### TIME and WORK

We are aware that this chapter is important for competitive exam from work and time chapter  question are asked in various exam like ssc, upsc, banking etc. that work and time are time consuming chapter.

If a person A can do a piece of work in 'a' days, then working at the same uniform speed A will do 1/a  fraction of the work in one day.
If B can do a work in 'b' days, then work done by him in 1 day is 1/b.
Then, in one day, if A and B work together, then work done by them in 1 day, = (1/a + 1/b);
For example,
 Days taken to complete 'a' work 15 10 1/3 Fraction of work done in a day 1/15th 1/10 3

Relationship between Men and Work
More men         ------- can do ------->             More work
Less men         ------- can do ------->             Less work

Relationship between Work and Time
More work        -------- takes------>               More Time
Less work         -------- takes------>               Less Time

Relationship between Men and Time
More men        ------- can do in ------->          Less Time
Less men          ------- can do in ------->          More Time

·      If A can do a work in ‘x’ days and B can do the same work in ‘y’ days, then the number of days required to complete the work if A and B work together is

(X x Y) / (X + Y)

·      If 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

·      If Mpersons can do W1 work in D1 days and M2 persons can do W2 work in D2 days, then
M1D1 / W1 = M2D2 / W2

·       If Mpersons can do W1 work in D1 days for h1 hours and M2 persons can do W2 work in D2 days for h2 hours, then

(M1D1h1) / W1 = (M2D2h2) / W2

Note:  If works are same, then M1D1h1 = M2D2h2

Type 1: On Basic Concept

1.    1.A piece of work can be done by Ram and Shyam and Hari in 15 days, by Shyam and Hari in 15 days and by Hari and Ram in 20 days. Ram lone will complete the work in
(CGL-2013)
(a) 30 days                            (b) 32 days
(c)36 days                             (d) 42 days
Solution: (a) (Ram’s + Shyam’s) 1 day’s work = 1/12
(Shyam’s + Hari’s) 1 day’s work = 1/15
(Hari’s + Ram’s) 1 day’s work = 1/20
2 (Ram’s + Shyam’s + Hari’s) 1 day’s work = 1/12+1/15+1/20=(5+4+3)/60=1/5
Ram’s 1 day’s work = 1/10-1/15=(3-2)/30=1/30
Ram alone will do the work in 30 days.

2.    A and B can separately do a piece of work in 6 days and 12 days respectively. How long will they together take to do the work?
(CGL-2012)
(a) 9 days                               (b) 18 days
(c)6 days                                (d) 4 days
Solution: (d) (A+B)’s 1 day’s work  = 1/6+1/12=(2+1)/2=1/4
A and B together will complete the work in 4 days.

3.    A can do a piece of work in 10 days, B can do it in 15 days. In how many days both of these can together complete the job?
(a) 9 days                               (b) 18 days
(c)6 days                                (d) 4 days
Solution:: A's per day work = 1/10; B's per day work = 1/15
So per day work of these people together = 1/10+1/15 (3+2)/305/301/6
So A and B together 1/6 th of the total work perday. So total work can be completed in 6 days.

Alternate Method:
Ask yourself that why the same work done by A in 10 days will be done by B in 15 days?
The answer is due to their efficiencies.  We know that number of days are inversely proportional to their efficiencies. So we can solve this problem by converting days into efficiencies.

Step 1: Estimate the total work.  This can be done by taking LCM of the days given. LCM (10,15) = 30.  Assume there is 30 meters length of wall to be constructed.

Step 2: Calculate the efficiencies of A and B: This can be done by dividing total work by their days Efficiency = (Total work)/days
So we get A's per day efficiency = 30 meters/10days=3mts
We get B's per day efficiency = 30meters/15days=2mts
So total work done by A and B together = 3 + 2 = 5 meters
Total work of 30 meters can be done in 30/5 = 6 Days

Type II – They Work Together For n days and A/B leaves.

1.    A and B can do a job in 6 and 12 days respectively. They began the work together but A leaves after 3 days. Then the total number of days needed for the completion of the work is: (CGL-2000)
(a) 4                            (b) 5
(c)6                             (d) 9
Solution: (c) A’s one day’s work = 1/6
B’s one day’s work = 1/12
(A+B)’s one day’s work  = 1/6+1/12=(2+1)/12=1/4
(A+B)’s three day’s work = ¾
Remaining work = 1- 3/4=1/4
Required number of days = 1/4×12/1+3=3+3 days
= 6 days.

2.    A and B can together finish a work in 30 days. They worked together for 20 days and then B left. After another 20 days. A finished the remaining work. In how many days A alone can finish the job?
(CGL-2003)
(a) 50                          (b) 60
(c) 48                          (d) 54
Solution::  (b) (A+B)’s 1 day’s work = 1/30
(A+B)’s 20 day’s work = 20/30=2/3
Remaining work = 1- 2/3=1/3
Now, 1/3 part of work is done by A in 20 days.
Whole work will be done by A alone in 20 x 3 = 60 days.

Type-III Based on A Man, B Woman

1.    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 number of days is (CGL-2012)
(a) 17 1/2                    (b) 5 5/11
(c) 8                            (d) 22
Solution (a) 12 (3 men + 4 boys) = 10 (4 men + 4 boys)
36 men + 48 boys = 40 men + 30 boys
4 men = 18 boys , 2 men = 9 boys
4 men + 3 boys  = 21 boys who do the work in 10 days
and, 2 men + 3 boys = 12 boys
M1D1 = M2D2
D2=(21×10)/12=35/2=171/2 days

2.    If 3 men or 6 women can do a piece of wok in 16 days, in how many days can 12 men and 8 women do the same piece of work?
(CGL-2000)
(a) 4 days                               (b) 5 days
(c)3 days                                (d) 2 days
Solution:(c) 3 m = 6w
1m = 2w
12m + 8w = (12 x 2w) + 8w = 32w
6 women can do the work in 16 days.
32 women can do the work in (16×6)/32=3 days.

Type-IV Based on Fraction

1.    A can cultivate 2/5th of a land in 6 days and B can cultivate 1/3rd of the same land in 10 days working together A and B can cultivate 4/5th of the land in: (CGL-2002)
(a) 4 days                               (b) 5 days
(c)8 days                                (d) 10 days
Solution: (c) The part of field cultivated by A in 1 day = 2/(5×6)=1/15
The part of field cultivated by B in 1 day = 1/(3×10)=1/30
The part of field cultivated by A and B together = 1/15+1/30=3/30=1/10
4/5 part of field cultivated by A and B together in  = (4/5)/(1/10) days = (4×10)/5=8 days

2.    A does 4/3 of a piece of work in 20 days: He then calls in B and they finish the remaining work in 3 days. How long B alone will take to do whole work? (CGL-2002)
(a) 37 1/2 days                      (b) 37 days
(c)40 days                             (d) 23 days
(a) A can do the whole work in (20×5)/4=25 days
Remaining work = 1 -4/5=1/5
(A+B)’s 1 day’s work = 1/15 and A’s 1 day’s work = 1/25
B’s 1 day’s work  = 1/15-1/25=(5-3)/75=2/75
B can finish the work in 75/2 days i.e., 37 1/2 days

TypeV- Efficient Worker

1.    X is 3 times as fast as Y and is able to complete the work in 40 days less than Y. Then the time in which they can complete the work together is               (CGL – 2011)
(a) 15 days                (b) 10 days
(c) 7 1/2 days            (d) 5 days
Solution: (a) If X completes a work in x days, Y will do the same in 3x days.
3x- x = 40
x = 20
Y will finish the work in 60 days.
(X +Y)’s 1 days work  = 1/20+1/60=(3+1)/60=1/15
Both together will complete the work in 15 days.

2.    Kamal can do a work in 15 days. Bimal is 50 per cent more efficient than Kamal in doing the work. In how many days will Bimal do that wok?                       (CGL-2002)
(a) 14 days                (b) 12 days
(c)10 days      (d) 10 1/2 days
Solution: (c) Efficiency and time taken are inversely proportional Bimal : Kamal = 150 : 100 (work) 200 : 150 (Time) = 2 : 3
3 units = 15 days
2 units ----- 15/3×2 days = 10 days

Type VI Based on  Formula

1.    If p men working p hours per day for p days produce p units of work, then the units of work produced by n men working n hours a day for n days is              (CGL-2008)
(a) p^2/n^2                             (b) p^3/n^2
(c) n^2/p^2                             (d) n^3/p^2
Solution: (d)  P men working P hours /day for P days produce P units of work.
1 man working 1 hour /day for 1 day produce
P/P^3 =1/P^2  units of work  n men working n hours a day for n day’s produce n^3/P^2  units of work

2.    One man, 3 women and 4 boys can do a piece of work in 96 hours, 2 men and 8 boys can do it in 80 hours, 2 men and 3 women can do it in 120 hours. 5 men and 12 boys can do it in                                                                                                      (CGL-2012)
(a) 39 1/11 hours                  (b) 42 7/11 hours
(c)43 7/11 hours                   (d) 44 hours
Solution: (c) 1 hour’s work of 1 man and 4 boys = 1/160
1 hour’s work of 1 man and 3 women = 1/96
1 hour’s work of 3 women  = 1/90-1/160=(10-6)/960=1/240
1 hour’s work of 2 men = 1/120-1/240=1/240
1 hour’s work of 4 boys  =1/160-1/480
= (3-1)/480=1/240
2 men = 3 women = 4 boys
2 men + 8 boys = 12 boys
5men + 12 boys = 22 boys
By M1D1 = M2D2
D2 = (12×80)/22
= 480/11=437/11 hours

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