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 = I

Thus, from /IlfJ~.jWl = M1D2Wj

16 x I 0 = M 2 X 40

16xl0 .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. 10 men 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 ill

D2 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 be

done 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.

) 1Hint: Here x = 6,.y = 4 and z = 2" Now apply the

given 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 days3. .A can do a piece of work in 6 days. B takes 8 days. 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?

\ .... 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 JTx812 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 3

day'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.

25xl5A 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 italone?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 days

45c) 15 days, 45 days, "4 days

d) None of these3. Five men can do a piece of work in 2 hours, which 7

women 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 these

4. A takes twice as much time as B and thrice as much timeas 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 these

A. 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 z

or,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 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 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, 3days

45c) IS days, 4S days, '4 days

d) None of these3. Five men can do a piece of work in 2 hours, which 7

women could do in 3 hours, or 9 children in 4 hours. Howlong would 1 man, I woman and 1 child together take todo the work?

f1260 1270 ) 1221

a) m b) m c 260 d) None of these

4. A takes twice as much time as B and thrice as much timeas 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 these

A, 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 + xz

or,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 days

Herice, 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) 3

7days 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 in

2 '.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 ---- == 1 1,.'" 25 +,20 <}

days3, c

Rule 8Theorem: If XI men 0l.&,YI women can reap afield ill 'D'

days, then "x2 men and Y2 women take to reap it

lIIustr tive Example. Ex.: [f3 men or 4 women can reap a field in 43 days, how

long will 7 men and 5 women take to reap it?Soln: First Method:

, ", 1

3 men reap 43 of the field in J day,

I I '•• 1 man reaps '43x3 of the field in 1 day,

I4 women reap 43 of the field in I day.

1:. 1 woman reaps 43 x 4' of the fie Id in J day.

(7 5 ') 1

.', 7 men and 5 women reap -4" :; + 4,":;"-4-')::: i-), .J X.J JX ,_,

of the field in I day,.. 7 men and 5 women will reap the whole field in 12days,Second Method:3 men= 4 women

'··r. r.

4", 1 man == - women" 3

.'. 7 men = 28 women3

28 437 men + 5 women == - + 5 = -- women

3 3Now, the question becomes:1f4 women can reap a field in 43 days, how long will

433" women take to reap it?

The "basic-formula" gives

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 these3. If6 men or 10 women can reap a field-inSo.days, how

long 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 a piece of work in

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

1 man = [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)13 men 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?<:)36 days b) 3(: Jays c) 42 days d) 45 days

3. If:' meu and 2 boy> working together can do 4 times asmuch work per hour as a man and ri boy to~ether, com-pare the work of a man with that of a boy.a)'2:1 b)3:J c)4:1 dj Drua inadequate

4. If : must hire 2 men and 3 bo~s for 6 days 10 do the same

piece of work a!'. limen and 5 boys could do in 1~2

days, compare- the work of a boy with that of a man ..il}7:3 b)3:7 c)2:5 d)5:2

5. 8 children and 12 men complete a certain piece of work inS days. Each child takes twice the time by a man to finishrhe work. In how many days will 12 men finish the samework?a)8 b)9 c) 12 d)15

(Bank PO Exam, 1988)AnswersI. h; Hint: Applying the above theorem,

a man's worka boy's work

24x4-16x/5 =.!Y=2:I12><:5-4xI3 8 ..

2. a: Hint: Here relationship between men and boys isgiven.

5.3men =, 5 boys ,', 1 man = 3' boys.

30x5Now 30M ·~14B=--+14=64boys and, . 3

From the-formula, 64 x 21 :: 112 »cD;3 -

:. Dz = 36 days.

PRACTICE BOOK ON QUICKER MATHS

.1. a; Hint: Let 5 men and 2 boys can do the work in x days .Hence a man and a boy together can do the samework in 4x days.Now, applying the given rule. we have

4xx l-xx2the required answer = _ .._-- = 2:I. 01

xx5-4xx INote: Also see Rule-IS4. b; Hint: Applying the g'ven rule, we have

3Man -x5-6x3 72-_.= =-Boy 3 36x2--·x II

2

•. Boy: Man = 3 : 7., 5. c; Hint: 2 children = 1 man

., 8 children + 12 men = 16 menFrom the question,Since lemen can complete a certain piece of work in9,days

(16x9 I

.. 12 men finish the work in 12 = 12) days.

Rule 10Theorem: A certain number of men can do a work in 'D'days. If there were 'x' men less it could be finished ill 'd'days' more, then the number of men originally (Ire

lIIustr . ExampleEl: A certain number of men can do a work in 60 days. If

there were 8 men less it could be finished in 10 daysmore. How many men are there?

Solo: Usingxhe above formula, we have

, '. 8(60+ 10)r the osiginal number ofrnen = --1-0 - = 56 men.

Exercise.\. A certain number of men can do a work in 45 days. If

there were 4 men less it could be finished in 15 daysmore. How many men are there?a)28 men b) 16men c) 24men d)20inen

2. A certain-number of men can do a work in 30 days. Ifthere were 6 men less it could be finished in 20 days•more. How many men are there?a) 15 men b) 12 menc)I8men d)20men

3. A certain number of men can do a work in 50 days. Ifthere were 6 men less it could be finished in 12 daysmore. How many men are there?

Time and Work e • .I

a) 30 men b) 32 men c) 28 men':' ··d)u·1 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 i11WHitn·tneycan 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 <15 good a workman a~ B andis therefore ableto finish a work in 80 days less than B. Find the.time inwhich they ell'! do it working together.a) 30 days b) 20 days c) 24 days d) 25 days

369

(MBA.1983)

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

Rule 12Theorem: If a person can finish a work in d, days at hi

hours a (lay ami another person canfinish the same work

in d2 days at 112 hours a day, then the no. of days in whichthe,y canfinish the works workingtogether 'II' hours a day

. r·o (Il1lJ/X'!ldi) .]~ da sIS -(/f'i/- ) +' (II ~il") 1z" y .

L / J -,1 2

Illustrarlvg ExampleEX; -can finish a work in 15 days at 8 hrs a day. You can

/. 2_ fifnish it in ?3 days at 9 hrs a day. Find in how many

days we can finish it working together 10 hrs a day.Soln: Detail Method: First suppose each of us works for

only one houra day.Then I canfinish the work'ill 15 x 8 =- 120 days

, . 20and you can finish the work in --:;--x 9 = 60 days

J

Now, we together can finish the work in,~20x!50 k'40 da '05120+60 :>- .

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: Detail MethodeA 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-days

22 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 days3

d) '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 ~p¥e~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;= 14

orx=21:. A finishes the work in 2 I days and B finishes thework in21 x 2=42 days.

Time and Work •• <

Note:

Quicker Method I: Using the above theorem:. (2+1)xI4.' r ,

A finishes the work in 2 =:= 21 days.

B finishes the work in (2 + 1) 14 = 4!2 days ..Quicker Method II: Twice + One time = Thrice activeperson does the work in 14 days. Then one-time ac-tive person (B) will do it in 14 x 3 =42 days and twice

14 .active person (A) will do it in '2 = 21 days.

Efficient person takes less time. In other words wemay say that "Efficiency (E) is indirectly prqpor- \tional to number of days (D) taken to complete awork ". Then mathematically

J KEo. D or, E = D' whee K is a constant

or, ED = constant

or, E,D, = E2D2 = E3D) = E4D4 = -r-E"Dn' And we

see in the above case: E,D, = E2Dz = E3D3 ··or, 3 x

14'=2 x.21 = 1 x 42Thus. our answer verifies the above statement.

ExerciseJ. A and B together can do a piece of work in 7.days. If A

does twice as much work as B in a given time, find howlong A alone would take to do the work?

Ia)21 days b)20days c) 10 days d) 10'2 days

2. A andB together can do a piece of work in 8 days. If Adoes twice as much work as B in a given time, find howlong A alone would take to do the work?a) 10 days b) 12 days c) 14 days d) 16 days'

3. A and G together can do a piece of work in 9 days. If Adoes thrice as much work as B in a given time, find howlong A alone would take to do the work?a) 12 days b) 14 days c) 16 days d) 18 days

-4. A and B together can do a piece of work in 6 days. If Adoes twice as much' work as B in a given time, find howlong A alone would take to do 'the work?a) 16 days b)9days c) 18 days d)21 days

5. A and B together can do a piece of work in 3 days. If A, does thrice as much work as B in a given time, find how

long A alone would take to do the work?a) 4 days b) 10 days c) 14 days d) 12 days

Answersl.d 2.b 3.a 4.b 5.a

Rule 15Theorem: If Q, men and b, boys working together can do n

limes as milch workper hour as a2 men and b2 boys to-

371

f ., gether, then the comparison of the work of a man with that

Man· nb] -blof a boy is given by Boy = -;;1 -na 2 .

Illustrative ExampleEx: 5 men and 2 boys working together can do 4 times as

much work per hour as a man and a boy together ..Compare the work of a man with that of a boy.

Solo: Applying the above formula, we have

Man 4xl-2 2'-"---B =; 5 4 1 = -1 [Here, a2 = 1 and b2 = I]~ - x . .

That is, a man is twice as efficient as a boy.

ExerciseL 6 men and 3 boys working 'together can do 5 times as

much work per hour as a man and a boy together. Com-pare the work of a man with that of a boy.a)2:1 b)3:1c)3 : 2 d) 4: 1

2. 8 men and 4 boys working together can do 6 times asmuch work per hour as a man and a boy together. Com-pare the work of a man with that of a boy.&)2: l up: t c) 1 : I d) 1 : 2

Answers1. a 2. c

Rule 16Theorem: If A and B can do II work in x andy days respec-tively, they began the work together, hut A left after sometime and Bfinished the remaining work in z days; then theno. of days after which A left is given by

(~~)(~)daySx+y y. . .

Illustrati e ExampleEX:. and B can do a work in 45 and 40 days respectively.

They began the work.together, but A left after sometime and B finished the remaining work in 23 days.After how many days did A leave?

Solo: Detail Method: Bworks alone for 23 days.23

•• Work done by B in 23 days = 40 work

23 17. A + B do together 1-- = - work40 40

40x45 40x45Now, A + B do 1 work in 40 + 45 = ---gs days.

17 40x45 17. A + B do - work in --- x - = 9 days.. 40 85 40 .

372

Quicker Method: ApplYing the above formula,,

. ,(-40X45X40-23)the required answer = --- = 9 days., 40+45 40 .

ExerciseI. A and B can do a work in 40 and 35 days respectively.

They began the work together. but A left after some timeand B finished the remaining work in 10 days. After howmany days did A -leave?

a) 13f days b) 13 days· c) J3jdayS d) 14 days

2. A and B can do a work in 35 and 25 days respectively.They began the work together, but A left after some timeand B finished the remaining work in 15 days. After howmany days did A leave?

'f

55 .5

a) 5 days b) 5 days c) "6 days d) 6"6 days

3. A and B can do a work in 20 and 15 days respectively,They began the work together, but A left after some timeand B finished the remaining work in 8 days. After howmany days did A leaveva) 4 days b) 5 days c) 3 days d) 6 days

4. A and B can together finish a work in 30 days. Theyworked for it for 20 days and then B left. The remainingwork was done by A alone in 20 more days. A alone canfinish the work in:a) 54 days b) 60 days c) 48 days d-) 50 days

(Cenfral Excise 1988)

AnswersI. c 2. c 3. a4. b: lIint: In the given formula, we have

"

. xy "'Od--=.' ays.x+y

Now, from the question, .B left the work, ie y = time taken by A to complete thewhole work and z = 10 days. .'~ .Now, applying the given formula, Volehave

• ~. t :. • 4 f J rt

(Y-20) .30 x l-y-. - = 20 :. y = 60 .days.

Rule 17Theorem: If x/ men or x2 women or X3 boys can do awork in 'D' days, then tile no. of days in which 1 mall, Iwoman and 1 boy do the same work is given by tile follow-illg formula, number of required days

r DXX/XX2XXJ 1- . days.

LX/Xl +x/x3 +X1X3 J

PRACTICE BOOK ON QUICKER MATHS

IIIustr ve ExampleEx: I man or 2 women or 3 boys can do a work in 44 days

Then in how many days will I man, 1 woman and !

boy do the work?Solo: Applying the above formula, we have

44x I x2 )(3the no: of required days = Ix 2 + 2 x 3 + Ix 3

44xlx2x32+6+3

24 days.

ExerciseI. 2 men or 3 women or 4 boys can do a work in 52 days.

Then in how many days will 1 man, I woman and I boydo the work?a) 24 days b) 42 days c) 36 days d)48 days

2. 3 men or 4 women or 5 boys can do a work in 47 days.Then in how many days will 1 man, 1 woman and 1 buydo the work?a)40 days b) 50 days c) 60 days d) 45 clays

3. Iman or 3 women or 4 boys can do a work in 38 da) s.Then in how many days will I man, I woman and I b(1;-do the work?a) 24 days b) 12 days c) 18 days d)3ti days .

4. 1 man or 2 women or 4 boys can do a work in 56 cia s.Then in how many days will I man, I woman and I boydo the work?a) 24 days b) 28 days c) 20 days d) 32 days

AnswersI.d 2.c 3.a 4.d

Rule 18Theorem: A group of men decided to do a work in x days,but 'n' of them became absent. If the rest of lite group didthe-work in ~'/ays, then tile original number of men is

(flY)"given by --. - men.. y·-x/\ ·!n.tj~. ..~:\

Illustrative xampleEx: group of'rnen decided to do a work in 10 days, but

. five of them became absent. If the rest of the groupdid the work in 12 days, find the original number ofmen.

Solo: Applying the above formula, we have

. 5xl2 ..,the required answer = 12 _I 0 =.,.0 men.

ExerciseI. A group of men decided to de a work in 13 days, but 6 of

them became absent. If the rest of the group did thework in 15 days, find the original number of men.

Time and Work

. a)30men b)35 men . c)40'men d) 45 men2. A group of men decided to do a work in 12 men; but 8 of

them became absent. If the rest of the group did thework in 20 days, find the original number of men.a) 18men b) 20 men c) 22 men d) 24 menA group of men decided to do a work in 15 days, but 2 ofthem became absent. If the restof the group did thework in 25 days, find the original number of men.a)5men ·b)4men c) 7 men d)6men

..AnswersI.d 2.1? 3.a

Rule.:J9Theorem: A certain number (~J'l1lellcan do tl work in 'D'days. If/here were 'x' men more it could beflnished ill 'd'

[X(D-d)]

days less, then tile number of men originally are d·

or

No.of more workers x Number of days taken by tile seeo!,!! groupNo. of less days

Illustrative ExampleEx.: A certain number of men can do a work in 60 days. Ifr there were 8 men more it could ~e finished in 10 days

~ . less. How many men are there?Soln: Applying the above rule, we have .

original number of workers \.. .

= !-Jo. of more workers x No. of days taken by the second groupNo. of less days

_ 8x(60- 10) _ 8x50 _ 40 .- 10 -l()" - men.

ExerciseI. A certain number of men can do a work in 50 days. If

there were 3 men more it could be finished-in 5'days less.How many men are the f!?a)36 men b) lSrnen c)27 men d}30men

.., A certain number of men can do a work in 75 days. Ifthere were 6 men more it could be finished in 15 daysless. How many men are there?a) 20 men b) 24 menc) 28 men d) 32 men .

3. A certain number of men can do a work in 35 days. If. :f

there were J 0 men more it could be finished in' 10 daysless. How many men are there?a) 25 men b)20men c)15men d)30men

. Answersl.c 2.0 3.a

, .

Rule 20Theorem: A builder decided to build a farmhouse ill 'D'days. He employed 'x' men-in tile beginning and 'y' moreme." after 'd'days mid completed the construction in stipu-lated time If he IIad not employed tile additionalmen, thentile men in the beginning would have finished it ill

: .. '

[D(X+y)-.Yd] " [Y(D-iJ)]---x --.- days and it would have. been x

da~s b~hi~d ihe schedule. .01' "'- JJ.' t , -

Illustrative Example I.

Ex.: A builder decided to build a farmhouse in 40 days. Heemployed 100 men in the beginning and 100 more .after 35 days and completed the construction in stipu-lated time. Ifhe had ~10temployed the additional men,how many days behind schedule would it have beenfinished?

Soln: Detail Method: Let 100 men only complete the workin x days. ,Work done by 100 men in 35 days + Work done by:fOOmenin .(40-35'=) 5 days = L'

or 35 + 200~5 =1'x 100x . ,\

45or -=1:.x=45day· s, xTherefore, if additional men were not employed, thework would have lasted 45 -40 = 5 days behind sched-uletime.Quicker Approach: .200 men dq the rest of'thework in 40 -r- 35 ~ 5 days.

. 5x200100 men can do the rest of the work in JOO = 10

days.:. required number or days = 10 - 5 = 5 days.Quicker Method: Applying th~ above theorem, wehave

100(40-35)the required number of days =, 100 = 5 days .

ExerciseI: A builder decided to build a farmhouse in 45 days. He

employed 150 men in the beginning and 120 more after30 days and completed the construction in stipulatedtime. If he had not employed the additional men, howmany days behind schedule would it have been finished?a) 12 days b) 10 days c) 15 days d) 8 days

2. A builder decided to build a farmhouse in 50 days. Heemployed 50 men in the beginning and 50 more after 40days and completed the. construction in stipulated time.If he had not employed the additional inen, in ho";' many

373

374

days would it have been finished by the men in the be-ginning? .a) 80 days . b) 60days c) 40 days d) 75 days

3. A builder decided to build a farmhouse in qO days. Heemployed 150 men in-the beginning and 130 more after'45 days and completed the c~n~truction in stipul~teqtime. If he had not employed the additional men, ho~many days behind schedule would it have been finished?a) 10 days b) 23 days 'c) 13 days d) 15 days

4. A builder decided to build a farmhouse in 20 days. Heemployed 40 men in the beginning and 20 more after 10days and completed the construction in stipulated time.If he had not employed the additional men, in how manydays would it have been finished by the men in the be-ginning?a) 50 days b) 60 days c) 40 days d) 5 days

AnswersLa 2.b 3.c 4.a

Rule 21Theorem: A, Band C can do a work in x, y and z daysrespectively. They all begin together. If A continues to work

till it isfinished, C leaves after working. d} day» find B d,days before its completion, then the time in which/work is

[X(YZ+d1Z +d })I)]=: is given by xy + xz + yz days.

Illustrative ExampleEx: A, Band C can do a work in 8, 16, 24 days respec-

tively. They all begin together. A continues to worktill it is finished.iC leaving off 2 days and B one daybefore its completion. In what time is the work fin-ished?

Soln: Detail Method:Let the work be finished in x days.Then, A's x day's work + B's (x - 1) day's work + C's(x-2)'day's work = 1 .

x x-I x-2or - +-- +-- = 1 . x = 5 days'8 16' 24 .. .Quicker Method: Applying the above formula, wehave

. . 8[(16x24)+(lx24)+(2x 16)]the required answer = (8 x 16)+{16x24 )+(8x 24)

= 3Q72 + 448 = 3520 = 5 da s.704 404 Y

Exercise1. A, Band C can do a piece of work in 16,32 and 48 days

respectively,' they start working together but Cleavesafte~ working 4 < days ~d B, 2 days before the comple-

PRACTICE BOOK ON QUICKER MATHS

2.

tion of work. Find in how many days the work was fin-ished?a) 5 days b) 8 days c) 10 days d) 12 daysA, Band C can do a piece of work in 10, 12 and 15 daysrespectively, they start working together but Cleavesafter working 3 days and B, 4 days before the comple-tion of work. Find in how many days the work was tin-ished?

. 2 1 2' 2a) 6-days b) 5- days c) 7- daysd) 6·- days

15 ~ 15 5A, Band C can do a piece of work in 5, 8 and 10 daysrespectively, they start working together but Cleavesafter working 2 days and B, 1 days before the comple-tion' of work. Find in how many days the work was tin-ished? .

3.

a) 3 days

AnswersI.c 2.a· 3.d

Rule 22Theorem: There is a sufficient food for 'M' men for 'D'days. If after 'd' days 'm' men/eave theplace, then the restof the jood will last for the rest of the men for

[!?- d x M]days .M-m

Illustrative ExampleEx: /There is a sufficient food for 400 men for 31 days.

J After 28 days, 280 men leave the place. POI' how manydays will the rest of the food last for the rest of themen?

Soln: Detail Method: The rest of the food will last for(31 - 28) = 3 days ifno body leaves the place.

-;,,~f;!l\.l~,th~restofthe food will last for {~~~ ]days for

the, 140 men left.

I (400)'. Ans = 3 - = 10 days... 120Note: For less persons .the food will last longer, therefore, 3

400is multiplied by 120 ' a more than one fraction.

Quicker Method: Using the above formula, we have

. _ 31- 28 x 400-The required answer - 400 _ 280 - 10 days.

Exercise1. There is a sufficient food for 200 men for 36 days. After

33 days, 140 men leave the place: For how many days

Time and Work

will the rest of the food last for the rest of the men?a) 5 days b) 10 days c) 18 days ' d) 15 days

') There is a sufficient food for 116 men for 25 days. After:!I days, 100 men leave the place. For how many dayswill the rest of the food last for the rest of the men?a) 19 days b) 24 days c) 29 days d) 15 days

3. There is a sufficient food for 300 men for 32 days. After29 days, 210 men leave the place. For how many dayswill the rest of the food last for the rest of the men?a) 12 days b) 14 days c) 15 days d) 10 days

4. There is a sufficient food for 150 men for 1'5days. After10 days, 75mehleave the place. For howrnany days willthe rest of the food last for the rest of the' men? .a) 10 days b) 8 days c)l5(days d) 15 days

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

Rule 23Theorem: A takes as much time as Band C together take tofinish ([job. If A and B working together finish the job in xdays. C alone can do the same job in y days, then B alone

can do tile same work in (::.~) days ami A alone call do

(2xy )

tile same work ill -- days.y+x

Illustrative ExampleEx: A takes as much time as Band C together take. to

finish a job. A and B working together finish the jobin 10 days. C alone can do the same job in 15 days. Inhow many days can B alone do the same work?

Solo: Quicker Method I: Using the above theorem, B alone

2x 15x 10Can do the same work in 15 -10 = 60 days

Quicker Method II:

15xl0(A + B) + (C) can do in ~lo- == 6 days.

Since A's days = (B + C)'s days.B + C can do in 6 x 2 = 12 days.

J5x12.. B[B={B+C}-C]candoin 15-12 =60 days.

ExerciseI. A can do a certain work in the same time in which Band

C together can do it. If A and B together could do it in'l 0days. and C alone in 50 days, in what time could B alonedo it?a)25 days b) 30 days c) 24 days d) 20 days

2. A can do a certain work in the same time in which BandC together can do it. If A and B together could do it in 15

375

days, and C alone in 30 days, in what time could B alonedo it?a) 40 days b) 60 days c) 45 days d) 35 days

3. A can do a certain work in the same time in which BandC together can do it. If A and B together could do it in 12days. and C alone in 24 days, in what time could B alonedo it?a) 36 days b) 40 days c) 44 days d) 48 days

4. A can do a certain work in the same time in which BandC together can do it. If A and B togethercould do it in 10days, and C alone ·in·15 days, in how many days can Aalone do the same work?a) 12 days b}60 days c) 24 days d) 48 days

AnswersLa 2.b 3;d 4.a

Rule 24Theorem: A team of x persons is supposed to do a work in'Dt days. After 'd,' days, y' more persons were employed

and the work wasfinished' d 1 ' days earlier, then the num-

ber of days it would have been delayed if 'x' more persons

[Y{D-(tl/ +tll)}-d?X]

were not employed is given by x-

days ami the number of days ill whiclt the work would have

[(x+ yXD-dl)-d/y]

been finished is given by x ---- days

Illustrative ExampleEx: A team of30 men is supposed to do a work in 38 days.

After 25 days, 5 more men were employed and thework finished one day earlier. How many days wouldit have been delayed if5 more men were not employed?

Soln: Quicker Approach:35 men do the rest of the job in 12 days (12 = 38 - 25-I):. 30 men can do' the rest of the job in

12x35 = 14 days.30

Thus the work would have been finished in 25 + 14= 39 days that is, (39 - 38) = Iday after the scheduledtime.Quicker Method: Applying the above formula, wehave .

. 5{38-(25+1)}-lx30the required answer = -- 30

5xl2-30=ld30 ay.

Exercisel. A team of 40 men.is supposed to do a work in 48 days.

After 35 days, 15 more men were employed and the work

376finished 2 days earlier. How many days would it havebeen delayed if 15 more men were not employed?

i Ia) 2 days b) 28 days c:) 18 days d) I day

2. A team of25 men is supposed to do a work in 44 days.After 18 days, 2 more men were employed and .the workfinished 1 day earlier. How many days would it havebeen delayed if2 more men were not employed?a) 1 day . b) 2 days c) 1.5 days d), None of these

3, A team of20 men is supposed to do a work in 30 days.After 12 days, 5 more men were employed and the workfinished 2 days earlier. In pow many days would it havebeen finished if5 more men were not employed?a) 30 days b) 28 days c) 32 days d) 34 days

4. A team of27 men is supposed to do a work in 36 days,After 30 days, 9 more men were employed and the workfinished 3 days earlier, h1 how many days would it havebeen finished if9 more men were not employed? -a) 35 days b) 28'days' c) 34 days d) 39 days

AnswersI. b 2 ..a 3. c 4. c

Rule 25Theorem: A, Band C can do a work ill x days, y days and zdays respectively. They started the work together but after

d, days A left. If B left the work d2 days before the compte-tion of the work, then the whole work will be completed in

.Illustrative Example

Ex:. . . 4

A, Band C can do a work in 16 days, 125' days and

32 days respectively. They started the work togetherbut after 4 days A left. B left the work 3 days beferethe completion of the work . .In how many days was 'the work completed? •.DetailMethod: ,Suppose the' work is completed in x days,A's 4 day's work +B's (x-3) day's work + C'sxday'swork = I

4 (x-3)S xor --+---+-=1

, 16 64 32

Soln:

16 + 5x - 15 + 2x _ Ior, 64 -

or,7x+I=64 ".x='9days.Quicker Method: Appiying the above formula, wehave .

PRACTICE BOOK ON QUICKER MATHS

. . 32[64(16-4)+(3XI6)']5 .

. the required answer ,= T6 --··~~+32.j

.' . . = 2 x 4.5 = 9 days.

ExerciseI. A, Band C can do a piece of work in 12, 18 and 24 days

respectively, they work at it together, A stops the workafter 4 days and B i~called off2 days before the work is

, done. In whattime was the.work finished?a)12days ,bY.14 days c) 16 days d) 8 days

2. A, Band C can do a piece of work in 6, 9 and 12 daysrespectively, they ..work at it together, A stops the workafter 2 days and B is called off I day before the work isdone. In what time was the work finished?a) 4 days b) 6 days c) 7 days d) 3 days

3. A, Band C can do a piece of work in 18. 27 and 12 daysrespectively, they work at it together, A stops the workafter 6 days and B is called ofT3 days before the work isdone. In what time was the work finished?

. 6a) 6 days b) 8 days c) 10 days d) 613 days

4. ' A, Band C can do a piece of work in 24, 36 and 48 daysrespectively, they work at it together, A stops the workafter 8 days and B is called off 4 days before the work isdone. In what time was the work finished?a) 10 days b) 8 days c) 16 days d) 14 days

AnswersI. d 2. a 3. d 4. c

Rule 26Theorem: A started a work and left after working a I days.

Then B was called and he finished the work in b I days.

HadA left the work after working for a2 days, B would

huve flnished the remaining work in b2 days. Then, eachof them ie A andB, working alone finish the whole work in

[b2a/ -bja21 , .. ' [a2bj -a/b2]

b2

- b/ J days and a2 - a/days respectively.

Illustrative ExampleEx: A started a work and left after working for 2 days.

Then Bwas called and he finished the work in 9 days.Had A left the work after working for 3 days, B wouldhave finished the remaining work in 6 days. In howmany.days can each of them, working alone, finishthe whole work?

SoIn: Detailed Method: Suppose A and B do the work in x.and y days, respectively. Now, work done by A in 2days + work done byB in 9 days = 1

Time and Work

2 9 3 6or. _. + -- = 1 Similarly, -x + -y. == 1, x : Y J

.. 1 1To solve the above equation put - = a and '- = b ,

x y

Thus2a+9b= 1 .... ,(1) and3a+6b=.' .... (2)Performing (2) x 3 - (1) x 3 we have

I I5a = I 'a = - or x = - = 5 davs,., 5 ' a .

Iand y = b = 15 days.

Quicker Method: In such case: (Using the above theo-rem)

3x9-2x6 15A wi II finish the work in - 9 _ 6- := 3'= 5 days.

For B, we should use the above result.

2 3 . '.Bdoes 1-- = _. work III 9 days.

5 5

5B does 1 work in 9 x -::;-= IS days.

.)

ExerciseI. A started a work and left after working for I day. Then B

. twas called and he finished the work in 42' days. Had A

1 ~lefl the work after working for 1- days, B would' have

2finished the remaining work in 3 days. In how many dayscan each of them, working alone, finish the whole work'?,I) .5days, IS days b) 2.5 days, 7,5 daysc) 3.5 clays, S.5 days d) None of these

I i\ started a work and left after working for 3 days. Then

tB was called and he finished the work in \32' days. Had

. IA left the work after working for 42' days, B would

have finished the remaining work in 9 days. In how.manydays can each of them, working alone, finish the wholework?a) 7.5 days, 22.5 days b) 7 days, 9 daysc) 5 days, 15 days d) 23.5 days, 8.5 days

3. A started a work and left after working for 4 days, ThenB II'.:IS called and he finished the work in 18 days. Had Aleft the work after working for 6 days, B would havefinished the remaining work in 12 days. In how many

377

dayscan'each of them, working alone; finish the-wholework?a) 5 days, 20 days'c) IS days, 30 days

b) 10 days: 30 daysd) 5 days, 30 days

Answersl.b 2,a3.b

;Rule 27,I

Theorem: A Co/I do a workin x days and B call do tile samework ii, y clays. If they work togetlter for 'tl' days and Agoes ~way, then the number of days ill which Bfinishes the

work is given by y - (J + ~)d days.

IIlu ,rrative Example .A can do a work in 25 days and B can do the samework in 20 days, They work together for 5 days andthen ~ goes away. In howmany days will 8 finish thework? .

Soln: Detail Method

" [ I I]A + B can do the work in 5 days = 5 -+_.25 20

"5x45 9==_._-=-

25x20 20

9 IIRest of the work = I - - :=: ---

20' 20

B will do the rest of the work in = 1_ .?." L~.days.20 20

Quicker Method: Applying the above theorem, wehave

(20) .the required answer = 20 - . I + 25 x 5

= 20 - 9 = II days.

Exercise" 2 .

1. A can do a piece of work in 6-;;- days and B in 5 days ..)

They work together for 2 days and then A leaves B tofinish the work alone. How long will B take to finish it?

Ia) 12' b) 3 days c) 2 days d) I day

2. A can do a piece of work in 50 days and B in 40 days,They work together for 10 days and then A leaves B tofinish the work alone. How long will B take to finish it?a)11 days b) 18 days c) 22 days d)26 days

3. A can do a piece of work in 20 days andB in 15 days.They work together for 6 days and then A leaves 8 to

378

finish the work alone. How long will ~ take to finish it?

I 1a) 3 days b) 4 days c) 3'2 days d) 4'2 days

14. A can do a piece of work in 12'2 days and B in 10days.

I .They work ~ogether for ~ 2: days and then A leaves B

, to finishthe work alone. How long will B take to finish it?

11 11'a) 3" days b) 5 days c) 6 days

IId) Tdays

AnswersI. a 2. c 3.d 4.d ..

'Rule 28. a~ ..

Theorem: If A call complete b part of a work ill x days,

cthen the d part of the work will be done ill y days. We call

. x ycalculate the value ofy from the given equation al b = cl d .

No. of days workedNote: . = constant for a person

Part of work done

Illustrative Example

~ fu::;<C 3 ..> .A can do ~ ofa work in 12 ~ays.ln how ~any days

1can he finish 8" of the work?

(SBI PO Exam 1987)Soln: Using the above theorem, we have

12 Y 12 I.314 = 1/8 or Y = 3 x 4 x 8"= 2 days.

Exercise

2I. 'Ram can do '3 Of a work in 16idays. In how many days

1 .can he finish T2 of the work?

a) lday . b)3 days 1d) 2- days

2c) 2 days

. ..12. Vinay can do '4 of a work in 5 days. In how many days

PRACTICE ~OOK ON QUICKER MATHS

1can he fmish '5 of the work?

a) 20 days b) 5 days. c) 4days d) Data inadequate

43. Sudhir can do '5 of a work in 8 days. In how many days

. 1can he finish TO- ibfthe work?

a) I day 'b)2days c)3days d) None of these

Answers1.c 2.c 3.a

Rule 29Ex.: 38 men, working 6 hours a day can do a piece of work

in 12,days. Find the number of days in which 57 menworking 8 hrs a day can do twice the work. Assumethat 2 men of the first group do as much work in 1

. Ihour as 3 men of the second group do in 1'2 hr.

Soln: Detailed Method:. 2 ~ 1men of first group = 3 x 1.5 men of second group

" 01',2 men of first group = 4.5 men of second group

. 38 men of first group = ~x38=19x4.5'.' 2

.,' (19 x 4.5) mendo 1work, working 6 hrs/day in12 days.

. .'. I man does 1 work working 1 hr/day in(12 x 19 x 4.5 x 6) days.

•• 57 me,n.do 2 work working 8 hrs/day in

= 12xI9.?<4.5x6x2=27 days.57x8

Quicker Method:Ratio of efficiency of persons in first group to thesecond group

== E, :E2=(3x~.5):2xl=4.5:2 ..... (*)Now, u~e the formula:

MIDITIEIW2 = M2D2 T2E2 WI ..... (*)(*)

•• O2

= 38x~2x6x4.5x2_ = 27 days.57x8x2xl

(*) Less number of persons from the first groupdo the san~e work in less number of days,so they are more efficient.

(*)(*) M represents the number ofmen.D represents the number of days.T represents the number of working hours.

.. . E represents the efficiency,.r " .WTepresents the work and

the suffix represents the respective groups.

Note:

, ,

I

Time and Work

Exercisel.

2.

40 men, working 8 hours a day can do 'a piece 'o~fworkin15 days. Find the number of days in which 60 men work-ing 4 hrs a day can do twice the}world Assume that 3men of the first group do as much work in 2 hour as 4men of the second group-do in 3 hrs, _a) 60 days b) 40 days c) 80 days d) Non~ of these30 men, working 4 hours a day can do apiece-ofwork in10 days. Find the number of days in which 45,men work-ing 8 hI'S-a day can do twice the wqrk. Assume that 2men of the first group do. as mu~ '¥.orki.i!h~.bour as 4men of the second group do in 1 hr.

(;

61 62 53 \,t~, ,Ia) 3" days b) 3" days c) 6' days dy 3- days', 6

Answers '. I

l. c; Hint: 3 x 2 men of first group ""4 x 3 men of secondgroup•. Ratio of efficiency of persons-in first group to thesecond group = EI : £2 =2: I. Now apply the givenformula. ; ",

2,b

Rule 30 " t ,t,

Theorem: If A working alone takes 'x' days. more than Aami H, and H working alone takes y' days more than A andH together then lite number of days taken by Abnd B work-

ing together is give~ by [[xY] days. '-, :.::. 1

Illustrative ExampleEx; A alone would take 14 hours more to complete the job

than if both A and B would together.jlf'B workedI ' --,l'

, alone, he took 32 hours more to .complete the job

than A and B work~d together. What'iiJ11e,'~ould theytake if both A and B worked togeihef?' ..

Soln: Applying the above theorem, we have..!, I· •...

~the required answer = 1/-;2-2-. =,7 ho~rs,-!

Exercisel. A alone would take 8 hours more to complete the job

than if both A and B wouldtogether, liB' wOJ;kedalone,, ,I,

I ' ..', ~he took 4·- hours more to complete the job than A and2 -If)

B worked together, What time, wouldthey take ifboth Aand B worked together?! .,,:, 'a) 6 hours b) 5 hours c) ':]hoqrs.'qc,d)r8 hours

(Income Tax and, E,xci~eFv""m, 1985), ,'h -; j r·VA alone would take 16 hours more to'corriplete the job

, j 1

2.

()''''f," 379

than ifboth A im~If""oUI((togeth~r.l(B\v()rked alone,he took 4 hours more to complete the job than A and Bworked tog~thktt\What time', would they take jf both AandB'worked together?a) 5 hours b) 8 hours c) 9 hours ' 'cl)Norie of these

3. A alone would take 27 hours more to cornple : \h 'Q.bII~, thanjfboth ~al}d Bwouldtogether. IfB worked alone,

.'1,' hejook 3\~01,lr,smore to,c(nnpJ~te thi job-than If..and B-1, ,w,orkedr}:9ge,~herlWh~t,>tim,e,would, tl\e,Y'ta\<eif both A~. "an,d a W9f,kefi!l9g~t!1£(?!~I!"",. d p'''' ",. (I 'Jl

a) 8-hours b) 10 h?urs c) 9 hours ~. ,d),6'hQ!lr~Answers '{t' ,'" ;, i ,I t··, '~J', i

I. a 2. b 3. c)'1 .

"Rule 31- L i ·It'.

'Flieorein: If A, B"dnd (i'l can do a'jolJ' afone indays mid z (laysirespectively.·" , ) 11(' be.'. alone time for A =x days' ."f' I ')' ",,:1

J" alone time for B =Y days'alonesbne for 0'= zJiiiys f"" '"

..Now co'lrsldefthefolio 'iitltr/ases,Case [:,Iq'o firiCt,itlie'arfiounttof work 'do'i1eby'~: B and C, " .separately, r' I ),'f(,'

. Using the'formula,"

Am r ; t f kL(.tIumber of-davs actually workedoun 0 wor _, ' v· , ,•\ ;,.,(,1 I, ,. alone time t '!

-and assuming, that" A;( B 'and C have. worked ford, days,' 'd1

2 "days' and !iNi}days resp~~ti~ely, then. !" ., '.. ~

'I , • "'dt'I,; i '

, : amow:t or work by, f ~()'oc .' arJ}p.ur:{of.,work by',' r (~~~:~." ('l(~ -t , ,,:,,;,.

:.' I I~ Ii .•

,d2 'd f '~k b dJ ~ "I~?B = -' -"an amount 0 wor y C = -- :'''"Y" ' 'j Zl

Case II: If the job is complete, 'th~niadd the amot)nt' of work. ,.done by A, Band C and equate it to' I : ~

, "J I. I " . l •..) , J r. -:'"dl• d~ d, " ',: r • e-Ie - + - + - = I , If the Job IS half complete the

,'''' d f Y Z

c"following equation is obtaih~d,)i t, ... • i~ iLh Ii" ,I

d, dz <d3 ,L .....) ~+-+-'-= ...:.-'!:."• r xi· ..ell:' ,z )::["~r"~ ,1'(

iliustrative:Ex.~r"~iel· .' iif' ' , ",. r '.

Ex: A man, a woman 91'a boy capdo ajob. irt)Ojdays, 30pays or, 60 'days. respe~tiv.~'Y"How.many boys mustassist 2 men and 8 women to qR,tpe,work i~:2.'days,

c, '" • ; , ,,' .; , • ~{MB 1992)SQlp:, L:e\the required number of boys be x.

Now, using th~ above theorem, ,) .'(2 men's work for 2 days) + (8 women's work for

380

2 days) + (x boy's work fs>r<tdays) = 1

~I o~,'(2X2X-1 )+(8X2X_1 )+r xx2x-1 )= 120 '30 ~ . 60

'll . ;

:.x=8boys."Exercise1. ,A and 8 together-can -do a piece of work in 12 days

which 8 and ,C' together can do in 16 days. After A hasbeen working at itfor 5 days-and 8 for 7 days. C finishes.it in 13 days. In how many days could each do the workby himself?' ~a) 16,48 and 26 days respectivelyb) 16, 48:and 24 days respectivelyc) 26, 48 '~nd 24 days respectivelyd) 16,46 and 24 da:y$respectively

,2. J:.. can do a,job)n 20 days, 8 in 30 days and C in 60 days.If A is helped on every 3rd day. by Band C, then in hCi),wmany days, the job is finished?

[11119891a) 20 days b) 15 days c) )..8 days d) 24 days

3. A can do a job in 12 days, 8 in,'15 days. They work-together for 2.days. Then B leaves and A alone contin- '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 days 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

, . 'I'5. A and 8 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 8 hasworked alone? ' [BankPOJ9891a) 8 days .b) lO days c) 4,days d) 5 days.

6. A man, a .woman or a boy can.do 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?

1MBAI992)a) 8 boys b) 10.boys c) 12 boys d) 16 boys

7. A can do ajob in 3 days less time than 8. A works at italone for 4 days and then 8 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 day~c) 15 days, t'2 days " d) 15 days,' 18 days

8. Acan do a piec~ of work in 24 days, while B alorie can doit in 16days. With the help ofC they finish the work in 8days. Find in how many days alone C can do the work?

• .t [MBA 1988]a) 48 days b)-36days c) 40 days d) 5Qdays

, P~CTICE BOOK ON QUICKER MATHS

Answers1. b; ,Hint: Let the whole work be 1

I:' f A and B in Iday 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'swork = 1Or, A's 5 day's ork + B's 5 day's work + 8 's ~ day'swork + C's ~day's work + C's II day's work = 1

" 152+ I~:+C' s ) 1 day's work = 1

(5 2) 1I

.:. C's 1I day's work= 1- 12+16 = 24, ,

I .

1· C's·1 day's work = -.. 24

1 I I· 8's 1day'swork = ---=-. .. 16 24 48

1 1 I· A's day's work = --- =-.. 12 48 16

) t

,'. A"B and 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 1day

,1' 1 1 1-x3+-xl +-+1 =- C, BandChelponlyon, . .20; ,30 60 5

• 3rd day], :. Total time for the job = 3 x 5 = 15days.

3. c;" Hint: Let C do'it alone in x daysA's amount ofwork +B's amount of work +C's amountof work = 1

(J 'f -) 1 ( 1) ( 1)or 2+1+5 -+ 2x- + 5x- = I, . 12· 15 x

4.c;

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

:. Ccan do it al'one in 25 days..Hint: In this problem, total time for the work is not'known and also it is not to be found out. Hence total

..• 'ti " ."time for the work is not to be considered.If A leaves after x days ie A works for x days and Bworks for x + 5 days,then/applying the given rule, we have

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

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 8

8 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 B working 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 12

Stepll: 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.

1In 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 = 15 days.

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!· -' 6QiJ

1 8 xor -+-+-=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 B·alone takes (11'2 + 3 =

15) days to complete the work.8 8 8

8 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 altermu«days, 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) = ~---=--'---::...<..

. X

(b) When, x - y = ±1,± 3 , then apply the following

formula"

T (Total job completion time) =, xy-dx- y)y

381

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

7 days respectively. They work at it alternatelyfor aday. IffA1sFU1S the ~~rf..'. ~md,~nhow many days thejob will pe fmished?

Soln: Applying the above theorem:

, ,'xy '5x7 35 . ~ .i·· ''J. , '

StepI" P = -- = --'= - ~3 (nearest integer value)• x+y 5+7 12

-Step Ih x- y = 5 -'7'=-2, Here, formula (a) 'will be applied:. Total time to fmislf the job if A starts.the work

"~l ~;'+~i(X ~ ;)' . 51~'7:'3(5':' 7jx .5" ,d'

29 4="5=55 days.

Ex. 2: ~d B working separately can do a work in 9 and 12days respectively. A starts the work and they workon alternate days. In how many days will the work becompleted? II

Soln: Applying the above theorem,.~..

, 12x,?, 108 5 'Step I: P = -- =- ~ (nearest integer value), ' , J,2 +9 ,2·1, '," 'Step ll:.x-y=9 -12 =-3, Here formulafb will be applied .

.'. Total time to finish the job if A starts the work;y- p(x-y) ; (9xl-2)~5(9-12)-'

= Y = h'. = 108+15 = 4J =10L days

12 4 4Now we try to solve the above examples by DetailMethod,

Ex •!: Detail Method: '

1In ~e first day A does '5 of the work

1In the second day B does -;; of the work

(1 1 12)., in the flrst 2 days 5'+ -;; = 35 , of the work

12 24. in 4 days - x 2 =- of the' work.. 35 35

(24) 11_,

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

1In fifth day A does '5 of the work

, (11 1) 4:.Swill finish the work 35 -5 = 35 ofthe work

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

, •• " , r (4' I 4) 5 4 . ,· 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-.~!)=31

6 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 alonecan 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) 18·days, . 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~n••Q~\~e;·same warin 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"<; wO,~kin 36. Qily.-s ~\lq ~, alo,ne}:an do :the samework i~?4 days. Find in what tirne-C alone can do thatwork? I., ' • •• • . " .,

a) 9 days b) IS days c) 24 days d) 27 days',' ..•

Answersl.a 2.c ·3.c 4.~

Rule 34Theorem: ifA and B can do a work ill x andy days-respec-

.,.\ .,t . I (, \

tlvely and A leaves the work after doing-for 'a' (lays,then B. 1 . } ( •. ,

" ." " ;>,. " ,,[(x-a)y] ,do~ the remaining work III x (lays..

I ~.

IlIus,tr rve ExampleEx:. A can c01JlpJete·atw9rkllin 25 days and B can do the

. ";i same work.in-l O\ql:\Y~;.lfA after. doing.d.days, leavesthe work, find in how many days B will do the remain-

, ing work? 'J . Ii

Soln:. Applying the above formula, we-havethe required ansWer " .

., (25.:.4)xlO _ 2t'klO _ 42. _82- ----,-- - - days25 25 5 5 .

Exercise1. A can complete a work in 20'days andB can do tlie same

work in 25 days. If A after doing 5 days, leaves the work,~~d in how ~aryy days a will do the remaining work?

3 3a) 18- days b) S- days4 4

3c) 17'4' days "P, d) None of these

: ' r' ; ~ . ' . , '...2 A can complete a work in 35 days and B can do the same. _ w6x:k~in ~8 dllys:.f(b after doingI 0 days, leaves the

work, find in howmany days B will do the remaining, 1<?'wor ;." ...:1 ,) " , " .

a) 25 days b) 20 days c) 27 days d) 24 days3. A can complete a work in 24 days and B can do the-same

work in 18 days. If A after-doing 4 days, leaves the work,ofmdm:h'UW::f'I'I'ItH1 ~~~u,ttfe~ainml;~?a) ]0 days b) 12 days c) 15 days d) 16 days

AnswersI. a 2. b 3.c

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 = 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-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 ., Ia) '2pm b) '2 pm c) -''2pm d) one of these

2. 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' j

0) ~.!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 'alonecan 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. -<I)24 days '3. A, Band C together can.dq a work in 4 days. A alone can

do the wo~l,<;\ipJL-2day'sa'rid:a~l~tw:~~I1.•QPI\h~,samewcrkin IS days. Firld in wbat time C alone can do that work?a) S days b) 27 days c) 9 days: d) IS'days '

4. , A, a;and C togethercan do a wOl'k)n l2 days. Aalone,,c~t.t4oJht; ~o~kin, 3~.~JlY.s~!lq~.al~ny can do the samework il1;?4,d~ys. Find in what.dme-Celone can do thatwork? 1,_ ' " ••

a) 9 days b) 18 pays c) 24 days d) 27 days

Answers1.a 2.c 3.e 4.•d

I,

Rule 34Theorem: If A and B c~n do a work ill x andy days-respec-tively and A leavt!J'i/,ewb~kafte~dbing'Jor 10' days, then B

r.. : •• . I" [(x~a)y] ., ',.does the remaining work ill .x .days ..

I ~.Illustr rve ExampleEx: A can complete-aiworkjin 25 days and B can do the

, ,;' same work invl 0*days. If A after doing 4 days. leavesthe work, find in how many days B will do the remain-

, ing work? J' I!' '

Soln: Applying the above formula, we-havethe required answer

, ' ' ., (25':'4 )x'lO _ 21'~10 _ 42, _ 2- 25 '-25-'5- -S'5 days.

Exercise1. A can complete a work in 20'days and'B can do tlie same

work in 25 days. If A after doing 5 days, leaves the work.find in how rya~y days a will do the remaining work?

3a) IS- days

4

3c) 17'4 days,,~ d) None of these

. r -v: ., ",. ,', It, ••..2: A can complete a work in 35 days and B can do the same~ ," . "', \" ~. , . . ...., _ work-in ~S days. Kb after doing 10 days, leaves the

work, find in howmany days B will do the remainingwork?'

;d. (., I 1'. ~) t,' . r, .a) 25 days b) 20 days c) ~7 days d) 24 days

3. A can complete a work in 24 days and B can do the-samework in 18 days. If A after-doing 4 days, leaves the work,find in how many days Bwill do the remaining work?a) 10 days b) 12 days c) 15 days d) 16 days

AnswersI. a 2. b 3.c

A.oII "m" "01' '0 ffth.l ':.Ruleb30'fI':!) .. III ;['10<1 ,';,

Th~tll'emr1f;4'.andr.Bftall'(Jo'a'wOr'k'ili 'x:andy"daysrespec-lively, andilfieaves'lhe work aftei: doingfor 'a' days, then

i,» i, .••·.. cl·" '.' ';t"[CV~~)X1 \ ) 'f"', '.A-doestl;~\te'hiaiiiihg work in] I. '.:'" ' lady's.:! '1ft· / •

.j _ .It·-f\"i~:~· J1".,d(q rj t\j I J![(.. ,,.;v ,', (lJ

llluative(Exaniple l i c•.• /,:,,! \, f

'of I,N'c~n ~d:AWork-in 15 days-and B'alone can do that'.1 i wor~in Q5 daYs~tfBfhfter doiJig5 days leaVe~the job,, 'i' 'fino in How manyIaays 'A wilra<i the·rertlainihkwork.

Soln: Applying the above formula, we have, '/' '~/i:,r": 1 ~ ro

.;.., 7'!.;A :~~ I' •••.J~

the required answer == (25:' 5)x 1525

==~Ox 15 = 12 daysf}125

.'

Exercise , ., " ,....•~;"'~ .f .} t h} ~:1 h ,1 \' .•• -'I \' t U l 1-' L' . Uti"

l.it \Aa.~9,Bt w~rkil{g t<?&t~~.\a.n.f10!'.~!ecJe" f~O,I; in 6"days, B alone could. do it in 8 days. Supposing B worksat it for 5 days, in how many days A alone could finishthe remain:ingwork? ' i\ I' •• ",\HI ml1 ,'..j;',\.

a) 9 days b) 8 days c) 6 days. d) 12 days2. A and B working together can do apiece ofworklfn.lld~

r'days, B alate could do itiin ZO days ..Supposing B works"at it for 4 days,' imhow many days lA'alone/coulo finishthe remainingwdrk? t .•• 0(, •. ,:!",j ,:' .' 'I ·.1!

a) 9 days b) 12 days c) 16 days 'd)JlO days3. A and B wcrkirig-togetherrcan-doisspiece ofwotk in BO~' .

days, B alone could do it in 50.daysfS'uPPbSill'g.-B'worksat it for 10 q!l-ys,r!ph,o~v1ll~nYt4jlys~alonecould finish

. the remaining work?. e) 12 days b) 60 Clays c) 16 days d) 18 days

4"1,I A and.B. werking-together can do a-piece ,ofwofk:in 7 ~,"t'l .J • f. fl" • if - j" ,i',"

days,g';a'I'orfe')~6ujd do' it i~12~/'days. Supposing B

works at it £rt)~i(~s, i~ how m,aiiY'~ay~~ Mone

could finish the remaining work?a) 5 days . . b) 8'days.. c) 7 days .d) 15 days

5. A can complete ajob in 9 days. B in 10 days and C in 15f' ) _,,.,~ !'{> ft~; It.f' I,)' ,~Ir,," '

. days, B and C sfa'n the work and'are forced to leave after;'. 2· Gays. Thetime talSen to co~plrite tlle [~maihing'{yorkI·' 'is: ., r •• 'I • '-' w l' • \rl(NDA'E~1h19871

a)~]} ?ay,.s bn~ ?~~~}<;),9dft~~ j ~)6 ~ays·rAnswers r'l' ~; ?":.~ t,' .1:a;, Hint: ·,Eirst~p'plyt~e Rul«!'-6"and find the no r of days

.in which ,A ..alo~~I(jOuld do the, whole work ie. ""

1>1 6x8 (., ·a'i' " "8-6 = 24 <fays.

Now, applying the given rule, we have

PRACTICE BOOK ON QUICKER MATHS

. ,(8-5)24the required answer = --8 - = 9 days.

2.c 3.b 4.d15xlO

5:d;i 'IHint: Band C together can do the work in 15+ 10 =6days. (See Rule-S)Here, y = 6 days, and x = 9 days.Now applying e given rule, we have

'\ . (6-2)'x9the required answer = 6 = 6 days,

Rule 36Theorem: A and B can do a piece of work ill x andy daysrespectively and bot" of them starts thework together.If Bleavesthe work 'a' days before the completion of work,thenlthe total time, 'in which the whole.work is completed,

= [(:++a: ] days.

llIus ative ExampleEx: A and B can do a piece of work in 15 days and 25

days. Both starts the work together for some time,but B leaves the job 7 days before the work is com-pleted. Find the time in which work is finished.

Solo: Applying the above 'formula, we have

L . \ ihe.i~guired ~swer ;",(25 + :')15 = 12 days." \,\ i' <,>\.» ';;~, .,.• ' I. ,'" 25 + 15

"

~;'~'''' ,I~i' ~,.1:' 1'.t .t

Exercise .1.' A and B can do a piece of work in 20-days and 30 days .

Both starts the work together for some time, but B leavesthe job 5 days before the work is completed. Find thetime in which work is finished.lA) 7'dayJ b)'li'day's' 1 c) 14 days d) 16 days

2:' A and p·cah do a piece of work in 25 days and 35 days,t' Bbillrst{rts1fie'worl( together for some time, but B leaves

the jQb' 7 days before, the work is completed. Find the• • .01'{ r C' ~l ":t'.l....... :' "time In whlcliworK is'finished. .

. . 1 'a)17d~y.s b) 17idays c) 18 days d) 20 days

3. A and B can do a piece of work in 30 days and 45 days.Both starts the work together for some time, but B leavesthe job '15 days before the work is completed, Find thetime in which work is finished.

, ',. af,24, daYS'ir;;~b)28 days c) 20 days d) 16 days4. A andB can do a piece of work in 16 days and 24 days.

Both.starts the work together for some time, but B leavesthe job 6 days before the work is completed. Find the,time in-which work is finished.a) 18 days b) 14 days c) 12 days d) 8 days

5. A arid B can do a piece of work in 17 days and 33 days.

rime and Work

Both starts the work together for some time, but B leavesthe job 7 days before the work is completed. Find thetime in which 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; 1£4 ~leaves lite work 'a' days before the comp/~f/l!n, OJ,t!'f1 worly',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. He works 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 samework 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 time

required 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 10 men 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 42

pages?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 ~a¥s. 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 P¥t.qf,wqrkin 12"

days32 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<f\.CTIC~ ,tjUUK Ul'l VUICK~.t< MATH:::;

Now, 'in 2 days ~ part of the work is done by24 '

2.d;

16 5, 24x-x-=40 men-, 2 24

25.men and 15 women can complete, a piece ofwork{ii"! " ,in 2 days. '

8 2:. work done by them in 8 days = 12 ="3'

Remaining work is completed by 25 men in 6 days.:. Time taken by 25 men to complete the whole work

3x6 18= --= days.1

From the question,Time taken by 25 men to complete the whole work

36= 3-2 =36 days

3.d;

1 1 1[.,' 12-18 = 36 work is completed by 15women in".

one day.] .One man alone finishes the work in 100 days.:. '10'men finish the work in 10 days.·r,}l{.p n·~f'"r,..n·~·1' - .From the question,

~ - 1 1 I15 women finish in one day, '6 -TO = Is work

:. l S'women finish the whole work in 15 days.:. ' 1women finishes the whole work in 15 x 15

=~25 days.

1-2 M >IrS=-12'W x 18 xi. w = ~ M, 3" 4

, 3'lOM+8W=10M+8x - M=16M

4I L ~ "nJ3fr! "lit .:i jj~

:. 16 'men can complete the same work in

5. b;

12x18 27 116=2'=13'2 days

6. a; 5 I!len -I' 3boys can reap 23 hectares in 4 days (i)3 men + 2 boys can reap 7 hectars in 2 days (ii)• 0.;,,' ~

:. from (1);14'(5 Irien'+'3 boys)' can reap 23 x 14hectares in 4 days

, ....(iii) , ,Now, from (2)23 (3 men + 2 boys) can reap 7 x 2 x 23 hectares in 4-days ....(iv):. 14(5 men -l: 3 boys) =23 (3 men + 2 boys):: 70 men + 42 boys = 69 men + 46 boys

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 = 7 men-+ 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 field

is 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 L

21+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.:. 1 man 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 = 16 men can complete '3 of the work in

72 116 = 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 .