5Rule 1

Theorem: If M, persons can do Wj works in D, days and

M 2 persons call do Wz works in D2 days then we have f!.very general formula ill the relationship of

MjDjW2 = M2D2Wj.

Illustrative ExamplesEx.y 16men can do a piece of work in 10 days. How many

'-../ men are needed to complete the work in 40 days?Soln: Detail Method: To do a work in 10 days, 16 men are

needed. or, to do the work in I day, 16 x 10 men are

ySoln:

16xl0 =4needed. So to do the work in 40 days, 40

men are needed.

QuickerMethod: MjDjW2:= M2D2Wj

M, =16,Dj =IO;Wj =1 and

M2 = 7, D2 = 40, W2 = IThus, from /IlfJ~.jWl = M1D2Wj

16 x I 0 = M 2 X 4016xl0 .

or, M: =---=4 men... 40

40 men can cut 60 trees in 8 hours. If 8 men leave- thejob, how many trees will be cut in 12 hours?Detail Method:40 men - working 8 hrs - cut 60 trees

60or, 1 men - working 1 hr - cuts 40 ;-i trees

60x32x12Thus, 32 men - working 12 hrs - cut 40 x 8 = 72 .trees.Quicker Method:

At] = 40, D, = 8 (as days and hrs both denote time)

Time and Work

Wj = 60 (cutting of trees is taken as-work)

M2 =40-8=~2, D2 =.12,W2 =?Putting the values in the formula, .

MjDjW2 = M2D2WjWe have, 40 x 8 x W2 =32 x 12 x 60

32x12x60or W2 = = 72 trees, 40x8 .

Ex.3~can do a piece of work in 5 days. How many days~ will he take to complete 3 works of the same type?

Solo: Quicker Method:

MjDjW2 := M2D2WjAs 'A' is the only person to do the work in both the

cases, so M, := M2 := 1 (Useless to carry it)

D, := 5 days, WI = 1, D2 =? and W2 = 3Putting the values in the formula we have,S x 3 = D2 X lor, D2 = 15 days.

Exercise ..1. 8 men can do a piece of work in 5 days. How many men

are needed to complete the work in 10 days?a) 8 men b) 4 men c) 2 men d) None ofthese

2. 15 men can do a piece of work in 6 days. How many menare needed to complete the work in 3 days?a) 30 men b) 25 men c)35 men d) 40 men

3. 20 men can cut 30 trees in 4 hours. If 4 men leave thejob,how many trees will be cut in 6 hours?a) 30 trees b),36 trees c)40trees d)Noneofthese

4. 10men can cut 15 trees in 2 hours. If 2 men leave the job,how many trees will be cut in 3 hours?a) 15 trees b) 20 trees c) 16 trees d) J 8 trees

5. A can do a piece' of work in'o days. How many days witlhe take to complete 2 works of the same type?'a) 12 days b) 10 days c) 6 days d) 3 days

Answersl.c 2.a ,3 ..b 4.d S.a

'Rule 2Theorem: If MJ persons call (/0 WJ works in Dr days work-

ing :r; hours a day and M 2 persons call do W;, works illD2 days working T2 hours a day then we have a very gen-

eralformula in the relationship of MJDJI;W2 = M2D2T2W1

IIJustrative ExampleEx: /sinen can prepare 10 toys in ,6 days working 6 hourso a day. Then In how many days can 12 men prepare 16

toys working 8 hrs a day?Soln: By using the above theorem

5x6x6xJ6 = 12x Dl x8xlO

D ;;;:5x6x6x16 ~3 d"~2 12x8x IO ay

Note: Number of toys is considered as work in the aboveexample.

Exercise1. The work done by a ..,,'ohalfin 8 hours is equal to the

work done by a man in 6 hours and by.a boy in 12 hours.Ifworking 6 hours per day 9 men can complete a work in6 days then in how many days can 12 'men..li women

, and 12 boys togetherfinishthe same work'work1ng 8hours per day?

I 2 Ja) 12days b) 3)- days c) 3 days d) 1

2days

(BSRB Patna PO-200J)2. to men can prepare'20 toys in 3 days working J2hours a

day. Then in how many days can 24 men prepare 32 toysworking 4 hrs a day?a) 2 days b) 3 days c) 4 days d}6 days

3. 20 men can prepare 4.0 toys ii124 days working 18 hoursa day. Then in how many days can 36 men prepare-txtoys working 16 hrsa day? 'a) 16 days b) 12 days c)2ldays, d) 18 days

AnswersI.d; Hint:8WQm~n,=QM~l.1=.J2Boys,

12M + 12W+ J2B~ 12M,+9M+6M=2-7M ,Now, applying the above formula, we have

9x6x6=27x8x D29x6x6' J

. Do = --- ,= 1'- days.. - 27 x8, 2

, ,

2. a 3.d

,Rule 3Theorem: 1f A and Bean do a piece of work ill x days, Band C iny days, C and A in z days.then (A + B + C) working

[2xyz ]

together will do tile same work ill - + + ' days.xy yz xz

(, 2xyz )

Let xy+yz+xz be=r+then

'A' alone will do the same work in (yr ),

-- (Jays ory-r

(2xyz )

xy + yz - zx days,

'B' alone will do the same work ill (zr ) days or,z-r

('2XYZ) ,yz + zx - xy days and

'C' a/one will do the same work in .(-~) days or,X -r '

Cz :;;~ yz) days.Illustrative ExampleEx: nd B can do a piece of work in 12 days, Band C in

15 days, C and A in 20 days. How long would each, take separately to do the same work?

Soln: Using the above theorem;

2x12x15>;20 ''r = ---,-------'----- =10 days,

12x 15 + 12x20+ IS x20

, JOxJ5Now, A can do the work in 15 -10 = 30 days,

IOx20B can do the work in 20 _ I0 = 20 days.

. IOx12 _ 60,Ccandotheworkm 12,-10 - days.

ExerciseI. A and B can finish a piece of work in 30 days, Band C in

40 days while C and A in 60 days. How long will theytake to finish it together?

2 2a) 26- days b) 16-da)'s c)25 days d)24 days

, 3 ' 32, AandBcandoapieceofworkin 10 days,B andC in 15

days and, C and A in 20 days. They all work at it for 6days, and then A leaves, and Band C go on together for4 days more, If B then leaves, how long will C take tocomplete the work?a)20day,s b) 25 days c) 10 days d) 15 days

Time and Work 3(> '[

A and B can do a piece of work in 6 days, A and.C in Illustrative Example/1

I . . Ex.: A can do a piece of work in 5 days, and B can do it in52" days, Band C In 4 days. In what t!IJ1~_,c9uldeach do. ~,./" 6 days. How long wi II they take if both work together?'t? ~I. ..]

4 16 t 9 16 4 '19 Soln: Detail Method: 'A' can do '5 work in 1 day.a) 20- 8- 7- b)8- 20- 7-

13' 31' 35 31 ' 13' '35

J.

7~ 8~ 20~c) 35' 31' 13 d) None of these

4.1 '"

A and B can mow a field in 32' days, A and C in 4 days,

Band C in 5 days. In what time could they mow it, allworking together?

3 75a) 104

2 74b) 103

3 74c) 103

AnswersI, a2. c: Hint: A, Band C together can do the work in

2xlOx15x20 120------- ='- days

IOxI5+20xI0+15x20 30

13work done by all in 6 days = 20

4work done by Band C in 4 days = 15

(13 4) J

Remaining work = 1- 20 + 15 = 12 ' which. is to bedone by C.Now, from the question,

120 x 10 .C alone can do the whole work in _1_3 = 120 days

12010---

13[See Rule-6)

1 . 120.'. 12 of the work is done by C in 12 = 10 days.) 1

Hint: Here x = 6,.y = 4 and z = 2" Now apply thegiven rule.

4.b

Rule 4Theurern: [fA call do a piece of work ill x days and B cand, r' in ; days then A and B working together will do the

sante W,IT/. ill (~) days.'X+ y

. I'B' can do '6 work in 1 day.

-Thus 'A' and 'B' can do (~+~) work in I day.

:. 'A' and 'B' can do the work in T'T days--+-5 6

30 8= -=2- days.

r , 11 J JQuicker Method: Applying the above theorem,

5x6A + B can do the work in 5 + 6 days

30 8= -=2- days.

11 II

Exercisel. 10.men can complete a piece of work in 15 days and 15

women and complete the same work in 12 days. Ifall the10 men and 15 women work together, in how many dayswill the work get completed?

2 1c) 63 d) 63(SBI Associates PO-1999)

A can do a piece of work in 20 days and B can do it in 30days. How long would they take to do it working to-gether? .. a) 12 days.' b) ) 0 days c) 15 days d) 16 days

3. .A can do a piece of work in 6 days. B takes 8 days. Ctakes as long as A and B would take working together.How long will it-take Band C to complete the work to-gether?

\ .... I"; ?

a) 2sday b) 2i days c) 6 days

a)62

b) 7-3

2.

2d) 4'-'days

5

74. A does TO of a piece of work in 15 days. He does the

remainder with the assistance of B in 4 days. In whattime could A and B together do it?

1a) 133 days b) 12 days

2c) 12"3 days d) None of these ,

5. A can do a piece of work in 16 days, B in t () days. A andB work at it together for 6 days and then e finishes it in3 days, in how many days could e have done it alone?a) 40 days b) 80 days c) 90 days d) 120 days

6. A can do a piece of work in 4 hours, Band e can do it in3 hours, A and e can do it in 2 hours. How long would Balone take to do it?a) 14 hours b) 12 hours c) 10 hours d) 16 hours

7. A can do a piece of work in 30 days while B can do it in40 days. A and B working together can do it in

3 1 1a) 70 days b) 42'4 days c) 27"7 days d) 17"7 days

(Railways 1989)8. A can do (1/3) ofa work in 5 days and B can do (2/5) of

the work in 10 days. In how many days both A and Btogether can do the work?

4c) 8S-days d) 10 days

(Railways 1991)

AnswersI. c; Hint: x = IS days, y = 12 days, Now apply the above

rule.2.a

(6X8 24)

3. b; Hint: e completes the work in 6 + 8 =-;;-days.. Band e together complete the work 'in

(

24 JTx8 12 2--=-=2- days24 +8 5 5 . .7

( 7 3) .4. a; Hint: ,1- 10 = TO work is done by A and B in 4. . ,~ '.' Vf (.' . ; ,

days

4xl0.'. The whole work is done by Aand B in -3-

40 I= '3 = 13"3 days.

5. d; Hint: A and B together can do the whole work in

16xlO 8016+ 10 = 13days.

13 6 39:. In 6 days A and B together can do 80 x = 40

PRACTICE BOOK ON QUICKER MATHS

work.

Now, remaining work (I - !~= 410) is done by e in 3day's:. The whole work is done by e in 40 x 3 = 120 days.

6. b;. Hint: A, Band e together can do a piece of work in

(4X3 12)4+3 =7 hours.

B alone take to complete a piece of work

[

12 x2 J7 -12

= 2 _ 1; - hours.

[See Rule-6).7.d'8. b; Hint: A can do the whole work in (5 x 3 = 15) days.

Bean db.~e whole work in Co; 5 = 25) days.25xl5

A and B together can do the work in 25 + 15

75 3= 8=9'8 days.

Rule'STheorem: If A, B andC can do a work in x, y and z daysrespectively then all of them working together call finish

the work in ( +xyz+ ) days.xy yz xz

Illustrati xampleEx.: can do a piece of work in 5 days, and B can do it in

6day's. If e, who tan do the work in 12 days, joinsthem, how long will they take to complete the work?

Soln: By the tbeorem:: A, B ahci Gcan do the work in

5x6x 12 360 2-=2- days.

5x6+6xI2+5xI2 162 9

Exercise .1. A can do a piece of work in 5 days, B in 4 days and A, B-. and e together in 2 days. In what time would C do it

alone?a)25d~y's, l?)12days c)15.days d)20days

2. A takes half as long to do a piece of work as B takes, andif'C does it in tlie same time as A and B together, and ifallthree working together would take 7 days, how longwould each take separately?

Time and Work

a) 21 days, 42 days, 14 days40

b) 20 days, 40 days, """3 days45

c) 15 days, 45 days, "4 daysd) None of these

3. Five men can do a piece of work in 2 hours, which 7women could do in 3 hours, or 9 children in 4 hours ..Howlong would 1 man, Iwoman and 1 child together take todo the work?

I1260 1270 1221

a)m b) 23l c) 260 d) None of these4. A takes twice as much time as B and thrice as much time

as C to finish a piece of work, working together they canfinish the work in 2 days, find the time each will take tofinish the work.a) 12,6,4 b) 18,9,6c)24, 12,8 d) None of these

5. If A does a piece of work in 4 days, which B can do in 5,and C can do in 6, in what time will they do it, all workingtogether?

a) 2 days50

b) 27 days

6.

60c) 37 days d) None of theseA. Band C can do a piece of work in 6, .12 and 24 daysrespectively. They altogether will complete the work in

~"..>a) ">'1 days 7 4 5b) 24 days c) 45 days d) 24 days

(Clerical Grade Exam, 1991)

AnswersI. d: Hint: Applying the given rule, we have

'. _ 5x4xz -2the required answer - 5 4 5 2 4 -x: + x + x zor,20z=40+ 18z '!,or, 2z= 40 :. z= 20 days.

2. a; Hint: Let A takes x days to complete the work.:. B takes 2x daysFrom the question, C takes to complete the work =

2xxx 2---=-x days2x+x 3

363

xor - = 7 . x = 21 days. ' 3 . . ..Herice, A completes the work in 21 days, B in

(21 x 2 =42) days andC ill (%X21:-::14) days.

3. a; Hint: 1men can do a piece of work in (2 x 5 == 10) days1 woman can do the same work in (7 x 3 = 21) days.I child can do the same work in (9 x 4 = 36) days.Now, applying the given rule, we have

, IOx21x36the required answer = lOx 21 + 21 x36+ IOx36

7560 1260= 1326 =m days.

4, a; Hint: Let A takes x days to complete the work.

x xB takes 2' and C takes '3 days.Now, applying the given rule, we have

x xxx-x-

___ -=2'---3::C..... __ = 2x x x xxx-+-x -+xx-2 2 3 3

:. x= 12 days.

:. A takes 12 days, B takes ( = 6) days and Ctakes (I: = 4) days 'to' complete the work.

5. c 6. a

Rule 6Theorem: If A and B together can do a piece of work ill xdays and A alone can do it in y days, then B alone can do

xythe work in -- days.y-x

ow, applying the given rule, we have

/ lIIustr . e Example(See Rule 4) Ex: A and B together can do a piece of work in 6 days and

A alone can do it in 9 days. In how many days can Balone do it?2

xx 2xx-x3 =7

. 2 2xxx2x+ 2xx--x+ xx--

3 3

1S?ln:- Detail Method: A and B can do '6 of the work in I

'Time and Work

a) 21 days, 42 days, 14 days40

b) 20 days, 40 days, 3days45

c) IS days, 4S days, '4 daysd) None of these

3. Five men can do a piece of work in 2 hours, which 7women could do in 3 hours, or 9 children in 4 hours. Howlong would 1man, I woman and 1 child together take todo the work?

f1260 1270 ) 1221

a)m b) m c 260 d) None of these4. A takes twice as much time as B and thrice as much time

as C to finish a piece of work, working together they canfinish the work in 2 days, find the time each will take tofinish the work.a) 12,6,4 b) 18,9,6c)24,12,8 d) None of these

5. If A does a piece of work in 4 days, which B can do in 5,and C can do in 6, in what time will they do it, all workingtogether?

a) 2 days50

b) 27 days

6.

60c) 37 days d) None of theseA, Band C can do a piece of work in 6, .12and 24 daysrespectively. They altogether will complete the work in

,,3 7 4 Sa) "'7 days b) 24 days c) 45 days d) 24 days

(Clerical Grade Exam, 1991)

Answers!. d; Hint: Applying the given rule, we have

Sx4xzthe required answer = -S-4-5 2 4 = 2x + x + xzor,20z=40+ 18z 'I;or, 2z = 40 ., z = 20 days,

2. a; Hint: Let A takes x days to complete the work.:. B takes 2x daysFrom the question, C takes to complete the work =2xxx 2---=-x days2x+x 3

363

4/3xl -7or, 4X2 -

xor, 3~7 :,x =21 daysHerice, A completes the work in 21 days, B in

(21 x 2 = 42) days and C in (~x 21 = 14) days.3. a; Hint: 1.men can do a piece of work in (2 x 5 "= 10) days

.i 1 woman can do-the same work in (7 x 3 = 21) days.1 child can do the same work in (9 x 4 = 36) days.Now, applying the given rule, we have

. lOx 21 x36the required answer = IOx21 +2Ix36+IOx36

7560 1260= 1326 = ill days.

4, a; Hint: Let A takes x days to complete the work.

x xB takes "2 and C takes 3' days.Now, applying the given rule, we have

x xxx-x-

2 3 = 2x x x xxx-+-x-+xx-2 2 3 3

:. x= 12 days.

.'. A takes 12 days. B takes (I; = 6) days and C

takes (I: =4) days to complete the work.5. c 6. a

Rule 6Theorem: If A and B together can do a piece of work in xdays and A alone can do it in y days, then B alone can do

xythe work in --- days.y-x

ow, applying the given rule, we have

/ llIustr 'e Example(See Rule 4) Ex: A and B together can do a piece of work in 6 days and

A alone can do it in 9 days. In how many days can Balone do it?2

xx2xx-x____ . 3 = 7

2 2xxx 2x + 2xx --x + x x ---

3 3

. 1Soln:" Detail Method: A and B can do "6 of the work in I

364

day.

1A alone can do 9" of the work in 1 day.

, (I 11 1.', B alone can d~ , 6 - 9) = -)-8 ofthe work i~,~day.:. B alone call do the whoie work in 18 days.Quicker Method: By the theorem:

, 6x9 54B alone can do the whole work in -- == - = )8

9-6 3days.

Exercise1. A, Band C can doa piece of work in 6, )2 and 24 days

respectively. In what time will they altogether do it?

4 . 2 3 3a) 37 days b) 37

days c) 37days .d) 27 days

2. A and B workingtogether could mow a field in 28 daysand with the help of C they could have mowed it in 21days. How long would C take by himself?a) 86 days b) 48 daysc) 84 days d) None of these

3. B can do a piece of work in 6 hours, Band C can do it in2 '

.4hours and A, Band C in 2-:;-,hours. In how many hours-'

can A and B do it?

",,-'a) -'7 hours

2c) 3- hours

7

,,5b) -'-hours7

d) None of these

4.4

A and B together can do a piece of work in 45' days, B

and C together can do it in 8 days, and A, Band Ctogether in 4 days. How long would A and C'togethertake to do it? In what time would B do it alone?a) 8 days, 12 days b) 4 days, 8 daysc) 6 days, 12 days d) 8 days, 16 days.

2A does -; .of a piece of work in 9 days, he then cal~s in B,

and they finish the work in 6 days. How long would Btake todo the whole work by himself?a) J 8 days b) 16 days cy'12 days d) 2 J days'A and B together can do a piece of work in 6 days, Balone could doit in 16 days. If B stops after 3 days, howlong afterwards will A have finished the work?

'4 4 4, I,a) 75'days b) 55'days c) 4~5days d) 45'days

5.

6.

PRACTICE BOOK ON QUICKER MATHS

7. A and B can reap a field in 30 days, working together.After. I I days, however, B is called off and A finishes itby himself in 38 days more. In what time could eachalone do the whole?a) 60 days each b) 15 days,30 daysc) 20 days, 60 days d) None of these

8. A, Band C together can do a piece of work in 6 days,which 13 alone can do in 16 days and Band C togethercan' do in J 0' days, in how many days can A and B to-gether do it?

240 242 241a) -_. b) .- c) --- d) None of these

31 31 3 I9. A and B together can do a piece of work in 8 days. 13

alone can do it in J 2 days, supposing B alone works at itfor 4 days: in how many more days could A alone finishit? 'a) 18 days b) 24 days c) 16 days d) 20 days

10. A and B can together do a piece of work in 15 days. Balone can do it in 20 days. A alone can do it in:a) 30 days b) 40 days c) 45 days 'd) 60 days

(Railways 1991)II. A and B finish a job in 12 days while A, Band C can

finish it in 8 days. C alone will finish thejob in:a) 20 days b) 14 days c) 24 days d) 16 days

(Hotel Management t 991)

Answersl.c

, 2. c;28x2J

Hint: Required answer = 28 _ 21 = 84 days.

Hint: Applying the given rule, we have

(6X4 )

C can do a the whole work in 6 _ 4 = J 2 days.Now from the question,

3. a;

8.' ' " 12 x - 96 24 3required answer = __ 3 = - = - = 3- days.

, 12-~ 28 7 73

4.c;

24----x 4Hint: C alone can do the work in _5__ = 24 days..,' , 24

--45

A alone can do the work in (:: ~ = 8) days.Now applying the Rule-4, we haveA and C together can do the work in

(24X8 _ 6)24+8 - days

366

Quicker Method:Number of days taken by B, t ,

== (Number of days taken by A + B + C) x (3 + I)== 10 (3 + 1) == 40 days

Similarly,Number of days taken by C == 10 (2 + I) == 30 days

I

days,

Exercise :1. A, B and Ctogether can finish a piece of work in 12 days,

A and C together work twice as much as B, A and B'together work thrice as much as C. In what time couldeach do it separately?

4 ,a) 28- 4248'5' ,

, 4b) 28-,36,48

5,

4c) 28,36-,48

5d) None of these

2, To do a certain workB would take 4 times as long as Aand C together and C 3 times as long as A and B to-gether, The three men together complete the work in 5days, How long would take Band C to complete thework?

I I 2 3a) 911 days b) 119 daysc) 263 daysd) 285days,

3. To do a certain work B would take 2 times as long as Aand C together and C 3 times as long as A and B to-gether, The three men together complete the work in 16days. How long would take Band C to complete thework?

2a) 27 -::; days b) 27 days

3 4c) 27- days d) 27 - days7 7

Answers ,1, b; Hint: B cOl~pletes the work if) U2(:? + I) == 36] days

andC completes the same work in [12(3 + 1) == 48] daysBand C together complete the work in

(36X48 144)36+48 =-7 days,

Now, from the question,A takes to complete the same work

14412x--- 144 4'--_-'-7_~ _' = 28- days,14~_12 5 57

PRACTICE BOOK ON QUICKER MATHS

2, b; Hint: B alone can do the work in (4 + 1) x 5 = 25 daysC alone can do the work in (3 + 1) x 5 == 20 days ,

25 x 20 IBand C tozether can do the work in ---- == 11,.'" 25 +,20

Time and Work

434x43 =-::.;-XD2

.J

4x43x3or D = ---=12 days

'2 43 .Quicker Method:

Required n~mber of days = [_7__+~]43x3 43x4

43x3x4 -127 x 4 + 5 x 3 - days

Note:.I. The above formula is very easy to remember.

If we divide the question in two parts and call the firstpart as OR-part and the second part as AND-p~rt then

7 Number of men in AND!- part43 x 3 Number of days x Number of men in 'OR - part

Similarly, you can look for the second part in denomina-tor.

(. 43X3X4)

2. Second step of the quicker method re 7 x 4 +5 x 3 is

the form .of'formula given in the Rule 8. ,

ExerciseI. If3 men or 5 women can reap a field in 43 days, how long

will 5 men. and 6 women take to reap it?a) 15 days b)25 days c) 18 days d) 12 days

) If2 men Of' 4 women can reap a field in 44 days, how long

'11~ d . 3,VI ., men an 5 women take to reap '4 th ofthe field?a) 10 Jays b) 8 days c)12days d) None of these

3. If6 men or 10 women can reap a field-inSo.days, howlong wilt 10 men and 12 women take to reap it?a) 30 days b) 35 days c) 25 days d) 40 days

4. If4 men or 6 boys can finish a piece of work in 20 days,In how. many days can 6 men and I I boys fmish it?a) 8 days b) 6 days c~:7 days d) 9 days

- (LICExam, 1991)5. 10 men can finish a piece of work in 10 days whereas it

takes 12 women to finish it in 10 days. If 15 men and 6'IWll10n undertake to complete the work how many dayswill they take to complete it?a)11 b)5 c)4 d)2

(Bank PO Exam, 1991)

AnswersI a 2. c 3. a 4. b5.b; Hint: 10menand 12 women can finish apiece of work

in the same no. of days ie 10 days. Hence we can say

367

that 10 men or 12 women finish the work in 10 days.Now we can apply the given formula .

Rule 9Theorem: If G] men and b] boys call do apiece of work in

X days and G2 men (IIUI b2 boys can do it in y days, thentire following relationship is obtained:

1man = [Yb2 - xbl ]bOYS. xaj - ya2 .

Illust tive ExampleEx: If 12 men and 16 boys can do a piece of work in 5 days

and 13 men and 24 boys can do it in 4 days, how longwill 7 men and 10 boys take to do it?Detail Method:12 men and 16 boys can do the work in 5 days (1)13men and 24 boys can do the work in 4 days (2)Now it is easy to see that if the no. of workers bemultiplied by any number, the time must be dividedby the same number (derived from: more workers lesstime). Hence multiplying the no. of workers in (1) and(2) by 5 and 4 respectively, we get

5 .. 5 (12 men + lti.boys) can do the work in '5 = 1 day

801n:

Note:

,:' i 44 (13 men + 24 boys) can do the work in - = I day

4or, 5 (12 m+ 16 b) = 4( 13 m+ 24 b)or, 60 m+ 80 b = 52 ll1 + 96 b ". (*)or, 60 m- 52 m== 96 b - 80 b01';8 m= 16 b.'. 1 man= 2 boys.Thus, 12 men + 16 boys = 24 boys + 16 boys = 40 _boys .

and 7 men + 10 boys = 14 boys + 10 boys = 24 boysThe question now becomes:"If 40 boys-can do a piece of work in 5 days how longwill 24 boys take to do it?"Now, by "basic formula", we have40 x 5 = 24 X D2 ..... (*) (*)__ 40x5_g2.

01, D~- ~4 - j days.During practice session (*) should be your first stepto be written down. Further calculations should bedone mentally. Once you get that 1 man = 2 boys,your next step should be (*) (*). This way you canget the result within seconds.Quicker Method: Applying the above.theorem,

24 x 4 - 16 x 5 161 men = = - =: 2 boys.

12x5-4x13 S

368

. Thus, J 2 men + 16boys= 24 boys + 16 boys = 40boysand 7 men + 10 boys = 14 boys + 10 boys = 24 boysNow. by basic formula, we have

40 x 5 =2.4 >:))2

40>:5 101' D =--=8-days I I

2 24 3

Exercisel. If! 2 me.i and J 6 days can do a piece of work in 5 days

an.l !3 men and 24 boys can do it in 4 days. compare thedaily work done by a man with that done by a boy.a)I:2 b)2:1 c)I:3 d)3:1

2. 1[30 men and 1'4 [jG;'S can reap a field in 21 days, in howmany days will 20 men and 4 boys reap it. supposingthat 3 men can do ,:15 much as 5 boys?

Time and Work e .I

a) 30 men b) 32 men c) 28 men':' d)u1 men.4. A certain. number of men can do a worklirr, 70 .days. If

there were 2 men less it could be finished.in 10 daysmore. How many men are there? ,;a)15men b)17men c)16men d)12men

Answers1.b 2. a 3.d 4.c

Rule 11Theorem: If A is 'n' times as fast (or slow) as B, ,(Inti istherefore able 10finish a work in 'D" days less (or more)than 8, then the time ill which they call do it working to-

-' ,

(Dn ) .

gettieris given by ,,1 -1 days.Illustrative ExampleEx: A is thrice as fast as B, and is therefore able-to finish

\...,./ a work in 60 days less than B. Find the time in whichthey can do it working together. \'

Soln: Detail Method.-, , .,. I, '

A i".thrice as, fast as B, means thatifA does a work inI dav then B does it in 3 davs.I-kn~e, if the differencebe'2 days, then A does thework in 1 day and B in 3 days. BULthe difference is 60days. Therefore, A does the work in 30 days and B in(jO days. -Now A and B together will do the work in

., - It'

/" 30)(90 45~;, -, ,;,./" --- davs =-- == 22.5 days

30 + 90 . . 2 ; ':! ..

Quicker Method: Applying the above theorem, wehave

60x3 60x3 45the required answer =--- =. --_. __..- = -_ '3" -I R 2

= 22.5 days. '.Exercise .\ \,l. A is twice as fast as B, and ,is therefore able to finish a

work in 30 days 'less than B. Find th'c:time i11WHitntneycan do it working together.a) 18 days h) 20 days c) 24 days d) 22 days

2. A is 4 times as fast as B and is therefore able to finish awork in 45 days less tha'1 B, Find the time in which theycan do it working together. 'a) 12 days b)16days cj S days d)20days

3_ A is thrice ,5 fast as B, and is therefore able to finish awork in -1Odays less than B_ Find the time iri1which theycan do it working together. .a) 16 days - b) 10 days - c) 15 days d)Noneofthese

4. A is thrice - .

But here we are given that we do the work 10 hrs aday, Then clearly we can finish the work in 4 days,Quicker Method: Applying the above formula, wehave

['15x8x?~ X9]- _ I 3 1_the required answer = \'-------" -- - 4 days .., _ 20 10

bxR+--x9 -... 3

ExerciseI. I can finish a work in 10 days; at 4 hrs a day. You can

finish it in 15 days at 5 hI'S a day. Find in how many dayswe can finish it working together 10 hrs a day.

50 70 60 40a) -2::: days b) r days c) -2--;:;days d). ")3" days

J _J .1 _

2. - 1 can 'finish a work i'l 16 days at 5 hI'S 11"day. You canfinish it in '12 'days at 4 hI'S a day. Find in how many dayswe can finish it working together 6 !li S a day, .,a) 5 days b) 4 (jays c) 6 days d) None of these

3. I can finish a work in 14 days at 6 hrs a day, You canfinish it in 8 days at 2 hrs a day. Find in how many dayswe can finish it working together 4 hrs a day.

Time and Work '/

a) 30 men b) 32 men c) 28 men .d) 3'] men4. A certain number of men can do a work irr.70 days. If

there were 2 men less it could be finished.in 10 daysmore, How many men are there?a)15men b)17men c)16men d)12men

Answers1. b 2. a 3. d 4. c

Rule 11Theorem: If A is '11' times as fast (or slow) (IS B, ,anti istherefore able to finish (I work ill '.0' days less (or more)tlian 8, then the lime ill which they call do it working to-

(DIl J .

getlieris given by ,,1 -1 days.Illustrative ExampleEx: A is thrice as fast as B, and is therefore able to finish

\./ a work in 60 days less than B, Find the time in whichthey can do it working together, \

Soln: DetailMethodeA i-:.thrice as fast as B, means that ':f 1\ does ~work in1 dav then B does it in 3 davs.I'kn~e, ir the difference 'be' 2 days, then A does thework in 1day and B in 3 days. But the difference is 60days, Therefore, A does the work in 30 days and B in0() days. . .Now A and B together will do the work in

~ 30x90 45 ~ ./' --- days = - = .n.5 days30+90' 2:' .....Quicker Method: Applying the above theorem, wehave

60x3 60x3 45the required answer =3~-1 =. -g-" = 2

= 22.5 days. I,Exercise .' \,l. A is twice as fast as B, and is therefore able to finish a

work in 30 days less than B. Find thc'time in'which,theycan do it working together.a) 18 days b) 20 days c) 24 days d) 22 days

2. A is 4 times as fast as B, and is therefore able to finish awork in 45 days less than B. Find the time in which theycan do it working together.a) 12 days b) 16 days c) 8 days d) 20 days

3. A is thrice LS fast as B, and is therefore able to' finish awork in ..H) days less than B. Find the tithe in'which theycan do it working together.a) 16 days b) 16 days c) 15 days d)Noneofthese

.f. A is thrice as good a workman a. B and is therefore ableto finish a work in 80 days less than B. Find the.time inwhich they can do it working together.a) 30 days b) 20 days c) 24 days d) 25 days

369

(MBA,1983)

Answers1. b 2. a '3.c 4.a

Rule 12Theorem: If (I person can finish a work in d, (lays at hihours a day ami another person cauflnish the same work

in d2 days lit ~ hours (I (lay, then the 110. of {/tlys in whichthey can fill ish the works working'together 'li' hours a day

. [.. (1'/(/1 X/t ]d]) ] ~ daysIS (h/(11)+'(h](/')) 11'.

Itlustranve.Example ,Ex: can finish a work in 15 days at 8 hrs a day. You can

/. 2finish it in 6 ~ days at 9 hrs a day. Find in how many-' . sdays we can finish it working together 10 hrs a day.

Soln: Detail Method: First suppose each of us works foronly one houra day.Then I canfinish the work ill 15 x 8:= 120 days

20and you can finish the work in ~-x9 = 60 days

.)

Now, we together CQ!1 finish the work in

~20 x 60 = 40 d. '05120+60 G}But here we are given that we do the work 10 hrs aday, Then clearly we can finish the work in 4 days.Quicker Method: Applying the above formula, wehave

roo15x8x~9 X91. I ." 1the required answer = \ .__ 2-.__., .- = 4 davs.. ')0 10 J.15x8+ ~.., x9

t 3

ExerciseI. I can finish a work in 10 days at 4 hrs a day. You can

finish it in 15 days at 5 hI'Sa day. Find in how many dayswe can finish it working together 10 hrs a day.

50 70 60 40a) -2::; days b) -2" days c) '2-= days d). ')3 days-' .) " -

2. I can finish a work in 16 days at 5 hrs a day. You canfinish it in 12 days at 4 hrs a day. Find hi how 'many dayswe can finish if working together 6 !:rs a day.a) 5 days b) 4 (jays c) 6 days d) None of these

3. I can finish a work in 14 days at 6 hrs a day. You canfinish it in 8 days at 2 hrs a day. Find in how many dayswe can finish it working together 4 hrs a day.

.:5/U

" 8a) .J 25 days3

b) 9 25 days

9d) 4 25 days

9c) 3- days

25

AnswersI. c 2. a 3:c

'Rule 13'Theorem: if A call d~ a 'work ill x days, B tak~s y days tocomplete it and C takes (IS long as A and B would takeworking togetlter, then Band C together take to complete

[xy J .

tile work = 2x ,+ y. days. A an,d C ~ogether take to corn-

'. [ xy Jplete the work = x + 2y days and A, Band C together take

[xy ].

to complete the work = 2(x + y) =. .Illustrative Eample .Ex:/ A can do a work in 6 days.' B takes 8 days' to complete

~ it. C takes as long as A and B would take working.together. How long Will it take Band C, A and C, andA, Band C to complete the work together?

Soln: Using the above formula, we have,

6xS(B + C) together take to complete the work = 12+ 8

48 12 2=, -- = - = 2 - days20 5 5 .

6x8(A + C) together take to complete the work = 6 + 16

48 2=-=2-days22 11 .

(A + B + C) together-take to complete the work

6x8 I 48 12 5= 2(6+8) = 2S-='7=I7'days

Exercise1. A can do a work.in 3 days. B'takes 4 days to.complete it.

C takes as long as.A and B would take working together.How long will it take B and C to complete the work to-gether?

s-a) -- days

66 4

b) 5 days C) '3 days 3d) '4 days

2. A can do a work in 4 days. B takes 5 days to complete it.C takes as long as A and B would take working together.How long will it take Band C to complete the work to-gether?

25b)-14

20d)-7

.3. A can do a work in 6 days. B takes 7 days to cc nplete it.C takes as long as A and B would take working together.

. How long will it take .,A and C to complete the work to-gether?

21 -. I " 1a) -days b)2days c) L--- daysd) J-days5 10 10.4. A can do a work in 8 days. B takes 6 days to complete it.

C takes as long as A and B would take working together.How long will it take A and C'to complere the work to-gether?

123 2a) 25 days b) 35 days c) 25 days d) 25 days

5.. A can do a work in 10 days. B takes 15 days to complete.' it. C takes as long as A and B would take working to-

gether. How long will it take A, Band C to complete thework ~pe~her? _a)6 days b) 3 days c) 4 days d) 8 days

6. A cando a work in 20 days. B takes 5 days to completeit. C takes.as long as A and B would take working TO-gether. How long will it take A, Band C to complete tilework together?a) 2 days b) 4 days c) 3 days d) 6 days

AnswersI~b 2.a '3.c 4.d 5.b 6.a

Rule -14Theorem: A is ntimes as good a workman as B. lf together,theyfinish the work in x days, then A and B separately CIIIl

d~ the.same.work in:( Il: /)x (lays and (11+ l)x days re-

spective!y:

Illustrative ExampleEx: ~is twice as good a workman as B. Together, they

~ finish the work in 14 days. In how many days can it be.done by each separately?

Soln: ,Detail Method:Let ~finish the work, in 2x days.Since A is twice asactive as B therefore, A finishes the work in x days.

2X2(A + B) finish the work in 3;= 14orx=21:. A finishes the work in 2 I days and B finishes thework in21 x 2=42 days.

Time and Work tim,e,would, tl\e,Y'ta\

380

2 days) + (x boy's work fs>r

Time and Work

x x+5Or - +-- =1or x =3. 15 10 '.'. A leaves after 3 days.

5. d; Hint: IfB works alone for x days; A's amount of work+ B'samountofwork = 1

or, I: + (\~4)= I .'.x =5'.. : '01 ,.' '.i

6. a; Hint: Using the given rule we'hilve(2 men'swork) +(8 women's work) +'(x boy's workp=I

or, (2X'2X 210)+(&x,2x 3~}+tl~~i.'~~y= b, .

1 8 xor -+-+-=1'5 15. 30

6+16+x -1 _'or, 30 -:. x - 8 boys.

7, b; Hint: Let A alone takes x days to finish the work andhence B alone takes (x + 3) days.Now, using the given rule, we haveA's amount of work +B's amount of work = 1

4 10or - +-- = 1 . x'" 12'x x+3 ...'. A alone takes 12 days-and B,aldne takes (11'2 + 3 =15) days to complete the work..

8 8 88 a.Hint: -+-+-=1" 24 16 x

, ,"toJ

.',,' f ,"

:. x ~ 48 days.,

Rule 32Theorem: Two persons A and B can finish ajob alone in xand y days respectively. If they start working on alte~na!edays, then to find the total job completion time,lollowing

, steps ate taken.Note: This formula is applicable only whenx andy are inte-gers,Case: If A starts the workStep I: First calculate the value of p; 'where p = nearest inte-

ger value to be consider~d =(2L)x+y

Stepll:

(a) When, x- y = 2, 4 then, apply the followingformula.

, . .. xy+ p(x- y)T (Total Job completion tune) = .. x

(b) When, x - y = 1, 3 , then apply the followingformula;

T (Total job completion time) = xy - p~x - y)y

381

llIustrative ExamplesELI: A and Bworking alone ~an finish ajob in 5 days and

7 days respectively. They work at it alternatelyfor aday. IfiA,star\S the ~~rf.'\fmd!tn ?0W,~i\flY days thejob will pe fmished?

Soln: Applying the above theorem:

. .xy I 5x7 35 .'..... ':... ,Step!" P = -- = -- = - Rl 3 (nearest integer value) x+ y 5+7 12Stepll: x - y =5 -7'= _2,1Here, formula (a) will be applied

Total time to fmisIl the jo'b if A starts the work. i} f~. ,I ~.c.',.. ,. I

'. xy'+ p{x ~ y)' '5'x 7 .;.3(5 - 7)= =

x ,5'. ,r!'29 4="5=5'5 days,

Ex. 2: ~d B working separately can do a ~o~k in 9 and 12days respectively, A starts the work and they workon alternate days. In how many days will the work becompleted's. , Ii

Sola: Applying the above theorem,

. 12~.Q 108 5 'Ste~I: P ~J.2+9 ~ 2,1 Rl (~ea~estintegerva~e)

Step ll:.x-y =9-12 =-3, Here formula fb) will be applied..'. Total time to finish the'jeb if A starts the work

xy- p(x- y) , (9x12)~5(9-12)l' '12

= 108+15 = 4J =10L days12 4 4

Now we try to solve the above examples by DetailMethod.

Ex.!: Detail Method: "

. 1In tpe first day A does '5 of the work,

1In the second day B does '7 of the work

(1 1 12) in the first 2 days 5'+'7 =.35 . of the work

12 24. in 4 days -x2=- of'the work.. 35 35

" ,.

(24) 11'.

Now, 1- 35 = 35 of the work remains to ,be done.1

In fifth day A does '5 of the work

. (11 1) 4 Bwill finish the work 35 - 5' =3s ofthe work

380

2 days) + (x boy's work f9.r.t days) = I

/ o~((2X2X-1 )+(8X2X~)+(~X2X_l ) = 1~, 2'0 30 " 60.,j , "

:. x= 8 boys,

Exercise1. A and B together can do a piece of work in 12 days

which B and.C'together can do in 16 days. After A has, been working at itfor 5 dayst'and B for 7 days. C finishes .it in 13 days. In how many days could each do the workby himself? '~ '; 0'a) 16,48 and 26 days respectivelyb) )6, 48 :and 24 days respectivelyc) 26, 48 '~d 24 days respectivelyd) 16,46 and 24 days respecti ely

,2. .f:. can do a,job,i!l20.~ays, B jn,30 days and C in 60,days.If A is helped on every 3rd day by B andC, then in he,wmany days, the job is finished?

[1'(11989)a) 20 days b) 15 days c) J8 days d) 24 days

3. A can do a job in 12 days: Bin 15 days. They work-together for 2,days. Then B leaves and A alonercontin- 'ues the work. After I day C joins A and work is com-pleted in 5 more days, In how many days can ,C do italone?a) 15 days b) 20 days ~c) 25 days d) 30 days

4. A and B can do a job in 15 days and 10 days respec-tively/They began the work together but A leaves aftersome pays and B finished.the remainingJob in 5 days.After how many days did A leave?a) 2 days b) 4 days c) 3 days. d) 6 days

'5. A and B can 'do ajob in 16 days'''and 12 'days respec-tively. 4 days before finishing the job, A joins B. B hasstarted the work alone. Find how many days B hasworked alone? . [Bank PO J 989]a) 8 days ,b)10day,s c)4days d)5days

6. A man, a woman or a boy can,Po a job in 20 days, 30days or 60 days respectively, How many boys must as-sist 2 men and 8 women to do the work in 2 days?

"[MBAI992)a) 8 boys b) ID.boys .' c) 12 boys d) I 6 boys

7. A can do ajob in 3 days less time than B. A works at italone for 4 days and then B takes over and completes it.If'altogether 14days were required to finish the job, howmany days would each of them take alone to finish it?a) 13 days, 16 days b) 12 days, 15 daysc) 15 days, t'2 days d) 15 days; 18 days

8. A'can do a piece of work in 24 days, while B alone can doit in 16 days. With the help ofC they finish the work in 8days. Find in how many days alone C can do the work?

, '[MBA 1988]a) 48 days b) 36 days c) 40 days d) 5Qdays

. P~CTICE BOOK ON QUICKER MATHS

AnswersI. b; .Hint: Let the whole work be 1

, .' 1.

I' ' !i', ;' "f. 1 "~" .Aand B .in I day do 12 .

1Band C in 1 day do 16 .

A's 5 day's work + B's 7 day's work + C's 13 day's'"I -work = 1

) J Or;A's 5'day's ork + B's 5 day's work + 8's ~ day'swork + C's ~day's work + C's 11 day's work = I

5 2' C' '" -+-+ S II dav's work = I", 12 16 ,.

, (52) II,:.C'stl~aY'swork=I-12+16 =24

I .

1 C's 1 day's work = -.. 24

1 I I B's 1 day'swork= ---=-, .. 16 24 48

1 1 1 A's day's work = - - - = -.. 12 48 16:. A, Band C can do the work in 16, 48 and 24 daysrespectively.

2. b; Hint: Since A is helped by Band C on every 3rd day.A works for 3 days while Band C work for I day

. l' 1 1 1-x3+-xl +-+1 = - [:. B and C help onlyon

. '20; . 30 60 5', 3rd day], :. Total time for the job = 3 x 5= 15days.

3. c; , Hint: Let C doit alone in x daysA's amount ofwork +B's amount of work +C's amountofwork = 1

( ~ " o 1 ( 1) ( 1)or 2+1+5 -+ 2x- + 5x- =1, ,12 15 x

4.c;

or .? = 1- (.!.+ ~) or .? =.!. . x = 25, x 12 15 x 5"

.', C,can doit alone in 25 days..Hint: In this problem', total time for the work is notknown and also it is not to be found out. Hence totalthhe for the work is not to be considered.If A leaves after x days ie A works for x days and 8works for x + 5 days,(ilien(applyihg the given rule, we have. ,

" No. of days A worked + No. of days B worked = 1.Alone time for A Alone time for 8

Time and Work

5. d;

x x+5 'Or - +-- =1 or x = 3. 15 10 '. A leaves after 3 days.Hint: IfB works alone for x days; A's amount ofwork+ B' s amount of work = 1

6. a;

or, I: + (\~4) = 1 .'. x = 5 . t ! .t ',1

Hint: Using the given rule we-have (2 men'swork) +(8women's work) +(x boy's workj=!

or (2X'2 x_I )+(8X'2X'!'\+(;~~'~~X= 1[",,,, 20 30~' t! -' 6QiJ1 8 x

or -+-+-=1'5 15 30

6+I6+x =1or, 30 :. x = 8 boys.

01

7. b; Hint: Let A alone takes x days to finish the work andhence B alone takes (x + 3) days.Now, using the given rule, we haveA's amount of work + B's amount of work = 1

4 10or -+--=1 . x""12'x x+3 ...'. A alone takes 12 days 'and Balone takes (11'2 + 3 =15) days to complete the work.

8 8 88 a.Hint: -+-+-=1., . 24 16 x

"';'lr'

:. x "7 48 days"

Rule 32Theorem: Two persons A and B can finish ajob alone in xand y days respectively. If they start working on altermudays, then to find the total job completion time, following

, steps are taken.Note: This formula is applicable only whenx andy are inte-gers.Case: If A starts the workStep I: First calculate the value of p; 'where p =nearest inte-

ger value to be consider~d = (~)x+y

Step Il:

(a} When,' x - y = 2, 4 then, apply the followingformula.

. . .. xy+ p(x- y)T (Total job completion time) = ~---=--'---::...

4 I 4in, J5 +7 .or ,5 ,days. ""&1.:

, " , r (4' I 4) 54 . , the total time 'required = +' + - = - days,.. , If' 5 5

.,i. 16;,, I I 7

Ex.2: Detail Method: (A + 8)'s day's work =- +- =:;-, ' 9 12 ,,6r , r ~

,Wesee that: 5-x, ?:.= ~,Uust less than l).ie (A1' B), ,36 36

: l~ I .

work f~r 5 pairs of days ie for 10 days.,. ~~ .,' .' ,;~

~~w re;t of the work (1-.~!)=316 is to be done by 'A c

d) k i I.' .1 d .:1Acan 0~36 wor In 9x~=- ay,1, ": 36 4',{ J i

.. \ '1 ,t tfi .~ i} '~'1 '~r'Pi' ' ~: ", . ,t Total days = 10+-= 10- days.v-.. . 4 A'"". {'

Note: Twopersons A and B can finish a job alone in x andy days respectively: If theystart working. on alter-nate days, Ihe':/to find the total job completion time,following steps .are taken.

Ca~e:,I(Bstarts the;work "Step I: First calculatethe value ofp; ,

where p = nearest integer value to be considered

StepU:a) when, x - y = 2, 4 , then apply, the following, ,

formula,

, xy- p(x- v)T (Total job completion time) = .

i Yb) wh:eii' x -y'= l, 'j, \h~nlapplyth~-'}ollowing

formula,. \. .

, -. ' t,,:~ iy + p(x _ y)T(Totaljob completion time) = x

Illustrative ExamplesEx.I: A and B working alone can finish a job in 5 days and

7 days respectively. They work at it alternately for a(Jay. IH~ st~rts~the'-wol'k, -find in-how marly days thejob will be finished?

Sola: Apply the above theorem:'r,

Xl' 35 .Step I: P = ---'-- = -- ",,;3 (Nearest integer value)

,r + y J2 ~ , .

Step II: x - y = 5 - 7 = -2; Here, formula (a) will be appl ied.:. Total time to finish the job ifB starts the work

= xy- p(x- y) = (Sx7)-3(5-7)y 7

35-3(-2) 41, 6= --, -7-- = '-7- = 57 days.

Ex. 2: A and 8 working separately can do a work in 9 and 12days respectively. B starts the work and they work onalternate days, In how many days will the work becompleted?

Solo: -Applyingtheabove theorem, we have ,1

I2x9 108' .Step I: p = --- = - ""5 (nearest Integer value)

12+9 21 .Step II: x-y= 9 - 12= -3, Here formula (b)will be applied,

:. total time to finish the job is 8 starts the workxy+ p(x- y) ,

.' ~ J x

,;",9,x12-t5(9:-12) = ~08-15 = 31= 10~. davs9 9 3 3 'J'

ExerciseL ~wQmen, Ganga and Jamuna, working separate I)' call

~ mow a field in 8 and 12 hours respectively, If they workfor an hour alternately. Ganga beginning at 9 am, whenwill the-mowing be finished?

/

6 1 8 1 ., 1a). -ipm b) 2pm c)"2pm dj None ofthese

2. R~Al 'aha 'M~han can do a job alone in lOamI 8 davs,c_!.;~ \~"'I, j.. f\. 'i~ ~' ,':,d ~ \ w

"'''resp'~cti'yelY. On 1st January. Ram starts the job and thenII'~ "'t~ ':" .

; theyworJ< on alternate days. When will the work be fin-ished?a) 8th January b) lOth Januaryc) 9th January d) None of these

3. A and B working separately can plough a field in 6 and,-""~'1~ . ..:.c , ';' ., ,. 'to BUrs respectively. At 8 AM A starts the work and

they work in sttctchesof one hour alternately, when wi II. '\ . ~, . 1,- l-

the ploughingbe completed?a)3:20PM b) 2.20 PMc) 12.3p,PM d) None of these

Answers

1. a;8x12 .

Hint: P=g-+12 ""5, Herex=-y=B> 12= .. 4

Hence appiy the formula (a).

96-20 76 19 1. reqd answer = --- = --':: -- = 9- hours.. 8 8 2 2 .Ganga started at 9 am hence she completes the work

Time and Work

I' Iat 9 am +92" hrs = 62" pm.

8x10. .2. c; Hint: P =18""~,here x - ':!= 10.- 8 -;-rt-1,hence

apply the formula (a). 'n ,) J I.J '}; :

. SO+ 10 9 "r '.. required answer = -1-0- = days:"

Since Ram starts on 1st January., work will be completed pn 9th day ie on 9th ofJanuary. '. .

3. a;6xl0

Hint: p=~,,"4 .. ', .,.. 1:_1 ~Il

Here x - y = 6 -10 =4, hence formula(a),\yiIl ~:app'Ii~..'. 6xI0+4(6-1O)

required time = 6 ...\

22 / 1= T = 73" hours.

. 1 .. required answer = 8 am + 73" hours ~ 3 :2R pm.

Rule 33Theorem: If A, Band C together can do a work Inx days,Aalone can do tlte work in 'a' days andB alone can do thework in 'b' days, then C will do tlte same work in

,. -, ,

[ab _Xx(: + b)] days.Illustrative ~xample . .I ' .-:";\',,..11'1]1 !~'l.Ex: A, Band C together can do a work in 6 days. A alone

can do the work in IS days and B alone .can do thesame work in 27 days. Find in wh,at time C alone cando that work?

Soln: Applying the above formula, we have

t

6x 18x27the required answer = I'S 2 6~ 2), x 7- 18+ 7

1= 13- days ..2

ExericseI. A, Band C together can do a work in 2 days. A alone can

do the work in 6 days and B alone can do the same workin 9 days. Find in what time C alone can do that work?

13,a) 42"days b) 6'4days c)9days d)INo~eofthe6e

2. A, Band C together can do a work in ~,days. A alone, can

383

do the work in 24 days'and 'S' alorte can do the same workin 36 days. F;ind in what time C'alone can do that.work?

,I a) 9 days, 'bt l5 days c) 18days, . d) 24 days'3. A, Band C together c~p.dq a work in 4 days. A alone can

. \ If, kdo the wor;I5:.!Jl.r1,2daysand,B~I~tw;,C~nQ~\~e;samewarin 18 days. Firld in what time C alone can do that work?a) 8 days b) 27 days c) 9 days: d) 18 days-.

4. , A, .~:and 9, togethe~.cC\.n~doa \~or~)n q ,days. A alone ..,,.c~~ 4...1"

414in 35';- -:; .or :5 ,days,

, '. '. , (4 1 4) 54 . ,.'. the total time required = -i-' + '5 = 5' days.

I I 7Ex. 2: Deta.il Met~od: (A + B)'s day's work = 9'+12 = 36

We see that 5,x.2.- =~ (just less than I) ie (A + B)36 36. ,

work for 5 pairs of days ie for 10 days.

(' 35) I

Now rest of the work 1 - 36 = 36 is to be done by

A

) . - I. JA can do 36 work In 9x~ = - day, .,,: -,6 4 ,( .

I ' , 1" ;'1. Total days = 10+-= 10- days.. 4 4 .

Note: Twopersons A and B can finish a job alone in x andy days respectively. If they start working. on alter- ,nate days, Ihe,/u) find the total job completion time.following steps are taken.

Case: If B starts the workStep I: First calculatethe value ofp;

where p = nearest integer value to be considered

StepU:a) when, x - y = 2, 4, then apply the following .:

formula,

, xy-p(x,-v)T (Total job completion time) = .

, Yb) wh:en' x-;'= l, :i', Jth~nl~pply thci'}ollowing

formula,

T ('T' '1' 'b ' I ." . ) xy + p{x - y)Iota JO comp etion time = xIllustrative ExamplesEx. I: A and B working alone can finish a job in 5 days and

7 days respectively, They work at it alternately for, a-day, If 8 starts the work, find in how many days thejob will be finished?

Soln: Apply the above theorem;r,

xy 35 .Step I: Ji = ---'-- == -- ",;3 (Nearest integer value)

,ny )2 ' .

Step II: x - y == 5 -7 = -2; Here, formula (a) will be applied, Total time to finish the job if 8 starts the work

= xy- p(x- y) = (5x7)-3(S-7)y 7

35-3(-2) 41 6= , 7 = -7- = S-:; days,

Ex. 2: A and 8 working separately can do a work in 9 and 12days respectively. B starts the work and they work onalternate days, In how many days will the work becompleted?

Solo: Applying the above theorem, we have

12x9 108' .Step I: p = -- = - '" 5 (nearest integer value)

12+9 21Step II: x-y= 9 - 12 = -3, Here formula (b) will be applied,

:. total time to finish the job is 8 starts the work= xy+ p(x- y) ,

'f x

Exercise1. 0 women, Ganga and Jamuna, working separately can

mow a field in 8 and 12 hours respectively, If they workfor an hour alternately, Ganga beginning at 9 am, whenwill the mowing be finished?

6 1 81 ., I

a) '2pm b) '2 pm c) -''2pm d) one of these2. Rain and Mohan can do a job alone in 10 and S days, - respectively. On ) st January Ra1n starts the job and 11](;n

they-work on alternate days, When will the work be tin-ished?a) 8th January b) l Oth Januaryc) 9th January d) None of these

3. A and 8 working separately can plough a field in 6 and. ";ieO 8u~~re'spective'ly. At 8 AM A starts the work and

they work in stretches of one hour alternately, when wi IIthe' ploughingbe complete'd?a)3 : 20 PM ' b) 2.20 PMc) 12.30, PM d) None of these

Answers

1. a;8x12 '

Hint: P=g-+12 :;:;S,Herex--y=8--12=-4

Hence apply the formula (a).

96-20 76 19 I, rcqd answer = --- = -, = - = 9 - hours.. 8 8 2 2 'Ganga started at 9 am hence she completes the work

Time and Work

J' 1at 9 am +92' hrs = 62' pm.

8x10. '2. c; Hint: P = -1-8- ""5 , here x - y = 10,- 8 ""'+1, hence

apply the formula (a). ' T! " r: ,lio' j0) ~.!t.'" 1"

. . . _. 80 + 10 _ 9 ".. required answer - 10 - days.

Since Ram starts on 1st January 'iI' OJ ..i'lt., work will be completedon 9th day ie' on 9th ofJanuary.

, 'J

3. a;6x10

Hint: p=16""4 .,'. . } ,t

Here x - y= 6 -10 =4, hence formula(a) will beapplied.

.'. 6xlO+4(6-IO):. required time = 6 'J,(

22 '71= 3= '3 hours.

, ' 1 ':. required answer = 8 am + 7'3 hours= 3:2.Qpm.

Rule 33 ~ . r:\,Theorem: If A, Band C together can do a work inx days,Aalone can do lite work ill 'a' days and B alone can do thework in 'b' days, then C will do the same work in

[x ab ]

ab - x(a + b) days.

. ,~ .., ,

Illustrative ~xample ":.,, ',l" .h'irj;H'm !)~1,:"Ex: A, Band C together can do a work in 6 days. A 'alone

can do the work in 18 days and B alone .can do thesame work in 27 days. Find in what time C alone cando that work?

Soln: Applying the above formula, we have

6x18x27the required an~wer = 18 x 27 - 6118+ 27)

'Co',1= 13- days ..2

Exericse1. A, B and C together can do a work in 2 days. A alone can

do the work in 6 days and B alone can do the same workin 9 days. Find in what time C alone can do that work?

1 3 .a) 4-days b) 6-days c)9days 'd)iNoneofthese

24.2. A, Band C together can do a work in ~I.days.A alone can

383

do the work in 24 days 'and B alone can do the same workin 36 days. Eind inwhat time C'alone can do that work?

. a) 9 days , .b}) 5.dllYS c) 18 days .r. -

A.oII "m" "01' '0 ffth.l ':.Ruleb30'fI':!) .. III ;['10

rime and Work

Both starts the work together for some time, but B leavesthe job 7 days before the work is completed. Find thetime inwhich work is finished.

5a) 3- days13

3b) 5- days\ 13

3c) 13- days

5

Answers.c 2.b 4.c 5 01 C

d) None of these'

3.a

Rule 37Theorem: A ami B can (/0 a piece of work in x and y daysrespectively and both of litem starts the, H?o.rktogether; 14 ~leaves lite work 'a' days before the comp/~f/l!n, OJ,t!'f1worly',then the total time in which lite whole work is completed

I I'

(x + (I)Y= (x + J~Tdays.Illustrati ExampleEx: and B can do a piece of work in 10 days and 20 days

J respectively. Both starts the work together but Aleaves the work 5 days before its completion time.Find the time in which work is finished. I.

5010: Applying the above formula, we have

the required answer = (10+ 5}20 = 10 days.. 10+20

Exercise!. A can do a piece of work in 14 days and B in 21 days.

They begin together. But 3 days before the completionof the work, A leaves off. In how many daysis the workcompleted? ,

l' 1a) 10 days b) 5 days c) 5- days d) 10- days5 , 5A can do a piece of work in 15 days and B in 25 days.They begin together. But 5 days before the completionof the work, A leaves off. In how many d ys is the workcompleted?

a) 12~days b) l3~ days c) 112. days d) 25 days2 2 4',

. ' .A can do a piece of work in 20 days and B in 40 days.They begin together. But 10 days-before the completionof the work, A leaves off. In how many days is the workcompleted?a)IOdays 'b)ISdays c) 20 days d)25 daysA can do a piece of work in 5 days and B in 10 days.

1Ihey begin together, But 2'2 days before the cornple-IIOIl of the work, A leaves off. In how many days is the

385

work completed?a) 2 days b) 4 days" c) 5 days

Answersl.d 2.a 3.c 4.c

d) 8 days

Rule 38Theorem: A can do a piece of work in x days. If A does thework only for 'a' days anillhe,remalnlng work is done by B~ ,

In ''b1days: the B afonJ'cair"'do ;h~work In ( ',~ .) d~ys.~~ J l \ ,~ ,OT ~ x - a

III S" ative'Example' ,," " . ,t ~

Ex: A can do a piece of work in 12 days. A does the workfor 2 days only and leaves the job, B does the remain-ing work in 5 days.. Inrhow many days B alone can dothe work?

Soln: Applying ,tl)e~b9v;/-9r"J~la, we h~v~

'h . J.I '\5 I ',12x5. 6 dt e reqUlreu anSwer = -- = ays.. 12-2

Exercise1. A can do a piece of work in 15 days. A does the work for

3 days only and leaves the job. B does the remainingwork in 8 days. In how "many days B alone can do thel.work?a) 12 days b) 10 days. c) 15 days d) 8 daysA can do a piece of work in 25 days. A does the work for'5 days only and leaves the job. B does the remainingwork in 4 days. In how many 'days B alone can do thework?a) 5 days b) 15 daysc) 9 days d) None of theseA can do a piece of work in 23 days. A does the work for11\days only and 'leaves the job. B does the remainingwork in 9 days. In h,~':-Vmany days B alone can do thework?

3.

, '.i I -3.a) 17 days b) 18 day~ .c) 174' days d) 174' days

4. A can do a piece of work in 22 days. A does the work for12'clays only and leaves the job: B does the remainingwork in 5 days. In how many days B alone can do thework?

~ , a) 11 days b) 10 days c) 12 days d) 14 days.5. A can do a piece ofwork in 80 days. Heworks a i~for 10

days and then B alone finishes the work in 42 days. Thetwo together could complete the work in:a) 24 days b) 25 days c) 30 days d) 35 days

(Clericai'Grade Exam, 1991)

AnswersI

1.b 2. a 4..a3.c

(80X42 l- '

5. c; Hj~,t:,Balone can do the work in 80 -10 = 48) days.

The two together could complete the work' in

(48X80 '\80 + 48' = 3C) days. . " (See Rule-4)

Rule 39 i, ."Theorem: A and R can do a piece of work ill x ami y daysrespectively. Both starts the work together. But flue to someproblems A leaves the work after some time, and B does theremaillillgworkill.f!.days.thelltheti.;.eafterwhi~h.l1

J ' .

. . ' 'i(y-a)x] .leaves tile l~orJ..is given by l .!. days.

'J, , ' x+.y, , .

IlIust tive Example ' . A andB can 'do a pie~Jlat'work in 45 'days 'arid'40 &iys'

respectively. They start the work too-ether but after .,,' , " b .I

some days, A leaves the job. B alone does the re-maining work in 23 days. Find after how many daysdoes A leave the job?

Soln: Usin'g'the'llbove'thlorem, we have1

the required answer' = (40 - 23 )45 = 9 'days.45+40

ExerciseI. A and ~ cary do.a piece of work in 20 days and 25 days

respectively. They start the work together but after somedays, A leaves the job. B alone do~s the remaining workin 10 days. Find after how many days does A leave the

. job?

-: 62 , I 2a) '3 days b) 6'3 days c) 6 d~ys d) 53' days

. 2.. A and B can do a piece of wdrk in 25 flays and 30 days,respectively. They start the work together but after some. days, A leaves thejob, B alone does the remaining workin 8, days: Find' afJ~rl how mal y' days does N leaY6' the, b? 'Jo. . ,-,'

, ,'" ," I

a) 12 days .~),.8 day,s " 9) 10 days ;d) 16 days3, . A and B can do a piece of work in 14 days and 21 days

respectively. They start the work together but after someda~s, A leaves the jo~. B alone does the remaining workin 6 da'y~. Find afte fhow many days does A leave the-job? - II . ;' I

a) 7 days ~)_:6d~~s . c) 8 days d) ~ days4. A an? ~ ca,n do a piece of work in 22 days and 23 days

respectlve,Iy. The .start the work together but after some. days, A leaves the job.' B alone does the remaining workin 8 days. Find after how many days does A leave thejob?

"'"'C. -------~ -.-~--~.--

:z 2 Ia) 6 days (b) 63' days c) 7'3 days d) 73 days

Answers1. a 2. c 3.b 4.d

Rule 40Theorem: A completes a work in 'x' days. B completes tilesame work in 'y' 'day . A started working alone and after'a' days B joined him. Then lite time in which, thev 1IIi11take together to complete the remaining work 'is given by

. [(x-a)y]x+y .

lIlustrati xampleEx: am cOniplete~ a work in 10 days. Shyam completes

, Hie' s~me workin IS days. Ram starts working aloneand after 5 days B joins him. How many days will theynow take together to complete the rernai,ing work?

Solo: Applying the above rule, we have

(10 - 5)15the required answer = 25 = 3 da~s

Exercise l',1. A completes a work in 12 days. B completes the same

work in 15 days. A started working alone and after 3days B joined him. How many days will they now taketogether to complete the remaining work?a)5 'b)8 c)6 d)4

(BSRB Calcutta PO 1999)2 A completes a work in 20 days. B completes the same

.work, in 25 days. A started working alone and after 2days B joined him. How many days will they now taketogether to complete the remaining work?a) 12 days b) 10 days c) 8 days d) 16 days

3., A completes a work in 12 days. B completes the same, work' in 13 days. A started working alone and after 7

(. /l:Iays'B~oirled him: How many days will they now take,I together to complete the remaining work?

. fl3 3 3

a) 15 days b) 35 days c) 2 days d) 25 days

4. A completes a work in 21 days. B completes the samework in 24 days. A started working alone and after 6

,ld'ays'B j inell him. How many days will they now take I togetlfet t complete-the remaining work?

a) 6 days ...b)-8 (jays c) 10 days d) 12 days5. A completes a work in 15 days. B completes the same

w6rk in 20 days. A started working alone and after I dayB joined him: How many days will they now take to-!?ether to complete the remaining work?a) 8 days - b) 7 days c) 6 days d) None of these

Time and Work

6. Mahesh and Umesh can complete a work in 10 and 15days respectively. Umesh starts the work, and after 5days Mahesh also joins him. In all, the work would becompleted in:a) 9 daysc) 11 days

b) 7 daysd) None of these

(Clerclal Gra~e 1991)~ r)

Answers._ J.~i ';'., ~.

l.a 2.b .3.d 4.b 5.a6. a; Hint: Here A = Umesh, B = Mahesh ,I'

.. x= 15 days andy= 10 daysNow, applying the given rule, we have the time takenby A and B together to complete the remaining work

_ (15- 5)10 = 4 .- 10+ 15 'c' days..

.. total time consumed to complete the work= 5 +4 =9 days.

Miscellan eo us1. Twenty-four men can complete a work in sixteen days.

Thirty-two women can complete the same work intwenty-four days. Sixteen men and sixteen women startedworking and worked for twelve days ..How many moremen are to be added to complete the remaining work in 2days?

. [Bank ofBaroda PO, 1999]a)48 b) 24 c)36 d) None of these .

2. 25 men and 15women can complete a piece of work in 12days. All of them start working together and.after work-ing for 8 days the women stopped working. 25 men com-pleted the remaining work in 6 days. How many days willit take for completing the entire job if only 15women areput on the job?' [Guwahati PO, 1999Ja) 60 days b) 88 daysc) 94 days d) None of these

3. 10men and 15 women finish a work in 6 days. One manalone finishes that work in 100 days. In how many dayswill a woman finish the work? ":

[BSRB Hyderabad PO, 1999)a) J 25 days b) 150 days c) 90 days d) 225 days

4. A can do a piece of work in 12 days, B can do t~e same, ,.4

work in 8 days, and C can do the same job i~ '5 th timerequired by both A and B. A and B work together for 3days, then C completes the job. How many completedays did C work? [NABARD,I999)a) 8 b) 6 c) 3 d) None of these

5. 12men take 18 days to complete ajob whereas 12women. 3

in 18days can complete 4' of the same job. How many

387

days will 10men and 8 women together take to completethe same job? [BSRB Delhi PO, 20001

, I'I

b) 13'2

6.' rf 5 men and 3 boys can reap 23 hectares in 4 days and if3 men and 2 boys can reap 7 hectares in 2 days, howmany boys must assist 7 men in order that they may reap

,. 45 hectares in 6 days?'''a)2boys "J(b)6~f~s'.' p)4bo~s ,d)5b9,Ys

7. 25 men can reap a {ieJ~in 20 days. Whe'n'should,l5 ~enleave the work, ifthe whole field is to be reaped in

I

a)6 c) 12 d) None ofthese

137 - days after they leave the work?2 . .a) 6 days b) 4 daysc) 5 days d) None of these

. 8. Acari copy 75 pages in 25 hours, A and B together can.. copy 135 pages in 27-hours. In what time can B copy 42pages?a) 21 hrs b) 5 hrs 36 seesc) 18 hrs d) 24 hrs

9. 15 men would finish a piece of work in 210 days. But atthe end of every 10days, 15 additional men are employed.

'. In how many days will it befinished?"! ~ l a) 30 days b) 70 days c) 35 days d) 60 days'

10; A piece of work was to be completed in 40 days, a num-ber of men employed upon it did only half the work in 24days, 16 more men were then set on, and the work wascompleted in the specified time, how many men.wereemployed ~t fiht?' .'a) 16men b)32 men c)2~me(l d)48 men

11. Ramesh can finish a job in 20 days. He worked for 10days alone and completed the remaining job workingwith Dinesh, in 2 days. How many days' would bothDinesh and Ramesh together take to complete the entirejob?a)4 . c) 10 d) 12 .

[BSRB BankPO Exam, 1991J12. A can do a piece of work in 12 days. B is 60% more

efficient than A. The number of days, it takes B to do thesame piece of work, is:

b) 5

1a) 7-

2

. 1b)6-

4c)8 d)6

; [CBIExam, 1991)'13. 12 men can complete a work within 9 ~as. After 3 days

they started die work, 6 men joined diem to replace 2men. How many days will they take to complete the re-maining work?

a)21

c)4 d) 4-2

[BSRB BankPO Exam, 1991)

b)3

vOO'

14. A can do a piece of work in 5 hours, B in 9 hours and Cin 15 hours. If C could work with them for I hour only,the time taken by A and B together to complete the workis:

1a) 2 hours b)?'~oursc) 3'2 hours d) 4 hours, [Clerical Grade, 1991)

! '15. A does half as much work as B in three-fourth of the 1time. If together they take 18 days to complete a work,how much time shall B take to do it?

. a) 40 days b) 35 daysc) 30 days d) N~ne of these

[LIC AAO Exam, 1988]16. Two workers A and B working together completed ajob

in 5 days. If A worked twice as efficiently as he actuallyI

did and B worked "3 as efficiently as he actually did, the

, work would have.completed in;3 days. Find the time forA to complete the job alone.

(MBA~ 1982)

1 1 3 1a)' 6'2 days b) 6"4'days c) 6"4days d) 12'2 days

17. Mohan can mow his lawn in x hours. After 2 hours itbegins to rain. The unmoved part of the lawn is _

2 " :~2'-x'" x x-2 "',a);;. , .!?) -2- ,cc),2" d)-;-,

. [ITI,1988J18. Iffactory A turns out x cars an hour and factory B'turns

out y cars every 2 hours, the number of cars which bothfactories turn out in 8 hours is '. "

, i)a) 8(x+y) 'b) 8x+~, 2

f "

cn6(x+.0 d)(2x+y)4

[MBA,198S)Answers1. b; 24 men complete the work in 16 days

f,~ t ~ i!'

, , j'" , (16 12.G),t: ..,hl L:. 16me,ncompl~~ 241X'16~\"2 Pt.qf,wqrkin 12

"days

32 women complete the wotk in 24 days

16 14 7:. 16womencomplete 32x24 = 24 partofworkin(12+2=) 14da1l~.iil;' , ',. _., Y-

1"

So, the remaining part of the work which is 4pne bysixteen men + sixteen women and the reqd additionalno. of men in 2 days "

= 1-(..!.+2.) =..!._2. = ~ (part), ,2 ~ 24 2 24 24

.I:'.t

Time and, Work

7, c;

1men = 4 boys 'Now, 5 men + 3 boys = 23 boys 23 boys can reap 23 hectares in 4 days:, 1 boy can reap 1 hectare in 4 days.'. 4 boys can reap 1 hectare in 1 day:,'4 x 45 boys can reap 45 hectares in 1 day

4x45:.-6- boys~an reap 45,?,7~t~~~i~?i.6~ax~

:. 30 boys can reap 45 hectares i9 6 da>,~But 30 boys = 28 boys + 2 boys = 7men-+ 2 boysHence 2 boys must assist 7 men.25 men can reap the field i~20 days,' - I

, 20x25 '.. 10men can reap the field In -1-0- or50days.

when 15men leave the work, 10 men remain and:1'. l.',

_ I 37~, 3these can reap In 37 - days _2_ or - of the field.

2, 50 4

Hence all men must work till (1 - ~) ?r ofthe fieldis reaped,

8. a;

1 20'Now 25 men reap - of the field in- or 5 days.

4 ,4In 25 hours A can copy 75 pages'

",

9,d;

75In 1 hour A can copy 25 .= 3 pages

In 27 hours A and B can copy 135 pages

135In I hour A and B can copy 27 = 5 pages

:. In 1 hour B can copy (5 - 3 = 2) pages.. B can copy 42 pages in 21 hours.

10 110days' work by 15 men = 210 = 21

At the end of every 10 days 15 additoonal men areemployed ie for the next 10 days we have15+15=30men,

. 2:. Next 10 day's work by 30 men = 21

, '(1 2 3) .Hence, in 20 days only 21 + 21 = 21 work is com-

,pleted.To complete the whole work we have to reach the

(21)value of 2i work

389

Now,

(1 2 3 6) .; 2 L21+2i+21+ .... 21 =21=1

Hence total time to complete the whole work = 10+ 10+ 10 + 10+ 10 + 10 = 60 days.

10. b; Let x men are employed at first.

1 " . I., ' ~. \ X men do '2 ofthe work in 24 days:. I man do the whole work in 24 x 2 x x = 48x days.Now, frorn'the question, ,

(x+ 16)meri do the remaining work (I-=) in(40

- 24 = 16) days.:. 1man do the whole work in 16 x 2 (x + 16) days.or, 48x ~ 32(x + 16) ., x = 32 men.

I11.a; Ramesh alone finished 2 of the work in 10 days ..

, . 1 " Remaining 2 of the job was finished by Ramesh and

Dinesh-togetner-in 2 days:Therefore, they both'together can finish the completejob in 4 days: ,- ,

112. a; A's 1 day's work = 12

1 1 2B's 1 day's work = -+60% of-=-.

12 12 15

15 1:. B can do the work in '2 ie 72 days

113. d; 12 men can complete '3 ofthe work in 3 days and the

2remaining '3 of the work in 6 days.

. . 2 .1 man can complete '3 of the work in (12x 6) = 72 'days.

2., 12 - 2 + 6 = 16men can complete '3 of the work in72 116 = 42 days.

(1 1 1-) 17'14.a; 5+'9+15 ie. 45 work is finished in I hour.. . . -1-.!2.- 28.. Remaining work - 45 - 45 .