गणित कक्षा ४

ul0ft

sIff $

k/LIf0fsf nflu

gk] fn ;/sf/
lzIff, lj1fg tyf k|ljlw dGqfno

kf7\oj|md ljsf; s]Gb|
;fgfl] 7dL, eStk'/
@)&*

k/LIf0fsf nflu

ul0ft
sIff $

g]kfn ;/sf/
lzIff, lj1fg tyf k|ljlw dGqfno

kf7\ojm| d ljsf; sG] b|

;fgf]l7dL, eStk'/

k|sfzs

gk] fn ;/sf/

lzIff, lj1fg tyf kl| jlw dGqfno
kf7\ojm| d ljsf; sG] b|

;fgfl] 7dL, eStk/'

© ;jfl{ wsf/ kf7\ojm| d ljsf; s]Gb|
o; kf7\ok':ts;DaGwL ;Dk0" f{ clwsf/ kf7o\ jm| d ljsf; sG] b| ;fgfl] 7dL, eStk/' df
lglxt /xs] f] 5 . kf7o\ j|md ljsf; s]Gb|sf] lnlvt :jLsl[ tlagf Jofkfl/s ko| fh] gsf
nflu o;sf] k/' } jf cfl+ zs efu xa' x' k|sfzg ug{, kl/jtg{ u/]/ ks| fzg ug{, sg' }
ljBt' Lo ;fwg jf cGo kl| jlwaf6 /s] 8{ ug{ / k|ltlnlk lgsfNg kfOg] 5}g .

k/LIf0f ;:+ s/0f M lj=;= @)&*

d"No ?=

kf7o\ k:' ts;DaGwL kf7sx¿sf sg' } klg ks| f/sf ;´' fjx¿ ePdf kf7\oj|md
ljsf; sG] b,| ;dGjo tyf k|sfzg zfvfdf k7fOlbgx' g' cg'/f]w 5 . kf7sx¿af6
cfpg] ;´' fjx¿nfO{ s]Gb| xflb{s :jfut ub5{ .

xfd|f] egfO

kf7o\ jm| d lzIf0f l;sfOsf] d"n cfwf/ xf] . kf7o\ k:' ts ljBfyLd{ f ckl] Ift bIftf ljsf; ug{] Ps d'Vo
;fwg xf] . o; kIfnfO{ bl[ i6ut ub} kf7o\ j|md ljsf; s]Gbn| ] ljBfno lzIffnfO{ Jofjxfl/s, ;do;fkI] f
/ u0' f:t/Lo agfpg] p2Z] on] kf7o\ j|md tyf kf7\ok':tssf] ljsf; tyf kl/dfh{g sfo{nfO{ lg/Gt/tf
lbbF } cfPsf] 5 . cfwf/et" lzIffn] afnaflnsfdf cfwf/et" ;fIf/tf, ul0ftLo cjwf/0ff / l;k Pjd\
hLjgfk] ofu] L l;ksf] ljsf;sf ;fy} JolStut :jf:Yo tyf ;/;kmfO;DaGwL afgLsf] ljsf; ug]{ cj;/
kb| fg ugk{' 5{ . cfwf/et" lzIffsf dfWodaf6 afnaflnsfx¿n] kf| sl[ ts tyf ;fdflhs jftfj/0fkl| t
;rt] eO{ cgz' f;g, ;bfrf/ / :jfjnDag h:tf ;fdflhs Pjd\ rfl/lqs u0' fsf] ljsf; ug{'k5{ . o;n]
lj1fg, jftfj/0f / ;r" gf k|ljlw;DaGwL cfwf/et" 1fgsf] ljsf; u/fO{ snf tyf ;f}Gbok{ |lt cle?lr
hufpgk' 5{ . zf/Ll/s tGb?' :tL, :jf:Yos/ afgL Pjd\ l;hg{ fTdstfsf] ljsf; tyf hfthflt, wd,{ efiff,
;+:s[lt, If]qkl| t ;Ddfg / ;defjsf] ljsf; klg cfwf/et" lzIffsf ck]lIft kIf x'g\ . b]zk|d] , /fli6«o
Pstf, nf]stflGqs d"NodfGotf tyf ;+:sf/ l;sL Jofjxfl/s hLjgdf k|ofu] ug,{' ;fdflhs u'0fsf]
ljsf; tyf gful/s stJ{ okl| t ;hutf ckgfpg', / bl} gs hLjgdf cfOkg{] Jofjxfl/s ;d:ofx¿sf]
klxrfg u/L ;dfwfgsf pkfosf] vfh] L ug{' klg cfwf/e"t txsf] lzIffsf cfjZos kIf xg' \ . o;
kIfnfO{ bl[ i6ut u/L ljBfno lzIffsf] /fli6o« kf7\ojm| d kf| ¿k, @)&^ sf] ddc{ g¿' k bz] sf ljleGg
ljBfnodf k/LIf0f ug]{ k|of]hgsf nflu cfwf/e"t lzIf -sIff $_ sf] ul0ft ljifosf] kf7\ojm| dcg¿' k
of] k':ts ljsf; ul/Psf] xf] .
>L xl/gf/fo0f pkfWofo, >L ;'hg sfˆn], >L ;/nf b]jsf6] f, >L /Lt' >i] 7, >L huGgfy clwsf/L /
>L /fdrGb| 9sfn /x]sf] sfob{ naf6 tof/ ul/Psf] kf7\ok':tsnfO{ o; ¿kdf Nofpg] sfod{ f o;
s]Gb|sf dxflgbz]{ s >L c0fk|;fb Gof}kfg,] tTsfnLg dxflgb{z] g >L sz] jk;| fb bxfn, 8f= /fdhLk|;fb
kl08t, >L nIdLgf/fo0f ofbj, >L j}s0' 7k|;fb vgfn, >L kl| dnf avtL, >L s[i0fk|;fb kf]v/n] ,
>L ufd] f >i] 7, >L clg?b|;fb Gof}kfg] / >L /fhsd' f/ dfy]dfsf] ofu] bfg /x]sf] 5 . o;sf] efiff
;Dkfbg >L lrgfsd' f/L lg/f}nfaf6 ePsf] xf] . o; k:' tssf] nc] fp6 On:' 6];« g l8hfOg >L v8f;]
;g' 'jf/af6 ePsf] xf] . o;sf] ljsf;df ;+nUg ;Dk"0fk{ l| t sG] b| xflb{s st[ 1tf ks| 6 ub5{ .

o; kf7\ok:' tsn] ljBfyL{df lgwfl{ /t ;Ifdtf ljsf;sf nflu ljBfyLn{ fO{ ;xofu] ug]{ 5 . o;n]

ljBfyLs{ f] l;sfOdf ;xof]u k'¥ofpg] Pp6f dxŒjk"0f{ / cfwf/e"t ;fduL| sf ¿kdf sIff ljm| ofsnfkaf6

xg' ] l;sfOnfO{ dha't agfpg ;xof]u ug]{ 5 . To;n} ] of] lzIfssf] l;sfO lj|mofsnfksf] ofh] gf

geO{ ljBfyL{sf l;sfOnfO{ ;xofu] k'¥ofpg] ;fdu|L xf] . kf7o\ k':tsnfO{ ljBfyL{sf] l;sfOdf ;xofu]

k¥' ofpg] Pp6f dxŒjk0" f{ cfwf/sf ¿kdf afns]lGb|t, l;sfOs]lGb|t, cge' js]lGbt| , p2Z] odn" s,

ko| f]ud'vL / lj|mofsnfkdf cfwfl/t agfpg] ko| f; ul/Psf] 5 . l;sfO / ljBfyL{sf] hLjGt cg'ejlar

tfbfTDo sfod ub{} o;sf] ;xh ko| f]u ug{ lzIfsn] ;xhstf{, pTk]|/s, k|jw{s / vf]hstf{sf ¿kdf

e"ldsfsf] ckI] ff ul/Psf] 5 . o; k:' tsnfO{ c´ kl/is[t kfgs{ f nflu lzIfs, ljBfyL,{ cleefjs,

a'l4hLjL Pjd\ ;Dk0" f{ kf7sx¿sf] ;dt] ljz]if e"ldsf /xg] x'Fbf ;Da4 ;a}sf] /rgfTds ;´' fjsf

nflu kf7\oj|md ljsf; s]Gb| xflb{s cg/' f]w ub5{ .

lj=;=+ @)&* kf7\oj|md ljsf; s]Gb|

;fgfl] 7dL, eStk/'



ljifo;"rL ki[ 7;ª\Vof
PsfO ljifoj:t ' !–!*
!= Hofldlt !
!=! /v] fx¿ / sf0] fx¿ ^
!=@ ;dtnLo cfsl[ t !!
!=# 7f]; j:tx' ¿ !(–&%
@= cª\sul0ft !(
@=! ;ªV\ ofsf] 1fg #$
@=@ ul0ftsf cfwf/e"t ljm| of ^@
@=# leGg, bzdnj / kl| tzt &^–!#@
#= gfk &^
#=! ;do (!
#=@ db' |f (*
#=# b'/L !)(
#=$ Ifdtf !!&
#=% tf}n !@%
#=^ kl/ldlt / Ifq] kmn
$= ;dx" !##–!$^
%= tYofªs\ zf:q !$&–!^#
%=! lan / ah6]
%=@ tYofª\s !$&
^= aLhul0ft !%%
!^$–!*@



Hofldlt (Geometry)

1.1 /v] fx¿ / sf]0fx¿ (Lines and Angles)

1.1.1 kg' /jnfs] g (Review)
tnsf cfsl[ tx¿ x]/]/ 5nkmn ugx{' f];\ / o;sf] lgisif{ sIffsf7] fdf k|:tt' ugx{' f];\ M



i. dflysf cfs[ltx¿df sxfF sxfF sf]0f ags] f 5g\ <

ii. sg' cfsl[ tdf ;aeGbf 7n' f] / ;aeGbf ;fgf] sf]0f ags] f] 5, x/] ]/ eGg'xf;] \ .

iii. o:t} cGo ;fdu|Lx¿ ;ªs\ ng u/L lrq sf/] ]/ o;sf zLif{laGb', e'hf / sf]0fx¿
bv] fpgx' f];\ .

1.1.2 rfbF (Protractor)

ljm| ofsnfk 1
rf/ rf/ hgf ljBfyLx{ ¿sf] ;dx" agfO{ Hofldlt afs;df ePsf] tn lbOPsf h:t}
;fdu|L x/] ]/ o;df sltbl] v slt;Dd ;ª\Vof pNn]v ul/Psf] 5, cGo efux¿ s] s]
5g\, 5nkmn ug'x{ f;] \ .

o:tf] HofldtLo ;fdu|LnfO{ rfbF elgG5 .

ljm| ofsnfk 2

sG] bl| aGb' cfwf/ /v] f

Hofldlt aS;df ePsf ;fduL| x¿dWo] dfly lrqdf b]vfOP h:t} ;fdu|L rfbF (Protractor)

ul0ft M sIff $ 1

xf] . o;sf] cwj{ [Qfsf/ 3/] fnO{ 180 a/fa/ efudf ljefhg ul/Psf] xG' 5 . x/]s Ps
efu a/fa/ 1 l8u|L x'G5 . o;nfO{ ;ª\s]tdf 1o n]lvG5 . rfbF df b'O{ tl/sfaf6 0o b]lv
180o ;Ddsf :sn] x¿ pNnv] ul/Psf] x'G5, lsg xf]nf < o;sf] ko| f]u s] ss] f nflu
ul/G5, 5nkmn ug{x' f;] \ .

rfbF df cwj{ Q[ fsf/ efusf] tnlt/sf] l;wf /]vfnfO{ cfwf/ /v] f elgG5 . o; /]vfsf] bj' }
5p] df 0 / 180 n]lvPsf] xG' 5 . 90 nl] vPsf] /v] f l;wf tn cfwf/ /]vf;uF hf]l8Psf]
:yfgnfO{ sG] b|laGb' elgG5 . sf]0fsf] gfk lng / sf0] fsf] /rgf ug{ rfFbsf] k|of]u ul/G5 .

pbfx/0f 1

(i) HofldtLo ;fduL| rfFbsf] cfsf/ ============== x'G5 .
(ii) rfFbdf 0 b]lv ============== ;Dd nl] vPsf] xG' 5 .
(iii) rfFbdf tnsf] l;wf /v] fnfO{ ============== elgG5 .
;dfwfg
(i) cw{jQ[ fsf/
(ii) 180
(iii) cfwf//]vf

1.1.3 rfbF sf] ko| fu] u//] sf0] fsf] gfk / /rgf (Measurement and construction

of angle using protractor)

-s_ sf0] fsf] gfk A

Z

X lrq (i) Y BC

lrq (ii)

ljm| ofsnfk 1

dflysf sf]0fx¿sf] gfk s;/L kQf nufpg] xf]nf <

(i) rfFbsf] s]Gbl| aGb' / sf]0fsf] zLifl{ aGb' ldNg] u/L rfbF sf] cfwf/ /]vfnfO{ lrq (i)
/ (ii) sf sf]0fsf] cfwf/ eh" f;uF vlK6g] u/L ldnfP/ /fVgx' f;] \ .

2 ul0ft M sIff $

A
Z

XY BC

(ii) cfwf/ eh" fdf sf0] f agfpg] csf{] /]vfn] rfbF df b]vfPsf] :sn] n] sf0] fsf] gfk
hgfpF5 .

sf0] fsf] gfk lnbF f cfwf/ /]vfsf] 0 af6 sf]0f agfpg] /v] flt/ uGb} hfg'kb5{ .

∠XYZ df zLifl{ aGb' Y / cfwf/ /]vf XY 5 . X ePsf] lt/af6 uGb} hfbF f ZY n]
rfFbsf] 130 df bv] fPsf] 5 . To;n} ] ∠XYZ = 130o eof] . To:t} ∠ABC df zLif{laGb'
B / cfwf/ /v] f BC 5 . C ePsf] lt/af6 uGb} hfFbf AB n] rfbF sf] 60 df b]vfPsf]
5 . To;}n] ∠ABC = 60o eof] .

-v_ sf0] fsf] /rgf

ljm| ofsnfk 2
rfFbsf] ko| f]u u/]/ sf]0fsf] /rgf s;/L ug{] xfn] f <

BC

rfbF sf] k|ofu] u//] sf0] f ∠ABC = 40o sf] /rgf •A
BC
(i) ?n/sf] ;xfotfn] BC /]vfv08 lvRg'xf;] \ .
A
(ii) laGb' B nfO{ rfbF sf] s]Gbd| f kg]{ u/L l7s BC
ldnfP/ /fVgx' f;] \ .
A
(iii) C af6 uGb} uP/ 40 df ;ªs\ ]t u/L A gfd
lbgx' f;] \ .

(iv) rfbF nfO{ x6fP/ ?n/sf] ;xfotfn] AB
/]vfv08nfO{ hf8] g\ x' f;] \ . sf0] f ABC tof/
eof] . ∠ABC = 40o eof] .

ul0ft M sIff $ BC

3

pbfx/0f 2 P
rfFbsf] k|ofu] u/]/ ∠PQR sf] gfk kQf nufpg'xf;] \ .

P

QR QR

;dfwfg

∠PQR df cfwf/ /]vf QR / zLifl{ aGb' Q 5 . R af6 uGb} hfbF f PQ n] rfFbsf] 50 df
bv] fPsf] 5 . To;}n] ∠PQR = 50o eof] .

ljm| ofsnfk 3

cfcfˆgf] sfkLdf rfFbsf] ;xfotfn] 600, 700, 1100, 1500 sf] sf0] fx¿ /rgf ugx'{ f;] \ /

sIffsf]7fdf k:| t't ug'x{ f];\ .

LP

∠LMN = 60o ∠PQR = 70o

M N Q R
X A
∠ABC = 150o
∠XYZ = 110o
C
Y Z B

pbfx/0f 3

rfbF sf] k|ofu] u//] tnsf sf0] fx¿sf] /rgf ugx{' f;] \ .

(a) 30o (b) 30o

(c) 120o (d) 120o
4
ul0ft M sIff $

cEof;

1. rfFbsf] k|of]u u/L tnsf kT| os] sf0] fx¿nfO{ gfk/] ltgLx¿sf] gfk l8u|Ldf
n]Vg'xf];\ .

(i) (ii) (iii)



(iv) (v) (vi)



2. tn lbOPsf gfksf sf]0fx¿ rfFbsf] ;xfotfn] lvRgx' f];\ .

(i) 60o (ii) 20o (iii) 50o (iv) 100o (v) 120o

3. tn lbOPsf cfsl[ tx¿sf] leqL sf]0fsf] gfk rfFbsf] ;xfotfn] gfKgx' f;] \ .

PS
AX

B C Q R Y Z
P Q
4. bfofFsf] lrqdf slt 7fpdF f sf]0fx¿ ags] f 5g,\ RS
kQf nufpg'xf;] \ . kT| os] sf]0fnfO{ gfk]/ sfkLdf U
nV] g'xf];\ . T

kl/of]hgf sfo{ (Project work) VW

1. tkfO{‘sf] 3/, ljBfno jl/kl/ /x]sf] sf]0f hgfpg] j:t'x¿ ;ªs\ ng u/]/ lrq
sfg] 'x{ f;] \ / pSt j:td' f ags] f sf]0f rfFbsf ;xfotfn] gfk]/ sIffsf]7fdf k:| tt'
ugx{\ f];\ .

2. kfFr kfrF hgfsf] ;dx" agfO{ x/]s ;dx" nfO{ 6-6 cf]6f l;Gsf lbg'xf];\ . ljBfyLx{ ¿nfO{
l;Gsfx¿ e'OdF f hf8] /] aGg] sf]0fx¿ rfbF sf] ;xfotfn] gfk/] sIffsf]7fdf k|:tt'
ug{ nufpgx' f];\ .

ul0ft M sIff $ 5

1.2. ;dtnLo cfs[lt (Plane Surface)

1.2.1 k'g/fjnfs] g (Review)
ljBfyLx{ ¿ pko'St ;ªV\ ofdf ;d"xdf a:gx' f];\ . h;' sf kfOk÷l;Gsfx¿ ;ªs\ ng
ugx{' f];\ . pSt l;Gsfx¿sf] k|ofu] u/L ljleGg HofldtLo cfs[ltx¿ agfP/ sIffsf7] fdf
k|:t't u/L 5nkmn ug'{xf];\ .

1.2.2 ;dtnLo ;tx ePsf cfsl[ tx¿ (Plane surface figure)
ljm| ofsnfk 1

tn lbOPsf cfs[ltx¿ cjnf]sg ugx'{ f];\ . o;sf sltcf6] f lsgf/fx¿ 5g\ < s'gfx¿
slt 5g\ < Pscfk;df 5nkmn u/]/ vfH] g'xf];\ .

(i) (ii)

(iii) (iv)

ljm| ofsnfk 2

rf/ rf/ hgfsf] ;d"x agfpg'xf;] \ . x/s] ;dx" n] s]xL l;;fsnd lng'xf];\ . l;;fsnd
hf8] /] km/s km/s vfnsf lqe'h / rt{e' 'h lgdf0{ f u/L efux¿sf af/]df ;dx" ut
5nkmn ugx{' f;] \ .

eh' f sg' f

e'hf lzifl{ aGb'
6
ul0ft M sIff $

ljm| ofsnfk 3

tLg tLg hgfsf] ;d"xdf a:g'xf;] \ . lbOPsf ;dtnLo cfsl[ tx¿ cjnfs] g u/L tnsf

k|Zgdf cfwfl/t eP/ 5nkmn ugx'{ f;] \ .

(i) (ii) (iii)

(iv) (v) (vi)

(i) kT| o]s cfs[ltdf sltcf]6f e'hfx¿ 5g\ <
(ii) dflysf kT| o]s cfs[ltdf sltcf]6f zLif{laGbx' ¿ /x]sf 5g\ <
(iii) pSt cfs[ltx¿sf] gfd s] s] xf]nf <

;dtnLo cfs[ltdf /x]sf /]vfv08nfO{ e'hf / bO' { eh' f hfl] 8Psf] sg' fnfO{ zLifl{ aGb'
elgG5 .

pbfx/0f 1

tn lbOPsf] ;dtnLo ;txsf] cfsl[ tsf] gfd nv] L eh' fx¿ / zLif{laGbx' ¿sf] gfd
n]Vgx' f];\ .

(i) A (ii) P S

B C Q R
P T
(iii) M Q

(iv)

N O RS

;dfwfg

(i) lqeh' ABC df (ii) ju{ PQRS

eh' fx¿ AB, BC, CA e'hfx¿ PQ, QR, RS, SP

zLifl{ aGb'x¿ A, B, C xg' \ . zLif{laGb'x¿ P, Q, R, S x'g\ ..

ul0ft M sIff $ 7

(iii) cfot MNOP (iv) rte' {h' QRST

e'hfx¿ MN, NO, OP, PM eh' fx¿ QR, RS, ST, TQ

zLifl{ aGbx' ¿ M, N, O, P xg' \ . zLif{laGbx' ¿ Q, R, S, T x'g\ .

pbfx/0f 2

lbOPsf] lrq cjnfs] g ug{'xf;] \ . ;dtnLo cfsl[ tx¿sf] e'hfx¿ / lzifl{ aGb'x¿sf]

gfd nV] g'xf;] \ . ;dfwfg

AD P (i) ju{ ABCD df

e'hfx¿ AB, BC, CD, DA

zLifl{ aGbx' ¿ A, B, C, D x'g\ .

B CQ R (ii) lqeh' PQR df
L O e'hfx¿ PQ, QR, RP
zLifl{ aGb'x¿ P, Q, R x'g\ .
E

M N (iii) cfot LMNO df
T
S F I eh' fx¿ LM, MN, NO, OL
U Z
zLif{laGbx' ¿ L, M, N, Ox'g\ .

VY G H (iv) k~re'h EFGHI df
e'hfx¿ EF, FG, GH, HI, IE
WX

zLif{laGbx' ¿ E, F, G, H, I xg' \ .

(v) if8e\ 'h STUVWXYZ df

e'hfx¿ ST, TU, UV, VW, WX, XY, YZ, ZS

zLifl{ aGbx' ¿ S, T, U, V, W, X, Y, Z xg' \ .

1.2.3 ;dtnLo cfsl[ tsf] eh' fsf] gfk (Measurement of side of plane figure)

ljm| ofsnfk 1

tn lbOPsf h:t} ;dtnLo cfs[ltsf] /rgf ugx{' f;] \ . ?n/sf] ko| f]u u/L e'hfsf] gfk

lnO{ sIffsf7] fdf k:| tt' ugx'{ f];\ .

(i) A D (ii) X (iii) C H (iv) P
T
Z GQ
D

B C Y EF RS

8 ul0ft M sIff $

lj|mofsnfk 2
kfrF kfrF hgfsf] ;dx" df a:g'xf];\ . tn lbOPsf 7f;] j:t'x¿ 6]«; u/L ;dtnLo
cfsl[ t lvRgx' f;] \ . pSt cfsl[ tx¿sf] aflx/L 3/] fsf] gfk s;/L kQf nufpgx' 'G5,
5nkmn u/L k:| tt' ug'x{ f];\ .

(i) (ii)

(iii) (iv)

pbfx/0f 1

vfnL 7fpF egx'{ f];\ M

(i) lqe'hdf cf6] f zLifl{ aGb' x'G5g\ .

(ii) ju{sf ;a} e'hfx¿ x'G5g\ .

(iii) cfotsf ;Ddv' eh' fx¿ x'G5g\ .

;dfwfg (ii) a/fa/ (iii) a/fa/
(i) 3

pbfx/0f 2

tnsf] ;dtnLo cfsl[ tsf eh' fx¿sf] gfd / gfk n]Vgx' f];\ M

(i) (ii) (iii) D

AD P
EH

BC QR FG

;dfwfg

(i) eh' fx¿, AB = ... cm, BC = ... cm, CD = ... cm, AD = ... cm
(ii) eh' fx¿, PQ = ... cm, QR = ... cm, PR = ... cm
(ii) eh' fx¿, DE = ... cm, EF = ... cm, FG = ... cm, GH = ... cm, HD = ... cm

ul0ft M sIff $ 9

cEof;

1. ?n/sf] ko| f]u u/]/ ljleGg lsl;dsf ;dtnLo cfs[ltsf] /rgf ugx{' f;] \ .

2. tnsf cfs[lt x/] ]/ o;sf eh' f / zLifl{ aGb'x¿sf] gfd n]Vgx' f];\ .

-s_ -v_ -u_ -3_
R
A CA C M
P

P

B B Q S N O

3. rfFb / ?n/sf] ko| f]u u/L e'hfsf] nDafO 5 cm ePsf] ju{sf] /rgf ug{'xf;] \ .

4. tLgcf]6f e'hf ePsf km/s km/s k|sf/sf ;dtnLo cfsl[ tx¿ lvRgx' f];\ /
?n/sf] ;xfotfn] e'hfx¿ gfKg'xf;] \ .

5. tn lbOPsf cfsl[ tsf] e'hfsf] gfk ?n/sf] ;xfotfn] gfKgx' f;] \ .

-s_ -v_ -u_

A EH I
K
D

B C F G J

-3_ P -ª_

L PS

M O QR

N

kl/of]hgf sfo{ (Project work)

tkfO{‘sf] 3/df k|of]u ePsf s'g} kfrF cf6] f ;dtnLo ;tx ePsf j:t'x¿sf] ;ªs\ ng
u/L sfkLdf 6]«; u/L o;nfO{ sIffsf]7fdf k|:t't ugx{' f];\ . 3/ j/k/ jf ;fjh{ lgs
:yndf o:tf cfs[ltx¿ sxfF k|ofu] ePsf] b]Vge' Psf] 5 < 5 eg] tL ;fduL| sf] gfd
l6kf]6 ug{x' f;] \ .

10 ul0ft M sIff $

1.3 7f];j:tx' ¿ (Solid objects)

1.3.1 k'g/jnfs] g (Review)

tnsf j:t'x¿sf] cjnfs] g u/L sg' sg' df s:tf s:tf ;dtnLo ;tx 5g\ <
sIffsf]7fdf 5nkmn ugx{' f];\ .

(i) (ii) (iii) (iv)

1.3.2 7f]; j:ts' f cfsf/x¿ (Shapes of solid objects)

ljm| ofsnfk 1
tn lbOPsf 7f]; j:t'x¿nfO{ p:t} p:t} cfs[ltsf] ;dx" df 56' \ofP/ /fVg'xf];\ . ;dx" df
b'O{ bO' {cf6] f cfs[lt yKg'xf];\ .
(i) (ii) (iii)

(iv) (v) (vi)
ul0ft M sIff $ 11

tnsf 7f;] cfs[lt / o;sf ef}lts gdg' f xg] 'x{ f];\ M

cfsl[ tsf] gfd ef}lts gdg' f -pbfx/0f_

3g

cfotsf/ if8\d'vf

3gsf] ;a} ;tx ;dtnLo / jufs{ f/ x'G5g\ .

ljm| ofsnfk 2
pkoS' t ;ªV\ ofdf ;d"xsf] lgdf0{ f ug{'xf];\ x/s] ;dx" n] Pp6f Pp6f 7f];j:t' lng'xf];\
o;nfO{ x]/]/ o;sf efux¿af/] 5nkmn u/L lgisif{ lgsfNg'xf;] \ .

(i) (ii) (iii) (iv)

n'8fs] f] uf6] L lrofsf] a66\ f ld7fO{sf] afs; rssf] a6\6f

dflysf ;a} 7f;] j:tx' ¿df 6/6 cf6] f ;dtnLo ;tx 5g\ . n'8f]sf] uf]6L / rssf]
a66\ fsf ;a} ;tx jufs{ f/ 5g\ . To;n} ] oL j:tx' ¿ 3gfsf/ j:t' x'g\ . lrofsf] a6\6f
/ ld7fO{sf] afs;sf ;a} ;tx cfotsf/ 5g\ / ljk/Lt ;tx a/fa/ 5g\ . To;}n] oL
j:t'x¿ cfotsf/ if8\dv' f xg' \ .

12 ul0ft M sIff $

pbfx/0f 1

tnsf 7f]; j:ts' f] cfsf/ s:tf] 5, x]/]/ eGgx' f;] \ .

(i) (ii) (iii) (iv)

lj|mofsnfk 3
1. tnsf kT| os] 7f;] j:t'sf] gfd / cfs[lt s:tf] 5, nV] g'xf;] \ .

(i) (ii) (iii)

(iv) (v)

2. 3gfsf/sf bO' {cf6] f / if8d\ v' fsf/sf bO' c{ f6] f :yfgLo 7f]; j:t'x¿sf] gfd
nV] g'xf;] \ .

ul0ft M sIff $ 13

1.3.3 7f;] j:ts' f] ;tx, lsgf/f / zLifl{ aGb' (Faces, edges and verties of solid objects)
lj|mofsnfk 1

tkfO{s‘ f] 3/df n'uf w'gsf nflu k|ofu] ug{] ;fag' sf] cjnfs] g ug{'xf;] \ . o;df sltcf]6f
;dtnLo ;tx 5g\ < lsgf/f / sg' fx¿ slt sltcf6] f 5g,\ n]Vg'xf];\ / sIffsf7] fdf
k:| tt' ug{x' f];\ .

lj|mofsnfk 2
lzIfssf] ;xfotfdf h'; kfOk jf uxFs' f] 5j\ fnL jf
l;Gsfaf6 3g / if8d\ 'vfsf vfj] |mf] gdg' f lgdf0{ f ug'x{ f;] \ .
pSt 3g / if8\dv' fsf] ;txdf km/s km/s /ªsf] sfuh
6f;F L ;tx 56' \ofpg'xf];\ . 3g / if8d\ v' fdf sltcf]6f
;tx lsgf/f / zLif{laGb' 5g\, sIffsf]7fdf k|:t't ug'{xf];\ .

h'; kfOkx¿ hf]l8Psf] s'gfnfO{ zLifl{ aGb' elgG5 . h'; kfOkx¿ hf]l8P/ ag]sf] aGb
;dtnLo cfs[lt ;tx xf] eg] bO' c{ f6] f ;txx¿ hfl] 8Psf] :yfgnfO{ lsgf/f elgG5 .
14 ul0ft M sIff $

pbfx/0f 1
tnsf 7f]; j:ts' f] ;tx, zLifs{ f0] f / lsgf/f bv] fpgx' f];\ .

;dfwfg

lsgf/f

zLifl{ aGb'

;tx ;tx

zLif{laGb' lsgf/f

cEof;

1. tnsf cfsl[ tnfO{ lgdf{0f u/L o;sf zLifl{ aGb,' lsgf/f / ;tx bv] fpg'xf];\ .



2. tnsf 7f]; j:t' jf o:t} ;fduL| sf ;tx, zLifl{ aGb' / lsgf/f bv] fpgx' f;] \ .
(i) (ii) (iii)

(iv) (v) (vi)

ul0ft M sIff $ 15

3. tnsf 7f;] j:ts' f ;tx, zLifl{ aGb' / lsgf/f s] s] x'g\, nV] gx' f];\ .

A D
B
;tx

C

zLifl{ aGb'

E F lsgf/f

HG

;tx
zLif{laGb'
lsgf/f

kl/of]hgf sfo{ (Project work)
tkfOsF{ f] 3/df ko| f]u ug{] kfrF kfrF cf6] f 3gfsf/, if8d\ v' fsf/ j:t'x¿ ;ª\sng u/L
ltgLx¿sf] gfd;lxt sIffsf]7fdf k:| tt' ug'{xf];\ .

ldl>t cEof;
1. lbOPsf 7f];j:t' s:tf] cfsf/sf 5g,\ nV] gx' f;] \ M



2. HofldtLo ;fdu|L rfbF sf] k|of]u ss] f nflu ul/G5 < n]Vg'xf];\ .
3. tkfOn{‘ ] k|of]u ug]{ bO' c{ f]6f ;dtnLo ;tx ePsf] j:ts' f] gfd n]Vg'xf;] \ .
4. ?n/sf] ko| fu] u//] nDafO = 6 ;]=ld= / rf}8fO = 2 ;=] ld= ePsf] cfot

agfpg'xf;] \ .

16 ul0ft M sIff $

5. rfbF sf] k|of]u u//] tnsf sf]0fx¿sf] gfk gfKg'xf];\ / nV] g'xf;] \ . Y
-s_ -v_ -u_ Y

A

X

P

B C Q R Z

-3_ -ª_ -r_

I LN

J K M WX

6. rfbF sf] ko| f]u u/]/ tnsf sf0] fsf] /rgf ugx{' f;] \ .

-s_ 60o -v_ 110o -u_ 30o -3_ 150o

-ª_ 90o -r_ 70o

7. tn lbOPsf cfs[ltx¿sf] zLif{laGb' / e'hfsf] gfd n]Vg'xf];\ .

-s_ -v_ -u_

AD P G
D

B Q R E F

C K

-3_ -ª_ -r_ 17

M

O SI
P

N Q R J

ul0ft M sIff $

8. ?n/sf] ;xfotfn] tnsf cfsl[ tsf] e'hf gfk lngx' f;] \ .

(i) (ii) (iii) (iv)



9. lbOPsf 7f];j:t'sf] zLifl{ aGb,' lsgf/f / ;tx 56' o\ fpg'xf;] \ .

(i) (ii) (iii)

10. tn lbOPsf cIf/x¿sf ljleGg efudf ags] f sf0] fx¿ rfbF sf] ;xfotfn]
gfKg'xf];\ .



18 ul0ft M sIff $

kf7 @ cªs\ ul0ft (Arithmatics)

2.1 ;ª\Vofsf] 1fg (Concept of Number)

2.1.1 k'g/fjnf]sg (Review)

tkfO‘s{ f] kl/jf/df kl/jf/ ;ªV\ of slt 5 < tkfO‘{sf] 6f]ndf ePsf] hg;ªV\ of slt
xf]nf < tkfO{‘sf] lhNnfsf] hg;ªV\ of slt xfn] f <

g]kfnsf] hDdf hg;ª\Vof slt xf]nf < s] tL ;an} fO{ uGtL ug{ ;Dej 5 < cj:o
5, To;/L ;a} kmnk"mnsf] ;ªV\ of / ;a} hg;ª\Vof uGtL ugs{ f] nflu xfdLn] uGtLsf
;ª\Vofx¿ k|ofu] u5f{‘}, h:t} M 1, 2, 3 ... . xfd|f kv' f{x¿n] uGtLsf ;ª\Vofx¿sf] ljsf;
x'ge' Gbf cufl8 j:t'x¿sf] u0fgf s;/L uy]{ < ;f] ;DaGwdf lzIfs;Fu 5nkmn ug{'xf;] \ .

xfdf| kv" fx{ ¿n] ;ªV\ ofsf] k|of]u ;'? xg' e' Gbf klxnf klg j:t'x¿sf] u0fgf ub{y] .
;'?df cfˆgf kl/jf/sf ;b:ox¿ / cfˆgf 3/kfnj' f hgfj/x¿sf] ;ªV\ of yfxf kfpg
eO' ‘d{ f wsf{ tfGg], leQfdf wsf{ tfGg] ub{y] . lbgel/ vfg]s'/f vfH] g 3/af6 lg:sbF f
kl/jf/sf ;b:o / wsf{ Ps Ps ;ª\ult x'g] u/L e'O{d‘ f tfGy] . a]ns' L kmsk]{ l5
k'gM kl/jf/sf ;b:o / eO' {d‘ f tfg]sf wsf{ Ps Ps ;ª\ult x'g] u/L d6] \by] . olb
e'O{d‘ f wsf{ afFsL /x]df kl/jf/sf] ;b:o x/fPsf] yfxf kfP/ vf]hL uby{ ] . eO' ‘d{ f wsf{
sf]bf{ dl] 6g] ePsfn] leQfdf wsf{ tfGg yfn] . kl5 kl5 9ª' u\ f jf 5:] sfnfO{ Ps Ps
;ª\ult ug,]{ n6\7Ldf lrxg\ nufpg], 8f]/Ldf ufF7f] kf//] j:t'x¿sf] u0fgf ug{] h:tf
lj|mofsnfk u/s] f] kfOG5 .
ljsf;jm| d;Fu} dflg;x¿n] j:t'x¿sf] u0fgf ug{ ;ªV\ ofx¿sf] k|ofu] ug{ yfn] .
;ªV\ ofx¿sf] ljsf;j|mddf al] jnf]lgog, OlhlK;og, /fd] g, hfkflgh, rfOlgh, lus| ,
lxGb' c/]lasx¿sf] dxìjk0" f{ e"ldsf /xs] f] 5 . xfdLn] ko| f]u ug{] ;ªV\ of k0| ffnL lxGb'
c/]ljs ;ªV\ of k0| ffnL xf] . o; k|0ffnLdf 0, 1, 2, ..., 9 ;Ddsf 10 cf]6f cª\sx¿sf]
k|of]u x'G5 . of] ;ª\Vof k|0ffnL bzdnj ;ª\Vof k|0ffnL xf] .

ul0ft M sIff $ 19

2.1.2 ;ft cª\s;Ddn] ag]sf ;ª\Vofx¿ (Number formed by seven digits)

(A) :yfgdfg (Place Value)

lj|mofsnfk 1
xfdLn] sIff # df g} kfFr cª\sn] ag]sf ;ªV\ ofx¿ nV] g / k9g\ l;ls;ss] f 5fF} t;y{

oxfF xfdLn] ;ft cª\sn] ag]sf ;ªV\ ofx¿ n]Vg / k9g\ l;Sg] 5f}F . To;sf nflu tnsf]

tflnsf cWoog ugx{' f];\ .

slt cª\sn] ags] f] ;ª\Vof ;fª\lVos gfd

Ps cª\sn] ags] f] ;a}eGbf ;fgf] ;ªV\ of 1 Ps (one)

bO' { cª\sn] ag]sf] ;ae} Gbf ;fgf] ;ªV\ of 10 bz (Ten)

tLg cª\sn] ags] f] ;a}eGbf ;fgf] ;ª\Vof 100 ;o (Hundred)

rf/ cª\sn] ag]sf] ;ae} Gbf ;fgf] ;ª\Vof 1000 xhf/ (Thousand)

kfrF cª\sn] ag]sf] ;a}eGbf ;fgf] ;ªV\ of 10000 bz xhf/ (Ten Thousands)

5 cªs\ n] ags] f] ;a}eGbf ;fgf] ;ª\Vof 100000 nfv (Lakh)

;ft cª\sn] ag]sf] ;a}eGbf ;fgf] ;ªV\ of 1000000 bz nfv (Ten Lakhs)

1000000 nfO{ :yfgdfg tflnsfdf /fVbf,

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

1000000

ljm| ofsnfk 2

;Fu} lbOPsf] Ans cjnfs] g u/L tnsf] :yfgdfg tflnsfdf cªs\ eg'{xf];\ .
xhf/ ;o bz Ps

xhf/ ;o bz Ps
ul0ft M sIff $
20

lrqdf 4 cf]6f Ps, 3 cf]6f bz, 2 cf]6f ;o / 3 cf]6f xhf/ 5g\ t;y{ :yfgdfg
tflnsfdf cª\sx¿ ebf{ lgDgfg;' f/ x'G5 M

xhf/ ;o bz Ps

3 2 34

cª\sdf M 3,234
cIf/df M tLg xhf/ bO' { ;o rfl} t;

ljm| ofsnfk 3

;Fu} lbOPsf] cafs; x]/L :yfgdfg tflnsfdf cªs\ x¿ eg'x{ f];\ .

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

lrqdf lbOPsf] cafs;df 7 cf6] f Ps, 3 cf6] f bz, 5 cf6] f ;o, 7 cf6] f xhf/, 4

cf6] f bz xhf/, 1 cf]6f nfv / 4 cf]6f bz nfv 5g,\ ;a}nfO{ lgDgfg';f/ :yfgdfg

tflnsfdf k:| t't ug{ ;lsG5 M

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

41 4 7537

ul0ft M sIff $ 21

lj|mofsnfk 4

;ft ;dx" df a:gx' f];\ . k|To]s ;d"xsf] kl| tlglwTj bz nfv 5
ug{] u/L Ps÷Ps hgf kfn}kfnf] cufl8 cfpgx' f;] \ 2 ;o
/ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 nl] vPsf] ;ªV\ ofklQ
lnP/ km/s km/s efF8fdf Ps÷Ps cf6] f 8
;ªV\ ofklQ /fVg'xf;] \ . ;a} ;d"xn] k|Tos] ef8F fdf nfv
/fv]sf cª\sklQx¿ x]/L pSt cªs\ x¿af6 aGg]
;ª\VofnfO{ :yfgdfg tflnsfdf k|:t't u/L sIffdf bz xhf/
5nkmn ugx'{ f;] \ . 3

6
xhf/

h:t} M ljBfyL{x¿n] jm| d;} uF Ps, bz, ===== bz nfv 1 9
nl] vPsf] ef8F fd] f 1, 9, 6, 5, 3, 8 / 2 v;fn5] g\ . Ps bz

ca :yfgdfg tflnsfdf k:| tt' ul/;s]kl5 tnsf

kZ| gx¿sf] pQ/ lbgx' f;] \ .

(i) nfv nl] vPsf] ef8F fdf sg' ;ª\Vof 5 <

(ii) 3 slt nl] vPsf] ef8F fdf 5 <

(iii) 8 k/s] f] ef8F fsf] hDdf dfg slt xf]nf <

(iv) efF8fdf v;flnPsf] ;a} cª\sx¿n] agfpg] ;ªV\ of slt xf]nf <

o;/L ef8F fdf v;flnPsf ;a} cªs\ x¿n] agfpg] ;ªV\ of 2835691 xG' 5 . ;ª\Vofdf

/x]sf cªs\ sf bO' {cf]6f dfgx¿ xG' 5g\ . Pp6f bl] vg] dfg / csf]{ :yfgdfg . 8 sf]

b]lvg] dfg 8 g} xG' 5 . nfv nl] vPsf] efF8fdf 8 5, To;}n] 8 sf] :yfg nfv x'G5 .

8 /xs] f] efF8fsf] hDdf dfg 8 nfv x'g] ePsf]n] 8 sf] :yfgdfg 1,00,000 × 8 =
8,00,000 xG' 5 .

;ªV\ ofdf /xs] f cªs\ x¿sf 2 cf6] f dfg xG' 5g\ . klxnf] b]lvg] dfg (face value)
/ bf];|f] :yfgdfg (place value) . s'g} klg ;ª\Vofdf /x]sf cªs\ sf] bl] vg] dfg
Tof] cªs\ xG' 5 eg] pSt cªs\ / pSt cªs\ /x]sf] :yfgsf] dfgnfO{ u0' fg ubf{ ;f]
cªs\ sf] hDdf dfg xG' 5, h:t} M ;ªV\ of 2832510 df 5 sf] bl] vg] dfg 5 g} xf] eg]
:yfgdfg ;o xf] . To;n} ] 5 sf] hDdf dfg 100 × 5 = 500 x'G5 .

22 ul0ft M sIff $

pbfx/0f 1
;ª\Vof 56,33,431 nfO{ :yfgdfg tflnsfdf k|:t't u/L 6 sf] b]lvg] dfg, :yfgdfg /
hDdf dfg n]Vgx' f];\ .
;dfwfg
oxfF ;ªV\ of 56,33,431 nfO{ :yfgdfg tflnsfdf k:| t't ubf{,

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

56 334 3 1

6 sf] bl] vg] dfg Ö6
6 sf] :yfgdfg Ö nfv
6 sf] hDdfdfg Ö 6 × 1,00,000 Ö 6,00,000

(B) lj:tfl/t ¿k (Expanded Form)

sg' } klg ;ª\VofnfO{ pSt ;ª\Vofsf] x/]s cª\s / ;DalGwt cª\ssf] :yfgdfgnfO{

u'0fg u/L of]ukmnsf ¿kdf k:| t't ug'{nfO{ pSt ;ªV\ ofsf] lj:tfl/t ¿k elgG5 .

pbfx/0f 2
;ªV\ of 57,34,298 nfO{ lj:tfl/t ¿kdf n]Vg'xf;] \

;j{k|yd :yfgdfg kQf nufpFbf,

5 7 3 4 2 9 8 :yfgdfg hDdf dfg

Ps (ones) = 8 × 1 8

bz (Tens) = 9 × 10 90

;o (Hundreds) = 2 × 100 200

xhf/ (Thousands) = 4 × 1000 4,000

bz xhf/ (Ten Thousands) = 3 × 10,000 30,000

nfv (Lakhs) = 7 × 1,00,000 7,00,000

bz nfv (Ten Lakhs) = 5 × 10,00,000 50,00,000

ca,

57, 34, 298 = 5 × 10, 00, 000 + 7 × 1,00,000 + 3 × 10,000 + 4 × 1,000 + 2 ×
100 + 9 × 10 + 8 × 1

ul0ft M sIff $ 23

cEof;

1. vfnL 7fpF egx{' f];\ M
(i) ;ªV\ of 23,19,891 df 2 sf] b]lvg] dfg ============= xf] .
(ii) ;ªV\ of 18,79,312 df 8 sf] :yfgdfg ============= xf] .
(iii) ;ªV\ of 71,83,294 df 3 sf] :yfgdfg ============= xf] .
(iv) 10,53,216 df bz nfvsf] :yfgdf cª\s ============= 5 .

2. tnsf k|To]s ;ª\Vofx¿nfO{ :yfgdfg tflnsfdf k:| tt' u/L 5 sf] :yfg dfg
nV] g'xf];\ M

(i) 93,45,281 (ii) 51,68,719 (iii) 12,67,815
(iv) 34,51,229 (v) 27,35,869

3. tnsf ;ªV\ ofx¿df nfvsf] :yfgdf /x]sf] cªs\ sf] :yfgdfg n]Vgx' f];\ M

(i) 52,34,564 (ii) 10,05,301 (iii) 9,75,608

(iv) 5,63,204

4. tnsf k|To]s ;ªV\ ofx¿df ufn] f] 3/] f nufOPsf] cªs\ sf] :yfgdfg nV] gx' f;] \ M
(i) 1 2 ,45,389 (ii) 2 4,68,135 (iii) 24,5 3 ,289
(iv) 89, 3 4,208

5. tnsf kT| o]s ;ªV\ ofx¿nfO{ lj:tfl/t ¿kdf nV] g'xf;] \ M
(i) 13,25,614 =
(ii) 19,82,543 =
(iii) 67,89,123 =
(iv) 5,00,230 =

6. tnsf] lj:tfl/t ¿knfO{ 5f6] s/Ldf nV] g'xf;] \ M
(i) 5 × 1,00,000 + 3 × 10,000 + 2 × 1000 + 6 × 100 + 9 × 10 + 4 × 1 =
(ii) 2 × 10,00,000 + 0 × 1,00,000 + 0 × 10,000 + 4 × 1,000 + 3 × 100 +

2 × 10 + 6 × 1 =

24 ul0ft M sIff $

kl/ofh] gf sfo{ (Project work)

1. cfˆgf cleefjs jf lzIfs;Fu ;f]w/] tkfOs{‘ f] :yfgLo txsf] hg;ªV\ of kQf
nufO{ :yfgdfg tflnsfdf k|:tt' ug'x{ f];\ .

2. dfly pNn]v ePsf] ;ªV\ ofeGbf km/s vfnsf uGtL ug{] ;ªV\ of k4lt
tkfO{Fx¿nfO{ yfxf 5 < cfˆgf cleefjsnfO{ ;f]wL tL ;ª\Vofx¿ sg' sg'
:yfgdf ko| fu] ePsf 5g,\ ;r" L tof/ kfgx'{ f;] \ .

ul0ft M sIff $ 25

2.1.3 ;ft cªs\ n] ags] f ;ªV\ ofx¿ nV] g / k9g\ (Reading and writting number

formed by seven digit)

lj|mofsnfk 1
tnsf :yfgdfg tflnsfsf] cWoog u/L ;f]lwPsf kZ| gsf] pQ/ lbg'xf;] \ .

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

7 1 2 6 530

(i) hDdf slt nfv 5g\ <
(ii) hDdf slt xhf/ 5g\ <
(iii) hDdf slt ;o 5g\ <
(iv) clGtd bO' { cª\sn] ags] f] ;ª\Vof slt xf] <
(v) dflysf ;ªV\ ofnfO{ s;/L n]Vg] / k9g\ ] xfn] f <
dfly lbOPsf ;ªV\ ofnfO{ k9\bf PsxQ/ nfv 5lAa; xhf/ kfFr ;o lt; eg/] kl9G5
eg] pSt ;ªV\ of nV] bf 71,26,530 nl] vG5 .

;ªV\ ofnfO{ cª\sdf n]Vbf bfofaF f6 afoflF t/ klxnf] tLg cªs\ cufl8 To;kl5 bO' {
b'O{ cªs\ cufl8 cw{lj/fd (,) lrxg\ /flvG5 .

pbfx/0f 1

/fli6«o hgu0fgf 2068 lj=;= sf cg';f/ sf7df8fF} lhNnfsf] u0fgf hg;ªV\ of
17,44,240 5 h;dWo] 9,13,001 k?' if / 8,31,239 dlxnf /xs] f 5g\ . ca oL
;ª\Vofx¿nfO{ cIf/df n]Vgx' f;] \ .

;dfwfg

dflysf ;ª\Vofx¿nfO{ tflnsfdf k|:t't ubf,{

bz nfv nfv bz xhf/ xhf/ ;o bz Ps hg;ªV\ of

1 7 4 4 2 4 0 hDdf hg;ªV\ of

9 1 3 0 0 1 k'?ifsf] hg;ª\Vof

8 3 1 2 3 9 dlxnfsf] hg;ª\Vof

26 ul0ft M sIff $

oxfF,

hDdf hg;ªV\ of = 17,44,240

hDdf k?' ifsf] hg;ª\Vof = 9,13,001

hDdf dlxnfsf] hg;ªV\ of = 8,31,239

cIf/df nV] bf,

hDdf hg;ªV\ of Ö ;q nfv rjfln; xhf/ bO' { ;o rfln; (Seventeen lakhs forty

four thousands two hundred and forty)

hDdf k?' ifsf] hg;ª\Vof Ö gf} nfv t]/ xhf/ Ps (Nine lakhs thirteen thousands

and one)

hDdf dlxnfsf] hg;ªV\ of Ö cf7 nfv Pslt; xhf/ bO' { ;o pgfGrfln; (Eight

lakhs thirty one thousands two hundreds and thirty nine)

pbfx/0f 2
>Lsi[ 0fn] ufO{kfng / s'v/' fkfngsf nflu gk] fn ;/sf/n] lbg] s[lif cg'bfgcGtu{t
s[lif ljsf; a}ªs\ af6 k;} 6\7L nfv lqkGg xhf/ kfrF ;o tLg ?lkofF C0f lnP5g\ .
>Lsi[ 0fn] lnPsf] /sdnfO{ cwl{ j/fd (,) lrx\gsf] ko| fu] u/L cª\sdf n]Vg'xf;] \ .

;dfwfg

oxfF >Ls[i0fn] lnPsf] C0f,

65,00,000 + 53,000 + 500 + 3
= 65,53,503

:yfgdfg tflnsfaf6 b]vfpFbf,

bz nfv nfv bz xhf/ xhf/ ;o bz Ps

6 5 5 3 503

ul0ft M sIff $ 27

cEof;

!= vfnL 7fpF eg{x' f];\ M

(i) ;ªV\ of 1215230 df hDdf ============= nfv 5 .
(ii) ;ª\Vof 2505110 df hDdf ============= nfv 5 .
(iii) ;ªV\ of 2000010 df hDdf ============= nfv 5 .
(iv) ;ª\Vof 7000005 df hDdf ============= nfv 5 .

@= tnsf ;ªV\ ofx¿nfO{ cw{lj/fd (,) lbO{ n]Vg'xf];\ M

(i) 2520344 (ii) 1000230

(iii) 6524000 (iv) 5820525

#= cª\sdf nV] gx' f;] \ M

(i) Ps nfv Ps;6\7L xhf/ kfFr ;o 5
(ii) kfrF nfv ;QfO; xhf/ cf7 ;o
(iii) aofnL; nfv tO] ; xhf/ kfFr ;o aof;L
(iv) rfx} Q/ nfv gf} xhf/ ;ft ;o ;QfO;
(v) c7f;L nfv krkGg xhf/ rf/ ;o aofln;
(vi) la; nfv kfrF xhf/ kfFr

$= cªu\ |]hL / g]kfnL b'j} dfWodaf6 cIf/df n]Vg'xf];\ M

(i) 6,24,005 (ii) 25,27,003 (iii) 26,18,598
(iv) 48,02,212 (v) 24,00,005 (vi) 92,51,000

kl/ofh] gf sfo{ (Project work)

1. cfˆgf] 3/sf sg' } kfrF cf]6f ;fdfgsf] dN" o cleefjs;Fu ;fw] /] pSt dN" onfO{
:yfgdfg tflnsfdf k:| tt' ug'x{ f];\ .

2. cfˆgf] 3/ tyf ;db' fodf /fd] g cªs\ ko| fu] ePsf j:tx' ¿sf] ;r" L tof/ ugx{' f;] \ .

3. dfly pNnv] ePsf ;ªV\ ofeGbf km/s vfnsf uGtL ug]{ ;ªV\ of k4lt
tkfOF{x¿nfO{ yfxf 5 < cfˆgf cleefjsnfO{ ;f]wL tL ;ª\Vofx¿ sg' sg'
:yfgdf ko| fu] ePsf 5g\, ;"rL tof/ kfgx{' f];\ .

28 ul0ft M sIff $

2.1.4 ;ª\Vofx¿sf] z"GofGt (Rounding off numbers)

ljm| ofsnfk 1
dfgfF}, tkfOs{‘ f] ljBfnoaf6 3/ 100 m 6f9f 5 . tkfO‘{ ljBfnoaf6 3/ hfFbf olb
kfgL k¥of] eg] ljBfno / 3/sf lardf cft] nfUg] 7fpF 5g} . 3/ hfbF f kfgL k¥of] t/
tkfO‘{;Fu 5ftf 5g} eg] tnsf] cj:yfdf s] ugx'{ G' 5, lsg < 5nkmn ugx{' f];\ .
(i) ljBfnoaf6 20 m lxF8k] l5 kfgL k/]df
(ii) ljBfnoaf6 50 m lxF8]kl5 kfgL k/d] f
(iii) ljBfnoaf6 60 m lx8F ]kl5 kfgL k/d] f
(iv) ljBfnoaf6 85 m lx8F ]kl5 kfgL k/d] f

lj|mofsnfk 2

;a} ljBfyLx{ ¿n] hDdf 2 cf6] f ;d"x agfP/ ljBfnosf] rp/df hfgx' f;] \ . ToxfF xfdL
Pp6f vn] f vN] g hfbF }5fF} . rp/df tkfO‘{x¿n] tnsf] h:t} ;ªV\ of/]vf b]Vg'xg' ] 5 .

;'? clGtd

1 2 3 4 5 6 7 8 9 10

;'? clGtd

Ps ;d"xsf ljBfyL{x¿ ;ªV\ of/v] feGbf dflyk6\l6 / csf]{ ;d"xsf ljBfyL{x¿ tnk6l\ 6
a:gk' g{] 5 . ca kT| os] ;dx" af6 1/1 hgf v]n v]Ng cufl8 cfpg] 5g\ / lhTgn] ]
1 cª\s kfpg] 5 . cGtdf ;a}eGbf a9L cª\s Nofpg] ;dx" nfO{ ljht] f 3fl] ift u/L

k'/:sf/sf] Joj:yf ul/g] 5 . v]nsf] lgod o;k|sf/ x'g] 5 M
(i) ;ª\Vof /v] fsf] dfly / tn bO' { ;d"xaf6 1/1 hgf ;?' sf] laGb'df pleg] 5g\ .

(ii) Ps hgf dfbn ahfpg] JolSt xg' ] 5 / p;n] dfbn ahfO/xbF f cfˆgf If]qk6l\ 6
/xs] f] aS;df km8\sf] df/L lx8F g\ k' g]{ 5 .

(iii) dfbn aHg aGb ePkl5 hf] hxfF 5 p ToxL plegk' g{] 5 . olb 5 jf 5 eGbf dflysf
cªs\ x¿;uF } /xs] f] aS;df plePsf] JolSt clGtd aS;df uP/ plegk' g{] 5 / 5
eGbf ;fgf] cªs\ sf] aS;df plePsf] JolSt ;?' sf] aS;df uP/ plegk' g{] 5 .

(iv) h;n] lgodsf] ;xL kfngf u/L a9L cªs\ Nofp5F p;nfO{ ljht] f 3fl] ift ul/g] 5 .

ul0ft M sIff $ 29

ljm| ofsnfk 3

lbbL, tkfO;‘{ Fu d;uF ?= 50 hlt
slt ?lkofF 5 < 5.

;Gtf]ifn] ug/] x/] ] t/ uGbf hDdf ?= 48 /x5] .

lbbL, tkfO{‘ t s:tf] dnfO{ 10, 20, 30, 40, 50 cflb cGTodf z"Go
9F6'jf xg' ' xFb' f/] x]5 Û ePsf ;ªV\ of uGg ;lhnf] 5 / glhssf]

;ªV\ of 50 ePsfn] ?= 50 eg]sf] xF' .

6f9f glhs
4 0 41 42 43 44 45 46 47 4•8 49 50

;ªV\ of/v] fdf 48, 50 sf] glhs 5 t/ 40 af6 6f9f 5 . o;/L lbOPsf ;ª\VofnfO{
10, 20, 30, ... 100, 200, 300 ... cflb clGtddf z"Go ePsf] glhssf] ;ª\Vofdf n]Vg]
tl/sfnfO{ ;ªV\ ofx¿sf] zG" ofGt elgG5 .
glhssf] bzdf z"GofGt ul/Psf ;ª\Vofx¿ M 10, 20, 30 ... 120, 150 ... 2350 cflb .
glhssf] ;odf z"GofGt ul/Psf ;ª\Vofx¿ M 100, 200 ... 2300 cflb .

pbfx/0f 1
37 nfO{ glhssf] bzdf zG" ofGt ug'{xf;] \ M

;dfwfg
6f9f glhs

30 31 32 33 34 35 36 3•7 38 39 40
37, 40 sf] glhs kg{] ePsfn] 37 nfO{ glhssf] bzdf z"GofGt ubf{ 40 x'G5 .

30 ul0ft M sIff $

pbfx/0f 2
563 nfO{ glhssf] bzdf zG" ofGt ugx{' f];\ M

;dfwfg
glhs 6f9f

560 561 562 56•3 564 565 566 567 568 569 570
563, 560 sf] glhs kg]{ ePsfn] 563 nfO{ glhssf] bzdf zG" ofGt ubf{ 560 xG' 5 .

pbfx/0f 3
45 nfO{ glhssf] bzdf z"GofGt ug{'xf;] \ M

;dfwfg
a/fa/ a/fa/

40 41 42 43 44 4•5 46 47 48 49 50
45, 40 / 50 af6 a/fa/ b'/Ldf kg{] ePsfn] 45 nfO{ glhssf] bzdf z"GofGt ubf{ 50
nV] g] ul/G5 .

pbfx/0f 4
3573 nfO{ glhssf] bzdf z"GofGt ugx{' f];\ M

;dfwfg
glhs 6f9f

3570 3571 3572 35•73 3574 3575 3576 3577 3578 3579 3580
3573, 3570 sf] glhs kg]{ ePsfn] 3573 nfO{ glhssf] bzdf z"GofGt ubf{ 3570
x'G5 .

pbfx/0f 5
4655 nfO{ glhssf] bzdf z"GofGt ug{'xf];\ M

;dfwfg
a/fa/ a/fa/

4650 4651 4652 4653 4654 46•55 4656 4657 4658 4659 4660

ul0ft M sIff $ 31

4655, 4650 / 4660 sf] l7slardf kg{] ePsfn] 4655 nfO{ glhssf] bzdf zG" ofGt
ubf{ 4660 xG' 5 .

pbfx/0f 6
550 nfO{ glhssf] ;odf zG" ofGt ug'{xf;] \ M

;dfwfg
a/fa/ a/fa/

500 510 520 530 540 55•0 560 570 580 590 600
550, 500 / 600 sf] l7slardf kg{] ePsfn] 550 nfO{ glhssf] ;odf zG" ofGt ubf{
600 xG' 5 .

pbfx/0f 7
4833 nfO{ glhssf] ;odf z"GofGt ugx{' f;] \ M

;dfwfg
glhs 6f9f

4800 4810 4820 4830• 4840 4850 4860 4870 4880 4890 4900
4833, 4800 sf] glhs kg{] ePsfn] 4833 nfO{ glhssf] ;odf zG" ofGt ubf{ 4800
x'G5 .

pbfx/0f 8
9285 nfO{ glhssf] ;odf z"GofGt ugx{' f;] \ M

;dfwfg
6f9f glhs

9200 9210 9220 9230 9240 9250 9260 9270 9280• 9290 9300
9285, 9300 sf] glhs kg{] ePsfn] 9285 nfO{ glhssf] ;odf z"GofGt ubf{ 9300
xG' 5 .

32 ul0ft M sIff $

sg' } klg ;ªV\ ofnfO{ glhssf] :yfgdf z"GofGt ubf{ z"GofGt ug]{eGbf Ps :yfg 36L
-;o eP bzsf] :yfg bz eP Pssf] :yfgdf_ x]/L pSt :yfgdf /xs] f] cªs\ 5 jf
5 eGbf a9L ePdf zG" ofGt ug{] :yfgsf] cª\sdf 1 ykL k5fl8sf] cªs\ nfO{ zG" o
kfg'k{ b5{ . o:t} 5 eGbf sd ePdf zG" ofGt ug{] eGbf k5fl8sf cª\snfO{ zG" o kf/L
n]Vgk' b5{ .

cEof;

!= vfnL 7fpF egx{' f;] \ M
(i) 15 sf] tnsf] bz ============= / dflysf] bz ============= xG' 5 .
(ii) 426 sf] tnsf] bz ============= / dflysf] bz ============= x'G5 .
(iii) 4824 sf] tnsf] bz ============= / dflysf] bz ============= xG' 5 .
(iv) 832 sf] tnsf] ;o ============= / dflysf] ;o ============= x'G5 .
(v) 7542 sf] tnsf] ;o ============= / dflysf] ;o ============= x'G5 .

2. tnsf ;ªV\ ofx¿nfO{ glhssf] bzdf z"GofGt ug'{xf;] \ M

(i) 16 (ii) 78 (iii) 245 (iv) 561

(v) 1680 (vi) 7825 (vii) 2456

3. tnsf ;ªV\ ofx¿nfO{ glhssf] ;odf z"GofGt ug{x' f;] \ M

(i) 390 (ii) 505 (iii) 450 (iv) 5820

(v) 4850 (vi) 8270 (vii) 9234

kl/of]hgf sfo{ (Project work)

tn lbOPsf lrqdf sf7df8fF}af6 kf]v/f;Ddsf] b'/L lbOPsf] 5 . sf7df8f}aF f6 kf]v/f,

sf7df8f}aF f6 dl' Unª / sf7df8faF} f6 8d' |;] Ddsf] b/' L slt slt lsnfl] d6/ xG' 5 <

glhssf] bzdf zG" ofGt u/L nV] g'xf;] \ .

108 ls=ld= 135 ls=ld= 200 ls=ld=
kf]v/f
sf7df8fF} ••

d'lUnª 8'd|]

ul0ft M sIff $ 33

2.2 ul0ftsf cfwf/et" ljm| ofx¿ (Basic Operation In Mathematics)

2.2.1 k'g/fjnf]sg (Review)

tnsf k|Zgx¿sf] cWoog u/L 5nkmn ugx{' f;] \ M

(i) tkfO{‘;uF ?= 115 lyof] / cfdfn] tkfO{n‘ fO{ ?= 225 ylklbg'eof] eg] ca
tkfO{‘;uF hDdf slt ?lkofF xG' 5 xf]nf <

(ii) tkfO{‘ sg' } k;ndf Pp6f ;fdfg lsGg hfge' of] / pSt ;fdfgsf] d"No ?= 295
/x]5 . tkfOn{‘ ] k;n]nfO{ ?= 500 lbg'eof] eg] ca k;nn] ] tkfOn‘{ fO{ slt /sd
lkmtf{ lbg'knf{ <

(iii) tkfO{n‘ fO{ aa' fn] ?= 50 sf 12 cf6] f gf6] lbge' of] eg] tkfO{‘;Fu hDdf slt /sd
x'G5 xf]nf <

(iv) tkfO{‘;uF 20 cf]6f rsn6] 5g\ ca 5 hgf ldNg] ;fyLx¿nfO{ a/fa/ efu
nufpgk' bf{ hDdf Ps hgfsf efudf sltcf6] f rsn6] k5g{ \ xf]nf <

ul0ftsf cfwf/e"t lj|mofx¿cGtu{t hf]8, 36fp, u'0fg / efu kb5{ g\ . tnsf cEof;x¿
ug{'xf;] \ M

1. tnsf vfnL sf7] fdf cfjZos cªs\ eg{x' f;] \ M

(i) (ii)

4 43 8
+ 57 4
+ 23
63

9 8 7 5

(iii) (iv)

69
504
- 21 5
- 238

7 5 3 1

(v) (vi)

24 × 57 = 42 × 56 =

34 ul0ft M sIff $

2. tn lbOP h:t} u/L Pssf] hf]8nfO{ k'gM bz / Pssf] ;dx" sf] ¿kdf JoSt
ug'x{ f];\ M

h:t} M 6 Ps ± 7 Ps Ö 13 Ps = 1 bz + 3 Ps
(i) 5 Ps ± 9 Ps Ö
(ii) 8 Ps ± 9 Ps Ö
(iv) 5 Ps ± 5 Ps Ö
(v) 2 Ps ± 8 Ps Ö

2.2.2 hf8] (Addition)
lj|mofsnfk 1

lbOPsf] Ans cjnf]sg u/L :yfgdfg tflnsfdf cª\s e/L hf8] ug'{xf];\ .
xhf/ ;o bz Ps

±

dfly lbOPsf lrqdf klxnf] ;d"xdf Pssf Ans 3 cf]6f, bzsf Ans 2 cf6] f, ;osf]
Ans 3 cf]6f / xhf/sf Ans 1 cf6] f 5 . To;} u/L bf;] f| ] ;d"xdf Pssf Ans 6
cf]6f, bzsf] Ans 3 cf6] f, ;osf Ans 2 cf6] f / xhf/sf Ans 2 cf6] f 5g\ . ltgnfO{
:yfgdf tflnsfdf k:| t't ubf{ lgDgfg;' f/ x'G5 M

ul0ft M sIff $ 35

xhf/ ;o bz Ps

1 323

±2 2 3 6

dflysf Ansx¿ Ps 7fpdF f ld;fpFbf,

o;/L ld;fpbF f Pssf Ansx¿ 9 cf]6f, bzsf Ansx¿ 5 cf6] f, ;osf Ansx¿ 5
cf6] f / xhf/sf Ansx¿ klg 3 cf]6f 5g\ . ca oL s'/fx¿nfO{ tflnsfdf ebf{,

xhf/ ;o bz Ps

1 323

±2 2 3 6

3 559

o;/L Pp6f ;dx" df ePsf j:t'x¿nfO{ csf{] ;d"x;uF ld;fpg,] hDdf ug]{ h:tf
zAbx¿n] hf8] nfO{ hgfpF5 .

pbfx/0f 1
hf]8 ugx{' f;] \ M
8 3 5 7 6
+ 2 3 4 1 2
10 6 9 8 8

pbfx/0f 2
;'hgnfO{ k|f]hS] 6/ / Nofk6k cfjZos eP/ OnS] 6«fl] gS; k;ndf lsGg uP5g\ .
k;ndf k|f]h]S6/ / Nofk6ksf] dN" o cªl\ st ul/Psf] 5 . ca ;'hgnfO{ b'j} ;fdfg

36 ul0ft M sIff $

lsGbf hDdf slt ?lkofF ltgk'{ 5,{ kQf nufpgx' f];\ .

;dfwfg kf| ]hS] 6/sf] dN" o Nofk6ksf] dN" o
?= 28,345 ?= 39,676

hDdf dN" o

?= 28,345 + ?= 39,676 = ? ?= 39,676
:yfgdfg tflnsfdf /fv/] hf8] f,F} ?= 28,345

bz xhf/ xhf/ ;o bz Ps

2 8 345

±3 9 676



bz xhf/ xhf/ ;o bz Ps

1 1 11

2 8 345

±3 9 676

6 8 021

hDdf ?= 68021 ltg{'k5{ .
pbfx/0f 3

;'hnn] ?= 28,538 sf] 6]lnlehg / ?= 26,628 sf] /l] km| h/] ]6/ lsGof] eg] hDdf slt

dN" o k¥of] xf]nf <

;dfwfg 1 1 1
hDdf dN" o ?= 2 8 5 3 8

28,538 26,628 + ?= 2 6 6 2 8

6l] nlehgsf] d"No /]lkm| h]/6] /sf] dN" o ?= 5 5 1 6 6

∴ ;'hnsf] hDdf vr{ ?= 55,166 eof] .

ul0ft M sIff $ 37

pbfx/0f 4

k;nn] ] ?= 25,560 df 6l] nlehg / ?= 4,280 gfkmf lnP/ aR] g slt ?lkofdF f aR] g'

knf{ <

;dfwfg 6l] nlehgsf] dN" o gfkmf

6]lnlehg lsgs] f] d"No Ö ?= 25,560 ?= 25,560 ?= 4,280

gfkmf /sd Ö ?= 4,280 hDdf dN" o
hDdf a]rs] f] dN" o Ö ?= ?
1

2 5 5 6 0
+ 0 4 2 8 0
2 9 8 4 0

∴ k;n]n] pSt 6]lnlehg hDdf ?= 29,840 df aR] gk' b{5 .

pbfx/0f 5

Pp6f sDkgLn] klxnf] dlxgfdf 34,580 cf6] f sfkLx¿ ar] 5] / bf];f| ] dlxgfdf klxnf]

dlxgfsf] eGbf 10,255 cf6] f a9L sfkLx¿ a]r]5 eg] bf];f| ] dlxgfdf sltcf]6f sfkLx¿

a]rs] f] /x]5 <

;dfwfg klxnf] dlxgfdf a]rs] f]

34,580

10,255

bf;] f| ] dlxgf a9L ar] ]sf]

bf;] f| ] dlxgfdf a]r]sf]

klxnf] dlxgfdf ar] ]sf] hDdf sfkL ;ªV\ of Ö 34,580

bf;] |f] dlxgfdf a]rs] f] hDdf sfkLsf] ;ª\Vof Ö 34,580+10,25 5 1

3 4 5 8 0

+ 1 0 2 5 5

4 4 8 3 5

bf];|f] dlxgf 44,835 sfkL a]r]sf] /x]5 .

38 ul0ft M sIff $

cEof;

1. tnsf ;ox¿ hf]8L k'gM xhf/ / ;osf] ;dx" df JoSt ugx'{ f;] \ M
(i) 8 ;o + 7 ;o = 15 ;o = 1 xhf/ + 5 ;o
(ii) 2 ;o + 9 ;o =
(iii) 9 ;o + 7 ;o =
(iv) 5 ;o + 5 ;o =

2. tnsf xhf/x¿ hf8] L k'gM bz xhf/ / xhf/sf] ;d"xdf JoSt ug'{xf];\ M

h:t{ M 5 xhf/ + 6 xhf/ = 11 xhf/ = 1 bz xhf/ + 1 xhf/
(i) 4 xhf/ + 9 xhf/ =
(ii) 6 xhf/ + 8 xhf/ =
(iii) 3 xhf/ + 7 xhf/ =

3. tnsf jfSox¿df l7s eP (√) / al] 7s eP (×) lrx\g lbg'xf;] \ M
(i) Ps ;odf hDdf 10 cf6] f bzx¿ xG' 5g\ .
(ii) 10 cf]6f Ps a/fa/ 1 bz x'G5 .
(iii) Ps xhf/df hDdf 8 cf6] f ;o xG' 5g\ .
(iv) 10 cf]6f xhf/ a/fa/ 1 bz xhf/ x'G5 .

4. hf8] ugx{' f];\ M (ii) 2 8 3 2 5 (iii) 8 8 3 2 7
+ 6 8 7 3 5 + 2 1 6 7 3
(i) 3 4 5 8 0
+ 1 0 2 5 5

(iv) 1 2 4 5 6 (v) 37256 + 28645 (vi) 85647 + 24632
+ 9 7 8 5 4

(vii) 1 2 4 8 5 (viii) 4 2 6 3 9
2 3 4 2 6 8 8 4 2 1
+ 1 7 3 2 5 + 5 6 7 8 2

ul0ft M sIff $ 39

5. tnsf k|Zgx¿sf] pQ/ lbg'xf];\ M

(i) ;ljgfsf] kl/jf/sf] k|To]s dlxgf sf7] fef8fdf ?= 15,550 / vfBfGgdf ?=
25,780 vr{ xG' 5 . pgsf] kl/jf/sf] sf]7f ef8f / vfBfGgdf u/L k|ltdlxgf
hDdf slt vr{ x'g] /x5] , kQf nufpgx' f;] \ .

(ii) ;Demgfsf] k;naf6 dfl;s ?= 48,950 / sl[ ifaf6 ?= 15,280 cfDbfgL xG' 5
eg] pgsf] k;n / sl[ ifaf6 hDdf slt cfDbfgL xg' ] /x]5, kQf nufpgx' f];\ .

(iii) l;d/gn] Pp6f 3/ 7,52,000 df lslgg\ / pSt 3/ ddt{ ug{ ?= 25,650
vr{ ul/g\ eg] pgsf] hDdf vr{ slt eof] xf]nf, kQf nufpg'xf;] \ .

(iv) Pp6f k;n]n] sDKo'6/ ?= 25,680 df lsg]/ ?= 5,320 gfkmf /fvL ar] ]5
eg] p;n] hDdf slt d"Nodf a]Rof] xf]nf, kQf nufpg'xf;] \ .

(v) Pp6f k;n]n] Pp6f ;fdfg ?= 15,280 df aR] bf ?= 3420 3f6f eP5 eg]
p;n] sltdf lsg]sf] /x]5 <

(vi) Pp6f gu/kflnsfdf 58760 dlxnf / 75280 k?' if /x]5g\ eg] dlxnf /
k?' if u/L hDdf slt hgf /x]5g\, kQf nufpg'xf;] \ .

(vii) tnsf] tflnsfdf Pp6f ;x/sf] hg;ª\Vof lbOPsf] 5 . pSt tflnsf
cWoog u/L ;fl] wPsf k|Zgsf] pQ/ lbg'xf];\ .

j[4 hg;ªV\ of oj' fx¿sf] hg;ª\Vof afnaflnsfsf] hg;ª\Vof

28,250 42,000 36,445

(a) j[4 / oj' fx¿sf] hDdf hg;ªV\ of slt xfn] f <

(b) oj' f / afnaflnsfsf] hDdf hg;ª\Vof slt xf]nf <

(c) pSt ;x/sf] hDdf hg;ª\Vof slt xf]nf <

(viii) ;z' fGtn] cfˆgf] a}ªs\ vftfdf klxnf] dlxgfdf ?= 75,780 hDdf u/5] g\ .
bf;] f| ] dlxgfdf klxnf] dlxgfsf] eGbf ?= 15,233 a9L /sd hDdf u/5] g\
eg] pgsf] bO' { dlxgfsf] hDdf slt /sd a}ª\s vftfdf hDdf eof] xfn] f <

(ix) sg' } kfs{df laxfg 11 ah];Dd 1726 kj| ]z l6s6 laj|mL eP 5 . To;kl5
a]ns' f 5 ah];Dd 8003 cf]6f kj| z] l6s6 laj|mL eP5 eg] pSt lbg hDdf
slt l6s6 laj|mL eP5g,\ kQf nufpgx' f];\ .

40 ul0ft M sIff $

2.2.3 36fp (Subtraction)

ljm| ofsnfk 1

tn lbOPsf] ;osf] Ansaf6 10 cf6] f Ans x6fpFbf
sltcf6] f afFsL /xG5g\, n]Vgx' f];\ .

dflysf] ;osf] Ansaf6 bzcf6] f Ans x6fpbF f ca
Ansx¿sf] ;ªV\ of 90 x'G5 . o;nfO{ ul0ftLo jfSodf
nV] bf,
100 - 10 = 90 n]lvG5 .
90 + 10 = 100 x'G5 . o;/L hf]8 36fpsf] ;DaGw
bv] fpg ;lsG5 .
To:t} bfofF lbOPsf] ef8F fdf 40 cf]6f uR' rf / pSt
ef8F faf6 30 cf6] f u'Rrf lemSbf sltcf6] f afFsL /xG5g\ <
dfly lbOPsf] efF8faf6 30 cf]6f u'Rrf lemSbf hDdf 10
cf6] f afsF L /xg5] g\ . o;nfO{ ul0ftLo jfSodf nV] bf,
40 - 30 = 10 nl] vG5 . 30 + 10 = 40 xG' 5 .

sg' } klg ;d"xaf6 lemSg], x6fpg], lgsfNg] h:tf zAbx¿n] 36fpnfO{ hgfpF5 . 36fp
u/L ;sk] l5 ldn] gldns] f] hfrF /] xg] { ;lsG5 .

pbfx/0f 1

;/nfn] ?= 58,437 lnP/ ;lkª ug{ uOg\ . pgn] ?= 45,326 sf] ;fdfg lslgg\ eg]
ca pgL;uF slt ?lkofF afsF L /xG5, kQf nufpgx' f;] \ .

;dfwfg

;/nf;Fu ePsf] /sd ?= 58,437

?= 45,326 afsF L /sd
;lkªdf vr{ ePsf]

ul0ftLo jfSodf nV] bf,

?= 58,437 - ?= 45,326

ul0ft M sIff $ 41

:yfgdfg tflnsfdf /fv/] 36fp ubf{ 7 Ps – 6 Ps = 1 Ps
3 bz – 2 bz = 1 bz
bz xhf/ xhf/ ;o bz Ps 4 ;o – 3 ;o = 1 ;o
8 xhf/ – 5 xhf/ = 3 xhf/
5 8 437 5 bz xhf/ – 4 bz xhf/ = 1 bz xhf/

-4 5 326

bz xhf/ xhf/ ;o bz Ps

5 8 437

-4 5 326

1 3 111

hfFr]/ xb] f{,

45,326
+ 13,111

58,437 xG' 5 .

pbfx/0f 2
98763 af6 26132 36fpFbf slt x'G5g\ <

;dfwfg 9 8 7 6 3
- 2 6 1 3 2
pQ/ hfFr/] xb] f,{
7 2 6 3 1
2 6 1 3 2
∴ t;y{ xfdLn] u/]sf] lx;fa ldNof] . + 7 2 6 3 1

9 8 7 6 3

pbfx/0f 3

xl/gf/fo0fn] a}ª\saf6 Ps dlxgfdf tna / cGo eQf u/L hDdf ?= 68,325 kf| Kt
u5{g\ . pgn] 5f/] f5f/] Lsf] lzIff / oftfoftdf u/L k|ltdlxgf ?= 39,576 vr{ u5g{ \ eg]
pgsf] Ps dlxgfdf slt art xG' 5, kQf nufpg'xf];\ .

42 ul0ft M sIff $

;dfwfg ?= 68,325
xl/gf/fo0fsf] cfDbfgL

?= 39,576 afFsL art /sd
xl/gf/fo0fsf] vr{
ul0ftLo jfSodf n]Vbf,
?= 68,325 - ?= 39,576

:yfgdfg tflnsfdf /fv]/ 36fp ubf{

bz xhf/ xhf/ ;o bz Ps

6 8 325

-3 9 576

bz xhf/ xhf/ ;o bz Ps

5 17 12 11 15

6 8 325

-3 9 576

2 8 749

5 af6 6 36fpg ;lsFbg} . bzsf] :yfgsf] 2 af6 1 ;fk6L lnFbf bzsf] :yfgdf 1 afsF L
/Xof] eg] Pssf] :yfgdf 10 yk eO{ 10 + 5 = 15 xG' 5 . ca b;sf] :yfgsf] 1 af6
7 g36\g] ePsfn] ;osf] :yfgsf] 3 af6 1 ;fk6L lnFbf bzsf] :yfgdf 10 bz ylkg
uO{ 1 + 10 = 11 bz xG' 5 . o;}u/L ;osf] :yfgsf] 2 af6 5 36fpg ;lsbF g} . To;n} ]
xhf/sf] :yfgsf] 8 af6 1 ;fk6L lnbF f ;osf] :yfgdf 10 ;o ylkg uO{ 2 + 10 = 12
;o x'G5 . ca xhf/sf] :yfgsf] afsF L 7 af6 9 36fpg ;lsbF g} . To;sf/0f bz xhf/
sf] :yfgsf] 6 af6 1 ;fk6L lnFbf xhf/sf] :yfgdf 10 xhf/ ylkg uO{ 7 + 10 = 17
xhf/ xG' 5 / bz xhf/sf] :yfgdf 5 afsF L /xG5 .

ul0ft M sIff $ 43