pipe is a worker whose job is one tankful: filling pipes add their rates, and emptying pipes and leaks take theirs away.

All Time and Work notes

Why this matters

Fifty-two questions in the bank, tied with work rates as the biggest page of the chapter, and thirty-four of them come from the 2024 shifts. Twenty-four are MODERATE and three HARD: inlets and outlets open together, a leak, one pipe so many times as fast as another, and pipes opened or closed partway.

Concept 1 of 4: Inlets and outlets open together

Each filling pipe adds its share of the tank every hour, and each emptying pipe takes its share away. What counts is how much is left in the tank at the end of each hour: the net rate.

Definition

An inlet that fills the tank in $a$ hours adds $\frac{1}{a}$ of it each hour. An outlet that empties a full tank in $b$ hours takes away $\frac{1}{b}$ each hour.
Net rate $=$ all the inlets' rates $-$ all the outlets' rates. Time to fill $= 1 \div$ net rate.
$n$ identical pipes together have $n$ times one pipe's rate.
LCM units: take the tank as the LCM of the times and give each pipe plus or minus its units an hour.
One inlet and one outlet, fast route: the tank fills in $\frac{ab}{b - a}$ hours when $b > a$ . If the outlet is the faster one, a full tank empties instead.
Watch the unit asked: hours $\times 60$ gives minutes.

Net rate of pipes

\text{Net} = \sum \frac{1}{a_i} - \sum \frac{1}{b_j} \qquad T = \frac{ab}{b - a}

$a_i$ hours each filling pipe needs alone
$b_j$ hours each emptying pipe needs to empty a full tank
a, bthe same two times for a single inlet and a single outlet
Thours to fill the tank with both open

Worked example

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

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 3 · Q55Easy

Example 1 · Time and Work · Pipes and Cisterns

Pipes A and B can fill a tank in 20 minutes and 30 minutes, respectively. Meanwhile, a drainpipe C can empty the tank in 60 minutes. If all three pipes are opened simultaneously, how long will it take to fill the tank?

Adding an outlet's rate

An emptying pipe works against the others, so its rate is taken away. Pipes of 12 and 15 hours with a 20-hour drain fill the tank in 10 hours; adding the drain gives a wrong 5 hours.

Answering in the wrong unit

If the options are in minutes, convert the hours:

\frac{9}{4}

hours is 135 minutes.

Drill 18 more on inlets and outlets open together

Concept 2 of 4: Finding a leak from the slower fill

If a pipe should fill a tank in 8 hours but takes longer because of a leak, the leak drains the difference between the two rates every hour.

Definition

Leak rate $=$ the pipe's rate $-$ the rate with the leak $= \frac{1}{a} - \frac{1}{b}$ .
Fast route: the leak alone empties a full tank in $\frac{ab}{b - a}$ hours.
"It takes $k$ hours extra" means $b = a + k$ .
A leak that lets out a part of the supply: if $\frac{1}{4}$ of the water runs out, only $\frac{3}{4}$ of the pipes' rate stays in the tank.

Time for the leak to empty the tank

T_{\text{leak}} = \frac{ab}{b - a}

ahours the pipe needs with no leak
bhours it needs with the leak

Worked example

A pipe can fill a tank in 10 hours. Because of a leak it takes 15 hours. How long would the leak take to empty the full tank?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 11 Sep 2024, Shift 2 · Q74Easy

Example 2 · Time and Work · Pipes and Cisterns

Pipe A usually fills a tank in 6 hours. But due to a leak at the bottom of the tank, it takes extra 2 hours to fill the tank. If the tank is full, then how much time will it take to get emptied due to the leak?

Taking the extra hours as the leak's time

If a leak makes a 10-hour fill take 2 hours longer, the leak does not empty the tank in 2 hours. It takes

\frac{10 \times 12}{12 - 10} = 60

hours.

Drill 2 more on finding a leak from the slower fill

Concept 3 of 4: One pipe so many times as fast as another

If one pipe is 3 times as fast as another, call the slow pipe's rate r and the fast one's 3r. Together they are worth 4 slow pipes, and the time with both open gives r.

Definition

"k times as fast": rates $r$ and $kr$ . Together $(k + 1)r = \frac{1}{T}$ , so the slower pipe alone takes $(k + 1)T$ and the faster takes $\frac{(k + 1)T}{k}$ .
"One and a half times as fast" is a rate ratio of $3 : 2$ . Write every ratio in whole numbers first.
"Takes $\frac{4}{5}$ of the time" means the rates are $5 : 4$ , because time and rate are inverse.
Times that differ by a fixed amount: call them $x$ and $x + d$ , write $\frac{1}{x} + \frac{1}{x + d} = \frac{1}{T}$ and solve the quadratic. Testing the options in product over sum is often quicker.

Pipes in a speed ratio

T_{\text{slow}} = (k + 1)T \qquad T_{\text{fast}} = \frac{(k + 1)T}{k}

khow many times as fast the faster pipe is
Ttime to fill with both pipes open

Worked example

One pipe fills a tank 3 times as fast as another. Together they fill it in 30 minutes. How long does the slower pipe take alone?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 26 Sep 2024, Shift 1 · Q71Easy

Example 3 · Time and Work · Pipes and Cisterns

One pipe can fill a tank four times as fast as another pipe. If together the two pipes can fill the tank in 48 minutes, the slower pipe alone will be able to fill the tank in:

Multiplying the joint time by k instead of k + 1

If one pipe is 3 times as fast and together they take 30 minutes, the slower pipe alone takes

4 \times 30 = 120

minutes, not

3 \times 30

. The two pipes together are worth

k + 1

slow pipes.

Keeping the wrong root

\frac{1}{x} + \frac{1}{x + d} = \frac{1}{T}

one root makes a time negative or shorter than the joint time. Only the root that keeps every time positive and longer than the joint time is the answer.

Drill 8 more on one pipe so many times as fast as another

Concept 4 of 4: Pipes opened or closed partway

This is workers joining and leaving with pipes: split the time into stretches, find the net rate in each, and add what each stretch fills.

Definition

Each stretch fills net rate $\times$ its time. The stretches add to one tank, or to the level the question asks for.
A pipe closed $k$ minutes before full: let the total be $T$ ; that pipe runs $T - k$ . Solve as for a worker who leaves before the end.
"After how long should B be closed?" The pipe that runs the whole time fills its share; the rest is B's, so B's running time $=$ the rest $\div$ B's rate.
Clock times: count hours from the first opening, then add them back to the clock.
Alternate hours: use cycles as for alternate days, but with an outlet in the cycle the tank can fill partway through an inlet's hour before the cycle ends. Stop at the first inlet hour that reaches full.
Rates that change: after the change, use the new rates for what is left.
A wall built by one worker and knocked down by another works the same way: the one knocking it down is an outlet.

Worked example

Pipes A and B fill a tank in 12 hours and 18 hours. Both are opened together, and after 3 hours A is closed. How much longer does B take to fill the tank?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 12 Sep 2025, Shift 1 · Q68Moderate

Example 4 · Time and Work · Pipes and Cisterns

Three pipes, A, B, and C are capable of filling a tank in 6, 8, and 12 hours, respectively. When all three pipes are opened together, they operate for 2 hours before pipe C is closed. How much additional time will it take to completely fill the tank after that?

Running whole cycles with an outlet in them

With an inlet adding 3 units an hour and an outlet taking 2, an 18-unit tank fills in an inlet hour once the level reaches 15, after 31 hours. Dividing 18 by the cycle's net 1 unit gives a wrong 36 hours.

Drill 20 more on pipes opened or closed partway

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Inlets and outlets open together
Net rate of pipes
$\text{Net} = \sum \frac{1}{a_i} - \sum \frac{1}{b_j} \qquad T = \frac{ab}{b - a}$
Finding a leak from the slower fill
Time for the leak to empty the tank
$T_{\text{leak}} = \frac{ab}{b - a}$
One pipe so many times as fast as another
Pipes in a speed ratio
$T_{\text{slow}} = (k + 1)T \qquad T_{\text{fast}} = \frac{(k + 1)T}{k}$

Watch out for (6)

Adding an outlet's rate
→ Inlets and outlets open together
Answering in the wrong unit
→ Inlets and outlets open together
Taking the extra hours as the leak's time
→ Finding a leak from the slower fill
Multiplying the joint time by k instead of k + 1
→ One pipe so many times as fast as another
Keeping the wrong root
→ One pipe so many times as fast as another
Running whole cycles with an outlet in them
→ Pipes opened or closed partway

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