Time & Work

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Time and work — along with its twin, pipes and cisterns — appears in every police recruitment paper. Stop thinking in fractions of a job and start using the LCM method: it removes almost all the arithmetic and makes the wage-sharing variants trivial.

The LCM Method

1. Take the total work as the LCM of the given times.

2. Each person's daily output = Total work ÷ their time.

3. Add outputs for people working together; subtract for outlet pipes.

Example: A takes 18 days, B takes 12 days. Take total work = LCM(18, 12) = 36 units. Then A does 2 units a day and B does 3 units a day — whole numbers, no fractions to manage.

Key Relationships

RuleStatement
Combined rateIf A takes a days and B takes b days, together they take ab/(a + b) days
EfficiencyEfficiency is inversely proportional to time taken
Efficiency ratioIf A is n times as efficient as B, A takes 1/n of B's time
WagesWages are shared in the ratio of work done, which equals the ratio of efficiencies when the time worked is equal
PipesAn outlet pipe is simply negative work — subtract its rate
Wages are shared by work done, not by days present. A person who works twice as fast for the same number of days earns twice as much. This is the detail most candidates get wrong under time pressure.

Worked Examples from Real Papers

TS Police SI 2022 — Q13

A can complete a work in 18 days and B in 12 days. Both begin together, but after some days B leaves. If A finishes the remaining work in 3 days, for how many days did B work?

Total work = LCM(18, 12) = 36 units. A does 2 units/day, B does 3 units/day.
Let B work x days. Then A works x + 3 days.
2(x + 3) + 3x = 36 → 5x + 6 = 36 → x = 6 days.

TS Police SI 2022 — Q29

Pipe A fills a tank in 30 minutes, Pipe B in 45 minutes, and Pipe C empties a full tank in 15 minutes. A and B are open for 15 minutes, then C is also opened. How long from then to empty the tank?

Total capacity = LCM(30, 45, 15) = 90 units. A = +3/min, B = +2/min, C = −6/min.
In the first 15 minutes: (3 + 2) × 15 = 75 units filled.
After C opens the net rate is 3 + 2 − 6 = −1 unit/min.
Time to drain 75 units = 75 minutes.

TS Police Constable 2022 — Q39

A is thrice as efficient as B. The difference in the days they take to finish a job alone is 60. If both work together to complete one-third of the work and B alone does the rest, how many days in total?

A is 3× as efficient, so if A takes x days, B takes 3x days. Then 3x − x = 60, giving x = 30.
So A takes 30 days and B takes 90 days.
Together: 1/30 + 1/90 = 4/90 = 2/45 of the work per day.
First third: (1/3) ÷ (2/45) = 7.5 days.
Remaining two-thirds by B alone: (2/3) ÷ (1/90) = 60 days.
Total = 7.5 + 60 = 67.5 days.

TS Police SI 2022 — Q15

A alone can do a work in 10 days, B alone in 15 days. They agree to complete it for ₹30,000. With C's help they finish in 5 days. What is C's share?

In 5 days A completes 5/10 = 1/2 of the work and B completes 5/15 = 1/3.
Together that is 5/6, so C did the remaining 1/6.
Ratio A : B : C = 1/2 : 1/3 : 1/6 = 3 : 2 : 1.
On ₹30,000 across 6 parts: A = 15,000, B = 10,000, C = ₹5,000 — which is one-third of A's share.

Pipes questions often ask "to empty", not "to fill". Read the final line of the question again before choosing. The SI question above gives a filled fraction partway through and then asks for draining time, and the options include the fill-time answer as a distractor.

Also see: Percentages · Profit & Loss · Simple & Compound Interest

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