Chapter 15

Time and Work

Rule 1 Theorem: If M , persons can do Wi works in D, days and

M2 persons can do W2 works in D2 days then we have a

very general formula in the relationship of

MlDlW2 = M2D2WX.

Illustrative Examples Ex. 1: 16 men 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 1 day, 16 x 10 men are

16x10 40

needed. So to do the work in 40 days,

men are needed.

Quicker Method: MXD{W2 = M2D2Wl

M , = 16, Z), = 10, Wt = 1 and

Mj = 7, D2 = 40, W2 = 1

Thus, from MXD,W2 =M2D2WX

16xlO = M 2 x 4 0

,V 1 6 X 1 0 A or, M2 = —71— - 4 men. 40

Ex. 2: 40 men can cut 60 trees in 8 hours. I f 8 men leave the job, how many trees wil l be cut in 12 hours?

Soln: Detail Method: 40 men - working 8 hrs - cut 60 trees

or, 1 men - working 1 hr - cuts 60

trees

Thus, 32 men - working 12 hrs - cut

40x8

60x32x12 = 72

40x8 trees. Quicker Method:

U - 40, £>, = 8 (as days and hrs both denote time)

Wx = 60 (cutting of trees is taken as work)

M2 = 4 0 - 8 = 32, D2 =\2,W2 =? Putting the values in the formula, MXD,W2 = M2D2WX

Wehave,40x8x w 2 = 32x 12x60

32x12x60 or, W2

40x8 • = 72 trees.

Ex.3: A can do a piece of work in 5 days. How many days wil l he take to complete 3 works of the same type?

Soln: Quicker Method:

M,DXW2= M2D2W, As 'A' is the only person to do the work in both the

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

£>, = 5 days, Wl = 1 , D2 =? and W2 =3 Putting the values in the formula we have, 5 x 3 = D 2 x l o r , D 2 = 1 5 d a y s .

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)4men c)2men d) None of these

2. 15 men can do a piece of work in 6 days. How many men are 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. I f 4 men leave the job, how many trees wil l be cut in 6 hours? a) 30 trees b) 36 trees c) 40 trees d) None of these

4. 10 men can cut 15 trees in 2 hours. I f 2 men leave the job, how many trees wil l be cut in 3 hours? a) 15 trees b) 20 trees c) 16 trees d) 18 trees

5. A can do a piece of work in 6 days. How many days will • he take to complete 2 works of the same type?

a) 12 days b) 10 days c)6 days d)3 days

Answers l .c £ a 3.b 4.d 5. a

3 6 0

Rule 2 Theorem: If M] persons can do Wx works in £>, days work

ing Tt hours a day and M2 persons can do W2 works in

D, days working T2 hours a day then we have a very gen

eral formula in the relationship of MlDlTyW1 = M2D2T2Wr

Illustrative Example Ex: 5 men can prepare 10 toys in 6 days working 6 hours

a day. Then in how many days can 12 men prepare 16 toys working 8 hrs a day?

Soln: By using the above theorem 5 x 6 x 6 x l 6 = 1 2 x D 2 x 8 x l 0

5x6x6x16 , , . . £), = = 3 days

12x8x10 Note: Number of toys is considered as work in the above

example. Exercise 1. The work done by a woman in 8 hours is equal to the

work done by a man in 6 hours and by a boy in 12 hours. I f working 6 hours per day 9 men can complete a work-in 6 days then in/how many days can 12 men, 12 women and 12 boys together finish the same work working 8 hours per day?

•I , , 2 t L Q . ^ i HXV It, a) 1 —days b) 3— days c) 3 days d) 1 — days

(BSRBPatnaPO-2001) 2. 10 men can prepare 20 toys in 3 days working 12 hours a

day. then in how many days can 24_men prepare 32 toys working 4 hrs a day? a) 2 days b) 3 days" c) 4 days days

3. 20 men can prepare 40 toys in 24 days working 18 hours a day. Then in how many days can 36 men prepare 48 toys working 16 hrs a day? a) 16 days b) 12 days c) 21 days d) 18 days

Answers 1. d; Hint: 8 Women = 6Men =12 Boys

J2M +,12^+ 12^=42A/+£A/+6M=27M Now, applying the above formula, we have

9 * 6 * 6 = 27 *8-x ' D 2

9 x 6 x 6 1 / s

t •• 2 27 x j 2 Q a y S

2. a\d

Rule 3 Theorem: If A and B can do a piece of work in x days, B andCinydays,CandAinzdays,then(A+B + C) working

together will do the same work in 2xyz

xy + yz + xz days

PRACTICE B O O K ON QUICKER MATHS

Let 2xyz

xy + yz + xz be 'r'then

'A' alone will do the same work in

2xyz

f \

y-r days or

xy + yz-zx, days,

B' alone will do the same work in

2xyz days and

^yz + zx-jcy_) *

C alone will do the same work in

2xyz

days or

xr x-r

days or

xz + xy-^z-days.

Illustrative Example Ex: A and B can do a pi^ct of work in 12 days, B and 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,

r = • 2x12x15x20

12x15 + 12x20 + 15x20 = 10 days.

10x15

Now, A can do the work in ——— = 30 days.

10x20 B can do the work in ——— = 20 days. C can do the work in

20-10

10x12 12-10

= 60 days.

Exercise 1. A and B can fin ish a piece of work in 3 0 days, B and C in

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

2 2 a) 26- daysb) 16 - days c) 25 days d) 24 days

2./ A and B can do a piece of work in 10 days, B and C in 15 days and C and A in 20 days. They all work at it for 6 days, and then A leaves, and B and C go on together for 4 days more. I f B then leaves, how long will C take to complete the work? a) 20 days b) 25 days c) 10 days d) 15 days

Time a n d W o r k 361

3. A and B can do a piece of work in 6 days, A and C in

5 — days, B and C in 4 days. In what time could each do

it?

™ 4 8 ^ 19

20 — 8 ^ , 7 13 ' 31 35

^19 20 4

7 — 8 — 20 35 ' 3 1 ' 13

16 19 M 8— 20— 7

; 3 1 ' 13 ' 35

d) None of these

A and B can mow a field in 3 — days, A and C in 4 days,

B and C in 5 days. In what time could they mow it, all working together?

* 75 „ 74 „ 74 • > 3 i 0 4 b > 2 ! o 3 - *3W3 d > 2

47 103

Answers l a

2. c; Hint: A, B and C together can do the work in

2x10x15x20 120 10x15 + 20x10 + 15x20 30

13

days

work done by all in 6 days = 20

work done by B and C in 4 days = —

. . 1 3 ' 4 Remaining work = 1 I . . + 20 15 i ~ 12 > w n ' c n i s t 0 D e

done by C. Now, from the question,

120

C alone can do the whole work in xlO

10 120 13

120 days

[See Rule-6]

3. a;

4.b

. — of the work is done by C in - — = 10 days. 12 J 12 '

11

Hint: Here x = 6, y = 4 and z = — . Now apply the

given rule.

Theorem: If A can do a piece of work in x days and B can do it in y days then A and B working together will do the

same work in f \ xy

x + y days.

Illustrative Example Ex.: A can do a piece of work in 5 days, and B can do it in

6 days. How long will they take i f both work together?

Soln: Detail Method: 'A' can do 7 work in 1 day.

' B ' can do — work in 1 day. 6

1 1 Thus 'A' and ' B ' can do I J" 1"^ j work in 1 day.

1 'A' and ' B ' can do the work in

1 1 - + — 5 6

days

30 „ 8 = T T = 2 T T d a y s .

Quicker Method: Applying the above theorem,

A + B can do the work in 5x6 5 + 6

days

3 0 - 7 8 A - T T - 2 T T d a y s -

Exercise 1. 10 men can complete a piece of work in 15 days and 15

women and complete the same work in 12 days. I f all the 10 men and 15 women work together, in how many days will the work get completed?

a) 6 b ) 7 | d)6~ 3 - / 3

(SBI Associates PO-1999) 2. A can do a piece of work in 20 days and B can do it in 30

days. How long would they take to do it working together? ^ \a) 12 days b) 10 days c) 15 days d) 16 days

3. 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 B and C to complete the work together?

V'Wfif ; 2 2 a) 2 - d a y b) 2~ days c) 6 days d) 4—days

Rule 4 ftrrfes [ X ^ d A does — of a piece of work in 15 days. He does the

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

1 a) 13— days b) 12 days

362 PRACTICE BOOK ON QUICKER MATHS

c) 12— days d) None of these

5. A can do a piece of work in 16 days, B in 10 days. A and B work at it together for 6 days and then C finishes it in 3 days, in how many days could C have done it alone? a) 40 days b) 80 days c) 90 days d) 120 days

, can do a piece of work in 4 hours, B and C can do it in 3 hours, A and C can do it in 2 hours. How long would B alone take to do it? a) 14 hours b) 12 hours c) 10 hours d) 16 hours

can do a piece of work in 30 days while B can do it in 40 days. A and B working together can do it in

3 „_1 ' 1 . a) 70 days b) 42—days c) 27 y UdyS d) 1'ydays

(Railways 1989) can do (1/3) of a work in 5 days and B can do (2/5) of

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

3 3 4 a) 7 —days b) 9-days c) 8—days d) 10 days

4 8 5 (Railways 1991)

Answers 1. c; Hint: x = 15 days, y = 12 days, Now apply the above

2. a

3. b;

rule.

( 6 x 8 _ 24^) Hint: C completes the work in I ^ + g J days

.-. B and C together complete the work in

f 24 x8 x8

7 24

+ 8 + 8 , 7

1^ = 2 2

5 5 days.

J

4. a; Hint:

days

_7_

10 10

4x10

5.d;

.-. The whole work is done by A and B in

40 1 = - r - = 1 3 j days.

Hint: A and B together can do the whole work in

16x10 80

16 + 10 - d a y s .

13 . 39 .-. In 6 days A and B together can do — x 0 = —

80 40

work.

d 3 9 - 1 1 Now, remaining work I 1 40 _ 40 I ' s c ' o n e ^ 3

days .-. The whole work is done by C in 40 * 3 = 120 days.

6. b; Hint: A, B and C together can do a piece of work in

( 4x3\ 1 2 \

4 + 3 ~ 7 J hours.

B alone take to complete a piece of work

f 12 x2

12

7

12 hours.

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

' 1 0 x 5 - ^ l : days. .

B can do the whole work in

A and B together can do the work in

7 5 - 9 3 A T~ s d a y s -

25x15

25 + 15

work is done by A and B in 4 Soln:

Rule 5 Theorem: Jf A, B and C can do a work in x, y and z days respectively then all of them working together can finish

r 1 xyz

\xy + yz + xz \

I l lustrative Example A can do a piece of work in 5 days, and B can do it in 6 days. I f C, who can do the work in 12 days, joins them, how long wi l l they take to complete the work?

the work in

E x :

days.

By the theorem: A, B and C can do the work in

5x6x12

5x6 + 6x12 + 5x12

360 = 2

162 9 days.

Exercise 1. A can do a piece of work in 5 days, B in 4 days and A, B

and C together in 2 days. In what time would C do it alone? a) 25 days b) 12 days c) 15 days d) 20 days

2. A takes half as long to do a piece of work as B takes, and i f C does it in the same time as A and B together, and if all three working together would take 7 days, how long would each take separately?


Thus, 12 men + 16 boys = 24 boys + 16 boys = 40 boys and 7 men + 10 boys = 14 boys + 10 boys = 24 boys Now, by basic formula, we have

40 x 5 = 24 x D 2

n 4 0 x 5 Q 1 ^ or D2 = —^- = 8-days.

Exercise 1. I f 12 men and 16 days can do a piece of work in 5 days

and 13 men and 24 boys can do it in 4 days, compare the daily work done by a man with that done by a boy. a) 1:2 b ) 2 : l c) 1 :3 d) 3 : I

2. IfSOmenand 14 boys can reap a field in 21 days, in how many days will 20 men and 4 boys reap it, supposing that 3 men can do as much as 5 boys? a) 36 days b) 30 days c) 42 days d) 45 days

3. I f 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. a) 2 : 1 b) 3 :1 c) 4 :1 d) Data inadequate

4. I f 1 must hire 2 men and 3 boys for 6 days to do the same

piece of work as 11 men and 5 boys could do in 1 1

days, compare the work of a boy with that of a man. 8)7:3 b)3:7 c)2:5 d)5:2

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

(BankPO Exam, 1988)

Answers 1. b; Hint: Applying the above theorem,

aman'swork 24x4-16x5 16 ± aboy'swork 12x5-4x13 8

2. a; Hint: Here relationship between men and boys is given.

3 men = 5 boys .-. I man = — boys.

Now, 30A/ + 14£ = ̂ ^ 5 - + 14 = 64boys and

20M + 4B = ~ boys.

112 From the formula, 64 x 21 = x D

.-. D, = 36 days.

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

4 x x ] - x x 2 the required answer = :— : r = 2 : 1 ,

Note: Also see Rule-15 4. b; Hint: Applying the given rule, we have

Man Boy

- x 5 - 6 x 3

6x2 — x] 2

5.c; .-. Boy: Man = 3:7. Hint: 2 children = 1 man .-. 8 children + 12 men = 16 men From the question, Since 16 men can complete a certain piece of work 9 days

12 men finish the work in 16x9

12 12 days.

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

x(D + d)~ d

Illustrative Example A certain number of men can do a work in 60 days. Iff] there were 8 men less it could be finished in 10 day*| more. How many men are there? Using the above formula, we have

Ex:

Soln:

the original number of men _ 8(60 + 10)_

10 56 « .

Exercise 1. A certain number of men can do a work in 45 days.

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

2. A certain number of men can do a work in 30 da>f there were 6 men less it could be finished in 20 d?w more. How many men are there? a) 15 men b) 12 men c) 18 men d) 20 men

3. A certain number of men can do a work in 50 da> s there were 6 men less it could be finished in 12 more. How many men are there?

Time and W o r k 369

a) 30 men b) 32 men c) 28 men d) 31 men I A certain number of men can do a work in 70 days. I f

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

Answers : b 2. a 3.d 4.c

(MBA, 1983)

Rule 11 Theorem: If A is 'n' times as fast (or slow) as B, and is therefore able to finish a work in 'D' days less (or more) ;han B, then the time in which they can do it working to

gether is given by Dn

n2-l days

Illustrative Example Lv 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 which they can do it working together.

Soln: Detail Method: A is thrice as fast as B, means that i f A does a work in 1 day then B does it in 3 days. Hence, i f the difference be 2 days, then A does the work in 1 day and B in 3 days. But the difference is 60 days. Therefore, A does the work in 30 days and B in 90 days. Now A and B together will do the work in

——— days = — = 22.5 days 30 + 90 2

Quicker Method: Applying the above theorem, we have

60x3 60x3 the required answer = 3 2 - l

45 2

= 22.5 days. ? Exercise

A is twice as fast as B, and is therefore able to finish a work in 30 days less than B. Find the time in which they can do it working together. aj/TS days b) 20 days c) 24 days d) 22 days

1 A. is 4 times as fast as B, and is therefore able to finish a >* work in 45 days less than B. Find the time in which they

can do it working together. a) 12 days b) 16 days c) 8 days d) 20 days

1 A is thrice as fast as B, and is therefore able to finish a work in 40 days less than B. Find the time in which they can do it working together. a) 16 days b) 10 days c) 15 days d) None of these

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

Answers l .b 2. a 3.c 4. a

Rule 12 Theorem: If a person can finish a work in dx days at ht

hours a day and another person can finish the same work in d2 days at h2 hours a day, then the no. of days in which they can finish the works working together 'h' hours a day

is (h,d,jh2d2)

( M ; ) + ( M ; ) days

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

, 2 finish it in 6— 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 hour a day. Then I can finish the work in 15 * 8 = 120 days

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

Now, we together can finish the work in

120x60 —— = 40 days 120 + 60 '

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

the required answer =

20 15x8x — x 9

3 20

15x8 + — x9 3

1 10

4 days.

Exercise 1. I can finish a work in 10 days at 4 hrs a day. You can

finish it in 15 days at 5 hrs a day. Find in how many days we can finish it working together 10 hrs a day.

50 70 60 40 a) — days b) — days c) — days d) — days

2. I can finish a work in 16 days at 5 hrs a day. You can finish it in 12 days at 4 hrs a day. Find in how many days we can finish it working together 6 hrs a day. a) 5 days b) 4 days c) 6 days d) None of these

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


a ) 3 25 d a y S

c) 3— days

b) 9— days

d) 4 — days

Answers l .c 2. a 3.c

Rule 13 Theorem: If A can do a work in x days, B takes y days to complete it and C takes as long as A and B would take working together, then B and C together take to complete

the work = xy

2x + y • 9 * . A and C together take to com

plete the work = x + 2y d a . v s and A, B and C together take

to complete the work = 2{x + y) days

Il lustrative 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 wil l it take B and C, A and C, and A, B and C to complete the work together?

Soln: Using the above formula, we have,

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

12 + 8

48 12 2 — = — = 2 — days 20 5 5

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

6 + 16

= — = 2 — days 22 11

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

6x8 48 12 ,5 . = —, r = — = — = \

2(6 + 8) 28 7 7 '

Exercise 1. 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 wil l it take B and C to complete the work together?

a) — days b) — days c) — days d) —days

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

20 25 22 20

a)17 b ) u c ) n d ) T , 3. A can do a work in 6 days. B takes 7 days to complete I

C takes as long as A and B would take working togethe* How long wil l it take A and C to complete the work to gether?

a) _ 1 2 — days b)2 days O 2 ±

10 daysd) 3 —da\

4. A can do a work in 8 days. B takes 6 days to complete n C takes as long as A and B would take working together How long wil l it take A and C to complete the work together?

1 , 2 -3 a) 2— days b) 3— days c) 2 — days d) 2—days

A can do a work in 10 days. B takes 15 days to complett it. C takes as long as A and B would take working to-gethcr. How long will it take A, B and C to complete the work together? a)6 days b)3 days c)4 days d)8 days

6. A can do a work in 20 days. B takes 5 days to complete it. C takes as long as A and B would take working together. How long wil l it take A, B and C to complete the work together?

a)2 days b)4 days c)3 days d)6 days

Answers l .b 2. a 3.c 4.d 5.b 6.a

Rule 14 Theorem: A is n times as good a workman as B. If together they finish the work in x days, then A and B separately cam

{ — ) '

\ J spectively. I l lustrative Example

A is twice as good a workman as B. Together, the> finish the work in 14 days. In how many days can it be done by each separately? Detail Method: Let B finish the work in 2x days.Since A is twice as active as B therefore, A finishes the work in x days.

(A + B) finish the work in

do the same work in

Ex:

\x Jays jjjjjJ+ l)x4aysrm

Soln:

= 14 3x

orx = 21 .-. A finishes the work in 21 days and B finishes the work in 21 * 2 = 42 days.


Quicker Method: Applying the above formula,

the required answer = 40x45 Y 40-23 40 + 45 40

= 9 days.

Exercise 1. A and B can do a work in +0 and 35 days respectively.

They began the work together, but A left after some time and B finished the remaining work in 10 days. After how many days did A leave?

, , 1 c) '3 — days d) 14 days a) 13— daysb) 13 days

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

5 5 5 c) days d) o— days

6 6

4.

a) 6 days b) 5 days

A and B can do a work in 20 and 15 days respectively. They began the work together, but A left after some time and B finished the remaining work in 8 days. After how many days did A leave? a)4 days b)5 days c)3 days d)6 days A and B can together finish a work in 30 days. They worked for it for 20 days and then B left. The remaining work was done by A alone in 20 more days. A alone can finish the work in: a) 54 days b) 60 days c) 48 days d) 50 days

(Central Excise 1988)

Answers l .c 2.c 3.a 4. b; Hint: In the given formula, we have

- ^ - = 30 days. x + y

Now, from the question, B left the work, ie y = time taken by A to complete the whole work and z = 20 days. Now, applying the given formula, we have

30 x :20 y = 60 days.

^\e 17 > Theorem: If x, men or x2 women or x3 boys can do a work in 'D' days, then the no. of days in which I man, 1 woman and 1 boy do the same work is given by the following formula, number of required days =

Dxx, xx2 xx3

X,X2 + X,X; +x,x 1*3 days.

Illustrative Example Ex: 1 man or 2 women or 3 boys can do a work in 44 days.

Then in how many days will 1 man, 1 woman and 1 boy do the work? Applying the above formula, we have Soln:

the no. of required days 44x1x2x3

1x2+2x3+1x3

44 x 1 x 2 x 3 2 + 6 + 3

•24 days.

Exercise 1. 2 men or 3 women

Then in how many do the work? a) 24 days b) 42

2. 3 men or 4 women Then in how many do the work? a) 40 days b) 50

3. 1 man or 3 women Then in how many do the work? a) 24 days b) 12

4. 1 man or 2 women Then in how many do the work? a) 24 days b) 28

Answers l . d 2.c 3.a

or 4 boys can do a work in 52 days, days will 1 man, 1 woman and 1 boy

days c) 36 days d) 48 days or 5 boys can do a work in 47 days, days wil l 1 man, 1 woman and 1 boy

days c) 60 days d) 45 days or 4 boys can do a work in 38 days, days will 1 man, 1 woman and 1 boy

days c) 18 days d) 36 days or 4 boys can do a work in 56 days, days wil l 1 man, 1 woman and 1 boy

days c) 20 days d) 32 days

4.d

Rule 18 Theorem: A group of men decided to do a work in x days, but 'n' of them became absent. If the rest of the group did the work in 'y' days, then the original number of men is

given by ny

y-x men.

Illustrative Example Ex:

Soln:

A group of men decided to do a work in 10 days, but five of them became absent. I f the rest of the group did the work in 12 days, find the original number of men. Applying the above formula, we have

the required answer 5x12

12-10 = 30 men.

Exercise 1. A group of men decided to do a work in 13 days, but 6 of

them became absent. I f the rest of the group did the work in 15 days, find the original number of men.

Time and Work

a) 30 men b) 35 men c) 40 men d) 45 men 2. A group of men decided to do a work in 12 men, but 8 of

them became absent. I f the rest of the group did the work in 20 days, find the original number of men. a) 18 men b) 20 men c) 22 men d) 24 men

3. A group of men decided to do a work in 15 days, but 2 of them became absent. I f the rest of the group did the work in 25 days, find the original number of men.

373

a) 5 men

Answers l . d 2.b

b) 4 men c) 7 men d) 6 men

3.a 7 Rule 19

Theorem: A certain number of men can do a work in 'D' days. If there were 'x'men more it could be finished in'd'

lx(D-d) days less, then the number of men originally are .

or

No. of more workers x Number of days taken by the second group

No. of less days

I l lustrative Example Ex.: A certain number of men can do a work in 60 days. I f

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

Soln: Applying the above rule, we have original number of workers

No. of more workers x No. of days taken by the second group

No. of less days

8x(60-10) _ 8x50 _ 4 Q

10 10 men.

Exercise 1. A certain number of men can do a work in 50 days. I f

there were 3 men more it could be finished in 5 days less. How many men are there? a) 36 men b) 18 men c) 27 men d) 30 men

2. A certain number of men can do a work in 75 days. I f there were 6 men more it could be finished in 15 days less. How many men are there? a) 20 men b) 24 men c) 28 men d) 32 men

3. A certain number of men can do a work in 35 days. I f there were 10 men more it could be finished in 10 days less. How many men are there? a) 25 men b) 20 men c) 15 men d) 30 men

Answers l .c 2.b 3.a

Rule 20 A Theorem: A builder decided to build a farmhouse in 'D' days. He employed 'x'men in the beginning and 'y' more men after'd' days and completed the construction in stipulated time. If he had not employed the additional men, th en the men in the beginning would have finished it in

D(x + y)-yd days and it would have been

y(D-d)

days behind the schedule.

I l lustrative Example Ex.: A builder decided to build a farmhouse in 40 days. He

employed 100 men in the beginning and 100 more after 35 days and completed the construction in stipulated time. I f he had not employed the additional men, how many days behind schedule would it have been finished?

Soln: Detail Method: Let 100 men only complete the work in x days. Work done by 100 men in 35 days + Work done by 200 men in (40-35 =)5days=l .

35 200x5 , or, — + — = 1

x 45

lOOx

or, — = 1 •'• x = 45 days

Therefore, i f additional men were not employed, the work would have lasted 45 - 40 = 5 days behind schedule time. Quicker Approach: 200 men do the rest of the work in 40 - 35 = 5 days.

5x200

.-. 100 men can do the rest of the work in - - 10

days.

.-. required number of days = 1 0 - 5 = 5 days. Quicker Method: Applying the above theorem, we have

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

100 •• 5 days.

Exercise 1. A builder decided to build a farmhouse in 45 days. He

employed 150 men in the beginning and 120 more after 30 days and completed the construction in stipulated time. I f he had not employed the additional men, how many 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. He employed 50 men in the beginning and 50 more after 40 days and completed the construction in stipulated time. I f he had not employed the additional men, in how many


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

2. There is a sufficient food for 116 men for 25 days. After 21 days, 100 men leave the place. For how many days will 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. After 29 days, 210 men leave the place. For how many days will 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 15 days. After 10 days, 75 men leave the place. For how many days will the rest of the food last for the rest of the men? a) 10 days b)8 days c)5 days d) 15 days

Answers l .b 2.c 3.d 4.a

Rule 2 3 7 Theorem: A takes as much time as B and C together take to finish a job. If A and B working together finish the job in x days. C alone can do the same job in y days, then B alone "/ t ->.... \ can do the same work in

the same work in 2xy

2xy y-x

days.

days and A alone can do

Illustrative Example Ex: A takes as much time as B and C together take to

finish a job. A and B working together finish the job in 10 days. C alone can do the same job in 15 days. In how many days can B alone do the same work? Quicker Method I: Using the above theorem, B alone Soln:

2x15x10 15-10

can do the same work in

Quicker Method II:

(A + B) + (C) can do in , , 15 + 10

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

60 days

15x10 6 days.

15x12 £ n

.-. B [ B = {B + C } - C ] c a n d o i n 1 $ _ 1 2 = 6 ° days.

Exercise 1. A can do a certain work in the same time in which B and

C together can do it. I f A and B together could do it in 10 days, and C alone in 50 days, in what time could B alone do 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 B and C together can do it. I f A and B together could do it in 15

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

3. A can do a certain work in the same time in which B and C together can do it. I f A and B together could do it in 12 days, and C alone in 24 days, in what time could B alone do 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 B and C together can do it. I f A and B together could do it in 10 days, and C alone in 15 days, in how many days can A alone do the same work?

a) 12 days b) 60 days c) 24 days d) 48 days

Answers l .a 2.b 3.d 4.a

Rule 24 y Theorem: A team of xpersons is supposed to do a work in 'D' days. After 'dt' days, 'y' more persons were employed and the work was finished' d2 ' days earlier, then the number of days it would have been delayed if 'x' more persons

\y{D-{dl+d2)}-d2x were not employed is given by days and the number of days in which the work would have

~{x + y\D-d2)-d,y x been finished is given by days

Illustrative Example Ex: A team of 30 men is supposed to do a work in 38 days.

After 25 days, 5 more men were employed and the work finished one day earlier. How many days would it have been delayed if 5 more men were not employed?

Soln: Quicker Approach: 35 men do the rest of the job in 12 days (12 = 38-25 - 1 ) .-. 30 men can do the rest of the job in

= 14 days. 12x35

30 Thus the work would have been finished in 25 + 14 = 39 days that is, (39-38)= 1 day after the scheduled time. Quicker Method: Applying the above formula, we have

5{38-(25 + l )}-1x30 the required answer = 30

5x12-30 30

1 day .

Exercise 1. 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


finished 2 days earlier. How many days would it have been delayed i f 15 more men were not employed?

, 1 , 1 a) 2 days b) 2 — days c) I — days d) 1 day

o o 2. A team of 25 men is supposed to do a work in 44 days.

After 18 days, 2 more men were employed and the work finished 1 day earlier. How many days would it have been delayed if 2 more men were not employed? a) 1 day b) 2 days c) 1.5 days d) None of these

3. A team of 20 men is supposed to do a work in 30 days. After 12 days, 5 more men were employed and the work finished 2 days earlier. In how many days would it have been finished i f 5 more men were not employed? a) 30 days b) 28 days c) 32 days d) 34 days

4. A team of 27 men is supposed to do a work in 36 days. After 30 days, 9 more men were employed and the work finished 3 days earlier. In how many days would it have been finished if 9 more men were not employed? a) 35 days b) 28 days c) 34 days d) 39 days

Answers l .b 2.a j . c 4.c

Rule 25 y> Theorem:A,B and C can do a work in x days, y days and z days respectively. They started the work together but after

dx days A left. If B left the work d^ days before the completion of the work, then the whole work will be completed in

y(x-d,)+d2x

y + z days

Illustrative Example

, . 4 Ex: A, B and C can do a work in 16 days, »2— days and

32 days respectively. They started the work together but after 4 days A left. B left the work 3 days before the completion of the work. In how many days was the work completed?

Soln: Detail Method: Suppose the work is completed in x days, As 4 day's work + B's (x - 3) day's work + C's x day's work = 1

4 (x-3)S x , or — + - — + — = 1

' 16 64 32 16 + 5JC-15 + 2 X ,

64 or, Ix +1 = 64 .". x = 9 days. Quicker Method: Applying the above formula, we have

the required answer:

32 16

y ( l 6 - 4 ) + ( 3 x l 6 )

64 + 32

= 2 x 4.5 = 9 days.

Exercise 1. A, B and C can do a piece of work in 12, 18 and 24 days

respectively, they work at it together, A stops the work after 4 days and B is called off 2 days before the work is done. In what time was the work finished? a) 12 days b) 14 days c) 16 days d)8 days

2. A, B and C can do a piece of work in 6, 9 and 12 days respectively, they work at it together, A stops the work after 2 days and B is called off 1 day before the work is done. In what time was the work finished? a)4 days b)6 days c)7 days d)3 days

3. A, B and C can do a piece of work in 18,27 and 12 days respectively, they work at it together, A stops the work after 6 days and B is called off 3 days before the work is done. In what time was the work finished?

• 6 a) 6 days b) 8 days c) 10 days d) 6— days

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

Answers l . d 2.a 3.d 4.c

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

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

Had A left the work after working for a2 days, B would

have finished the remaining work in b2 days. Then, each of them ieA and B, working alone finish the whole work in

b2a, -b,a2

days and a2b, a,b2

days respectively.

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

Then B was called and he finished the work in 9 days. Had A left the work after working for 3 days, B would have finished the remaining work in 6 days. In how many days can each of them, working alone, finish the whole work?

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


2 9 _ 3 6 , or, ~ + - 1 Similarly, ~ + y -•'

To solve the above equation put — = a and _ 0 .

Thus 2a + 9 b = l (1) and 3a + 6b =1 ....(2) Performing (2) x 3 - (1) * 3 we have

1 1 ' • 5a = 1 • a = — or, * -> days.

5 n

1 , c and y = T days.

Quicker Method: In such case: (Using the above theorem)

3 x 9 - 2 x 6 15 _ A will finish the work in — ~ — — = .5 days.

9 — 6 3

For B, we should use the above result.

2 3 B does 1 — = - work in 9 days.

5 5 3

:. B does 1 work in 9 x — - 15 days.

Exercise 1. A started a work and left after working for 1 day. Then B

. . 1 was called and he finished the work in 4 — days. Had A

left the work after working for 1— days, B would have

finished the remaining work in 3 days. In how many days can each of them, working alone, finish the whole work? a) 5 days, 15 days b) 2.5 days, 7.5 days c) 3.5 days, 8.5 days d) None of these

2. A started a work and left after working for 3 days. Then

B was called and he finished the work in 13 — days. Had

A left the work after working for 4— days, B would

have finished the remaining work in 9 days. In how many days can each of them, working alone, finish the whole work? a) 7.5 days, 22.5 days b) 7 days, 9 days c) 5 days, 1~5 days d) 23.5 days, 8.5 days

3. A started a work and left after working for 4 days. Then B was called and he finished the work in 18 days. Had A left the work after working for 6 days, B would have finished the remaining work in 12 days. In how many

days can each of them, working alone, finish the whoie work? a) 5 days, 20 days c) 15 days, 30 days

Answers l .b 2. a 3.b

b) 10 days, 30 days d) 5 days, 30 days

Rule 27 Theorem: A can do a work in x days and B can do the same work in y days. If they work together for'd' days and A goes away, then the number of days in which B finishes the

work is given by y-\ + days.

Illustrative Example Ex: A can do a work in 25 days and B can do the same

work in 20 days. They work together for 5 days and then A goes away. In how many days will B finish the work?

Soln: Detail Method

A + B can do the work in 5 days = 5 25 20

5x45 25x20

_9_ 20

Rest of the work = 1 _9_ 20 20

9 11 B will do the rest of the work in = 1 = — days.

20 20 Quicker Method: Applying the above theorem, we have

the required answer = 2 0 - ( l + ^ - j x 5

= 2 0 - 9 = 11 days.

Exercise

1. A can do a piece of work in 6 y days and B in 5 days.

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

a ) l 1

b)3 days c)2 days d) 1 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 to finish 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 and B in 15 days. They work together for 6 days and then A leaves B to


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

1 A a) 3 days b) 4 days c) 3— days d) 4— days

4. A can do apiece of work in 12— days and B in 10 days.

They work together for 2 ^ days and then A leaves B

to finish the work alone. How lone will B take to finish it?

i : n a) — days b) — days c) 6 days d) — days

Answers l .a 2.c 3.d 4.d

Rule 28

Theorem: If A can complete — part of a work In x days,

can he finish — of the work?

a) 20 days c) 4 days

b)5 days d) Data inadequate

3. Sudhir can do — of a work in 8 days. In how many days

1 can he finish — of the work?

a) 1 day b) 2 days c) 3 days d) None of these

Answers l .c 2.c 3.a

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

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

then the — part of the work will be done in y days. We can

x _ y calculate the value of y from the given equation a ^ ~ c ^ .

No. of days worked Note: = constant for a person

Part of work done

Illustrative Example

3 Ex: A can do — of a work in 12 days. In how many days

can he finish — of the work? 8

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

12 y 12 1

374" = W 0 r > ' = T > < 4 X 8 = 2 d 3 y S -

Exercise

1. Ram can do — of a work in 16 days. In how many days

can he finish — of the work? 12

a) 1 day b) 3 days c) 2 days d) 2 — days

2. Vinay can do — of a work in 5 days. In how many days

hour as 3 men of the second group do in 1 — hr.

Soln: Detailed Method: 2 x 1 men of first group = 3><1.5 men of second group or, 2 men of first group = 4.5 men of second group

38 men of first group :

4.5 x38 = 19x4.5

y (19x4.5) men do 1 work, working 6 hrs/day in 12 days.

.-. 1 man does 1 work working 1 hr/day in (12 x 19x4.5 x 6) days.

.-. 57 men do 2 work working 8 hrs/day in

12xl9.x4.5x6 57x8

2 = 27 days.

Quicker Method: Ratio of efficiency of persons in first group to the second group

= E, : E 2 =(3x1.5):2x1 = 4.5:2 (*) Now, use the formula:

M ^ . T . E . W j = M 2 D 2 T 2 E 2 W , (*)(*)

38x12x6x4.5x2 _„ , .'. D , = = 27 days.

57x8x2x1 Note: (*) Less number of persons from the first group

do the same work in less number of days, so they are more efficient.

(*)(*) M represents the number of men. D represents the number of days. T represents the number of working hours. E represents the efficiency. W represents the work and the suffix represents the respective groups.

Time a n d W o r k

Exercise 1. 40 men, working 8 hours a day can do a piece of work in

15 days. Find the number of days in which 60 men working 4 hrs a day can do twice the work. Assume that 3 men of the first group do as much work in 2 hour as 4 men of the second group do in 3 hrs. a) 60 days b) 40 days c) 80 days d) None of these

2. 30 men, working 4 hours a day can do a piece of work in 10 days. Find the number of days in which 45 men working 8 hrs a day can do twice the work. Assume that 2 men of the first group do as much work in 2 hour as 4 men of the second group do in 1 hr.

,1 .2 .3 .1 a) 6-days b) 6 —days c) 5 7 days d) 3— days

3 3 6 6

Answers 1. c; Hint: 3><2 men of first group = 4 x 3 men of second

group .-. Ratio of efficiency of persons in first group to the second group = £, : £ 2 = 2 : 1 . Now apply the given formula.

2. b

Rule 30 Y Theorem: If A working alone takes 'x' days more than A

and B, and B working alone takes 'v' days more than A and

B together then the number of days taken by A and B work

ing together is given by \[xy \

Illustrative Example Ex: A alone would take 14 hours more to complete the j ob

than if both A and B would together. If B worked

alone, he took 3— hours more to complete the job

than A and B worked together. What time, would they take if both A and B worked together?

Soln: Applying the above theorem, we have

the required answer = ^—-— = 7 hours.

Exercise 1. A alone would take 8 hours more to complete the job

than if both A and B would together. If B worked alone,

he took 4— hours more to complete the job than A and

B worked together. What time, would they take if both A and B worked together? a) 6 hours b) 5 hours c) 7 hours d) 8 hours

(Income Tax and Excise Fram, 1985) 2 A alone would take 16 hours more to complete the job

than if both A and B would together. If B worked alone, he took 4 hours more to complete the job than A and B worked together. What time, would they take if both A and B worked together? a) 5 hours b) 8 hours c) 9 hours d) None of these

3. A alone would take 27 hours more to complete the job than if both A and B would together. If B worked alone, he took 3 hours more to complete the job than A and B worked together. What time, would they take if both A and B worked together? a) 8 hours b) 10 hours c) 9 hours d) 6 hours

Answers l .a 2.b 3.c

Rule 31 ^ Theorem: If A, B and C can do a job alone in x days, y days and z days respectively. .-. alone time for A =xdays

alone time for B=y days alone time for C = z days

Now consider the following cases, Case I: To find the amount of work done by A, B and C

separately. Using the formula,

Number of days actually worked Amount of work =

alone time and assuming that A, B and C have worked for

d, days, d 2 days and d 3 days respectively, then

I d, amount of work by A = — , amount of work by

B = — and amount of work by C = — •

f * 8 « o 7 * 3 & » •"• r ' '?W<4 fester* Case II : If the job is complete, then add the amount of work

done by A, B and C and equate it to 1.

d d d ie — + — + — = 1, if the job is half complete the

x y z following equation is obtained,

x y z 2

Illustrative Example Ex: A man, a woman or a boy can do a job in 20 days, 30

days or 60 days respectively. How many boys must assist 2 men and 8 women to do the work in 2 days.

(MBA 1992) Soln: Let the required number of boys be x.

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

380

2 days) + (x boy's work for 2 days) = 1

or, f 2 x 2 x — l + f 8 x 2 x — l + f x x 2 x — I 20) { 30) { 60.

.-. x = 8 boys.

Exercise 1. 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 it for 5 days, and B for 7 days. C finishes it in 13 days. In how many days could each do the work by himself? a) 16,48 and 26 days respectively b) 16,48 and 24 days respectively c) 26,48 and 24 days respectively d) 16,46 and 24 days respectively

2. A can do ajob in 20 days, B in 30 days and C in 60 days. If A is helped on every 3rd day by B and C, then in how many days, the job is finished?

[ITI1989] a) 20 days b) 15 days c) 18 days d) 24 days

3. A can do a job in 12 days, B in 15 days. They work together for 2 days. Then B leaves and A alone continues the work. After 1 day C joins A and work is completed in 5 more days. In how many days can C do it alone? a) 15 days b) 20 days c) 25 days d) 30 days

4. A and B can do ajob in 15 days and 10 days respectively. They began the work together but A leaves after some days and B finished the remaining job 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 respectively. 4 days before finishing the job, A joins B. B has started the work alone. Find how many days B has worked alone? [ Bank P O l 9891 a)8 days b) 10 days c)4 days d)5 days

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

[MBA 1992] a) 8 boys b) 10 boys c) 12 boys d) 16 boys

7. A can do ajob in 3 days less time than B. A works at it alone for 4 days and then B takes over and completes it. I f altogether 14 days were required to finish the job, how many days would each of them take alone to finish it? a) 13 days, 16 days b) 12 days, 15 days c) 15 days, 12 days d) 15 days, 18 days

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

[MBA 1988] a) 48 days b) 36 days c) 40 days d) 50 days

A and B in 1 day do

B and C in 1 day do

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

J_ 12 '

J_ 16 '

As 5 day's work + B's 7 day's work + C's 13 day's work = 1 Or, As 5 day's work + B's 5 day's work + B's 2 day's work + C's 2 day's work + C's 11 day's work = 1

5 2 J •. — + — + C'5 11 day'swork= 1

12 16

C's 11 day's work = h 12 + 16 24

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

.-. B's 1 day's work =

24

_ L _ - _ L - _ L 16 24 " 4 8

1 1 .-. A's day's work = — - — •

12 48 16 .-. A, B and C can do the work in 16, 48 and 24 da\respectively.

2. b; Hint: Since A is helped by B and C on every 3rd day A works for 3 days while B and C work for 1 day

1 , 1 , 1 , 1 2 0 X 3 0 X 6 0 + 5 [ v B a n d C n e l P o n l y ° n

3rd day] .-. Total time for the job = 3 x 5 = 15 days.

3. c; Hint: Let C do it alone in x days A's amount of work +B's amount of work +C's amour: of work = 1

° ' . ( 2 + , + 5 ) 7 r K ) + K ) = '

it

or, A = , - f - 8 - + ^

12 15 5 1

or — = 7 .-. x = 25

.-. C can do it alone in 25 days. 4. c; Hint: In this problem, total time for the work is not

known and also it is not to be found out. Hence tota time for the work is not to be considered. I f A leaves after x days ie A works for x days and B works for x + 5 days, then applying the given rule, we have

No. of days A worked No. of days B worked _ ; Alone time for A Alone time for B

Erne and Work 381

x x + 5

o r ' T I + l r r = l o r ' x = 3

. . A leaves after 3 days. Hint: I f B works alone for x days; A's amount of work + B' s amount of work = 1

° r ' l ? + ~ i 2 ~ - 1 • • X = 5

Hint: Using the given rule we have (2 men's work) + (8 women's work) + (x boy's work) = 1

or, 20 2 x 2 x — 1 + 1 8x2x — ) + f xx2x — U- !

30 60

1 8 * , or, - + — + — = 1

' 5 15 30

6 + 16 + * r, — - 1 .-. x = 8boys.

30

: Hint: Let A alone takes x days to finish the work and hence B alone takes (x + 3) days. Now, using the given rule, we have A's amount of work + B's amount of work = 1

4 10 , or, - + r = l *=12

x x + 3 .-.A alone takes 12 days and B alone takes (12 + 3 = 15) days to complete the work.

8 8 8 . Hint: — + 77 + - = 1 .'. * = 48 days. 24 16 x

Rule 32 )rem: Two persons A and B can finish a job alone in x

i y days respectively. If they start working on alternate then to find the total job completion time, following

weps are taken. ce: This formula is applicable only when* andy are inte-

• w : I f A starts the work Hep I : First calculate the value of p; where p = nearest inte

ger value to be considered

n :

ar+y*

(a) When, x - y = ±2, ± 4 then, apply the following

formula.

T (Total job completion time) _ xy + p(x-y)

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

formula,

T (Total job completion time^ = — — — —

I l lustrative Examples Ex. 1: A and B working alone can finish a job in 5 days and

7 days respectively. They work at it alternately for a day. I f A starts the work, find in how many days the job wi l l be finished?

Soln: Applying the above theorem:

Step I: P = xy

x + y

3x7 35 - 4

^ + 7 * j2 (nearest integer value)

Step I I : x - y = 5- 7 = -2, Here, formula (a) wi l l be applied .*. Total time to finish the job i f A starts the work

_ xy + p(x-y) _ 5x7 + 3(5-7)

x ' 5

29 . 4

= T = 57 d a y s ' Ex. 2: A and B working separately can do a work in 9 and 12

days respectively. A starts the work and they work on alternate days. In how many days wi l l the work be completed?

Soln: Applying the above theorem,

Step I: P = -12x9 108

x 5 (nearest integer value) 12 + 9 21

Step I I : x - y = 9-12 = -3, Here formula (b) wi l l be applied.

.-. Total time to finish the job i f A starts the work

_ s y - p ( j t - y ) ^ ( 9 x l 2 ) - 5 ( 9 - 1 2 )

V 12

108 + 15 41 1 = — — = - = 10- days

Now we try to solve the above examples by Detail Method.

Ex.1: Detail Method:

In the first day A does 7 of the work

In the second day B does — of the work

1 1 _ + _ : 5 7

12

35J in the first 2 days

12 ,

in 4 days 35 x 35" 0 1 m e work

of the work

24

Now, 1 2 4 1 _ H

1 ~ ~ J - ~ of the work remains to be done.

In fifth day A does — of the work

B wi l l finish the work 11

35 5, 35 of the work


at 9 am + 9— hrs = 6— nm. 2 2 v

c; Hint: P 8x10

18 5 8 5 , here x - y = 10 - 8 = +2, hence

apply the formula (a).

required answer;

80 + 10 10

: 9 days.

Since Ram starts on 1 st January .-. work will be completed on 9th day ie on 9th of January.

6x10 „ 5. a; Hint: P = — — * 4

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

6x10 + 4(6-10) .-. required time = —

6 22 1

= — - / • - hours.

.-. required answer = 8 am + 7 - hours = 3:20 pm.

Rule 33 T h eo rem: If A, BandC together can do a work in x days, A ilone can do the work in 'a' days and B alone can do the *ork in 'b' days, then C will do the same work in

x ab ab - x(a + b) days.

lustrative Example c A, B and C together can do a work in 6 days. A alone

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

: i n: Applying the above formula, we have

6x18x27 the required answer = c

H 18x27-6(18 + 27)

- 13 i days.

.ericse 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 work in 9 days. Find in what time C alone can do that work?

a) 4— days b) 6 —days c) 9 days d) None of these

A, B and C together can do a work in 8 days. A alone can

do the work in 24 days and B alone can do the same work in 36 days. Find in what time C alone can do that work? a) 9 days b) 15 days c) 18 days d) 24 days

3. A, B and C together can do a work in 4 days. A alone can do the work in 12 days and B alone can do the same work in 18 days. Find in what time C alone can do that work? a) 8 days b) 27 days c) 9 days d) 18 days

4. A, B and C together can do a work in 12 days. A alone can do the work in 36 days and B alone can do the same work in 54 days. Find in what time C alone can do that work?

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

Answers l .a 2.c 3.c 4.d

Rule 34 Theorem: If A and B can do a work in x andy days respectively and A leaves the work after doing for 'a' days, then B

~{x-a)y~ does the remaining work in days.

Illustrative Example Ex: A can complete a work in 25 days and B can do the

same work in 10 days. I f A after doing 4 days, leaves the work, find in how many days B will do the remaining work?

Soln: Applying the above formula, we have the required answer

_ ( 2 5 - 4 ) x l 0 _ 21x10 42 _ 0 2 Tt —zrz— - — - 8— davs. 25 23 5 5

Exercise 1. A can complete a work in 20 days and B can do the same

work in 25 days. I f A after doing 5 days, leaves the work, find in how many days B will do the remaining work?

a) 18— days

c) 17— days

b) 8— days 4

d) None of these

2. A can complete a work in 35 days and B can do the same work in 28 days. I f A after doing 10 days, leaves the work, find in how many days B wil l do the remaining work? a) 25 days b) 20 days c) 27 days d) 24 days

3. A can complete a work in 24 days and B can do the same work in 18 days. I f A after doing 4 days, leaves the work, find in how many days B wil l do the remaining work? a) 10 days b) 12 days c) 15 days d) 16 days

Answers l .a 2.b 3.c

384 PRACTICE B O O K ON QUICKER MATHS

Rule 35 Theorem: If A and B can do a work in x andy days respectively, and B leaves the work after doing for 'a' days, then

A does the remaining work in days.

Illustrative Example Ex: A can do a work in 15 days and B alone can do that

work in 25 days. I f B after doing 5 days leaves the job, find in how many days A will do the remaining work.

Soln: Applying the above formula, we have

_ (25-5 )x l5 25

the required answer

20x15 25

= 12 days

Exercise 1. A and B working together can do a piece of work in 6

days, B alone could do it in 8 days. Supposing B works at it for 5 days, in how many days A alone could finish the remaining work? a) 9 days b)8 days c)6 days d) 12 days

2. A and B working together can do a piece of work in 10 days, B alone could do it in 20 days. Supposing B works at it for 4 days, in how many days A alone could finish the remaining work? a) 9 days b) 12 days c) 16 days d) 10 days

3. A and B working together can do a piece of work in 30 days, B alone could do it in 50 days. Supposing B works at it for 10 days, in how many days A alone could finish the remaining work? a) 12 days b) 60 days c) 16 days d) 18 days

4. A and B working together can do a piece of work in 7 1

days, B alone could do it in 12— days. Supposing B

v 1 works at it for 2— days, in how many days A alone could finish the remaining work? a) 5 days b)8 days c)7 days d) 15 days

5. A can complete a job in 9 days. B in 10 days and C in 15 days. B and C start the work and are forced to leave after 2 days. The time taken to complete the remaining work is: (NDA Exam 1987| a) 13 days b) 10 days c)9 days d)6 days

Answers 1. a; Hint: First apply the Rule-6. and find the no. of days

in which A alone could do the whole work ie 6x8 „ ;

Now, applying the given rule, we have

2.c

5.d;

the required answer

3.b 4.d

(8-5)24 _ 9 days.

15x10 15 + 10 Hint: B and C together can do the work in

days. (See Rule-5) Here, y = 6 days, and x = 9 days. Now applying the given rule, we have

(6 -2 )x9 the required answer = 0 days.

Rule 36 Theorem: A and B can do a piece of work in x andy dan respectively and both of them starts the work together. Ifi leaves the work 'a' days before the completion of work, then the total time, in which the whole work is completed.

x + y days.

Illustrative Example Ex: rt and B can do a piece of work in 15 days and 3

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


the required answer = (25 + 7)15 ' 25 + 15

12 days.

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

Both starts the work together for some time, but B leav

the job 5 days before the work is completed. Find rae| time in which work is finished, a) 7 days b) 12 days c) 14 days d) 16 da\

2. A and B can do a piece of work in 25 days and 35 az Both starts the work together for some time, but B lea the job 7 days before the work is completed. Fine time in which work is finished.

x-l^jiii-. ••• -rtitrr hTiriftun-Wit mW a) 17 days b) 17 - days c) 18 days d) 20 day

3. A and B can do a piece of work in 30 days and 45 Both starts the work together for some time, but B I the job 15 days before the work is completed. Fine time in which work is finished, a) 24 days b) 28 days c) 20 days d) 16 da\

4. A and B can do a piece of work in 16 days and 24 Both starts the work together for some time, but B I the job 6 days before the work is completed. Find time in which work is finished, a) 18 days b) 14 days c) 12 days d) 8 days

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

Time and Work 385

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

a) 3— days

c) 13 j days

Answers l .c 2.b 3.a

b) 5 J J days

d) None of these

4. c 5.c

Rule 37 Theorem: A and B can do a piece of work in x andy days respectively and both of them starts the work together. If A leaves the work 'a' days before the completion of the work, then the total time in which the whole work is completed

_ (x + a)y

: {x+y)

Illustrative Example Ex: A and B can do a piece of work in 10 days and 20 days

respectively. Both starts the work together but A leaves the work 5 days before its completion time. Find the time in which work is finished.


the required answer = x_ + ^ ® = jo days. 10 + 20

Exercise 1. A can do a piece of work in 14 days and B in 21 days.

They begin together. But 3 days before the completion of the work, A leaves off. In how many days is the work completed?

a) 10 days b) 5 days J Mm c) •>— days d) I U — days

A can do a piece of work in 15 days and B in 25 days. They begin together. But 5 days before the completion of the work, A leaves off. In how many days is the work completed?

1 1 3 a) 12—days b) 13— days c) 11— days d) 25 days

A can do a piece of work in 20 days and B in 40 days. They begin together. But 10 days before the completion of the work, A leaves off. In how many days is the work completed? a) 10 days b) 15 days c) 20 days d) 25 days A can do a piece of work in 5 days and B in 10 days.

, • • . - 1 They begin together. But 2— days before the completion of the work, A leaves off. In how many days is the

work completed? a) 2 days b) 4 days

Answers l . d 2.a 3.c 4.c

c) 5 days d) 8 da\

Rule 38 Theorem: A can do a piece of work in x days. If A does the work only for 'a' days and the remaining work is done by B

xb \ in 'b' days, the B alone can do the work in

X-M days.

I l lustrative Example Ex: A can do a piece of work in 12 days. A does the work

for 2 days only and leaves the job. B does the remaining work in 5 days. In how many days B alone can do the work?


12x5 the required answer

12-2 6 days.

Exercise 1. 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 remaining work in 8 days. In how many days B alone can do the work? a) 12 days b) 10 days c) 15 days d)8 days

2. A can do a piece of work in 25 days. A does the work for 5 days only and leaves the job. B does the remaining work in 4 days. In how many days B alone can do the work? a)5 days b) 15 days c) 9 days d) None of these

3. A can do a piece of work in 23 days. A does the work for 11 days only and leaves the job. B does the remaining work in 9 days. In how many days B alone can do the work?

1 3 a) 17 days b) 18 days c) 17 — days d) 17 — days

4. A can do a piece of work in 22 days. A does the work for 12 days only and leaves the job. B does the remaining work in 5 days. In how many days B alone can do the work? a) 11 days b) 10 days c) 12 days d) 14 days

5. A can do a piece of work in 80 days. He works at it for 10 days and then B alone finishes the work in 42 days. The two together could complete the work in: a) 24 days b) 25 days c) 30 days d) 35 days

(Clerical Grade Exam, 1991)

Answers l .b 2.a 3.c 4.a

Time and Work

6. Mahesh and Umesh can complete a work in 10 and 15 days respectively. Umesh starts the work and after 5 days Mahesh also joins him. In all, the work would be completed in: a) 9 days b) 7 days c) 11 days d) None of these

(Clercial Grade 1991)

Answers

l .a 2.b 3.d 4.b 5.a 6. a; Hint: Here A = Umesh, B = Mahesh

.-. x = 15 days and y = 10 days Now, applying the given rule, we have the time taken by A and B together to complete the remaining work

(15-5)10 _

= "loTl5""4days-.-. total time consumed to complete the work

= 5+4 = 9 days.

Miscellaneous 1. Twenty-four men can complete a work in sixteen days.

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

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

2. 25 men and 15 women can complete a piece of work in 12 days. A l l of them start working together and after working for 8 days the women stopped working. 25 men completed the remaining work in 6 days. How many days will it take for completing the entire job i f only 15 women are put on the job? [Guwahati PO, 1999] a) 60 days b) 88 days c) 94 days d) None of these

3. 10 men and 15 women finish a work in 6 days. One man alone finishes that work in 100 days. In how many days will a woman finish the work?

[BSRB Hyderabad PO, 1999] a) 125 days b) 150 days c) 90 days d) 225 days

4. A can do a piece of work in 12 days, B can do the same

4) ... »^4Qha«S^t ta t t f t l i ^£ ; 4

work in 8 days, and C can do the same job in — th time required by both A and B. A and B work together for 3 days, then C completes the job. How many complete days did C work? [NABARD, 1999] a) 8 b)6 c)3 d) None of these

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

3 in 18 days can complete — of the same job. How many

days wil l 10 men and 8 women together take to complete the same job? [BSRB Delhi PO. :0»>v

a) 6 b) 13 1

c)12 d) None of these

6. I f 5 men and 3 boys can reap 23 hectares in 4 days and if 3 men and 2 boys can reap 7 hectares in 2 days, how many boys must assist 7 men in order that they may reap 45 hectares in 6 days? a) 2 boys b) 6 boys c) 4 boys d) 5 boys

7. 25 men can reap a field in 20 days. When should 15 men leave the work, i f the whole field is to be reaped in

37^- days after they leave the work?

a) 6 days b) 4 days c) 5 days d) None of these

8. A can 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 sees c) 18 hrs d) 24 hrs

9. 15 men would finish a piece of work in 210 days. But at the end of every 10 days, 15 additional men are employed. In how many days wil l it be finished? 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 number of men employed upon it did only half the work in 24 days, 16 more men were then set on, and the work was completed in the specified time, how many men were employed at first? a) 16 men b) 32 men c) 24 men d) 48 men

11. Ramesh can finish ajob in 20 days. He worked for 10 days alone and completed the remaining job working with Dinesh, in 2 days. How many days would both Dinesh and Ramesh together take to complete the entire job? a) 4 b)5 c)10 d) 12

[BSRB BankPO Exam, 1991] 12. 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 the same piece of work, is:

, 1 b ) 6 - c)8 d)6

[CBI Exam, 19911 13. 12 men can complete a work within 9 days. After 3 days

they started the work, 6 men joined them to replace 2 men. How many days wil l they take to complete the remaining work?

a) 2 b)3 c)4 „ 1

d ) 4 -

[BSRB BankPO Exam, 1991 ]

3 9 0

1 1 14 (A + B)'s 1 hour's work = 7 + - = — .

5 9 45

i i 45

work is done by A and B in 1 hour.

28 (45 2 8 , , — work will be done by A and B in I y j x 7̂ " I - -

hours.

15. c; Suppose B takes x days to do the work.

3 '4

, 3 ] 3x A takes | z x A x j ie — days to do it.

Now, (A + B)'s 1 day's work = 77 . 1 o

1 2 1 .-. - + — = 7 r o r x = 30.

x 3x 18 16. b; Efficiency is proportional to work done per day and

work done per day x number of days worked = amount of work done Considering efficiency of A and B initially as 1 Suppose A alone can do the work in x days and B alone can do the same work in y days.

5_ 5 Then, ~ + = total work done = 1

.(ii)

PRACTICE BOOK ON QUICKER MATHS

Since efficiency of A and B are 2 and 7 respectively,

. — x2x3 + — x — x3 = 1 " x y 3

6 J _ _ 1 1 1 or>x + y~ •-® m d x + y~5 Now, subtracting eqn (i) from eqn (ii) we have

25 e 1 x = — = 6— days.

4 4 17. d; Mohan mows the whole lawn in x hours.

2 .-. Mohan mows, in 2 hours, — of the lawn.

x . 2 x-2

.-. Unmowedpart= P a r t -18. d; Factory A turns out x cars in one hour. Factory B

turns out — cars in one hour . 2

In one hour both the factories A and B can turn out

cars.

.-. in 8 hours both factories turn out ST* + ^ I c a r s

ie 4(2*+y) cars.

Chapter 15

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