Qa by arun sharma

apptitude

1. The McGraw-Hill Companies

TATA MCGRAW-HILUS (we) SERIES

EDITION

Quantitative
d DULLES te

ARUN —

2. |

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7 West Patel age, New Dei 10 008

Sth reprint 2008
ERAS

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“Tis don cas be expe fm Ii ey by the pubes,
Toa Meal Pais Company Lie.

ISBN: STGO
ASE 1: 007081993.

General Manager abi: UAL an School VB Kamer
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ter Ta McGee Ml nar its ho pasamos the acuy or compltenss fay ferme peed hei. and
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Type Te Compose, W2 391 Mad New Deli 11062, and pd opus Papers Lin À & 3, Sei 6.

3. CONTENTS

Prato to SeconsEten
Pilato o Frs ton

“xs

Er]
To Check hate a Nunberis Phnaer Net 1.

A Blok Why This ees? 16

‘Th Concepto! GOO (Gets Cenar Dior or Hrs! Common Fact) _18
a
Tea on Coran m

Rule rFndingie LM 2 nembarsn and 18
rra

HOF andLcM 19

Pacs Ersse 19

Dióeity_24

Thearonsel Duby 22

HumterolZevss nan Exprcden_24
FindngtneNamberelZersesineFactralVahe_ 24

4. xiv | omen

anne 31
Menu Pres 22
CET

Lave iti L00)1_ 37
Lane tity LOO) 42
Lori Os LO) 48

Ansmerkay 55
2 pangaessiqus

Amel Presses _58

na Sun haha Nate Tes an Arret rogeen_58

ToFnanaAnnmoschsantetusenan, Twa Given Ouais _ Se
also Gen ambre Menos etre Ta Ge Qui 58
GeometiePngrussen 61

hdi anti ea etage Teo iv Queres 81

Taler. Gen Mandar Gear Mens hetynen Tino Ga Dustin. 62
bide Sunol a NenberolTemsina Geonete Progression _ 62
Hamon Pagreesen 83

1 int Hama nanbetwans Ti Gian Cuties 69

Toten tad Progr 65

Ventas et

Wits Patios

Leiden non) 6

\eruisibetem one 10

Lave Lenin 79

Answerer 50

Bock réa Test 2

tol ey LOO)! 99
Level ip LOD)_104

107
Aranerkey_ 15

5. T5

4 auumanans. M
neuen 1
Theo ne
The Alien Staton 116
taco RapacrtatenctAsgaon 119

‘Tha Saa Approach 120
Soma Typca Saca nor Algas CanteUssd 121

Soma Koy toSpet A, A.andAwand rent ose om and, 122
ArpcaPraiem 122

Pa
e ii
BLOCK 3
aia
ee
ne u
mm
Uy 40
propicie RN
us
ae nsnnpann 1
aa mn

EtectotaPercetChargointhe Numero ona Rat lve 143
Parcartage Change Gran and pplcaons 149

ra lr Data proton 1
FraclonoPrcartage Canvas Tale 147
SomoUistonsclte Tale 147

(Cetin gente by ben 1.21,08andencn 148
Works Probes

Levelot Day LOO)! 150

LevelotOticty LOO}! 154

Lolo! Oty (OO) 189

CT

-PRORT AND LOSS.

a

1

aa

cken 124
Thon 12a

rota Less incas neal Transactions 174

Bron ed ie y

6. i Joss
alain one Bass ua o Ame Spat andthe Amount Esmas
CarseotofMaisUo 177
Moscas tems 177
Lovet fuer")
ovale itty LOO) 1 185
Let Di 190
Anar 199
7 AMTEREST. am
motion 202.
Tine Vas lens 202
Poraiinglolnoreis 202
Simpelnerest 200
lc Among Pca, Tee, Rate Percent nest Por Armand Titan 203
Gempoundinterost 203
Fomus 200
Depraciatoncivaho 204
Popdaton 204
‘AppleabcnscniaressinD |, 204
WotesoutFrsiens 207
Levloloereuty soon 210
ve of Deut LODU 213
Arsmarkey 217

1 RAI PROPORTION AND VARIATION a
lation 210
Baño 218
SomatngotaniPegerasofisios 219
Mineral Us ofFaton 221
Proportion 222
Wien 223
WorlesouPrblens 224
Lolo Day LOL _ 225
Lier Date (LOO 229
yee fey (LOO) 280
Angmerkoy 209

7. opt ol Padus Cartney Tablet Ti and Work. 244
Work Eivalnca Method (Te See Tera ad Werk Probs] 245

ee Se
‘The Concept Een 208
Se
ment
CET
inetd} 2
le D
A2, Peu SINE E) AMD APLCAIN OF Tepe oran m
pr
‘Theory of TSD_ 285
re
TT
‘Applications of Time Speed and Distance 270
mn
= a
Pre
rr
one
Brock 4 ”
ere
en
1 GEOMETTANDACASUnAON an
ees
gr
Be
Polragas 374
re
Equitateraltrangles (o sige al 317
scscales Triangle 317

pt Anglo Trang _ 317.
Importar Term vi Rospettoa Trang 318
Ce

Tipes Ouen 319

Regular Hexagon 321

Giles 321

“nous 929

8. AVI] ome

Elose_329
Sa 2

Modan Pts 26
Howto Tian Gocmetyandensuraten 228
Spada Nota Aout CAT Questons 33
Geometry: Laval DISC (LOD)! 233
Nensuaion: Level Doy (00)! 395
Geometry: Levelt Dt (LOO) 339
eration Level ofDPFeuty LODI 341
ExtaPracion Em 246

Ancuerkay 259

Bock aren Tes 356

BLock 5 w

BacktoSchos 385
PrassesamanTest 374

12. FUNCTIONS om

Basset RapesntngFunetons 377
Material 377

Ta oproseraon ol Furctons 377

Graphical Rareseraboncl Futons 378

Even andOsstncions 378

ven Functions 378

OssFunctons 378

Into of aunction 378

Sirga 300

Conaringloverens 20

reales 381
‘TheLogistGaphicalPrcessforSuvnghequises 81
gets ser

Graphical Vew cl Logic Funcion S31
Vias Plana 384

Aamerkey_ 205

Lolo Duty (.00)1_ 384
nett Dey Loojı 287

stot Dey (OO) 200

‘INEQUALITIES. se

Precisa llegues 356
Prpertes clogs 369
Cotannpota recae 559

9. Coros exh

18, COONDMATEOEOMETAY m
Casa Coca Sen sp
Roctanquiar Coordinate Anes 522
AniopecartRacst 515
Levee Ofiaxy LOO}! 515
Levelt (on 517
Levelt De (LOO) 518
Ansneckey 521

10. BLOCK 1

A ce R
CHAPTERS

+ Number Systems.
* Progressions

INTRODUCTION >)

ck One contes the mos important por fe entire portion forthe preparation

of QA fr all compertie examinations. I he conte of the CAT, lock One has
always had a weishage of benveen 12-15 mark: Seen fr he perspective ofthe act
‘that the etfs require 1 clear the QA section ofthe CAT paper have as hen in
‘the range of 12-15 marks ou of O marks, the importonc of his tection con be clearly
angel.

is due 10 is reason that his Mock of chapters assume a high level of iporance
ln your preparation forthe CAT.

Hence. you are advised 1 50 draus tis block in grea dete und ry 1 create
complete concept clarity as well ax question exposure for developing your ability 1 do
el a this area.

eo first mat you 1 go through he sample pre aessmentac based extaively
‘on quesos taken from this lock ef chapters in onder to gauge your starting lve.
Note: There imo time lini tat you nee 10 follow while you are solving thi tet The
objective f hs et 1 gage yon starting knowledge level. Hence ry 1 think troupe
‘cach quon for as long as you wantnee to, Try 0 she the question ti tet o
he best of your abi

A the end ofthe tet we have proved a Score merpretaton too. Use the sore
“nerpretaton tool order o define your objectives whe Sou sus ad po though the
individual chapters ofthis block. =

11. 2] Hort Pepe arta Ai rt CAT

BACK TO SCHOOL

+ Chapter in is lock: Number Systems and Progrestoms
+ Block impertance—12-1S marks out of 0.
As alreay emphasised inthe preface. is book is oe ofthe most crucial aes, when you come to tink of

(CAT preparations Im fact, sev inthe content of the fact thatthe sectinal quan score in QA is jas ard
12-15 mart ot of 5, e ral imporace comes ot.

ln cooking atthe CAT papers over the lst few year alone wil give you a fair den ofthe importance of his
‘ck in the content ofthe CAT.

“The following table wll make things clearer for you:

Fer Dame af marks From MAT Fetal quan wre
Caras Temas B=
Cara, maso EY)

As ose se fom te tbl abo th Hoek loe can get yu othe eve ua sor rete

doler al impor QA seco fre CAT

| eme, uadertanding the concepts involved in ese capis ppl and
solving apre mip goa long vay tart pag yu ls e CAT.
Before we mone nto the ivi haps fin bh, fi organi or

thinking by ooking the or concept that we had rat out dung er sche,

12. REASSESSMENT TEST

‘Tis test consists of 25 questions based am the ehapers of
BLOCK ONE (Nurnher Systems and Progression) Do yout
‘best in eying o solve each question.

“The ime Lint to be followed for this tests 30 mite
However, fer the 30 mins is over continue solving ll
you bave spent enough ine und pad sufficient ion to
‘exch question. Ale you fish thinking about each and
‘every question of the es, check your scores. Then 0
Aro tie SCORE INTERPRETATION ALGORITHM given
at the end of the test to understand the way in which you
need to approch the chapter imide this block.

1. The number of integers satisfying m +2 2 00d

Mai
wo wt
@2 CE
2 The som of two integers is 10 and the sum of tir
reciprocalsS/12. Then the large ose integers is
m7 ©.
os os
2. Misa pose integer such ha 2 + 12 perfectly
visible by x, thea the mumber of posible values ol
CE LE
@ 6 on

A Let bo positive imeper such hat is visible

by 7. Then the sales poste iger, greater dam

2 sch tha + 2 à
ws
CE

5 D 227 A ie he same as
om CE
on CES

6 Tere nes de fit of re conserve old integers

{3 more then twice be hi. Wat ithe third
ow?

ws CE

on ws

17. x yar ze thw postive liners sah that >> 2

‘Which ofthe folowing is closest the product a?
@ o @ a6-De
CECI @ a6+ De

AA postiv inegeris sido be a prime number itis

ro vise by any postive integer ter thn self

nd 1. Le p be à rime number greater dan 5, den

Can

0

2

Der
(a) never disse 6.

(©) aways divisible by 6 and may or may nat be
divisible by 12.
(6) alas divisible by 12 and mayor may not be
divisible by 2,
(62 aways divisible by 24.
Tahal dal some cars 0 Mashing and himself rm
à fll pack of playing cards and lid the rest si
‘abel en sad to Mashing “Ifyou give me a cra
marier of our cards wil ave four mesas many
‘ond a you wil ave I give ya sam number.
“of cada, ill have tre as many card as pou will,
have" Of the given chics, which could represent
the number of car with Lea?
on DE]
on ws
In ia, each bay’ quota of match ks oil into
bones snot more han 200 per session Ie reduce
the umber of ticks per box by 25, heal 3 more
‘ores with he ol umber fsck assigned ohm.
Which ofthe following isthe possible mumber of
sticks assigned tw each boy?
a 200 ® 150
ons CR
Alor! got an oder from a garment manufacturer for
480 Denim Shins. He bought 12 sewing machines and
point some exper or to de he job However,
many ida’ eport fo ty. As ares each ofi
‘wo di had 10 sch 32 mor sins than originally
plano by Al, wih equa isrbuion of werk
How many lors hd ben appointed exe and how
any had not reported fr work?
wa @ 103
ons (6) None of bese
ow many 3 digh eve numbers can you form such
that fone of the ig iS, he Following digit must
ber?
ws DE.
CET os
"decide whether number of gts divisible by
7. e ea define a process by which is magne is
reac follo: care Be digs he
number, sting from the mos sigifcat digi.
igo a te te
eg 29 > see
Ulémately the resalta number wil be seven afer
‘repeating the above process acerlo number of mes.

13. E] rower Ouais Ata CAY

Alte bow many such stages, des dhe number 203
reduce 07?
(2 CE
©. y)
1M, A third standard teacher gave a simple mulóplcadon
exercise oe bith, at one Kid reversed the digits ol
both the numbers and carried ost the multilicadon
a ound ha the prodac was exact the same 28
the one expected by the teacher. Only coe of tbe
following pis of martes wil in he deck of
the exercise. Which one is hae?

@ 42 Re
@ 19.3 CET

15, AFS 12 22,74 = en 10+ 1
@ 0 es
@6 on

16. Find the minimum itera vale of m sch à

vision S80/2$ leves no romaine

wm @ 13
oa we

17, Was the value of for which de following system
of equations as mo sol
De By Band hrs dy 10,
CE on
CE CE
18. À postive integer is sad o be a prime i is mot
‘vile by any positive Ineger tr han self and
one Letp be a rime number si greater than 3.
‘Then, when 17 divided by 12. the remainders
@s OT
oo as
19. Aman sel chocolates that come in bores Eier fll
bates or bala box of chocolates cn be bou fom
Hm. A caster comes and buys half the uber of
nes the seller has plas half a bat. A second eu.
Homer comes and bays half the remaining number of
tones plus hal bon Ale tithe aller is ef with
o chocolates box. How many chocolates bone did
fhe seller have before the firs cstmer came?
(2 os
oa ws
A. X and Va playing a game. Thee ae eleven SO pice
(ins 0 the able and ech player must pick up at
lest one coin but not more than five. The een
picking up the last coin loss. X sans. How many
shold be pickup a the sut ensure a win ao mater
what strategy Y employs?

a

ws CE

@2 os
Aa < be which ofthe following Is always tue?

(a actor /2<b

© ace

(o acia

(9 acaben
“The money onder commision elle a fll.
From Rs. X tobe sent by money une, surat DD
and dive by 10. Get the quotes and add to if
the esas Ye mony ender commision is RS. OY.
Aa person sends two money andes to Arrangabod
“and Brindo for Rs 71 and Ra 48 respectively, the
tota commission wilde

(9 R70 © R650

© 8.600 @ 8.70
‘The ao fue is Alma as he following formula
bused upon the meter reading. The mete reading is
‘une ap to the nex higher maple of 4. Fo ne
stance, the racer radio 637 pais ls roanded
upto 40 pas, The esla is mul by 12. The
final result is rounded off o peste multiple of 25
pase 153 pase isthe mer reading what willbe the
tol fare?

(a) Rs.695 ©) 650

© Roars @ R750
ai ed Bhagyasvee were playing sine athena
‘al pazaes, Job wrote a o digit umber and asked
‘Bhayesvee to guess i Juhl also indicated tht the
amber is ex ice produc ofits digits. What
swat he number hat Juhi wrote?

(a 36 y

tar wis
is dro exact matin perf ran 24,
herentandn-1}.(n- 2). 321. What vil bee
exponent of 32

ws as
on @ 0
5 @
mo
150
ao
zo

14. E] terrane truest spice tsa

Step Five: Move to LOD 2 and repeat the proces that
ou followed in LOD Ist Inthe chapter of Number
Systems. then with he Chaper on Progressions, Concen-
trae on understanding each and every question and is
dering concept

‘Step Sixt Got the cond review in given athe end
ofthe lok ad solve it Agua, wile dings measure your
score within the provided ine Kimi ist nd then continue
ose he ts further without time limi andy o rate
e improvement tht you have bad in your score

Step Seven: | Move 1 LOD 3 only alter you have solved
and under each of the questions a LOD 1 and LOD 2
Repeat the process tht you followed in LOD 1 ~fistin the
Evene Se eh Ce os

If You Scored: 7-15 (In Unlimited Time)

Although you are bere than the person following the
instructions above, brio} there salt of scope forthe
development of your score Yeu will ne 10 work both or
your concpis as well as seed. ndalyempluslze mer on
the concept development spec of your preparation, thee
move your enpasis onto speed development. The ll
process is recommended for you

‘Step One: Go thoogh he black one Back to School
Section caeflly. Revise each of the concepts explained in
that part Going tough your Ab, 9 and 101 standar
books wil be an optional exercise for you. I will be
recommended incase you sored a single digi, whe i
your score is in two digits, leave the choice to you

Step Two: More into he fs chapter ofthe block. Viz
Number Systems. When you do so, comen on clearly
understanding each ofthe concepts explained inthe chapter
theo

"Then move oo e LOD 1 exes, These exercises will
provide you withthe fi level of challenge. Ty o solve
ach und every question povided under LOD 1 of Nbr
Syaers. Once you nth solving LOD revs he quemen
and thee solo process,

Step Three: After finishing LOD 1 of number systems,
‘move ino Chapter 2 0f is block, rogresins) ad repeat

the proces, vz: Chapter theory comprehensively follomed
by solving LOD 1 questions.

‘Step Four: Gotoite fit review test sven atthe ead of
the lock and solve i. While doings, Fist look he sere
you get within e me limit mentioned. Then continue to
ole the test fabor without ae mit andy to evaluate
the improvement in your sore.

‘Step Five: Move to LOD 2 and repeat the procesa
you followed in LOD If in the char on Nomber.
‘Systems, then with the Chapter on Progressions

Step Six: Go wo the second review test ven at the end
of be Bock ar sv. Again wile ding so meature your
score within the provided Sine limi ist ad then cosinor
nalen fuhr witout ime limit ety to erate
the improvement tht you have had in yur score.

Im case the growth your score s not gras, go
mack the theory of otk the chapters and resolve LOD 1
‘and LOD 2 ofboth he chopers While doing so concentrate
‘more on the LOD 2 questions

Step Seven: MovetoLOD 3 ad repeat proces thst
ou followed in LOD Ist in the chapter on Number
Systems, then withthe Chapter on Progreso.

If You Scored 15+ (In Unlimited Time)

(Obviosh you ate much hier han the fs two categories
of stdeots Hence unike them. your focus should te on
‘developing your speed by picking up the shore presea
explained inthis book: Besides, you right alo need pick
up concepts that ght be zy in your mind. The fllosing
process of development is recommended fr you:

‘Step One: Quickly review te conrps given inthe block
one Back 10 School Section. Only go deeper nto a concept
{in ete yon find it new. Going back to choo evel books is
ue require fer you

Step Two: More imo the frst chapter of tbe Block:
Number Systems. Go hr the theory explained there
castle Concert specifically on early understanding
ie concep which ae ew to you. Work ut the dor cus
adi fact uy expand you thinking by yng wo think of
stematve (and expands) linet of questioning with respect
1 the concept you sre saying

15. E] Heroes tr ues ap ter CNT

‘CORE CONCEPTS,

|. The concept ofthe number in ne e he mos crc
‘concepts Quantitative Apte.

"The number line line ls start from zero and goes
towards positive infinity when it moves tothe right and
cowards megane ini whem it moves to the If

AS

“The difference beten the values of any to pais om
lhe number line aso gives the distance between the país.
“Tus, for example if we lok athe distance Between the
points + 3 and ~ 2 it willbe given by thelr différence.
3e Des+2es
Types of numbers ~
‘We wil be looking atthe types of numbers in dei, again
nen we 0 inthe chape of uber ste. Let us fit
‘woth oat in ou nnd the arcas pes of martes. While
oir u do ot il tne thal most of cr ber types
ocean pars Ge. deiniin lose of then defines the
ber atoms

Natural Numbers and Whole Numbers: Natural
number, socalled cooing umbers are he ist aber
we lear They ae the number et 1, 2,8 4 une.

(On the ober hand, whole sures ste set of natural
umber pas. the number cr.

“Tas, 0,1 2, 3a in the wet of whole aber

Integers and Decimals: All unbersthl do nave
“decia inthe ae calle inepen Ths 3-17. 44.413,
“+1473, 0 tae al meers

vii, decimal numbers are number whieh ave a
eva valu stacked to them. Thus, 13, 146, - 2 te
ae al decimal numbers, since the have cena ales aer
‘he decimal poi,

‘Before me mel lus pause a brief vil, father
understand decimals. As you shall sc. the concept ol
decimal le clowly cac othe concept of division and
visti. Suppose, have pieces o bread eich Y want
vo divide equally pen wo people li easy for me todo
this, since can ive to whole pieces to ech of then.

However, we alter the situation in such away. tom
ave S pics of ead eue equally amongst people
‘What do Io?

1 give two whale pieces each, to each of hem. The Sth
picos has to be divided equally berween the two. I can no
longer do is, witht in some way breaking the Sth pise
imo 2 pans. This i the clemesary station at gives rise
toe need for decimals in mathematics

Going back othe sittin above, ny only option isto
vide the Sh lee no two equal pars (which in quant are
called salves).

“This concept has hog implicalons for problem solving
«specially once you recoge that a Ralf (ie. a 5 inthe
ecm) oly comes when you divide a whole in two parts

Tin in ac al standards decimals emerge ot certain
fixed divisors

ence, for example the divise 2, ives se tte decimal

s

im tbe visor gives seo the dial 3333 an

és.

Prime numbers and Composite numbers Arnot
ara numbers, there are tee brad visions

Unity Wisrepresceaive ofthe number

Prime numbers These ses numbers which have no
viser Factors apart (rom 1 and is

Composite numbers On he ate hand, are number,
Vich ave a leas one mare vise apt frm nd ef

[Note À brief wer about icon’ division = Amumber X
is cito vie Y ore said he a divicor factor o Y)
Whe the division of VX lanos no remite

All conposte abs have Ue property at Uy cante
writen a a product of her prime faces.

"Thus fr instance, the number 4 can be represented as
DADAS or PS

ig is called he standard form of the

The difference between Rational and Ireationl numbers:
“Tin ference soe ofthe etal bt unfortnatly one of
the less well andentoed difeences in elementary Mat

‘The definition of rational numbers: Numbers which cante
‘expressed inthe orm pq bere 9 O ae called rations
unter.

‘Obvicaly, oumbecs which cannot be represented in the
form plg are called irtonal numbers.

However, ono ofthe les wel under is in thi
regard i what does this mean?

“Te ference cores clear when the vals of decima
are examined tie

16. nse te following numbers
aa

Den

0) TS …

What isthe ference between ci values ofthe
thee numbers above?

Tout simply, int umber hss aca be dee
safe decimal vale Such robe canbe expremedn
te torm pig ee. Since an eit tens 210 and
ten covered 215.

Sir mam ie 4.5732 ane pc 4573
1000. Ths, members having a ie emiastngéeinal
valo rato

Nov er osier th decimal ae: 433...

Such decima vale wil continue end Le. Boy have
o end Hence. they ae cle lite deals (or ncn-
teminadog desis.

TUL we can xs ae hat the member 4.333. can be
represented at 13. Hence this ome i ak aoe In
fac all bes wich have fit decia) vales, the
any reciting form within he can be rprseted in ie
Platon.

RS
sai

(What mean o sy a es whenever you have any
securing devil moter even the vl of might ot
‘vious, da it wil aways exit)

Than, we ca coche la number whove decimal
values are infinite (nn-teriating) but which Reve a
recuing pte within hem are aer! number

“This lemen us with he third Kind of decimal vat, i
Toflle nowrecuring decimal value. These decals
ete have arecumngputem. nor do they have an end
they go onerdlely, For soc number ase peso

which ca be waist
to eprset them apg Hence och mbes called at
‘naa! numbers

"day today matas, we come ro numbers ke
3.18.3071, ac ubico malonlnmbers sine ey
oot havea pig representa

Note V3 ca ao derepente s 3% usas IT can

be present as 7!)

An important Tip:

SE

appeared tie solving a question, it will remain il the end
‘ofthe solucion. can only te removed fom the ein if
Hs multiplied or divided by the same ational number.

(Consider an example; The ares o am equilateral tangles
ven by the formula (I) >a where ais the side he
quite mange). Since, N3 de an lations member, i
remains inthe answer til he end. Hence, the aes of an
tel ing wil always have Vas patof he ans.

Before we move ahead we need 1 understand oe Final
thing about reci decimals.

As have already mesones, recuring decimal have he
propery of being able ob represented inte ig form. The
urn that arses ini there any process 10 conven a.
recurra decimal int a prope fraction?

Yes there i. I ac, in order o understand how thie
pers, you fst need to understand that there are two.
kinds of rcuring devils. The process for converting an
inf rearing decimal ino a fraction basica varies or
oh ofthese types. Le ook at these one by one
‘Type 1-Pure recurring decimals: These ae recaing
ccs whore the recumenee stats Immedltly ater the
decimal pot

Forexample 05855, =03
32004. «az
same. 536

‘Toe proves or converting these decimal 1 factions
an be lat a:

us -50

3200 #340459)

SO 252 (6200)

Ai io inosptom will you thal wa we he
ove is nothing but topar dows the rearing par ofthe
decimal a it is and dividing I by a group of 9s. Also the
umber of Sin his group pas the amber of dit in the
recuring par of he decimal.

“That. inthe second case, the faction ls derived by
viding 24 y 99. (4 being he ecuming pa of he decimal
rd 99 having 2 ins because he number af gh in 24 is
2)

ases
2

‘Type 2-Impure curring decimals Unite pure recurring
decimal, in hee decimals the recunence oct ar &
ea numberof digs inthe decimal. The process 10
covert these imo faction I alo est used by an

cap

Sim, S787...

17. TO | Ho Pogure uri ance he CAT

(Comer he Social 0.542422
sous
“The fractional vale ofthe same wll be given by: (43542
— 435999000, This canbe understood la two eps.

Sup 1; Sabre the mo-recuria ntl pur ofthe decimal
(in his exe, itis 439) from te number formed by wring
‘down the staring ts ofthe décimal vale upo the digit
‘where the recuring decal are writen forthe fist ie;

Esparding the meaning

Noe: or 0.438.2:242 sober 435 from 4342
Step 2: "The umber thas tin, has to be divided by a
number formed as follows; Write down as many 9 a the
‘numberof its ia the rearing par of the decimal (ins
a, ico the ering pen 2 has igs, we wie down
129%) These tines have w be followed by as many ere
‘tthe number of digit in the non recurra par ofthe
decimal value. (m this case, the nom recurring par of the
decimal value is BS" Since, 485 has 3 dig, nach re
Bees tothe two ines 1 get he mumber to divide the seul
ofthe first step)

ence divide 43542 435 by 99000 to gehe fraction.

ama 61-46,
Sina. fo 3496213213 ne gt SEBS
Let us now move onto our net opi

‘Tables and thelr visualization: Imagine the number ie and
a ro siting point zero te mue ire. Let ws say tha
the frog always jap an equal distance (53 2 ais.)
Imagine he ro Sting at he cio (pon 0) of he
umber line start jumping to gh hoop equis
jumps of exscly 2 units. wil fst and on ee point repre
send by the number 2 on he nube ine. ' en jump will
‘ake it land onthe number ihn 6, then and so om
This how you shoald visualise he tale o any umber.
“Thus, a fog stating fom 0 and Jumping 7 unis o the
righ wal land on 7. be 14,21, 28 and so foi. This os
jump represent he table af the rumber 7.
Tre meaning of and n+ 1: Za means a number which
mile of th amber 2 Since is can be visualised as
fog sain fom te origin and jumping 2 unio eh
inver jump, you can also sa ha this frog represents 24
(Note Muliplso17,aeevencumers Here, 2 also
sed lo denote even number.)
So, what does 2a +1 mean?
‘Wel simply pt if you place he above frog on he point
represented bythe number onthe nsmber ine then the fig,

il reach pies such as 3, 57,9, 1 and so 08, This
escri mean that the pot the frog now reaches are
&spicedby 1 unio the rig of te 2 og. In mabeatied
terms, his is epesenied as 2 +1

Ta other words, Za + 1 also represents numbers which
leave aremainer of 1, when vies by 2 (Note This isso
he definition of an 0 number. Hence, in Mathematics
Qu Dis usd o denote eno number. Also note that
‘akenrogetier Jn and 2n + | denote Ie entire se inter.
Le. al niger from = 1 + = eth number ine can be
denoted by eher 2a or 2a + 1. This happens because when
ve divido any npr by 2, there are only two res posible
‘with respec th remainder obcined vie: À remainder of
ero Qo aremainderofone (2a+ 1,

concept cn be expanded to represent integer

respecto any number. Thus, ners of 3, we can only ave
three types o integer e, Sa +1 0 +2 (depen om
habe th integer leave aremeinde 0, 102 respectively
when divide by 2, Similar, wid respect to, me Rave 4
posiiiies- d, de + Lan 2004 +3,

“fem of epesentaton fgets extremely rai
logically solving QA.
ales Hdicer dices means be power on number. Many
‘maternal states require us to be bl o use dh les
of indices Hence, understanding these rules and thie
appropiate use might go along way towars helping you in
developing yoar sls in Quan

“The following ues apply or indices

QD dd = Ts P92 = 2

ET 20! =

0) de aoe aM = Ha. Tin 3

= Thee, =H

(9 d°= Voc all ales of «Ths, 7? = 1

Besides, the flowing principles apply for indices:

QD Ie men an

CO ai = (ah or ve ven
6 d # de, Tux 2° = 2" and not 2°,

Squares and square roots When any numbers
‘upd by el, iis called as the square of te under
‘This 3232 29
gares ave a very important role play in mama.
the context o preparing for CAT and other Management
exams, I might be a good ides o be able wo reel he
squares of 2 digi numbers.
Les us now go tvough the following Table 1.1 cael

18. Table 1a
Fer Sue Number Square Naber Sore
a ar e. m
2 4 % D 2 40
5 0» 7 wen si
4 6 RB M 2 sm
sos yom BoD
6 % o 160 A s%
7 pt 1 À es
s 6 are M STH
oo 2 Be 7 os
CR a
um s ms D es
po % 26 ® 6w
E HG
MOSS 2 © 64
so» 2 ur
M 26 OS
CEE TES
nom 2 Mo 7%
vo. S209 9 7%
2 m 5 26 #74
a m sw» m
z m % 15 © sw
2» 7 2 m
II]
> o 9 Mi a)
x m om
z 7” aa % os
ame
> m SO OM
AA
a sm» œ
RO 4% 10 100
a 6 #4»
MNS m

So, how docs one gt these numbers ono one's finger
tips? Does ane memariz hese valves ors there a simpler
war

Yes indeed! There a a very convenient process when
comes Lo memriing de squares of the ft 109 numbers.
Fi fa you are expected to memoriss he squares of
{heft 0 narber In my experience, have normal teen
‘hat most students steady know this The problem arses
with number ar 30. You donot acto worry abot tat.

oes TI

ust flo: the following processes and you'll now all
squares uo 100,
Trick 1; For squares from SI to $0 ~ (Note: This method
depends on your memory ofthe fis thirty squares.)

“The process est explained through an example.

Suppove, you have to get an answer forthe vale of 67
Look 67 25 (50+ 17). The git answer vi have wo pts
as follow

wy Co
Dre

The st vo digs willbe he same as de las wo digits
ofthe square ofthe number 17. (I value 17s derived by
looking athe difference of 67 wi respect 1 50)

Since 17° = 289, yu cn say thatthe st digit of 67°
lb 89. (eth last 2 digs of 29.) Also. ou will need
Lo cary over the 2 in he hundreds place of 289 theft
pat of the number

‘Theft (wo digits ofthe answer wi got by adding 17
(bic sg from 67 ~ $0) and adding the carry over (2 in
this cas) othe mum 25. (Standard number to be ated i
al ases) Herce, the ist too digs of Be answer wl be
ven by 25417420 4.

Hence, te answer is 67? 2450

mili. suppose you have 0 find 76

Sep 1:76 50 625

Sup 2:26 676. Hence, the lat 2 digits of he aver wil
te 6 and we villa over.
Sin 3: Te ist (wo digits of th answer willbe 25 +2646
a

Hence, the answers 3776

‘This technique wil ake cr of squares from $110 80 (£
you remember the square fom 1 O). You are assed
the tis process and sce the answer for youre

‘Squares for Numbers from 31 to 50 Such numbers
an be treated the fora (50 ~ x) andthe above process
modified to get the vales of squares fre 31 1 5. Again,
Lo explain we will use an example, Sopose you have 0 nd
he square of 1

Sup I: Look a 41 a (60-9)

‘Agnn, similar to what we did above, realise thatthe
answer his two pars—the fist two and the ast fo ip
Sip 2: The as two gts are go by the Last wo digits in
the vale of 97 m. Mene, 8 vil represen the las o
dvi of 48

19. ‘Stop 1: Wie down the number 70161 podes of ts

Prmefscten. WIG =2X2X2X2X21X21
=P

Step 2: Te rege square ro ls One by halving the

Vales of de powers.

Hene, Vs max ait

CUBES AND CUBE ROOTS

‘When nomberis multiple viii times, we gettbe
cate ofthe number,
Thus eo ed

‘Method to find out the cubes a 2 gt numbers: The answer
aso const of part, eae of which has o be calculated:
separately.
"The fist pat ofthe answer will be given by the cube of
| net's digi

‘Soppose you have t find the cab of 28
“Toe fist sep i 1 find the cube of 2 nd write it down,
“enr part he mb wie derived flows,
Derive the vales 32,128 und $12.
(by cresting 4G. Pal tem withthe fst cr intense
5, and a common rato got by calculating the rio of the
uni git ofthe number with sen dig a his case the
rulo ED 4)
No, re the trs ina rt ine as below, Alo, 1.
the mile o terms ad double the valor

8 mon os
+ “26 E

| TAN

HD) RHEIN) (1284256451 535)
Cm Ca)
Hence, 28° 21952

Properties of Cubes

1. Wen aperecteube is writen in ts standard form the
‘ales ofthe powers on each prime factor wil ea
mile.

2. Inordrto Find the cube roto mame, fs rei
In ts standard form and ten ide al powers by 3.

a

“Tas, the cube oot of 38 5 x 17x 2s
sven by 3817 2!

2 The cube of all number (integen and dia)
rater ihn as greater ibn the number if
40-00 snd ml Tecate

nly three sance where the cae of the numbers
‘eal othe number ine
4. The vata ofthe cubes ofa mamier between and I
ls over an he number sel Tas, LS? <0.S 05.
6 Thee ola number between and Li rater hen
the number ine. (= 0.2)" >=
7. The cube of any nunberles tha =
tha the mime. Thus, CIS < (15)

‘The BODMAS Rule: Icisusd forthe owering of mah
nati operations ina mathematical sation
la any eathematial sitio, the fist thing so be
considered is Bracks followed by Division, Mulan,
‘Aiton and Subtation in that order
DÉPENS
Aa 345-623
A. 306-043

3
EN

Operas on Odd and Even numbers

Dos EVENS
COTE ECTS
Joss sod = Evee| | Even + Even = Even
Joat-edt = Even! | Even even Even
Lost +odd_ ode | [Been even= Evenor ot

‘ops & EVENS
DRE = Even]
OM Even Ou
dE = Oud | O44 Even > Not dvisble
Even sold Even
SERIES OF NUMBERS

In many inmances in Mathematics wear presented with a

series of number formed simply when a group af numbers

is writen together The following ae examples of series:

CERCLE

(RD 3,7, 11,15, 19..(Suh seres where the ext er is
eve by ada een Fined valet previous
umber are called as Aritmeti Progressions.

20. NUMBER SYSTEMS

INTRODUCTION

{The chapter of number systems is amongst the mostimpor-
lant chapters in the whole of mathematics syllabus forthe
(CAT examination. Students are sed to go tough this
chaper with umos care undersanding each and every
question type on this ope. The CAT has consistent st
Between 10-15 marks had on he concepts ofthis hap
ence, going through this chapter and its concepts properly
is very imperative fr you. Sen nthe context ofthe qual
fying score i the CAT beingin he range of 12-14 marks (or
(QA). ths eaves ws with fir idos of he rial importance
‘ofthis chapter It would hea goed idea 1 fist go through
the basi definitions of al types of mes someting have
found be surprisingly very less known abou). The student
Is aso advise to go through the solutions ofthe various
question usted inthis chapter. Ress, ile soning his
te ty o maine your ing eine with ven
Problem tat yo solve. To start of the following poll
represen ol he ype frum vil ep you improve
er quality o comprehension of diferen types of amen.

DEFINITIONS:

Natural Numbers These ar the munber (1, 2.3 ec)
that reused or counting. In cher words, ll postive eee
ters are nal mum
‘Thee are efi natural number ar the umber isthe
Test natural urbe
Eagles of satu numbers 12,4, 8.32.25 4321 andsoon.
“Te olowing numbers ar examples of number th are
1-238, and won.

Based on dvisibli her could be two types of natural
numbers Prine and Compost

Prime Numbers À scr umber larger han unity isa
prime mamber if it doesnot have eer divisors except for
sel and uty

‘Notes Unity ie. Dis nota prime number

[Some Proj of Prime Numbers

= The lowest rime number 2
2 ab the only even prime umber
2 The lowest odd prie umber 3.

21. TE | one Prop Queue iaa CAT

ah render when a prie number p 2S is vided
by bie or. Howener fa number on being divided
by 6 gives aremainderof | ot Sth number eed pot
terme

|= Toereminder ofthe vision ofthe square of prime
amber p 2 vied by 241

‘= For prime meme p > 3, p Li he by 24

|= Prime Number been Ito 10e: 2,3.3.7.11 13,
ODIN SARA SN SD IIA
u.

= Pine Naben beine 10010 200 ae 107 03.107;
00, 13,127, 131, 137,139,189, 11, 15716516717,
179,181,191, 193, 197,198.

= a nd b re any two al primes then — 8 ig
composite. Als, a+ 1? is composite

=. Thererainder of the division of the quae of aprime
umber p25 divided by 12s.

SHORT CUT PROCESS

To Check Whet
Prime or Not

To check whether number Nis pms, ado on

process

(a) Tae the square too ofthe mame.

0) Round of th square toot othe immediately lower
‘eget. Call his numbers, For example if you have to
check for 18 xs square oot willbe 13. Hence, te
salue of, is cae wil be 13.

(6) Check for hat of the name N by all prime
amer terio prime arto the
value of which divides tren the number will be
ine.

Toile

‘The vale 0 VHD lis Bean 1 o 16 Hence, abe he

value of as 16,

Prime miners es then 16 ar 2.4.5.7, 11 and 13.299
isnot divisible by any of these, Hence yo can onclade
at 239s pine suber.

A Brief look Into why this works?

‘Suppose you are asked to find the actors of he member 40.
An uiid mind wil find factors as :1,2,4,5,8, 1,
2021440.

“The same tsk willbe performed by a tained mind as
follows:

1 x +
2 x 2
4 x 10
and 5 x 8

Le. Te discovery of on factor wll automatically yield the
‘aber factor. In ober words factors wil appear In tes of
‘what can bo called as factor pis The locating of one co,
vel oral prs eo ne for yas. Tas, the
‘example above, wea you ind Sas a factor of 40, you will
automatically get 8 too a a fcr.

Now take ook again a ib ais inthe example above
you compare the vals in cach pir withthe uae ot
AD (Le 6__) you wl find ht or ech parte number
in ie ll sum is lower ha the square rot of 40, while
the member the rig cour is higher than the square rot
of40,

“ris proper fr all numbers and is always true.

ence, we ean now phrase this as: Whenever. ou have
1 ind he factors of any narber N, you wil get the factors
in pi (factor pain Farber, he trp will be sch
that in each pir of factors, one ofthe factors will be lower
than the quar oot of Nil the cbr wil be higher than
the square rot of N.

“As rs ft fact one ned at make any fort Find
the factors of number above be square oo ofthe number.
‘These come astomatical, All you ned todo isto id the
Factors below the square rot ofthe number.

Extending ths Loge, we can sy that f we are nt able
10 find factor ofa number upto the value ads square rat,
vee wll not be able 0 find any foto above he square root
‘and the number under consideration will be a prime
‘umber. This isthe reason why when we need o check
weber a mmberis prime we have to check for factor only
below the square oc.

Bat, we ves tht you nc to check fr dvi
aly with he prime numbers below (and including) the
quae roof number What gle wi explo is:

Letus look aan example to understand why yon nod to
took only at pine numbers below the square root

Ups new, we ave decada order to check wher
a ramber is prime, we just eed o o a factor search below
(and inctadin) the square rot.

“Ths, for example. in oder ind whether 181 ia prime
umber we pedo check with he number =2,3,4,5.6.,
8,9, 10,11, 12,4 8

22. ‘The fist thing you wil els, when you ist ok the
ka above is that all even number will ge eliminated
“automatically (since 20 even number can divide an odd
ruber and ofcourse yeu will check a number for being
prime oy it sofa)

"is will ave you wih he mumbers3,5,7,9, 1120013
check 181.

‘Why do we not need to check with compete numbers
an ie square rot? This wil aa bo endestood best
explained in the contest ofthe example above, The only
compo number in the it above i 9. You do no eed to
heck with 9, because when you checked fo vii
with 3 you would get either of two eases:

(Case: EN sve by 3: In sch case Nil ato
cally become non-prime and we can stop ou checking
Hence, you wil ot need o check frie dvi o the
munber by 9.

(Case I: Nis not divise by not vibe by 30
is vista wil ot be divisible ty 9. Hene, you wi
ot ed o check or the dvi ofthe number by 9.

‘Thus, in either cate, checking for divistility by a
composite number Jin is ease) wil become nee, This
ile tre forall compis numbers.

Hence. when we have 1 check wheter a number
prime or ot, we need o oly check or is ir by
Prime factors below the square rot of À,

Integers A set weich consists of art numbers, epa
tie integers (-1,-2,-3..n..Janderoiskoowaasthe set
integer. The number belong 1 this et re own as
inegen.

Composite Numbers tis natural rumor ht hast
ens one disor diferent fom wat ad tl

Every compose number can be factored it is rime
factor. (This is sometimes calle the casonial form o a
under)

Inmachemaikal ems: = pp pl herp PP
sxe prime numbers cll faces and mare raul
‘umber.

ES

‘Ths representation of a compote number ls known as
(he standard fre ofa compost number. Iisa externe
ful frm of sein a composite number a we bl se.

(Whole Numbers The se of numbers tht includes al
astral numbers and he number zero ae called whole num.
er Whole number ae a called ws Non negative ter.

(Chart: Number pions pz

‘The Concept of the Number Line Thenunserlineis
tight line between negative ini, onthe ft 1 iii
tothe right

EEES +

“The dstance betwee any two pois on the number
ot by subir the lower value from the higher vl
Altea}, weca lo stat with the lower umber and ind
‘the required aon to teach the higher number

For example: The dance between he points 7 apd wil
bee

Real Numbers All numbes that can be represented on
the number line re called real numbers. Every real number
‘can be epprosiraey replaced witha terminating decimal
‘The following operations of ation, succión.
"pican and vision are vali for both whole numbers and
real numbers [For any real or whole number a, band cl.
(9 Communatve property of addition: u + b=b+ 0.
10) Associative propery of add: (a+ + € = a
+40,
(6) Commaatire propery of aplication a: b= a
(8 Associa propeny of multiplican: (a-D)+€
ibe
(6) Distro propery o mien with respect to
ads (a+ D) en ac + be.
(0 Sebteaion and division are define as te inverse
‘pertons onion and mulipiaton respeivel.
Thai a à ne ben = 2 and qe afb hee bg
a (here b 40)
Divin by zero is ot possible since there is o number
for which bg equal non zero number

Rational Numbers A nei mers define rum-
ec of th form alb where a an! bare integers and
bro.

"These of rational numbers encloses he se of mens
id fractions. Te rues given above fo addon, subire»
tion, mation and division also apply om rational
umber.

Rational eumbers that arena itera wil ave decimal
vales, The vales ean be oft Les:

(9 Terminating (or nt) decimal fractions: Forcxampl.

17462 425,215 24.2 ano forth,

23. TE Jresnontuomunnnunscn

(0) Now-termioating decimal fractions: Amongst nom
terminating decimal factions here ae two types of dec
smal valse

© Nomtermivaing pera fractions: These re n00-

termi deci factions of the ype x ays,
16

ameaaı- gerne. Foe example À =

383, 1825200805, 14287428762 TE. ann

Nomterminatng non period rotons: These ore

athe orm bab. brie, For example
SALES.

Of ie above categories. eminsting decimal an non.
terminating price decimal fractions Belong to the set of
racional sum

Irrational Numbers Picas tats cooing,
nero aon, ae ira urbe,

‘Sorc cumplo von mentor YE, ain
cabe word al suas ae cube roots of tral bers
At ot ss and ees featur mer ae ia:
Anal Ode roa umber incide ad 0.

Every pose aon urbe hs a epa rana
sente conespoacing wit

‘Allerton of adn, action, main and

sion applicable to ana! numbers re also applicable to
‘atonal numbers

‘As rely sted inthe Back wo Scho! section, whenever
“an expresion coats rational and a national number
together, he two have o be cam together til he end. Ln
er words, a iatonal namber once it appears in Ihe
soin of à gestion wi continue to appear il the end of
‘he question. This conept is particu seul in Geometry
For example: fou ar eked find th ato of te ees of
circle lvl an quiera ange. youcanexpctt0see
22/4 nthe amener, This because the are faire will
always have ap component it, while that of an equllsterl
tangle vil always bave 3.

You should ease hat once animation number agen
the solton ofa question, I an ony disappear i is
molle or dviged by te same Ilona number.

‘THE CONCEPT OF GCD (GREATEST COMMON
DIVISOR OR HIGHEST COMMON FACTOR)

(Consider wo natural numbers, andy.

the numbers, and; ore exactly divisible by ho same
umber then xin a common diviarof n aná 1

“The highest of all he common divisor of and m is
led the GOD othe HCE Ts dentes GOD (m,

Rules for Finding the GCD of
[Two Numbers m, and m,

(6) Find the standard form ofthe mamber and nz

(0) Write cu al prime factors that ate corn 1 the
standard forms ofthe numbers y amd

(© Raise each ofthe common prime acord bone
to the lemerof the power in whic appear i the
‘andar forms ofthe numbers y amd

(8) The product ofthe resus fhe previews ep will be
the GED of and ns

strain: Fi he CD Of 150,210,373

¡Step 1: Wetting down the standar form of number

130.

O
Stop 2 Wing Prim factors comment dl tros rem
tenia S13"
Sep: This wil ive te same rl, 13!
Step de Hence, the HCF willbe 832 15
For pacto, find the HCF of the folowing

arms
Dee
CRE

THE CONCEPT OF LOM
(LEAST COMMON MULTIPLE)

Let m and m be to rata numbers disc from each

‘ter The sles toral member tb exactly vie

‘by and scaled te Lest Common Multiple (LCM) of
and yan is desigaated as LOM (em.

24. 20 | pomor crite itt car

© Pas es! 0 Nene of hose
Answer. () Use HCP x LCM = prod of the num
ters. Hence (2331 103 OMAN"
310322 073 «N.
10. Two eguilrl eagle have tb sides of legs 34
and 85 excl,
(6) The greatest eo of ape that can measure
Bot f them ext iss
Answer: HCP of 4 and 85 17.
(©) How many such equal parts canbe measured?
Answer 3417 +8917 =245=7
1. Twonurbers ae the eto 17.13. IF ele CS 15,
‘what are the numbers?
Aer Ty Sand 1315 Le 25508 19 respete
12. A forester wants to plant 4 apple trees, 6 ba
tees and 110 mango tes in equal rows (items of
uber of tee) Als, be wants o make dsc fs
of tes (Le. only one typeof te in one now). The
umber of ows (minimum) that are required ase

(m2 CE
on @

‘Answer: (6) 4422 + 66422 + 110722 (Since 2 isthe

cH,

13, Tease insert runing around a circular tack, ca
complet one evolution in, nd 5. howe espec
tively. When wil they meet atthe string poi?

on DE

on (0
(re nene wi be the LOM oC 2. end 11. Tas ul
ive you 4 as ho ansnen)

14, The HCF and LCM of two members ar 33 and 264
respectively. When te frst number is vided by 2,
the quotient is 33. The eter number is?

os 2
om cr
(Answer: 39% 264 = 6630, Hence,n = 132)

15. The gress number hich will vide 4009, 126008

249, leaving the same remainder i ech ease:

as va
os (0 Nove ofthese

‘The answer willbe the HCF ofthe thre numbers. (81

da is casey

16. Which ofthe following represents te largest git
number which can beaded to 7249 in order io make
the derived umber divisible by each of 12, 14,21,33,
ist,

mE DES
om (6) None of these
Answer; The LCM of the numbers 12, 14,21, 33 and
815 836. Hence, in onder forthe cndion 19 be
sthiod we need to gt the number a:
7289 4 = EX 2
Hence, Rune,
17. Finde greatest number of 5 digs, ka wil give us
‘remain of, when divided by Bad 8 respectively
a 95831 De
ons (0 None of these
Answer The LOM of and ie 72. The longest digi
rie of 72 99936, Hence. he eq answers
sa.
18 The least perfect square number which divisible by
3.4,6,8, 10404 lie
Solution: The number should havea atone 3, re
2s, eS and one 1 frit o be divisible by 3,4, 6,
8 10snd 11
Further, ach ofthe prime actos shold be having an
even power. Thus, he conec answer willbe: 33%
224 22«5%5% UL
19, Find the greatest number of four digs which when
‘vide by 10, 1, 15 and 2 leaves 3.4, 8 and 1525,

reines respective
wor ©) 903
CES (6) None of ese

Answer: Fis find the greatest 4 digit multiple of he
{LOM 10,1, 1 and 2. nes 9900, Then,
sac 7 fom 0 give dhe anar

22. Find the HCF of 1) and °° D.

Answer: The soltion of is question is based onthe
res

‘The HER of Gé = 1) and (e — 1) is given by
wre)

“Tas, im his question be answers: 3" = 1). Since 5
ds e HCP of 5 and 125.

2. Went wil be the lest possible mue ofthe planks,
tie ices liber 42m 49 mand 63 mang have
to be divided into planks ofthe same engi?

m7 os
“2 (9 None ofthese

22 Fie the reses number, wich wi ide 215, 167

and 13530 asoleave Li same remainder in ech ae.
ws“ LE
on (9 16

25. 22 ] romo motero pince bn CAT

‘Theorems of Divisibility

da) Ia is divisible by then oe is also visite y 2.

(0) Weis divisible by band bi divisible by then 2
divisible by

(6) Ha and bare natural seers such hat is divisible
by band bis divisible by ben a =

(A) Unie divisible by dand mis visible by the (m
m and im = m) ae bach divisible by d. This has a
important implican. Suppose 2 and 742 are both
seb 7. Then (722-28) 38 velas (742 28 ae
¿visible by 7 (and in Fat is + 28 742).

(©) Mais divise by band cis divisible by d then acs
visible by I.

10, Tuehighes power ofapime munberp which divides
exactly is given by

{ep + Ep +t
sere a emotes tho gratos integer es than or qua 10:
As we have already seen caer ~
‘Any composite mumber ean be writen down asa product
ofits prime Factors. (Also called andar form)
‘Tha, for example the number 1240can be writes 2° x
sx
‘The standard form of any munber has à bugs amount of
Information stored in it. The best way to understand the
infomation sored in estará form uf a number stock
at concret examples, Asa reader | want you to understand
‘eich ofthe grocese defined below and use he 1 solve
similar questions given inte exes fellows and beyond:
1. Using the standard form ofa number to de sum and
the number of actors ofthe number:
(a) Sumof factors of a number:
Suppose, we have to find the sum of factors and te
under of factor of 240.

A
“The sum o actors will be given by

E)
Beeren

Note: Ts a standard process, wherein you create
the same number of brackets athe namber of ii
prime factors the number contains and thon each
bracket filed with the sum ofa the powers f the
respective prime number starting rm O1 the bigest
power of hat prime number cortsined in estandar
fam

“Tus, for 1240, we create brackets —one each or 2,
‘and . Further a the back corresponding o 2 we
wee 24242542424.
ence, for example forte urbe 40-25! the sum
of facos wil be given by: 42 42 PASAS
(2trackets since 40 has? dint rie fotos 2 and
a

(0) Number factors ofthe number:

Let us explore the sum of sors of Ai a fret
CRETE
ERA
Sas ads
s1ese2e 10934209 seem

sn

A clear ook atthe number above will make you
realize that its nothing bat the aon ofthe actos
o
ence, we realise thatthe number of terms In Ue
expansion of (2+ 2! +2! PASAS!) will ive us
the number of factors of 0, Hence, 4 has 4 2 = 8
factors,

[Note:The moment you raise thar 40 = 2 x! the

answer for he nunber of lacio can de got by (3 +

Dds

2. Sam and Number of even and odd factors ofa number.

‘Soppos, you are ying to find othe number of factors
‘of number represen in th andar for by: 2) 3"
rear
As you are already aware the answer the question is
G+ N+ EE IX ES + and is based one logic
that the numberof ters wil be the tame asthe number
fiers im he erpanie: (2 +2!42°+ 2804 3! +3"
APIS
Now, suppose you have 10 find out he sum ofthe even
factors ofthis ume. The ony change you aed to do
in this respect willbe eves below. The answer wil be
aiven by:
AAA
rn
Note: That we have limited 2° fom the orginal a
see By timiating 2” from the expression forthe sum
ofall actos you are envia that yeu have only ever
‘umber in the expansion ofthe expression.
CConsequety the number of ven faces will te given
TON

26. Line Ps elimina, ne do ot ab inte racket
companies to 2.
Les py o expurdosr ing y haut
the number o 48 fair lora number
In is cs ws jus have odo th oposite of what we
fr even eunbes. The Folle tp wil make it
sear
(2 estore member whose stand frm is: 2?
xSP
Sam of od co
Sheehy
ie: oral poner of. The eto ti expan af
the above expresion wl be he complet se of od
facto ofthe number. Consequently, be number of 044
(str forte number wil begin y he number of
Les expo le above expression
‘Tha be number of 08 fects fe the mar 2 4 +
IN
3. Sum and number factors sang oe conce
for ny compte munber:
“Tes atest lil ough expe:
Fiat sum andthe munde of fais of 120 such
Ut the acto visible by 15.
Selation #1200 2x4.
Fora acto tobe divisible by 18 sould compalso-
Fe 3 nin Tas, sum flecos vise
ie
msegeny be number of fc wi given by
Seat = 10
Cat we ve done iene hin every advil
term ofthe expansion, er Isa maimum of 5!
“hiss dene by moving power of and which ae
below 1)
Tak forthe den: Pay verify ih answer
queno aber and ry o comer the gi into
ment lor
NOTE FROM THE AUTHOR—The need fr thought
er.
have olen observed at be Key diles betwcen
undesandig a concept and actully applying under
sami pressure be prscse or absence of ena
‘Hoop elgoritim ich life the concept 1e you out
ind The hugh agri ia peal epseniaion of
à concept—and any concept tht you endunderand in
Ai book (rt) wil rein an este concep it
remains in someone ces werk. The moment e bot

1. Forthe umber 2450 ind.

(0 The sum and number ofl (cin.

(@ The sur and number of ven factor.

(8) The sum and number of od fc.

(9 The sum and numberof factor div by 5

(9 The sum and number of actors divisible by 35.

(6) The sum and rubor of actos divisible by 245.

I. Fortheneraber 7200 find
(0) Toe sum and number of al con.
9 The sum and number f even fates.
(©) The sum and number of 0 facon.
(00) The sum and number of actos divise by 25.
"Tae rem and number of factor divisible by 40,
“The som and nomber of factor divisible By 150.
“Te sum and numberof Factor not divisible by 5.
‚Te sum and numberof factors na divisible by 24.
Fd the rer of divisors of 1728,
ws DEI
CE op

16. Find the nunber of divisors of 109 enclin the
‘troophoutdiviors, whlch are perfect square.

ws CE
CE wa

17. Fade monter fin of 4 excluir 1 and 4
wn os
on @ 0

18. indie number of vars SH which ae gree than
3

ws CN]
on @) None of ese,
19. Finde em of divin of SAS exctading | and 44.
wo DES
Cr) ons
ZU Find the sum of divisors of $44 which are perfect
saurer.
CE we
CE aa
21, Fit he sum of od vss ot 344
ow mE
oe ws

22. Fie the sum of even dir of 4096

27. Howto Papo aa do eb CAT
24

CES van
(o suo @ oe
ZU Find be sum the sums of divisor of 144 and 160.
E] om
CRI (9 None of hese
24, Find the som ofthe sum of ere divisors of ate
sum of od iin of 600.
oo om
os os
Answers
so KH no RH BE
a0 10 20 BO 240

NUMBER OF ZEROES IN AN EXPRESSION

‘Suppose you ave 1 find the nub of zeroes in a produc:
DM 3D 172319. 2!) SA

As you can voice, this product will have no zeroes
Because as no Sin it

Honever.if ou have an expression ke: 8x 1523317
van

“The above expression can be rein i the andar
forme:

PRA AS

Zero ate formed by a combination of 2x3. Hence the
umber of zeroes will depend on he mamber of par of 2's
and 5 tha can be formes.

Inthe above prot there ae four ros and two fives.
enc, we sul be able frm only ro pais of (2 x 3)
ence, there wil be? zeros inthe product.

(Refer to Solved Exareple No. 1.1 for another example ol
is)

Finding the Number of Zeroes Ina Factorial Value

Suppove you had 1 fed the numberof eros in 6
616 x5 4x3 2x1 DAS ACADA)A
ext

“The above expresion wil have only ane par of S 2,
since there is only ne and an abundance of 2's.

is clear tat in any factorial value. he number o's wi
says bo lesser than the sunber of. Hence, al we need
Lo doi 1 count the numberof Su. The process fr thie
explained in Solved Examples 1.11013.

“E EXERCISES FOR SELF PRACTICE m

Find the member of zeroes in the following eases:

Lam
238

à x 15 22% SAI
A
som
SS IS
71

am

a a

10 dues

A special implication: Seppe you were to fndthenamber
‘of zeroes inthe value ofthe following factorial vals:
AS em 481,491

‘What do you notice? The number of eres each ofthe
cases willbe qual to 10. Why does this happen? Li not
fut to understand tht the mamber of fives in any of
thes actas equal 10, The sube of eres nil cel
change at 0! twill become 12)

o ft, is wl be te foal ctrl alos Between two
consecutivo prodact ofS.

‘Tas, 0, 51,521, $3! And 54 will have 12 eres (since
bey all have 12 fives.

‘Slay, 336,71 S81 An 39 will each ve 13 eos

Apart fom this fact did you motive another thing? That
ti ee ae 10 zeroes ia 9! thee are die 12 ers
SOL Thi meant ther is nove o factorial which wi
give 11 zeroes. This cur case to ge SO! we mail he
value of 491 by $0. When yon do so, the resul hat we
nous two inthe product. Hence the umber of eros
Sepa by two (since we never had any pacity of twos)

[Note a 1241 you wil get 24+ 4 =>28 eos.

‘Ac 128 you wll pet25 + 8 + 1 = 31 zeron (Ajump of 3
aves)

"EU EXERCISES FOR SELPPRACTICN m

1. ats 2 aes, Whats he maximo possible vale
am

2 al hs 13 zes. The highest dut values of are?

3 Finde number of zeroes inthe prt 2232
HAS KO ETS

28. 4. Find the number fer
100! x99 98°97
£. Fad the number of roca
A 10
6 Find the number of zeroes in the vale of
ERSTEN
7. Whats he numberof zeroes inthe following:
(3200 + 100040000 + 32000 SONO
(9) 3200 1000x 400032000 1500000

Solution:
1. This can never happen
2 Sand SS respecte.
5 The fives wil be les than the twos. Hence, we eed
19 count only the fives
Than: $4 10% 159 29° 25830335
ass
ve a 5 + 1D + 1542042929 04164404
45 fives Ths, product has 280 eres
4. Again te key ere 10 count he naher of ve
“This can gt done by:
100 9550" BS ET. SM
(14611 +16 421 425431636651 446%.
+56) (1 #2645176)
520485 443385
‘sind in the net chapter)
970+ Sts 1124,
$. The answer willbe di nunberof Home, le
sie
& The numberof ives is agin ese an the number
of eros.
‘he manber of 5 wil be given bythe power of $i
be prod:
SO 20!" 10825"
m4+8+12+16+ 18+ 40098,
7. A Tie munhero res inthe or wile uo, sic:
san
vom
40000
2000
15076200
3200
“Ts in sch caes mamie of zero il he
east number of zeroes amongst e numbers
sora: 3200 + 18008 000 (es zero mt vo.

(Using sum of AP. exe

RS [BE

B.Thenamber of reves willbe:
2344246218,
A etes ofthe proces or ling the number of rere.
Consider he flowing quences:
1. Find the highest pomer of S which is contained inthe
she of 177
22 When 127! is divide by 5 the resol I an integer
Find the highest posible value for m
3 Find he number of zeroes in 127!
In each of the above eases, the value of the mn will
be given by:

DAS
2284541031
“Tis process can be entend 10 questions rad to.
cer prime numbers. For example:
Find the highest power of:
L 3 which comple divides 381
Soon: [383] + BSD = 124441 =17
2 7 which is contained in 574
CURE
‘This process changes when the divisor is not pine
umber, You ae frst advised lo go through worked ext
problems 14,15, .6and 1.19.
‘Now try 1 solve the flowing exerce:
1. Find the highest power of 7 which divides 81!
2 Find ih highest power of 42 which dvies 122!
3 Find the highest power of 84 which dives 3421
4. Bind the highest power of 15 wich vides 3441
5. Find he higher power of 360 which vies 5201
Solution Hints:
1. You will beck ony with 7.
2. You wi nd check with only. (Sine 42 = 733
xa
A BA=TXIAZAZ You il ne chek ore number
of scaly
A 1S=S x 7 Inthiseme you ned bear. You
will acd io check the number of $s andthe amber
ot
s

x 3°. Im is ease you wil need o check
Tor ie number of 2, the number of 3 and the
umber of 3"

“WIEXENCIEES FOR SELF-PRACTICE m

1. Fndtherraximan vals sucht 157s pere
Alvis by 10.

29. ‘The rest will ever be nepal if the Ivo denominators
avon pine

Note: This holds tue even for expressions ofthe stare
AN Bie,

‘This as huge implications or problem solving expecially
in th ase of solving linear equations related to word Isa
problems. Students are advised 1 uy 10 ss these through
out Bleck 1,2 and 3 of this bok.

Key Concept 2:
cope
Example: ind all five-digit numbers of te form 34 X SY
that are vile by 36,

Sollen: 36isa pret of wo copies 4 and. Hence,
AA X Sr divisible by 4 and, will ako be divisible by
36. Hence for iris by 4, we have thatthe vale of Y
can be 2 ce 6. Alo, i Y is 2 the number becomes
3432. Forthisto be divisible by O he ation of 3 + 4 +
4.5 4 2 should be divisible by 9. For hs X canbe 4

ence the umber 34452 ls divisible by 36.

Also for 6, the number 34% 5 wil Be dvb by 36
when the ation ef the digits is divisible by 9, This will
hagpen when Xisor9. Hence, th nares 34056 and 34956
willbe vse by 36,

"Two consecutive integers are always

"EXERCISES FOR SELF-PRAGTICN

Final numbers ofthe form 36 x 3ÿ that ae vise by 36,
Fil ne ofthe form 72 Ut are vite by 45.
Find al number ofthe for 1352 that are vile by $5.
Fin al murs ofthe form S17 xy ht at divi by 89

viii by Zr A pame ih vice hy Dor the
las iii divide by 2 ors

Ii by 3 (or 9): A such umber tbe of whose
Ais ave divide by 3 (or 9) are divisible by 3 (or 3)
Div by 4: A number dise by if tbe ast 2
ist are isle by 4

Divi by 6 A names visi by Fi simul
eos divisible by 2 and 3

Débit by 8: member is dvs by 8 if lat
hits ofthe mer se dil by 8

comer: nat AE

Dinky y 11 A amer ve by Wie ter
enc ofthe sum ofthe dig inthe ed places ad the sum
(ofthe digits ia the even place is zero ors divisible y 1.
Din by 12: Al number dvb by 3 and are
divi y 12

Diviibilty by 7, 1 or 13: The inter a divisible by 7,
11 or 13 if and only ifthe difference of the number os
thousands andthe reminder of lts division by 1000 i
visible by 7, Loc

Example: 475312 is visible by 7 ine the frere be-
Agen 73312 = 161 is visible by 7.

"Even Numbers All integers tht are divisible by 2 are even
rumbers. They ae alo dented by Zu.

Example: 2.4,6,12,122.-2.-4,-12. Ali note that 20 is
an even number

2s he lowest postive even number.

(044 Numbers Alliegers hare or divisible by ar 08
number. Odd number leave a remainder of 1 on being
vide by 2. They are dena by 2 +1 or 201,
Lomestposive 084 number is

Example: -1,-3,-1,-35,3.11 et
Complex Numbers: The aihmeic combination of ral
number andiominny uber ae call comple numbers
Alternately: AL numbers of the form 04 ib, where
(2 ¥-Larealle complex number

Ti Primes: A pair of pme numbers ar ai o be twin
prime when they fer by 2

‘Example: 3 and 3 ae Twin Primes, o alo ave 1 and 13.

Perfect Numbers: number ns said be a prfect number
ifthe sum fall the vise fr inching) is equal to 2m.

Example: 6= 1x2 %3 um ofthe divisors
2x6

424346

DAT 14,28, 56

Tas for sunt Find ah perfect numbers below 1000,
Mixed Numbers. A number that has bo a nigral and à
fractional partis kro as a mined number
Téangaler Number: nunber which can be represented as
the um of consecutivo natural numbers sai with 1 are
called stampa number.

eg:l+2+3¢de10

30. Learning Point: newer od e remand 17x23
‘when divided by 12, you nosé to look atthe individual
rmainden of 17 ond 2 wen divided by 12 The respete
remainder (5 and 11) will give pow the reiner of he
‘iia expresion when divided by 12
Mathematical, scan Be writen as
“The remainder of he expre [AMB C4 DEY,
we same athe reader othe expression LA x
KG + Dax EM
‘Where Agi the remade whe As died by,
‘inthe reminder when Bs vided by M,
Cats te reminder whee Cis dived by Mt
‘ps the remalnde whens vied by M and
is einer when Bis idl A
‘We call this wansformuton a he remaider sem
anse and dene it by he ign E+
“Thos the remainder of
MI DIX 1425 when vid 12 cn e vena
MES ST, 308
[2 zn
110
m,
2 pres elder of 3
Inthe ove question, we nerd ei ofremainder
or wansormaions (dot by 4) ad cua
transformations o tanto fica loking expresion
into simple expression.
“iy to solve the flloning questions on Remainder
bere:
Find th remainder in each ft folowing ea:
1.190200 126.38 vied by
2 248 2452472495251 died 12
m2

a

248249250
s

y ana

FABS, 1971
> 9
USING NEGATIVE REMAINDERS.

Conse he following question
Find the remainder when: 14% 15 is dived by.

cor traga BS

‘Tae obvious approach inthis css woald be

Mots 4,922 ay acumen

y ce
However there is another option by which you ca solve
the same question:

Wien 1 ivi by 8 he reminders orally seem
2 4 6. However, there might be times aben sins the
negative value of the remainder might give vs more
«convenience, Which y you sould know the following
process:

Concept Note: Remainder by definition am aways on
egtive Hence, even when we divide a umber ke 27 by
$ we say thatthe remainder i 3 (and not ~ 2. However,
Soaking a he negative vale of he remalnder has is en
vantages in Mathematics a it rotin racing calla
Ths. when a man ik 13s divided th einer
being 5. he negative reine is 3
(Note: isin scout tht we mention rumbers Ike 13,
21, Wee 2584+ So 3 numbers)
mas ea
3 wit give ws 2%
Consider the advantage hs proces wil give you in the
fotowing question:
sin a
nun
(Te terme wi involv on calls. Hence,
principle is that you shoul use negative remainders
‘wherever you ca, They can make life mach simpler!

a

‘What If the answer comes out negative?

Bue, we kos hata remade of -2, eu remain
42 when divide by 66. Heece, the anener 42.

OF eoere sothing sos you from sing postive and
esse remander 1 de sume tne in erer to alv the
same question =
DAC) pea

ne DED pu

Tes

ETES

Dealing with large powers There are tu tos which
ao efectivo oderto deal wid large powers =

31. 30 | Howe repre teatre Acer be CAT

(A) If you can express the expresion in the form

such a exe. no mater ow large the value of the
power ni, the remainders

om) (en
cu 3
In sach cae the vals ol power des nat mat

Fer isa,

o) CRY ta such a ee using

it sill erden atthe romande willbe + mis
‘even nd it will be —1 Bence a 1) when ns.

she reminder

ME, en
en
ANOTHER IMPORTANT POINT

‘Suppose you were asted1o find the reainde of 14 divided
y & Is clearly visible thet the answer should be 2

ut consider te following process

AA 722 —E 1 (The answer has change)

Wat has happened?
‘Weave atome 16! io 72 by diving the ameno
and the deooriostor by 2. The resalto thatthe original
remaloder 2 is also divide by 2 ving us | athe remainder
Tower to ke cx ofthis problem. we ned to revere the
elec of the ion of he remainder by 2. This done by
‘mukipyig the fal eminde by 210 gee come aser
No: In ary question on remainder theorem, yo should ry
o cance au pass of the aerator und denominator as
much as you com since ively reduces the cleaatons
required.

AN APPLICATION OF REMAINDER THEOREM

Finding ie last o digs of an expression:
‘Suppose you bad find be st 2 gis o Ue expression:
BHM AS ZII
‘The remainder the above expresion wil give when tis
vide by 10D isthe answer tothe above quests
ence answer the questo above find the minder
ofthe expression when it ic vided by 100.

AIX ITAITXES

Solution:

2 BIT Ad

4 (on diving by 4)
2, BXOKIx2K12«I8 _ Bx22K216
a = >
2, mx

+=
ESTA

IA

‘Ts the reraader being 14, (aer division dy 4), The
tua reminder shoal e 56.

Don’ forget to malt by 4 1

Hence, he lst 2 dips ofthe answer willbe $,

Using negative remainders here would have belped
‘ort
Nor Sisal fading the lst hee digits of a expression
reas ining the renader when the expression is divided
by 10m.

XERCISES
1. Find be remainder when 73+ 75+ 78+ 57-+ 1971
ivded by 4.
@ 2 E
os on
2. Find the rerinder when 73 75 X78 57 x 197 is
divided by 4.
mE CE
os CE

A ia the remainder when 73X 732 78257197

3 0 x
CH CE

Find reader when 43 is divided by 7
m2 we
os a

|S Fd the remainder when SE diidedby7.
ms CE
CR @ 6

{6 Find be remainder when 39" i vided by 7.
m2 we
os CH

1. Find the remainder when 67 is divided by 7.
m2 wa

las ot

32. Find be remainder when 759 divide by 7

ws CE
@2 as
A. Find the remainder when 417 ie divide by 7.
(02 1
(06 we
10. Find he remainder when 217° i vided by 17.
ws on
(o 16 wo
IN. Find he remainder when S4™ is vide by 17.
wa os
on os
12. Find the remainder when 83s divide by 17.
on & 9
os @ 2
1%. Find the reminder when 25" is ivid by 17.
on os
os 2
‘Answers
LO 2@ 10 dw sw
6® 710 8M 20 DO
no 20 Be
Units Digit

(A) By the logic of what we have just seen above, the
nits dig ofan expression willbe gotby geting de
remainder when the expressions divided by 10.
‘Ths fr example we have to find the wots digi of
the expression:

ARIAS TD KES
We ty o find th reminder ~
17225368627 «3165,
To

py PRA

10
XD y, 1x0
IAE]

Hence, ih required answer is 6
"This coul have been dry got by mulpying:
ZM GANT | Band only accounting forthe
ani digit

compart: Mme ET

® Units in the coments of powers =
‘Study te following ble carefully

Unit's digit when "N ie raleed to 6 power

Peer
Eng due of power
wet

In the table above, if you look at the coloma.
corresponding tothe power 5 or 9 you wil realize ha the
uni dig fr al uber i repeated (Le. is for 12 fer,
3for3..9for9)

‘Tis meas hat whenever we have any number whose
ani dti and it red a power ofthe frm da +
Inthe valse ofthe units gt of the answer wil be the same
he orginal uns digit.

Hsraons:
ZT wil give awit cpt of 3. 1547} wil gives uit,
igi of 7 and 0 Forth,

“Toes, the above table can be modifi imo the form =

Value of power
Kader
w
emdagin tab me? ed
7 1 1 1
2 4 4 5 .
3 3 5 7 \
4 4 6 ‘ .
6 6 6 o .
74 5 3 1
98 1 9

(Remember, ahi pois that we had sad Gn buck 0
school section of Block 1) a al narra numbers can be

33. 32 | non rage er Quart he eta AT

expreso forma. Hence, wth the lp of ts os
‘het helps us Bild his tbe, we can ely derive the ete
ii a any number when it i asd 0 a power.)
Aspect

Question: .

‘Wha wil be the Unit's digit of (1273)

Solution

122 mumber ofthe form a. Hence, be answer should be
1 Note: here serie by thinking o ita 3 (orn + D.
Serán + 23,7 or +31, Mord

TM EXERCISES FOR ALUPRACTICESS m

Find the Units dii in each of the following cases:
Le
2 xx Peas xo.
BAT K23xS1 x32 4 15% IX 16x22
SIRT xR A IA
sp
6

x 1008"

67x37 Ax ADH IBA
(92 wm
ws wi
2, INIA ADA
wt ws
as ws
8 6735 xAS ON ADH STAT
(02 A
oo ws
9, 673545 491 042% 525082
ws or
oo ws
10 axes
(2 ws
os ws
A
(97 CE
(5 wo
ENT am”
(02 vs
os os
13 BDO
wo m6
CE wa

14 82 ast BAT BOAT

CE we
4 ws
IS 32 53 ASI 4 5372531 417997
ms ms
CE ws
Answers
sw 120 10 90 We
no no BO no Be
WORKED-OUT PROBLEMS
o inne number otzeres inthe atria of
Gear 18.
BOMBER” 141 Cousin 15205, which combined wins

even numer give zeroes. Als, 10 ls also contained in 18
which wil give an ational zero. Mene, 8! Contains 3
‘eres andthe at digit wi always be era,

PEZ Fe 0 notes es in 20!
SRB manie
berri

income y cnt ny amero
ing aged yy ern mer Sly even à
tube ening ins onde pr, adn
aca a Ft pte ce 25-53 Sl pe
amena ens sabe ety 15108 Hee
iran sees
Storch Nantes ei 2-128 27281
ter mé urge Ju lover Dan acon
‘Hence, [27/5] = 5 and (27/5'|= 1, 6 zeroes.
EEE Fis name in I
SoHMON”. (13775) +1137) + 11378")

mates las}

ice esc Be ner aes ue he
mera)

120% 1S NO

‘WT EXERCISE: FOR SELPPRACTICN m

Find the number of eres la

wu em [m

34. 34 | tort repre treo apna CAT

WERERCISES FOR SELF PRACTICE m

(a) Findihenunber of number beten 14010259, both
included, wich ae divisible by 7

(0) Fin member of numbers beiwsen 10010200, thet
ae divisible by 3.

Fd number of umber between Mo
200 oh tr anti by 2, 3,405

Solution: Toul nembers: 101
Step ts Not divisible by 2 = Al even numbers rected: 51
aber ie 50.

Sup 2: Of whic: divisible by 3 = Ast umber 300 lot
number 389. But even numbers have already been removed,
ones coun ut nly dd number Between 300 and 400
¡divisible by 3. This gives us tat:

Fist amber 303, last number 399, common difference 6

So, remove 1399-3036] 1=17.
80-17 = 39 number et

‘We do pt ned remove sto ems or ivi

by 4 since this would eliminate only even nunbers (mich
ave steady been limited)

‘Step 3: Remove fom 38 mue fal od murs at se
ise by $ and not divisible by 3.

‘Between 30010400 the fst odd number divisible Sin
305 ad st i 395 (ins bo ends ae coun, we have
10 such numbers as: (395 - 3059/10 + 1 = 10),

However, simo ofthese 10 numbers have ares been
removed et 13) suben

Operation ef Of hese 10 number 30, 31.295, rece
ll numbers tt are ao divisible by 3. Quick peral
‘hows thatthe number sr wi 315 ad have common
een 30

Hence (Last number - Fist aumberYDiffeenee + 1]
(075-1500 123

These 3 numbers wer aleady removed fome origina!
100. Hace, fr number divisible by 5, we need en reme
only ose members that ae od dise by $ but ot by
3. There are 7 sch number between 300 ad 400.

So mans et ae: 33 7 = 26

E EXERCISES Fon SELF PRACTICE m

Find the number of numbers Between 10010 00 which are
visible by cr 23,5 and 7

SNA id nue freres in te folowing

Amal. 510315320325 30X25 4045350,

Solution. The numberof zerue depend on he number
fives and he number of wos. Here, close sen shows
thatthe number of wor ithe costa The expresion can
be orita
SUDAR SAS
IDAS EADAD MEN IE IX( XXI)

Number of 5s 12, Numer of 25-8.

Hence 8 zeroes

(DESIRE Fost emainder tor 797981910),

‘Soliton The ems forthe expression: (737981
11} willbe he same asthe reins for (72 AV]
That is, S611 = remainders |

Hindi reiner for)
Salon. AA

(999..<280tmesy8

remain for above expression remain fr.
‘ize remainder |

FREIND Fo the reminder when (228 +
ses,

1.00

Sebi ss fe me (222297 + ss
We now proceed 1 find the individual remainder of
7. Letteremanderte
‘When 2222 vide, eves remainder 03
Hnos. fo remainder purpose 222217 = C7)

EN BETT RETTET
en
BET) = (27) Remainders
Simi, ST AE «(=a =

CAN NY e
raie,

Hees, ARZT à (S527 a(S à 297 Render
o

Finde GCD andthe LCM of he numbers
128 sand 630

‘Solution’ Te snarl forms of the mue are

35. BE] pontornoota arrete rec

sont] 130] [1]

=P)
tem]

$004 HIBS Te IS 4412134441

Silay we find the umber of as

BB]

2 1657 4208-6 29.64 = 1058.

Tha, weave 1698 evens, 247 es and 397 egos
contained in 10200:

The regie ral of willbe given by the lowest of
tee Wve [Te det expec to cl why tis
apes]

Short Cut We wit look ony fore unter of Tin is
cas. Reason: 7 > 3x2. o, the number af Ts ast always be
les than the numberof 2

And 7>243, 0 the narber of 7 mus he es tha he
umber of,

Recall ihat carr we had talked about the Ending of
[powers when the divisor only had pine factor, Thre we
ad sen tt we needed to check only forthe highest power
tthe resto hat Ni there

In eases of he divisors having composite factors, we
have to he alight cael in nai the actor tat will,
reflect te esti. In he above example, we saw à case
‘where eventhough 7 was the Jones factor (in relation 108
and 9) the estiction was sil placed by 7 aber tan by 9
(as would be expected Rused on the previous process of
‘aking the highest runden).

(REED Fi tc uns digit of expression: 78°
xx 7,

Bolton. We can gt he unit iis inthe expression by
looking atthe pateras lowed by 78, 6 and 97 when they
ae rise ti

Infact. fr teas gt we Just eed to consider the mis
ii ofeach part ofthe prod.

A number ike 78) having 8 asthe units dig wil el
un digit as

was men [sis

Was was |:

Wr Wr | Hana 7a wi itd

W366 | fouras the ens git

Sindh. 5 Sich
266 each
6

Sei

mur [ror

m LES

3703 | Hence, 9 wil il a ont if.

a

en, o required units digit a given by 469 4 6
men.

EME Fs tne Go aná te LCM ofthe members
Pend Q where PR DAS P and Q=3? St

Bolton. CCD er HCF a given by the lowest poner of
the commen factors.

Thus, Ces.
[LOM given bythe highest power of factor available
Du MED

SIN à cho ts 78 gants en 675 oy
¿ten The chal vide o art boys o cy

serios. A secs in the school have the sme
ber of wadens. Gives Informa, what ae th
unbe of seston In he ach

Solution. Toe answer willbe given by the HCFof378 and
as

Max x7

CRE
Nene HO ae moi =27. 5
ene th number of sections given hy: 28
4252 3 sections E

Bay

36. Level of Difficulty (LOD) Il

1. The ist igi ofthe number obtain by mulptying
‘the numbers 82 83x 84852 8687 BEE
winbe
@o
CE

2 Thesemof de digs ofa modigi urber 10 while
‘when tb digit ae reversed, he number decreases by
4. Find the changed number
8 09 OF
or

2. When we mal a certain two-digit number by the
sum ofits digits, 45 Is achieved I you muiply the
ber wien ia revere ode ofthe se ii by the
sum ofthe digi. we get 486, Find te number.

© @7 ma

ord

wa ws
os os
(6) None of these

4. Thesumof two numbers LS dir gomme meta
is 20% lomera hir aime mean id the num-
ert,

wind CES
@ 132 ons
“26

5. Te älerenee between two number in 48 and the
lern between be are mean and ie geomet
Fi meas is two more han af of 13.01 96 Find the

mer.
wa CET
(a 02 CET
(6) None of hese

(6 HEAD Ib, fre vale of tbe amas
aural member A?
ws
(97
Le) None of these
7. ANA is is by 9 find the valve of smallest
satura amber A?
ws
(97
ws

os
ws

ms
oo

o

m

e [137

‘Wa willbe th emainde bined whea (+1) will

be divided by 82
wo M3 @r we
wi

Find the rai been the LCM and HCF of 5, 15 and
m

ws
on
Find the LCM of 52.89, 11/14,

vn O2 we

wo 0 30
om CET
(6) None of ese

be and A a ich Roi wil bee?
(6) 3A wil always be vise by 6

(©) 3A 4 vil alays be visible by 11

LU + DU aile divisible y 7

(8) Allof these

(6) None of these

A five-digit umber ic take, Sum ofthe fit for digits
Ceci the amber whe uns igi eques sum of
alle five is. Which ofthe following will not divide
this number césar?

wo» 42 04 ws
(6) None of these
‘Auber 158s diviible by 6 Which of these wil be

tive about the postive integer D?
(0) Bwilbeeven

©) Bil bea

1) B wi be divise by 6

4 Bou (e)ané(e)

(2) None ef these

Teo mmben Pa 2125 and 0= 25317! ae gen
Fiad the OCD of Pend.
(2357

CES
CEST

Find the anis digit ofthe expression 255 4 365%
ur

wa
on
iad the unis digit of the expression 55° 4 73904
za

CES
CES

wo @é ws

ws
os

oo ws ws

37. BE] toro Ppantrcsseteacntrtn cat

17. Find he uns digit te expression 1 + 127+ 1+

Was,
@! 08 07 wo
CE

18 Find the units digit of the expresion 1111251.
sia
ws CE
@7 wo
(9 None of se

19. Find he number of eres at end of 1090!
@ CE
CET om
(© None of ese

20. IFLAG is Avis by 5° then fad the maxima value
An.

ox CE
os on
(© None of these

21, Fiod the mue of divior of 1420,
ow os
eB we
(Nove of ese

22. indie CF ans LCM othe pornos 2-56)
oo.
WED
© Wa
© G-ME-DE- DH)
(dada?
(©) None tes
icons for Quins 23-25: Given wo it pine
runes Pan Q, od be numberof dio of the
King
am
(2
@s
a Po
(2
ws
rg
m2
os
25 The sides fa pentagon fel (es regula) ar 1737
(me 260 metes, 235 mes, 1422 pes and 2214

we @8 ws

we os 08

vi 00 wR

a

mets respectively. Find the pra ength ofthe tape
by which the five sides may be mesure completely?
07 60D on 09
on

1. There ne 576 boys and 448 gr ina schoo hat eo

be vied into equal sections of either boys ar grs
one. Find he total mer of tions tt formed?
& 24 DE

@ 16 on

(© None ofthese

‘A milan has the different quale of milk 403
allons ft pl: 465 aos of 2nd quay and 496
alos of dd quality. inthe es posible numberof
bodies equal size in chien milk of diferent

quais can be filed without ising?
WA 06 0% wa
CE

‘What is egreso numberof 4 igi hat when di-
vided any fermes 9, 1,17 aves a remain.
derof 1?

wm wo
os co 9087
wo

Fin te least number hat when divide 16, 18 and
20 eaves a remainder din each case bots completely

‘visible by 7.
CES was
(o 2964 CES
CE

Four ing at als 68, [Zand 18 seconds
‘Tey sat ringing tops at 12°0 clock ler how
many seconds will they ing tpeier again?

on ou OO was
CET

For quesion 31 find how many tines wil hey sing
together ring the nent 12 lasts? incluia the 12
‘inte mu)

os

on

(0 None of these
‘The unis dit of be expresion 1254 x 552" x
ni

on
on

wa CE
oo CE
(©) None of se

38. 34. Which of he following is not a perfect square?
DIT © 325137

(9 94572 Allo hese

le) None ofthese

35. Waich of the following can nove be in the ending of
perfect square?

“os 00 ~om wi

wo
34 The LCM 5, 812,20 will sobe a maple of
DE we
© @s
(6) None ofthese
37. Fit manber of vis. 729 (cating Hand 720)
O23 03 03 0»
on
SR Te LCM Of (16 2) and + 2-6) ie
(@ DAA
© 418-2) (043)
© A
Cr)
(e) Nöne of tes
Dort sand de x 68
@ 242 @ 2-2
@ 2-2 @ 242
(@) None of these
40. ThemmberA isnot dive 3. Which te follow
ing wl tbe divisible by 3?

) 9x4 @ 2x4
© (2a

10 None of these

41. ind the reminder when the member 9! y divided
CT]

WI 2 00 we
@2

2. Find he mainder of 2 when vied by 3?
@! 02 @4 @6
oo

23. Desompos te umber 20 ito two tens such ht
their roduc sd ges.

Wan | yEsy
Omnia Men
(© Nove of se

4. Fina te suber of sere at heen Sat
on» OH 05 wR
on

15
2

COTES IE

45. Which of the following can be amb iii by 24?

CET 0) 2541284
© RRs @ Al of de
(6) None ofthese

46. Fora mumber to be divisible by 88, e shouldbe
(@) Divisle by 2 aná
0) Divisbieby 11 and
(6) Divisible by 11 and hice by 2
(0 Both Wand)

(6 All of these
42. Find the number of visor of 10800.
os oo
on os

(9 None ofthese

48. Find he GCD ofthe polynomials (c+ IM 17
and (e+ + D+ 4)
@ AND
0) (e+ 3-2) er erh
CRETE
@ we hess?
(6) None ofthese

29. Pod de LM Se
[ver
WR
CANNES)
© DE
(9 NE HGs44)
€ None of these

0. The prod of thee consecutive natural numbers, the
fist of which isan even number, is alvays divisible by

on os
os © Alber tase
€ None of these

SL. Some birds sealed on he branches ofa ee. Fist hey
sat one oa Branch and thee was cae bird oo many.
[Nest they sat a branch ad thre was oe branch
100 many. How many branches wee there’?
WI Mf OF 06
CE

52. The square of» number grater dan 1000 at is nt
visible by tec, wben divided by thre, leaves a
reaainderof

(0 always 0) 2atw
oo @ Sher 12
16) Cannot be said

39. n.

ws ws
os wr

Inthe famous el Air Amen in Ranchi, hee are
three watchmen meant o protect he precious fat in
the campus. However, one day a tit got ia without
being noticed ud sol some precios mangoes. On the
vay out homever, be was cononted by the he
Wachen, theft to of whom asked hie topar with
113% fie mit ad one mor. The las std him 10
art with 15° ofthe mangos and 4 moe. As result

oe
(0 None ofthese

A ende and twenty dig number i formed by writ
{ng the st x mural numbers for ofeach the as
12343678910111213.. Find te reminder when is
umber is vided by 8

ws m7

(02 wo
‘Atesthas 80 questions. There isone mak fora corset
answer, while hee isa negativo penalty of -12 fora
wrong anever and 1 for an uraterped question.
What she munber of questions answered correct, if
the suet has scored a nt ttl of M.S mar.
ws Dr

os 19) Cannot be determined
For ie question 72, ft is kon that hes ef 10
questions unanswerd, Ih number af ect answers
was œ 4

CE (9 Cama be determined
“Thee mangocs four guavs and five walemeiom cost
Rs, 750. Ten watermelons ix mangoes 2069 guavas
¿os Ra 1580. What ls Ue cost of ax mangos, Lem
‘watermelons aná 4 guavas?

@ 1280, © 10

(o 1080 (0 Cannot be determined
‘From number Msberac |, Te the reciproca ofthe
result gt De vale of. Then which the foo
ing i necessary tae?

(a sete? wars

© es (0 lets

“Te cos o our mangoes, six guavas and sateen wa
temelons is Rs, $00, wile he con of seven mages,
to uavas ed lgteenwateeloes is Rs. 620. What

Car meen [AT

in the con of ene mango, one gueva and one water
‘cio?

@ 2 oo
os (4) Cannot be determine
For the question above, what i the con of a mango?
mE) y

os (0) Camot bo determino

The following is known about tye el number, x.y
ante

-45 154,-85)520m0-85:52 Thentheraege
of values hat Mx) can ta i bet represented by
Wire (o -I6sx8

© 85158 CPP

A man el 38 pieces of clin (combined in oes
of hrs, trouser and is) he soldat ent 1 pese
ofeach cn and be old more ait han rowen and
mor rowers than tie, then he number of tis at he
must ave sod is:

(a) Rusty © Atleas 11

@) Atlest 12 (4) Cannot be determined
For the question 79 find the number of shire must
ave sold?

(0 Atleast 13 © Atleast 14

(©) Atleast 15 @ Amos 16.

ind he lest number which when vid by 12,1,
er 20 aves in esch cate remainder 4

@ ©

om (@) None of these
hat is the feast uber by which 2300 should be
mul so that ch rodact may bea pete square?

CE OT
ow (0 None of hase

“Te es unbe of digit which a perfect aque i
wu oo

om @ 02

“The leat mall of 7 which leaves a eniade of 4
when divided by 6,9, 15 nd 186

ws m

CET on

‘Win isthe est 3 digit ruber tha when vided by
2.3.4, 5 0 6 leavesaremaindrof 1?

ou @ 161

om GO None ofthese

The highest common factor of 0 and 245 is equa 10
os os

os os

40. CE © 02
CE own
© Bom

10. ya umber that is divide by ob where xy < ab and
ies a esl Oy then ab equal
On DS OB We
CE]
Amber mpi by ance number bande
result comes as par where ra 29, 932 y)and p=
Ze here x,y €, 920, The value of ab may be

wu os
on CE
(6) None of these

12. Le denotes the pestes inter vla jus below and
A its factions vale The sam off)? and 2]?

-291-Fiads,
Der a
(9 297 (9 17
(None of these

E 16542 diva by
wa OB on ws

on
Is. ITAB+XY =IXP, where #0 and alles ig
fret digi rom 0109. hen the valve o As
on OT
© os
16) Any value above 6

Divecions for Questions number 15-16: Find the posible

imepral values of x
CRETE meer

mt »

ME] (02

(e) There are many seutions
CETTE

on CA

oo @ 10

LE

17 11409 amd are vice by S, ring am even
integer, nd the leas vale of x

wı m2
© 3 wo
(6) None of these

che: here [43

the sum of he numbers (225) and eis vs by
9, hea which ofthe fllowing may be a vale fora
CE m7

(09 Os

(6) There is 00 value
MLx=41+1y~41=4, hen how many integer values
an the et have?

(a) Tato

os

CET

11501 gives remainder and {. denotes ie fractional
pat of The fracc partis fh er (be) The
alos of x ou be

ws
CE

@2 ws
os os
(6) None of thee

“The sumoftwo numbers 20 and thse geomeric meat
520% lower than their attic mean lod aio
ofthe punters

wa
as
sn

“Tos highest power on 990 ar wil xy vie 10001

CE
@ 0:3

om om QI 10
@ 10

1 146i visi by ten inthe maximum ra
ern.

WA on 0% WIS
on

"Telas gis in the maltpleatienof 38-339
32.31.3020 28.27 26

om © 4

(3 @ 10

(o) Nowe of these

‘The cxpresion 335" + 555" is divisible by

to 2 @ 3
os on
(Allo ese

Tel dentes be greats integer value just below and
da) racional valo. Te im [sand 125.16
Finds,

CET © e

€) Bosh (ayand (b) (0) 4H

(@) Cannot be deemined

41. 44 | Hormonen ter

7 Ie dé the square ofthe digit in the ens place ofa
potitve rod umber to the product of th dig
‘ofthat numer, we shall ge 2, and if we athe squat
‘ofthe digit inh is plac to he same prac of he
dis, we sal get 117. Find the wo digi number
OB 0» O8 we
on

2 Find mamen sach tat their sah rout and
the diferences of thei squares are qual

=.
|
ae
(9
(ls)
ee

zul

(0 Alto
(e) None ofthese

3. The sum of he dighs ofa teo gt aber ls 17, 25d
the sum ofthe soars fs gis 109.1 we abet
495 rom ht rer, we sal get a aber using
ofthe same digits write in ho reverso order. Fad the
aber

am vo
© 68 ws
oo

30. Fite amber of ero in the produce 1852 x 3°
ET
(o 1200 (o 1300
o 1050 @ ins
(0) None ofthese

1. Find the pairs of a aural number whose greatest
commen divs is and he least comsnan multiple
sus,

Sand 105 or15and3s
(©) 6 and 105 or Wand 35

(9 Sand 15015 and 135
(@) Sand Wer 18 and 35

(6) None of dese

‘The denominator of an ieducibe faction is greater
than the numeraor by 2 I we Fede the mern of
the reciprocal fraction by 3 and subir the given fac
oa fom the resuling cre, we get 115. id he given
fraction,

2 a s 2
et m2 03 2

A two mumber exceeds by 19 he sum of the
gares ofits iis and by 4 the double predict of ts
gis. Find the rade.

an De 02 wR
os

‘The sum of te squares of he ii conti a 0.
igi positive number 25 tines as re sto sum of
‘a igi and is larger by ay than the bled product
ofits digits. Find the number,

@ Banta 0) Lada
CEE (@) Wanda
16) None ofthese

‘The units digit ofa no ii umber is greater han is
‘ens igi by 2, amd the product of hat mumber bythe
sam fs digi is 144. Find te member

WU DA 06 08

(a 20

Find the number free inthe product 5 x 10x25 x
40% 50355 65 125380

wos 09 on wD
on

1. The power 5 tat will exactly divide 123 is

a2 OY ON OB
oD

‘Tee numbers ae such that the second is as much
lesser than the tint as the frst lever than the
second the product ofthe two smaller numbers 85
andthe product of ro larger mamber is 115 find the
ride under

W9 DS @R ww
on

Find the walls url name such hat ivi
ibleby 990.

DE CE

42. ou wr
(@) None of these

40. Le fe Jay is te only when
@ >0y>0 WM x>0andy<0
(o r<0ady>0 — (8) Allofthee
(0) None of these

Directions fr Questions 41-60: Read ve instructions be-

low and solve the questions based on this

lo an examination stato, always solve the following
‘ype of questions by sobminsing the given options, to ave
a te solution.

However, ou cn se, there ar 0 options vena the
questions here sine these are mean o be an exercise
genen writing (which cire sa primary skill required
do well in split exam teng mathematical apte).
Indeed, hese questions had options fr em hey wot
be at as LOD 1 question. But ine the po bed
lutin technique is removed hee. Cave placed thes |
{Me LOD 2 exe.

4, Find the two-digit mumter that meets the following
cite. I the amber in the ua place excods, the
rmber ne tens by 2 and the product ofthe requir
aber with the sun of gis is equal o 148,

© Theprodhctof the dgisetatwordigtmumberietwice
ax age athe um ofits digits. we subtract 27 rom
the required mbr, we get number cosising of the
sare digs writen inte reves order. inthe uber

AN The proéuet fie digits of two-digit numbers one
hit number we ad 18 tothe required number,
we get number coming ofthe same digits writer
ln de revere order, Find the number?

AA The sum of the squares of the digs of a two-digit
numbers 1. i we rberastO fom hat umber, ve get
ume consisting of Ue same il writen in the
severe one. Find the number?

4S. A two-digit number rico as largo as the sum of ts
‘igs ard the square of tha sur Is equal oe bled
eid uber. Fd he wunder?

46. Find a rue ig number hat exceeds by 12 he sum of
the squares fs its and by 16 the doubled product
of is digits.

©. The samof te squses ofthe digit coming two
ig mamber is 10, and the product of the requires
number by the number consisting ofthe same digits
writen in the reverse oes 405, ind the 2 numbers
that susy tes conditions?

«

sı

a

ut hotes TE

wo divide voit member bem of ts ii,
wege ds queen and 3 ss aremainde Now, if we
‘divide tht two-digit amber y the radar of it,
‘wo get a quent and Sasaremeinder Fido:
gi numer
“Thee i a natural umber that becomes equal tothe
square of aratrl saber when 10D is added ot and
to he aquer ef another natura number when 169 is
added 10h. Find he number?
Find wo seal numbers whose sum is 85 and whose
lees common mail 102
Find wovtre dig numbers whose sum ia aile
of S08 andthe quen is a maple of 6.
‘Te difereae berne the digits ina mo gi oumber
‘i goal o 2, std the sum ofthe squares ofthe same
is is $2 Find ll he posible number?
ve vide a given two-digit number bythe product of
iii, we obtain 3 asa quiet and asa remainder
we surat be prodect of the igs certuin the
umber, from te square of te sum of digs, we
‘btn the given mami, Find the mame,
nd the eig number if itis known that ie sum.
ofits digs is 17 andthe sum ofthe suas oF ts digits
is 109, ne ber 495 rom hs number, we ob
number consisting of te sane digits weiten a revene
onder,
‘The sum ofthe cubes ofthe digits cost a to
ii number is 248 and the product ofthe sum of is
igs bythe act fis digit is 162. Find he ew
ro igtmamber?
‘Tre free between two meri 16. Whatcaa be
ssi abut the ttl narber divisible by tht can ie
in between these two numbers.
Arrange he fllowing in descending order:

sy 110108208107, 10n.10112113,
35x Sand y 27. nd the genes valve of ay
ad the lest vale of 29:
Which ofthese greater:
() 200° 0 3002” or 400!
0) 5 aoû 27
(©) 10 anda
‘Toe sum of the two numbers Is qua o 15 and thee
are means 25 percent pete an its promet
‘mean, Fi the ruben
Defines number K sch that itis the mum of he wares
of heft Maul nurberie.Km 1 + 24.0)

43. EEE

bere MS. How many values of Mena ch Oat K

in divisible by 47
on o
on (0 Nove ofthese

(© Mis atmo digit number which has the propery that
"Te producto tical of dit > umo trae

ofits digs

How many ves of M exis?

ws wo

© 8 (a) None of these

GR Anawal number when increased by S0% has name
ber of factor unchanged. However, when th vale of
be numbers reduced by 75%, the number of aos is
reduced by 6.6%. Ose such number could be:

CE o
(9 126 (9) None ofthese

(4, Find he 28989" em ofthe seres: 1245679101112.
ws @ 4
os (67

6 Uf you form a subset of sieges chosen fom beincen
110 300, such that no two integer a up toa ple
Of ire what can be he maximum number of elements

fn the sib.
ww om
© 23 wi

Theses of amber IRA u)
As taken. Now evo numbers are ae fom his seres
(the fist two) say x,y. The te operation 2 +34 xy
is performed to gi aconscidtsd number. The process
As repas. What willbe the vale ofthe se afer al be
numbers are consolidated into one number.
wm om
om (9 None ofthese

7, Kin a thre digit amber such that he ato of the
urbe to the sum ofits git sas? What isthe
‘ference been the hundred ad the er digit of
ra
ws os
(97 None ofthese

(nthe question 67, what can be said about the ie
‘ence been te fens andthe units git?
wo wt
(2 (8) None of these

(@. Forte above question, for bow many vales of X will
the rato be he highest?
ws os
@7 8) None of these

AR A wiangular number i defined aca umber which has
the propery of being expressed a stm of conse
tive mul numbers waring with How many wang
ar mombers ls tha 1000 have the propery tha they
ae ie difference of ques of to comacutve natural
numbers?

a» CE
(o 2 (03

Ti. xand,y ae wo posive integers. Then what lb e
‘sum ofthe coefficients ofthe expansion ofthe expres-
son + Answer 2!

mE mae
CEA CE

72. What isthe remainder when 9 +9498 + 5
divided by 6?
on @2

CE ws
TA The remainder when the somber 123486789101112
AS cided by 16
DE) wi
os os
TA, Whats the highest power of available in be expres
sion SH 38!
on on
©» (4) Nove ofthese
75. Find the reminder when the number represented by
225M raed othe power (I + 2 .+ 6s divided

bys
w2 wm.
@ 0 (9) None ofthese

7%. What he ttl numberof visors the amber 12°
x32?
wa CET
(o 24 (8) None ofthese

TR. Fer tbe question 76, which ofthe Following wll repre
‘ent the sum of factors ofthe number (ach that only
(dd factors ae counted)?

o 3

A)

w 5

y 2

ofS (0) None ofthese
TA What isthe remainder when (1 2? +99)? 4
(SP divided by 11527
on vw
CE @ 2
TA st Sis formed y including some ofthe fit One
‘ound ral numbers 5 cousins the maximum.

44. E | Hort Parla ignorar

questions in thie levels of difculg—LODL.
1003.

‘The following table gives ih deals ofthe postive and
nepative marks atched to ech question type:

Lov? and

“Dien eed Ponte mars Jor Nesave mat or

wen the “raver the
‘cern cory gran wrongly

ror 3 2

LoD2 3 1

Lop 3 2 1

“Te est ad 200 quesons wit Mon LOD 1 and 60 ash on

LOD 2ansLOD3,

A a sadent bas solved 100 questions exacly and
scored 120 mars, be maximum number of incorrect
questions that bei might have marked is
o. ©) %

oo (4) Nowe of these

1 Ami temp he east uber of questions and gt
tol of ras ef tic ran that he ater
8 st one of every type the the member of questions

he mast have atemptd I
wa os
©» (@) Nove of these

9 In the above question, whats the least number of
‘questions he might have got incor?
wo @ 1
@2 (0 None ofthese

100. Arab has cenit mumber of toffee, such that ithe
“states them anceps en children he has ine et,
‘fhe into amangt child he would have lef,
fhe dites amongs eile he woul ave 7 eft

‘ed 10 0 unt if he distrib amongs 5 hiere

e should have 4 Tell. What ds Ub second highest
umber oftafees e cold have with hit?
ET y
om (0 None of these

Level of Difficulty (LOD) m

1. What ii mumber is les han the sum of the
square of is digs by II and exceeds their doubled
product by 5?

(15.93 CE

(9 Bosh (a and)
(6) Nove of hese
Find the lower ofthe two success aural numbers
‘ifthe square ofthe sum of hose numbers exceeds the
sum of their square by 112.

@6 oT @8 ws
on

Fin we Increase ih denominator ot a posiive fac»
tion by 3 and then we decresed iby 5, The um ofthe
‘esting fraction proves 1 be equal to 23. Find the
‘enorinator of the ran fs nameratrs 2.
@7 05 on 09
on

Fd he st og 153.37 63 197 17,
OS O8 OS 08
os

Lets consider ction whose denominators smaller
tan the square ofthe numerator by ul we add 2
tothe numerar ad the denominar, e fica will
exce 13 IF we soba 3 from the numerator ande
“denomino, the Faction will be pose bt smaller
than 1/10, Fed the vals?

(0 15,95 and 12545

3 4
w vi

6
of ws
0 None of these

Find the sum of al rei t numbers that give à
reminder of when they are divided by.

one © m0
© 1200 (9) 90210
(© None of these

Fine sem fall ten dpi nombra give emir»
er of 3 when they are divide by 7.

ww &
© es as

(6) Nove of hese

Find the sum of all odd tree digit murbers hat ae
ive by 5

(9 0500 mn

o same @ 950

(5120

"Tas produtofatmo-digh mer by number comic
ing ofthe same digits writen in the reverse order is
‘halo 2430. Find the lower number.

wos os 00 Ws
os

45. ED | torrente yours

om os
(6 Bodh and (>

2 A tee peste Integer abe is sch tht! + 8
+207 ais ul tothe double sam ofthe digit in
be tensand unis places Find be nanber iskaowe
ste difference betwen at number andthe uber
‘writen by the same digits nthe reves oder 495,

@ 33 CE
oo on
(6) None ofthese

DA Repeset the number 1.25 2 a product of tree pos
tive factors so thatthe product f the fist factor bythe
square ofthe second is equal to Si we ave ge the
ouest possible sum of be three factors.

(0 5y=228.n5 85,2002

© asi. =40=45

© x= 125,220

(@ n=1 Re
(e) None of tse

25. Find uber such tha the ur of that umber and
its uae isthe least.
wo mas
CE

24 When 2223" + $555" ide y, th remainder
is
wo M2 we ws

Ti. lisa numberof dis which whee divided by 8,
12 15 and 20 leaves respectively 5, 9, 12 un IT as
rinda then nd x such ha isthe lowest such
umber?

(a) 10017 @) 10057 (6) 10097 (0) 10137
16) None ofthese

2 Pe 1 is divisible by 24 or

os wis

wı or
(03 ws
© None ofthese

29 100 (54 VF is divise by 2° for what whole

bec value of
@2 03 07 we
(e) None of hese

BEER eater emi
vor @1 we

31. Find the remainder thal the number 1989-1950 1992°

ives when vid by 7
wo ml @s 02
@2

32 Find the remainder of 2% when divided by 3

W3 mo @l @2
(©) None of these

A Find the remainder when the member 3 is divided
wr

WI 05 ©6 ws
CE

JA Find he Ist digit of dhe acter 1° 42° 4.4 9%
mo wl 02 ws
CE

36. Find god (21,209 1,

@ 2-1 @ 2-1
la Pa CES
or
3% Find the ged (MI... Bonded ones UL... sity
ones)

(9) IL..fonyonss (8) 11 twenty iveco
(© WL tweety ones (8) LIL say ones
(€) None of thane

37. Fite the lst digit of the number 1+ 21 3° +47... +997.
wo 1 or ws
os

3 Fd te GCD of te ers +13 adn +.
ot ®2 @3 ws
os

ya
7
ms 62 @1 ws
os

(AL The remainder when 10! + 10° + 10" 4,4
gf ive by Ti
wo @1 @2 ws
os

A. nl a number, such at Zu as 28 factors and Ja has

CE

(0) Nove ofthese

42. Suppose the product of n consccive integers is 2
G+ C+ DIE N. (09 (n= 1) = 1000. en wich

46. of the following cannot be tus about the number of
1. The mme of ers canbe 16
2 The number of tra can be S
3 Tre umber of terms canbe 25
A The number of terms can be 20

AR The remainder when 226-22" « 2207 6 2002" 6
222.29 won] vide D:
w2 os
as (97

4. N 202 anon x 200000002» 20000000000000002
200000000... (3 ere) The sam of gts inthis

rica wile:
@ 12 wm
om (a) Cannot be determines

8, Tuemy Ave se of problems on Das Irc
‘one each for the DI sections of 25 CATALYST tex
ere prepare by the AMS research eam. The DI ec.
tion of cach CATALYST contained $0 questions of
which exactly 35 questions were unique, Le, ey had
o ben usd inthe DI section of any ol te or 24
CATALYST. What could be the maximum ponsble
‘amber of questions prepare For the D sons ofa
the 25 CATALYST pu together?

Par} wo
om ws

44 ln ho above question, what could be the minimum
possible number of questions prepared?

@ 80 ME
on (0 None of these

Directions for Queront 11-13: A paticla time nthe

‘twenty fit cet there were seven bowlers inthe Indian

riet tea is of 16 players short sed pla the neat

‘orld cp, Sails discovered tha hat you Tooke a

‘he number of wickets taken by ay ofthe 7 bowler ofthe

care dia cricket ea, be marber of wickets ken by

Aer had a strange propery. The numbers were such har

any team selection o 1 players having 1107 bowlers) by

sing be number of wickets lake by each Bowler a a

tachng coeficietsof 4.0, or each vals avale and

“dig the estat values, any number rom 101093, both

Include could be formed. we dente WWW. Wa M.

W, nd W, ath ales inthe atcending order what could

be the answer to the following guesins

1. Find the valu of W +2) + 3Wy = AW 4 y=,
@ 2 (by 1935
om (0 None ofthese

Ou: times [SA

“Fld he inde ofthe gen omer of 3 contained inthe
product, WA, Wa WW,
ws on
oa ws

8. Ifthe cum of seven coefciens iO, find the mal
umber that cu be obtained.

lo 1067 739
CET ®
Directions for Questions 50 and SI:

sions the beis of he information given Below
Inte ancient game of Horololo the sk involves solving
puzzle asked by the chief of he ibe. Angy answering
the pare comet i given the baad ofthe mos beautiful
maiden ofthe ie. Unfortnaely. othe youth ofthe tte,
solving the puzzle snot a cakewalk since te che isthe
gres mathematician ofthe tbe.
In one such competion the chief elle everyone wat
tention and announced opel
A Credit mamber mp isa perfect square and the
umber of factor thas is ls a pere square. Ks alo
Foot git on and ara ininet Now antwer.
my questions and wi the man's und.
50. Mo np isso peo que, whats the nanber
‘of factors of the snip number prey?
CE on
os (8) Cannot be termine
51 Ifthe far poner of he product f the digits of the
umber nap isnot divisible by, what the umber of
factor ofthe nine digi umber, mapmnpry?
@ 2 on
os 1) Cannot be determined
82. Inacrckettouramest organise he ICC, a total of
15 teams paricipaed. Australia, as usual won he tour
amet by Scoring the maximo mue of prs. The
oumamen is organised as single round robin tour
rent her each tes playa wi very ter tum
exact coe. pois ae avs fora win 2 pins are
add fora efvashed oat mach and 1 point is
nando for losa Zimbawe had he lowest sore in
cms of pois) ue cal of wurnanenı Zimbabwe
seoreda total 2! pois. Alle LS national tam por
à sine score Gin terms of pis scored. is lso
Lown that at least ne mach played by the Ansan
team was Gedhvashed out. Which of he follwing I
alas re for the Australian ea?
(0) (had al Jess u tease
(©) Had a maximem of ose.

47. 52 | Norrie Quand

(©) Rnstamauinun of 9 wins
@ Año Ue sone

$3, What iste reader when 1280 rie y 158
oo Os om Oe

54, Find the remainder when 80" s divided by

os OF or 03
SS. Find the remainder when 32°” is divido by
ws OF OM wt
56. Find the remainder when 307 is divided by
@s 9 66 Wi
$7. Find the remairder when SO” ls ide by 1
@7 0509 au
8. Find the reminder when 39% ie vide by 7
@s as 06 m2

39 Lt Sy dente the sum of the squares of the Fist m
‘url numbers. For how many vales fm < 100,55,
amakipleofs?
won OS 0% wx

Ga. For he above question, for how man valss wil the
sum of aber ofthe tortura members a male

of Site 509?

oD wa

(o 2 D None ofthese

How many integer valves fx and any the expres

sion tr Ty 3 where < 1000 sey < 1000,

@ DEI

(o 26 (© None ofthese

Hints and Solutions

L Ha and are two numbers en hir AM = (a + BV
2 ond GMs (abP3. Use the options to answer the
‘question.

2. Use ie principal of coantng given nthe theory of
‘he cher Sta with 101 numbers and rc al he
ubers which ar divisible with 2,3 and 5. Ease
‘hat there ls no double counting In this process.

Fa the nis digits individually and sobe.

3 (109-7) ~(10™+ 3)" 1010-1) 209 fortis
expression to be divisible by 3, the vale of as o
be.

6 Solve using opis.

7. Solve using epions

12%! 2349 4 as un ple.

Similar, Sue te 2° — 6 athe unis place.
Hence, O is the answer.
9. Usethe formula forthe som of aGR, witha 1 andr
2

10. This is property ofthe number 99.

11. The vate of hast be 2 since, r= 2s, Hence, option
isthe nly choice.

12. Solve tough opos 10 ge integra vale and rc
‘onal vale, Remerbertha7.9 as an gr ae
A,

13. 1654250 ED) Hence ie
CES

14. For A + X be X te only pomibl situation i that
he vale of a should be either 0 o 9.

15816. Une the method of physical oui fall possible
vales of

17. Use opions and ty to fin the values one by one.

18. Check tenga opis

19. The to modus values have 1 al wp 0 together
“This an happen by (09 4). (1+3),24248+ Dor
(460). Find the inv ale fx and fr wich
these set of vales are sated.

22 Thelasdigtof 3s 1. Abo, any umber having he
units digit as 1 wil give 2 at the Las place i he
quotient.

2, Use standard fornla (or Am and GM.

25. Tis isa propeny ofthe number 37. Besides, the
tone +o = ven Hee it willbe wie
by2 Youd’ t need ocak foo mark the answer
pro

26.38 Solve through open.

36. The nunber of zeroes depends onthe numberof 5's
andthe number of 7%, whichever i less. Her, be
consi isthe omber of 2s ae not J's the sual
ea,

31. Check the powers of 5 and 3 contained in 123 The
lower vale amongst these vil be the ansner.

38. Selve though options.

39. Find the largest prime factor contained in 990 and
check fcr vale for iii by 90.

2. Check for diferent pesiv and negative values of «
sed according to he options,

56 Wie simple equations for each ofthe questions and
solve.

51. 110.100.108.107 is obviously smaller han 109.110
1213 anda In“

Men, ie lowes. Also, 11> 109.110.112.113

propery of squares)

48. SE Both andy shouldbe highest foray tobe maxims.
Sims x bool be minima andy shoal be ma
‘mum for te minimım.

9. Resolve 10 à coamon base for comparing.

Hints and Solutions

1-5. Solve trough options

‘6 Use AP with fist ter 104 and It tere 949 and
common diferen,

7. Fledhe fs 2 digit mmber which gives a remainder
‘of 3 when cvied by 7 and then ind the largest sch
umber iad 94 respectively). Use Arithmetic Pro
session formule to ad the numbers.

& Use AP ih At term 105 and last tr 998 and
common differences 10.

10, Thea ofthe numbers sres—3,x+ Zend 43 Use
plo and you wil se that (a) i the answer.

1. P=) 48 Le. 9) 45. The facto 45
posible ae, 15, 3:9, 3nd a5,
ence, the nombre ae 9 an 6, 07 and 20 23 and
2

12-18 Use options 1 check the given conditions.

19. Theme willbe Since, 1256122 wl give Oase
last D digs.

22 The remainder theorem isto be wo.

21, Theat digit will be 1 X 5 = 5 (on cary over) The
sens digi willbe (4°1 +522) 4 (any over). The
nés gi vil be (391 +22 + 3°1) m6 (eared
over) 27 Hence, answers 745,

2-38 Une option to solve

25. Use the nue of indices and remainder theorem.

7. Options are not provide as is an LOD 3 question,
IW they were there you should have used options.

28 Use wil and roe

2. Use options.

30. Use remainder theorem and lock pate by app
ing he rules ef indices.

We get the value as:
mm... Rune
sul... Tine,

JR. times

PN... Wim
HR. Sofie,

apr: torera [53
232. time
122 ine,

> Looking ste pate we wil get a he Fil
remand

31. Use the remainder theorem and et the remainder as:
CLAN IMAN = 124 = 2istheremainder 32

2 22041,

22. Use the remainder theorem and ty finding the pat

34. Finduhetas git of e number pot adding 1° 2°
+9 (ou will ghee). Then maki by 1010 et
eo as the amet

8. Use reader theorem and lok a patersby ey
{ng he rules of indices.

‘Thought Processes, Solutions and
Short Cuts to Select Questions

top

2 The two numbers shouldbe actors of 40. A factor
search will yield the factors lok only fo 2 digit
facto of 48 with sum of dis Between 1101).
Aso 40S #53, Hence: 1827

45 x9 are the only tuo options
roe these actor par only the second par gives us
the desired rel.
is, Number um of digits = 40,
Hence the anne is 45.

26. The de ofthe pentagenteing 142, 1737, 2160, 214
and 2358, the est ference beracen any two num-
ess 1. Hence, the core answer will be à ator
of 54
Farther, since there ar some od numbers inthe Hist
the answer shoold be ano factor of $4.

Men, check ith 27,9 aa nn re Yo wil pe
Das de ICR

SL. When the bind sa one on a branch, there was one
sta bird, When hey sat 20a barch one bart was
To find he numberof branches, go through ptons.
Checking option (a)
he were 3 branches, thee would be isis
os eave one bind win branch as per the ques:
tion)

49. Cugat: Neusten [55

ANSWER KEY
Number Sytem LOD
La EN PET
En EN rn
he += ners
La
Sa au HCF + 30
m ae -
1 [e
m) aa
st aa
ne =
Me ae
me —
ne ae
we 3
= EM
me EE
ay En
me she
De Sa
EN =
EN oa
EN
En:
ed
oa
©

3

»
E
fi
é
»

2
2
EJ
E
2
D.
E
=
El
En
El
EX
El
=

2

Je ls[sls ls le [ala lalalelalsl=la ls la [ala lolo [ola [=[=]8 [ala [a

50. Number System LOD I

CS

57

EN

EN

EN

me

ma

EN

Er

aa

ae

aa

51. PROGRESS

IONS

‘The chape on progress essentially yields common:
sense based questions In examinations

‘Questions in the CAT and ater apie exams mostly
appear fom ether Arithmetic Progressions (more common)
oF from Geometric Progressions

“The chapter of progeision i logical end natal en.
tension ofthe chapter on Number Systems, since there is
such alo of commonality of lg between the problems
sociated with these two chapter. Ac already tated lock
1 ofthe sl lock of Chapter in QA account for anything
between 12-18 maka ia the CAT. This pli has been
coontenly observed ver the pt decade,

ARITHMETIC PROGRESSIONS

Quant are sid 10 be in arithmetic progresion when
they increase or decrease by a common difference
‘Thos each ofthe following series forms an arithmetic
progreso:
THIS.
#2 4-10,
aa+da+ a+ M.
The common diferen is found by subreting any term
ofthe seres fom the net tem.
‘That ix common diferente of an AP = (y~ fy
In te fist of the above examples the common dir
ence iin the secon ii 6, in the Whe ii
I we examine series aa +d + 2d, à + 34,
we notice that any em the cogficen of is aways ess
by one than the pasion of tha tern he series
‘Ths the rh ter of an amet progression is given
by Tease

IE be the number of ems, and if L denotes the las
term or the ma ter, we have

Tear anti

To Find the Sum of the given Number of
‘Terms in an Arithmetic Progression

Le à dente the ist ter the common difference, and
‘the total numberof tems A, let denote the ast ea,
and She required sum: then

ee, w
Laas (01M a
seh Ban Dil o

any vo fran artnet progression b given
the eres can be completely determined; fr ths dita re
salts in two simultaneous equation, he solution of which
will give the fist tem and the common irene.
‘When thee quantities are in arithmetic regression. he
mie oe to be the arithmetic mean ofthe deriva.
‘Ths isthe armenio mean between ad and a + de
So, be ci reqied to arbitral consider thee numbers
in AP ake a=, a and a + das ihe tree number his
‘edaces one unknown thereby making the slution easier

To Find the Arithmetic Mean between any
Two Given Quantities

La e ad b be two quantiles and À be cr ariete
mean. Then sine 0. À ae In AP. We must have

52. PETER
ach being equa ie common difference:
Tin gious [4e SAP

7

Berneen two given quantities I de always posible 10
insert any numberof terms such thatthe whale seis thus
Ford sal bein AP The ers dus ase ar called the
arthmete means

To Insert a given Number of Arithmetic
Means between Two Given Quantities

Le aan be te given quantities and m be the munber of
Inching the extremes, the number of tems wll the be
1 42:50 that we have fo finda src of 4 2 tems in AP.
of whi ais Ue fest, and i he Les em.
Las d be the common difference;

den OUTRE em
satay ii

60
wen

Hesse,

and the required moans are

INT
TW ow ve have id Ain ci mate con
text is wa po fo Jo o enana the se
tara iat of AP Homer, oe
nd m on AR e rl nv on a
‘mera pd in ps ae
Jon ae ely oe una de CAT sad or apie
‘Sto In sch tin ember! pc
‘chet popes bcd suri: Maly 1 al
ensuring lowing aga mp on
‘mete Popes iy o hp 0 wie qe.
ne bcd on APs comi ef ope xa
Tet wk he oss by ome
Proc fr Mag the wh ter wh A
Seppe ys fave oid Fo of
pene
Treen aha pom is quo
Se ive gd oda
Paar
Th ete tm ne we do
Fyn de) Dedede 164067

ou guns [BQ

Most students wos mechanically insert the values For a
mad d and get this answer.

However. if you replace the abone process with a
sought algci, you will get ih answer much Fase
‘The algorithm goes like this:

order wo fr the 17h er of the above sequence ad
the common différence to the fist er, salen times.
‘ote: Sixteen, since Ht ee less lan 17).

il order fe the 375 erm of the AR 3,11
All you nce 1 doa the comen eifeeace (Bin this
ase), 36 tines,

‘Tas, the answer 285 +3 = 29
‘ote: You ultimate end up dong tbe same thing bat you
ate at an advange since the cie solution press I
restons)

2. Average fan ALP. and Corresponding terms ofthe
AP:

(Consher he AP, 2, 6,10, 14, 18,22. you uy to ind he
average of be at numbers you will got: Average = 2
6410 184 184 206 12

Novice that 12 lalo he average ofthe fit and the last
terms of he AP. ln fc, its so the average of 6 ud 18
ich correspond oth second and th terms ofthe AP).
Furs, 12s aso the average of the Ju and th ems of
de AR

(Note In this AP of sn tems, the average wes the same
she average ofthe Ist and 6th ems. T was lso given by
the average ofthe And and the Si ems, as well that of
the Sd and th term.)

We cun call cac ofthese pain as "CORRESPONDING
TERMS in ae AP

‘What you need Lo understand ls tha every AP has a
average.
And for any ALP, the average a any par of correspond-
ing terms wi also be the average of the AP.
you ty 1 nie the sum ofthe tem numbers of the pair
‘of comesponding tms given above:

stand ih (so that 1 + 6 = 7)

And and Shen, 24 5=7)

Sed and th hence, 3 + 47)
Note thot: In each of these ees, the sum of the
numbers fo the ers in a coresponing paris ne greater
‘han the number of terme of the AP.

‘This mie will old te for all APS.

For example an AP, has 23 es then or insane, you
‘an predict ht Ub Ts erm wil have the 170 ern a is
correspon term, or for that mater the 9 tem will

53. ED | How w Powe tr Quran Asado lr CAT

ave the Ih cm ss corresponding tem (Since 24 is
one more than 23 and 74:17 = 9 + 1S = 24)

3. Process for Finding the Sum of an
Once you cn find pair of comespoding terms for any
AP you can esl And the sum ofthe AP. by using the
open of averages:

fe, Sum = Number of terms x Average

lo fac, this isthe Bet poes or Finding the sum of an
A: It mach more superior th the process of flo

the sum of am A ing the expresion ain -1d.

4. Fling the common difference of an Ad, given 2
terms ef an AP.

Suppose you were give ha an AP hats Anl rm as
Sand St em as 28. You should vimalize his AP as

Sms ES

Prom te above gore, you can easly vials that 10
move from the ind ter othe eighth er (8 1028) you
eed o ad the common deren Five times. The net
ction being 20, he comman ference should he 4
‘lastaton: iad the sum ofan AP. of UT terns whose 3b
toe is 8 and Sb term is 28
Slaton: Since we ko he thi term rd he eighth em,
‘we can find the common difrense as 4 by the process
lysate above

Te ttl = 17 x Average ofthe AR
(ar bte now sits into the Finding of de average al
te AR. In onder o do o, we need to identity cier the
OU term (which wil be the corresponding erm Fo the Su
tem) or üb LS tenn (whieh will be the corresponda
ten fr the Sed im)
Again: Since the Sh tees 28 and =
become 28 + 4 2 4 = 36

Tran, ie average of the AP. = Average of th and

kh tern
= 084 36)2 = 2

ence reir answers sum ofthe AP. = 1732

“Te logic that has appli here is thatthe erence in
the ern numbers will give you te nuber of times the
commen dierense Is wed 10 get fom one 1 the ober

For sanos if you know tht he diferen beten the
cr and 12th tern of an AP 8-80, you shold redire
that $ times tbe sommoe diffrense wil be equal to -30.
(Since 12-7 = 9,

the 100 em

Heres, d = -6.
"Note: Replace his algorithmic thinking neo he math
matical thinking of
12% em mas td
PA er 2 à + 64
Hence, difrence = (a + Ha) =
een
de.

5. Types of APctncreasing and Decreasing APS.
Depending on nlaher “is positive or negative, an AR:
an be increasing or decresing

Let us explore these wo pes oF AS father:

(4) Increase APs:
Every term ofan increasing AP is greater hen e previous

Depending on the value of the fst em, we can con-
struct two pr foe sum of an increasing A
(Cas 1e When he Fit er ofthe increasing AP pose
tive. Insuch a cae the sum ofthe AR, will show a continu
‘uy increasing graph which will ook like the one sown
in the figure below:

Marta ces.

Case 2: When he int em ofthe increasing À Pix cp
tive In wach a ess, the Sum ofthe AP. pleted aint he
umher of terms wl give the Following figure:

|

54. 52 ] He Pape lo Garenne Acto lr CAT

To Insert a given Number of Geometric.
Means between Two Given Quantities

Leta and bbe the given quan and m the required num
er of cans 0 be inserted. ll here wil bem + 2 terms
so that we have o ind a seis of à 2 terms in GP of
ich othe fit ae the I

Let be the coramon ratio:
Thee a te im + Atem
2

rl)

ence he routed number of means ae a, 97. af,
where has de sal found in (>

o

To Find the Sum of a Number of Terms in a
Geometric Progression

La a be the fin em, rth common año. he member
of tems, and, be the sum 0 eras.
Wer> L Ben

San

o

2

[Note 1 wl be convenient to remember bot forms given
above for $. Number (2) willbe used in all eases except
‘en e postive and greater than one.

‘Sum ofan info geome progression when r < 1

es

‘Obviculy, this formula is used only when the commen,
rai of he GP is es than ne.

‘Similan APs, GPs cn ala be logically viewed. Based
‘on the value ofthe common ratio and is fis tem a GP
sight have ote ofthe following sucess
(9) Increasing GPs type I:

AGE with fst tem poste and common rulo greater
‘han L This isthe mot commen typeof GP.

2653.6, 12,25..(A GR wit fit em 3 and common
rado 2)

“Tae plot ofthe sum ofthe serie with pest to the mar
oF tere in such à case will appear as falls

ano

(2) Increasing GPs type 2:

A GI wit fis em negative and common ao les han 1.
SEA,

As you can ace inthis GP a tem ae pestes ha their

previous ters

Te following figure will illastate the reltionbip be-

teen the somber of terme and he um on terms inthis

sel

notte!

ns

(9) Decreasing G.Ps ype
These GPs hae hir fist tes pois and common aio
Jess han 1

eg12,6,3.13,075

recen
Regen
aan ‘amar
(4 Decreasing GPs type 2:
Pi tema negative and common rato greater han 1
ce 2-618

la this cate the relationship looks ike,

55. (BA ] How Psu ts arto Ate la Be CAT

Cynder the Following question which appeared in CAT
201.
ind the infinite um of the series

4,9,
BE 22 27 2

wm AB (6) 40127. ao
Solon: Such questions have 100 alemative widely di
Veen processes Lo solve them.

“The first relies on maihematis using algebra solving
Unfortunately this process being overly matematica re.
gies lo of writing ai henc is ot advisable to be ud
in an anne exam

‘The other proces is one where we ty o pd he
approximate vale of the sum by taking ino account the
fis few significant tens. (This approach is pesible wise
because ofthe fat tat in sch series we invariably reach
the point where the vale ofthe nex tem become ris.
fiar and docs nt dd suba to the sum). Ar
‘ing the significant terms wares a poston 10 guess
the apprximate value ofthe sum ofthe series.

‘Lets lok the above question in oder o understand
the proces

In the given senos the vals ofthe terms ae

int term
Second tenn = A7 = 057
“Tied term = 9163 = 014
Fourth emm = 160343 = 0108
FA om = 292401 = 0.01
Addin uo the fi term is approximately 1.76

‘Options 2 en re smaller than 176 in valu ad hence
cannot be core
That leaves

options L and 3
of | 92 approximately while option

176 (sam of de fir terms)
‘Looking the pate we can predica the sh term
vil be

367? = 3616807 = 0.02 (approx)
And the seveih term would he 487" = 49/1769 =

0004 (approx.

“The eighth tem wil obviously become much smaller
Tecan be cry visualized that the residual terms in the

seis ae highly insignificant. Based on his judgement you

resize tha the answer will no ech 1.92 and will be =
stricted 1 1.81, Hence the answer wil be option 3

‘Ty using this proces 1 solve ether questions of is
au whenever you come actos them. (There ae few
such questions men in the LOD exercises of his Chapter)

USEFUL RESULTS

1. Mb same quam be ade o, or sut from,

a the temas ofan AP, Ab roi terms il for
an AP. but with the same common difference as
efor

al the terms fan AP be mali or dido by

Ihe same quantity, the resulting tems wll form an

AP bt with «new common deren, Which wil

be the muliplcatondtivsion of the old common

difference. (asthe case may te)

3. If all dhe cms ofa GP be mp or divided by
Ihe same gant, he resting ema wil form a
wid the stn common ratio as before.

4. ah 6 dre a GP they are aso in continued
proportion, since. by definition,

ath = Be = od

Conversely seres of quant in continued pro
poi may be represented by x, ar, a,

5. low have to assume 3 tems In AP. assume tem as
andador maa+de a+ 4

For asuming 4 es of an AP we we: o,
da + dd a+ M
For aisaming $ term ofan AP, tke her as

eer

‘Thee are the ment convenient in ems of problems
solving
$. For assuming thre terms of a GP asume then as
ar and ae or mt alr aan or
7. To fied the sum ofthe fist natural numbers
Let te sum be denoted by 5; then
s:

+23 mis ren by

min
2

8. To fed the sum of the squares ofthe fist astral

56. Let the sam be dent by 5: then
Seren

| Dr
nip evens}

9. To nd te sum of he ees of be fist mal
ober.
Lee mam be decd by; hen
Sa Pa Da Pa
CANE
po]
Tus, su ofthe abe ole at sal
aber ul to re ol de vm ofthese
unten.
10.7 ed esr the fink wo sail man.
Suede en Or Dr
Tod sun ofthe ft ven ara naan.
5=1+4+68..+21+n0+ D
„nen

12 To Sind the sum of od numbers Sm where mis
tral number.

Cae A Unsold a + AP
Case Be An seven [nA

13. fed dhe mu of even numbers Sn where nina
ral number.

Care A: WE mis 084 + [CAL + 11
Care Mis even = [n= 12 + 12}

14. Namber of temas In a count
2 we ae courting in eps of À fo my à my

aching both the ead pots, we et (m =) +1

umber,

(© AF we are coursing in eps of 1 fom my
including only on end, we ge (m, 1) number

= If we ae couning ln steps of 1 fom m, 0 m
excluding bth ems, we gt (m, -m)-1 nunken.

‘Beample: Between 162nd25 both inladad ter are

D + 1= 10 somber,

Between 100 and 200 both exclue there are

100~ 1 = 9 uber.

1 If we ao counting in steps of 2 fom 1m
‘nating both te en pies, me get - m]
41 oumbers.

One repens [GE

11 ve are coming in steps of 2 fromm m
incluia only an ead, wo ge (nm num
ten.

16 we are conti in sep of 2 from mb m,
excl both nds, we gt nm V2 num
ten.

angle: Number of esen numben bruce 20 and

100 (od inde is 10020 =» (BYE) +1 =

UF we a counting in eps of 3 from a, m
inch bth he ed poi, we em
#1 ramen.

= we we coating ia sep of from m;
inclinan nd, no gt sm) um.
en.

Me ar cod in pe 3 fom m, © ms
exclding th ends. we get fg = RDA =
omer

sample: Number aber bomen 100 a 200,

vibe by tre.

Solon: Tb is mumbers 102 nd he ast rum

teri 198, Hose, answer = 905) 4139 nce

Be 102 and 198 are inch)

‘Avene hgh mer Blow 10 a I di

ine by 3 1 9, andthe Invest mame above 200

hic indivi by 3 à 20.

Hence, 201 -99 = 102 > 1028 = 34 -> Answer =

H=1 233 ine th nds ar et nce)

In general

= Awe are cogi sep of frm mt incl

ing ech the ed pots, we to; ml 1 u

ten.

fe courting in ep of fm

luting où one ed, we et; = me tuber

Wwe an coming in tes of fon y o y €

iin bth end, we a I, = 11 member.

Fer inane if we have fod how may em re

there te seis 17, 2. 24, we

have

OSI ION + = HAN + à 2 +1222 eae

Pre

Of os, en appropriate asen wil have Be

‘made hen es ot fl o sei. Ts wi

Le dre a folios:

Fer inte, ve have 1 rd Row me ters of

the sve IGT 18, 21, 128 a below 25%, then

we ave by the formal:

57. 66 | Hot pue ts Que Aster a CAT

(25810707 + 1= 1517+ 1 22151 ST
‘This willbe ads by Ling the lower integral
Salbe = 22. —» The number of terms in the series
Below 258

‘The student is advised to ty and cxpriict thee

primes pe clear picture

WORKED-OUT PROBLEMS

‘Two persons Ramo Dhob and Kat Mochi
have joined Duakey-work Associates. Ram Dhabi ard
Kalu Mochi started with a ntl salary of Rs. $0 and Rs.
640 respectively with annual increments of RS. 25 and Rs.
20 each respectively In which yer will Ram Dob star
ani more slay tan Kala Mochi?

Solution The cure difference bemece he salaries of
the two is Rs. 140, The anual rate of reduction ofthis
erence is Rs. $ per year, Ar tise, it wil tke Ramu
Dob 28 years 0 elise his salary with Kale Dots
salary

‘Tous, in the 2h year he will car more

This problem should be soled while reading andthe
Abou process shoul! be HS = 26. Hence, answer is
2h year

REE oe ie vale e expresos

1269 2-723-=8 4. lo terms

(0) 250. (9) 500 (o) -480

Solution The series (1692279380000 10
10 terms) can be roten a
a (424 34 0 SD tem) (6974841030
tems)
Bal these are APS with valves of and das >
anime SO and d= land ant me Wander!
respectively.
Using the forma or sum ofan AP we get

250 + 49) 25112 +49)

> 2551 61) = 250

@ 300

Altemaivey we can do this far by considering
(1 =6), = 7), and 50.688 one uit oF one tem,
16227 wm... = -S. Thus the shove sees is
equivalent te a sels of fly 84 added wo exch cr
Sa. (1 6) + (2 7 += 81 + $0 ters = 8 #0
--250

PELIS Fn ne sof a nantes ave 6
in bein 10 2 400

‘Solution Here Item = = 102 (which she It term
ester than 100 that is divivbl by 6

“The ne term es than 400, which is ie y 61396
‘The number of tems In he AP: 102 108, 114.396 I
ven by (396 10246] +1 $0 unbe.
‘Common difereace = d = 6

So. SCA 12450

Aa za in GP, thea VAL + logo,
A ogy) amd M4 og) il Be i

(ae
our (4) Cannot be said
‘Solution Go trough the options.

‘Checking option (2), the tre wil bein AP if the 2
‘expression ithe average of the It and A expressions
‘This canbe mathematically writen as

2X Hopp) HAL + Fogel + TA log)

[elegante
LEE TE)

Gel
Ton red

Anping jdgemen remo no nilo

tha ee gi op sain

Checking pin ©)

AU à eo = LK à oo UN 1)
FIA opt D + lo)

‘Asin we ar rapped ad any soln i ati St.

Checking open.

AA + eg VA + log and log a

IP en opts Jo and oem ein AP

So lg gy ad logy wl ab e in A

Here. 2 ona = lott + an

= Pa she pen

Sn (is te come option

‘Aertel yo cole slved ough e oo

ing pes

stad zu sve logit nam.

Aus = 1, y= Wand 5 = 108,3, 5 ein GP

Sa eg L$ laggy 2and 1 gp

58. (Thus we find that since 1, 2 and 3 are in AR we can
assume that

Le logit 1+ logy and 14 loge ar in AP
> Mence, by definition of an HP we have hat I +
Log MU + loja and VA + ogg re in HP. Hence,
‘option (6 is the required answer.

Autor Notes la my experience have always fund
hat the toughest equations and Fctoriacons ge solved
ver easly when her are option, by asuming values
in place of the variables i the equation. The value of
the variables shoul deta in such a manaer thatthe
ase restnctions put on the variables should be re
specie. or example, am expression in thee variables
band cis given and it mentioned tha a + b+ € =
thea te valo hat you assume or band sha
‘any ths resto. lence, you shoul oak als
Mel. 2 and 3 or, =I, = et.

“This process especialy wef in he case whore the
question a wel a the options bth contain expression.
Factrstion ad advanced technique of mals are thea
ot regu This process wil be very Reni for
Anders who are weak at Mathematics}

SE Fes no nd 5, forte following seres:
ETS

‘Solution, This is an AP with int tem 1 and common
irnos 7.
tea bin Dde 149% 7e 64

+ a= Da)

Sa
7

a +

71
GO ans

Atem, she numberof term 6 small, you can
count it ety

Fin ty en Si forthe Following series:
Au.

alien" This is a GP wih fst tems 2 and common
mio
PRE"

CET

zu

a | BZ
ERIE ts de series 1.4... 10 n terms an AP. oF a

(GP, or an HP, or series which canot be determined?

Solution. To determine any progression, we shoud have
a leas thee ers,

{he series isan AP thn the nex term of
7

Ain. ifthe nex term is 16, den his will be a GP series
W410.

‘0, we cannot determine he maar ofthe progression of
is ses

MEN ine he won o 200 terms of the seis
Te4e6e54il 46s,

) 30200

©) 30200

Solution. Spor tat the above series ls a combination of
wo APS.
The It AP is (1464 11 4. andthe 2d AP is (4
5465.)

Since the terms of the two series allemate. $= (1+ 6+
A1 4 10 100 terms) 4 (845 + 6 4 1 100 ter)
OX + 995] | 1001254 +991)

. es 7

the formula fo he sum of an AP)

= 501497 [phone]] = 30200
‘Alteatvely. we can tea every two consecutive terms
So we will have a ou of 100 terms of the nature:

nr.

es

© 200
10) Nowe ofthese

sing

UFOs O94

Now aus. da Gand n= 100
Hence the sum ofthe given seres i

Se xs em xd
«mn

IIED tos mary termo of be seres =12, 9.
6... mun be en hat the sum may be 54?

Solution Here $ » 54,2 -12, d = 3, ic unkaoun and
as 1 be callate. To do so we use the formula for the
um of an AP aod ge.

Pr!

+6
7

Da

59. 0

B

1

16

3

Gs. 500) 0) Rs 900
€ Rs. 1150 (0 Rs 1000

(e) Rs 1200,

‘A mur 15 is divided into ls pars which are in

A sd the sum of thei sr sE, Fin th sal
es number.

@s
CE
‘The sum of be Bt 16 ems of an AP ve ft

os 06 (ws

term and ind dem are $ and 15 respectively e
W 40 GTS (60 60
(o @0

‘The numberof terms ofthe sees 54 + 51 +48 +.
such thatthe sum i 513

@ 18 & 1

(© Bad amd (IS

(6) Nowe of these

“Te least alu of for which ln sum ofthe seis $
$8 + Mas esis ne es tha 670 is

wa Oo o2 u
on
A man vsiven Rs 0 for he fit weck and Rs 3

more eich week thin the prceting Weck. How much
does he eam by the 201 week?

(a) Re 170, (o Rs 1020
(9 Re 1890 (9 Re 1790
(6) Nove of these

ow many ters ae ter nthe GP 5,20, 0,30.
20580

ws
os
A bey ages to work the at of one rupee onthe
is day two rape te second day, fou ropes on
the third day and so on. Hew much will the boy gti
e stats working om the Ist of February and finos
con the 20 of Fra?

os 07 ws

o> vr
@ 2-1 mE

(0) None ofthese

the fitter of GP i 1 and fst terms 16, what
Wil be the th tem ofthe GP?

@% OB @s wm
on

“Tae seventh fem of GP i & times the four tern.

‘What il be the fist term when if tees i 487

a

Cep? pegue | 59

wi 03 05 2
@6

‘The sum ol ee number ina GP is 14 ad the sum
oftheir squares is #4 Fl the Ingest number.
ws ws

@ 4 on

(e) None of these

‘The tte ofan arme propresion suit and
the common difference is 4. Which ofthe following
vill be a em ofthis APP

(a) 4551 109 10091
(9 788 as

(© 13337

ow many natural numbers between 300 to 500 are
pes of 72

w» w 2

on @ 9

(e) Nowe of tese

“Tee num of he firs andthe thr erm fa rame
progreso is 20 andthe sum ofits fist thre terms
1526. Fi the progression

@) 2.6.18, 1) 18,6
(0 Bor of ese (8) Note of ese
16) Canne be determined

a man saves Rs. 4 meee cach yar tha he did the
year before and A he saves RS. 20 i he Fit yea,
after how many years will his savings be more than
Rs. 1000 aogier?

(@) 19 years m 2 yeas
(©) 2iyears (8) 18 years
(o) 17 yeas,

A mars salar is Rs. 800 pee month Be fra year
le has Joined in the sal of 00-40-1600. After bo
many years wil hs savings be Rs. 64.8002
Deren 0 7 years

©) Gea 14) None of these

(0) Canoot be determined

“The 4th am {Oh tem ofan GP are 13 and 243 re
spectively ind the 204 term.

@3 wl [m
ws

Te Mhand 2st terms of an AP are and -22 respec:
vel. Find te 260 term.

WDR |! -2
(o 16

ow

(9 10

60. 2

(o 2550
€) None ofthese
Find the sum of al integers of 3 digs that are dvi
ibe by 7.

(0 69336
omas

€) Nove of these
“The fit and the Ist terms ofan AP are 107 and 253,
there are fiv terms in this sequence, find the sam

(a) 2650

© 71536
m na

of sequence
€ 1080 © 70
© « 60
4) 1020

Find the value of 1 =2-342-3,

+... pe 100

wo © -06
ns CET
€ 676

a vil be he sum to terms of the serie + 88
Bevor:

m „tm
[DEE BA — 10)

(e) Nove of these

1 Wa, Bein GP then log a og. log cate in
war wor

© HP 14) None of these

(6) Cannot be sait

16.

Alter striking the oo, rubber bal ebounds 1 ih
‘ofthe height fom which i has fallen. Find the total
tance thai tele Before coring to es if a
been gent dropped from a hegh of 120 metes.

(0) 540 mates 9 960 metes
(©) 1080 mes (0 1020 mes
(0) 1120 metres

abe the fist tem, y be the nth tem and p be the

prac nts o GP thn the alo wil Be
or So

© wy! or

(6) None of hese

‘The sum of an init GP whose common rai is
eric ls than 1 32 and be sum of he fest.
fe term a 2, What wil be he hid tr?
@2 01 8 wR
ls

20.

a

2

Guen Pepe [Fi

What willbe the vale of a 1 infty

@s A
or CE
(e) None of tase

Fin the sum o tema of the serios
123423463454...

G) (a+ hn + Din + 390

© an + D + Din + 24
Omnia

8) hn + In + BY + 344

(6) None of hese

. Detensie the fit term ofthe promet progression,

the sum of ab fs eon an thin ter e MD nd.
(he sum ofthe second term and foun erm a 80.
on O6 @8 0.
os

Fd the second ern of am AP if the sum of Fst
five even terms qua o 15 andthe sum of the fis
re tems is equal

03 02 @4
CN

“Tae sum ofthe second and the fifth term of an AP is
‘and tit of te hed and he seventh er is 14, Find
the eleventh tem.

os on
CE

How many terms an AP must be taken fr ter sum
10 be equal to 120 if ts third em is 9 andthe fer
nce between the seventh nd he seco tem is 277
ws DS ©7 ws
CE

Four numbers re Insert beeen the numbers 4 ad
39 such that am AP rel. Find the biggest ofthese
our mame.

oo

os om

on OM OR OW
CE

Find sum of al ue in nates ri, ich oo
ing divided by 5, ave a ronde qual 4.
@ 51270 0) 96780
CET (@ sam

€ 0270

“The sum ofthe fs re ter ofthe arnet pro
reisen is 30 und the sam ofthe garen of the fit
term and the second term ofthe same progression ls

61. 72 | vow er tr arto paa bem CAT

116, Find the seventh tern of the progression fis
Sits term is known o be exactly divisible by 14.
os A WA
on

26. A and et out to meet each eer from two places
ES ko par A travel 15 km theft ay, 14 the
second day 13km the shit day and so on. tels
10 km he fst day. 12 km the second, 14 am the
thi day amd o on, After how many days will they
meet?

Per © 5 days
(a 6 days @ Tan
(a 9d

27. Ira man saves Rs. 1000 each ear and invests a he
ea o the peu a 5% compra interes, ha much
wil the amount he a be end of 15 year’?

(a) Rs 21478 (0 Rs. 21578
CRE @ Re 22478
(e) Re 22078

28. Hunt tem ofa seres is given by (+
its second term wil be given dy
@ 10 oo
ws wi
(6) None ofthese
29. FA isthe sum ofthe toms ofthe seis 1418 +

164... a Bis the sum of 2 tems ofthe sexos
Lo UD + VA + then find the valve of A.
an on om wa
was

30. Aman receives a pension ating Rs 109 forthe
int year Bach yea he reis 0% of what here
‘cived the previous year. Find the maximum total
mount he can rei eve I e Hives forever,

ai Rs. 1100, Rs. 1000
(o) Rs. 1200 A Rs. 900
€) Rs. 1230

“The sum ofthe series represented a:

MS + SO + 1... + RAS
i

(a) 29221 (sont

do $5225 (9 Nowe of these

32. The sum ofthe sees
AO + VD + M2 + V9 +.
van ic
tar 10
on

+ tan +

on
18) Nowe ofthese

30. Find the infinite sur ofthe series +173 + U6 +

wenns.
al CES
CE CE

34. The sum ofthe eres 58.48 11411 um
toes wl be

D me DSC + Pam O
8) A+ Da HP + Gt + Dr 1 + 10
Om Den + 2) + Gt + + O
8) + DBO + 1) + 6in + D + 1] + 10

3S, The sum of the seis 4 M6 + 1124 10.1186

are
(o 13 (an
@ 140 (4) None ofthese

36. For the above question 38, what Is de sum of he

series if ake to infin terms:
a on
ou (0) None ofthese

Directions for Questions 37.39: Answer the questions

based on the following information

There ae 290 integers js OLA of hem nec

sarily een. Lette greatest integer ofthese 250 in

teger be refer 10 as Max, and be smallest iger be
refed to as Min. The neg» OUP oa Form se-
quere A, apd the ret form sequence B. Each member of

À is es ha or equal each er ol.

37. All values in A are change in sign. while those in 0
remain changes Which of the following sumen
ds me?

(2) Every member of A is greater than or equal to
every member of 8.

© Mac isin À

(6) Mal umber riialy in À and had be same
sian, iben after De change of sign. the largest
number of A and Bs À.

(@) None of these

38. Elements of A are in asco eres, and se of B
arin descending order a andy, ar merchanged
“Then which ofthe following statements i ve?
() A contesto be in ascending order.

(0) 1 coetiones to be in descending order,

(6) A continus 1 be in ascending order and 2 in
descending oe,

(4) None of the above

39. Every clement ofA is made greater than or equal to
‘every element of B by adding 0 each elemento A an
integer x Then, capot be Is han

62. CE
0) de smallest vale of B

(6) te ages vale of 2

(0 Mami)

40. Rohit drew a rstangular grid of 52 cells range in
23 Rows and 25 clara, and filled each cell wit à
umber. The numbers wih which be filed euch cel
were such tht the numbers of each row taken fom
left wo right formed am arme seres andthe num-
‘bers of each column taken from to to atom lso
formed an armee series. The seventh and the ser
encens sumer ofthe fh row were 47 and 63
respectively, while the seven 2d the seveatcath
numbers of te cer row were 53 and 77 respec»
ively. What isthe sum of ll he number in the id?
DES 65586
(9 290 (9) Nowe ofthese

1. How many tes digit numbers have th property that
hei is then rom eft a ight for an Armee

ot à GeameticProgesion?
wis os
om (4) None ofthese

Directions for Questions 42 ond 43: These quetoes are
‘based on the following dt.

‘At Burger King—a famous fst food cone om Main
Steet in Powe, burgers are made only on an automatic
burger making machine. The machine cemiuoaily makes
fret sets of burgers by sing diferent sets of File
Ings on a common brea. The machine makes de berges
atthe rte of urge por alfa minute. The vaicus ings
are added the burgers in he following manner. The st

Sth, ih... burger ae filed with a chicken pay: the
nd. ob. Vi... bupers with vegetable pay: the Ist,
Sth, A, … burger with mushroom pay; and the rest

‘with pli cheese ard tomato filings.
“The machine makes exactly 660 bases per day
42. How many burgers pr day are made wth cheese ant

tomar a filings?
was © 26
om (a) None ofthese

43. How many burges ae made wi
Chicken, vegctebe and ruso?
wa mA
(o 25 (026

44, An ihre progression Pcondst om er From
‘he progression thee deren progressions P, Pd
Py a rete noch tat P 5 obtained by the I, th

the flings

Cue Pros [73

A. es OÙ PP as the nd, SU eras
SEP and Py hs the Bd, 6h the sers of Bis
found tha of P,P ad Py two progressions have De
property that cir average is el term ofthe orig
ul Progression P. Which ofthe following can be à
pasible value of a?

(o 20 CET

os (9 Both 1 and 2

For the above question. if the Common Difference
between the tems of P, is 6, what is the common
leen of #7

(2

(96

ws
KA) Cannot be determine

Level of Difficulty (LOD)

+ Min any decreasing amet progreso, sum of all

its toms, exept or he ister is qual o 36, the
sumo al asters excep forthe laste is eo, nd
the diference of the tenth and the sith vr e equal
do = 16, then wht wil be Fit term of thie tenes?
EEE rare
eu

‘The sum of all terms of the atte progres
having ten temas except or the fist term. I 9, and
‘xcept forthe sx tem. 89. Find he id tem the
progression if the sum of the it and the ith er is

gu © 10.
ws ws
ws @ 10
(e) None of these

Product of the fot er tthe fifth term an
arithmetic progression à 456, Division of the in
ter by the fourth tem ofthe progression gives quo
Hint as and be remainder as 10. Find the int em
of the progreso.
@-2 D-8
oo

A namber of saplings are ying ata ple by the side
‘fasta wad. Tee ar tobe pled in se
lie al a distance intel of 10 meters beween two
‘sonsecutive saplings. Mitesh, he county’ retest
forester. can cary only one siping ata me and has
to move Back othe engl pet 0 ge he net sapo

(92.5 0-6

63. ow Papa o One Ap hr a CAT

ling. In is manner be cones atta distance of 1.32
Aus. How many saplings docs he plat in process
ie ends tte sting pom
os Du on
on

5. A geomet progression comio of $00 terms. Sum
‘ofthe terms eccoping the odd places is and the
sumo the tees occupying the ven places is P, Find
the common rio.

(a es,
ETA
(e) None of thse

6. The sum of the frst te teme of the geome pro
Bresson is 5, nd he sum ofthe nxt en terms Ith
through 204) s 5; Find the commen rao

on

0) ee,
(0) P+ Pies

ca ys" © = 6/9"
CES & ss
(e) None ofthese

7. The fest and the thin ers ofan wine pages
son are equal. respectively, othe Fst and the ind
tem ofa geometric progression, and the second tern
ofthe aime progression exceeds the second tm
ofthe rome progression by 025, Calculate the
sum of the fist ive terms of the amet progres.
som if fs tem is equa to 2

(a) 225 0028 m 2590275
@ 15 CE
(e) None ofthese

B24 44 6 4. 80 mm à 3 454 teams)
= SU. then fd the valve of m

@2 DE 09 wm
on
D. (666... digits + (88... m diga) ic equal to
4 4
ar ed
PISTEN BETEN
(o HE Auen

(e) None of these

10. The imeror angles fa polygo ae in AP. Te mal
st angle is 120° ard the common difference is 5°
Find the numberof ses ofthe polyon.
@7 ws 09 wo
om

11, Find the sum wo teas ofthe serios 11 + 103 + 1005

e fun,
fo WUT en
(6) None ofthese

‘The sum ofthe fi em ad he fifth er ofan AP
15126 and he producto the second tem by he fourth
ver i 160, Find the sum ofthe fit seve term of
is AP.

(a 110
lo 10
“Te sus fe hed an he ninth tem of en AP is 10,
Finds possible um of theft 1 terme of thin AP.
ws va

© 6 CRI

16) Nene ofthese

‘The sum ofthe quarts of the fia and the eleventh
term ofan AP is 3 and the proto the second and
the foarte term is xual cP, Find the product of
the fit andthe Fica om ofthe AP.

(a GAP 39y48 (D) CRF + 3992

ou [m Wis

(6) GP 190 (a RP + 39950
16) None ofthese

tbe rato of harmonic mean of tao numbers to ir
‘geometric means 12:13, father he numbers.
(a) 49 or CET

©) 25 052 won

©) Moras

Find the sum of te series 124-2284 32 4 à
100, 2",

wma ay 99.242
BR 1002" +2

(6) None of these
. The sequence [Ji GP with x x= IM and +44
= 108, What wil be the vale of 3?

Wa 08 04 0%
CE

1

3

A . care in OP and oH and ae qual, en a,
ewe

war wor

on (0 None of these

(©) Cannot be determined

Find the sum of al possible whole number divisors of
mm.

64. 76 | How so rr e une As wen CAF

32. Inacio colony of cancerous cells, cach ell repro
duce by giving Da two new cell every hor, I

process continues fr bas, how any cell will tbe
Colony ave at the end of 9 hours tis ow thatthe
Ie of an individual cells 20 hour and only 50% of
the cells ae capable enough of prodacin next pere

i cells?
@ 2-1
or
CEE

33, Find e sam of al three dig whole numbers less
than 500 tha eave à emainder of 2 when they are
died by 3

a) 49607 © 7
(e 49634 w 00
CET

Ua be the armee mean and b,c be the two geo-

etre means been any two postive numbers, ih

+e loe uals

(a ay"

(o eo

(e) None ofthese

38. Up. . rar ree comscaiv divine url numbers
then the expresion (q +r pp +r 9 Xp +47)

e

wt
CE

(a) Positive (0) Negative
(6) Nompodive —— (9) Nowaegaive
(e) Either (0) or)

Hints and Solutions

Larter d + (+ 20 + la + mem
ad ta + 34) = (a + da + 2

4. Calculate the sum of an AP with fst erm com
mon diferen | ad ls 12. Mull his sum
by 4 for 2 days

5. The masi sum wil our when the lt wer i
ciber 2 or

6. Visell the AP as 7, 13.196.

9. The AP is 105, 112.90.

10. The common dienes LÉ «292

m

15
1
m

See the tems of the stes in 33 backs of 3 each
“Ts wil pve the AP 24,8, 633 Fur, the
Dundes ten will be 34,
Solve trough options.
The fist drop ls 120 mes. Ae hs the all il
ise by 96 menes afl by 96 mei. Ths process
sl coimas a the for fan fie GP wit cor
on ato amd fi erm 96.
‘The required answer will be got by

120-2 96+ 1.2562
Tole any GP and sue by using vales.
Sole hy using vales to check op
‘The dire between the seventh and thi term ie
pen by

(a+ Gi (a+ d)

ony

-1

The required answer wl be by adding 20 tems of
the GP seting wih e fs em 36 1000 and the
‘omen rai as 1.08,

Visuals an infinite GP with son rio 0.9,

M

Diese bebecen the tenth andthe sath frm =
16
(a+ 94) = (a+ Sd)
dead

Sum ofthe ist term and he ff tema = 10

Hints and Solutions

6

oe a+a+ its 10
e. asus w
nd, the sam fallt o e AP except fr te Ist
em = 99
Pi Da + 454 = 0

esa o

Solve (1) and (2) to get the arson.

The second statement gives the equation as a + Bi =

War x46
a a-u=6

Now, ae Le opis to Find the vale old, ad pt
these vaine o check te equation obtained rome
fir sement.

de lar 2a + Sd = 406

To plant he Is sapling. Mies wil cover 20 m:
10 plat the 2nd sapling be will cove 40 m and so

65. ZB | How Pape br usas Arh lr e CAT

While the second series is 3,69. 240
Hence, the ast common tm is 207.
“Ths our answer becomes seo

ri

7. vins Option (ah
“We get less ler $ ad large tem 30 (ire the
Tages term I 6 times he leat term.

The average af the AP becomes (5 + 3012 = 17.
‘Ths, 175 x 105 gives us

Lo tata of 15 we ned = 6 terms in this
A Th meas the ALP. shou oak ike

5 20.

Ir can be exil seen that he common difference
should be $. The AB, 5,10, 15,20, 25,30 fs he
‘The same process uses for option (b) ges us e
AP 10.35,60.(10435 +40= 105) and inthe hind
‘option 15, 90415 + 90 = 105)

Mere, alte three opions ae corset.

9. The dire Between te amounts a he end of 4
yeas and 10 years wil be the simple ies o the
inal capital fo 6 year.
enc, 3006 = @ =(inpe erst)

Also te Simple Incest or 4 years when add 10
the sum gives 1240 as the amount.
ene, he original sum must be 100,

16. Sum of GP wih st cr | and comman ratio 2

and no of tems 20.

Lee

a=

18. Inthe cose of a GP th Th term I derived by ml.
"ping the fourth term hie by the comme rin
(ete: tio vecy slr o what we had cea the
use of an AR)

Since the seventh rm I derive by muhipying the
Four te by 8, tbe eiii

a 8 mus be te,

Henes, r= 2

Ihe A ern i 8, the series In revere fom the
Fi othe fst tem wil ook ike:

48,24, 12,6, 3. Hence, option (b) à comet.

21, The seres will be 301, 308, 497
Hence, Answers DE 4 1 229
25, Similar to what we sw in question I,

term x

The answer wil given

“The th tem re 3 lan he elo tele
Merce Su
Gives ww: r= 3.
Here, the second tem

ts)

vil be given by (Second

te,
[Note Togo forward in a GP. you map bythe
‘commen rao 10 go backward In à GP. yeu divide
by the common ato.

OFM 124 un + $0) [169 28
“rmx

D = 160 = 1070.

+1

Think He this

“The average ofthe frst tens 7, while the aver
age ofthe fs ers mas be
Now visalze his
m And And ci Si Gin ih Sin
avenge 7 average = 11
ence, dm 47 = 2 [Note understand his as à
propery ofan ARI
ene, the average othe 6s and Tb ers 15 0d
the average of the Be and th tenn = 19
Berths (19) also represents the average ofthe 16
term AP.
Hence, required mm = 16 x 19 = 308,
Go trough options. The cores open should give
sale as, when 3 and as $ won = 8
Only option () Satis both conditions.

2

[deny an A which satis the given condition.
Suppose we are ting bout the second ad hed
terms ofthe AR

Then an AR with soond sema 3 and third term 2
sass the contin.

‘tines the ah fem ties the berm

{a this ease the vale of a = 2 and b= 3.

Hence, foe the (a + BP tm, we have to id the
fifth tem.

{tis lear thatthe ft tem of his AR mus be zero,
‘Check the ether thee options to see wheter any
option giver O when a = 2 and b= 3.

Since none of he options bc Fd gives zen for hs
parier value, the option a is comet

66. 1

Mew 1445-8 49-12... Storms as
(4) 46829) +49 = 12). Dim,
Hence, 3 + 23 +23. 25 terme
TRS

Since this is a decreasing AP with fet term pou
tive, the masiman sem will cor wp the point

Menos, 23 tems x 22 = 506.
A lie number Jogging woul! give you 20d term ls
8 and Sed term e 4 à possible ston tht
satis the condition

Te AR vil become

16, 1D, 12, 29,5%. 1

rin decimal term, 0186, 0383, 05,0655, 0833, 1

Sum to 6 rms = 3.5
‘Check the option with m m2 and n= 3. Only option
Le) gives 35. Hence, must e th answer,

‘The fr 100 tems ofthis series can be viewed at
(2-34 A -34-35)48
‘The fis 33 terms of the above seis (ea in-
sde de brackets) wil give an AP: 4-5, 6-16
‘Sum of his AR

Answer

‘The path ofthe rubber bal is

In the figure above, every bounce is 4/50 of the
previous drop.

In the above movement, here ae tuo infite Cs
(The GP representing the fling distances and he
GP representing e Rising distances.)

ue roms | TE

‘The role ansner (Using fl) formula)

For, sum should be 6 Option (be) and
(a) al give 6 5 the answer
For a = 2, he sum should be 30.
Only opin d gives this vale. Hence must be the
Fd sana of he seis:
104, 109,118... 999
‘Average x = 3513 x 180= 99270
Since, sam to tems is ven by (n + 8
‘Som to À terme = 9
Sum to 2 terms 10
“Thos, the 2d dr must be.

3

For I term, he vale shouldbe:
er
‘ny pen () gives 4 form = 1
Tre solution (rom de opos) as got something to
do wit ie 2" or 2 fr 100 ems. Hence, for
5 teams recreate the options and crsscheck with he
ati sum.
For à terms: Sem 2 4 8 +24 2 34,
2) 100 x 29 + 2 for 100 terms becomes 3 x2 +
2 for 3 tems
48+ 22 50 # 4, Hence not comes.
19 99 x 28 à 2 or 100 ems becomes 242° +
2 for terns
Bat this does ot pve 4. Hence i nat core.
[DE PEST EEE
(8) 100 2°62 9 342

ence, opio () is coma.

67. 30 | Hari Paper Dunn Are rw CAT

ANSWER KEY
Lot
he wa
ha EN
[xe = Be
on me
on EN
Ge
Te
he
on
1.
ma
he
the

ie

68. 8

5

69. fen Tess [BS

BLOCK REVIEW TESTS

REVIEW TEST ONE

Lasa has the se number of sisters ssh has br
er, br her brother Shyars has nice as ay sisters
ashes brother, How many children are herein he
Sail?

@7 6 OS ws
How many tines does the digit 6 appear wen you
count foes 1 o 100

wo. ew
Im €, den
(omen
@ mms nn
Omnnenmm

on om

Ommmenn.n
A square den by Jim midpoint ofthe ide
‘ofa given square A third square sdrawa in side he
second sere in the same way and tis process is
común idee. I a sie ofthe fra square is
em, he sum ofthe acaso te square (ins, rs)
lo 18 w no
(o % 16) Nove ofthese

. Find the Feast number which whe divide by 6,15,
{leaves a remainder but when divided by 7 eaves
no rminder
MAL Wim ww
‘The mumber of positive integers no reste than 100,
which ae nt divisible by 2,3 oF Sis
ws Dis 9
(8) None of thee

The sulle number wich when dire by 4,6017
leaves à reauinder of 2 is
wu Oe OM WH
[Aa imllience agency decides on a code of 2 digits
Selected fom 0, À, 2...9. Bu the sip om which the
(ode Isha-wrten alu confusion between tp nd
om. esate bee ar iedsinguishable. Ths, for
example tb code 91 cold be confusa wth 16. How
‘any codes ar there sch ha thre is mo posibili
of any confusion?
wo»
©”

m7
(8) None ofthese

9. Suppose ose wishes o ind diia postive integers
2 3 su that Cx sp is als a pose integer
emi the cores erative.

(a) Ths never posite
(0) This is posite and the pt 3) satin the
sted condition is unique
(6) isis posible and here eit more tan one bat
nite member of mayne cboesing the pai,
(4) This spout ad he par (9) an be chosen
ile vs
10, A young gt count in the following way où the
Fingers her et hand. Se sre cling the am
1. eh inde finger 2, mide finger 3 sig finger 4
lise Her, hen revere direction, cli the ing
ing 6, mide finger 7 index finger, mb, then
oc 10 the iden finge for 10, wil ger fr I
nd 30 on. She counted up o 194, She ended on her.
@) tn 9 inden finger
(©) mide Anger A sing Anger

A1 199 persons have signed up for an eliminan ours
men. AN payes ae 1 be paired up forthe int
round, bt cause 139 isan dd number one player
ets bye, wich promotes him 1 the second round.
boat actually playing inthe fst round. The pain
continues he nest round wit à bye w any player
leftover the scbeele is planed so hat a minimum
uber of matches required 1 determine the cham-
ion. the number of matches which mas Re payed ie
wm RS BY
12. The product fall ntegers rom It 10 wil have the
following number of oros athe end
OD HM OB wR

13. Tre are en SO pais coins placed on a table, Six of
these show tails four show hea. A com chosen at
‘ano and fipped ove (at os} Tas operation
ls performed seven Umes. One of the coñas i hen
‘over Of the remaining nine coins five show tails
‘a four show heads. The covered coi shone
(@) bead au
(6) more ei a head (8) mor el ni

14, A ive digit number le formed ing digit 1,3, 5,7
‘and 9 without rpesting any ame of hm. What i he
samy of all such posible numbers?
(a) 666600 6565680
(e) 6666666 (u None

70. PQ and R we tee comective od number in a
omg order If the sale of tee times Pi tee
Jess han to tes R fd he value of.
@s 67 @9 au
ABC a ved number in which A > 0. The value
{ABC sequal tothe am of the factoria of ts ee
igs. What isthe vale of |
oe 07 @4
A, Band Ca defined a follows
A = (2060004) = 12.000008 + (4000005)
= (000003) + 03.000009? + 1.000609)
Cm (6000002) + [LUGO + (8090002)
‘Which o he following st aa he vale ofthe
ove three expressions?
(a) All of them ie Between 0.18
a is eee €
(6) Cis he smallest
(8) Bis the smallest
Latx<050,0<y € LL Given a set numbers.
‘the middle number, when hey aranged in ascending
Order i called the median. So te media f the num-
es y and 3 would be
(a) les than ene (D) between O and 1
(©) greater than one (d) canst say
let a bed, and eb integer such at a= 6b Le,
‘and 2b = 9 = 12. Then which of the Following pa
‘contains a number tha is pe an integer

we

om

Ya à + 2, end @ + 4 ae prime number, en the
numberof possible souios for ke

(a) Ore 10) Tre

(©) Tire 14) Mor tam tree

Le xml y be postive itegers such at x pre
aod y is compost. Then.

9 y — à canoe bean even Seger

(0 ay cannot be an even integer

10) (e+ y) cant be an even integer

(0 None ofthe above statement is te

Leta GI) be composite natural number such hat the
gr ro of ais tot an integer Comer he follow
ing statement:

fei tos [85

‘Az mba factor whichis greater han andes than
quae neun

1: bas factor which is greater than square oot
bat es than a

Then

(a) Bah A and B ae fase

© Boch A and Bare mes

(6) A is ale bat Bis ine

(4) Both A and B ae tue
18, What the remainder when 4% i divided by 67
wo 2 ©3 we
19, Whats the sum of all wot number that give a
remainder of 3 when they ae divided by 7?

wor a) 197
20. The infinite sum 1949 À “
TE

a o
© CE

21. He prt of» positive ral number is u
Air sum is necessarily:

(a) amine of 9 () equal n+ À

(6) vever ess ban n (0) Nene of ese

22, How many hee digit positive iger, with ii 4. y
and inthe hunden, tens and unit place resp

ls nl sch hat € 3.2 < yan À 202

(28) MM 300

23, How many even integers, where 100 £ 5200, ae
‘sible nee by seven vor by nine?
DEE da

24. A postive whole number AF los tbn 100 ls ep
seated in base 2 notation, buse 3 rotation, and base 5
alos. His found that al re cases the ast git
1. while in enacly two out ofthe thre cates the

Healing digit in. Then M equal:
an De on wer

25, Ina eosin examination paper, thee ate queden
Foc/=1.2...n tere are 2" students who answers
{ox more guesions wrongly. I the tol number of
‘wrong answers is 095, thea the vale of ni
on mn om we

71. BE | How» Ppa unie em ota CHT

ANSWER KEY

Review Test One

72. rin Toe [BZ

EIS

73. BACK TO SCHOOL

+ The Relerance of Averages
Average sone of the most Imporant matematcal concep tha we use in ou day to day if. In fost. even the
‘most aon macbematcal individuals regularly wiz the concep of averages on a day to day bass.
So, we use average in all the following and many more imtances
+ How a clas of students fared in an exam i assessed by looking athe average acre
+ What isthe average price of items purchased by an individual
+ A person might be intrested i knowing his average telephone expeadiue electricity expen, pero!
espendiure et.
+ A manger might be interested a ing out the average sles peremlony or even he average prob ae
‘month 9 mom
+ Clearly there canbe immense aplication of averages that you might be ale wo visualize o your ou.
+ The Meaning ofan Average
"The average is test sen as a representative value which can be wed to represent te alu ofthe general tem
na gro of values.
For intanes, uppone thats ticket tum had 10 potreros as follo:

Vice Pt wiket 42
viet 12 iaa
SP wicket So wieker25
wider #1 wicket 18
oF wicker 9 1 wicket 15

On adding the en vales above, we get total of MO-—which gives an average of |
3: uns pe wicket, the average partnrship ofthe team was 4 u

74. DO | tenor tet CAT

In other words, if we were w replace teva of all he ten pans y Bins, we so perth same ta. []
score. Hence, 3 represents he average partnership value fo he tam. |

‘Seppose, in a cricket series ofS matches between 2 comes, you are given that Team A had an average
anne of 8 Runs per wicket while Team B hada average pares im per wicker. What cochon
«an you daw shout the performance ofthe two teams, given ha both he eus played complete text matches?

‘Obs Team B would have performed mech worse than Team A: Ferhat mae. if cell you that he average
aye high ermperatue of Lucknow was 18" C fora area month, you can aly daw some kid of conclusion
in your mind about the month we cool possibly be talking about

“Thus, you shoul realize hat the Deut average isin heft thet ts one ingle manier tha ells you

out the group of numbers ~heace I one number that represents an ete group of number

‘ee of the ey coaceps tht you need to understand before you move iat the chape of ths block ie
the Coney of WEIGHTED AVERAGES,

As ala, the conos is best expand though e concret example.

Seppose 1 had 1 buy a hi and a oser amd les say tol Ihe average cos fa shit was RS 120 while
that of tour was Ra. 900.

In such a case, ih average cos ofa shi and trouser would he given by (1200 + 90012 = 1050.

“Tis can de whale onthe number line as

Aidit) amer
so 10 00
a ++

‘As you can cay see in be figure, the average occur he midpoint ofthe (no numbers.

Now, let u ty 1 maby the stanton

‘Seppo, wer to buy 2 trousers ad | shin. In such a ns wosid endup spending (900 + 900+ 1200) Ra.
3000 in baying a toa of tem, What would be my average in ths case?

Obvisly.20003 = Rs 100011

Clearly the average has sited”

‘On the member ine we cull visualize thie follows:

sr]
nn toa
<<
ick ie that be verge hs sid towards 00 (ue we cnrs fie oe age
pacas)
aya salte ay pan weighing blanc on weights bing pt on The blanc

sits wards he pan containing the lager weigh
‘Sima, this as, he comet average (100) ose to 900 than isto 1200, This as happened because

‘the numberof lement in the group of average 900 is greater than the number of elements in the group having

areas 120 Sic, le slr othe sem of weg, we alls as a ace sion)
Ati stage you shout realize that weighed averages ar no solely reiten D

‘wo groups We ean also come up with a weighed average station for thre prs) |

(although in such a case the representation ofthe weighted average onthe number In | |

right ots ely pose fact i isthe number Inerprsetaionofa weighed ||

average station hit fs defined as allignion (ben 2 prope are involved)

75. 92 | norbPeauet Que mtr

‘The number of educated employees working inthe
rico ae:
ws on
on CE
16. Mr Kilos Bapa wile going from Lucknow to
Sanshedpar covered hal the distance by tin a the
seed of 96 kas hal he resto the distance by his
Scooter atthe speed of 60 km and the remaining.
‘stance athe sped of 40 mr by ct. The average

seed a which he completed is Jamey i:
(o Sem Sohn
(e) ¿Ot (0) km

1. Ther are four types of candidates in AMS Leaving
Systems preparing for the CAT. The number of st»
‘ents of Enpineiag, Science. Commerce aná Ha-
malts is 400,600, 500 and 200 respectively atthe
respective percentage of sudets who qualified the
CAT is 80%, 75%, 60% and SOS respectively the
‘overall percentage of sccestal casas in oar

@ ane Dr
CA (@) None of these

18. Me agota ku the menge of 10 “Tae
sigh number. Bt ie o misa e een ie
ig member ds his arg ceed y
36. The dfn beeen he unit ii ad br
Arc digit of tat member
w. CE
@2 {0 cont be emia
Directions for gustos 192: Answer be pesos il
co e lowing infers.
Fron pen for ube of is In bc et et
er
Das [HEIEIENEIKIE
Number ont [18010 120 so [i 200)

Fra track that an cary 2.000 cols fc, iting cos pot
fay i Rs. 1.000. Storing cost per cubic ees Rs. per day.
‘Any reia) materia tthe en ofthe seventh day has
to be taste.
19, Tall ie units sould be set to the markt then on
which days should he rks heise 1 mire the
nn

2 the sorge cos reduce to Rs. 09 per cube ee
par day. then on which daylday, hou he tek be

iret?
oe or
(OT 0 None of thee

Answers (Block 2 Preasesment Test)

L@ 20 3@ 4% 10
6m 7H ko 9 Mme
1® 20 RO Ke KE
ba MO Ro Be 20

SCORE INTERPRETATION ALGORITHM,
Block 2 Preassessment Test

IM You Scored: <1

fin Unlimited Time)

Step One Go trough the rs chpter oh Doc. Vie
Averages. When you do so, eoncentte on ceatly
ndersardig ech ofthe concepts explained in the chapter
theory.

“The move ono the LOD | exercise These exercises il
provide you with the first level of challenge. Try to solve
ach and every question provided under LOD 1. While doing
o donot thin aboot the ms requirement Once you ish
solving LOD 1, revise the questions and ther solution
presses.

Also at tis stage stoy the concept of averages fom
your schol tet books (Clas 8, 9 & 10) and solve al the
‘uetins which ae available 1 you in hate books.

Step Two Alter nsting LOD 1 of Averages. move nto
{LOD 2 and then LOD 3 ofthis chape.

Step Three Go tothe copter allions andy the
shorts provided cre. Understand the ate of the
allguon proces as clay ax posible.

Then ove oe LOD I esersise ofthe same. (Now: The
tapes of alguns doesnot have an LOD? & LOD 3
ete)

‘Step Four Goto ie practice est given at te endo the
Block and solve it Wal oi so, ist ok tthe core you
et within he tne lit mentioned. Then cocine to solve

76. ‘he et fre witout aime Hit nd y 1 evaluate the
impeonenc in you unlimited ie ser.

Tncasethegrowthin your scores ot siria, go back
to the theory of each chapter and review each of the
questions you have solved fr both the chapters.

POE]
If You Scored:> 12 (In Unlimited Time)

Follow the same process as aka. Te only ference shat
the schoo book werk i optional — do iter if ou fel you
ced to. However, your concentran during the solvia of
the two chapters has to be on developing your speed at
solving questions 0m this char

77. AVERAGES

INTRODUCTION

‘The concep of Averages important ine questions based
‘this chapter egal ppea inal age exams While
‘most ower level MBA ex aswell as Bank PO, NLS, NIFT
exam tet the concept of averges through dct qui,
be CAT and the XLRI examinations have used the concept
to st concepial problems that will no be Found in most
textbooks. Such problems have ben given a this chapter in
{LOD I a LOD tt questions.

‘While studying thie chapter, make sure that you under
stand the techniques of se late computation in the
text. These ecmiques ao esten] bas on cena com
eps and understanding them wil in tum, help you pet à
beter grip onthe cone.

‘THEORY

‘The average ofa number isa mesure ofthe centrale
ofa setf number. nother words iti an estimate of where
the center pot ofa st of number Les.

The bi formula forthe average ofn numbers 11

sis

DETTE
numbers

This alo means A, x = eal of he st of numbers

‘The average is always callo fra set of numbers.

+ an = CT of st of

Concept of Weighted Average: When wehave2or
more groups whose Individual averages are know, then 1
find the combined average of all te elements of all the
groups we use weed average: Ths if we ave groups

ih averages A, yA and having elements
then the weighed verge ds given by the formu:

„Am Atm Atom A

‘Another Meaning of Average The average [also
Ava es ahnen mean (AN) ofa St of numbers can
tho be defined as the mumber by which we can replace uch
Anderen umber of the st witht changing the tal ofthe
set of number

Properties of Average (AM) The propeniesef aver.
ases aime mean] can be laide by the following
pie:
Example 1: Thesvergeof number 12.13.17 and Ihe
Solution: Required average = (12 + 13 + 17 + 194
ET
‘This means hat ich of the 4 manie of the set were
replaced by 18 each, there would be no change in the sal
‘This is an Important way 10 Took a averages. In fact,
whenever yu come actos any sition where the average
‘of a group of a numbers is give, you should visualise at
there are numbers each of when vale isthe ara of
the group. This sie isa ery important way to vise
vengo.
This can be visual as
nass
Bars
42915
15623515
ETE

78. 96 | Homann Apr emo CAT

Esemple2: In Example visuals ado fa mo
ber. which increases th average by 1.

Sete 16

setae

15+1=16

15+1=16

"The + appearing mes de othe ith mame, which
sable 1 ain the average of 16 int and then ‘ve one"
10 each of te inc,

Hence, fila number in his case i 20
Example 3: The average always Ties above the lowest
umber ofthe se and below the highest number ofthe se.
come de The net defi eto he morir below the
average always quals the et surplus de to the number
above the average.

Exemple Ages and averages: the average age of a
up of persons is x years today then afer years Weir
average age wll be (4.

AA, years ago ber average age would have been
(a — 1). This happens due to te fet tat fr a group of
people, 1 your sade to each person's age every year.
Ecomple 6 A man rc a 60 mph onthe journey from
‘Ato Band cars at 100 kph, Find his average speed for
the journey.

Solution:
Average spec = (otal distance Mica im)

we assume distance between 2 poets tobe d

Then

Average speed = 2 (4) + (100) = (2% 60 100%

(6+ 100) 260% 100Y150 =75

average peed 125 SMS
1S: sa Sate speeds)

‘of going and cong back respectively.

‚Short Cut The average speed villa come out by the
following process:

‘The rai of speeds is 60: 10O = 3: 5 (sa 7): 1)

‘Ten, divide the ference of speeds (40 in this case) by
+13 45 = 8 in sc) o get on par 408 = 5
this ea)

“The required answer wl be the parts a Le. parts
ana from the lower sped

‘Check out how this works withthe following spect:
5)=20 and

Stop 1: Ratio of speeds = 20:40

Step 2: Divie difference of 20 int 3 pas (n + 7) >

20 macs

Regard average speed «20 + 1686
ai: Ths process is escenaly based on aigaons and we
shall se apa in de next hate

"EXERCISES FOR SELF PRACTICE,

Find he average spend for he above probes if
ms S,=200
pers
S50
1

WORKED-OUT PROBLEMS

o AA
vrs 56 os priainge aer 2 ning ha vea
Increase y 2 runs then tr mas i sor ne 26h
ining?

SOREN Normal roces

ins a2 eg = Ras tl tr 6 Inia Ras
tal af 2 nino

22658-25555
Fue met ets sc:

(9542) 26~36%25,
226+ (5626-5625)
=52656= 108

Short Cut Since the average increases by 2 uns per
innigs iin equivalent to 2 uns being aed to each score
in the fs 25 ining. Now. since dese rans can only be
‘ed bythe rons scored inthe 2640 inlng, the sore in
‘he 260 inning mut be 25 x 2 = 50 uns higher nen the
average afer 26 insngs (i. new average = 58).

‘Hence ran coed in 26h ining = New Average + Olt
Innings Change in average

Er ee)

79. Visas his as

Average in int 25 imings Average fer 26 innings
ES s
ES =
ES =
ETS ET

Ditfercce ina vo 25 tics and 58 once, at, 8
+22

‘Tae average age of cs of D tuent and
‘teacher reduce by OS years fe exclede the teacher.

A the ita average i 1 yeas, ld he age of the class
teacher

Solution Normal process:

Age of teacher = Toul age of (dems + wachen)
= Total age of atente
= 3114304135
cas
29 ya

Short Cut The acter ale filing te average of 13
(othe group to which he belonged) slo able give.
yeas to the ae of eah ef the 30 sets, Heace be as
30% 05+ 15 years to ge over and above maiiaining his
own verge age of 14 yeas

‘Age of escher = 14 +3005 =29 years

(Wore: Tis problem should be viewed as chang of average
rom 1531014 when teacher inc.

TEESE th serge mr ofa gronp of 20 sndents
na tet in reduced by 4 when th topper ho see 90
‘aks replace new des How many mars ide
sew sde have?

Solution Normal proces:
Latina weruge be.
“Then the nial tal 520%
New average will be (e 4) and the ne etal wil be
20 (2-4) = 20-80.
‘The reduction of 0 ls created by the rplacemen.
ence, he new sten RO marks es han the sale
te replaces. Hence, be must have scored 10 marks.

cos hae HE

Short Cut The placement has the fes redocng the

ancrage mark for ech ofthe 2 students by 4. Hence, he

replacement mun be 20 x= 80 masks teow th original
Hence, answer = 1 mas.

“Tre merge ge ofS ene, Band Cis
AS mato. Aner der D ja the rep ad the on
average becomes M mas fant dent E bo has à
mars mor D. joins the grap. he average of he 4
Stet .C.D and Eecamen 9 mues. ind bow many
musa gotintbe exam.

Solution Solve while reading, Th int sentence gives
you ata of 14 for A, Band Cs mais Second sentence:
‘When joes the group, the cal becomes 444 = 176
ence Dus get 32 marks.

Alteretivly,yua canceach his pin by considering the
fax 2 statement ger a:

{jing the group educ the average from 48 1048
mas (ie mas)

“This means tht mini average of 4 marks, Dias
totake 4 mack from A, rum # and. rom € = À to of
12 murks. Hence he mast have got 32 mus

From ber:

“Te fist par ofthe thin sentence gives u infomation
soot E geting 3 marks more tan 32 =» Hence, get 35
maño.

Non ti fre td ar whe 4 placed by Ene
average marks ofthe stents reduces by 1 03,

Mathematically és can be shown a

ABS CoD eddxse 176 while, 89094 E
ETES

Sabina he two equations, we get À B= 4 maso,
ence, A would have got 39 mks.

Aliratvely, you ca thik of his a:

“The replacement ofA with Eres in he rection of |
mark fom each ofthe people who belong 10 he group,
enc, he ference sd ark, Hence, A wou get mara
more tas Eie.A pet mau

RARE me meer cerpersure of Monday o
Wednesday was 27 °C and of Tuesday to Thursday was
24°C. fe temperature on Thursday was 23 of the tem-
erature on Monday. what was the temperate on
Thursday

80. As soon as you get It the second ine ofthe question
Et beck o the fit sentence and get the tt number of
‘passed sadents = 2 +64 12 and you ae though withthe
problem.

Level of Difficulty (LOD) Il

1. The average age of 24 students and the pricipal
13 years, When the principal age ls erde the
erage age decreases by | year What isthe age of the

princi?
(38 oo
©» ws
(@) Data inadequate

2 The merage weight of 3 men A. B and Cs 84 ky,
AAnctber man D join he groop ad the average now
becomes BD ¡Fama mas E whose weighs 3g
more tha that of D replaces Athen the average weight
LB, €, Dand E becomes 78 kg. The weight of A

© Toke CE
(9 kg CET
CET

The mean temperature of Monday to Wednesday was
37 °C and of Tuesday to Thunday was 34 °C. Ifthe
temperature on Thursday ws 4/5 that of Monde, the
temperature on Thunday vas

ES @ 36€
o ac ww
(6) None of these

4. Thre yea apa, the average age of A, Band Cwas 27
ea dt of Band CS yeas ago was 20 yeas. A's
Prsem age i
(o yews
(40 years
(6) None of ese

$. Aj ende las cain serge for9 mings. ne
tenth eng. he scores 100 rus hereby creasing bis
average by 8 nu. His nen average is

(0) 35 years
© 48 ans

@20 OM OB wR
CET

The average ofthe ft five male of?
@ OA 02 wD

on

”

rent o [E

ce ci A, Bad CCA Ea

= and average of A. Band Cis 112. What isthe
value of?

win CE

© 18 an

(e) None of these

“The marks tired by Hare Rama In Mathematics,
English and Biology ae respectively 93 ou of 100,
780101150 and 177 out of 200. Fd bi average sone
in percent.

ES
ons

“The average monthly expendiure of a family was
RS 2750 foren 3 mom, RS. SO fore nese
‘mont ad Ra. 6750 forthe ext tc mente. Fa the
average income ofthe family or the 9 moat, if they
save Rs. 650 per om.

rn BH

(a) Ana CES
(o ao nr}
(e) None of dese

‘The average age ol family 6 member is 22 ea.
le age othe younest member be? yea, what was.
the average age ofthe family athe bit ofthe youngest
ment

as on
(e) None ofthese
‘The average age of 8 persos ina commit sin
creased by 2 yca when two men aged 35 yar and 5
yess ae subsite by (wo women. Find the average
age ofthe two women

ws was
46

“The average temperature for Wednesday. Tray and
Fay was 40°C The verge for hunda, Fly and
Satu was °C the tempera o Sted was
42°C, what was the temperature on Wednesday?
We wur @ ME WAC
(e) None ef hese

The speed of the tula in going from Nagpur 10
Alla 100 kdb wile when coming back from
‘Allaabod to Nagpur, is speed is 150 nhs Fin the
Average speed during the wok joerey.

ws DS ORS Wh Ko

te 120

oa wr

os wa

81. TOO] Horo rro Ose dtc xt CAT

A. The average weigh ofa class of 29 dent is 40k.
2 the weight ofthe teacher be incl, the average
ies by 500 ge. Wha is Ihe weight ofthe teacher?

la 45e CE
CET CES
CET

15. Theavergeof3 numbers is 1 end ha of he fs wo
1516. ied the tid number.
os on on
os
16. The average weight of 19 ren in à hip s incre
by 3.5 kg when one of the men, who weighs 79 kg. is
replaced arew man. Find de weightofihe new man
pte 2 decimal places
(a) 10525
(a 15.90
(6) None of these
17, The age of Shaya and Kauraki sn he ratio 2: 6.
After 5 yeas the aio dci ages will become 6:8.
Find the average of heir ages ar 10 yea.

w »

0 10755
(9 10050

on OB OF wm
(6) None of these

18. Find the average ofthe fist 97 mutual number.
on DT 08 04
(a 495

19. Financ verge ofl prime mers benne 0 ná 50.
PEE
(o as

22 we take four umber. e average ofthe fs three
is Und that of the ast re 1S te lst mbr
is 18, de ist number is
an on
(2

AL. The average of consecutive numbers in. Ifthe nent
two numbers ae ako ich, the average wil,
(increase ty 1) ems the me
(©) increase by 14d) increase by 2
(© None ef these

22 Travers of SO mener is 8 wo ruben, aly,
AS and 5S are discarded, ih average othe remaining
ember is
CET
(o ys

0301

on om wm

ZU The average often numbers is 7. IF each number is
malipli by 12, then tbe average ef the nen set of
number is
ar 0»
os

24. Ina fay of 8 malos and few ais, he average
monthly consumption of rain er heals 105 kg. Ifthe
verge monthly consumption pr headbe 15 i tbe
se of aes ee 6g ithe ase of males, fi the
aber of females in the fay.
as O7 Ho ws

on os

os
25. Average marks ohne by a student in 3 papers ls 52
din he fourth paper he obras 60 marks. Find is
few average
ws

On os gas

Re. 1200. hic average ering forthe acd and sh
months Rs. 1300 find his ering in the ist om?
OR ODO 1200,
om

27, Ina hoe where rooms ae numbered fren 1011 130,
ach 100m gives an caming of Rs. 3000 for the fist
fifteen days of month ad for he ae half, Rs. 2000
per room. Fin! the average earning pe oom per day
over the month. (Assume 30 day month)

@) 250 ©) 20 (2750 HGS
lo 2683.33

28 The average weight ofS men isdecrcas by 3hg when
one of em weighing 150 kg is replace by another
peros, Find te weight ofthe new pero?

@ 1652 Da
Ca @ 1x
(o 16528

29. The average age ola group of men Is increase by 5
year wena person aged 18 years replaced by anew
Person ofaged 38 year How many mena erin e
rep?

@3 4 ws 6
(97

30. The average score of a che thee mus is 22
runs and in tuo otber males, is 17 rns. Find the
average inal he five matches,

82. TOR] ow Peeve cure Apto CAT

(o) Gyan
(o Syn
(6) None of these
45. The average salary of20 workers ir an ofi 1900
et month Ihe manage” lan sad, he average
salary becomes Rs, 2000 per month, What is the
nager anal salary?
(y REZAGO
€ Raso
16) None of hese
46 tha, bye dende we five coscctve ol pubes ben

© 48 jean
(8) years

0) 25200
WR.

Weir average is

@) ash) © ehedens
ars

@ Sarbrerdra @ tt
(6) None of these

47. The average of fa five males of 3 is
63 9 on ws
4) None ofthese

48. The average weight ofa clas of 40 dent i 40 bg.
the weight ofthe teacher be included, he average
eight intesses by 500 gm, The weight ofthe teacher

CPS © is
CE OS
(o 64g

49. Inamanagement cancel sud scores aks
For every onect answer nd oe 0.5 mark for every
Wrong answer. A rude tempts he 10 quesicns
and scores 120 marks. The number ef quesions be
enter cometly was
PEE
(e) None of these

$0. The average age of Four children is 8 year, which

ceased by à yeas when the age ofthe uberis
led. Find the age of be father

oo os

@2 GA | wm
CET

SL. The average of the fist en natura mens is
ws OSS 06 HS

52 The average of te fist ten whole numbers is
as OS 085 wa
2. The average of the fs on even mues s
ms 42 as @

The average of e Fist in 08 romber a
on 9 07 @9

5. The average fe fin ea prim members à

os 05 00 @ RD
‘Te aveage ofthe ft en compost numbers is
or mn om wm
‘The average ofthe fis en prime tubers, which are
olds

om OBS |] IS
‘The average weight of class of 30 sets is 40k.
however the weight ofthe tsche is nude. tbe
sverage become 41 Ag, The weight af he teacher is
U DEE ON EN
Rara bought 2 toys for Rs. $.50 each, 3 toy for
Rs.368cach and Stop for Rs, 1.833 each The average
price per ty i

RI MR ORAS REY
30 ranges a 7 apples were purchased for Rs. $10.
Ie pice per apple was Rs 2, hx he everage pice
of oranges was

RC DREI CORIO) REIS
“The average income of Sambbu and Ganesh ds
Rs. 3.00 ad tha of Arun nd Vinay is Rs. $00, What
is the average losome of Sam, Ganesh Arun and

Ver
(3 Re 1750 Re
OR @ 200
A atimn ade an vea 40 us Is. bat

in the ih inning, be was ou on zero, What ls e
average afer fit inning?

OR 2 O* WH
‘The average weight of schoo! of 40 each ls Oh.
however the weight of he pricipal be cluded. he
verge decreases hy bg. Whats the weigh ofthe

rép?
(0) 10% o
BET, (8) None thse

‘The average temperature of It, 20 and 310 December
as 244°C. The average temperature ofthe frst two
days was 24°C. The temperature om the rd of Dever
eros

(200 CES

(o 520 (0 Nove ef these

‘The sverige age of Ram and Shyam 20 year. Tee
verge age years hence wil be

83. a

a

=

(0 25 years CE
(© 2 yews (9) 20 yeas
The average of 20 ress 30 and ta of 30 more

read 20, For all the results taken tog, the
erage is

03 00 on ma

‘The average ofS consecutive narber I I. The igh
est of these members will be

Ou 08 00 wR

The average of 6 stndent ie 1 yan. more students
of age H and 16 yeas Jin. er average vil become
lo 2yews © yes

CET (© 19 year,

‘The average of £ numbers ic 12 1 each rumber is
Increase by 2, the new average wl be

DE OH Ob wis
Three yeas go, he merage a of fail of Sem.
bers was 17 years À Baby having heen Bor, the aur-
ge ofthe Fry athe same todas. What she age of
the baby?

Dar? @ 2 yeas

(©) mondo 16) 9 mons

Samblu’e average daily expedite ie Re 10 during
‘May, Rs. 14 doing June and Rs 1S daring July. His
appris daly expenditure forthe 3 mor à
(a) Rs. 1S epproximtely © RZ
(0) Re approximate rn
A ship ails cut markt ie rat of Sk per bout
and sl back thereto 20 or, What te verge
tate of sling?

(a) 16554 ©) 12m

© 78h (@ 180

‘The average temperature on Monday. Tuesday and
Wednesday was 41 °C and on Tuosdiy, Wednesday
and Thursday it was 40°C: [fon Thursday it was excl
39°C. hem on Monday. the temperate ma
WRT HHT OBC ww
“Tha sers of 20 rests SD ot of which the fit 10
rests are having an average o 10. The average of the
et Wem is

an 4. on ms

A man had seven children, Wien theiraverage ape was
12 year child aged 6 years ed. The average age of
evening 6 ibn is

(a 6 years © 13 years

© Tyee (8) 15years

Gex az [108

6 The erage income of Ram and Shyaris Rs. 200. The
average income of Rahul and Ro Rs, 250. The
average income of Ram, Shyam, Raul and Rois
(a R275 RS
(o Rs 880 (9 R280

TI. The average weight of 33 stent is 35 hg. the
teacher is lo inched, he average weight increases
10 36 kg. The weight of he teases is
Gi Mi OO

TA The average OP, and Sa much more han
the average ss ys less than he average. Find th value
ES
was

os 08

os

3 A

ok om

‘SL The average salary per bend of all the workers in a
company is R 98. The average salary of LS fice is
Ra 525 amd average sary per ead the rest Ra
85. Find the nal number of workers in the workshop,
Gé @ CH WHO

BL. Thesverageage of men increased by 2 ycan when
one of bem whose age is 24 years is reply a
woman. What i the age of he woman?
da) 35 yeas D 28 yeas,
€ yen (years

2 The average monthly expendiure of Ravi was
Ra 1100 daring the ist mono, Rs 2200 darin the
ext monts and Rs. 620 orig the bs five
moots ofthe yee Ite total saving during the year
as Rs, 2100, find avis average monthly income.
(o Re 1888 Dee
© Ra (9 None of these

5% Shgam bought 2 ailes for Rs. 550 ech, and 3
“rice fr Rs. 350 cach and arce for Ra 5:0 cach
nd. acts for Re 1.0 each The average prie or
ve ailes
@) Rea © R310
(9 R390 (0 Ra?

34. In dag. ee ae 150 cons of Re 1, $0 p aml 25 p
denominan, Ifthe ttl ale of cone i Re. 150,
then find how many rupes can De consticd by

3
ot

50 pois.
ws mo
oz (0 Nene of thse

84. OA] non Popu te meta Bruce PONT

Level of Difficulty (LOD) Il

1. With an average spend of 40 Am, a ri reaches its
destinada intime. ft pos han average speed of
35m is ate by LS minutes. The length ofthe wal
oumey is
(@ km ©) O
(0 Om

2 Inthe month of Jul of acenain year. the average dal
expenditure of an crgatistioa was Re 6, For the fit
15 daa of he month the average al expenditure was
eR and forthe ast 7 ays, Ra SI. Fed the amount
spent by the oracisic on he 15th ofthe month.

© Mm O Wim

(o RA 0) RAS
ORM RAD
(o R40

A I 1919, W. Rhodes, he Yorkshire rater cor 891
ru for bi county st am average of 4.7: in 1920, be
scored 929 ns a an average oF 28.75: in 1921, 1329
una atan average 1287 and 192, L101 rasan
average of 36.70, Wha was ht county bang average
fer the four yeas?

CCE)
(6) None ofthese
4 tin ayes wih a spend of 20 mis inthe Fit 10
tries, gros. kin th est 10 mites, 1 kin be
‘ext 10,8 km inthe ext 10 ad Glam in he nen 10
minutes. What is the average speed of he tin in

Filme per hou or he joe) describe?

CERTES

6) amp CET
(552m CET
CES

One out ein ourey is covered be rte of
2S En, one ind art 30 kon nd the ret at
SD km, Find the average pe forthe whol journey.
(9 ASIA) LOSA
A GO MODS
(©) one of these

& Typist 4 can ype a sheet In 6 minutes, pt Bin 7
Ines and typist Cin 9 minutes. The average munber
ff sheet typed pe hour per typist forall thre tis
is

CE] CES

©) se
(©) None of hese
Find the average increase ate if increase in th poe
asii it year 20% and tha inthe soon year
ss

wa
ee»
“he average income of person fr the fia 6 days is
Ra 29, or ner 6 daysitisRs.25 fr na IO days
Iris RS. 32 a forthe emaining days ofthe month it
ls R630. Find the average Income per day.

fa) Rae (o Rezo

(e) Re2926 @ RID

(6) Cannot be determined

Labo Jasmin he roms ae member rom 101 o
130,8 fi oo, 221 2 en second or
30610345 onthe hd flor I the mort Jus 2002,
‘te room ocupan was 60% o he fit or. on
second foo and 7% onthe thi Noe Fit also
known thatthe room charges ae RS 200, RS. 100 and
Rs. 180 on ech ofthe Mors, the find the average
Income per rom forthe moat of June 2002.

CE

os on Oe

(0) RsASIS © RIT
(o RS (Re 47
(6) None of these

A salesman gets» bonus according to the following
toc: he sels articles woth Rs ben be gts
han of Rs. (100 1). a the man of Jana. his
soles value was o. 10, in Febuary was Rs. 200, rom
“March to November ic was Ra 300 for every moat nd
In December it was Rs 1200. Apart fom this. he also
receives basi salary of Rs. 30 per mark from his
employer Fi his average income per meth dig the
yea

(a RA3125 Der
CFE (ROMS
CRE

‘Aman cove hal of his ey by tin at OR al
ofthe eine by hu 30km and heresy cycle
22 10 keh Find his average speed daring the entire

cures.
EC CET
CE 10 IS
CES

85. LOG] nur cars apto car

wm
om

25. The average slay of the emir staf in an ofc is
Re S000 perro The average slay lice Rs
(00 an tht of sonia RS 2000. ewer
office Sn ind the number fon fics in.
the office?
as 02 os ws
on

26. A person travels es qual distances u a speed of a
An Amand km respectively. What willbe the
average speed during the whole joue?
A @ Gy +32 + age
CETTE (0 None ofthese

Directions for Questions 27-30: Res lowing panago
and answer the questions tha lon,

Ina Fri of fe persons A,B. C.D an E, uch and
everyone loves onc another very much Their itéays are
Inder oat and on different dts. remerber tat
is Bray is horucen 250 and 300,0 is eres 20%
nd 29h, ol Cis beten 14h and 200, of Dis between
Sth ad 10 and of E ti between It 10 Sth ef he mon
‘The sum ofthe da ofthis define as headin f the
(ie an he mont. for example 12: Janay willbe writen
1512 and will ado wu of he ai 3, Berween 25%
ar 30 ete ach 25 and 30).

27, What may be the maximam average of ir sm of the
zs of bia?
was 152
os on

28. What may be be minimum average ofthe sum of he
ses of bins?
@ 246 o 12
CE w

2 Matisse thatthe dates of bin of tee of them ae
ven numbers hen id maximum average of his
‘tthe dass of Bt,
was win
ons w 2

30. the dat of bin of for them ae prime mante,
then in the maximum average o o sm ofr dates
ol tit,
(o 72 CET
on 19 None of these

on om ww

231 The average pe of a mp of prams ging for a
en is 16.75 year. 20neu peso wih an average
age of 13.28 year join the group onthe spr due o
sich the average ofthe group becomes 1S yea. Find
the number of persons iria puig forthe pul?
Ox OD & on

2 A school has only our class ht coman 10,20, 30,
08 20 students respectively. The pass porentoge of
beso cases a 20%, 20%, 60% and 100 espe
tively. Find the pass of he eine school,
© OH TE TT |

A Find the average JU 560. D, a= 10
Ash +2, ga) 503 Ko og and fe)
sane
om y ms | ADs wm

34. Find the verge fx) = tes) A) Mi) A),
40-10
@0 23 (099

w 23

3 Êmrrateeren
genes à
2

n=? e

CES CE

26e enge of nun iz erde re
replaced by the number 2’, thea the average becomes.
22. Pind the relation between m, , 2 and x

©

37. The average salary of workers In AMS careers Is
Rs. 2.000, the average salary of cal being Re 4.000,
Andıhe management trainees being RS. 1.250 The ul
umber of workers could de
a m m ws“

Directions for Questions 38-41: Real the following and

answer the questions that fallow.

Daring arte mach India playing gast NZ scored ia
the fllowing manne:
Pormerikip Ran scored
ticket 2

86. 2a wicket
Antwicket
déchet
‘Ss wicket
Gin wicker

Fie the average runs scored by the Fist fost basen.
lo 5 09 65

© 668 8) Cannot be determined
‘The maximum average runs scocd by the fist five
batsmen couk be

Pen wos

(076 1 Cannot de determina.
"Te minima average uns score y thes e at
men o get out could be

DET us HE |O

oe ith done batsmen get ou ar dock th ind
‘he average runs sored by the fis six batsmen
ma was

lo 485 (8) Cannot be determined
"The weight of body a culata by the average of
Tre experimente is 5335 gm. Te verge of he
fin dee experimens 54.05 gn, fe Four is 0008
meer than the ft, bio he average af the sath
and seventh experiment was 0.010 gra less tha the
average ofthe it thse. Fed the weight ofthe body
‘btn bythe our espere

@) 933 pn © SL7I2gn
CET @ Stage

Beene

A mas average expendiie forthe Fit $ maths of
the year was Rs. 251.25, For the net $ months the
average monthly expenditure was Ra 2627 me than
‘vi twas during the fit month pes rent
Rs, 760 inal during rain mont of he ea,

(o) sé
13 Nose of ee

A certsin numberof rocks were oque o tramos
(60 tons of «eel wire from the TISCO facto) in
Jarabe. However. it was found that since euch
truck could ke Stons ef caro ls, other Suche
were reed, How many trucks wee ni planed to
be used?

on ms

on» wx

gr: Au [107

45. One collective frm pot an average harvest of 21
of wheat and anther olive farm ha ha 12 ares
of land ee given to wheat, gi 25 ons fom scare
As a result the second fama harvested 300 tons of
‘whet more han the ir, How many ors of wheat did
sh fam bart?

(o) 31503450
(o 180.2890

aso
(9 None ofthese

Level of Difficutty (LOD)

Directions for Questions 1-28: Read the flowing
‘Ther ae lanes having 20, 25 nd 30 sons ope
tively having average mars in an examination as 20, 25 and
i respectively. Ihe hoe cles ar represented by A.B
tod C and you have the following information about the
thee clases, answer the questions hat follow:
Am Highest score 2, Lowest score 18
134 Highest score 31, Lowest sce 23
Cm Highet score 33, Lowest sore 26
Ue fie sant ue andere fren A 1 8.
1. Wat wil happen 1 be average seo of B?
(a) Definitely increase) Definitely decease
(6) Remain consam (4) Camo sty
2 What wil happen othe average sore of A?
(6) Defimlyincrese (9) Definiely deceo
(6) Romain consi (8) Cant say

Ina rame ofS students rom Ato €

3. What vill happen othe average soue of CP
(4) Definitely erase (1) Deich decrease
(6) Remain cons (8) Cant say
4 What wil happen to he average son of 42
(6) Definitely increase (0) Definitely dereme
(6) Remain constant (4) Canna sy
Ana transfer of 5 tnt from Bo € (Querons 5-6)
5. Waat wil happen t the average score of CT
(a) Definitely increase (0) Define decrease
16) Remain cons (0) Cannot say

6 Which ofthese can be sail about the average score
am
(a) lores € decreases

87. 2

2B

2

2

2

© 36
(©) Cant say
‘The minimum posible average of group B ater this st
of operation it

was

(o 18

(0 Cancer say
‘The minimum posible average of roupA aer the sc
ofS operation is

CE)

@ 2 ma
© 24 18
(© Cannot say

Which of these will definitely ot consiute an

operan for geting the minimum possible average

vale for group

(0) Transfer of five 3s rom B10 À

© Tasse of five 26 from Ci $

16) Transfer of five 2s from A to B

(@) Transer of five Ds fom Co.

(0) None of these

For geting the lowest possible vale of C's average,

the sequence of operations end be

(6) Transfer five 3 rom Cw ie 234 fn B 0A,
five 18 fom Ato B five 185 fom BC

(0) Transfer five 33 from Co E ls fram 804,

(9 Bor a and b

(0) None of he above

© Camot say

we st the ges posible average af ass Cas the

primary objective and want to achieve the highest

sible vale for lass Bas the secondary objective,

‘whats the maximum vale of las D's average thats

aile?

on

CE

Fer question 26, e secondary ebective l changed

16 achieving the minimum poschie average vale of

lass Us average, the lowest vale of cas B's average

at col he aie ie

os 05 x

om 02 0m 12
ou
For question 27, what can be sald about class A's
enger

(6) Wülbe determined auomclly st 2225

2

Ones. sue [108

(©) Wi have maximum posible value of 22.25,
(e) Wilhaveamixinenposcbl vale 0122.25

(© Wilde dermined automaticaly 122.5

(6) None of these

A team of miners planned to mine 1800 tors of
ore daring acen number of days. Due 0 technical
icles la oneabin of the planned number of
dys the team was able to achieve an ouput of 201008
‘of ore less than the planned ouput. To make
19 für this, the tara veracheved forthe rest of
‘he day by 20 toms The end resll ws tha the team
‘completed the tsk aoe day ahead of tne. How many
tons of re id he tam inal panto ore per day?

(m Dim (0) 100 ons
(© 1501008 CET
(e) 2501008

According toa pan tes of woodeoters decided to
Invest 21610 of wheatin several days Inte ft ree
days, the team failed the dally amigos, and then
it harvesed 8 m? of wheat over und above the plan
everyday. Therefore, a day before the planned dat,
‘they had already harvested 232 of wheat How many
cubic metres of wheat à day dd the team have to ut

according 1 the plan?
won on 02 (92
(928

On an average tuo les o milk andone tro water
ar nerd abe mis o nl 1g of stand.
‘of ype A, amd 3 es of milk and 2 Her of water are
needed tobe ised to make kg 0 sade sad of
‘ype Hom many lograms ofeach ype of sind
Was maricas loma 130 in of mi nd
80 ie of wae were sed?

(8) 20 of ype À and 30 of ype B

(©) 30 of ype A and 20 of type B

(62 15 of ype A and 30 of ype B

(9) 30of ype A and 1S of ype B

(0) None of thee

‘Ther se 30 sein Minerva Crema, Marne, pce in
similar own. Afr the rsontrcion ofthe bal the
total number fest became 10% lett. The number of
roms was reduced by at each row contained sets
‘more than before. How many rows and bow many sets
Inarow were thee idad in he Rll?

(0) 20 ows and 25 seats

(©) 20 rows and 20 sea

88. LID] tow Peroote ose toc opa CAT

(62 10 ms and $0 seas
(@) SO rows and 10 seas
(e) 40 rows and 20 seas
AL One fasion house has to make 810 dresses and
the ene 9 dresses daring the sme pio fie
In he fist house, the ceder was ready 3 days sed of
tinea in he econ House, days head of tine: How
many dress di each fashion Rowse make a dar if the
second house made 21 dieses more a day ihn the
fine
(a) Stant7s
© sates
(e) None of these
‘SLA shop soli 6 Lets of wo different capaces. The
malle cos ape les than the large one. The
hop made 100 pees fom these of age kes end
“pees fom tbe sale of sll nes. How may tes
‘tether capacity id he shop sel and what was the
price ofeach Lette?
(6) 20keesfor2Sropees each and A kets for LS
rupees each
(D) sO ketles for rupees each an 24 ets oe 23
ees each
(©) AOkees for 2 8 rapes each and 24 ets for LS
rupees each
19 either ad
de) None of these
35. An enterprise got a bonus and decided to share it in
‘equal pars between the exemplary werke. turned
ou however. tht there were 3 more mp workers
than i had teen assured. I tht cas, each of thes
would have got sopa less. The adminis bad
found the possibly 10 Increase the total sum
the bonus hy 90 recs and asa seul cach exer
lary worker got 25 ropes. How many people got
the bonus?
“ws 08 ws
on

Directions for Questions 36-39: Rea the (allowing and
ansner the question dat follows.

nthe ala of Hola Bols Mooka de habitats ave
a strange proces of celulas their verge incomes and
expenditres. According to an old legend. prevalent
fen tha island, the average monthly income had o be
‘act on the bass of mont in acendar ear while

0) 2and4s
(6) Oand25

ws

the average monthly npendiure was 1 e calculated on the

ass of moe per year. This would lead people Rang

an underestirtion of hei savings since there would be an

‘underetinatio of the income aad a oxerestimaion of ie

expendio per mont

3% Ifthe minister fr soie ali decided to revere
the races of caution of average income and aer

age expedir, what will app othe estimated sa
figs of a person living. on Hoole Boola Moola
ans?

(a) Hvillineease

9) teil decrease

(6) Keil remain constant
(6) Wil depend on the vale

37. Ati Mr Magoo Hocl Boca eines his
avi at 10 Moolas and is fuer known hts
cal expe is 258 Moolah nan yar (Moolah,
for dose who are at aware, ithe oficial amen of
Tool Boola Moot), then what wil happen wise
timated savings ihe suddenly calculate on he bas
‘of 12 month calendar year!

16) Willineense by 10
(6) Wie

38 Me Boogie Woogie comes back from the USA o
ola Boca Moola ad convie his comunity com
sing $46 families osa acalating he average in
come und average expenditure on the hass of 12
‘months per calendar year. Now if tis known tha he
average eximatediceme on te islands acceding lo
the old sytem) 87 Moola per month. then wha will,
be the change inthe average estimated savings for the
island of HoelaBoola Moola (Assume hat ter iso

er change
(0) 25160Moolbs (0) 585.5 Moolah
(6) 6255 Moolshs (0) Cant be determined

38. Me Boogle Woople comes hack from the USSR and
‘convinces bis community comprising 273 fares to
‘sat calling tbe average income onthe basis of 12
mothe per calendar year Now if i i Laon that
the average estimated income in his community is
(according 10 the old system) 87 Moolabs per ment,
‘then what wil be the change inthe average estimated
savings forthe island of Hoola Booka Moola. (Ass
‘hot there a o oer chang).

89. 4) 25140 Mod) 20278 Moos
(@) 31275 Moclhs (Cannot be determined

Dietlons fer Questions 40-44 Read he following and
answer the questions tht follows.

“The Indian cricket teu has to cor 360 runs the ast
day fa test match in 90 overs, to win te test atch This
‘sth tet se by the opposing captain Brian Lara ater
e declared is innings closed a he overnight sore of
4160.

“Te odia team ach has the following information
boat the bang rates (in terms of uns per over ofthe
teren men:

sue tht theron ste ofa porter isthe weighed
average of the individual bang rates ofthe batsmen in
‘volved in the partnership (onthe bass ofthe ado ofthe
stk ech basan gts Le, rn ie ofa panes is
defined asthe weighted average of he ru rates of he 10
batsmen involved weighed bythe rato of the number of
ali faced y each batman)

Since decimal fractions of rors are mot possible for
any fosa, assume that the estimated rus scored by a
Votan in an inning (on the basi of is run rate andthe
number of over Face by hi) is oad olf he next
Higher integer immediatly above the esiated alae of
the ras sored daring the ining

For enumpe ifs besman scores a un average of 3 runs
per over for 2.1685 eves, then he will be estimated to have
Scored 21666 x 3 65 rue in his Sings, bat since
this is not possible, he actual number of runs scored by
‘We buno wi be taken a (the nen higher ner shove
EN

Als, is round off cn ak place oy one for ne
innings of ban.

[Runs scored per over indifferent batting styles
Fan gun Dire Noma hewn
Paso
“ear
Lam
seas
Ganga
Rae

es
Sins
man

Ena

ou Ama [TE

Asume no eures unless eherwise stated,
same hate tk sql shared unless otherwise
stated.

A If eft wicker par of Das and Dasgupta bas for 22
‘overs and during this pare Dos as started ble
ting only and tod aggresive afer IS ove vale
Dasgupta sated of defensively but sifted gear o
‘ot ramal afer bing for 20 ones athe expected
score afer 2 overs
ws on OR we

41, OF the fistvicker parsentip between Das and
Dasgupta per the previous questo, the aio ofthe
umber of rant cored by Dat to hose scored By

Dasgupta ic
(a 46:25 0) 96:46
Cre 16) Cannet be determined

42. The test time by which Tendulhas an come o bu and
sill in be game assuming tht hen rt ato tine
‘ot hic walking th wicket limo 2: rns per over, e
‘assuming be shares vu equly wih hs partner and
Abe ets the maximum possible support tthe ober
nd rom hs hating parer and both play ih ast
tah
) After $0 008
DETTES

0) Alter 55 vers
(9 Cannot be determined
43. For question 42, where Tendulkar bated aggresively
and assi tat itis he Teuer Laxman air at
trina he gun or ada (a Tendalkar walk no bat
with the caret rate at 2 per over and athe lest
possible tine for Maso win the game wi maaan
possible suppor from the oppose end) what wl be
Tendula’s score for the imings asm equal tite)?
wis om
ns 16) None ofthese
44. For questions 42 and 43, it was Laxman who bated
wth Tendulkar For his ee innings. en how many
rn would Laxman sore nthe innings?
wis m7
om (9 Canet be determined

Directions for Quentons 45-49: Read the following and
answer the questions thet follow.

I Sachin Teather walks ito bat afer the fall of the
‘ity wicket and has to share parmesipo with Gangaly,
Kobe Harbhajan, Sind nd Yohasean who ave bed
‘onal, defensively, defensively, defensively and defen-
sively respectively while Tendlkar has bated normally.

90. 16 Pat of the runs scored in he WA innings will go
towards increasing the average of th fit 6 innings
to he new avenge and the remaining par ofthe rns
il go towards muta th new average for the
Wit inninge The eal cosraa inthis problem i
that there an crete inthe average by a whole
amb of rns Thi is possible fr ll re opens.

17. Awsome x i he average expenditure of 19 people.
‘Then, 191 13979 + 4 4),

‘ML The average weight per bal ss. Hence the ag
oes not ave to be count asthe diem,

21, TAG ED TENTE EEE
Alternate tis an be solved by ulng the concepts

of surpluses and dei as:

2159 (defi) - 4 * Llunas) + 48 average co

be maintained by Ue GO number) = 18-44 + 48

2

‘Solve through the same process as the Q

chape

(14933326 04712,

CRT
188

‘Use aligaionto sche 2032 68 Tas, 5

comespnd to 12, hence for 6 the answer wil be 1S.

26 Letibe equal distances be each Then Ji + d

edie dy.

17-30 You have t ak Between 25th and O0 o mean tat

Both thse dates are alo inode

22, The maxi average wil car when the maximum
possible values ee used. Thus:

‘A should have been bom om 3, 8025, Con
201, D on 106 and £ on Sh. Fate, the months of
‘is ia random oder will Rave tobe berween August
1 Decentbero maximize the average
Here the tal willbe 30425 +20+ 1045412411
+ 104943 140, Hence average 28.

28. Te ii average willbe when weave L +5 +10
4204254 14243-4445 = 76, Hence, averages
152

23 This doesnot change anything. Hence the answer i
the same as Q.27

30 Tre primo des mas be 29th, 238,190 and 30,
enc, he mariman possible average will dace by
AIS 208. Hence, anne willbe 272.

31. Solve using aia. Sine 15 Is he mid pine of
1325 ax 1675, thertois 1:1 ad hence there sre 20
peopl who were ging forthe pice italy,

Sort

BEB OR

oo vor [TTS

32 The number of pss candidates ae 2 + 6+ 18 +40
fout af tol of 10. Hee, 65.
SRM Put x= 10 in the given uations and find the
average ofthe rsa values.

23S, Solve tough options.

36 mex + «ne Simplify to get Option (6) comet.

37. By align the rio is 3:8. Hence only 1105 pos.
ES

qua

3%. You don’ know who gt ont when. Hence, cannot be
termined.

38. Since posible ax asked abont you will ave to
consider all posites. Assume. the sath and se
rh batimen have score ero. Only then wil the
possibly ofthe int 5 batsmen scoring e highest
possible average rc. nhs case he maximum pos:
be average forthe fit $ tases cou be UNS =
me

40. Again itis possible tht only the fr btıman bas
scored ane.

41, We cannot find ou the numberof ran scored by the
‘Pi besman, Hence ame

42. You cun take 53 athe base lo rece your calla
tices. Otherwise the question wil become highly cl

‘ation intensive.

2512804 + 715245 +760"

‘Selveesing opone. 20 the only possible value,
43. Check though options to solve,

RS

Hints and Solutions

1. Defisitely decreas, sine the highest marks in las
A es than the lowest mark i Cee 8.

2. Cannot sy sine there sno indication ofthe values
ofthe numbers which are ransfoned.

le vi defintely decrease since the highest possible
rufe i lower than he lowes vale in C.
44 The let 50 À wil depen onthe ro of he pope
‘who are wasfered. Hence, ang can happen.
5. Cannot say since there i possibly at he nr
en transfer ae such tha the average can either
Increase, decrease or rai constant.

6 Mineras, then the average of Coes fom 30
Fortis to happen itis define thatthe average of
sould drop.

91. Tid] totor Dors Apia CAT

7. Themaxioum posible average far will sou Fall
the tamferes from A have 22 mas.

A. Theaveragof Gp A flr be taster in. 7 hove
(Gon SISSI

3. 40-22¢5)15 219.33

10, 40042345 S15, Average» 51828 =206

1. 400+ 31°55, Average 2 35925 222

12. Will always ocres sine the net vas eager
from Bw wil lg indent vl rae
from Ato.

11. Since the lowest score in Cats 223 which is moro
than the highest score of any stent in Ciss A.
Herc, A'savergs wil aways ierese

14. The mauimum possible vale fr 8 will happen when
A toner has the mation psi vl and
the reverse uansfer has the minimum posible value.

15, Forthe minima possible vale of we will ed the
A B water 1 be the lowest posible value while
the Bin A transfer must have the highest possible
value, Thus, À 1 Basler +18 5 we Bm A
transfer willbe 3195 ence atone 224.

16. Themusimun vale for wilhuppenin thease Q
15. Then the increment for group e
3103 189523003 189065.

"Tas main possible value is 4652022325

17. Minin posible average vil happen or trar
‘we sin 14 Ths anne willbe 408202 2023.

‘The maximum posite vale for € wil bs achieved

eben tens fram Cin! Give 25 andre

‘buck from Dis of ive 31's Hence, Greta

‘wll be 425 Hence, max. average = (90025900 =

pr

to Per that 900 has como dy 20130]

19. Forte manie pose vals of Clas he fellows
ing et of partons wil have to held:

Five 39% ee tested from Cto B, whatever
goes fram B 10 À comes bok fom À 10 A, then
Five 2 ae tanufer from 0 € This leaves us
vite

Increase of $0 marks - average Increase y 240
a

2, Amilo mim vu ive 234 coma fae
Cihrough Maná ie 18% eve A, ln ch ct the
ret res is gong 0 be a change of +75. Ths the
average will go up by 7520 375 w 2828.

2066

2123, Wl be slve by the same rem as above ques.

24 Only option wil gives the ered stain soc

the taster of five A’ scree the value of the
average of group A.

2524 Will be solved by the same pate as above ques
dons.

295, These ae standard questions sig the concept of
averages. Hence analyse each and every sentent by
itself and ink the interpreto I you are peng
‘wok the only rason i that you have no wed he
infection i he questions filly.

36. Monty cxlmats of income Is reduced as the de
minor ere from 1210 14 athe sae time
the monly esate of expedite is increase as
he denominator reduced from 12 109. Hence, he
savings vil be underestimated.

3799, Use the average formulas and common sense 10

0:9. The questions are commansensical with a tof al.
ators and assumptions vole. You have 19
solve these using all he infomation povided.

40, Dass soe = 156247925 2475-48,
Dasgupta’ score = 201 + 241523

AL. From the above the answers 48:23 = 96:46,

2-4, By maximum poste sara forte her end os
are anne ta he has Laxman or Seta bating
aggressively forthe ate tenure atthe cease. She
as 1 be shared ec.

22 Through options Aer overs, sore woul be 150.
“Then Tendulhar can sore @ 4 runs pe ver having
te trike and bating aggressively) and get maximum
supp © 3 runs per over Thos in 20 over et he
target il e achieved,

48. Tendlkar's score forthe Inning wil Be 30°4 = 120.

4. We donot know hen Laxman would ave come nto
a, Hen this cannot be determine,

4548. Buin ech of Ue conditions in the per Lom

stale ike:

Partnership Partner Overs fed Tender score
Pannes sore

Gin vices Ganguly 12 Gorenx6 Gore

Tin wicket and 0 on

Shia

wicht

tO wicket

92. TB] Horner casas pere CAT

‘Averages LOD

Le

SHEEEIEEE

=

a

8

EX]

93. Cara anges [117

‘Averages LOD IIL
Lb Ra
23 EN
ae En
aa EN
sa En
“> Er
2 Ex)
aa ae
Be ae
wa ae
Le ae
2» EN
En ae
ue 4
ie cn
Ke wa
ma om
CE

we

ms

E

slelelslelelelele

En

94. LLIGATIONS

INTRODUCTION

‘The chape of aligtion i nothing Dt a fair chi
of solving problems bused on the weighed average situa:
tion as apple wo the case of two groups beng mixed 1o-
‘eer. 1 ave often cen sudents having aft of ety
in solving questions on ligation. Pleat remember that all
problems on algun canbe solved though the weighted
Average method. Hence, te stents advise 1 revert 19
the weighed average formula incase of any confusion.

The Use of the techniques of this chat for solving
vel average problema will help you ln saving vela
bl time wherever a diet question based onthe mixing of
vo groups is asked. Besides, inthe ease of questions that
tie the concept ofthe weighed average as à part ofthe
problem, you wil gain à signifiant de if yon are abe to
‘oe the techniques rad hee.

‘The relevance of this chapter forthe CAT is the ue of
be aligation technique for solving problems where the
once is used to solve a prof the whole quetion. Bo-
ses, question ofthis chapter ae diet relevant forthe
‘xan like CET (Mabarasir), NMIMS, NIF, Syioss,
MAT and al ober level management entrance exams as
wells forthe bank PO exams andthe MCA exams, which
‘ae based on the pater of an apt test

‘THEORY

In the chapter on Averages ue had seen the use of the
weighted average formal. To recollect, the weighted aer
age is used when a number of males groupe are mixed
{gether 0 form ene lager group.

tbe average ofthe measred quantity was

A for group containing ny cements
Ar oe group 2 conning m cements
44s foe group 3 mal m clement
‘Actor gro comtining m cements

We sey dat the weighted average, Aw is given by:

Aes mie RAE MASS
met

‘That ls, he weighted average

FMA atm:

a Sum total ofa
"rol manero! elemens al group I

Inthe case of the situacion where Just two groups are
being mixed, we ean write this as

(más + Ay Mm +)
Rewriting this equation we gt (m + m) Ave mA
tis

aw

mA A = my = Am)
cx ml = (Ay= AN ~ A) > The aligation equation.

‘The Alligation Situation

“Two groups of elements are mined together to frm a hind
gros containing the element of both the groups.

the average of the ia group eA and the number of
element im nd average ofthe second grup is A nd
{he numberof elements sy hemo find he average of he
‘ew group formed, we can me ee the weighed average
exuaion or the aliation uation,

95. 29. Two sion of 90% and 97% pay are mixed se
sulting in 21 les of mixture of 94% party. How
such sche quantity ofthe frst solo in the reste
ing mixture
(15 hres 0) 12 tives
19 9 tes @) 6 lives

30, In he Singapore 200, there are deers and here are
ducks If the heads are counted, there are 180, while
the legs are 48, What ill be the number of den in
Le
wi 06 | wR

LA bonus of Rs. 985,000 was divided among 300
‘workers of a factory. Bach male worker ges 5000
rares and each female worker ges 2500 rupees. Find
the number of male workers in the factory.
os 07 ON GB

32. What will be the ratio of pero and kerosene in the
fia! solution formed by mixing pel and Lersese
that ae preset inte veses the ratio 4: 1,5:

and 6: | respectively?
(a) 16:22 032
CRE (6) Nove ofthese

33. A mire wort Rs. 325 a kg is formed by mining
vo types of flour, one coming Rs. 3.10 per kg while
the other Ra. 3.60 per kp In what proprton mast
they have been mixed?
ws 0 7:10
(9 10:3 @7:3

34. A gin percent ol 20 is mude by sling the mare of
uo ype of her at Re, ABO per Ifthe ype coming
‘610 pe kg was mined with 126 kg ofthe uber, how
many kilograms of the former was mixed?
DET MENA
© 61% (8) Cannot be determined

35, In what propio mast water be mined wits milk so
as gain 20% by sling the mixtre ate cos price
ef the milk? (Assume Ua water i cy avila)

ES

a.

Es

».

“0.

@1:4
@ 1:
A barender sole champagne from a bot tht co
sind 50% of spirit and he replaced what he had
stole wi champagne having 20% spirit. The boxe
then contained only 25% sprit How much of te
ete di he stent?

wur © 933%

© 857% CES

A bag contains total of 15 ins of Rs. 1. $0 pad
25 pdecominatione. Find the al namber of coin of
Re | if there ae ttl of 50 rupees in the bag and
its Keown that the number of 25 pls coins we
133.338 more than the number of 1 nupse sois
ws 2

on (8) None ofthese

A man possessing Rs. SEDO, leat a pat of it at 105
‘mpl intrest andthe remaining 7.5% simple in-
(eres is tal Income after 3
Find the sum Tet at 108 rates.
(a) Ro 1260 @ Rs 1709

(Bs. 1360 4) None ofthese

a man decides to tal 80 kilometes in 8 hours
party by foot and pay om a icyle his spe on
foot hing kw ad that on bicycle being 16 An,
what distance woul e wave on for?

yea was Rs 1908.

(@) 20 1m META
© im @ 60 km
‘Two vestes contain a rite of sit and vtr lo

the fiat vee ai o spit o wer I 83 an
in te seconé von the ratio is 5 1. À 35 Ke case
ls fille from these veses so as to cola a mitre
of spt and water in the rado of 4 |, How many
res ae taken trom te fit esse

(6) 1 tes CET

(6) 165 ties (9) 175 tes

96. 1128] ven Pop o Ort ae Be CAT

ANSWER KEY
Alligations LOD 1

he Rd
he we
he EN
a 36.8
3 whe
és me
ka Be
te Oe
Be
ie
ma
me
Ba

a
ie |
ma
me
pu
me
Be
2b
EM
2
Be
250
Te
EM
Be
EJ

97. pi

BLOCK REVIEW TESTS

REVIEW TEST ONE

aks bought 19 eases or Rs 10. He paid 20 paíse
ror foreach white eraser han foreach brown ese,
Wha could be the price ofa white eraser and how
trary white erasers cold be have bout?

(a) 60 pase, 8 © 60 pue, 12

(30 pais, $ (0) 50 pase, 10

fer payng al ou Bl, yu Find hat yo have Rs
720 in your pocket. You have equal number of 0
pune and 10 pase coins; bat no ether coins nr any
‘ter sumeney notes How many cons do you have?
ms DA En Mm
Suresh Kumar went to he markt with Ra. 100 be
boys thee pens and u pencils he ater op all hie
money. On the aber had if he bays three pens and
six pens he would fall shor by 20%. IP he waats wo
boy equal number of pens & pencils, how many
‘pencils canbe buy?

ms 05 @6 wT

For be above question. what ste amount of money
he woul save fe were to bay 3 pes and 3 pencils?
Rs. 90 ©) Rss

(6) RSIS (4) Reo

Al goes 10 the markt in buy bananas, If be can
again and reduce the price per doren by RS. 2 be
can buy 3 dozen bananas instead o 2 dozea with the
money he his. How mach money does he hare?
(0 RU MR G) Re IS Gd) Rs 4
‘Two oranges, tree bananas and four apples cost
RS, Tree eranges, (no bananes und one apple ost
R10, I boaghe 3 oranges, 3 bananas and 3 apples.
How much did 1 pa

(a Rs10 CET

CE (9) comer be determined
John tought five mangoes and en oranges together
for fry rupees. Subsequerly, he eure one mango
and got two oranges in exchange. The pre af an
‘range would be

WRI ORD (RSS (6) Red
‘Two towns A and B are 100 ke apa. À scosl sto
eb fr 100 students of Town B and 30 students of
Town A. The Expendiure on transport is Ra120 per
Am per person. Ifthe tral expenditure on tapon by

1 130 students isto be as amall as ponible, then the
‘schoo! shouldbe bul at
(a) 33 Am rom Town A.
19) 33 km fom Town B
(e) Town A
48) Towa B
9. A person who has a cersin amount with hi goes vo
‘the market. He can buy 0 anges or 40 mangoes. He
tin 10% ofthe amount for taxi fare and buys 20
mangoes and of Ue lance he purchases oranges.
Number of oranges be can purchase is
os OO EIS am
10. 72 bens costs Rs_967.. Then what does each en
cost, where numbers af 7 are not visible or are
‘writen in eb hand?
er @ Rss
(o Ras (0 R622
Directions for Questions 10-12:
‘There ar students in cas. These stents ae vided
mo tre groups A, B and C of 15,20 and 25 students
‘each. The groups A and C are combined to form group D
11. What isthe average weight of the stdens in group
m
(2) more than ce average weight of A
(o more than We average weight of C.
de) less than he average weigh of €
(8) Cannot be determined.
12. If one sun: from Group A is sited to group B,
‘which ofthe folowing wil be tre?
a) The average weight ofboth groups increases
(©) The menge weight of bo groups decrees
40) The average weight efthe clas remain the same
(9) Cannot be termines.
13. If all the sde ofthe cass have the same weight
then which ofthe Falling is fle?
A) The avenge weight ofall the four groups isthe
(0) The total weight OF À and € is twice the a
eight of B.
(6) The average weight of D is gremer than the
average weigt of A
(6) The average weight of ll he props remains the
same ever If à number of students ar shied
‘for one group to soother.

98. — [ia

ln such a case go back in te problem and uy to deny each statement and see wheter you have willed
A ormot Ifyou have already wo ll the infomation, you night be inerte ia Knowing wheter you have mcd
the information given in the probe in the coret order. If bth of these have bee done, Jou might want 10
explore the next reason for geting sock.

‘Reason 2: You are stuck because esca though you might hve vd ll the informativo given inthe problem,
ou have ot ted some ofthe infomation completly.

In such à ee, you need o review eich ofthe part ofthe information given inthe question and look a
whether any onal del canbe derived cut of the same informara, Very oc, in Quant, you have
situations wherein one semence might have moe has ne contain. Ifyou have used that sete only in.
one perspective ben tig it in the eter perspective wil sohe the question.

Reason 3: You are stuck because the problem does ot have à solution.
In such a case, check tbe question unge and is correct go beck o Reasons 1 and 2. Your solatin has to

‘THE LOGIC OF THE STANDARD STATEMENT

‘What have been trying o Il you tt mos of tet, you il ge suck In a problem only when you are
ot able fo it Namen in he problem, Hence, my aise to student (especially tone who are Wek
in these ate

Concentrate on developing your abiliy to decode the mathematical meaning of a sentence in a problem,

‘Todo this ven in problems that you ae able o sole (easily or with ic) po back ito the language
‘ofthe question and work out the mathematical resction that you should hove with each satement.

miga aot be a had idea to make à ht of standard tement along with thee mathematical reaction for
‘each chapter inthis loc of spe. You wil aie that in almost po tim, you wi cometo assi where
ou wil only rarely encounter new Language

{Coming back to te sc of linear equations

near pa are expresos shout arabe mighhep us gt vale ofthe variable if we ca soe
the equation.

‘Depending upon he number ol variable ina problem, iar aan might have on variable, we variables
even thee or more variables. The onl thing you shold know is hat in onder to ge the value of «variable,
‘the numberof eqistions needed is always equal 1 the number of variables In eter won, if you have more
variables in system of equations han the number of equation, you cannot solve for the individual values of
the variables.

‘he basic mathematical principle goes lite thi:

Fora syatem of equations ta be solvable, the number of equations shold
‘qual othe numberof variables lo the equations.

"Tas for tan, you have two variables, you nos Imo equations to get te |
values cto vaa, wil iyo have varas ou wl ce dre equations.

‘This station is best exemplified by the situation where you might have the
folowing equation. x + = 7 itis known dat both ly ae natura members,
it es se of posts forthe vales of x & y a follows: (1, 6), 2,5). 8.4)

(4,3, (5.2) (6 D One ofthese possiblities ha to be the answer.

99. HA] mercer carreteros

In ua, i might he a good eat hin o al near equation situations in is fasion Herce else yon go
stead 1 ad aout the neu equation. you shuld set up this set of possible tus on the fist equation.
‘Consider the following situation where a question yields a set of possibilities:

Four enemies A, Cand D gather together fora pen ina park wid thee wives As wife consumes
times es many gles of jie as A. B's wile consumes à dines as many glsses of juice as 8. Cs
‘wife consumes 3 ties as many glasses of juice a Cand D' wife consumos 2 times as many plates
of jo aD. In ta, the wives ofthe far enemies comume a a of 4 late of jie, If A
consumes atleast glass of juice while ech ofthe other men have at least one plas, id te least
‘number af chinks tht could have been consumed bythe 4 enemies eier.

mo on on wm
In the question above, we have 8 vaiables—A, D, C& D and a 6, d= the number of gases consumed
ty the four men and the number of glass comme by the fur wives.
‘Also, he quesion gives ws five Informations which canbe summarized ino 5 equations a flows,

cate
4-20
wad as becados
Alsa, 45
‘Under this condition, ou do où have enough infomation to gt al values and Hence you will ge set of
posites
Since the minimal value of A ls 5, a can take the values 25, 30,38 and AD when A tes the vals 5, 6, 7

and respectively. Based en these, and onthe realization tht b has o be a mulóple uf 4 a multiple ef 3 and
da malle of 2 the following pont emerge
ma=s
= EE Bs
+ a 8 7 7
© (outil of 3) 3 9 3 9
4 (mie of 2) 4 2 2 $
asbl CE “a
= Fe en
CIE A 7
© (muleta 6 3
4 mine A 2
breed 4 ss

Inne ano MS sc BE oa Dt. a also CD
D nn sn ner os nen ce 7
that os pss de unter lp ed 1
Soc lant tar gan ae ey comme la CAT a a eel
aptitude examinations. |
Cortes
He ra pe te 7
were outa be pues fo dea! pou |]
“hs con e roo 0 tf dca va ad un
rgd y 20% i ew pecas vb

100. PREASSESSMENT TEST

1. Tiros rumen A, Hand C un a race, with nner A
fishing 24 mes abend of ranar B and 36 metres
head of rurerC, we runner B Fishes 16 meres
abad of runer €. Each runne als the ere
distance a a constant speed. What was Ing of

the nce?
(0 mews 0) 96 meus
(©) 120 metes (0) 144 metres

2. A dealer buys dy fits Rs 100, Rs, 80 ad Rs, 60
por logres, He mixes them inthe eto 45.6 by
‘weight and sells at a profi 0%, At what price pee
logra does he ell the dy fas?

@) Rus 09 R106
€) Rls 10) None of thee

3. There are two conines the fra consis 00 mi of
cohol while the second consis 00 ml of wate.
Five cups of albo] fm the it container is akon
‘ut ani zed well in th record contre, Then,
five cops ofthis mixte is ten out from te second
‘comainer and put back ino te Fi container. Let X
and Y dene the propor 0” alcool i the fst
‘nd the proportion of water inthe second coins.
‘Then what isthe relationship between X & Y?
‘sume te size of the cups tobe denia)

wer 1 xr
© (6) Cannot be determined

4. Aides ook five papers in an examination, where
‘exch paper was of 200 mars. His marks in these"
papes were inthe proportion of 7:8: 9:10: In
a papers together, the candidate tained GO ofthe
{otal mark. Then, be number of papers in which be
ot more than SO marks i

wı ws
©. ws

5. Aad B walk up an escalator (moving sia) The
‘calor moves at a constant speed. A takes six steps
for every four of B's tps. À gets 10 the op of the
‘escalator after having taken 50 steps while 1
use Bis slower pace les the escalator do lithe
rm ofthe work) takes only 0 steps o reach the
Hop. Ihe escalator were ted ff, Bow many steps
‘Would they have to take 0 walk vp?

ws © 100
om © 10

(i

6. Filly perm ofthe employees ofa eran company
ae men, and 80S of the men cam rare han RE, 25
lacs per yea, I 60% of the company’s employers
cue mare than Rs. 23 lacs per your, then what
fiction of 1e women eaphyed by the company
‘urn mote thin Rs. 25 Ines per year?

wm @ 14
ow CE

7. A piece of string 8D cemimetrs long is ut into.

three pieces. The longest pios i 3 mes as long as
the middlesized and the shortest piece is 46
cemimeters shores tha the longest piece. Find the
length ofthe shot pies (in cm).
an wo
os wis
8. Taree members of à family A. B. and C. work
together to pt al owscholé ches done The ime
ttes ther to doth work togethers ix boars less
than A vould have taken working loc, one hour
Jess than B veu have taken alow, andl he ine
€ would have tea working loe, Haw long did it
take them to do these chores working togster?
(0 20 minus
(04 minas
9. Fresh grapes contain 90% water by weight while
ris gras contain 20% water by weight. What i
the weight of dry grapes availble from 20 kg of
fies grapes?
CES CES
«2H (0 Nove ofthese
10. At the end of the year 2008, a shepherd boupht
twelve den gots, Henceforth every year he aed
Df of ie out at he beginning f 1 yer ds
(1% ofihe oat a the endothe year where p> Dam
420.1 be shepherd hal twelve doren goats atthe
nd ofthe year 2012, (ter maki the sale foe hat
eH), which ofthe lounge?
@ pea © per
© pa @ pean

Directions for Questions 11-12: Answer the questions

asd on the following information.

‘An Indian company purchases components X and Y
from UK and Germany, respectively. X and Y form 40%
and 30% ofthe total produc cos. Current ain is 25%.
Due to change inthe international exchange rate scenario,
‘the com of he German mark rented by SO and trol

101. 136] votos te Oct Ag taa

UK pound increased by 25%. Due 19 tough compte
marke conditions, tb selling pice cannot be ineesscé
beyond 10%.

A Win isthe maximum cure gain posible?
(o 10% © 125%
© 0% @ 13%

12. If te UK pound becomes cheap by 15% over ls
orinal cost und the cos of German mark increased
ty 20%, what wil Bethe gain if the selling price

ot led.
€) 10% CES
(o) 25% (@ 75%

13. A college has rained 80% ofthe amen it need for

cw building by receiving an sera donation of
Rs. 80 from be people already sliced. The people
already soi represent 50% of the people, the
college will ak for donatios. I he college is 10
raise exactly the amount need forthe new bling,
whut should be the average donation from he

remaining people tobe solicita?
(o 300 (by 200
(9 200 ww

I A stent ges an aggregaie of @fmarks in ive
subjects ln the ratio 10:9: 8: 7: 6. If the passing
mas are 43% of te maximum marks and each
subject asthe sare maximum marks. in how many
subjects did he pass the examination?

@2 CE
wa os

15. After allowing a dissout of 123% a under sl
takes gain of 405. Ar what per ont above the
‘oat price dos he mark on his goods?

(0 45% or
CES (6) None ofthese

16. The owner ofan at shop conduct his busines in the
following manner Every once in à while he ses
his peices by X%, then 2 while aer he reduces al
the new prices by Xf. ARer one such up-down
cycle, the price ofa punting decreased by Rs. 441
‘After second up-down cycle the pining was sold
for Rs, 1944.31. What waste oiinal price ofthe

painting (e Rs)?
(a) 295625 0) 25625
€) 2500 (a) 2000

17. Maras, Mirza, Shoy and Japa bough a mot
fer 60000, Manas pai SO% ofthe anus pid by
the other thee boy, Mirza paid oe third ofthe sum

of the amounts paid by the eter boys; and Shorty
pai one fourth ofthe rum of the amounts pai by
Ibe eter boys. How much dd apa have to pay?
ta) $15000 © 513000
(© $1700 (8) None ofthese
1%. A main X departs from station A at 1.00 am. foe
statin D, whichis 180 km away. Another tin Y
pans from station B at 10am. for sion A.
“Train X travels at an average sped OO ans ad
does fot sop anywhere until aries a station B.
“Tai Y travel at an average sped of 50 kr, et
as 1 stop for 10 mite at satin C, which is 60
Ams away ftom sation B earoule 10 son A.
Tnering the legs of the wales. what is the
distance, o the nearest km. rom son A 10 the
pont where the ins eros each lar?
wo @ 12
(9 16 (4) Nose of these
19. ln a survey of pial preferences, 81% of those
A were in favour of at le one of the thre
gear proposal A Band C. 50% of those asked
avowed poposal A30% favoured proposal B and
20% favoured proposal C. If 5% of those asked
favoured all the thee proposal, what percentage of
Hove asked favoured more then one of the dee

proposals?
(@ 10% @ 12%
(9 9% @ us

Directions for Questions 20.21: The pel consumption
rate ofa new model car Pal” depends on ls speed and
may be described by the graph below:

a

20. Manas makes the 240 kp from Mumbai o Pane
ata steady speed of 60 km per hour. Wht is tbe
amount of pol ceased forthe joue?

(a) 12S lies @) 16 tires
(©) 15 ties © 1925 tes

Cu »

102. 21. Manasa would ik to misimiz the ue consumption
forthe ip by diving ath aproprite speed Hom
shld bo change the spect?

(0) Increase the speed

() Decrease the speed

(€) Maintain the speed at kr aur
(6) Cannet be determined

Directions for Questions 22-23: Answer the questions

based on the ellowinginforsatin:

There a ive macines-A, B,C, D, and B- stated on
‘sight line at distances of 10m, 20 1,301, 40 m and
‘Som respectively from the origin of te ie. A robot is
stutioned atthe origin of the line. The rbot seven the
machines with raw mei übenever a machine becomes
ide. Al he raw materials are lected at ie engin. The
robot is in an ile state atthe gi at he beginning of a
day. As s00n as one or more machines become ile, they
sed messages 1 the robot ao aná e rbot stats and
serves ll th machines from which it received messages.
message is receive athe ation while the rbot say
‘rom i the rbot takes notice ofthe message only when it
runs othe sation. While moving, l serves the machines
in ie sequence in which they are enccunterd, end en
curas to the origin. If any message are pending al the
station when i remres, it repeats the process agan
Othernse. it remans jäle at the origin Ul the nent
message(s) i (ae) received.

22, Suppose on cen day, machines À and D have
seat the fast two messages to Le origin atthe
beginning ofthe fist second, Chas sent message
Ub beginning ofthe Ah second, Bat the begining
‘ofthe th second and E atthe beginning of the 10th
second. How much distance has the rbot tele
since te beginning ofthe day when it notices the
message o? Assume hat the speed of movement
ofthe robot is Las,

@ Hom © 50m
© 40m © 0m

23, Suppose thee iz asocood satin with rw materia
forthe robot a e other enteme ofthe ne which
is 60 m from the orgia, Le, 10m fen Aer
Finishing the services In a up, te robot sens to
the nearest station. I bah stations are equidistant, it
chooses the origin as the station to rem to.
Assuming tht bot statins receive the messages
Sent by the machies and that all the ether data
‘remains the same, what would be the answer Yo the
shove question?

Sect | 137,

@ 10 © 160

(o 160 om
24. One bacteria plis into cipht bacteria of the next
generation. But due to environment, only 50% of à

¡enero survive. If be eighth generation munber
ls 8192 millon, what is De number inthe fast,

generan?
6) I million 0) 2 milion
(©) 4 milion (0 Sion

25, 1 bought 10 peas, 14 pencils and 4 eres. Rav
ought 12 pens, eraers and 28 pencils for an
mount which was half mor what had pi. What
percent of the al amoan pai by me was paid for

the peas?
ES mas
CES 8 None ofthese
Answers (Block 3 Presssessment Test)
LO 2@ 30 40 50
6@ 26 50 XO We
LO LO BH B® 150
bo 70 Be BO
LO O 20 240

SCORE INTERPRETATION ALGORITHM
Block 3 Preassessment Test
If You Scored: <7: (In Unlimited Time)

‘Step One: Go th th block thee Back o School
Section carey. Grsp each ofthe conceps explained in
‘hat pr arf. would recommend tha you go back Lo
our Mathematics school books (ICSE/ CBSE) Class 49 e
Tost you fel you sed it.

‘Step Two: Move ito cach of the chapkers ofthe block
thee by one.

‘When you do o, concerts on ce understanding
ach ofthe concepts explained in te bapter theory

Step Three: Then move omo the LOD 1 excises.
“These exercises will provide you wih the fit level of
challenge. Try to solve each and every question provided
under LOD 1. While doing o d not ak about the time
equrement. Once you finish slving LOD 1, revise the
questions an thei soon processes. |

103. 12] How raro tr art Agee re CAT

¿on know the previous years sales ie (though on
the face oft Company B seems 10 have grown more).
we had father information saying that company Abad
als rover of Ra. 1 core in the previous year and
compa B had a sles tumover of Rs 100 ror in the
previos year. we can compare growth rates and ay at it
ls company A that has grown by 100%, Heoce, company
A has a higher row rt, eventhough in tems af bso.
lue value inca of als, company hs grows much

IMPORTANCE OF BASE/
DENOMINATOR FOR PERCENTAGE
CALCULATIONS

Mathematically, the percentage valve can only be caes
Tai for ratios that, by definition, mest have a denominaer.
Men, ns of most real aspects ole percetge is
the denominator, which in ocr words also cae the
bse value ofthe percentage. No percentage calculan is
possible without Knowing te base wo which the erentage
be eile.

Hence, whenever faced withthe question “Wha isthe
peerage ..7 alaay ty fst find oot the answer to
the question "Percentage 10 what base?”

CONCEPT OF PERCENTAGE CHANGE

Whenever the value of s measured quantity changes, the
change can be cape though

(0) Absolute value change or

(0) Percentage change

oth these measurement have dcir own advantages and
usdvanger,
Absolute value change: 1 is the actual change inthe
measured quai For instance, als in year 1 à Ra. 2500
ror and the sale in year 2 is Rs. 2600 err, then the
solute value ofthe change is Rs. 100 crore.

Percentage change: Iti the percemage change got by
the om

LL inclu valve change „
Portage change = APRES 10

= 10 x 100 =
AU x 100» 4%

As seen ca, his fen gives ws a bete picture of he
fect ofthe change.

Si: The base ie for he sake of perce ame
skins ss the bina] ges ules bete
me Std

Example: The population o iy grew fom 20 lakh to

22 lakh Find he

(a) percentage change

(percentage change based m ena) value of pope

tation

Soliton: (2) percentage change = (2720) x 100 = 10%

(0) percentage change onthe final vale = (2
2100 = 909%

Difference between the Percentage Point
Change and the Percentage Change

‘The diference between the percentage point change and
the percentage change is best lasted through an ex-
ample. Consider this

“The savings ate as a percentage of the GDP was 25% in
‘heist year and 10% the second year Assume that here
is no change in the GDP between the two years. Then

Perentago point change in savings rae 30% - 25%
= 5 percentage pois

Percentage change in savings rate
2258

2.

PERCENTAGE RULE FOR CALCULATING
PERCENTAGE VALUES THROUGH ADDITIONS

strate below sa powerful method of cacusing per
centages In ny opinion be ability Wo calcule percentage
through this metbod depends on your abiliy to handle 2
ik ado. Unes you develop tb silo ad 2 digit
dis in your mind, you are always Lely o face prob
tems in calculating percentage through the method le
rated. below. In fact, tying this mebod without being
strong at 2-igtadtina/ubeactons (including 2 digit
ster decimal pol) would prove tobe a disadvantage in
our sept at calculating perertges fast

“This process, esemialy being a commonsense process,
ls bes ill trough few examples:

104. 144] vo par ste Ai Be CAT

3. Product constancy application
4. À 2 BVA application
5. Denomizator chang: to Rat Change application
8. Ue of PCG 1 calcular Ratio Changes
Application I: PCG Applied to Succeie Changes
‘Tis isa very common stain in mos questions.
Suppose you hae to solve a ques a which a number 30
as two successive percentage increases (20% and 10%

respectively).
“The sinon is handled in the following way wing
co,
wm ms

Miosiration

Ars slay increases by 20% and then decreases by 20%.
‘What is he net percentage change in 4% salary?

Solution:
nm rm

Heace, A's salary as gore down by 4%

Jilustration

A ter gives succesive discounts of 10%, 20% and 10%
respectively. Te percentage ofthe original com price he
il reaver i

Salto

eg 28s go te 79 man Gas

Hence the overall discount is 39.2 und the answer is
sue.

Iustration

‘A under maris up the price of his goods by 20%, et toa
pariculaly hagling csomer he end up giving a count
‘of 10% onthe marked prie. What i the perenage prof
he makes?

Solan:

AA

Meme, tb peremtage profit is 8%.
Application 2: PCG applied to Product Change
Suppo you ve a product af two variable say 10 x 10
ue Fist variable charges o 1) ade second variable
changes 1 12, what wil be the percentage change in the
produc? (Note thee is à 10% increase in one pur ofthe
product and a 20% increas in theater pet]

“The formula given for ths ion goes as (a+b + a
100)

ewe Raid tage 10:20 + LOIR
(her 10 20 a te pe pes cage
inh wo pf ep (ME ag la
Shoe mo Into ros e ay eae)
ome mich ese sole o is ncn be
va

100—2EL 129 HEE 32, Hen, ih nal poco

shows a 32% increase.
Similar seppos 10 x 1 x 10 becomes 11 x 12 x 13
1 such a use the following PCG wil be use:

100 2815199 ei EE TH
ence, the final produc sees à 7.6 percent increase
(Gives, the product changes from 10010 171.6)

‘Note: You wil ge the ame ret especie of the order

in which you use the respective percemage changes.
Also not that his proces is very sir o he one used

for calcula successive percetag change.

Applicaton for Diz

Suppose you have two pie chars as follows:

If you are asked to calcule the pereentge change in
the sles revenue of scoters forthe company from year
‘one 19 year two, what would you do?
‘The formula for percetage change woud give ws:
026551944356) - (02357 x 1734234) 100

WATTS

105. Now Saler Revenue - Original Sales Revenue 199
“ Original Sales Reverue

‘Obviously this calculan ds ser sid than done.

However, the Prode change eplicatin of POG allows
sto exec thi calculó witha et of ese compara:
ively. Consier the fllowing slain:

Produc for year one is: 02347 x 1734234

Product for year mo i: 02585 x 19443 56
‘These can be approximated ito:
234x173 and 265 «104 respectively (Nos tha by moving
int thre dit we do oot end up losing any accuracy. We
ave elaborate this pin inthe chapter on Ratio and Pro-
prions)

‘The overall percentage change depends on two ind
‘ia! percentage changes:
254 increases 10 268: À 4 change of 317284 = 132 %
‘pprox. Tis calculo bas 1 be done using the prcet-
ge rule for calculan the percentage vale of the cao

173 increases to 194~ A percentage chang of pena»
roue 12%.

‘Thos PCG wil give the answer as fellows:

too 2251 1132 BEE 12676

ence, 2676 % Increase inthe products vale. (Note
that the vale on the calcite fr ie fall auton sans
any approximations i 262 4%, and given he fat that we
ave come extremely clove 1 the ansner—ihe method is
8004 enough 1 solve the question with a resonable degree
of accurcy)
Applleation 3 of PCG: Product Constancy Application
nerve proportionality)
‘Seppose you have à situation wheria the pice of a com-
modi has gone up by 25%. Incase you are require to
Keep the total experdíure onthe commaliy const yoo
‘would obviously need o ut down on te consumptin. By
‘what percentage? Well, PO gives yo the answer as fol
dos:
30021 125 ——+ 100

Hence, the percentage drop in consumption to ose th
Price increase is 20%

leave ito the sde lo discover the percentage drop
required in the second part ofthe product If one part Ine
creases by 50 percent
[Note Product constancy is js another name for Inverse
proportonaly.

csm. pongo [TES

Table 5.1 gives you some sandal values or this kind uf
Application 4 of PCG: ABA.

Very often we are faced with station where we compare
‘wo numbers say A and. In such cases if we are given à
relationship from A B, ben the reverse rlaonsip can
be dremined by using PCG in mach the same way asthe
rt constancy ws shown above

lustration

Bs salary is 25% more than A salary. By what percent la
Bs salary ls than B's slay?

Van Lan — mn

A op of 25 on 125 gives a 20% drop
Hence As salary is 20% less than 2's,
Note: The values which eplid for Product Constancy
sho apply bere Hence Table 41 is uefa for bi situation
sho,
Applcaion § of PCG -» Eee of change In Denomina-
tor om the Value of the Ratlo
“The denomiato bas animen reladonship withthe value
of ri.
Hence the process used for product constancy (and ex-
plained above) can be used for exculting percentage
ange in the denominar.
For instance, suppose you ave o evaoate the die:
ence between two as.
Ratio 1: 1020
Ro? 1005
‘Asis evident Be denominator ie increasing fom 20 10
25 by 25%,
we callate the value ofthe two ratios we will get
Ratio 1 =05, Ratio? = 04,

Wanna werent wenns 8 0200
at cto ec Pe
an HD Ha dog

Note: This is exactly the same as Product consancy and
works here because the numerator is constant.

Hence, Ry = NID, wd Ry = NID;
eR, x Dy = Nod R, x Dym Ne which isthe product
constancy sition.

106. TAG] vet rro or Ouro pte re CAT

Direct process for cleaation
To find ou the percentage change inthe ratio due 1o à
change I tb denemiatr follow the following process:
In oder ind he perecauge change from 1072010 10
25, calcule the perenage change in the denominaor in
the revere fashion.
is. The require perentage change from Ry 1 R; willbe
pen by calculating te percentage charge inthe denomi-
ors foe 251020 (ie. in a revere fashion) & or om
210%,

Table 5:1. Product Constancy Table, inverse Pro-
porlonalty Table, AB — A table,

Ratio Change to Denominator table

EEE
Los A= Be frase BAS areas
(rum, , Kemmer Femme
reported Y

[tano change eect of Derominanor (Rao decreta]

Danio change cms (6)
Ra nase

Application 6: Use of PCG to Calelate Ratio Changes:
Under normal situations, you will be faced wäh cios
where both namentor and denominator change, The gro-
ces lo handle and cakulte such changes is also quite
‘convenient if you go though PCG.

Illustration

Calcule the percentage change between de Ran.
Ratio = 1020 Ru 2 = 1525

“The answer in this case is OS — 0.6 (20% increase)
However, in most cates calling the values ofthe io
wil Wo be easy. The following PCG process can be used 10
set the answer:

‘When 1020 changes to 15725, the change occurs prima
iy due 10 100 reasons

(A) Change inthe numeros (Nemerator eet)
(0) Change in the denamintor (Denominator effet)

y sogregaing the two effects and calling the effect
due 10 each separately, we cam get the anower easily as
follows
Numerater Effect Th numerator fc a the value of the
rai isthe same athe change in the numerator. Hee,
‘acu the numerator eet, jst calcula the percentage
change in the muertos:

a is case the numero s clay changing from 1010

15 (ie a 50% increase.) This iris that the romero
at is als SOS.
Denominator Elec As we have jst een above, the effect
of a percentage change in the denominator on the value of
ratio is sen by callao the denominatos percent
age change inthe reverse order,

In nis ease the denominator i changing from 20402,
Hence the denominator effect will be sen by going reves
from 28 10 20 ie. 20% drop.

‘With these two values, he overall pewenage change in
Ihe Rai in een by

no m
‘Tis means tht the aio has increased by 208.
"eave ito we student to practice such calalaons with
smote complicated vales forthe ratios.

Implications for Data Interpretation

Percentage is perhaps ane of ih mont rca links been
QA and Data Interpretation.

In the chapter teary mentioned above, the Percentage
Role for Percentage Calculations and te PCG applied 1
product change and ratio change are the most erica.

As already shoe, the use of FCG to calcule the per
cage change in a prou (as cable through be pie
chart example above) as well a te ws of PCG to calle
rato changes ar two extremely useful apliaios ofthe
conceps ol perceatages into DI.

107. FEB] veu ane nen reece

sting tne decimal point by two places o he left
‘Tas, 83.33% = 0.8333 in decimal vale.

1 secon leasing from thi table in the process of
‘vison by any ofthe numbers such as 23,5, 5,6,
7.8.9, 11, 12,15, 16,24 and so o, students ner
rally ace problems in cleoating the decimal val
cs of hee evans. However, I one get sed lo
‘he decimal vales that appear in he Table $2. cal
‘uation of decimals i divisions will become very
simple, For sance, when an Itper bs vided y
7, the decimal values can only be 1, 28, 2.57,
71.83 or 00, (There ae approximate values)

+ This also means But te difference between two

aos the 2 an be rl nd ely xis
hve by bo 467

“hs price i very wel a an advanced shot
ca fer opts bs con of som quo 1
ave If De sde o dior apli of
is rine

Calculation of Multiplication by Numbers
like 1.21, 0.83 and so on

In my ainia, he caution of mukipleaon of any em
ter by u number of the form Oay or ofthe form Lab
sold be viewed as à abimctionadéition sittin and
oras a muleplicadon sation. This can be explained as
follows

Example: Calelte 123 x 473.

Solution: 1 we wy 10 calcul this by mukipying, we
will end up gag rough very ne thing proces, which
ill ye the Final valu at the end bet noting before th
(Ge. you will have no cue about he answers range til you
reach the end of he calalton.

Tasted, on shold view Dis multiplication as an di
tion of 25% o the eii) number This means, th answer
an be gm by adding 23% of the number to se

“Thus 573% 125-473 + 29% of 473 2 73 4985438
Of 473 2 $676 + 14.19 = 8179

(Te percesage rule can be ud 0 caca the add
iow and get the answer)

“The similar proces aa be wie forthe calculation of
mation by a number such as 087

Cane can be got by tbc 13% ofthe number rom
sl an this calculation can again be done by percentage
ne)

Hence, te suden is advised 10 become thorough th
the persenge als, Percentage calculation & sions of
2a 3 cp numbers

WORKED-OLIT PROBLEMS

REISER A sets is goods 30% cheaper than Band
30% dearer than €. By what percentage i the cost of C's
goods cheaper than Fs good.

Shin. There are two atematve processes or sol
ing this question:

1. Assume the prie of C's goods asp Then As goods
treat 1. pand B's goods are such hat A's goods ae 30%
cheaper thin Is goods. Le. A's good a pried at 70% of
Bs goods,
Hen, 13 p = 70
Brs price => 100
B's pice = 130 PO = LAS71 p

“Then, the percentage by which C's price i cheaper than
Bs pice

CLS pp) x IOUT) = CONS 46.15%
Learning ta for student Couk you answer the ques.
ti: Why did we asume Cs price asa variable pa ben
work ur the problem on its bass. What would happen
if we assumed Bs peice as p or if we asserted A's price
02
2. Instead of assuming the price of one ofthe three asp,
assume the price as 100.

Le B = 100. Then A = 70, which is 30% more tan C.
Hence Ca 23.07% less han A (rom Table 41) = apro
S344. Hence answer is 46.15% approximately.

“ais cauto canbe done mentally i you ae able to
work through he calculations bythe sc of perce al.
‘The sadents ae chic 10 uy 10 assure the value of 100
for eich ofthe variables A, Band C and ce what happens
tothe calelatons involved inthe problem. Since the vale
‘of 100 is assumed for a varble to mise the requie-
ments of culaons to salve the problems, we should
ensure hat the variable assed as 100 should have the
maximum calelaoas associated with it)

108. Me Abhimanyu Banerjee is word about he balance
of is monty budget. The pice of perl has in-
crease by 40% By what pers dal he ete the
‘consumption of perl so that eis ble o balance his

buster”
ES (o 2355
CE] 10 1428
(e) None of these

In quesion $F ME. Banerjee wanted to limi be in.
rene in his expenditure to $5 on hi bac expendi
‘ure on petro then what should be the corresponding
decrease in consumption sn tha expendier exceeds

‘only by 52
was 0) 2856
(925 CE]

(e) None of tise
Ram sels his goods 25% chesper than Shyam and
25% dearer han Bram. How much perselage is
Bram’ goods cheaper than Shyam’?

wa © 50%
© 6 (9 00%
(6) None ofthese

Lo an election between 2 candids, Di gos 65%

ofthe toral yall votes. If the tll votes were 6000,
what ste number of vai votes that he other an-
date Mare get if 25% ofthe otal votes were de.
lard invalid?

(0 105 (0) 1575
1 None of ese
In a medal eerfkae, by mike a candidate gave
his eight as 25% mor Uan normal In the interview
panel. he lcd tat his height was 5 feo inches.
Find the percemage conection made by the candidate
rom his stated eight to his actual height

(© 169 (0185

@ 2 (0 2856
(9 25 (9 1666
(6) None ofthese

Ast Sharma generally wea his father's oa. Unfors

una, bis comin Shaurya poked him ene day tht
he was wearing a cout of length more than his height
by 15% the leg of Ari's father’ conti 120m
then find the actual length of his coat.
WR 1043 (0) 10272
on

A number is mitkenly divided by $ intend of being
mulipid by 5. Find the percentage change inthe
rest dueto this mistake.

n.

".

16

owes nan [ET

wo 0 sa
© 2200% CES
(0) None of de

Hur waned to aubiat $ from a number. Unfors:

na, be added $ Instead of subrcing. Find the

percentage change in ıh res

o 300% © ur

(©) 50% © sam

(©) Cannot be determined

MESA of x= 13% of y then find the value of x ify

= 200.

PENES

(0 None of ese
à mixer 80 los o milk nd water, 256 of te

‘mixture emi: How much water sould beaded to

the stare o hat milk becomes 20% of the maitre?

© @ 500

(4) 20 les © 15 lives
(0) 25 les CET

(© None of these

SO of tcf bis TS of HR af Which of the
following is e?

(0) 150
(©) 1560
A landowner increased the Keogh and the bredt of
‘a recempla plot by 10% and 20% respectively ind
‘he percelage change in the cost ofthe pl assuming

(0) 06570 (©) 05a (9) 1250

land prices ae wnifrm thoeghoet hit plot
w@ Be ©

(o 22% CES

(0) None of tose

‘The eight of ange isinresed by 40%, What an
o the maximum percentage increase in length of te
bate 0 thatthe increase in are rite o mari
mm 607

SR 0) 20% (0) 14285 (D 256
CES

‘The length, breadth and high of room inthe shape
of à cuidar incre by 108, 20% and 50% 1
spectively. Pd the percentage change I he volume
of the cuboid

or 7%
©

‘The salary of Amit is 308 more han hat of Var
Find by what percentage isthe salary of Var less
than that of A?

PET

on ans

CET

109. 152] Howe Prep tr ata Asco re CAT

(9 223%
(e) None of these

20, Tbepicol sugar is reduced by 25% but pit of the
dicas, Any em op increasing his expendiure
‘on sugar by 20% What isthe percentage change in his
moutly consumption of sugar ?
(a) 460% (0) 10% (6) 433.338 (8) 50%
(e) None of dese

21, The rice of ie falls by 20%, How much ic canbe
bought now withthe money tat was sufficient bay
20 kg of rice previously?
@ Ske ET
(o) kg

22. 30% ofa number when shit fom 9, gives the
number If, Find e number

wo rare

CETTE

(a 6 98
om CE
(e) None of hese

23, When (0% ofs mame Ai ed 10 ancber number
1,8 becomes 175% oi previous vale. Ten which
of the following is e regarding the values of A
m
(048
@ 454
(©) 2A
(4) Eier (a or (9) canbe true depending upon the

sabes of A a 5

(©) Noxting con be sid

25, Ataneleetioa, the candidato who got 55% ofthe votes
cast won by 14 votes. Find th toa arber of votes
on the ving ii 80% people cast their ve and
there were no invalid votes,
a Mr (0) 1800
o 1500

25. The popaljo of a vila is 1.0000. The tate of
Increase I 10% per an. ine population athe

co 1500

rt ol be ind year?
(a) 132100 © 1210
de) 118800 (9 120000
(0) None of thse

26, The population ofthe village of Gava is 1,000 a is
moment. It increases by 10% in Be fs year: Hower,
in he second year, de o mignon, the potion
tops by 5% Fin the population athe eral ol Ue thin
yea iin he td year the population Intense by 20%.
(o 1240 ©) 1250

& 127500
e 1320

27. A mas invest Ra. 10,000 in some shares ind rado
2:3: which pay dividend of 10%, 25% and 206
(on is investment) for that yer respectively Find his
divided income.

CE

4 1900 4 2000
CE 48) 1990
de) 1880

28. In an examination, Mobit obtained 20% more than
Sens but 10% less han Rajesh the mas ob-
taie by Sshant is 108, find tb percentage marks
obtained by Rajesh ifthe full mars is 2000.

(a) 86.66% & Re
(o) 783% on
(o) Nove ofthese

29. ln a clas, 25% of the stent were absent for an
exam. 30% fied by 20 marks and 10% just passed
became of grace mark ofS. Find the average score of
the clas if the remaining student sored an average
of 60 marks and the pass mars ae 33 (counting the
Sina scores of the candidates).

(0) 37266 & 376
(9) 378 CES
CE

30. Ram spends 20% of hs many income on hi house-
bold expendinze, 18% ofthe rest on books 30% of
the reson clothes and aves the rest On couig, be
comes lo Know that be has ily saved Rs, 9520
Find his monthly income.

(a) 10000, © 15000
(e) 20000 (9 12000
(6) None ofthese

31. Hans and Bhaskar have sais that Joly amount to
Rs, 10,000 per month. They spend the same amount
smoothly and then it is found tat the rai oftheir
savings 6: 1 Which ofthe following canbe Ha
salay?

(a) Re 6000 Rs. 5000
1) Ra 4000 {Rs 3000

32. The population ofa village I 500. Ifthe number of
males incresses by 11% and the number uf females
increase hy 20%, the the population becomes 6330.
Find the population of females inthe town.

(a 2500 © 3000
(o 2000 (a) 3800

110. TBB] sue arme nero

(©) 100 (or and 200 (agains)
16) 150 Car, 30 (aio)

(9) 200 (ot and 300 (agains)

40. OF the adsk population in Nagpur. 485 of men end
25% of women are med, What percentage of the
ol population of ads is married (assume tha mo
man marie more an one wornan and vice vena)?
() B36 DEN
CE (8) Nowe of thse

41. The weight of an ion bucles increases by 33.13%
ben filed with war to SOR of its capacity Which
‘of tse may be SOS of the weigh of he backe when
its led with water (assume the weight of bucket
ands capacity in i o be inter?

Wr Wo (SKE Wh Rte

42, Asuras scored ttl of run in $0 overs. India
Wid the sores in 20% leis over. I India’s average
run mie had teen 23.33% higher the sores would
ave been ed 10 overs eurer. Fi Bow many runs.
were sored by Australia?
wm 20 09 220
«© 20 (4) Cannot be determined

43. Due lo à 25% hike in the price of ice pe Kilogram.
18 person is able to purchase 20 kg les fr Rs. 400.
Fin the increased price of ree per Kilogram?
WKS ORG (RID (8) Rea

4. A slam sapped on the base salary of RS. 1200
pez month and the coco that fer every sles of
Rs, 10.000 above Rs. 10,00, he wil gt 305 of basic
‘olay and 10% ofthe sales as reward This incentive
bea does ot operate for tbe fit Rs. 1000 of
tale. What shouldbe the vale of ales if he wart 0
‘ara Rs. 7600 in a potelar ment?

(a Rs. 60000 © Rs. $0.000
(0 Rs 40000 (© None of these

45. In question 4, which of the folowing income cannot

be achieved in a mon?

a) Rs. 6000

© Rs. 9000

(6) Hoth a and D

(0 Any iecome can be achieved

46. In question 4 despie 5 percentage point increment
on the commision from 20%, the total commision
remained unaltered. Find tbe change inthe volume of
the Vaneo?

(a 10% CES
CES ES

7. la an assembly election at Sr, ie total mat as
180% out o which 16% of the toa voter en the vo
ing ist were declared ivi. Find wich of he fol:
lowing can be the percentage votes got bythe winner
ofthe election if the candle who came second ot
120% of he ttl votes on the voting it (There were
ou tree comestans, only ee ane and he il
mbr of vers on the ven ist was 20000)
we 0) 466%
or (0 None ofthese

48, A wach gains by 2% per hour when the temperature
‘sin the range of CSIC ad it oss a he same
rate when the temperstor i ia the ange of 20°C-
30°C. However, th watch owe is tante sce it
uns on time in all ober temperature ranges. On a
sunny ay, te tempentur stated sonne up from 8
am. in the moming atthe uniform rate of 2°C per
hour and sometime daring the afemoon it stared
‘coming down athe same rue Find what ee wil it
tbe by the watch a 7 pum. i at 8 am. the temperate
was 32°C end at pam. i was OC,
la 6:55pm) 6:55: pm
(6235228 pm. (0 None ofthese

(Questions 49-50: Sty the folowing able and answer the
cine that follow.

A Cope
Siete Mineral, Men gung Ro)
nm » ,
cas mn 1
Se 5 o “ 7

Which ofthe folowing beverages contains te maxi

‘mum amour of vii?

(a) Pepsi worth RS 16

(0) Coke worth RES

(6) Sprite wont RS 8

(0 Al the dee worth Rs. 125 (125 grams of exh)

0. Which ofthese i the cheapest?

(0) 200 prams of Pepsi + 200 grams of Coke

(© 300 rams of Coke +100 gras of Pepa

16) 100 grams of Coke + 10 grams of Peps + 100
rans of Sprite

(a) 300 grams of Coke +100 grams of Sprite

111. Level of Difficulty (LOD)

1. The price of am materi as gone up by 15%, labour
cost has ako increase from 25% ofthe cost of a

serial to 30% ofthe cost of raw materia. By how

mach percentage shoul her be a redaction i the

sage of raw materials 0 a o kcp the cost sarc?

@ 1% DE

(o 28% (a) 25%

(e) Cannot be determined

Me Alsa computer program. Hei signed three

Jobs for which time aid is in the rio of

5:4: 2 Gas ae needed to be done evil). Bat

due to some technical ana 10% of the tine alloted

‘foreach job gets wasted. Three, owing tothe lack

of ines, he inves ony 20%, 30%, 20% of the

hows of what was actually all te do the ee jobs

individually. Find baw much percentage of the total

‘ime alloted isthe Eme invested by 47

(o 38336 CES

(o 272% CET

(e) Cannot be determined

3. In ibe MOCK CAT paper at AMS, questions were
Add in five sections. Oat of the total sent, 5
candidats cleared the cuff in ll he sectors and
5% cleared none. Of he rest, 25% cleared only one
section and 20% cleared Four sections. I 24.35 ofthe
mie candidaies cleared o sections and 300 eo
dates cleared tee sections, ind out how many can-
les appeared atthe MOCK CAT at AMS?
(a) 1000 @ 1200 (e) 1500 ca) 2000
(e) 1800

A. There ae thre galleries in a cal mine. On the frst
day, to galeries ae operative and afte some time,
‘he thin gallery is made operative. With this, he 0%
put ofthe mine became Hal as are gsi, What isthe
capacity of the second gallery as à percentage ofthe
fir, it given tha four-month Copa ofthe fit
and the tind gales was the same as the annual
‘output of the second galery?

war mr
© cor wo”
(6) None ofthese

ups: purgas [158

5. 10% of salty sa water cetined ina Mask was poured
ut ito a bear. Aer this. apart of the water con.
tained inthe beaker was vapeurised by heating and
ne 1 this, the perenage of sal in the beaker ia
creased AF ines. fi is know tha fer the content
‘ofthe beaker was pedo the Mask percentage
‘of sl inthe flak increased by 2%, ind the origial
‘gant of ca wate inthe ask

DA OM + Das
DE CE

ON = 13% OM +x
oT or
(6) Nove ofthese

6. In an election of 3 candidatos A,B and C À gts 30%
more votes than À also beats C by 1,0000 vote. I
‘nis known ta ges 5 perceatage pont more vores
tan Ca find the numberof votes on the voting lit
(eisen 90% ofthe yours on the voting ist voed and
o votes were ilegal

(a) 72000 09 81,000
(e) 50900 (e) 100000
© 1.0900

7. A chock i st right 12 non on Mond I loses
12% onthe comet lime in the fica week bat pairs
AR on te tra time daring the second week. The
‘ome shown on Monday aer two wecks willbe
Wen wu
2:0 WR
(6) None ofthese

8. The petro prices shot up by 7 asa eso he hike
in the price of crudes. The pis of pol before the
hike was Rs. 28 per live, anal wavels 2400
Islometes every mon and his ca gives a mileage of
1 almetes to a ie. Find the inreate in the expen-
ture hat Vals to ner dae tothe increase the
pce of pt ( ie pare rape)?

(0) Rs. 270 Q Ra 262
€ Rs. 276 CRE
(o Rs. 20

9. For question 8, by how many kilometres should Vasa
reduce his travel wants o main his expeni-
ture at the pri evel (pro to the pee increase)?
(0) 157 kn (5) 7 km
(©) 168 km (4) 180m
(6) None of those

112. TED] tm rut aras At ls a CA

10. te question 8 if Vaal wm 1 mil the increase in

‘expenditure to Rs. 20, what strategy should he adept
ith respect 1 his ave?
a) Reduce travel to 2350 Kleines,
(0) Radice travel to 240 klometres
€ Reduce wave to 2360 kilometres
8) Reduce tve to 2370 lometres
(e) Nove el these

I. A shopkeeper announces a discount scheme as for
Tows: for every purchase of Rs, 30001 Rs. 6000. the
casame gris a 15% discount or ticket ht entes
im 10 get a 7% discount on a further purchase of
goods coin more than Rs. 6000. The customer
eve, woul! have te option of eating bi ight

the sbopkecper at 4% of his fil purchase value
(por the ight refer to the 7% encour ticked) nan
enthusiastic response 1 tbe scheme, 10 people pur-
‘hate goods worth Rs 400 each. Find the maxima.
Possible revenue for the shopkesper

(o) Rs 38400) 60) Re 38000
© Ra 3940 (4) Ra 39000
(© Rs 40000

12. For question 1), find the maximum possible discount
a be shopkeeper wea have offer the customer
(a) Bs 1600 09) Rs 2000
©) Rs 6000 (Rs 4000
(e) None ef these

Directions for Questions 13-16: Read he following and
‘ane he questions that follow.

‘wo fends Shayar and Kailash oom two vesons of.
ca. Shayam owns the diesel version of the car, while
Kai owns dhe peto version.

Kia car gives an average that i 20% higher han
Shays (in ers of ies per Kilometro), II known tat
‘evel cous 60% of its prize higher than diel
13. The rato ofthe cost per Kilometre of Kult car to

ayant eu e
as wis
un a

(e) Cannot be dterived
1. I Shaya’ car gies an average (20 km pr re,
then the difference in be cos ol vel pe Kilometro
betmonn the tuo car is
(0 Aaa CRE
(R25 (6) Roa
(e) Cannot be determined

15. For question 14, be rat of the cos per lamer of
‘Sbayan’s rave o Kai rave is
m3 @1:3
@ 1:12 ma
(©) Cannot be determined

16. lee cos Rs. 125 per ie. ben the ifference in
the cos of travel per kilometre between Kash’ and
‘Shayan's is assume a average of 20 km pe ir for
Shayarv's car and als assume tat peta 50% of its
‘own pie higher than dese)

(@) Rs 175 © Rs 0875
(@) Rs 125 (@ Re LBs
(© None of these

Directions for Questions 17-23: Read the fllowing und
answer the questions that follo,

Inthe island of Heol Boola Moola, the inhabians have
stage process of elclting their average incomes and
‘expenditures. According t an old legend prevaen on tat
island, th average mori income hal 1 be calculated en
‘he bass of 1 montas in a calendar yea wile he average
monthly expendiure was 1 be calculated onthe basis of 9
month per year. Tis wool’ esd to people having an un:
‘cretion of thir savings ine here would be an un-
eresimatin ofthe income and an overestimation ofthe
‘expenditure per mach
17. Me Boogle Woogle comes back rom the USSR and

comvinces his community comprising 273 families to
stat elelaling the average income and the average
‘expenditure onthe bass of 12 months per calendars
yea. Now it is kaoıa thatthe average existed
Income in his commarity is (according 1 the old
system) 87 moolas pe month then wha willbe he
percentage change ia te savings of the cummaniy of
Me. Boogle Woogle (assume tht here e mo ote

change)?
(o 123% CE
(o 131% CES

(e) Cannot be determined
18. Por queen 17, if iis known lat ie average es
reed monty cxpendiure à 19 meolebs per mont
forthe island of Haols Rooks Moola en hat willbe
the percentage change inthe estimated caving of he
comme?
(o 20% CET
(o 25236 (a) 26.66%
1) Cannot be determine

113. 19. For question 18, If is own tat be average estr
‘mated monthly expenditure was 22 moolhs pe month
for the community of Boople Woogl (having 273
familie) then what wil be be percentage change in
‘he estimated saving ofthe orme?
DE CES
(o) 25:23% CES
(e) Cannot be determined

20, For question 19, na wll be the percent change in
the estimated average incre ofthe commanly (al.
alto on the bass ofthe new eximatod average)?
(a) 1328% increase (0) 14285 decease
(6) 1666% increase (€) 166% decease
(e) None ofthese

21. te finance mialmer of the isla Me. Bhola Ram
cars an encens the average monhly income
aso be estimated onthe basis ef 12 mouths pe yer
while he average monthly expendiur i 10 be ese
rated on the basis of 11 mont othe year, what wil
happen to the savings in the economy of Hola Boola
Moola?
(a) Increwe (9) Decrease
(6) Remain constant () Either (D) or (6)
(0) Cannot be determined

22, For question 21, wha willbe he percentage change in

saving?
31% © 152%
© 23% @ 33%

1) Cannot be determined

23, For question 22, what willbe the perenne change in
the estimated monthly expendiure?
(a) 22.22% decrease — (0) 22.22% increase
(6) 18.18% decrease — (6) 18.18% increase
(6) None ofthese

24. Abbimanyy Banerje has 72% vision in is eft eye
and 68% vision in is sight ye. On comte therapy.
estas wearing contact lense, which augment his
‘isos by 15% inthe eft ye and 11% nthe right ye.
Find out he percentage of normal vison tat he pa.
eses after conctive therapy. (Assume that à
pensons eyesight is a malpicatve contre ofthe

sep of M et and sigh eye)
(0) 525% (9 25% (9 725%
(6) None of hese

wor

ome: rece [TST

‘cuage profit of 30.05% oa the CA Pin the vale of
the percentage profit tha the shopkeeper would hive
‘cared if had given discounts of 10% an 15% ony
@) 3S) 25% (0756 (685%
te) 0.2%

26, If the third discount in question 25 was Rs. 229,50,
then id the original marked price of the item.

(a) Rs. 1.00000) Re 125000
(@) Rs. 200.000 (4) Re. 250,000
(e) Re. 225000

27. Krishna Iyer, a moi uses 24% o his fet in cow
ering te first 20% his total jay (im city ving
‘conditions. Ihe Knows ta he has to cover anther
25% of his al journey in city diving coditioas,
hat shouldbe the minimum percentage increase in
the fe effceney or poi dieing over the ey
ving fe ficiency, so that he is jst able o cover
is ene journey witout having to ref? (Approx:

mately)
(a 392% ©) 458
(0 458% CRE

Directions for Questions 28-30: Read the following and
ser the questions a follow the BSNL announced eat

the STD rates on 27 December 2001. The new ates and
labo are given in dhe able Below and ae to be imple-
mented fom the 14 January 2002.

Sieb Detalla
EZ

Dinner Peak aes C2
ou

TE] 12

wm Hé 48 von

m 00 as as

m 890 6 s

28. The maximum percentage redaction In costs wil be
experienced fr calls over which ofthe following
Asancısı
(a) 50-200 do 500-1000.

(e) 1000 (4) 200-800

29. The percentage difference in ib cost of a et of tele
phone calls made o the hand 14 Jury having
‘aration 4 minutes over a dance of 350 km, 3
‘inte fora distance of 70 km and 3 minuten for a

114. 162] to put art Apo be CAT

distance of 1050 kn is al Ue tre als ae made

in peak tines)
@ S128 (o 51.76%
© 98% (9) Cannot be determined

30. 1 one ofthe thee cal in question 29 were made in
an off peak time on both days, then the percentage
eduction in the ttl cost ofthe calls between 13th
and 14th January will
(a) Definite ree
(9) Def increase
16) Willdepend un which partial call was made in

a of peak time
(9) Cannet be determine

Directions for Questions 31-15: Read be following casco

and answer he question tha flow:

‘The circulation of the Deccan Emerald newspaper is
273000 copies, while its closes competitor ae The Timer
Of Hindusen ee Indias Tes. wich sell 247,00 and
20% more than that respectively (round off othe higher
isa). All the newspapers cost Rs. ch. The rar
comision offered by the thee papers are 205, 25% and
30% respectively (hese commissions are calculated on the
sale pie ofthe newspaper. Also ts known that news
apes ear primarily rough alos und advertising.

31. Taking the base as the net revenue of Deccan Ber.
athe percentage difference of the net revenue (uv.
mes = commission disbursed wo hawkers) between
Deccan Emerald aná Indie Tre e
@ 240% CES
(o 262% 10 Nove ofthese

32. The ati of the percentage ference in the ttl net
revenue between Deccan Emerald and Indias Times
10 the percentage diference inthe ttl revenue be-
tween Decean Emerald and Inia» Times is
do 1488 om
lo 06720 (0) Cannot be determined

39. IF cost ol priming te newspaper is Rs. 8, 7.3 and
respectively per day for Decean Emerald, Ties of
Hinds and nia Times respectively and on any
day the avaiable advertising spe inthe Deccan
Emerald newspaper is 800 ce (column centimes)
and de adverisig mie für Deccan Emerald is
Rs, 2000 per ce ten the percentage o he adverse
space that mat be wild o ensure the al cover
ofthe days cost for Deccan Emerald is

(o 95.83% DE
or (8) Cannot be determined
34. Based on th dat In the previas question and the
aééiioal information dt the space avalabily in
India’ Times a 1000 co and du in the Times of
Hindustan I 100 ce, fio the percentage pit ier
ne in the percentage of advenising space 10 be
‘lied da India Toner and that which mast be
ised in Times of Mirdason so thal both ewsppers

Jat breakeven
(o 45 0 52
@ 10 (@) Cannot be determined

8. For the das in questions 3 and if know that
the averti rt in Times of Msn is o. 1800
et cand thi the Jia Timer Ra. 2100 per ce,
then what isthe percentage point diference in the
perentage of adveising pace tobe sed by Tines
ef Hinducen sa dia Tre 10 at bo of tema
are jist able 0 breakeven?
wa © 36
(9 109 (@) Cueca be determined

36. On a tral journey, there ae 5 kinds of ches AC I.
AAC Il, AC Il 3er, nd general. The relationship
between the sis ofthe tickets for the Era iv
AC His 20% higher than AC I and AC is 705 of
‘AC II's value higher than the AC II ücke’s valo.
‘The Baier ticket i 25% ofthe AC T's ick con and
the gene lets 13 the rise ofthe AC like
‘The AC iker cous 78D euros bern London and
Paris. The ference in the rue of 3 trad general
ticket is
@) 4125 euros (0) $58 euros
(6) 4875 euros (A) 52.78 euros

37. For e above question, the tual cout of ane ticket of
‘och clas will be
(@ 23375 383325
CE 19) 373325

Dictons fer Question 38-40: Read the following and
answer the questions tht follow.

‘A Barailepress win has 2 AC 1 bogeys having
24 berths each. 3 AC I bogeys having 45 berths each, 2 AC
Ul bogeys having 6 bers each and 12 Sir bogeys hay
Ing 64 bers each. There ar no general bogeys the a.
17200 caros ish asf an AC 3er be rom Landon
vo Glasgow, answer he fllowing questions: