**Concept:**

Work is the quantity of energy transferred from one system to another. Work is defined as the amount of job assigned or the amount of job actually done. Problem on work are based on the application of concept of ratio of time and speed. Work is always considered as a whole or one.

Above mentioned definition of work throws light on three important points.

Work = 1 ( as it is always measured as a whole)

Rate at which work is done = Efficiency of Work

Number of days required to do the work = No of Day and Hour

**FORMULAE**

1. Work from Days:

If A can do a piece of work in n days, then A’s 1 day’s work = (1/n).

**2. Days from Work:**

If A’s 1 day’s work = (1/n), then A can finish the work in n days.

**3. Ratio:**

A is thrice as good a workman as B, then:

Ratio of work done by A and B = 3 : 1.

Ratio of times taken by A and B to finish a work = 1 : 3.

4. If A is ‘x’ times as good a workman as B, then he will take (1/x)th of the time by B to do the same work.

5. A and B can do a piece of work in ‘a’ days and ‘b’ days respectively, then working

together, they will take [xy/x+y]th days to finish the work and in one day, they will finish [x+y/xy]th part of work.

6. Definition of Variation: The change in two different variables follow some definite rule. It said that the two variables vary directly or inversely. Its notation is X/Y = k, where k is called constant. This variation is called direct variation. XY = k. This variation is called inverse variation.

7. Some Pairs of Variables:

i. Number of workers and their wages. If the number of workers increases, their total wages increase. If the number of days reduced, there will be less work. If the number of days is increased, there will be more work. Therefore, here we have direct proportion or direct variation.

ii. Number workers and days required to do a certain work is an example of inverse variation. If more men are employed, they will require fewer days and if there are less number of workers, more days are required.

iii. There is an inverse proportion between the daily hours of a work and the days required. If the number of hours is increased, less number of days are required and if the number of hours is reduced, more days are required.

**8. Some Important Tips:**

More Men – Less Days and Conversely More Day – Less Men.

More Men – More Work and Conversely More Work – More Men.

More Days – More Work and Conversely More Work – More Days.

Number of days required to complete the given work = Total work/One day”s work.

Since the total work is assumed to be one (unit), the number of days required to complete the given work would be the reciprocal of one day”s work. Sometimes, the problems on time and work can be solved using the proportional rule ((man*days*hours)/work) in another situation.

9. If men are fixed, work is proportional to day (time). If work is fixed, then day, hour, efficiency is inversely proportional to men therefore,

(M1*D1*H1*E1/W1) = (M2*D2*H2*E2/W2)

**SOLVED EXAMPLES:-**

1. Worker A takes 8 hours to do a job. Worker B takes 10 hours to do the same Job. How long should it take both A and B, working together but independently, to do the same job?

sol:-

A’s 1 hour’s work = 1/8

B’s 1 hour’s work = 1/10

(A + B)’s 1 hour’s work = (1/8) +(1/10)=9/40

Both A and B will finish the work in 40/9 days.

2. A and B together can complete a piece of work in 4 days. If A alone can complete the same work in 12 days, in how many days can B alone complete that work?

sol:-

(A + B)’s 1 day’s work = (1/4).

A’s 1 day’s work = (1/12)

B’s 1 day’s work =((1/4)-(1/12))=(1/6)

Hence, B alone can complete the work in 6 days.

3. A can do a piece of work in 7 days of 9 hours each and B can do it in 6 days of 7 hours each. How long will they take to do it, working together 8 (2/5) hours a day?

A can complete the work in (7 x 9) = 63 hours

B can complete the work in (6 x 7) = 42 hours.

A’s 1 hour’s work = (1/63) and B’s 1 hour’s work =(1/42)

(A + B)’s 1 hour’s work =(1/63)+(1/42)=(5/126)

Both will finish the work in (126/5) hrs.

Number of days. of (42/5) hrs each =(126 x 5)/(5 x 42)=3 days

4. A and B can do a piece of work in 18 days; Band C can do it in 24 days A and C can do it in 36 days. In how many days will A, Band C finish it together and separately?

(A + B)’s 1 day’s work = (1/18) (B + C)’s 1 day’s work = (1/24)

and (A + C)’s 1 day’s work = (1/36)

Adding, we get:

2 (A + B + C)’s 1 day’s work =¬(1/18 + 1/24 + 1/36) =9/72 =1/8

(A +B + C)’s 1 day’s work =1/16

Thus, A, Band C together can finish the work in 16 days.

Now, A’s 1 day’s work = [(A + B + C)’s 1 day’s work] – [(B + C)’s 1 day work:

=(1/16 – 1/24)= 1/48

A alone can finish the work in 48 days.

Similarly, B’s 1 day’s work =(1/16 – 1/36)=5/144

B alone can finish the work in 144/5=28 4/5 days

And C’s 1 day work =(1/16-1/18)=1/144

Hence C alone can finish the work in 144 days.

5. A, B and C are employed to do a piece of work for Rs. 529. A and C are supposed to finish 19/23 of the work together. How much shall be paid to B ?

Work done by B = 1 – (19/23) = 4/23

(A + C) : B = (19/23) : (4/23) = 19:4

Bs share = RS.529*(4/23) = 92

6. A is twice as good a workman as B and together they finish a piece in 18 days. In how many days will A alone finish the work?

(A’s 1 day’s work):(B’s 1 days work) = 2 : 1.

(A + B)’s 1 day’s work = 1/18

Divide 1/18 in the ratio 2 : 1.

:. A’s 1 day’s work =(1/18*2/3)=1/27

Hence, A alone can finish the work in 27 days.

7. A can do a certain job in 12 days. B is 60% more efficient than A. How many

days does B alone take to do the same job?

Ratio of times taken by A and B = 160 : 100 = 8 : 5.

Suppose B alone takes x days to do the job.

Then, 8 : 5 :: 12 : x = 8x = 5 x 12 =x = 7 1/2 days.

8. A can do a piece of work in 80 days. He works at it for 10 days B alone finishes the remaining work in 42 days. In how much time will A and B working together, finish the work?

Work done by A in 10 days =(1/80*10)=1/8

Remaining work = (1- 1/8) =7/ 8

Now,7/ 8 work is done by B in 42 days.

Whole work will be done by B in (42 x 8/7) = 48 days.

A’s 1 day’s work = 1/80 and B’s 1 day’s work = 1/48

(A+B)’s 1 day’s work = (1/80+1/48)=8/240=1/30

Hence, both will finish the work in 30 days.

9. A and B undertake to do a piece of work for Rs. 600. A alone can do it in 6 days while B alone can do it in 8 days. With the help of C, they finish it in 3 days. find the share of each.

C’s 1 day’s work = 1/3-(1/6+1/8)=1/24

A : B : C = Ratio of their 1 day’s work = 1/6:1/8:1/24= 4 : 3 : 1.

A’s share = Rs. (600 *4/8) = Rs.300, B’s share = Rs. (600 *3/8) = Rs. 225.

C’s share = Rs. [600 – (300 + 225) = Rs. 75.

10. A and B working separately can do a piece of work in 9 and 12 days respectively, If they work for a day alternately, A beginning, in how many days, the work will be completed?

(A + B)’s 2 days’ work =(1/9+1/12)=7/36

Work done in 5 pairs of days =(5*7/36)=35/36

Remaining work =(1-35/36)=1/36

On 11th day, it is A’s turn. 1/9 work is done by him in 1 day.

1/36 work is done by him in(9*1/36)=1/4 day

Total time taken = (10 + 1/4) days = 10 1/4days.

11. 45 men can complete a work in 16 days. Six days after they started working, 30 more men joined them. How many days will they now take to complete the remaining work?

(45 x 16) men can complete the work in 1 day.

1 man’s 1 day’s work = 1/720

45 men’s 6 days’ work =(1/16*6)=3/8

Remaining work =(1-3/8)=5/8

75 men’s 1 day’s work = 75/720=5/48

5/8 work is done by them in (5/48)/(5/8)=6 days.

12. 2 men and 3 boys can do a piece of work in 10 days while 3 men and 2 boys can do the same work in 8 days. In how many days can 2 men and 1 boy do the work?

Let 1 man’s 1 day’s work = x and 1 boy’s 1 day’s work = y.

Then, 2x+3y = 1/10 and 3x+2y = 1/8

Solving, we get: x = 7/200 and y = 1/100

(2 men + 1 boy)’s 1 day’s work = (2×7/200 + 1 x 1/100 )

So, 2 men and 1 boy together can finish the work in 25/2 =12 1/2 days

13. 12 men can complete a work in 18 days. Six days after they started working, 4 men joined them. How many days will all of them take to finish the remaining work?

(12 * 18) men can complete the work in 1 day.

1 mans 1 days work = 1/216

12 men’s 6 days work = (1/18)*6 = 1/3

remaining work = 1 – (1/3) = 2/3

16 mens 1 days work = 16/216 = 2/27

2/27 work is done by them in 1 day.

2/3 work is done by them in (27/2)*(2/3) = 9 days.

14. If 12 carpenters, working.6 hours a day can make 460 chairs in 24 days, how many chairs will 18 carpenters make in 36 days, each working 8hours a day ?

(Carpenters 12: 18) : (His/Day 6:8 ) : (Days 24 : 36) : : 460 : x

12 * 6 * 24*x =18 * 8 * 36 * 460 or x = 1380

15. Some persons can do a piece of work in 12 days. Two times the number of such persons will do half of that work in

(Mens 2x : x) : (work 1:1/2) :: 12 :y

2x * 1 * y = x * 1/2 * 12 or y = 3