In this section, we are going to learn about the concept of time and work (men, women and children) in which there are different type of people eg. men, women and children with different efficiencies. 

Here are some special points which you should remember:

• Work and Person Directly proportional (more work, more men and conversely more men, more work to be done.
• Time and Person Inversely proportional (more men, less time and conversely more time, fewer men).
• Work and Time Directly proportional (more work, more time, and conversely more time, more work).
• If a person can complete a work in ‘n’ days/hours then that person’s 1 day/hour work would be = 1/n. 

E.g. Vandana completes a work in 35 days. What work will she do in 1 day?

⇒ Let the work is done by Vandana in 35 days = 1 unit

     Work done by Vandana in 1 day would be = 1/35 unit (Ans.)

E.g. Pcan do work 3 times faster than Q and therefore takes 40 days less than Q. Find the time in which P and Qcan complete the work individually.

⇒Time is taken by Q — Time taken by P = 40

Let the time taken by P be ‘K’

3K – K = 40 ⇒  2K = 40

K = 20

The number of days required by P = 20 and the number of days required by Q = 60.

Let’s try to understand this with depth

• If A can do a piece of work in x days as the same work B can do in y days; then (A+B)’s 1 day’s work ⇒1/x + 1/y = (x + y)/xy. So time taken would be ⇒ xy/(x + y)

This can be extended to ‘n’ number of people.

E.g.  A can do a piece of work in 10 days and B can do the same work in 12 days. How long will they take to finish the work, if both work together?

⇒ A’s one day work be = 1/10

⇒ B’s one day work be = 1/12

According to given formula time taken would be = (10 × 12)/(10 + 12)

⇒ 120/22 = 60/11 (Ans.) 

Till now, we tried to solve some basic question, now we will look into some traditional questions of men, women.

E.g. If 6 men or 8 women can reap a field in 86 days, how long will 14 men and 10 women take to reap it?

As we know that 6 men = 8 women

As 6 men can reap the field in 86 days

1 Man will reap the field in = 86 × 6 days

1 Man will reap the field in 1 day = 1/(86 × 6) part

Similarly

As 8 women can reap the field in 86 days

A woman will reap the field in = 86 × 8 days

A woman will reap the field in 1 day = 1/(86 × 8) part

⇒ 1/[14 × 1/(86 × 6) + 10 × 1/(86 × 8)]

By simplifying

⇒ 43 × 24/(28 + 15)

⇒ 24 days (Ans.)

A quite tedious task?

NO PROBLEM we have an alternative formula

If ‘a’ man or ‘b’women can finish a work in D days, then the time taken by them be

D(a₁b₁) / a₂b₁ + a₁b₂

Where

D = no. of days to complete the work in the first case

a₁ = no. of men in the first case

b₁ = no. of women in the first case

a₂ = no. of men in the second case

b₂ = no. of women in the second case

Let’s back to our previous question

no. of days = D(a₁b₁) / (a₂b₁ + a₁b₂)

⇒ 86 (6× 8)/(14 × 8 + 10 × 6)

⇒ 86 × 6 × 8/(128 + 60)

⇒ 86 × 6 × 8/188 = 24 days (Ans.)

E.g. 4 men and 6 women can complete a work in 8 days, while 3 men and 7 women can complete it in 10 days. In how many days will 10 women complete it?

By the first condition

4m + 6w = 1/8      (i)

By the second condition

3m + 7w = 1/10   (ii)

By simply solving both equation

4 × (ii) – 3 × (i)

12m + 28w – 12m – 18w = 4/10 + 3/8

10w = 40 (Ans.)

E.g. 3 men, 4 women and 6 children can complete a work in 7 days. A woman does double the work a man does and a child does half the work a man does. How many women alone can complete this work in 7 days?

Let 1 woman’s 1 day work = x.

Then, 1 man’s 1 day work = x/2 and 1 child’s 1 day work x/4.

So, (3x/2 + 4x + + 6x/4) = 1/7

28x/4 = 1/7 => x = 1/49

1 woman alone can complete the work in 49 days.

So, to complete the work in 7 days, number of women required = 49/7 = 7.

E.g. If 10 men or 20 boys can make 260 mats in 20 days, then 8 men and 4 boys will make how many mats?

10 men = 20 boys

∴ 1 man = 2 boys

∴ 8 men + 4 boys

= (16 + 4) boys = 20 boys

Hence, 8 men and 5 boys will make 260 mates in 20 days.

E.g. 8 days are taken by one man and one woman together to complete the job. A man alone can complete the work in 10 days. In how many days can one woman alone complete the work?

Work done by 1 woman in 1 day

= 1/8 – 1/10 = (5-4)/40 = 1/40

∴ One woman will complete the wok in 40 days.

E.g.  45 people take 18 days to dig a pond. If the pond would have to be dug in 15 days, then the number of people to be employed will be:

A case of inverse proportion in which more people would take less day

Number of people required = (45 x 18)/15

= 54 people

Let look at more example questions in which walls have to be built by laborers with their efficiencies,

E.g. 12 men can do a piece of work in 24 days. How many men are needed to complete the work, in half days?

APPROACH: Here question states that 12 men would take 24 days

⇒ Half days means how many men are required to complete the task in 12 days.

Let the work done by one man on one day be = 1 unit

So work done by 12 men in one day = 12 units

Work done by 12 men in 24 days = 12 × 24 = 288 units (total work)

As now we want to make the work happen in 12 days

⇒ No. of men required = 288/12 = 24 men (Ans.)

E.g. Rajan can build a wall in 30 days. Somu can demolish that wall in 60 days. If Rajan and Somu work on alternate days, when will the wall be completed?

This question is very simple just do little bit mental calculation

As ‘Rajan’ can build a wall in 30 days

⇒ Work done by Rajan in 1 day = 1/30 units

Similarly,

 ⇒ Work done by Somu in 1 day = – 1/60 units

On the first day, Rajan made 1/30 wall but on the second day Somu demolished 1/60 wall

⇒ Total wall created in two days = 1/30 – 1/60 = 1/60 units

Total days should be,

inverse of 1/60 = 60 days

On the 59^th day, Rajan already builds the wall so the answer would be 59 (Ans.)

E.g. A can build a wall in 16 days while B can destroy it in 8 days. A worked for 5 days. Then B joined with A for the next 2 days. Find in how many days could A build the remaining wall?

A’s one day work would be = 1/16 units  (i)

B’s one day work would be = – 1/8 units

As A worked for 5 days total work in these 5 days = 5 × 1/16 = 5/16

In next 2 days both have worked so work done = 2(1/16 – 1/8)

⇒  -1/8 units

Total work in those 7 days = 5/16 – 1/8

⇒ 3/16

Total left work = 1 – 3/16 = 13/16

 With the help of (i) A would take 13 days.(Ans.)

E.g.  45 workers can make a road in 12 days working 5 hours a day. In how many days will 30 workers working 6 hours a day to complete the work?

Let the work done by a worker in an hour = 1 unit

Work is done by 45 workers in 12 days working 5 hours a day = 45 × 12 × 5 units

Similarly,

Let in ‘x’ days 30 workers did 6-hour work = 30 × 6 × x

⇒  45 × 12 × 5 = 30 × 6 × x

⇒ x = 15 days (Ans.)

E.g. A man makes a wall in 9 days but a woman can demolish the same wall in 10 days, both have started working on day 1. How much work has been done on day 45?

The work done by ‘A’ in a day = 1/9 units

 The work done by ‘B’ in a day = – 1/10 units

So total work completed at the end of day 1 = 1/9 – 1/10 = 1/90 units

At the end of 45 days work done be  = 45 × 1/90 = 1/2

Half work has been completed. (Ans.)