Work and Wages

Rule 1 Theorem: A can do a work in x days and B can do the same work in y days. If the contract for the work is RsX, and both of them work togehter, then the share of A and B is given by

Rs X

x + y -xy and

X x + y -xx respectively.

Illustrative Example Ex: A can do a work in 6 days and B can do the same work

in 5 days. The contract for the work is Rs 220. How much shall B get i f both of them work together?

Soln: Method I: As 1 day's work :

B's 1 day's work =

ratio of their wages: F t

6 * 5 5:6

220 B's share = x6 = ftsl20.

5 + 6 Method II: As wages are distributed in inverse pro-portion of number of days, their share should be in the ratio 5:6

220 '' .-. B's share = " j y - * 6 Rs 120. Method I II : Applying the above theorem, we have

X 220 c B'sshare= J^y~xx = J ^ + 6 = Rsl20.

Exercise 1. Two men A and B, working together complete a piece of

work which it would have taken them respectively 12 and 18 days to complete i f they worked separately. They received in payment Rs 1492.50, find their shares. a) Rs 895.50, Rs 597 b) Rs 895, Rs 597.50 c) Rs 885.50, Rs 607 d) Rs 885, Rs 607.50

2. A can do a work in 10 days while B can do it in 12 days.

Both of them work together to do the work. I f the total amount paid for the work is Rs 440, how much is A's share in it? a)Rs200 b)Rs250 c)Rs240 d)Rs260

3. Ram can do a certain work in 15 days while Chandan can do it in 25 days. Both work together and finish the work. In what ratio should the total earnings be divided be-tween them? a) 3:5 b)2:5 c)5:2 d)5:3

4. A, B and C can do a work in 4, 6 and 10 days respec-tively. They finish the work together and earn Rs 310. What is the share of each? a) Rs 150, Rs 100, Rs 60 b) Rs 140, Rs 110, Rs 60 c) Rs 160, Rs 90, Rs 60 d) Rs 150, Rs 110, Rs 50

Answers l .a 2.c 3. d; Hint: Ram's wage: Chandan's wage

X x + y

xy: X x + y

XX

Note:

4. a;

or, y:x [i.e. the ratio in which the wages are divided is in inverse proportion to the time taken by them to do the work alone] = 25 :15 = 5:3 This rule can be extended to more than two workers also. Seethe Q.No. 4. Hint: A's share : B's share : C's share

= 1:1:^ = 15:10:6 4 6 10

A's share =

B's share

C's share =

15 15 + 10 + 6

X 3 1 0 = R S 150

x310 = R s 100

.15 + 10 + 6

Note: (Also see RuIe-2)

x310 Rs60

392 P R A C T I C E B O O K ON Q U I C K E R MATHS

Rule 2 Theorem: A, B and C can do a work in x, y and z days respectively. If doing that work together they get an amount

( Xyz * ofRs X,. then the share ofA=Rs xy + xz + yz

Share of B = Rs

Xxy

Xxz

Kxy + xz + yz and Share of C = Rs

yxy + xz + yz^

and ratio of their shares is given by A:B:C = yz:xz:xy

Illustrative Example Ex: A, B and C can do a work in 6, 8 and 12 days respec-

tively. Doing that work together they get an amount of Rs 1350. What is the share of B in that amount?

Soln: Detail Method:

1 A's one day's work =

6

B's one day's work = o

C's one day's work = 12

1 1 1 As share : B's share : C's share = ^ . ,

Multiplying each into ratio by the LCM of their de-nominators, the ratios become 4:3:2.

1350 B's share = -x3 =R S 450.

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

1350x6x12 1350 Share of B = = x72 = Rs450.

48 + 96 + 72 216 Quicker Method II: A's share : B's share : C's share = B's time * C's time: A's time x C's time : A's time x

B's time

= 96:72:48 = 4 : 3 : 2

1350 '

.-. B's share = x3 =Rs450.

Exercise 1. A, B and C together do a piece of work for Rs 374.50. A

working alone could do it in 5 days, B working alone could do it in 6 days and C working alone could do it in

7 days. How should the money be divided among them? a) Rs 147, Rs 122.50, Rs 105 b) Rs 157, Rs 122.50, Rs 95 c) Rs!47,Rs 112.50, Rs 115 d) Rs 137, Rs 132.50, Rs 105

2. A, B and C contract to do a work for Rs 4200. A can do the work in 6 days, B in 10 days and C in 12 days. I f they work together to do the work, what is the share of C? a)Rs2000 b)Rsl200 c)Rsl000 d)Rsl500

3. A, B and C contract to do a work for Rs 6500. A can do the work in 10 days, B in 15 days and C in 20 days. I f they work together to do the work, what is the share of B? a)Rs200 b)Rs3000 c)Rsl500 d)Rs2500

Answers l .a 2,c 3.a

Rule 3 Theorem: A person A can do a work in x days. With the help of a another person B, he can do the same work in y days. If they get Rs Xfor that work, then the share of A and

B is given by Rs Xy

and Rs respectively.

Illustrative Example Ex: A man can do a work in 10 days. With the help of a

boy he can do the same work in 6 days. I f they get Rs 50 for that work, what is the share of that boy?

Soln: Detail Method:

10x6 The boy can do the work in - = 15 days

(Recall the theorem) Man's share : Boy's share = 15: 10 = 3 :2

Man's share = ~ x 3 = 30 .

Boy's share: 50

x 2 = Rs 20

Quicker Method: Applying the above theorem 50x6

Man's share:

Boy's share:

10 = Rs 30.

50x( lQ-6) 10 = Rs 20.

Exercise 1. Ram can do a work in 20 days. Ram and Shyam together

do the same work in 15 days. I f they are paid Rs 400 for that work, what is the share of each. a)Rs300,RslOO b) Rs 200, Rs 200 c)Rs250,Rsl50 d)Rs350,Rs50

2. Suresh can do a work in 15 days. Suresh and Ramesh

Work and Wages

together do the same work in 10 days. I f they are paid Rs 1500 for the work, how should the money be divided between them? a) Rs 1000, Rs 500 b) Rs 700, Rs 800 c) Rs 1200, Rs 300 d) None of these

3. Sohan can do a work in 25 days. Sohan and Mohan together do the same work in 20 days. I f they are paid Rs 625 for the work, how much money should Mohan get? a)Rs400 b)Rs200 c)Rsl00 d)Rsl25

Answers l .a 2. a 3.d

Rule 4 Theorem: A and B undertake to do a work for Rs X. A can do it alone in x days and B in y days. If with the assistance of a boy they finish the work in'd' days, then the share that

A gets is Rs

Rs X

dX x

x + y xy

, B gets is Rs (dX)

y) and the Boy gets is

And the ratio of shares is given by A: B: Boy = dy: dx: xy -d(x+y)

Illustrative Example Ex: A and B undertake to do a work for Rs 56. A can do it

alone in 7 days and B in 8 days. I f with the assistance of a boy they finish the work in 3 days, then the boy gets Rs .

Soln: Detail Method: A's 3 days' work + B's 3 days' work + Boy's 3 day's work = 1

3 3 Or + + Boy's 3 day's work = 1

7 o

Or, Boy's 3 day's work = ' 3 3 11 7 + 8 j 56

3 3 11 _ 3x56 3x56 11x56 Ratio of shares = - > - : - : :

;. Boy's share:

= 24:21:11 56

x l l =Rs 11. 24 + 21 + 11

Quicker Method: Applying the above formula, we have

Boy's share * 56 7 + 8 7x8

foil

Exercise 1. Two men undertake todoapieceof work for Rs 200. One

alone could do it in 6 days, the other in 8 days. With the

assistance of a boy they finish it in 3 days. Find the share of the boy. a)Rs25 b)Rsl00 c)Rs75 d)Rs50

2. A and B contract to do a work together for Rs 300. A alone can do it in 8 days and B alone in 12 days. But with the help of C they finish it in 4 days. Find the share of C. a)Rs30 b)Rs60 c)Rsl00 d)Rs50

3. Sita and Gita undertake to do a work together for Rs 600. Sita alone can do it in 15 days and Gita alone in 20 days. But with the assistance of Rita they finish it in 5 days. Find the share of Rita. a)Rsl50 b)Rs200 c)Rs250 d)Rs300

Answers l .a 2.d 3.c

Rule 5 Theorem :A,BandC contract a work for Rs X. If together,

x A and B are supposed to do ~~ of the work, then the share

y

- f L% Rs X 1 - -

. - i y) of C is given by Rs

Illustrative Example Ex: A, B and C contract a work for Rs 550. Together A

and B are supposed to do of the work. How much

does C get?

Soln: Detail Method: (A + B) did work and C did

1 - 1 11 11 work

7 4 (A + B)'s share : C's share = : = 7:4

C's share : 550 11

x4 = fo200,

Quicker Method: Applying the above theorem, we have

C's share = 550^ 1 -^ j = fo 200.

Exercise 1. A, B and C undertake to do a work for Rs 660. A and B

8

together do of the work and rest is done by C alone.

How much should C get? a)Rs200 b)Rsl60 c)Rsl80 d)Rsl90

2. A, B and C undertake to do a work for Rs 707. A and B

394 P R A C T I C E B O O K ON Q U I C K E R MATHS

together do of the work and rest is done by C alone.

How much should C get? a)Rs202 b)Rs200 c)Rsl02 d)Rsl50

3. A, B and C undertake to do a work for Rs 480. A and B

1

together do of the work and rest is done by C alone.

How much should C get? a)Rs360 b)Rsl20 c)Rs240 d)Rsl80

Answers l .c 2.a 3.a

Rule 6 Theorem: Wages for x women amount to Rs X\ d\

days. If the daily wage of a man is n times that of a woman,

then the number of men that receive Rs X2 for the work of

7 X X 1^1 J A)

d2 days is

Ex: Wages for 45 women amount to Rs 15525 in 48 days. How many men must work 16 days to receive Rs 5750, the daily wages of a man being double those of a woman?

Soln: Detail Method:

1 5 5 2 5 115 Wage of a woman for a day = = Rs -rr

45x48 16

Thus, wage of a man for a day = 2 x 115 16

Rs [15

Now, number of men

Total wage No. of days x 1 man's 1 day's wage

5750x8 = 25 men. 16x115

Note: We should remember the relationship: Total wage = One person's one day's wage * No. of persons x No. of days

Quicker Method: Applying the above theorem, we have

5750 48 45 e l h e n o . o f m e n - x - x y = 25 m e n .

Exercise 1. I f the wages of 45 women amount to Rs 46575 in 48 days,

how many men must work 16 days to receive Rs 17250, the daily wages of a man being double than those of a woman? a) 20 men b) 25 men c) 30 men d) 15 men

2. Wages of 20 boys for 15 days is Rs 9000. I f the daily wage of a man is one and half times that of a boys, how many men must work for 30 days to earn Rs 13500? a) 12 men b) 20 men c) 16 men d) 10 men

3. Wages of 10 women for 5 days is Rs 1250. The daily wage of a man is twice that of a woman. How many men must work for 8 days to earn Rs 1600? a) 5 men b)8men c)4men d)6men

Answers l .b 2.d 3.c

Rule 7 Theorem: xx men and yx boys can earn Rs X\ d\

if x2 men and y2 boys can earn Rs X2 in d2 days, then the following relationship

men _ X[y2d2-X2yldl boys

is obtained

Illustrative Example E K 3 men and 4 boys can earn Rs 756 in 7 days. 11 men

and 13 boys can earn Rs 3008 in 8 days. In what time will 7 men with 9 boys earn Rs 2480?

756 Soln: (3m + 4b) in 1 day earn Rs - ~ = Rs 108

( l l m + 13b) in 1 day earn Rs 3008

= Rs 376 ....(2)

From (1), we see that to earn Re 1 in 1 day there should 3m + 4b

be persons. Similarly, from (2), to earn Re 1 108

in 1 day there should be

3m+ 46

11m+136

And also;

376

l l m + 136

persons.

108 376 or,m(3 x 3 7 6 - l l x 108) = b(108* 13-4*376)....(**)

m _ 100 _ 5 ' b ~ 60 * 3 Now, from (1) (3m + 4b) in 1 day earn Rs 108

or, 3m + 4 x - m in 1 day earn Rs 108

27m

or, j- in 1 day earn Rs 108

108x5 .-. 1 m in 1 day earns Rs = Rs 20

and Wages 395

Thus, we get that a man earns Rs 20 daily and a boy earns

v,;i*9gp DbnBffi ^fib.01 .-ni;QW?*ff na t t ia r^3-Rs 2 0 x - =Rs 12 daily.

V 7m + 9bearnRs(7x20 + 9 x 12) = Rs248in 1 day. 7m + 9b earn Rs 2480 in 10 days,

i *) Since both the LHS and the RHS denote the same quantity: "Number of persons earning Re 1 in 1 day". **) We can arrive at this step directly be using cross-multiplication-division rule. Arrange the given infor-mation as follows: Men Boys

| .3 4 x 11 13 x Now,

Earning 756 3008

Days 7

Men 3x3008 11x756^

8

Boys ^13x756 4 x 3 0 0 8 j

7 8

or,m(3 x 3 7 6 - l l x 108) = b(108* 13-4*376)

m 5 r' ~b=3 Quicker Method: Applying the above theorem, we have

men 756x13x8-3008x4x7 boys 3 0 0 8 x 3 x 7 - 7 5 6 x 1 1 x 8

-5600

-3360

756 Now, 3m + 4b in 1 day earn Rs ~z~ = Rs 108.

or, 3mx4x-m i n i day earn Rs 108

27m or, in 1 day earn Rs 108

I m in 1 day earns Rs 108x5

27 Rs 20.

Thus, we get that a man earns Rs 20 daily and a boy

3 earns Rs 20 x - = Rs 12 daily

5 V 7m + 9bearnRs(7x20 + 9 x 12) = Rs248in 1 day .-. 7m + 9b earn Rs 2480 in 10 days.

lercise I f 5 men with 7 boys can earn Rs 3825 in 6 days and 2 men with 3 boys can earn Rs 1050 in 4 days, in what time will 7 men with 6 boys earn Rs 22500? a) 15 days b) 20 days c) 25 days d) 30 days

2. I f 3 men with 4 boys can earn Rs 2100 in 7 days and 11 men with 13 boys can earn Rs 8300 in 8 days, in what time will 7 men with 9 boys earn Rs 11000? a) 16 days b) 18 days c) 14 days d) 20 days

3. I f 12 men with 13 boys can earn Rs 4893.75 in 3 days and 5 men with 6 boys can erarn Rs 3562.50 in 5 days, in what time will 3 men with 4 boys earn Rs210? a) 8 days b)7 days c) 10 days d)9 days

4. 5 men and 5 women earn Rs 660 in 3 days. 10 men and 20 women earn Rs 3500 in 5 days. In how many days can 6 men and 4 women earn Rs 1300? a)5 days b) 10 days c)6 days d) 12 days

5. 4 men and 6 boys earn Rs 1600 in 5 days. 3 men and 7 boys earn Rs 1740 in 6 days, in what time will 7 men and 6 boys earn Rs 3760? a) 6 days b) 8 days c) 10 days d) 12 days

Answers I d ; Hint:

Men 3 8 2 5 x 3 x 4 - 1 0 5 0 x 7 x 6 1800 900 Boys 1 0 5 0 x 6 x 5 - 3 8 2 5 x 2 x 4

1 men = 2 boys. (5m + 7b) can earn in 6 days Rs 3825.

3825 5m + 7b can earn in 1 day - 637.5

6 5m + m can earn in 1 day Rs 637.5

2 j 637.5x2

.-. Im can earn in 1 day KS is

= 2:1

and / b can earn in 1 day Rs

17 75

75 (7m + 6b) can earn in 1 day = 7 x 75 + 6 x = Rs 750

.-. Rs 22500 can be earned by (22500

2. a

(7/W + 66) in

3.b 4. a 5.b I, 750

= 30 days

Rule 8 Theorem: IfA, B and C together earn Rs x , in dx days, A

and C together earn Rs x2 in d2 days and B and C to-

gether earn Rs Xy in d3 days, then the daily earning of A

isRs

Rs

xfd} -x}dt dx.d.

x7

B is Rs xtd2 -x2dx

and C is

396 P R A C T I C E B O O K ON Q U I C K E R MATHS

Illustrative Example Ex: A, B and C together earn Rs 1350 in 9 days. A and C

together earn Rs 470 in 5 days. B and C together earn Rs 760 in 10 days. Find the daily earning of C.

Soln: Detail Method: Daily earning of A + B + C

/^ y 1350 Rs 150 ....(l)

to470 _ n . Daily earning of A + C = - = to 94

Daily earning of B + C : Rs 760

10 = Rs 76

...(2)

..(3)

From (1) and (2) daily earning of B = 150-94 = Rs56....(4)

From (3) and (4) daily earning of C = 76 - 56 = Rs 20 Quicker Method: Applying the above theorem, we have

1350 9

= 76 + 94-150 = Rs20.

760 470 daily earning of C - + ^

Exercise 1. A, B and C together earn Rs 2700 in 18 days. A and C

together earn Rs 940 in 10 days. B and C together earn Rs 1520 in 20 days. Find the daily earning of C. a)Rs20 b)Rs40 c)Rsl0 d)None of these

2. A, B and C together earn Rs 640 in 8 days. A and C together earn Rs 250 in 5 days. B and C together earn Rs 420 in 6 days. Find the daily earning of C. a)Rs60 b)Rs50 c)Rs80 d)Rs40

3. A, B and C together earn Rs 1500 in 10 days. A and C together earn Rs 800 in 8 days. B and C together earn Rs 900 in 9 days. Find the daily earning of B. a)Rs50 b)Rs60 c)Rs40 d)Rs30

4. A, B and C together earn Rs 750 in 5 days. A and C together earn Rs 400 in 4 days. B and C together earn Rs

. 1 * 450 in 4 days. Find the daily earning of A.

a)Rs40 b)Rs60 c)Rs55 d)Rs50

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