AFCAT Numerical Ability · Time and Work

When Workers Leave or Join Mid-Job

Take the man-days already done away from the job, give what is left to the team now on site, and compare with the plan only when the question asks.

All Time and Work notes

Why this matters

No AFCAT past-year question is on this page yet, but it is the second largest in the chapter and most of it is MODERATE. The method is one line; the marks go on the question's last words, because 'more days', 'days in all' and 'days beyond the plan' are three different answers.

Concept 1 of 4: Work left after the team changes

Before the change, the full team put in some man-days. Take them away from the job, and what is left must be done by the team now on site. The plan is not needed at all unless the question compares with it.

Definition

Work left = job minus (old team × days worked).
More days = work left ÷ new team. Days in all = days already worked + more days.
Several changes: take away each stage's man-days in turn.
A change by a fraction, such as 'a third of them leave': call the team 3 parts, so 2 parts stay.

Days after a change

\text{more days} = \dfrac{M D - M d}{M'}

Worked example

36 men can finish a road in 25 days. After 10 days, 12 men leave. How many more days does the road take, and how many days in all?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Time and Work · Workforce Changes Mid-Job

A project was scheduled to be completed in 100 days by 80 workers. After 40 days, 20 workers left. How many total days will be required to finish the work?

Take away the work already done

If 30 men need 16 days and 6 leave after 4 days, the 24 men share only the 360 man-days that are left: 15 more days. Dividing the whole 480 man-days by 24 counts the first 4 days twice.

Drill 37 more on work left after the team changes

Concept 2 of 4: Extra days against the plan

At the moment of the change, the plan still had some days to run. The new team needs more days than that, or fewer if workers joined. The extra days are the difference, and there is a quick formula for it.

Definition

Planned days left = planned total minus days worked.
Extra days = actual days left minus planned days left. If workers join, the same difference is the days saved.
Shortcut when $L$ workers leave a team of $M$ with $R$ planned days left: extra days = $\dfrac{L R}{M - L}$ .
Read the last words. 'Extra', 'beyond the plan' and 'after the planned date' ask for the difference; 'more days to finish' asks for the actual days left. 'Additional days' is used both ways, so check which reading is among the options.

Extra days

\text{extra} = \dfrac{L \times R}{M - L}

Worked example

40 workers were to finish a job in 30 days. After 10 days, 8 workers leave. How many days beyond the plan does the job take?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Time and Work · Workforce Changes Mid-Job

48 workers were hired to finish a job in 60 days. After 24 days, 12 workers left. How many extra days will be required to finish the work?

Extra is not the same as more

When 40 workers are planned for 30 days and 8 leave after 10 days, the job needs 25 more days, which is 5 beyond the plan. A question asking for extra days wants 5; one asking for more days wants 25.

Joining saves days

When

J

workers join, the days saved are

\dfrac{J R}{M + J}

: the team grows, so the sum goes below the line.

Drill 15 more on extra days against the plan

Concept 3 of 4: Keeping to the deadline

Sometimes the deadline is fixed and the question asks what must change to meet it: how many workers to hire, or how many hours a day to work. Find the work left and the days left, then divide.

Definition

Workers needed = work left ÷ days left. Hire the difference between that and the workers still on the job.
Hours a day needed = man-hours left ÷ (workers × days left).
'Will it finish on time?': work out the days the new team needs and compare them with the days left in the plan.

Team needed

\text{workers} = \dfrac{\text{work left}}{\text{days left}}

Worked example

10 workers were to finish a job in 16 days. After 6 days, 2 workers leave. How many must be hired to finish on time?

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The same idea in a practice question:

Practice questionModerate

Example 3 · Time and Work · Workforce Changes Mid-Job

A contractor has

20

workers to complete a construction in

30

days. After

12

days,

5

workers are transferred to another project. How many additional workers should be hired to complete the work on time?

Count from the workers still there

If 10 workers are needed and 2 of the original 10 have left, hire 2. Answering 10 hires a whole new team on top of the one already working.

Drill 5 more on keeping to the deadline

Concept 4 of 4: Hours, efficiency and team size change together

The hardest questions change more than one thing at once: the team shrinks and the hours grow, or new workers come in at a different efficiency. Count everyone as standard workers and every day as standard hours, and it is the same subtraction.

Definition

When hours a day change, count the work in man-hours: men × hours × days.
New workers at $p\%$ efficiency count as $\tfrac{p}{100}$ of a worker each. Overtime that raises efficiency by 25% makes each worker count as 1.25.
Unknown team size: write the job as $n$ × planned days, write it again from the story that happened, and solve for $n$ .
A fixed end date with stages: the last group works whatever is left of the schedule.

Standard workers

\text{effective team} = \text{regular workers} + \tfrac{p}{100} \times \text{others}

Worked example

20 men working 8 hours a day can finish a job in 15 days. After 5 days, 4 men leave and the rest work 10 hours a day. How many more days are needed?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Time and Work · Workforce Changes Mid-Job

28

workers can build a bridge in

21

days. After

7

days,

7

workers left and

14

new workers joined who work at half the efficiency of original workers. In how many more days will the bridge be completed?

Efficiency changes the team, not the job

Five trainees at 60% efficiency count as three workers. Adding all five as full workers makes the rest of the job look shorter than it is.

Each stage at its own hours

When the hours a day change partway, the work done so far was at the old hours. Only the work left is done at the new hours.

Drill 14 more on hours, efficiency and team size change together

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 (4)

Work left after the team changes
Days after a change
$\text{more days} = \dfrac{M D - M d}{M'}$
Extra days against the plan
Extra days
$\text{extra} = \dfrac{L \times R}{M - L}$
Keeping to the deadline
Team needed
$\text{workers} = \dfrac{\text{work left}}{\text{days left}}$
Hours, efficiency and team size change together
Standard workers
$\text{effective team} = \text{regular workers} + \tfrac{p}{100} \times \text{others}$

Watch out for (6)

Take away the work already done
→ Work left after the team changes
Extra is not the same as more
→ Extra days against the plan
Joining saves days
→ Extra days against the plan
Count from the workers still there
→ Keeping to the deadline
Efficiency changes the team, not the job
→ Hours, efficiency and team size change together
Each stage at its own hours
→ Hours, efficiency and team size change together

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