AFCAT Numerical Ability · Time and Work

Someone Leaves or Takes Over

When people join or leave partway through a job, cut it into stages, find the work done in each, and give what is left to whoever is still working.

All Time and Work notes

Why this matters

Three AFCAT past-year questions are on this page, from 2016, 2017 and 2019, and over a third of its questions are HARD. The marks go on the stage boundaries: counting the last stage from the start of the job, or forgetting that 'leaves 3 days before the end' fixes the last stage, not the first.

Concept 1 of 2: Work in stages: one stops, another finishes

Lay the job out as a timeline. Each stage has a fixed team, so its work is that team's rate × its days. The stages add up to the whole job, and the unknown is usually the length of the last stage or one person's rate.

Definition

Work in a stage = (sum of the rates of those working) × (days in the stage).
Work left after a stage = 1 minus the work done so far. Divide it by the rate of whoever carries on.
If the finisher's rate is unknown, the stage they worked alone gives it: work left ÷ days.
'How many more days' is the last stage alone. 'In how many days was the work done' is all the stages added.
With three people dropping out in turn, keep a running total. Taking the job as the LCM of the times keeps every stage a whole number of units.

Stages

\sum (\text{rate} \times \text{days}) = 1

Worked example

A can do a job in 8 days and B in 24 days. They work together for 3 days, then A leaves. How many more days does B take to finish?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 03-211Easy

Example 1 · Time and Work · Partial Work, Joining and Leaving

A

can do a piece of work in 10 days. He works at it for 4 days and then

B

finishes it in 9 days. In how many days can

A

and

B

together finish the work?

More days, or days in all?

If A works 5 days and B then needs 12 more, the job took 17 days. Check which of the two the question asks for before you answer.

Use the rate of whoever is left

After A leaves, the rest of the work goes at B's rate alone. Dividing it by the joint rate makes the last stage far too short.

Drill 12 more on work in stages: one stops, another finishes

Concept 2 of 2: Leaving a few days before the end

When someone leaves 'k days before the work is finished', you do not know when they left, only when the job ended. Call the whole time

t

. The one who left worked

t - k

days, everyone else worked

t

, and one equation in

t

does the rest.

Definition

Each person's work = their rate × the days they actually worked.
A leaving $k$ days early: $\dfrac{t - k}{a} + \dfrac tb = 1$ .
Shortcut: B works alone for the last $k$ days. Take that, $\tfrac kb$ , off the job, and the rest was done together.
Wages follow the work done, so once $t$ is known, each person's pay is their share of the work, as on Combined Work.

Leaving k days early

\dfrac{t - k}{a} + \dfrac{t}{b} = 1

Worked example

A can do a job in 12 days and B in 18 days. They start together, but A leaves 3 days before the job is finished. How long does the job take?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

AFCAT · 2017 · AFCAT (II) 2017 — Numerical Ability · 03-210Hard

Example 2 · Time and Work · Partial Work, Joining and Leaving

A

B

and

C

can do a piece of work in 20, 24 and 30 days, respectively. They undertook to do the piece of work for ₹5400. They begin the work together but

B

left 2 days before the completion of the work and

C

left 5 days before the completion of the work. The share of

A

from the assured money is

The one who left worked fewer days

If the job takes

t

days and A leaves 3 days before the end, A worked

t - 3

days. Writing

t + 3

, or giving A the full

t

, loses the mark.

Drill 2 more on leaving a few days before the end

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 (2)

Work in stages: one stops, another finishes
Stages
$\sum (\text{rate} \times \text{days}) = 1$
Leaving a few days before the end
Leaving k days early
$\dfrac{t - k}{a} + \dfrac{t}{b} = 1$

Watch out for (3)

More days, or days in all?
→ Work in stages: one stops, another finishes
Use the rate of whoever is left
→ Work in stages: one stops, another finishes
The one who left worked fewer days
→ Leaving a few days before the end

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