AFCAT Numerical Ability · Time and Work

Man-Days, Man-Hours and Groups of Workers

Measure a job in man-days or man-hours: more workers need fewer days, longer working days need fewer days, and a bigger job needs more of both.

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Why this matters

This is the largest page in the chapter, and over half of it is EASY. Two AFCAT past-year questions sit here, both from 2016. Marks go when a question mixes men, women and children: turn everyone into one kind of worker before you count anything.

Concept 1 of 5: Man-days: workers × days

A job needs a fixed amount of labour. 12 workers for 10 days is 120 man-days, and any team that puts in 120 man-days finishes it: 20 workers in 6 days, or 8 workers in 15. Ratio and Proportion met this as inverse variation; here it becomes the main tool.

Definition

Work = workers × days. For the same job, $M_1 D_1 = M_2 D_2$ .
More workers means fewer days. Halving the team doubles the time.
If only part of the job is done, scale the man-days: $\tfrac 35$ of a job in 300 man-days means the whole job is 500 man-days.
Anything used at a steady rate works the same way: food eaten, pages typed.

Man-days

M_1 D_1 = M_2 D_2

Worked example

14 men can build a wall in 30 days. How many men are needed to build it in 21 days?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Time and Work · Man-Days and Group Work

18 men can complete a task in 24 days. How many men will be needed to complete the task in 16 days?

More workers, fewer days

If 10 men take 12 days, 15 men take 8 days, not 18. Multiply men by days and divide; never scale the days up with the men.

Drill 30 more on man-days: workers × days

Concept 2 of 5: Man-hours: when the working day changes

If the length of the working day changes, count hours instead of days. 10 men working 9 hours a day for 4 days put in 360 man-hours, and that is the job, whatever the mix of men, hours and days.

Definition

Work = men × hours a day × days. For the same job, $M_1 H_1 D_1 = M_2 H_2 D_2$ .
The unknown can be any one of the three: men, hours a day or days.
Weeks work the same way: men × days a week × hours a day × weeks.

Man-hours

M_1 H_1 D_1 = M_2 H_2 D_2

Worked example

12 men working 10 hours a day finish a job in 9 days. How many days do 9 men working 12 hours a day take?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Time and Work · Man-Days and Group Work

8

men working

7

hours a day can complete a job in

18

days. If

6

men work

8

hours a day, how many days will they take to complete the same job?

Hours a day are part of the work

If the hours a day change and you count only men × days, the answer is out by the ratio of the hours. A team that halves its hours needs twice the days.

Drill 10 more on man-hours: when the working day changes

Concept 3 of 5: When the amount of work changes

When the job itself changes size, say 90 shirts instead of 60, find the labour for one unit of output first. Then multiply by the new output and share it out among the new team.

Definition

$\dfrac{M_1 H_1 D_1}{W_1} = \dfrac{M_2 H_2 D_2}{W_2}$ , where $W$ is the output: walls, metres, chairs, books.
Labour for one unit of output is $\tfrac{MHD}{W}$ . Leave out $H$ when hours are not mentioned.
An unknown output is the labour available ÷ the labour per unit.
Phases of a job split in a ratio: the labour for each phase is in that ratio.

Work with output

\dfrac{M_1 H_1 D_1}{W_1} = \dfrac{M_2 H_2 D_2}{W_2}

Worked example

15 tailors stitch 60 shirts in 8 days. How many shirts do 20 tailors stitch in 9 days?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Time and Work · Man-Days and Group Work

If 10 masons working 7 h daily can build 28 water tanks in 16 days, how many days will 12 masons working 6 h daily take to build 36 water tanks?

Output goes under the labour

More output needs more labour. Writing

\tfrac{MHD}{W}

keeps that straight: doubling the walls doubles the man-days instead of halving them.

Drill 13 more on when the amount of work changes

Concept 4 of 5: Converting men, women and children

'2 men or 3 women can do the job in 18 days' says that 2 men do as much as 3 women. Turn everyone into one kind of worker, and the question becomes man-days again.

Definition

' $p$ men or $q$ women' in the same time means $p$ men = $q$ women, so 1 man = $\tfrac qp$ women.
Convert the new team into women (or men). The job is then women × days.
Three kinds at once, such as '1 man or 2 women or 4 children': convert everyone to children, or give each a rate and add.
Half of one team plus half of the other makes one full team, which takes the same time.

Equivalent workers

p \text{ men} = q \text{ women} \;\Rightarrow\; 1 \text{ man} = \tfrac qp \text{ women}

Worked example

2 men or 3 women can do a job in 18 days. How long do 2 men and 6 women take together?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 03-212Moderate

Example 4 · Time and Work · Man-Days and Group Work

If 3 men or 4 women can plough a field in 43 days, in how many days can 7 men and 5 women plough the same field?

'Or' is not 'and'

'4 men or 6 women in 20 days' means either group alone takes 20 days. 4 men and 6 women working together are two full teams and take 10 days.

Drill 12 more on converting men, women and children

Concept 5 of 5: Two mixed teams: solve for one worker

When two different teams finish the same job, their total labour is equal. Write a man's daily work as

m

and a woman's as

w

. The equal-work equation gives

m : w

, and then everything is in one unit.

Definition

Same job: (team 1's daily work) × its days = (team 2's daily work) × its days.
That gives the ratio $m : w$ , and the job becomes a number of woman-days.
If the two teams take the same time, their daily work is equal.
From 'x men take a days and y women take b days', the efficiency ratio is $m : w = yb : xa$ .

Equal work

(p\,m + q\,w)\, d_1 = (r\,m + s\,w)\, d_2

Worked example

1 man and 4 women finish a job in 15 days; 2 men and 1 woman finish it in 18 days. How long does 1 man take alone?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 5 · Time and Work · Man-Days and Group Work

3 men and 4 boys can complete a piece of work in 12 days. 4 men and 3 boys can do the same work in 10 days. Then, 2 men and 3 boys can finish the work in how many days?

Equal work, not equal daily work

Two teams finishing the same job in different times have equal TOTAL work. Setting their daily work equal is right only when their days are equal too.

Drill 8 more on two mixed teams: solve for one worker

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 (5)

Man-days: workers × days
Man-days
$M_1 D_1 = M_2 D_2$
Man-hours: when the working day changes
Man-hours
$M_1 H_1 D_1 = M_2 H_2 D_2$
When the amount of work changes
Work with output
$\dfrac{M_1 H_1 D_1}{W_1} = \dfrac{M_2 H_2 D_2}{W_2}$
Converting men, women and children
Equivalent workers
$p \text{ men} = q \text{ women} \;\Rightarrow\; 1 \text{ man} = \tfrac qp \text{ women}$
Two mixed teams: solve for one worker
Equal work
$(p\,m + q\,w)\, d_1 = (r\,m + s\,w)\, d_2$

Watch out for (5)

More workers, fewer days
→ Man-days: workers × days
Hours a day are part of the work
→ Man-hours: when the working day changes
Output goes under the labour
→ When the amount of work changes
'Or' is not 'and'
→ Converting men, women and children
Equal work, not equal daily work
→ Two mixed teams: solve for one worker

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