15. TIME AND WORK

IMPORTANT FACTS AND FORMULAE

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

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

    3.  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.

SOLVED EXAMPLES

Ex. 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? (IGNOU, 2003)
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.

Ex. 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? (Bank P.O. 2003)

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.


Ex. 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 bours each. How long will they take to do it, working together 8 hours a day?

Sol. 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

Ex. 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?


Sol. (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.

Ex. 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?
Sol. (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.

Ex. 6. 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?
Sol. 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.

Ex. 7. 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?
Sol. 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.

Ex. 8. 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.

Sol :C's 1 day's work = 1/3-(1/6+1/8)=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.

Ex.  9. 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.
Ex 10 .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

Now,5 work is done by them in 1 day.
       48

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


Ex:11.   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?
Soln: Let 1 man’s 1 day’s work = x and 1 boy’s 1 day’s work = y.
Then, 2x+3y = 1 and 3x+2y = 1
10 8
Solving,we get:  x = 7  and y = 1
200       100

(2 men + 1 boy)’s 1 day’s work  = (2 x   7 + 1 x 1  ) = 16 =  2
 200   100    200   25
So, 2 men and 1 boy together can finish the work in 25 =12 1     days
2         2