4 Ratio, Proportion, And Variation

The basic applications of the concepts involved in this

chapter are comparisons of two or more quantities and changes

in their magnitudes, e.g., comparison of the ages, weights

income, savings, heights, volume, density, temperature etc.

So this chapter is very useful in solving the problems ofData interpretation. Also each and every year one or twoproblems from this chapter is/are asked in CAT, either directlyor application based for QA section. Last but not the least, the

concepts of ratio, proportion and variations are very useful insolving the maximum arithmetic problems. That's why these

problems are usually asked in most of the competitive exams

like FMS,IIFT, MAT, SYMBIOSIS etc.

RATIO

The comparison between two quantities jn tetms ofmagnitude is called the ratio, i. e., it tells us that the one quantity

is how many times the other quantity.For example, Amit has 5 pens and Sarita has 3 pens. It

means the ratio of number of pens between Amit and Sarita is 5

is to 3. It can be expressed as '5 : 3'.

nftll lt should be noted that in a ratio, the order of the terms is veryimportant. For example, in the above illustration the requiredratio is 5 : 3 while 3 : 5 is wrong.

So the ratio of any lwo quantities is expressed a s! or a : b.b

The numerator 'a' is called the antecedent ancl

denominator' b' is called as consequent.

Rule of Ratio

The comparison of two quantities is meaningless if theyare not of the same kind or in the same units (of length, volumeor curency etc). We do not compare 8 boys and 6 cows or 15

litres and 5 toys or 5 metres and 25 centimetres. Therefore, to

find the ratio of two quantities (of the same kind), it is

necessary to express them in same units.

NTEIE1. We do not compare I boys and 6 cows, but we can compare the

number (B) of boys and number (6) of cows. Similarly, we cannotcompare the number (15) of litres ancl the number (5) of toys etc.

2. Ratio has no units.

PROPERTIES OF RATIOS

l. The value of a ratio does not change when tire

numerator and denominator both are multipiied bysame quantities r. e .

'cl-

LfL.

bkblbmb36 9

e.9., --4 8 t22. The value of a ratio does not alter (or change) when

the numerator and denominator both are divided bvsame quantities r.i'..

e _alk _all _ol*.r"b blk bl I blnt

1 a,a a /aJ J]: JlJ JI+e.9., - =

- =

- =

- ... etC afe Same fatl0.- 4 412 413 4t4

3. The ratio of two fractions can be expressed in ratio ofintegers. e.g.,

314 _3 *4 =15t4 4 5 5

alb a dSince -- - = -- x t- (Rel'er to the fractions in

clcl b c

Fundamentals)4. When two or more than two ratios are multiplied with

each other, then it is called as compounded ratio e. g.,2 4 6 16. ^2 4 6-x-x isthe compoundedratio of-.'.'3 5 1 35 ls1

acek-V-V-bdf t

(compounded ratio)

When the ratio is compounded with itself, it is calledas duplicate, triplicate ratios etc. e.g.,

. ^()as dupllcate ratro of

is called as triplicate ratio of Ih

So,

5.

) .l0 a a- lct\x = -': =l -' I is calledb b b: \.bi

.a a a (a\3and-x-x-=l - |

h h b \h)

RATTO, PROPORTTON & VARTATTOIU

18?

Similarry,,ffi) =(1)' 'is caued as sub-duplicate

ratio and im =(;)''t is called as sub-triplicate

ratio of9.

Ratio, Proportion and Variation

r. !!t 9 if fo, every posiriv e k. I <t *rd o - k

<ob+k b ' '-'b-'*--b-k-b

g. o+'ro if 'rob+d b d b

lo. o +' ,.o if ' .ob+d b d h

-k

-k

!=!=g=g=b d f h "'a+c+e+g+...

a6.-h

.^c alI and onlv rt - = -'db

. 30 30+3m 33 36t,€., ^ = .-40 40+4n 44 4g

This properly is very useful when we compare twofractions

e.g.,tocompare between * *O *20 22

We see tttat 4 = 1202.^10 . l2 l2 l0+ m

Now tl - and - are eoual then20 22 22 22+2mNow putting m = 2, w e don' t get the required fiaction.

^ t2 10^So-*-toranv22 20 - vatue ot r,,z*fi t'n rerms of

ratio)

- a+k a.^^t., -<-

rtloreverv oositive k-9>tb+k b "' 'b

, a-k qano b-k b

b+d+f+h+...

12. Let + , :, : ,9 . . . t" some different rarios, then theb d' f 'h. ( a+c+e+s+...\

value ot I --------:-j1- lmust lies between the\ b+ d + f + h+... ) "-"- "-" "

lowest and highest ratios.13. If a : b and b : c are given, then

-qba : b : c = t".!= (a. b) :(b. b) :(b. c)

14. If the ratios between a : b, b: c, c; d, d : e are givenindividually, then the combined ratio of a : b: c: d : e.

a; b al bl bl bt btb:c bl tl .l .l .lc..o ,l'.1' .l' ol' old: e d{ d+ d+ d* e*i. e., a : b : c'. d 1 s = (a.b. c. d) : (b. b. c. d)

b

c+qmd+bm

11. If

then

;exnmptr t Findtheratioof 25to40. -j

lJio'urtrr 5 outof l44persoruworktngt orno6rr,s6qremen

soLUTloN 25_ 5x5_ 5 lndtherem*iningarewomen.Findtheraf{oofn:umber'afwamento40 5xg g numbe:rafmen ,,,'J

nfFfl To get the ratio, we rationatize the fractions by canceiling out the SOLUTION Ratio = # = + ('.' number of women =I44 - 56)common factors of numerator and denominator.

fl*$

* EXAMPLE 6 In a club hoving lAA memhers,.Z0 play,,ewom;'Z;4**e phy table,tennis and 16 play cricket and the remaining members do

not play any game. No member plays more than one game. Find.theratio of the number of members who play.

ff*^t tt,-q 3 The number of boys and girls in a sehaoL sre 576 qnd480 respectively. kpress the ratio of the number of boys to that of

${

( sze 96x6\\ +so 96xs)

d

$ eXlvlptE 4 Shukla earns Rs. 14,0AA per month and. Mishraecrns Rs. 18,000 per manth. Find tke ratio of Shukla's salary toMishra's salary.

SOLUTION Required ratio = !!'09? =718,000 9

* exRiltpts 2 Find-the rqtio of 9o cm to 1.5 m.

SOLUTION 1.5 m = 150 cm (units must be same)

So, rhe required ratio = 90 = !-150s

girls in the stmplest form.

SOLUTION Required ratio = 5,79

=0480 s

(a) Carrom to the number,of thosewha plajt table.tennis., ,.

, .{b) Aricket to the nttmb,er.af th,ose who play cqrrom.. . , . {c) Crfuket to the nurnber of those whs da not play any gqme.' :'

:'1i1 Table-'tennis ta the number af thase *ho io *, itiyl oigdme, :

,', I ' 1', '1 :

k) Some game'tdthe number of those who do not play anyg&me. . ,,,#SOLUTION Total members = 100

Carrom = 20Table-tennis = 24

Cricket = 16

No any game = 40

$

(q)!t6

SOLUTION

Arternativery: * *= +'+ (*'+,+)=B:6=4:3

nffii:t ln case of fraeticns, convert them to whole numbers bymultiplying each term by the L.C.M. of their denorninators.

(b) 2 1

: zr =7 .7 =71: =.2 or 2: 33 2 3 2 7/2 3

nfm ln case of two fractions if numerator$ are same, then the requiredratio is the inverse ratio of the fractions,

r.t !, !, s! =!, !, E =! xz+ :1 x2+:E xz+"6I 4 6 8 4 6 8 4


, 20 5 _.16 4(a) = (b)246205

-.16 2 ,.,24 3(c) -=-

(d) =-405405_.60 3(e)

-=-402fl*6 EXAM PLE 7 A person ecrns tu. 12a0 per day and spe:nds tu, 80O.

Find the ratio of his scvings to expenditure. *

soLUTtoN 400 - 1

800 2

1.4-L-Z

ffixlrupt f I Simptify thefollowingrdrlos .'

183

So A gets Rs. 10, B gets Rs. 20 andC gers Rs. 40 ifthe rario oftheir ages is 2: 4: 8.

EExnwrplr Ll An amaunt of Rs. r00 b being dtvtded rrmong twa. 1 1 ..persons in the ratio - : .. How much money does each get?' l01s J

SOLUTTON A,i=axso: !x go=3: 2 (here 30 isthe10 15 10 15 LCM of 10 and 15)

So, the ratio of amount of money = I x 100 : -2

x 100-55

= 60 and 40

fi fXAmptf 12 The lengths of sides of a triangle are in the rqtto2; 3: 4. If the perimeter of the triangle is 63 cm, find the lengths ofthe sides of the triangle j

SOLUTION Let the sides of triangle be 2x,3x and 4x, then2x+3x+4x=63

+ 9x=63 + x=7.'. The sides of triangle = 2x, 3x, 4x = 74, 21, 28

Alternatively:

63x 2 and 63* 3

u.rd(2+ 3+ 4) (2+ 3+ 4)

: 74,27 and 28.

* fXAmptg L3 Divide 1224 into three parts such that first part be

double thst of second part and second part bel- of the third part. s,3J

SOLUTION Let A, B, C be three parts respectively, then

(Savings = Inconie - Expenditure)

Anshul:2:74r2

(s+ 2)

:4

B:C2:4

t anzr,z! r.)!'-3,g38 " 3 2 61 4

..1 7 7/6 1 B 4(a) : = - X = or4i3"6 I 7/B 6 1 3

g

L63x '

(2+ 3+ 4)

= 20 :9 ;90 {'.' LCM of 6, 8 and 4 is 24}

r*E EXAMPLE 9 Divid:e 74 toffees among Anktta and Anshul tn the

ratio 5l2 jSOLUTION According to the question, if there are 7 toffees thenAnkita will have 5 and Anshul will have 2 toffees but since there are

14 toffees, which is twice of 7. So Ankita will have 10 toffees andAnshul will have 4 toffees.

SHORTCUT

and

A:B=2:1B:C =1;3

A:B:C=2:7:3

A =1224x? = 4oB6

B =7224 r! = zO+6

C =1224 11 = etz6

Alkita5

14x 5

(s+ 2)

10

iexnmpte L4 tf A: B =3: 4A: B'.C.

SOLUTION A: B =3: 4and B:C:5:2

B : C = 5:2 then find" the value af6J

3:+lI I,/l If5:2l

ffinmpH 1O Three boys are aged. 2 yeais, 4 years and B ymrs.,They want to divide se1|enty rupees in the rqia of their ages. Howmuch money would each get? *j

SOLUTION The ratio of theirages =A:B:C =2:4:8=7:2:4

1_2_470x : 70x7+2+4 7+2+4 I+2+410;20:40

A : B : C = (3 x 5) : (4 x 5) : (4 x 2)

or A;B:C=15:20:B

fIXnfUpLE 15 The ratio of A: B = 1 : 3, B : C = Z: S, C : D = 2: 3Find the value of A : B : C : D.

SOLUTION A: B =1:3B;C=2:5C:D=2:3

A ; B : C : D = (1 x 2x 2): (3x 2x 2) : (3 x 5 x 2): (3 x 5 x 3)

A:B;C:D=4:12:30:45

_iA:1:

184f*-i gXlm ptf 76 Ther e are two rypes of mixtures of milk and. water.In the first mixture, out of 12 litres of mixture, S litre is milk onlyand in the second mixture, 6 lirre is milk and l2litre is water.l4hichone mixture is betrer in terms of milk's strength? g


3x+4x=64 = 7x=6464x=-7

4 ,64 I92 .256_and_777

SOLUTION First Mixture

5

n(: r36

12

15

SOLUTION

Second Mixture

6

18 (6 + 72=18)6 (milk + water

r3618 "" : mixture)

12

Check for option (d)

Numbers u." 3 " 94 urrd7

(4) :

Again, 2400_ 600

Savings 1800

So the first mixture has more milk in comparison to water.

r**TEXAMPLETT The ratio of salary of A:B=I:2, B:C=3:4,C: D = 5:6andD: E =7 :8.ttlhatistheratioofsalaryof A andE?3

which are not the natural numbers.Hence option (d) is the required answer.

'EXAMPLE 2L Monthlyincomesof A and,Bareintheratioof 4:3

and their sqvings qre in the ratio of J : Z If the expenditure of eachwill be Rs. 600, rhen the monthly Incomes of each are..

(b) 2400, 1600(d) 1600,1200 $

A: B:C:D:E = (1 x3 x 5x7): (2 x 3 x 5 x/): (2x 4 x 5 x/) : (2x 4 x 6 x7): (2x 4 x 6 x 8)

or A'.8:C:D: E = 105:210:280:336:384So the ratio of salary of A: E = 105: 384 = 35: 128

nliffl ln every next step, we leave the left term and adopt right term.

f-- n 2'EXAMPLE 78 If : = !.. thenfind rhevalue olTa - 4b:3a+ b.-b +(a) 7 :7 (b) 5: 13(c) 12: I (d) none of these

C

*-eSOLUTION Simply substiture rhe value of a and b as 3 and 4 in thegiven algebraic ratio

as (7 x3-4x4): (3x3+4)=5:13

{rxnmpte t9 rf a:(a) 9 : 6:5(c)3:3r5

SOLUTION a:b=3:2D:c=b:5

(a) 1800, 2400(c) 2400. 1800

SOLUTION Income = Exp. + Savings

A-+ 4x = 3y + 600

B -+ 3r'= 2y + 600

Therefore, 4x - 3y = 600 and 3x - 2y = ggg

- 4x-3y=3x_2y + x-^y.'. 4.r-3x=600 = x=600Then, theincome of A=4 x600=2400and income of B = 3x600=1800Alternatively: Check the options. Consider (c)Income A B

2400 1800

A:B I:2B:C 3:4C:DD:E

5:67:8

(3) = correct1800* 6001200

b = 3 : 2andb'. c = 6 : Sthena: b : cls equalto(b) 9 : 6:7A(d) 3:6:5

$

I .,rlV,/ V

a: b : c= (3 x 6) : (2x 6): (2 x 5)

=18:12:10=9:6:.5

I EXAMPLE 2O The sum of two natural numbers is 64. It[hich of the

following can not be the ratio of these two numbers?(a) 3:5 (b) 7:3 (c) 7 :9

SOLUTION Let the numbers be 3x and 5x, then3x+5x=64 = Bx=64

- x = 8, which is possible

Thenumbers arc24,40.Take another oprion:

x+3x=64 > x=16The numbers are 16 and 48.Check for oprion (c) :

7x + 9x =64 > I6x =64

= x=4The nurnbers are 28 and 36.

. (3) : (2) = correctHence, option (c) is correct. If you check other options, the

ratio will nor march.

$EXAIVIPLE22 A,BandC hqlte 40, xandy balts with themrespectively. If B gives 2A balts to g he is Ieft with half as many balkas C. If together they had 60 mare balls, each of thern would havehad" 100 balls on an qverage. I{hat is value of x : y ?

(a) 3: 2 (b) C: 0 (c) 2:1 (d) 3:4 JSOLUTION From the lasr sratement

loo+x+/=1oo3

= x+y=200Again ftom lirst statemen t,

* -20 =v

= 2x_y =49Solving equations (1) and (2), we get

x = 80 and y =720Therefore, required rario of x ; y = 2t 3Hence, option (b) is correcr.Alternatively: After forming the equation

through the options. Ler us assume option (b).

x+y=200x:y=4i6

(d) 3:4 $.-_._i

.. .(1)

...(2)

(1), we can go


:+ x=80 and

Now from the first statement,

So80-20

L20

hence option (b) is correct.

).qd. theii, gp endings ar e tn tLie r,ario 1 5: : 9 i' 8,' It Asdipsi *Sok' oJ: hisi.ncome. What is the ratio of the savings of A, B and C? J

SOLUTION Income = Expenditure + Saving

A -+ 12x - 15y + 3x (3x = ZSarbof 72x)

B-+ 9x=9y+(9x-9y)C-+ 7x=By+Qx*8y)Therefore,

Therefore, savings = (income - expenditure)

A=72x-9x=3x

B=9x-T*=E*5524 11L=/x-T*=T*

i. e., the ratio of savings of A : B: C = 3x, f t, { *55=15x:18x:11x=15:18:11

ff{ EXAMPLE 24 There are tota-l 100 coiru consisting of 20 paLse, 50

patse anrl Re 7 ii the ratio ol7-':l 8 :.5 Wa1*:tfte no;,'af,eains, af-SA

:pa*e':if ihe,SiSe ebinueei the a&aini|i@,;bj:2!':patie,andRe., l,coirl is Jgf : r :, ,

:,',i,,;;; 6g1' 92,,.',;,', t',-. : :,' (b), 4A ',, (d)"56' :', : ;

18 = 1800 paise)

(140x)-(500x)=1800

iii**$"irilffi360x = 1800

x=5Therefore, number of coins of 50 paise = 8 x 5 = 40Alternatively: Let there be 40 coins of 50 paise

rlenominalion, thenThe no. of coins of A, B,C =7 x: 8x : 5x = 35: 40 : 25

Therefore, the amount from 20 paise coins = 35 x 20 = Rs. 7and amount from Re 1 coins = 1 x 25 = Rs. 25Hence difference = Rs. 18 (25 - 7)Thus the presumed option is correct.

ilfXamplr 25 There are 43800 srudenrs tn 4 schook of a city. IfItalf of the first, two-third of the seconQ three-fourth of the thtrd and

four-fifth of the fourth arc the same number of students, then find.the ratio of number of stuLdents of A and D if A, B, C and D be the

lir s t ; s e c ong t[rird", pnd ur th s ch ools r esi ectiv ely,i., :.' r.. ii,,i' r;

SOLUTIONA28T4D2345

A 4 -B 9 -C 16Therefore.'^ =' and " =a and'B 3 C 8 D 15

... A:B=4:3B:C =9;8C:D=16:15

... A:B:C:D =(4x 9 x 16): (3 x 9 x 16)

:(3x8x16):(3x8x15)A : B : C : D = 576 : 432: 384 : 360

Therefore, the ratio of number of snrdents ofAandD=576:360=8:5

Alternatively: 4=?t= l6 = 1o =u23 4 5

then A:B:C:D =2kt?k,4 kt5 k234= 24k:18k : 16k : 15k

.'. A: B :C : D =24: 18 : 16: 15

t =120x-20_1

y21 (verified)

: gXlfvlptf 23 The incames of A, B, C qre in the ratto of 12 9 :7

(8,,26

SOLUTION (2O x7 x) - (100 x 5x) = lggg 1Rs.

72x-3x=ISyr5y3

3xJ- _

-5

PROPORT!0N

An equality of two ratios is called a proportion and we say

that the four numbers are in proporlion.

i. e., if !- 9, o, a : h = c : d, thenwe say lhat a, b, c and d'bdare in proportions and written as a : b :: c: d, wherc the symbol'::' indicates proportion and it is read as 'a is to bas cisto d'.

Here a and d arc called 'oextremes" (or extreme terms)an b and c are called as 66means" (or middle terms). Thusfour numbers are said to be in proportion, if the ratio of the firstto the second number is equal to the ratio of the third to thefourth number. e.g.,2:3::4:6. Some important results a-re

derived from the discussion of proportion, which are veryimportant for clear understanding.

Properties of Proportion1. If four numbers (quantities) are in proportion then

product of the extremes is equal to the product of themeans and if these are not in proportion, then productof extremes is not equal to the product of the means.

i.e., ifthen

a:b::c:d,axd=bxc

Thus it is clear that if three out of four terms of aproportion are given, then we can find the fourth termby dividing the product ofmeans (or extremes) by theremaining term.

2. If a,bandc are three numbers such that a;b=b:cthen these numbers a, b, c are said to be in continuedproportion or simply in proportion.

186

i.e., a:b=b:cHere 6 is said to be the mean proportional betweena and c, and c is said to be the third proportional toa and b. e.g.,3:9::9:27Here 9 x 9 = 3 x 27 which are in continued proportion.

nEnll Sometimes the above idea is also expressed by saying that thethree numbers are in the ratio g : 9 : i7. Thus if tnnje drantitie.are proportionals, the Jirst is to the third as the dupticate ratio ofthe first to the second.

i.e.,itthen

a'.b'.'.b:c

Some Facts about proportion

(a) Invertendo: If a =

c

bdac


(b)Alternando: Ifa=c -) n-bbdcd

nnlGl Each of the two results (a) and (b) can be obtained by crossproduct.

(c)componendo: tf! =c=( q:! l=f tg)hdib)[a)(d) Dividendo: tf

q =

c =(

y-l ')=f !:l')bdl.b)[a)aic=a2..b2

l"ba2abaza'.'._ = -- *= aiolb c b2 bc-62

r*****^T-sEXAJvlPLEl Thefirst,.second-and'fourthtermsofaproportionare ffinurple 6 Theratioof thenumberofboystothatof girlsina5' 15 and 90 respectively' Find the third term. J school is 9 : rl. If the nu^b$ of girls in the school is 203s, find :SOLUTION Let the third term be x, then 5, 15, x,90 are in @)numberof boysinschoolproportion i. e., 5 : 15 :: x : 90 (b) number of srudents in school

*.i

(e) Componendo and Dividendo: Ifq =cbdI=a':b'l ( a+b) (c+a)

t_t-tt

lo-b)-1,4)maFfif The result (e) can be obtained by dividing result (c) by resutt (d)

9:11 = x: 2035

9 x 2035

11

r = 1665 (number of boys)(b) number of students

= number of boys + number of girls = 3700c*I EXAMPLE 7 lMhqt k the leqst possible nttmber which must be

subtracted from 16, 19 and 23 so'tni the r:rr"ir* "';;;r;;;r';contlnued proportion?(a) 2 (b) 4 (c)6 (d)7

*_jSOLUTION Going through options, we find option (d) is correct.

Alternatively:(16 - x): (19 _ x):: (19 _ x): (23 _ x)

= (r9_x)2=(16_x)(23_x)By solving the above equation, we get x = 7

tr! EXAMPLE 8 If (a+ b): (a- b) = 15 :t,thenthevalueof o2 _b2 is:(a) 56 (b) ts (c) 112 - i;;;

" _"j

SOLUTTON (a + b) - 15

(a- b) 1

aBD/

=) x=30

{ gxllrtptg 2 The ratio of length to width of a rectangttrar sheet ofpaper is 5: 3 If the width of the sheet is 18 cm, find ti length. g

SOLUTION Let the length of sheet of paper be ,, .m. fh.., tffratio of length to width = x : 1g

Thus x:18=5:3

= x=30Hence the length of paper = 30 cm.

f*reXemptE 3 f 81, x, x,2S6areinproportioryfind x. aI

sotuTloN

SOLUTION

:> x=4O (by the first property)

81:x::x:25681x256=xxx = x=744

fHxnrvrpr-e 4 The ratio between the number of men and women inan ffice is 5 : 7 . If the nuri (ber of women working in the office is 56,find the number of men working in the ffice. !

M

SOLUTION 5:7 = x:56 (suppose number of men = x)

Therefore, nurnber of men in the office = 40

E EXAMPLE 5 The age of Chandi and. Radhtka are in the ratio S : 3If Chandi's age is 2O years, find the age of Radhika. I**E

(by componendo and dividendo)

therefore a2 -b2 =64-49=7s

'Ihe mean proportional between g and.9g is :(b) ss (c) 112

Hence (d) is correc.

B:x::x:98x2=8x98x=28

5:3=20:x3x20x='--'-

5

= x=12Hence the age ofRadhika is 12 years.

ffiEmplr g(a) 16

SOLUTION(by the first properry)

(d) 28 *i


ilfxnffpfe 7A The students in three classa are in the rado of2: 3: 4. If 40 students are added in each clsss, fhc raris be$ma4 ; 5 : 6 Find. the tatsl number of students in all the rlrree clssses is :

ffiAfUerg 13 Aear rak* 7 stepsfar every Sr*p, "f*Aog,

#tTstgps af a dag are equal to 6 steps af cat. t$hat is the ratio.af speed ofcat to that af dog?

(a) 24 t 25 {b}'42:25

SOLUTION CAT

:1.:!i

{a) 27,0 (b} 180 (e) 125

SOLUTION 2x+40=4y3x+40=5y4x+40=6y

2x=4O

x =20Hence, total number of students = 2x + 3x + 4x = 9x

=9x20=180Alternatively: It can be solved through options also.

$ cxeuptr Ll The dimensions af *ph.otograph are 4 sftd l.g cnl'.If:the breadth of the enlarged,phato is 4,5 cm and k was elilargedproportionalty then what is the new knghgf ne-w photopraph?

(a) 6 (b) s.4 {c),10 (d),9, JSOLUTION 4:1.8=x:4.5

+ x=10Thus, the length of nerv photograph is 10 cm.

{ EKAMPIE \2 'Two equal contatners arefilledwith thetmixqre ofmilk and water. The concentration of milk in each of the containersis 20o/o snd 25Ya respectivellr. What rs rhe ratio pf water inboth the,container s r e sp ectiv ely ?

.{a)15:t6 ('&116:15 (e}4tE (d)s:+ JSOLUTI0N Milk 20a/o 25o/o

Water 8Oo/o 7So/o

Thereofre, required ratio = I = { or 16: 1575 15

[il,N.1T,*SV;,$rtGT,,flffi ,.

A method in which the value of a quantity is first obtained tofind the value of any required quantity is called unitary method.

In solving problems based on unitary method.

Direct Proportion

SOLUTION More nore book more cost; less note books, less cost 27 stu&4ts?,

Note books Cost64s1 45/6g 8t45=6o

6Hence 8 note books cost F,s. 60.

but the length of 5 steps of dog = length of 6 steps of cat

it means the ratio of length covered by dog i, to .ut = 9.5

Therefore, in each step a dog will cover I times distance than

that of a cat.

rhus the'""" "r":';i';'.";'=T:: is to dog

5

SHORTCUT Actualspeed of A: BGiven speed of A

no. of steps of A in terms of length '

Given speed of Bno. of steps of B in terms of length

= CAT : DOG

7s--:-=7:66s! EXAMPIE tn 4,ornsl rr.rtsl& an elephant qnd r;rkes S leaps furevery 7 leops of the elephant, but S leaps of elephant are equal to 3Ieapt al eamcl Whatis the ratia af speeds of eamel and elephant?j

soLU'oN *"'" "':T:;'_ !:i :1 "i::""'

8,5=-x15:-x1525:27

(work force) at work. More men at work, more workdone in same time. Less men, less work done in sametime.

lnverse Proportion

SOLUTION More students, less days; less students, more daysStudents Days

45 601 60x4527 uo+l!

= 1oo days,'7

{d) lss J. . .(i)

...(ii)

. . . (iii)

Given speed 7 steps

(c) 24:19 (d) 25 42 I

DOG5 steps

Therefore

+

Two quantities are said to be directly proportional if the , Two quantities are said to vary inversely if the increaseincrease (oi decrease) in one quantity

"urr.", tile increase (or (or decrease) in one quantity causes the decrease (or increase)

decrease) in the other quantity ty same proport ion. e.g., in the other quantity by same proportion e.g., The time taken to(i) The cost of articles varies directly to the nrirnU.. of furish a work varies inversely to the number of men at work.

articles. More afticles more "ori

l"r, articles less _ Moremenatwork, lesstimetakentofinishthesamework.cost. Less men at work, more time taken to finish the same work.

ffi:**"*-*'-'*'*"

notebooks cosr? J months, find for how matny days the same stock "f n"i i,ii t*in-,

{

r88E EXAMPLE 3 A man working I hours a day takes 5 days tocomplete a project. How many hours a day must he work to completeitin4days? J

SOLUTION 'More days, less hours; Iess days, more hours'

Days

5

1

4


Hours8

5x8lrg = 1o hours a day

4

When two or more quantities are dependent upon eachother and then if any one of them is changed, the othef(dependent) quantity is also changed.

For example :

(i) When the salary of a person increases, then itssavings/expenditure increases.

(ii) When the number of guests in a hotel/number ofstudents in a hostel/number of employees changes,their respective expenses increases.

Basically, as it happens in direct proportion and inverseproportion, there are two types of variation :

(i) Direct variation (ii) Inverse variation

Direct VariationA quantity I is said to vary directly ifthe increase (or

decrease) in B yields increase (or decrease) in A but not inproportion. It is expressed as

AxB = A=KB,where K is called proportionality constant

K=4B

lnverse VariationA quantity I is said to vary inversely ifthe increase (or

decrease) inB yields decrease (or increase) in I but not in sameproportion. It is expressed as

AnL = z={BBor K = AB, K is called as proporlionality constant.

nrFfi1. lf it is not mentioned that a particular quantity is inversely variable,

then it means ihe given quantity is directly variable.2. A quantity sometimes vary jointly r.e., directly on any quantity and

inversely on another quantity.

A*B and AnLC

,qnB- - A=fPCC

e.9.,

It means

e.9.,

Here A varies directly as B but inversely as C"Also it can vary as only directly or inversely as more than one quantities.

AIBQ * A=KBC

and A- I * A=KBC BC

&

€ EXAMPLE L Avaries directly as B and. inversely as C. A is L 2 when'B is 6 andC is 2. l{hat is the value of Awhen B is 12 andC is 3? _}

SOLUTION

Again

V nr2

V=Kr2

2= K x(7.5)2t

K =-2.2s

__8K --9

_8)5=-xr-9

" 5x9I

SOLUTION A a:B and A n\L

AnB- = e=rPCC

whenA=72B=6,C=Zthen

I2--KP = K=42

Again A =K 1= + *Y =rcC3

= 4=16

I EXltUptg 2 The value of a coin varies directly to the square of itsradius, when its thickness is constdnr. The radius of a coin is 1.5 cmo.nd i* vqlue rs tu. 2. IVhat will be the radius of a cotn if lts value is

= 1.5 x rD.5 = 1.5 x 1.6

r=2.4 cm

Hence, required radius = 2.4 cm

16 years and the age of her mother is twice etc. or the ratio ofthe ages ofthe father and son at present is 3 :1. 4 years, earlierthe ratio was 4:1. What is the present ages of the father and sonetc.

3,8'= t" li

R.s. 5?

tlBROBt.ffi,frugu.AU,AGE$ ' ,,,',,it,

i"''r.:'' ',

:, :,.,,,,,.,,',, ,.,'t,t,,,...,,::i :

This article is very suitable as an appendix ofRatio-nroportion, since most of the questions based on ages

involve the concept of ratio-proportion. e.g.,lhe age of Ravi is

J


B gXem ptg 1 The rotio of ages of Krishna and. Balram is 3 : 4. Fouryears earlier the ratio was 5:7. Find the present ages of Krishna and

189

= 8x=24 ) x=3Therefore the ratio of their ages 3 years after

3x + 3_ =12 =!11x+3 36 3

i EXAM PLE 3 The age of Sachin is 4 times that of his son. Ftve yearsago Sachitt was nine times as old. as hk son wos at thcLt time. The

Balrqm:(a) 15 years, 20 years(c) 16 years, 20 years

SOLUTION Let the presenr3x and 4x, then

(b) 24yeors,32years(d) 32 years, 24 years *j

age of Iftishna and Balram be

four years ago their ages be (3x - 4) and (4x * 4)(3x-4)_5(4x-4) 7

7(3x-4)=s(4x-4)21.x-28=20x-20

x=8Present age of Ikishna = 3x =24years

age of Balram = 4x = 32 years

t EXAMFLE 2 The ratio of age of Aman andher mother is 3 : 17. Thedifference of their ages is 24 years. What will be the ratio of thetragu after 3 years? i**i

SOLUTION Let the present age of Aman and her mother be3x and 11 x then 3 years later their ages will be(3x + 3) and (11x + 3)respectively.

Again I7x -3x =24

SOLUTION Let the age of son is x years, then the age of Sachinwill be 4x years.

.. (4x-5)=9(x-5)ex=8

.'. Age of Sachin is 32 years.

filXnm ple 4 The ratio of Varun's age and- his mother,sage is 5 : 1 1.

The dffirence af their ages is 18 years. The ratio of their iges after 5years will be :

(a) 19: 59 (b) 2:3 (c) 37 :75 (d) 70:19 tSOLUTION Let their ages be 5x and 11x.

11x-Sx=18 <) x=3So their present ages are 15 and 33 years. Therefore, ratio of

their ages after 5 years = 20: 38 = 10: 19.

So

present age of the Sachin rs :(a) 25 years(c) 32 years

(b) 36 years(d) 4syeqrs J

and

PIARTNERSFIIP ',, ,.:,.:.,., ':',,,, ;:,,: , t;:;,:

When two or more than two people run a business jointlyby investing their money/resources, then it is called a jointventure or the business in partnership.

All these people, who have invested their resources, arecalled as Partners.

Partners are basically of two types(i) Working partner: A partner who is directly

involved with day+o-day activities of business iscalled as working partner.

(ii) Sleeping partner: A partner who just invests hisor her money is called as sleeping partner.

General rules of partnership :

(i) If the partners invest different amounts for the same

period of time, then the profits of all the partners areshared in the ratio of their investments.

(ii) If the partners invest same amount for the differenttime periods, then the profits of all the partners areshared in the ratio of time periods for which theiramounts were invested.

(iii) If the partners invest different amounts for differenttime periods, then their profits are shared in the ratioof products of respective investments with the timeperiod for each partner, individually.Thus gain or loss is divided , in the ratio of'money-time' capitals.

Fffill Somerrmes olfferenr prootems are solved on the basis ofpartnership to find the expenses.

f*IEXAMPLE t Bhanu and Shafeeq started" a business by invating

Rs- 35,000 and tu. ffi ,AA0. Find the share of eacfu out of an annualprofit t:f k. 5.500.

E

*-"8SOLUTIONI Ratio r:f shares of Bhanu and Shafeeq

= 36000 :63,000= 4:7

.'. Share of Bhanu = 5500 ' 1 = Rr. ZoOO11

and Share of Shafeeq = 5500 ^L = SSOO.' 11

IEXAMPLE 2 A starts same busines.s with tu. 50,000. After 3months B joins him with k. 70,000. At the end of the year, in whatratio shoutrd they share the profits?

I

SOLUTION Ratio of amount of A and B = 50,000 : 7e000Ratio of time periods for A and B = 12: 9.'. Ratio of their money-time capitalInvestments = 50000 x 12: 70000 x g = 20 : 27

f*{ EXlfVlpte 3 Harsh Vsrdhqn started. a business by investing Rs.36,A00. after 4 months Gyan Vardhan joined him with someinvestment. At the end of the year, the total profit was dividedbetween them in the ratio of 9 : 7 . Ho,w much capital was invested. byGyanVordhan in the business? c:H

36,000x12_9xx8 7

x = 42000

SOLUTION

j

'. Gyan Vardhan invested Rs. 42,000 for 8 months onlv.

190

ffiXnfVfpf-g 4 A started some business wirh tu. 26,000. After 3months B,jotned htmwithRs. 16,0;;.Aft;-ro-J

^orc rimeC joined.

them with Rs. 25,000. At the end of the year, out of a,ont proit ofY: ,t:a-t3:C gets tu. 3B2S ss ltts shqri. now many

^onth, "irr;joined,thebusiness did C join? I

SOLUTION Ratio (of share) of profits

= 26,000 x 12 : 16000 x 9 : 25000 xC

= 372: I44: 2tNowc,s share = 2{ - 3825

456+ 2y 15453

= C=6Therefore C joined 3 months later than B joined.

{ EXAMPLE 5 A, B and C stsrted. a businws with their investmentsin the rati.o I : 2: 4. After 6 months A invested- the half amount moreas before and B invuted twice the amount as ieyore while Cwithdrew lth of the their investments. Find the ratio oy rnrf, proyf*

qt the end of the year.

SOLUTION Let us assume their initialx,2x and 4x respectively.

Therefore, ratio of their investments during the whole year3x

= 1x x b *, r 6): (2x <6 + 4x x6): (4x x6 + 3x x 6)

= 15x: 36x : 42x

=5x:I2x:I4x= 5:72:74

Ratio of their profits = S: 12: 14

1. lf 40 articles cost Rs. 1g0, the cost of 18 articles is :

Ratio, proportion and VariationTEXAMPLE 6 A storred a business with k. 52.000 and after 4

months B joined him with n .ii,oooi. er'r;; ,;;;ir;r";;;;;;;r;the total profits B received toral tu. 20.;;; ,nnukrn1o 25o/o of theprofi* as cammissionfar managing the business. vni

" l,oui iiiArecetve? "" "J

SOLUTION profit,s share of A and B

= 52000 x 12: 39,000 x 8 = 2: 1

Let the profit be Rs. x, then B receives 2lo/o ascommission formanaging business, the remaining 7So/o of the total profit x isshared berween A and, B in the rario 2 : l. Hence A will get 1rd part

Jof this in addition to his commission. Hence his total earning

=o.2sx*1^0.25*3

= 0.5x = 20,000

So, the remaining profit goes to A, hence the profit of A is Rs.20,000.

!EXAMPLE 7 Aworkingpartner gets 200/o ashis commrssio n of theprofit after his commisiion is -paid.

If th.e workirg porin"*icommission is k. 80A0, then what is the totoli proyit inU* A*i"*r3

SOLUTION Let the total profit be Rs. x. T,he remaining profit afterpaytng2}o/oworking panner's commission = (x _ 8000). Again since20% of this is working partner's commission,

therefore 20

roo *(t-8000)=8000

g

investments were

3 x = 49,000... The total profit in the business is Rs. 4g,000.

UNITARY METHOD

6.2.

7.3.

8.4.

lf 20 persons can do a piece of work in 7 days, then thenumber of persons required to complete the work in 28days :

(a) 4 (b) 5(c) 14 (d) 10lf 20- men can reap a field in 3g days, in how many dayswill 19 men reap the field?

40 men can build awall20 m high in 15 days. The numberof men required to build a similar wall2E m high in 5 dayswill be :

(a) 100 (b) 125

(a) Rs. 18(c) Rs. 81

(a) 21 days(c) 76 days

(b) Rs.36(d) Rs. 40.5

(b) 19 days(d) 40 days

(a) 10 weeks(c) 15 weeks

(c) 150

(a) Rs. 500(c) Rs. 100

@) ra

(b) 1 1 weeks(d) i6 weeks

(d) 200

(b) Rs. 75C(d) Rs. 1500

(d) 16

Cost of erecting a fence rouncj a square fielC cf GZIhectares at 15 paise per metre is :

(a) 130(c) 200

A rope makes 260 rounds of a cylinder with base radius20 cm. How many times can it go round a cylinder withbase radius 26 cm?

(b) 300(d) 150

A garrison of 750 men has provision for 20 weeks. lf atthe end of 4 weeks they are reinforced by 450 *"n, ho*long wiil the remaining provision last?

56 workers can reap a field in B days" lf the v;ork is to becompleted in 7 days, the extra worlcerr iteeded are :(a) 7 (b) B

lf 5 persons weave 1g0 shawls in 12 days, how manyshawls will 6 persons weave in 15 Cays?@) 26A (b) 370(c) 270 irJ) ?6o

5.9.


L0, 15 men take 42 days of 4 hours each to do a piece ofwork. How many days of 6 hours each would 21 womentake if 3 women do as much work as 2 men?

(b) 22(d) 30

L1. 10 engines consume 80 litres diesel when each isrunning t hours a day., How much diesel will be requiredfor 10 engines, each running 15 hours a day, whereas 6engines of the former type consume as much as 5

L. Amit is as much younger to Barkha as he is older toChaman. lf the sum of the ages of Barkha and Chaman is48 years, what is the present age of Amit?

(b) 75 litre

(d) 111| ritr"

?lf

f,th of a cistern is filled in 30 minutes, how much more

time will be required to fill the rest of it?

19r

(a) 15(c) 25

(a) 1B years(c) 24 years

(b) 36 years(d) 28 years

(b) Rs. 1981.25(d) Rs. 2007.75

engines of latter type?(a) 160 litre

(c) 80 litre

(a) 50 minutes(c) 15 minutes

(a) 3 years(c) 6 years

(a) 25 and2O(c) 36 and 9

(a) 1 month(c) 3 months

(a) Rs.3200(c) Rs.4000

(a) Rs. 1200(c) Rs. 1000

(b) 20 minutes(d) 45 minutes



(b) 35 and 10(d) 40 and 5

(b) 2 months(d) 4 months

(b) 10 months(d) 14 months

(b) Rs.4200(d) Rs.5400

(b) Rs.4500(d) none of these

IrurRooucToRy ExrRcrsr -4,2

Bipin is 6 times old as Alok. Bipin's age will be twice olChandan's age after 10 years. lf Chandan's Tthbirthdaywas celebrated 3 years ago, what is Alok's present age?(a) 15 years (b) 12 years(c) 5 years (d) none of these

3. Renuka got married B years ago. Today her age is 1 I-3times her age at the time of marriage. Her daughter,sage is 1/8 times her age. Her daughter,s age is :

4. Ten years ago B was twice of A in age, lf the ratio of theirpresent ages is 4:3, what is the sum of their presentages?


The sum of the ages of Aryabhatta and Shridh ar is 45years, Five years ago the product of their ages was 4times the Aryabhatta's age at that time. The presentages of Aryabhatta and Shridhar respectively are i

PARTNERSHIP

5J,

A company make a profit of Rs. 9,00,000 , 2O/o of whichis paid as taxes. lf the rest is divided among the partners

P,Qand R in the ratio of t :t | :2, then the shares ofz

P, Q and R are respectively :

(a) 2,40,000; 3,20,000; 1,60,000(b) 3,20,000; 2,40,000: 1,60,000(c) 1,60,000; 3,20,000; Z,4O,OOA(d) 1,60,000; 2,40,000; 3,20,000

We have to divide a sum of Rs. 13,950 among threepersons A, B and C. B must get the double of As shareand C must get Rs. 50 less than the double of g,s share.The share of A will be :

(a) Rs. 1950(c) Rs.2000

A started business with Rs. 45,000 and B joined after_ward with Rs. 30,000. lf the profits at the end of one yearwere divided in the ratio 2 :1 respectively, then Bwouldhave joined Afor business after :

4. Aand B are partners in a business. They invest in theratio 5 : 6, at the end of 8 months A withdraws. lf theyreceive profits in the ratio of 5 : 9, find how long gsinvestment was used?(a) 12 months(c) 15 months

Four milkmen rented a pasture. A put to graze 16 cowsfor 3 months, B 20 cows for 4 months, C 1g cows for 6months and D 42 cows for 2 months. lf As share of rentbe Rs. 2400, the rent paid by C is :

A, Band C subscribe Rs. 47000 for a business. lf asubscribes Rs. 7000 more than B and g Rs. 5000 morethan C, then out of total profit of Rs. 4700, C receives :

AGE RTTATED PROBLEMS

IrurnoDUcToRy ExrRcrsr -4.9

fiiifi+}rillird$## Ratio, Proportion and Variation

IrurnosucToRy Exrncrsr -4,4

2. Mean proportional between 17 and 6g is :

It a,b,c,d,e,f , gare in coniinued proportion, then thevaiue of a.b.c.e.f.g is :

(a) dl (b) d7(c) d6 (d) none of theself A and B shared Rs. 1300 in the ratio 1 : 12, how much

(b) 1200(d) 1000

16. Rs. 3960 are divided among A, B and C so that hatf of Aspart, one.third of B's part and one.sixth of C,s part areequal. Then C's part is :

(a) 720 (b) 2160(c) 1080 (d) 810

17. A sum of Rs. 21000 is divided among A, B,C such thatshares of A and I are in the ratio of 2:3 and those ofB and C are in the ratio 4 : 5. The amount received by A is :

(a) Rs.6000 (b) Rs.450O

ir!ririr.ii:.!,iia,:ituiji

did A get?(a) 12o(c) 100

(c) Rs.4800 (d) Rs.8400A certain amount was divided between Aand Bin the ratio7 :9. lt B's share was Rs. 72OO,Ihe total amount was :

(a) 3:4(c) 2I: 44

(a) 51(c) 4

(a) 64(c) 81

(a) 12:3 :6 :4(c) 6 : 12:4:3

(a)1+J3(c)J3+g

(b) 21 :55(d) 7 :5

(b) r44(d) 4e

14.

15.(b) 24(d) 34

3. Third proportional between 16 and 36 is :

4. lf a: b =2:3, then (5a + b): (3a + 2b)is:(a) 13 :12(c) 12 :13

5. a = 2b = 3c= 4d, then a : b : c:d is :

(b) 15 :17(d) 13 :11

6. The fourth proportional to 4,7 and 20 is :

(a) 28(c) 18

(a) 5 :13(c) 9:4

(b) 3: 4:6:12(d)12:6:4:3

(b) 12 :13(d) none of these

(b) 3 :5(d) I:2

(a) Rs. 1280(c) Rs.5600

(c) Rs. 1800

(a) 2.7 m(c) 6.0 m

(a) Rs. 63,000(c) Rs. 54,000

(a) 16 :56(c) 15 :36

(b) Rs.6300(d) Rs. 12800

(d) Rs.650

(b) 7.2 m(d) 5.5 m

(b) Rs. 45,000(d) Rs. 60,000

(b) 14:49(d) 16:72

(b) 54(d) 52

(b) 2r(d) 35

(b) J: -r(d) 2J3

... a+ b c+d(O) ------a- -

-a'

(d) ac = bd

7. ll J2 ; G + J3) :: J6 : x, then x is equat to :

rZ=i=ftn"n'*!o*"=,(a) 2 (b) 3(c) 4 (d) 5

tf9=9,then:bd

- a+b c+d(a)- .a-b c-d. a+b c+d(c) ---z- = ----u-a-(lfa:b=b:c=c:dthr a b c

=nb';'Aare:(a) in AP(b) in continued proportion(c) in GP(d) both (b) and (c)

lf (a + b):(a - b) = 3:2,the(a2 - F;:1a2 + b2)equats:

Rs. 11250 are divided among A, Band C so that Amayreceive one.half as much as g and C together receiveand 8 receives one.fourth of what AandC togetherreceive. The share of A is more than that of B by :

(a) Rs.2500 (b) Rs. 1500

A girl 1.2 metre tall casts a shadow 1.1 m at the timewhen a building casts a shadow 6.6 m long. The height ofthe building is :

The prices of Bajaj Scooter and Bajaj pulser are in theratio of 4 :9. lf the Bajaj pulser costs Rs. 30,000 morethan a Bajaj Scooter, the price of Bajaj pulser is :

11.What is the ratio whose terms differ by 40 and themeasure of which is 2/7?

12. Two whole numbers, whose sum is 64, can not be in theratio :

(a) I :7(c)5:11Two numbers are in the ratio 3 : 4. The differencebetween their squares is 28. Find the greater number .

(a) 12(c) 24

23. Two numbers are in the ratio 3 : 5. lf 9 be subtracted fromeach, then they are in the ratio o'f 12 :23. The secondnumber is :

(a) 53(c) 55

Consider the following staiements :

(1) lf both the terrns of a ratio are muiiipiied or dividedby the sanre natural number, then the ratio remainsunaltered.

(b) 8(d) 16

RATIO AND PROPORTION

13.


(2) A statement which states that two ratios areequivalent is called proportion.

(3) lf 4 quantities are in proportion, the product ofextremes is not equal to the means.

(a) The mean,profortion between any two numbers is' efiual to the square root of their product. The wrong,/one statements is,/are :

27. Suppose y varies as the sum of two quantities ",

-1,::one varies directly as x and the other inversely as x. lf

/=6 when x=4 and y=31 when x=3, then the3

relation between x and'y is :

25. ln a mixture of I2O litres, the ratio of milk and water is2:L l'f the ratio of milk and water is I:2, then theamount of water (in litres) is required to be added is :

The time period of a pendulum is proportional to thesquare root of the length of the pendulum. Consider thefollowing statements :

(l) lf the length of the pendutum is doubled, then thetime period is also doubled.

(2) lf the length is halved, then time period becomesone-fourth of the original time period.

The correct assertions are :

(a) 1 (b) 2(c) neither 1 nor 2 (d) both I and 2

26. A quantityxvaries inversely as the square of y. Given thatx = 4, when / = 3, the value of xwhen / = 6 is :

(a)x=y+4

(c) y= zx-9x

(b) y= zx*9X

(d) y= zr-!x

(a) 1

(c) 3 and 4

(a) 20(c) 80

(b) 3(d) 1 and 4

(b) 40(d) i20

(b) 2(d) 4

(a) 1

(c) 3

L. Four numbers are in proportion. The sum of the squares ofthe four numbers is 50 and the sum of the means is 5. Theratio of first two terms is 1 : 3. 'vVhat is the average of the fournumbers?(a) 2(c) 5

9. In the previous problem (no.possible no. of chocolates?(a) s2(c) ss

(b)(d)

8) what is the maximum

5360

2. A naughty student breaks the pencil in such a way that theratio of two broken pans is same as that of the original lengthofthe pencil to one ofthe larger part ofthe pencil. The ratio ofthe other part to the original length ofpencil is :

(b) 3(d) 6

(b) 2: (3+ G)(d) can't be determined

t-{ ^ = 2. the value of (x, y) is :(y+2)(b) (2, 8)(d) (8,2)

(b) s: 4(d) none ofthese

(b) 4,000(d) 12,000

(b) Rs. 1080(d) Rs.998

Mr. Teremere and Mr. Meretere have 5 chocolates and 3chocolates with them respectively. Meanwhile Mr. KhabbuSingh joined thern and all 8 chocolates were distributedequally among all these three people. In turn Khabbu Singhgave Rs. 16 to Mr. Teremere and Mr. Meretere, since KhabbuSingh did not has any chocolate. What is the difference ofamounts received by Teremere and Meretere? Given that theamount was shared in proportion of chocolates received byKhabbu Singh.(a) Rs.8(c) Rs.14

Rs, 4536 is divided among 4 men, 5 women and 2 boys. Theratio of share of a man, awoman and a boyisT : 4: 3. \l/hatisthe share of a woman?

lo.

3.'

(a) 1:2-G(c) 2: r/5

fix+Y=!u.rox-y 3

(a) (4,1)(c) (1, a)

(a) 2:3(c) 5: 2

(a) 1

(c) 3

(a) 22,000(c) 16,000

(a) Rs.980(c) Rs.1200

4, lf as + b3 : a3 - b3 = 133 :717; find, a: b :

11.

12.

(b) Rs. 12(d) Rs. 15

(b) Rs.498(d) Rs.256

(a) Rs.336(c) Rs.166

(a) 6 litres(c) 16 iitres

5. A student obtained equal marks in History and Sociology. Theratio of marks in Sociology and Geography is 2: 3 and theratio of marks in History and Philosophy is 1 : 2. If he hasscored an aggregate of 55% mark. The maximum marla ineach subject is same. In how many subjects did he score equalto or greater than 60% marks?

(b) 2

(d) none ofthese

The ratio of income of Anil and Mukesh is 2: 3. The sum oftheir expenditure is Rs. 8000 and rhe amounr of savings ofAnil is equal to the amount of expenditure of Mukesh. What isthe sum oftheir savings?

The concentration ofpetrol in three different mixtures (petrol

and kerosene) is 1. ! und 1 ,.rpe.tively. If 2 litres, 3 litres25 s :and 1 litre are taken from these three different vessels andmixed. What is the ratio of petrol and Kerosene in the newmixture?(a) 4: 5(c) 3: 5

(b) 12 seconds(d) 10 seconds

13. Time period (f ) of pendulum is directly proporrional ro thesquare root of length of string by which bob is attached to afixed point and inversely proporrional to the square root ofgravitational constant'g'. Time period of a bob is 3 secondswhen the gravitational constant g is 4 m/sec2 and length ofstring is 9 metre, what is rhe time period of a bob havinga string of length 64 metre and gravitational constant76 m/sec2?(a) 4 seconds(c) 16 seconds

In a milk shoppe there are three varieties of milk, ,pure', .Cure,

and 'Lure'. The 'Pure' milk has 100% concentration of milk.The ratio of milk is to water in the 'Cure' is 2 : 5and in the Lureit is 3: 8 respectively. Sonali purchased 14 litres of Cure and22 litres of Lure milk and mixed them. If she wanted to makethe concentration of milk in the mixture of purchased milk to50%. How many litres of 'Pure' milk she is needed?

(b)(d)

3:22:3

6.

7. Hutch and Essar entered into a partnership just 5 months ago.The ratio of profit claimed by Hutch and Essar is 6: 17. IfEssar had just started his business 12 months ago with Rs.7275, what is the amount contributed by Hutch?

14.

A child has three different kinds of chocolates cosring Rs. 2,Rs. 5 and Rs. 10. He spends total Rs. 120 on the chocolates.What is the minimum possible number of chocolates, he canbuy, ifthere must be atleast one chocolate ofeach kind?(a) 22(c) 17

(b) 1e(d) 1s (b) 8 litres

(d) 18 litres

8.


15. In the squadron of Indian Air Force the ratio of Sukhoi is toMig and Jaguar together is 5: 7 and the ratio of Jaguar is toSukhoi and Mig together is 1 : 2 Find the ratio of Sukhoi andMig:(a) 2:7(c) 3: 1

16. During our campaign against child labour we have found thatin three glass making factories A, B and C there were total 33

children aged below 18 were involved. The ratio of male tofemale in A, B and C was. 4:3, 3: 2 and 5: 4 respectively. Ifthe no. of female children working in the factories B and C be

equal then find the no. of female children working in factoryA:(a) 5(c) 8

17. The value of a diamond is directly proportional to the square

of its weight. A diamond unfortunately breaks into threepieces with weights in the ratio of 3 : 4 : 5 thus a loss of Rs. 9.4lakh is incurred. What is the actual value of diamond :

1ff5radds up 15 litres of pure water. Now the ratio of milk andwater is 5: 4. \Mhat is the new quantity of mixture?

(b) 3: 5

(d) 5: 3

(b) 2(d) 6

(b) 13.5 laktt(d) 18.8 lakh

26. A and B are two alloys of copper and tin prepared by mixingthe respective metals in the ratio of 5: 3 and 5: 11

respectively. If the alloys A andB are mixed to form a thirdalloyC with an equal proportion of copper and tin, what is theratio of alloys A and B in the new alloyC?(a) 3:5(c) 3: 2

27. Ahotel incurs two types of expenses, one which is fixed andothers depend upon no. ofguests. lVhen there are 10 guests,

total expenses ofhotel are Rs. 6000. Also when there are 25guests average expenses per guests are Rs. 360? lVhat is thetotal expenses ofhotel when there are 40 guests?

(a) 72 litres(c) 135 litres

(a) Rs.8,000(c) Rs. 15,500

(a) 3:1(c) 4: e

(a) 4.5 cm(c) 6 cm

(a) Rs.240(c) Rs.320

(b) 270 litres(d) data insufficient

(b) +: s(d) 2: 3

(a) 28.8laklh(c) 14.4lak*t

(a) 4:3(c) 5: 2

had:(a) 139(c) 278

(a) A(c) c

(a) 4:3(c) 8:15


(b) 5: 7(d) 7: 5

(b) 240(d) none ofthese

(b) B(d) can't be determined

(b) 5: 8(d) 10:15


orl,.,(#)

(b) ns. 12,000(d) none ofthese

O) 2:3(d) 1:3

(b) 5 cm(d) none ofthese

(b) 8:6(d) z: e

(b) Rs.280(d) data insufficient

o) rgrs

(d) both (b) and (c)

i0: 6: 5

data insufficient

28. The ratio of third proportional to 21 and 42 and meanproportional to 16 and 49 is :

18. In the Ruchika's wallet there are only Rs. 16, consisting of 10

paise, 20 paise and Re. 1 coins. The ratio ofno. ofcoins of 10

paise and 20 paise is 6 : 1. The minimum no. of Re 1 coin is : 29. The period of the pendulum is directly proportional to thesquare root of the length of the string. The period of such apendulum with string of length 16 cm is 52 seconds. Find thelength ofthe string ifthe period is 65 seconds :

(b) 12(d) 8

(a) 5(c) 4

19. There are two vessels containing the mixture of milk andwater. In the first vessel the water is 2,/3 of the milk and in the

second vessel water is just 40% of the milk. In what ratiothese are required to mix to make 241itres mixture in whichthe ratio of water is to milk is 1 : 2 ?

3O. For any two numbers m, n; (m + n) : (m - n): mn =7 : 7 : 6O

Find the rrulo" of 1, 1mn

(a) 4:3(c) 3:4

31. Rs. 960 were distributed among A, B, C and D in such a waythatC and D together gets halfofwhat A and_B together gets

and C gets one-third amount of B. Also D gets ! times as muchJ

as C. \,Vhat is the amount of A?

2O. Nehru Ji had 'n' chocolates. He distributed them among 4

children in the ratio or 1, 1' 1' 1. If he gave them each one2358a complete chocolate, the minimum no. of chocolates that he

21. The ratio of working efficiency of A and B is 5 : 3and the ratioof efficienry of B and C is 5: 8. Who is the most efficient ? ^2-^2Find the value of r;-{ , lf p : q: '. r : s.r'+ s'

rat 1 rb) -l"4 "9

,'(=)

Find the value of ':l{,if p: q :: r: s.r'+s'

32.

22. lf 4A= 58 and 3A = 2C,the ratio ofB : C is :

(c) 4rq

23. Equal quantities of three mixtures of milk and water are

mixed in the ratio of 1: 2,2: 3 and 3: 4. The ratio of waterand milk in the mixture is :

(a) 193:122 (b) 122:793(c) 67:97 (d) 137 : 178

24. The ratio of age of A and B is 8 : 9 and the age ofB is 2/3 of C's

age and age of C is -2^ times the age of D. If the age of B is13

18 years then the age ofC is : The speeds of riclshaw, car and.scooter are in the'ratio of3: 5: 6. 'vVhat is the ratio of time taken by each one of themfor the same distance?

33.

34.

25. A milk man has a mixture of milk in which ratio of milk and

water is 5: 3 He sells 40 litres of rrrilk i. e., mixture then he

(a) 6: 5:3 (b)(c) 1,2:7 :6 (d)

196

35. Divide Rs. 6940 in such a way thar ,4 gets ;2rd of what B SetsJ

.)

and B gets lth of what C gets? 'vVhat is the share of A and B".5


46. In a wallet the ratio of 25 paise, 50 paise and Re 1 ,coins are inthe ratio of 'J.2:4: t which amounrs to Rs. 600. Find the no.of coins of 25 paise :

(a) 200(c) 275

47. (x-c):(x-b):(x-c)=11:9:5 where x=a+b+c.\ /hat is the ratio of q b, c?

2

(a) 10: 8:7(c) 7 : 8:10

(a) 3:4(c) 9:19

(a) Rs.60(c) Rs. 100

(a) 777(c) 218

(a) 150 miles(c) 220 miles

(a) 13:29{c) 29:77

(b) 22s(d) none ofthese

(b) 3: 5: 9(d) 6:5:3

(b) 7 :4(d) 9:10

(b) Rs.8s(d) can't be determined

(b) Rs.80(d) Rs.120

(b) 133(d) 228

(b) 80 miles(d) 180 miles

35. The ratio of age of A and B is x : y. lf A's age is increased by 3years and B's age is increased by 2 years then new ratio oftheir ages becomes 24 : 25. Given that the sum of their actualages is 93 years. Find the actual ratio oftheir ages. 48. Petrol is 7 times heavy than Kerosene and Castrol mobil is 1g

times as heavy as Kerosene. What should be the ratio of petroland mobil in the new mixture to get the mixture which mustbe 11 times as heavy as kerosene?

together?(a) Rs. 1982(c) Rs.3470

(a) 21.:22(c) 45: 48

(a) 77:25(c) 28: 40

(a) 27(c) 32.4

(a) 6:7(c) 7 :54

8 days?

(a) 27(c) 48

43. The LCM of rwo numberssum o[ these numbers is :

(a) 210(c) 315

(a) 1

(c) 5

(b) Rs.1388(d) none ofthese

(b) 100

(d) 130

(b) 30(d) can't be determined

(b) 7 :6(d) can't be determined

(b) s4(d) 64

is 210 and their ratio is 2: 3. The

(b) 17s(d) can't be determined

(b) 3

(d) none ofthese

(b) 42:45(d) can't be determined

37. o" : b = 4 : 9 if 4 is added to both of the numbers then the newratio becomes 27: 46. What is the difference betweenqandb?

(a) B0

(c) 125

(a) 20 years

(c) 25 years

(b) 50 years

(d) 40 years

38. The ratio of ages of Rahul and Deepesh is 3 : 5. 10 years laterthis ratio becomes 5: 7. What is the present age of Deepesh?

49, A girl buys 2 pigeons for Rs. 182. She sells one at a loss of 50/o

and another at a profit if8010. But she neither gains nor loseson the whole. Find the price of pigeon which has sold at aprofit :

(a) Rs. 112(c) Rs.70

39. When 5 is added to the numerator and denominator both of a(positive) fraction, then the new ratio of numerator todenominator becomes 11 : 15. What is the original ratio?

(b) 3: 5

(d) none ofthese

4O. Five numbers q b, c, dand e are in the ratio of 2: 3: 5: 8 : 9and their sum is 162. Find the average ofall these numbers :

5O. The ratio of prices of Cello and Rotomac pens in 2000 were inthe ratio of 3: 5. In 2005 the price of Cello pen trebles itselfand the price of Rotomac pen is increased byRs. 100, then thenew ratio of prices of the same pens becomes 4 : 5. 'vVhat wasthe original price of the Roromac pen in 2000?

51. A goldsmith has 361 rings of gold. He sells some of them at aloss of 4% and rest at a profit of 75o/o making overall profit of80/0. Find the no. of rings sold at a profit of 7io/o :

47, 6 pumps of Kirlosker can fill a tank in 7 days and 2 similarpumps of USHA can fill the same tank in 18 days. lVhat is theratio of the efficiency of a Kirlosker pump and a USHA pump? 52. Michel travelled from New york to New Jersey covering total

distance of 250 miles in 8 hr. partly by car at 30 miles/hr. andrest by train at 35 miles,zhr. The distance travelled by car is :

142. 16 persons can reap :th field in 6 days. How many persons

(with same efficiency) are required to reap rest of the field in

44. What number must be subtracted from each of the numbers53, 2I, 47, 17 so thar the remainders are in propoition?

53. A rabit takes 22 leaps for every 1 7 leaps of cat and 22leaps ofa rabit are equal to lZ leaps of the cat. What is the ratio of thespeeds ofrabit and cat?(a) 1:1 (b) 484:289(c) 77 :22 (d) none ofthese

54. Rs. 771. are divided among four friends in the ratio of

+ , + ' + , l. wrru, is the amount of the person who got the3456greatest share?(a) 1+(c) 36

45. The angles of a triangle are in the ratio of 2: 3: 4.Find themeasurement of greatest angle :

(a) 30"(c) 100'

(b) 40(d) 60

55. l0years ago the age of Karishma *ur]rd of the age ofBabita.3

14 years hence the ratio of ages of Karishma and Babita willbe 5: 9. Find the ratio of their presenr ages :

(b) 77:27(d) 13: 25

(b) 60"(d) 80"


56. The ratio of numerator to a denominator of a fraction is I5

when x and 5x are added to the numerator and denominatorrespectively to the given fraction then the ratio of the newfraction will be :

(a) 1:1(c) 1:5

57. xvaries directlyasy and xvaries inverselyas the square ofz.When y = 75 and I = I then z = 5 Find the value of x when

I =24andz=4:(a) 1 (b) 2(c) 3 (d) 4

58. xvaries directly asQr2 + z21.Aty =1andz =2, thevalue ofx is 15. Find the value of 4 when x =39 and y =2:

,"1.gri59. Weight of a sumo is jointly varies as his height and his age.

When height is 1.2 m and age is 20 years his weight is 48 kg.Find the weight of the sumo when his height is 1.5 mete andage is 30 years :

(a) 60 kg (b) 72ks(c) 90 kg (d) s8 kg

6O. If(a+ b):{b+ c): (c+ a)=5:6:9anda+ b+ c= 10.Whatisthe value of c :

@) 1:25(d) 2:7

(b) 3

(d) 6

6t. A, B, and C have amounts in the ratio of 3: 4 : 5. First B gives

ltn to a urra 1Jh to C then C gives ]trr to a . Find the final44"6ratio of amount of. A, B and C, respectively :

(a) 2(c) 5

(a) 4: 3: 5

(c) 6:4:2

(b) 3

(d) 7

@) 5:4:3(d) 5: 2: 5

(a) 2(c) 4

1. A tin contains a mixture of Dew and Sprite in the ratio of 7 :3and another tin contains the Dew and Sprite in the ratio of5: 4. In what proportion should the solution of two tins be

mixed to achieve a perfect proportion of 2: 1 (in which Dew is

2 times that of sprite).

ratioT i 5was formed from the mixture of 7 :3by adding theKerosene in it. If 240 litres petrol is required in thereplacement method, what is the total amount of Kerosenewas added to prepare the mixture of7 :5?

(a) 10:3(c) 3:10

(b) +: t(d) 3: 1

(b) 21 line(d) can't be determined

(a) 100 litres(c) 50 litres

(a) \09462.5(c) 14400

(a) 20o/o

(e) 26.660/o

(b) 4OOlitres(d) 200 litres

(b) s8800(d) data insufficient

(b) 16.660/o

(d) 8.330/o

(b) 6(d) 11

2. There are two containers A and B filled with mentha oil withdifferent prices and their volumes are'J,4O litres and 60 litresrespectively. Equal quantities are drawn from bothA and B insuch a manner that the oil drawn from A is poured into B andthe oil drawn from B is poured into A. The price per litrebecomes equal in both A and B. How much oil is drawn fromeach ofA andB:(a) 40 litre(c) 42 litre

2:3t '.7

(a) 0.5 kg(c) 0.75 kg

O) 1kg(d) none ofthese

5. A man bought 9 mangoes for a rupee and sold them at 6mangoes for a rupee. What is the ratio of profit to the cost

7. Sometimes ago in a Cinema hall a blockbuster movie wasbeing shown. Due to excessive demand of the show, managerof the Cinema hall increased the prices of tickets in all thethree categories. 25o/o in first class, 72.50/o in special class and400/o in balcony. If the collection of one show was Rs. 218925,then find the collection from the balcony only given that theratio ofprice ofticket (increased price) for first class, specialclass and balcony is 5: 9 : 14 :

3. A three digit number is such that this number itself is divisibleby the sum of its digits. The sum of hundreds and unit digit is

6 while the sum of the tens and unit digit is 5. What is theratio of unit and tens digit :

In a mixture of petrol and Kerosene petrol is only 99 litres. ifthis same quantity of petrol would be presented in anothermixture of petrol and Kerosene where total volume would be198 litres less than the actual mixture then the concentrationof petrol in the actual mixture would have been 13.3370 pointless than that of the new mixture. What is the concentrationof petrol in actual mixture?

8.

(a) 1:2(c) 3: 4

4. Sachin bought 1.5 kg fresh grapes. The ratio of water is topulp was 4: 1. When his naughty child cmshed these grapes,

then some water get wasted. Now the ratio of water is to pulpis 3 : 2. What is the total amount of the crushed grapes? A drum of 20 litres is filled with milk. A milkman has only nrro

measuring vessels of 3 litres and 5 litres without anycalibration. He has to measure four litres of milk for a

customer without using anyother vessel. Minimum how manyoperations are required for this work, where an operation iscounted if the milk is transferred from one vessel to anothervessel?

(a) 5(c) 8

The ages of Vinay, Varsha, Veera and Vikram are in arithmeticprogression, but not in order. The ratio ofages ofVinay andVarsha is 6 : 5 and Veera is to Vikram is 7 : 8. Two years later

o)(d)

9.

price?3(a) --'10

(c) 1./2

b') 3

(d) none ofthese10.Half of the volume of Petrbl and kerosene mixture of ratio

7 : 5 is converted into a mixture of ratio 3: 1 by thesubstitution (or replacement) method. While the mlyture of

6.

1198:,',,,

the age of Varsha and Vikam will be 2: 3 Find the ratio ofages ofVinayandVeera:(a)7:o(c) 6:7

11. A container is filled with the mixture of milk and water. Theratio of milk and water is same. Bobby and Sunny increasesthe concentration to 60%. Bobby makes it by adding the milkand Sunny makes it by replacing the mixture with milk. \Mhatis the percentage of milk added by Bobby to that of milkreplaced by Sunny :

(al 100%(c) 133.330/o

12. There are two vessels AandB containing 25 litres each ofpure milk and pure water respectively. 5 litres of milk from Ais taken and poured into B, then 6 litres of mixture from B istaken and poured in A. What is the ratio of water in A and Brespectively :

(a) 4: 5

(c) 5: 4

13. The ratio of age between A and B is 6 : 5 and the age of each

C and D is 2 times that of B. Age of F is less than A but10

greater than B. The ratio of ages between I and E is 2 : 3 alsoage of A is 3 years less than E. 'vVhat is the ratio of ages ofA and F if all the ages are in integers?


November. Initially Hari's cows used for grazing then for theremaining days of the month Murli's cows grazed it. If Harihas paid tu. 3500 and Murli has paid Rs. 5000 for grazingthen for how many days Hari used the grazing field :

(a) 12: 11(c) 24 :79

(a) Rs.62,000(c) Rs.80,000

(a) Rs.36(c) Rs.45

(a) Rs.45(c) Rs.72

O) s: e(d) 8:9

(b) 7200/o

(d) none ofthese

(b) 1:4(d) 2: 3

(b) 9 :7(d) 12: 13

(b) Rs.72,00o(d) none ofthese

(b) Rs.10,800(d) none ofthese

(b) Rs.54(d) Rs.42

O) Rs.69(d) Rs.84

19. The speeds ofscooter, car and train are in the ratio of1 : 4 : 16.If all of them covers equal distance then the ratio of timetaken/velocity for the each ofthe vehicle is :

(a) 14(c) 21.

(a) 256:16:1(c) 16: 4: 1

(b) 16(d) 20

(b) 1:4:16(d) 16:1:4

(b) Rs.2000(d) none ofthese


Radhika purchased one dozen bangles. One day she slippedon the floor fell down. lVhat can not be the ratio ofbroken tounbroken bangles :

(a) 7:2(c) 2:3

21^. ffg=!bd-. ad(a) ,

DC

n'' #

(b) 1: 3(d) 1:5

= 9, then the value or( a'P + cnq+ e'r\t/"

f -'- " I a"p + d'q+ f'r)

b)#(d) none ofthese

14. The ratio of students in a coaching preparing for B.Tech andMBA is 4 : 5. the ratio of fees collected from each of B. Techand MBA student is 25 : 16. If the total amount collected fromall the students is 1.62 lakh, what is the total amountcollected from only MBA aspirants?

At a casino in Mumbai, there are 3 tables A, B and C. The pay-offs at A is 10 : 1, at B is 20 : 1 and at C is 30 : 1. If a man betsRs. 200 at each table and win at two of the tables, what is themaximum and minimum difference between his earnings canbe?

(a) Rs.2500(c) Rs.4000

22.

rf m= 4Pq,thentherrulueof m+2P *^*4,P+ q m-2p m-24

(a) 2

k) 2tnPq-' (p+ q)The cost of the marble varies directly with square of its

weight. Marble is broken into 3 parts whose weights are inthe ratio 3: 4: 5. Ifmarble had been broken into three equalparts by weight then there would have been a further loss ofRs. 1800. What is the actual cost of the original (or unbroken)marble?(a) Rs.3600(c) Rs.2160

23.

The ratio of volumes of two cubes is g: 27. What is the ratioof surface area of these cubes respectively?(a) 2:3(c) 8:19

(b) +: g(d) g:+

15 Directions for 16 and l7:, Fourfriends A I B,C and D havesome rnoney among them one day they decided to equate themoneyt so first A gave B what B had initiatty, then B gave Cwhat C had initially Again C gave D whatr D had initaiily andfinally D had doubled the mroney of A. fnius each of them hadequal sum ofRs. 4s.

16. 'vVhat was the initial amount of B?

17. What was the amount with C after second transaction?

Dudheri Lal has two jars. Jar A is completely filled with milkand another jarB is totally empty. Before selling the milk in atown he transferred some milk in to the empty jar B then hefiIl the jar A with water. Once again he transferred themixture of milk.from A to B so that B is completely filled.Which one of the following is correct?(a) Concentration of milk in B cannot be less than 7So/o(b) Concentration of milk in B cannot be greater than 75o/o(c) Concentration of milk is always 75%(d) none ofthe above

In ABC corporation there are some management trainees.These trainees are divided into 3 groups A,B andC for 3different projects in the ratio of 3:4: 5 respectively, whereP, Q, R are the projects-in-ch arge of A, B, C respectively. Thedifference between the no. of trainees in A and C is notgreater than 3. Also P, e, R belongs to the group of trainees.The no. ofassistant ofQ is less than the no. ofassistance ofRby:

Hari and Murli have 24 cows and 30 cows respectively. Bothof them together hired a grazing field for the whole month of


(a) 33.33%(c) 25o/o

27. tf

b+c+d+e+f1(a) -'- 81

n, 1

28. An engine can

(b) 2oo/o

(d) 16.660/o

find the value of(a) 140:744:747(c) 15: 27:28

(a) Rs.380(c) Rs.355

(a) 1

(c) 3/4

O) 40 : 44: 47(d) 252:245:240

(b) 12(d) 10

(b) 7:7:2(d) 2:1:2

(b) Rs.480(d) Rs.448

(b) 7/2(d) none of these

(b) 16 years

(d) 12 years

199

steps of A are equal to the 7 steps of B and B steps of C. What is

the ratio of their speeds :

de1e f 3

)l:)

abcbcdb+ c+ d+ eQ+ 33. In a family there were n people. The expenditure of rice per

month in this family is directly proportional to the 5 times the

square of no. of people in the family. If the elder son left the

family to study in USA there was decrease in consumption of95 kg rice per month. What is the value of n?

ft)I"27

wagon attached. Reduction in the speed of the train is directly

proportional to the square root ofthe no' ofwagons attached

to the engine. When there are only four wagons attached its

speed is l4) -zr. The greatest no. of wagons with which the' \eiengine can move is :

(a) 5(c) 9

34. One day in summer I wanted to chill me out, I went to a cool

corner. I gave him a note of Rs. 10 and asked for a coke

costing Rs. 5 per jar and he did so, but he returned me Rs. 5,

in the denomination of Re. 1, 50 paise and 25 paise. What

could be the ratio of no. of coins of Re. 1, 50 paise and 25

paise respectively :

(a) 2:3:1(c) 6:1: 3

(d) 1

move at the speed of f mZs without any

29. Mrs. Annapurna per day sells exactly four quintal sugar at Rs.

2000 per quintal getting the profit of Rs. 25%. Since she

mixes two varities of sugar, one costs Rs. 14 per kg and

another costs Rs. 22 per kg. One day due to huge demand in

market she had only 3 quintal of the required mixture so she

purchased the sugar costs Rs. 17 per kg at Rs. 18 per kg from

the wholeseller on that day and then she mixed 300 kg

mixture with 100 kg sugar costing Rs. 18 per kg to fullfill the

- demand of the market selling at the same price. How much

percent less does she gain that she would have gained, if she

had sufficient quantity of usual mixture of sugar?

36. The ratio of the density of 3 kinds of petrol Pt, P, and P. is9 :7 : 5. The densiry of Pr is 18 gm,/cc and Pr, Pr, P, are mixed

in the ratio of 6 : 5: 4 by weight. If a lltre of P. cost Rs. 40,

then find the cost of P, in 450 kg of mixture of Pt, P, and P, :

(a) 744(c) 1.2

(a) 72.5o/o

(c) 62.50/o

(a) 72(c) 18


(b) 18.18%(d) can't be determined

(b) 20(d) 15

3s. If a = b -1 ando+b+c c+a a+bb_1S:

a+b+c

ral I (b)"2(.) 1 (d)"4

b+ c+0 then the value of

1

;J

1

3O. Two vessels P and Q contain 'a'litres of petrol and 'b'litresof Kerosene respectively. ' c' litres of petrol and same quantity

of Kerosene is taken out and then transferred to Q and P

respectively. This process is repeated several times. If after

the first operation the quantity of petrol or Kerosene in either

of P and Q does not change. 'vVhat is the value of 'c'?

(u't ab rb) 2ab

'-' (a-b) ' ' (a+ b)

(c\ ab (d) l'g)'" (a+b) " (b,

31. Awind Singh purchased a 40 seater bus. He started his

services on route no. 2 (from Terhipuliya to Charbagh withroute length of 50 km). His profit (P) from the bus depends

upon the no. of passangers over a certain minimum number

of passangers 'n'and upon the distance travelled by bus. His

profit is Rs. 3600 with 29 passangers in the bus for a journey

of 36 km and Rs. 6300 when there are 36 passengers.

travelled for 42 km. What is the minimum no. of passangers

are required so that he will not suffer any loss?

37. Three persons Amar, Akbar and Anthony aSree to pay theirhotel bills in the ratio of 3: 4: 5 Amar pays the first day's billwhich amounts to Rs. 26.65, Akbar pays the second days billwhich amounts to Rs. 42.75 and C pays the third day's billwhich amounts to Rs. 53.00. When they settle their accounts,

which of the following happens?

(a) Amar gives Rs. 3 to C(b) Akbar gives Rs. 2 to Amar(c) Amar gives Akbar Rs. 1.95 and Rs. 2 to Anthony(d) none ofthe above

38. Findthevalueofxif(14x - 4): (8x - 1)= (3x + 8): (9x + 5):

32. Three cats are roaming in a zoo in such a way that when cat A

takes 5 steps, B takes 6 steps and cat C takes 7 steps. But the 6

39. Pooja, Shipra and Monika are three sisters. Pooja and Shipra

are twins. The ratio of sum of the ages of Pooja and Shipra is

same as that of Monika alone. Three years earlier the ratio ofage of Pooja and Monika was 2: 7. What will be the age ofShipra 3 years hence?(a) 21 years

(c) 8 years

40. A couple got married 9 years ago when the agb,of wife was

20% less than her husband. 6 years from now the age ofwifewill be only 72.5o/o less than her husband. Now they have six

"fo6nchildren induding single, twins and triplbts and the ratio oftheir ages is 2: 3: 4 respectively. \,Vhat can be the maximumpossible value for the present age of this family?


that she could purchase double the no. of toffees with thesame amount as she had spent on the ground floor. Also topurchase the same no. of toffees on the third floor she had tospend Rs. 2 less than *rat of on the first floor. How manytoffees did she buy?(a) 6(c) 1g

41. The price of a necklace varies directly as the no. of pearls in it.Also, it varies directly as the square root of radius of a pearl.The price of a necklace was Rs. 150. When it had 75 pearlseach ofradius 1 cm. Find the radius ofthe pearl ofa necklacehaving 100 pearls whose cost is Rs. 600.

46. A contracror deployed some men to plant 1g00 trees in a

certain no. of days. But in 1rd of the planned time 120 plants3'could be less planted so to ful fill the target for the rest of thedays every day 20 more plants were planted. Thus it savedone day out of the initially planned no. of days. How manyplants he planned to plant each day initially?


{a) 2(c) 3

(a) 1.s l/km(c) 2.81/km


(b) e(d) 4

(b) 9 years

(d) 3 years

(b) 2ukrrr(d) 201/I.n

(a) 180(c) 120

(a) 4s(c) 54

orphan gets :

(a) 90:75:42(c) 75: 42:9O

(b) 12(d) 1s

o) 100(d) 160

@) zz(d) 100

O) Rs. 18(d) none ofthese

(b) 180:150:82(d) none ofthese

42. The price of a bookvaries directly as the no, of pages in it andinversely as the time periods in years that have elapsed sincethe date of purchasing. Two books cost the same, however,the no. of pages in the first book is triple of the second book. ifthe first book is sold on 18 years ago, how long ago was thesecond book sold?(a) 54 years(c) 6 years

O) Rs. 7s00(d) Rs.9000

47. A and B have to write 810 and 900 pages respectively in thesame time period. But A completes his work 3 days ahead oftime and B completes 6 days ahead of time. How many pagesdid A write per hour if B wrote 21 pages more in each hour?'

43. Akbar and Birbal who purchased the shares for the cosr oftheir basic salaries which are in the ratio of 5: 6. later oncompany gave them 40 additional shares to each, due towhich the ratio changed to 7 : 8. If the worth of each share isRs. 75, what is the basic salary of the person who got lessshares?(a) Rs.10500(c) Rs.8800

Distance covered by a train is directly proporrional to the timetaken and also it varies directly as the square root of fuel usedand varies inversely as the no. of wagons attached to it. Affain coveres 192 kmjourney in 20 hours when there are 10wagons attached to it andrtotal fuel consumption was 256litre of diesel. Find the consumption of fuel per km when atrain goes 200 km in 25 hours with 15 wagons attached to it :

48. Three friends A,B andC decided to share the soda waterwith D, who had no soda water. A contributed 2 tumblermore than that of B and B contributed 1 tumbler more thanthat of C and then all of them had equal amount of sodawater. In turn D paid money, which was divided amongA, B and C in the ratio of their contribution to D. Thus A hadgotten thdce as much money as B had gotten. The price ofeach tumbler of soda water was Rs. 15 and each transactionwas integral in numbers either the sharing of money orcontribution of soda water. What was the sum of money thatB had gotten?(a) Rs. rs(c) Rs.22.5

49.

44.

45. At Sahara shopping centre, a person can purchase as mucharticles at a time as his or her age that is a person of n yearsage can purchase only n similar articles at a time. Amisha isyounger to her elder brother who has just entered into histwenties. One day Amisha went to the Sahara shoppingcentre, she purchased same toffees at a particular rate on theground floor. But when she reached on third floor she found

In Maa Yatri Temple every devotee offers fruits to theorphans. Thus every orphan receives bananas, oranges andgrapes in the ratio of 3: 2: 7 in terms of dozen. But the weightof a grape is 24 gm and weight of a banana and an orange arein the ratio of 4 : 5 while the weight of an orange is 150 gm.Find the ratio of all the three fruits in terms of weight, that an

(a) 11:3(c) 8:11

(b) 11: 8(d) none ofthese

(a) 120(c) 180

(a) 14 litre(c) 8 litre

(b) 1s0(d) 170

(b) 16 litre(d) +2 litre

3. 6 litre is taken our from a vessel full of Kerosene andsubstituted by pure petrol. This process is repeated two moretimes. Finally the ratio of petrol and Kerosene in the mixturebecomes 1701 : 27. Find the volume of the original solution :

In a zoo, there are rabbits and pigeons. Ifheads are counted,there are 340 heads and if legs are counted there are 1060legs. How many pigeons are there?

ffifu1. In two alloys the ratio of Iron and copper is 4: 3 and 6: 1

respectively. If 14 kg of the first alloy and 42 kg of,the secondalloy are mixed together to form a new alloy, then what willbe the ratio of copper to iron in the new alloy :

2.


4. In three vessels, each of 25 litres capacity, mixture of milk and' water is filled. The ratio of milk and water are 3: 1, 2:3,4: 3

in the respective vessels. If all the three vessels are emptiedinto a single large vessel, then what will be the ratio of waterto milk in the resultant mixture?

iii*siliii11. Dia and Urea are two chemical fertilizers. Dia is consists of

N, P and K and Urea conists of onlyN and p. A mixture of Diaand Urea is prepared in which the ratio of N, p and K is 260/0,680/o and 6010 respectively. The ratio of N, p and K in Dia is2Oo/o,7oo/o and 100/o respectively. lVhat is the ratio ofN and pin the Urea?(a) 27o/o and,630/o

(c) 35% and 650/o

12. The ratio of copper and nickel by weight in the two alloysX and Y arc 2:7 and 5: 4. How many kilogram of the alloyX and Y are required to make 42 kg of new alloy Z in whichthe ratio of copper and nickel is same?

(a) 1.79:247(c) 279:777

(a) 43:96(c) 348:962

(a) 1:3(c) 3: 4

(a) 31o/o

(c') 49o/o

b) 797 :214(d) 179 234

(b) +:z(d) 5: 1i

b) 438:962(d) 962:438

O) 2:1(d) 2: 3

(a) 6 kg and 36 kg(c) 7 kg and 35 kg

(a) A <B(c) A >B

O) 330/o and 670/o

(d) 70o/o and 300/o

O) i0 kg and 32 kg(d) none ofthese

(b) 3(d) s

(b) A =B(d) can't be determined

5. Two liquids are mixed in the ratio 4: 3and the mixture is sold

at Rs. 20 with a profit of n!;h.If the fust liquid is costlier

than the second by Rs. 7. Find the sum of costs of both theliquids :

(a) Rs. 11 O) Rs.29(c) Rs.35 (d) Rs.70

6. Two alloys made up of copper and tin. The ratio of copper andtin in the first alloy is 1 : 3 and in the second alloy it is 2: 5. Inwhat ratio the two alloys should be mixed to obtain a newalloy in which the ratio of tin and copper be 8 : 3 ?

There are two alloys made up of copper and aluminium. Inthe first alloy copper is half of the aluminium and in thesecond alloy copper is thrice as much as aluminium. Howmany times the second alloy must be mixed with first alloy toget the new alloy in which copper is twice as that ofaluminium?(a) 2(c) 4

(b) 56.25 litres(d) 36 litres

13.

(a) 3:5(c) 3: 8

7. Three vessels having volumes in the ratio of 2 : 3 : 5 are firll ofa mixture of water and milk. In the first vessel ratio of waterand milk is 1 : J in second 2:3and in third vessel, 2: 5. If ailthe three vessels were poured out in a large container, what isthe resulting ratio of milk and water?

14. There are 90 litres castrol and 150 litres CRB mobil oils. Theprice of castrol is Rs, 8€ per litre and price of CRB is Rs. 75 perlitre. Equal amount of castrol and CRB is taken out and thenCRB is poured out in the vessel ofcastrol and castrol is pouredout in the vessel of CRB. Now the rate of both the mixtures issame. \Mhat is the amount of mobil oil taken out from each ofthe vessel?

(a) 45 litres(c) 24.5 litres

There are two containers, the first contain, 1 litre pure waterand the second contains 1 litre of pure milk. Now 5 cups ofwater from the first container is taken out is mixed well in thesecond container. Then, 5 cups of this mixture is taken outand is mixed in the first container. Let A denote theproportion of milk in the first container and B denote theproportion ofwater in the second container then :

8. The number of oranges in three baskets are in the ratio of3: 4 : 5 In which ratio the no. of oranges in first two basketsmust be increased so that the new ratio becomes 5: 4 : 3 ?

15.

9. A vessel of capacity 2 litre has 25% alcohol and another vesselof capacity 6 litre had 40% alcohol. The total liquid of g litrewas poured out in a vessel of capacity 10 litre and thus therest part of the vessel was filled with the water. lVhat is thenew concentration of mixture?

(b) 71o/o

(d) 290/o

lO. Alloy A contains 4oo/o gold and 600/o silver. Alloy B conrains35% gold and 40% silver and2|o/o copper. Alloys A and B aremixed in the ratio of I: 4. \,Vhat is the ratio of gold and silverin the newly formed alloy is?(a) 20o/o and 30% @) 360/o and 44o/o(c) 25o/o and 35% (d') 49olo and 360/o

Mitthu Bhai sells rasgulla (a favorite indian sweets) at tu. 15per kg. A rasgulla is madeup of flour and sugar in the ratio of5 : 3. The ratio ofprice of sugar and flour is 7 : 3 (per kg). Thus

he earns 66 ?o/o proht.\ /hat is the cost price of sugar?3

(a) Rs. 10/kg O) Rs.9/kg(c) Rs.18/kg (d) Rs. 14lkg

16.


1 (c) 2. (b) 3. (d) 4. (c) 5. (a) 6. (b) 7.(d) 8. (b) 9. (c) 10. (d)11. (a) 12. (b)

1(c) I 2.(c) I g.cU I +.ral I s.r.l

1 (d) I z. (.) 3. (c) 4, (a) s. (d) 6. (c)

1 (b) 2. (d) 3. (c) 4, (a) 5. (d) 6. (d) (c) 8. (b) 9. (a) 10. (d)11. (b) 12. (d) 13. (b) 14. (c) 15. (c) 16. (b) iz.'(e) 18. (d) 19. (b) 20. (b)21. (c) 22. (a) 23. (c) 24. (b) 25. (d) 26, (a) 27. (c) 28. (c)

1 (b) 2. (b) 3. (d) 4, (c) 5. (b) 6. (d) 7. (b) 8. (c) 9. (b) 10. (b)11. (a) 12. (b) 13. (a) 14, (c) 15. (d) 16. (d) t7. (c) 18. (c) le. (b) 20. (a)21. (a) 22. (c) 23. (a) 24. (c) 25. (c) 26. (c) 27. (b) 28. (a) 29. (d) 3o. (c)31. (b) 32. (d) 33. (d) 34. (b) 35. (c) 36. (c) 37. (b) 38. (.) 3e. (d) 4O. (c)41. (a) 42. (c) 43. (b) 44. (c) 45. (d) 46. (d) 47. (c) 48. (b) 49. (c) so. (b)51. (d) 52. (d) 53. (a) 54. (d) 55. (a) 56. (c) 57. (c) s8" (b) 59. (c) 6o. (c)61. (d)

1 (a) 2. (c) 3. (b) 4. (c) 5. (c) 6. (d) 7. (d) 8. (a) 9. (c) 10. (c)11. (d) 12. (b) 13. (a) 14. (b) 15. (d) 16. (c) 17. (d) 18. (a) 19. (a) 20. (c)2'-. (c) 22. (c) 23. (a) 24. (b) 25. (a) 26. (c) 27. (c) 28. (b) 29. (a) 3o. (c)31. (d) 32. (a) 33. (d) 34. (b) 3s. (b) 36. (b) 37. (c) 38. (b) 39. (c) 40. (b)41. (b) 42. (c) 43. (b) 44. (b) 4s.o) 46. (b) 47. (c) 48. (a) 49. (a)

1 (d) 2. (b) 3. (c) 4. (a) s. o) 6. (b) 7. (d) 8. (b) e. (d) 10. (b)11. (c) 12. (c) 13. (c) 14. (b) 15. (b) 16. (d)

Now

Again

=)

b+ c=5a+ d=7a: b ='J.:3c: d=2:6

1. a: b:: c: d

a2 +b2 + c2 + d2 =Sob+c=S

and q.: b =7:3If consider a: b = 1: 3 as it is, then

c=2 (s-3=2)and d=6 (..a:b::c:d)

a2 + b2 + c2 + d2 =12 + 32 + * + 62 = 5oHence, the presumed values are correct.

Thus, the average of q b, cand d = a+ b + c + d

4

_1+3+2+6_Q4

Hence @) is correct.Alternatively: Assume option O)... a+b+c+d _a

4":+ a+b+c+d=l2

Again"=8=zy+2 2+2Hence option (d) is correct.

4. Best way is to go through options

qnA q3+b3 _53+23 125+g 133;515-53_2r=12s_8=1i7

Hence option (c) is correct.

5. H:S=1:1and S:G=2:3

H:P=7:2.'. H:S:G:p=2:2:3:4

(verified)

=2x:2x:3x:4xTherefore 2x + 2x + 3x + 4x

= 11" = 5544=t x =2OMarks in History = 49

Sociology: 49Geography = $Q

Philosophy = gg

Hehce, only in two subjects he scored 60% or above.Hence option (b) is correct.

Let the incomes of A and M is 2x and 3xLet the savings ofA be K, then the expenditure ofM be KAlso expenditure ofA =2x -KGiven (2-x-K)+K=8000 =e x=4000.'.Total income ofA and B = 2x + 3x= 5x = 5 x 4000 = 20000.'.Total savings of A and B = 20000 _ g000 = Rs. 12,000Profit of Hutch _ time period x amount of Hutch investedProfit of Essar ti*

6 5xK77 72x1,275

=+ * -6x12x1275 -1o'n17x5 -rvuv

Minimum numbei of chocolates are possible when hepurchases maximum number of costliest ihocolates.Thus 2x5+5x2=Rs.2ONow Rs. 100 must be spend on 10 chocolates as 100 = 10 x 10.Thus minimum number of chocolates = 5 + 2 + IO =77Maximum number of chocolates are possible only when hepurchases. minimum number of costlier chocolates andmaximum number of cheaper chocolates

2x5+1x10=Rs.20Now Rs. 100 must be spend on 50 chocolates as 100 = 2 x 50.Thus maximum number of possible chocolates

=2+I+50=53

6.

2.

Now verify that a2 + b2 + c2 + d2 = 50. Since it is correct.Hence option (b) is correct.

a _o-+ b

baa2=ab+b2

q2 -b2 - ab=oLetb=1,thena: b=a:1.'. q2-a-1=0

7.

:+ abF---j----

(byputting b = t1

_ 1 t G /Solving quadratic equation )=A= 2 lby sridhu.u.fruryu's for-uUJ

:."='*f (negative value can,t be considered)

o,6=1*^6,12

or a:b=(I+.u/5):2

Therefore, b22a+b i+Js+2 g+Ji

Hence option (b) is correct.

The best way is to go through tptions

x+y 8+2 10 s_=x-y 8-2 6 3

8.

3.

(verified)

9.

Meretere -+ M, Teremere -+ T and lGrabbu Singh + K

f+ = 2.66 (since all the 8 chocolates were shared by 3)a'J

It means M has given 5 - 2.66 = 2.33 chocolates to Kand T has given 3 - 2.66 : 0.33 chocolates to KThus M and T will receive the amount in the ratiodonations (i.e., share of chocolates)


Since in the required mixture the ratio of milk and water is

1 : 1so she has to add up 16litre of more milk (pure) to get it,for the fixed quantity of water.

3:04

10.

So the M receives fu. 14 and T receives Rs. 2Thus the difference : Rs. 12

Share of a man, a woman and a boy =7 x, 4x and 3xthen the share of 4 men = 4 x7 x = 28xthen the share of 5 women -, 5 x Qv = )gvthen the share of2 boys = 2 x 3x = 6x

Now, the share of all women =20x

x 4536(28x+20x+6x)

!x4536=Rs.168054

Hence, the share of one woman = 16-80

= 3365

Concentration of petrol inABC134tis

Quantity of petrol taken from A = 1 litre out of 2 litreQuantity of petrol taken from B = 1.8 litre out of 3 litreQuantity of petrol taken from C = 0.8 litre out of 1 liueTherefore total petrol taken out from

A, B andC = 1 + 1.8 + 0.8= 3.6 litre

So, the quanriry of Kerosene = (2 + 3 + 1) - (3.6) = 2.4litre

Thus, the ratio ofpetrol to Kerosene = - =:2.4 2

15. S:(M+J)=5:7 + 7S=5M+5J.I:(S+ M)=7:2 > 2J=S+M

By solving equations (1) and (2), we get

17.

Therefore,

S:M:J=5;3:4S:M=5:3Male Female

A4x:3xB3y;2yC5z:42

2Y=42Male Female

A4x:3xB6z:42CSz:42

7x+792=33but z can assume only one value i.e., z =7

Hence, 7x+79=33 + x=2Thus the no. of female children in factory A = 3x = 6.

The ratio of broken parts (by weighQ = 3x : 4x : 5xTherefore value ofbroken parts ofdiamond

= (3x)2 + (4x)2 + (5x)2 = 50x2

The value of original diamond = (3x + 4x + 5x)2 =744x2Therefore, loss in valqe =744x2 - 50x2 = 9.4 lak<h

+ 94x2 = 9.4lakh

= 94x2 =940000

Hence, the actual value of the diamond =144x2

= 144 x 10000

= L4.4lakh

Again since

l}x+20y +7002 =1600x:y=6iI

.'. 60y + 20y t 1002 = 1600

= uOy + 1002 = 1600

- 4y + 5z=80Puttingz =7,22 4,5...,wegetatz = 4,y = 15(aninteger)Hence, min. 4 coins of Re. 1 will be there

Concentration of water in first vessel = ? = OOro5

concentration of water in second lr"rr"t = I = 28.57o/o7

By alligation227... ,.5\r,'

.23'.(z-|)/ \rt 2)[s 3) |.5 r)

11

of

.. .(1)

...(2)

M2.33

o1J

T0.33

1

316.

ZJ

7

1

;J

1

but

11.

12.

18.

13. , * "l'i'* /-lot/6/o

rherefore, t=*l; or

+ r=kElg

s=r! = K=22

Again

Pure100%

1

T

r =K E =2,.8 =+1g v i6

T = 4 seconds

Cure Lure40o/o 37.5o/o

?!58

19.

14.

-CureI I l,Uret-bNew mixture

Milk4T

6lMilk101

161

261

1

Water101

161

Water261 _

I

I

26r I1

Mixture1_41

221

15

27

27

:157:5

Therefore, ratio of first mixture to the second mixture = 5: 7Required mixture


20. LCM of2,3,5,8 = 120295.

Therefore,

Therefore, minimum number of chocolates

= 60 + 40 + 24+ 15= 139

21. A:B =5:3B:C=5:8

.'. A:B:C =25:75:24So, A is the most efficient.

22. A:B=5:4 + 10:8A:C=2:3 + 10:15

.'. A:B:C=10:g:15

.'. B:C=g:1523.ABC

fProportion of milk) 1 2 3t-t'(inmixture ) z 5 ior 35 42 45

10s 10s 10squantity of milk in new mixture = 35 + 42+ 45 = 1221quantiry of water in new mixture = (105 x 3) _ 122= lgglTherefore, ratio of water is to milk = 193: 722

24. A:B=8:9B:C =2:3C : D =9:73

A : B : C : D =744x : l62x : 243x : 351xBut we need not to solve this, since we already know that

B:C =2x:3x.'. bc:3x::19:k + k=27years

(initially)

(after selling 401 mixture)

(after adding 151 water)

3 x=20

.'. New quantity of mixture = (5x _ 25) + 3x = 135litre26:AB

C:T C:T

!x?o=60

! x72O = 40

! x12O = 2a5

1x120=1s8

5:3 5:11

Concentration of copper in a = !8

Concentration of copper in g = !

By alligation(B) -r s (A)tu\r,,au

,.2'-(s-_\/ \rt s)18 2) lz-r6J

2316 16

2;3So,therequiredratioofA:B=3:2 (Since B: A=2:g

27. k+10x=6000k + 2Sx = 9000 ('.. 25 x 360 = 9000)

= x=200 and k=4000.'. k+ 40x=4000+ 40 x200=12000where k is the fixed expenditure.

28. Third proportional of 21 and 42 is g4 and mean proportionalto 16 and 49 is 28.

Therefore required 84 3ratro=2g=ior3:1

29. P*..,/i = P=k^[Is2= k,lt6k=13P=KJi6s=13d

l=25cm

BO. m+ n _7xm_n x n 3x

Again mn = 4x x3x =72x2and mn= 6OxSo 6Ox =I2x2

.(=J

m=20 and n=15Hence, 1,]=1,] =..om n 20'1s-"'-Alternatively: m - 4x

n3x11--:-=3:4tnn

31. !+_j:!+n., -1-and B:C and C:D+ + ;- :-3135

9:3:5Again A+B:C+D

U: ,3* I768

ThusA+B=16 =+ A=7 whenB =9.'. Therefore share o 7to=

^x9600=tu.2S0

Again,

Water

Ej

a-1si)--rbxri5l(+ 1st)-rEi4

5x-25_ 5

3x4

Milk

ry6rs)

+

l5*:%lI

J

l5"-2Bl.5

32. The best way is to consider some values in the proportion andverify the option, but care should be taken thdt in this type ofquestion you are required to verify a possible correct optionatleast with 2-3 tirnes with different values.Since, ifyou consider

' p: q:: r: S :+ 'J.,:2::3: 6

you will find answer (b) is aiso correct, but for onlyparticularvalues. So it is not the general solution. Finally you willrealize that option (d) is the most appropriate answer.

33. Solve by componendo and dividendo or as mentioned in theprevious solution.

34. Since for the constant distance time is inversely proportionalto the speed. So, rhe required ratio of time taken by each ofthe rickshaw, car and scooter is

Ratio, proportion and Variation1

42. ;th field can be reaped by 96 man days5

4.'.-th field can be reaped by 96 x4 = 3g4 man days5

Now, since there are only g days so reqired

number of men = 384

= 4g men8

2'l.O=2x3x5x7N,

= a. J.Z

N2 3kxSx7

N1:Nr=l;JNr = 70 and N, = 195

N1 +N2=70+105=175

43.

Therefore,

since,

Therefore,

35. A:B=2:3 and B:C=3:5+ A:B:C = 2:3:5.'. (A+B):C=5:5=1:1

Hence, Share of(A + B)= 1 x694O =34702

36. x+y=93and x+3_24

y+2 25

+ 25x-24y =(-27)From equations number (1) and (2)

x=45 and l=49, Alternatively: Go through options.

g7. I =4* and q+ 4 -2rb 9x b+4 46

+ a=80 and b=180

Hence,

b-c=100, R 3x+10 5anq D 5x+10 7

x=5R_15D25

aa. -(s9: ") = !1] - "] :+ x = 5(21- x) (77 - x)Alternatively : Go through options.

45. 2x+3x+4x=18o:+ x=20 ... 4x=80

46. (25 x 12x) + (50 x 4x) + (100 x 3x) = 996* = 60000

=) x =75.'. number of coins of 25 paise = 1.2x = 1"2 x7S = 900Altemadvdy : Go through options, choices (a), (b) and (c)are eliminated since neither of 200,225,275 is divisible by 12.Hence choice (d) is correct.

47. Go through options.

Alternativety : (x-; a) - (x

- b) - (x - c) - k11 9s

=) a.= x _],!kb= x -9kc=x-5k

... x _ (x - 11k)+ (x - 9k)+ (x _ 5k)2

2x=3x-25k =) x=2Ska=74k =) b='J"6k and c=20k

q.: b: c ='14:'J.6: 2O =7 : 8 : 10.

48. If the weight of kerosene be k kglunit volume, thenweight of petrol =7k/witvolumeweight of casffol = lSkzunit volumerequired weight of the mixture = 1lklunit volumeBy Alligation

7kr /.18k\----*/(11k)

7k/ \+r7:4

The required ratiois7 :4.49. By Alligation

_5\ /.8\//0

s/ \sThe ratio of price of pigeons soldrespectively are in the ratio of g : 5

.'. Cost price of profitable pigeon = - 5

x 1g2 = Rs. 70.8+5

1113'5'6 + 10:6:5

l-1.19, 1,9, l,! =#, *, !=ro, u, rl1310s66s

... (1)

...(2)

BB. R-3xD5x

Alternatively: Go through options.

39. Let the fraction be I thenv

x+5_11Y+5 G + 11Y-15x=20

Since, we have only one equation in two variables, so wecannot find the solution.

OO. 2x+ 3x+ 5x+ 8x+ 9x =! =ZZ.+

41. The number of days required by a single Kirlosker pump to fillthe tank= 6 x7 = 42days and the number of days required by, a single USFIA pumpto fill the same tanks = 2 x 1g = f e auyr.Now, since efficienry is inversely proportional to the nu;berofdays. Hence,

Efficiency of one K-pump _ 36 _ 6Efficiency;F Sump = a=V

at loss and profit


Again

C_3xR5xC_ (3x)3 _4R (5x + 100) 5

+ x=16.'. Price of Rotomac pen in 2000 was Rs. g0.

51. By Alligation(_4)\

,,.(rs)\^////t8{

7" : \12Now, we get the ratio of no. of rings sold at a loss and profit is7:12

.'. Number of rings sold at profit = P x36l = 22g^19Alternatively : Go through options.

52. Best way is to go through options.consider option (d)

#.rr4=6+2=8hrsHence, (d) is correct.

Alternativeb : a * 250 -x -,-3035+ x=18Omiles

Alternatively : Since the Average Speed

_ Total Distance 250totalrime =-mtles'/hr

.'.ByAlligation ^.3o-.r ..35

'250..t8".3/' : \1

Therefore, the ratio of time taken @ 30m/hr and @ 35 m,zhris in the ratio of 3 : 1. It means he has travelled @ 30 m/hr for6 hr.Therefore, the distance travelled by car is 1g0.('.'Distance : Speed xTime).22 77--: -' = 1 :1 61.22 17

1.1.I 1

3'4's'6

:t'ZAV,

55. Let the present age of Karishma and Babita be x and y then,x-10 1

y-rO=S

Again x+14-5y+14 9

By solving (1) and (2) we get x = 26 and,.y = 58

56. Let the fraction be 35k

then, k+x (k

57.

5/( + 5x

xccv

...(1)

...(2)

:)

X

k

(k+x)5(k + x)

and x

> Y=L

1

5

1q,

vz2

20 15 72 10-) :-.-60 76 60 60

xn+z'

6=kr75^ + k=2(5)'

Again *=2r24^ + x=3(4)"

xoc (y, * rt) = x=kbt2 + z2)

15=kG2 +22) = k=3again 39=3x(22+221+ 13=(4+22) => z=3

W rIIANow, 48=K x1.2x20 = K=2Again W =2x1.5x30:. W =90

(a+ b): (b + c):(c + a)= 5x : 6x : 9xand

or

A

3x

(3x + x)

=4x

(4x + x)

a+ b + c=70(a+ b)+ (b + c) + (c + a)= 20x

a+b+c=10xa+b+c=10

c=(o+ b+ c)-(a+b)c=10-5=5

BC4x 5r

2x (5x + x) (B gives 1- to A urrd I to C)44

6x

(5x) (c gives f-,o

al

s9.

60.

('.' x = 1)

Largest share =

(by taking LCM)

x77I

2x

2x

2X 5X.').tr

,J

5x

5

(20+15+12+10)

4 x77I = 60.57

:,,'r;'. €OQ Ratio, Proportion and Variation

@D:S7:3

1. 6)\_7D:S5:4

Cost price :

1

9or2:

Selling price1

6

J

DS2:1

proportion of Dew in tW, = L'10proportion of Dew in tW

" = !'9

proportion of new mixture = | treCuired)

By alligation rule/5ro\ 7 s\?,./

,r3 a(?_t\ // \rz _l)l.s e)' '[ro

3)

-1 1or930

or10:32. The ratio of quantities = 140 : 60 =7 :3

.'. The quantity to be exchanged

= 42litreIrrfr,Fff ln this question individual prices are nol required. you can solve

it by forming equalion.

Alternatively : Go through oprions by assuming someprices.

3. Let the number be 100x + 10y + z, then

x+z=6 and !+z=SAlso, from the given options only option (b) is suitable

1.e.,

or

CP:SP=2:3since profit = Sp _ Cp

.'. profit=3_2=1

.'. profit:Cp=1:26. Flow Chart: P:K

7x , t"-1

Water pulp

4xx3y 2y

but x = 2y since pulp remains contantWater Pulp

-.. ,---. lsy 2y5Y = loss ln water -J ILev 2y

but 10y.(= 3y 1 2y) = 1.5kg

5Y (= 3Y + 2y) = o]skg

+l rc.r"r"r.-l by addition

F,*il* ^ ' *l-I st.p i}_ 3* , " *--.J

bY rePlacement

By the replacement formula from alligation chapter1_ s r,r_?a)4 12\ t2x)3 (- 240\5 \ t2x)

= ?=2o5x

= x=50Therefore, half of the initial amount = (350 + 150) litresthen, the required amount of kerosene

= 5x _ 3x = 2x = 100 lirres

But for the whole amount required kerosene to be added= 200 litres

7. Data insufficient. Since we don't know that how manypersons bought ticket of all the categories individually1.4., 5x + 9y +'1.42 = 273925

but we don't know anything about x, y atd, zPetrol Kerosene Total Mixture99 x 99+x99 (x - 198) (x - 99)

Again -2 xloo- 99 xloo=13.33- (x -99) (x + 99)

or eesg(*.2y=:2'l = rr.r,I x" _99. )

or eegO (1eg) _ 40xt -992 3

= x' *gg2 =992 x75

Therefore the actual concentration of petrol99

= " = 2lo/i

(99 + 396)

1

30

J

2?

y+z=53+2=5

4.


9. Do it yourself.

1O. Varsha : Vinay= 5: 6= 5x : 6x

Veera : Vikram =7 : g =7 y : gy

But their ages are in A.P.

Therefore, 6x - 5x =8y -7y

Again,

AfterfirstoperationAftersecondoperation

Therefore, the ratio of water in A and B is 1 : 4

n=!e6

and c =o =2e10

arso "

=3tand E - A=3

From (i) and (iii) +=t * E =:A

i*,ffi,$E - A = 4 - e= 3from (iv) and (v)

4

+ A=]2 and E=15 and B=10AlsoC = D = 9 andF = 11, sinceB <F < AandF is integer

8x+2 3

= x=2Therefore, the ages ofVarsha, Vinay, Veera and Vikram are1.0,12, 14 and 16 years respectively.Therefore, the ratio of ages ofVinay and Veera = 6 : 7

11. Bobby sunnyMilk Water1:1

5Oo/o 5oo/o

By replacement method

2 2(- k)_-_tt- I

s-4[' 4)4 (- k\

= -=l I --ls\ 4)1k54,-4

- K=-

5

It means Sunny willA

replace -l litre of initial-5mixture by the samequantity of pure milk.

Hence, the percentage of milk added by Bobby to that ofreplaced by Sunny = + . I00 =725o/o" 4/5

AMilk Water Milk2510020t 0 ---------> 5t

1:5

A:F =72:17The ratio of fees collected from B.Tech : MBA

= 4x x2Sy : Sx x16y

= 100ry : 80ry

=Sry:4xy=Sk:+kThe amount collected only from MBA students

4= - x 1.62lakh

= Rs. 72,000

c nW)2 :

Wr:Wr:Wt =3; 4t 5.es1 = (3x)2 + (4x)2 + (5x)2 = S0 (x)2

Again }1( :W, :W, = 4 : 4 : 4 (when weights are equal)

.ss1 =14x)2 +(4x)2 +(4x)2=4gx2loss=50x2 -4gx2=2x2

1800 = 2x2

x=30.'. Actual cost of unbroken marble = (4x + 4x + 4x)2

= (12Jo2 =744x2

=744 x x2

= 744 x 900 = 129600

Solutions for 16 and 17 :ABC

Initially 69 45 42A-+ B 24 90 42B-+C 24 48 84C-+D 24 48 48D-+A 48 48 48

Solve it in reverse order (i.e., from the result side)

Ratio of price paid by Hari and Murli = 7 : 10

or (24x x):30 x (30 - x)= Z:10

or go through options.

x ="1.4

Since distance is constant. Therefore ratio of speeds ofscooter, car and train = 7:4:76and therefore, ratio of time taken = 76 4:'J.

Therefore, required ratio = + t it +=16:1'a=zs6rto,t

1.6

Since there are 'J.2 bangles, then the no. of broken tounbroken bangles can not be 2:3, since 5x =(2x + 3x) cannot divide 12 for any integral value of x i.e., all the sum ofratios which are the factors of 12 can possibly be the ratio iifbroken to unbroken.

t4.x=!5x+2 2

8y+2 3

5x+2_2

15.

Milk Water1:1

7 5Oo/o 500/o

l@ @I

IL 6o0/o 40o/o

@:@It means bobby will addup 1 litre of milk, in4 litre of initial mixture,to prepare 5 litresmixture in the ratio of3:2

D36363672

48

12. B

Water2sI251

201

18.

19.4Isl211

13. ...(i)

...(ii)

. . . (iii)

...(1v)

...(v)

20.

#ffiffi t2t.

25.

( anp + cnq+ e,r); _ 12n1,zn

[b"p * cnq+ f"r) \'! /

. (anp+cnq*r'r\"' , q c e

[bnp*d'q+f"r) " b d f

Hence (c) is the possible answer.

22. Maximum earning will be only when he will won on themaximum yielding table.

A-+L0:1B -+ 20:7C + 30:1

i.e., he won on B and C but lost on A

20 x20O + 30 x 200 - 1 x 200 = 9800minimum earning will be when he yion on table A and B andlose on that table 3.

10x200+20x200-1x2006000-200=5800

Therefore, difference = 9g00 - 5900 = 4000Alternatively the difference

=[(30 + 20 -7) - (10 + 20 _ 1)] x 200

= 20 x2OO= Rs. 4000.m+2p +m+4m-2p m-2qmm-

= 2p+ irrt comPonendo and dividendo)

= ^( e:t)

\zps )

=(lO)t+*l substituting the value of m\p+ q) I zpq )

Volume of a cube : (side)3 : (o)3surface area of a cube = 6 (c)2 (a -+ side)

Q:a|=s127

:> al:a|=419

4pqp+q

20 = 24 -kJ4

k=2when train will stop its speed becomes zero

0=24_Ni

x = Rs. 16.5 per kg

+ w =744since at 144 wagons train will stop, so at 143 wagons trainjust can move with its least possible speed having maximumpossible wagons.

Cost of 1 kg (mixture) of sugar = Rs. 16lkgsince selling price is tu. 20lkgTherefore, ratio of quantity of sugar costing Rs. 14 and Rs. 22perkg=3'1

,""16

,6'/ \23:1

So, the available stock of mixture costing fu. 16 = 3 quintaland the required stock ofsugar costing Rs. 1g : 1 quintalTherefore, the new price of mixture costing 3 quintal of sugar@ Rs. 16lkg and 1 quintal ofsugar @ Rs. 18 per kg

16. -18\*'t--a"\

3t:-1

ace-_=_=_=kbdf

on =rn =rn _b,

bn dn f"

= a'p

= rnq

= enr _ 7r,

b"p dnq f"r... {p + c'q + e'r _ ,rn, 27.

b"p+ dnq+ f'r \'' /


.'. A:B:C=3:4:5Thefefore, number of assistant trainee (except project incharge) ofQ = 3ntr6 assistant trainee ofR = 4

Therefore, required percentage = T . tOO

=! ,1gg = 2507o4

b=3ec=3b=9ad=k=9b=27a.e = 3d = 9c = 27b= 81aand f = 3e = Z+Uput the values and simplify.

or weknowthat!=c =e - a+c+ e

b d f"' b+d+fLet w be the number of wagons and s be the speed of enginewithout wagon = 4 ^ts = 24km/hr."3then speed of the train = s - kJi

f.'f,*zt :20Wl,/hrl

28.

29.

(': k=2)

23.

New price = Rs. 16.5 per kgNow, original profit = 20 - 76 = Rs. 4 per kgNew profit = 20 - 16.5: tu. 3.5 per kg

.'. Percentage reduction in profit = 1-$ x 100 = 12.520

30.

Clearly it can not be less than 7\o/o. For clarificationconcept consider some values and then verify it.

A: B:C = 3x : 4x: 5xC-A=5x-3x<3

C-A=2x<3 = x=1C-A=2

but

orb - c (kerosene)

c (petrol)

26.

of


Since there is no change in concentrationthird... etc operations.In the vessel P;

The fraction of kerosene in p is 9a

In the vessel Q;

The fraction of kerosene in q i, [b , t)\ b/

ih the second,

(b-c)l cI t_

l. b j-;c(a+b)=ab

ab"- a+b'

Since at this moment the concenuation in both vesseli is same.

31. The minimum number of passengers 4 at which there is noloss and number of passengers travelling = m and let thedistance travelled is d, then

P e(m-n)dor P=k(m-n)d;k is a constant.

whenP = 3600, m = 29and d = 36 then

3600=k(29-n)x36Again, when p = 630Q m = 36, d = 42llgrt

6300=k(36-n)x42DMding equation (2) bV (1)

6300 k (36-n)x423600

=i (2g -n)"36(36-n) 9(29-n) 6

= 3n=45 + n=15Hence to avoid loss, minimum number of 15 passengers.arerequired.

32. Frequenryof step of A:B:C =S:6:7but it terms of size of step , 64 = 7 B = 8C

.'. Ratio of speeds of A, B and C = Z, ,9 ti

=280:288:294='140:1.44:L47

33. Expenditure = 5 (no. of familymembers)2

ffiConsider option (b)

1 x 100 + 7 x 50 + 2x25= 500 paise, can_be truebut we also check the option (c) and (d) now when you checkthe choices (c) and (d) you will find wrong as choice (a).So, only choice be could be the best answ€r.

35. Bycomponendo and dividendo:ab

b+c= c+aa+b+c_b+c+aa-b-c b-c.-a

= a-b-c=b-c-a

similarily

a+b+c 3

36. Density of Pr, P, and p, are 18, 14 and '1.0 gm/cc

again since volume - Yieightdensity

now the weight of P, in 450 kg mixture - +s9: a = 120 ke

I.5

now the volume of & = 120

= lZlit u-10.'.The cost of L2 litre p, petrol =72x4O = Rs. 4g0

37. 26.65 + 42.75 * 53 = 122.40Proportionate amount of Amar, Akbar and Anthony in theratio of 3: 4 : 5is Rs. 30.60, Rs.40.g0 and Rs. Sl_respectively.Now Amar Akbar Anthony

30.60 40.80 51.00

53.0026.6s 42.75So, Amar pays Rs. 1.95 to Akbar and Rs. 2 to Anthony.

38. The best way is to go through optionsAlternatively; Solve by cross product rule andcomponendo.

39. Since Pooja and Shipra are twins so their ages be same. Lettheir ages be x and age of Monika be y, then,

x+x=! ...(1)

and (x - 3) =?(v-3) 7

+ 7x -2y =!5Now, from equation (1.),

7x-4x=15 = x=5So the age of Shipra 3 years hence will be 5 + 3 = g years.

4c,. H-9 =luraH+6 8W-9 4 W+6 7

.'. Thus the present age of Husband is 34 and present age ofhis wife is 29 years.

Now,._the maximum age of any child must be less than 9 years.Hende their ages can be 2,3 and 4 years or 4, 6 and g years.So the max. possible sum of age of this family

=34 + 29+ (1 x{ + 2'x6+ 3 x g)

= 103years

Hence (c)

. .. (1)

...(2)

...(1)

...(2)

a=ba=b=c

b1

again

but

4 = 5(n)2

E, = \n -t)2Et-Er=95

sln2{ (n - 1)21 = 95

51nz'-(n2 +L-zjl=9sn2-n2_ l+2n=192n=20 + n=10

34. Finally I have Rs. 5

Consider option (a)

2 x 100 + 3 x 50 + 1 x 25= 375paiseIts integer multiple can not give the value equals to 500. Socan not be truel

"'""'e$t41. pnnJr > p=knJi

where P is the price of necklace, n is number of pearls and risthe radius of a pearl.


Now you can see that the rate is being half from 4 to 2 so shecan purchase double number of toffees as she was alreadypurchasing on the ground floor. Again to purchase the samenumber of toffees she had to spend Rs. 2 less than thespending on the ground floor :

Rate Number of toffee,/Re Total Number of toffee4x 32x 6

12

72and if you check other options (c) and (d) theywill nor satisfythe given conditions.


Total no. ofplants No. ofplants DaysPrevious scheme > 1800 = 100 x 1g

1

Again after lrd days, remaining planrs-3

-1809x2 +720=73203

1320= (100+ 20)x11This shows that in the second case 1 day was saved than theplanned no. ofday.


810 900

s4 -E =3=(6-3=3)

15-12=33=3

Hence choice (c) is correct.i.e., if A writes 54 page, then B writes 75 (= 54 + 21)pages perhour.

44.Initial

' amount ofsoda waterfinalamount ofsoda water

A(x+3)

Now

again

150 = k x75xJlk=2

6O0 = 2x 100 xrF

"..F=3 + r=9cm42. Po.{ = P=K{TT

P -+ Price of a book, N -+ Number of pages, T -+ Time period

\=Pzr lL =r llaT7 T2

3NN18 -iT = 6years

45. 5x + 40 =7 = Akbar

6x+4O 8 Birbal

+ x=20.'. The actual number of shares of less salaried person

= 100

.'. The salary of Akbar = 100 x 75 = 7500

'.' (5 x 20 = 100)

agarn

6=1x62x33x26x1

It is not true.Again consider option (b)

72=7 xl2' 2x6

3x4-+ 3x4=124x3

6 x2-+ 6 x2='1,2'1.2xl

44. D *g#J -+ Distance, F -+ Fuel, T -+ Time, W -+ No. of

wagons

D =kJF "rw

L92= kJ2s6 x2o10

k=6

2gg =6 "'[F "2s15

= JF=zo + F=4oolitre

.'. tuel used per km = * = zUX^200

45. Just go through option and factorize the product into rwofactors such that the given conditions must satisfy.

B(x+1)

CD(x) o

(?r') (?.') (+.i (+.iTherefore to be the integral value x must be 4m; m = 1, Z 3 ...Soatm=1Hence option (a) is correct since B had contributed 1 tumbler.

ABCDInitialamount 7 S 4 0Finalamount J J J 4

3,--'1.-'o2 'l_

49. Ratio of fruits (by dozen) = 3 : 2: 7Ratio of fruits by weight =720 :150: 24.'. Ratio of fruits (combined) by weight

=3x120:2x150 7x24= 30:25:'1,4

Ratio, Proportion and Variation ,!:##$itliil:

,lst Alloy,,,: '',?n:d'Atrloy.

lf0a coppe! ,. UOBDef:..,1':,: ...3irt:, b I

Proportionof ironin thealloys

47

+8

14

67+36a

So in the new alloy total iron will be 44 kg and copper will be12kg.... Iron=8+36=44and Copper = (14 + 42) - (8 + 36) = 72

.'. Ratio of copper ro iron = 12: 44 = 3 : 11, hence (d)

2. Suppose there are all the pigeons then total no. ofheads are340 and total no. of legs are 680. Now since380 (= 1060 - 680) legs are extra, it means there will be

:rlo(=3101 .ullitr. As we know a rabbit has 2 extra legs\ 2)

than that of a pigeon who has only two legs.

Therefore, number ofrabbits = 190

and numberof pigeons =34O- 190=150

Alternatively : go through options and consider choice (b)Pigeons Rabbits

ffi&

Heads (340) 1s0 -Z) 190 I 340Legs (1060) 300 +:'- 760 I 1060

Alternatively : P + R = 340

and 2-P + 4R = 1060

(x+7)-15_3(15-x) 4

x=11'. x=11 and (x+7)=18Thus the total value ofboth the

6. 6)\i/CT13

7. Ratio of Wr M7

1:31

Proportion of water -1 5x7t4 5x7

1.

(x + 7) ..r

1.5

o/,\,

77+18=29

. ..(1)

...(2)

Solve these tvvo equations and you will get the answer.Alternatively : It can be solved through alligation rule.

3. By the replacement formulaDecreased amount

/ - , \no.oftimes(n)= original amount [ 1 -

rePJa-cmg amount ]

\ orlglnal amount /Now, since the ratio of petrol and kerosene is 1701 and27 itmeans initially there was (1701 + 27) = 172gunit of keroseneand the decreased amount of kerosene is 27 unit.

/ .\3.'. 27=77281r-91 = k=8litre\. k/

IEH Here 27 r 1728 iu$t show the iatio nbt the exa6t amount. : r

4. The ratio of milk in 3 vessels

3 5x7 2 4x7 4 4x5=4" s"7t s" qiti'-4rs:9, i9,i9r40 1.40 740

Remember, The value of 25 litre does not matLr, the basic

thing is that the amount of mixture in all the three quantitiesis same.

Sothetotalquantityof milkinmixture= 105+ 56 + 80= 241

So the total amount of water in mixture

35 56 40740 r40 1.40

Now since all these three mixtures are mixed in the ratio of2;3: 5

352lnerelore new ratlo14O 2 \40 x3

70 168 200= 280' 4i0' 7oo

Now, the amount of water = Z0 + 168 + 200 = 438

.'. The amount of milk = (280 + 42O + 700) - 438 = 962

.'. Ratio of milk to water = 962: 438

11:-77 4411:-744:7

2 4xS- x-7 4x5

w2 M2

2:322s7

2 4x75 4x7

56x3

ws M3

2:5

40x5140 ,5

Therefore ratio of water to milk in the new mixture

= 1,79 : 24I15. '.' Profit = 33- Vo, it means cost price : Rs. 153

Now, by alligation

prices =

6)CT25

^7^2Lopper+ - uopper-+-

.3Requ[ed copper -+ = -So, the required ratio is 4 i 7Since it is clear from the above values(7+2-+3and4+7-+11)Alternatively : By Alligation

12o tt

-t t'

" 11':

(z -s\/ \rs 1)t.7 rr) [-r-4.J

= t(3 x 140) - 24|l=779l1ire

xH#ffi##8. Br:Br:83=3xi4xisx

{gain Br: 82: B, = 5y : 4y :3ySince there is increase in no. of oranges in first two basketonly, it means the no. of oranges remains constant in the thirdbasket

Hence

and Sy : 4y: 3y = iSy : 20y :1.5y

Therefore, increase in first basket = 16

and increase in second basket = g

the required ratio = 2i I9. Amount of alcohol in first vessel = O.2S x 2= O.Slitre

amount of alcohol in second vessel = 0.4 x 6 = 2.4litreTotal amount of alcohol out of 10 litres of mixture is0.5+ 2.4 = 2.9litre

Hence, the concetration of the mixture is 290/ol= ?€ ,. too)\10 )

1O. Assume the weight of Alloy A is 100 kg.'. The weight of AlloyB is 400 kg.'. Gold Silver Copper

A 40kg 60kg OkgB 140 kg 160 kg 100 kg

.'. total -+ 180 kg 22O kg 100 kg

.'. Ratio of Gold and Silver in new alloy = 4 , ?99- 500 500

= 360/o: 440/o

DiaNPK

20o/o 70o/o 7Oo/o

Mixture YNPKI

260/o 680/o 60/o IThis 6% of K is obtained only from Dia.

Ratio, Proportion and Variation.'.By alligation #.. ,.#\g/

,r/1t\,.'. amounr *t = : x 42=7 kg

and amountof y=! x42=35kg

First AlloyCAI72

Second AlloyCAI31

.'. Required alloyCAI2t

.'. Copper in first allov = 1'3copper in second alov = !'4

copper in required alloy (mixturel = ?3

Now, by alligation13t \, //4

'.3.r(E- z\/ \rz 1)14 .3) [5

-5.]1 '1n'a

---)l:4Therefore, second alloy be mixed 4 times the first alloy.

14. Note in this type of question individual prices does notmater. To prove this solve it algebrically.

Exchanged amount - 3 x 150 + 5 x 9o2(3+ 5)

= # = 56.25litre

Here 3 and 5 are obtained from the ratio of amounts i.e., from90 and 150.

15. Here the ratio of mixtures (i.e., milk, water) does not matter.But the important point is that whether the total amount(either pure or mixture) being transferred is equal or not.Since the total amount (i.e., 5 cups) being transferred fromeach one to another, hence A = B.

5x=3y3x: 4x: 5x

9v 72v 15v

;,T: ----r- = ey:72y:t'y 13.

11. ureaNPKxy0

ffiffill. NPK

xy0

ffiNPK

120 420 60

r[ffi#]NPK

260 680 60

Ny+No=N, + Nr+120=260N -+ Nitrogen, p _+ phosphorus

:@: Pu+pr=pM = nr+420=6g0U, D, M -+ Urea, Dia and mixture.'. Amount of Nitrogen in Urea = 140and amount of phosphorus in Dia = 260.'.Ratio of N : p =7 :13 > 35: 65

t2.xtC N C N27 54

12510Copper-+:= 'Coooer-+

918918

(Flour) 3x..t ,.7x(Sugar)\n/,/ \,

(9-3x)=! = v=)Ax-9) s

.'. Price of sugar =7 x = Rs. 14 per kg.

JJJ

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4 Ratio, Proportion, And Variation