गणित कक्षा ४

pQ/ hfrF /] xb] f{,
3 9 5 7 6

+ 2 8 7 4 9
6 8 3 2 5

∴ t;y{ xfdLn] u/]sf] lx;fa ldNof] .

pbfx/0f 4

;+;bLo lgjfr{ gdf Pp6f lgjf{rg If]qdf v;s] f dtdWo] 59,700 ;b/ dt lyP . pSt
lgjf{rgdf ljhoL pDdb] \jf/n] 38,654 dt kfP5g\ eg] slt hgfn] pgsf] ljkIfLnfO{
ef]6 lbPsf /x]5g\ <

9 0 af6 4 36fpg ;lsFb}g . bzsf] :yfgaf6 ;fk6L
6 10 10 lng'kg]{ xG' 5 . bzsf] :yfgdf klg 0 ePsfn] ;fk6L
5 9 7 0 0 lng ldn]g . To;}n] ;osf] :yfgaf6 ;fk6L lnFbf 10
- 3 8 6 5 4 bz xG' 5 / ;osf] :yfgdf 6 ;o afFsL /xG5 . kml] / 10
Ps x'G5 / bzsf] :yfgdf ;fk6L lnbF f 10 Ps xG' 5 /
2 1 0 4 6 bzsf] :yfgdf 9 bz afsF L /xG5 . ca,

10 Ps - 4 Ps = 6 Ps
9 bz - 5 bz = 4 bz
6 ;o - 6 ;o = 0 ;o
9 xhf/ - 8 xhf/ = 1 xhf/
5 bz xhf/ - 3 bz xhf/ = 2 bz xhf/

dflysf] 36fpnfO{ ;fk6L lnO{ ul/g] 36fp elgG5 . olb ;fgf] cªs\ af6 7n' f] cª\s
36fpg'kg]{ ePdf Ps :yfg dflysf] cª\saf6 ;fk6L lnO{ 36fp ugk{' 5{ . slxn]sfxL‘
Ps :yfg dflysf] cªs\ af6 ;fk6L lng ;Dej gePdf 2 :yfgdflysf] cªs\ af6 eP
klg ;fk6L lnO{ 36fp ug'k{ 5{, h;nfO{ tnsf] pbfx/0fdf k|:tt' ul/Psf] 5 M

44 ul0ft M sIff $

pbfx/0f 5

Pp6f ;x/sf] hDdf hg;ªV\ of 68,432 5 h;dWo] dlxnfsf] hg;ªV\ of 38,865 eP
k'?ifsf] hg;ªV\ of slt xfn] f <

;dfwfg

hDdf hg;ªV\ of = 68,432 hDdf hg;ª\Vof
dlxnfsf] hg;ª\Vof = 38,865 68,432
k'?ifsf] hg;ª\Vof = ?
38,865
ca,
dlxnfsf] hg;ª\Vof
5 17 13 12 12 k'?ifsf] hg;ª\Vof
6 8 4 3 2
- 3 8 8 6 5 pQ/ hfFr/] x]bf,{
2 9 5 6 7
3 8 8 6 5
+ 2 9 5 6 7

6 8 4 3 2

∴ k?' ifsf] hg;ª\Vof 29,567 /x5] g\ .

pbfx/0f 6

k|ldnfsf] Ps dlxgfsf] tna ?= 42,500 5 . ;/nfsf] Ps dlxgfsf] tna ?= 35,800
5 eg] ;/nfsf] tna k|ldnfsf] eGbf sltn] sd /x5] <

;dfwfg kl| dnfsf] tna
?= 42,500
ul0ftLo efiffdf n]Vbf
?= 35,800
42,500 - 35,800 = ;/nfsf] tna km/s tna

36fp ubf,{ pQ/ hfrF /] x]bf{,

4 2 5 0 0 3 5 8 0 0
- 3 5 8 0 0 + 6 7 0 0
0 6 7 0 0 4 2 5 0 0

∴ ;/nfsf] tna kl| dnfsf] eGbf ?= 6,700 n] sd /x]5 .

ul0ft M sIff $ 45

pbfx/0f 7

hfjnfv]nsf] lrl8ofvfgfdf dªl\ ;/ dlxgfsf] klxnf] zlgjf/ 15,367 l6s6 laj|mL
eP5g\ . dªl\ ;/ dlxgfs} t;] f| ] zlgaf/ klxnf] zlgaf/sf] eGbf 1089 cf6] f l6s6 sd
laj|mL eP5g\ . t;] |f] zlgaf/ sltcf6] f l6s6 lajm| L ePsf /x5] g\ <

;dfwfg klxnf] zlgaf/ laj|mL l6s6
ul0ftLo jfSodf n]Vbf 15,367

15,367 - 1089 = 1089

36fp ubf{, sd laj|mL l6s6 ;ªV\ of t;] f| ] xKtfdf lajm| L ePsf
l6s6 ;ªV\ of
1 5 3 6 7
- 1 0 8 9 pQ/ hfrF /] x]bf{,
1 4 2 7 8
1 4 2 7 8
+ 1 0 8 9
1 5 3 6 7

∴ t;] |f] zlgaf/ lajm| L ePsf] l6s6 ;ª\Vof 14,278 /x]5 .

pbfx/0f 8

?= 48,000 kg{] ;fdfg ?= 5000 5'6 lbP/ a]lrof] eg] pSt ;fdfg sltdf al] rof]
xf]nf <

;dfwfg ?= 5000 ;fdfgsf] d"No
56' /sd ?= 48,000
ul0ftLo efiffdf nV] bf
ar] ]sf] dN" o
48,000 - 5000 =

36fp ubf{, pQ/ hfrF /] x]bf,{

4 8 0 0 0 4 3 0 0 0
- 5 0 0 0 + 5 0 0 0
4 3 0 0 0 4 8 0 0 0

∴ pSt ;fdfg ?= 43,000 df ar] s] f] /x5] .

46 ul0ft M sIff $

cEof;

1. ;'hg;Fu ?= 32,850 lyof] . ca p;n] ?= 28,225 kg]{ 6]lnlehg lsgk] l5 p;uF
slt /sd afsF L /xG5 xf]nf <

2. tn lbOPsf ;ª\Vofx¿df ;osf] :yfgaf6 bzsf] :yfgdf 1 ;o ;fk6L lnO{ kg' M
;o / bzsf] ;dx" sf] ¿kdf JoSt ugx{' f;] \ M

h:t} M 570 = 5 ;o + 7 bz + 0 Ps = (5-1) ;o + (10 + 7) bz + 0 Ps = 4
;o + 17 bz + 0 Ps

(i) 628 =
(ii) 324 =
(iii) 555 =
(iv) 280 =

3. tn lbOPsf ;ªV\ ofx¿df xhf/sf] :yfgaf6 ;osf] :yfgdf 1 xhf/ ;fk6L lnO{
kg' M xhf/, ;o, bz / Ps ;dx" sf ¿kdf JoSt ug{x' f;] \ M

h:t} M 2324 = (2-1) xhf/ + (10 + 3) ;o + 2 bz + 4 Ps = 1 xhf/ + 13 ;o
+ 2 bz + 4 Ps

(i) 5823 =
(ii) 6225 =
(iii) 9555 =
(iv) 7665 =

4. 36fp ug'x{ f];\ / pQ/ ldn] gldns] f] hfFr ug'x{ f;] \ M
(i) 8 7 5 4 3 (ii) 9 4 5 3 7 (iii) 5 4 8 3 2
- 5 4 2 3 1 - 2 8 7 5 8 - 4 7 9 5 1

(iv) 5 5 2 6 4 (v) 5 8 0 7 0 (vi) 3 3 0 0 0
- 4 8 5 8 5 - 3 9 8 9 3 - 2 8 5 3 8

(vii) 5 0 0 0 0 (viii) 8 0 0 0 0 (ix) 6 5260- 28870
- 3 8 9 7 6 - 6 9 8 7 3
47
ul0ft M sIff $

(x) 98010 - 79855

5. tnsf k|Zgx¿sf] pQ/ lbg'xf];\ M
(i) Pp6f ljBfnodf hDdf 1480 ljBfyL{ lyP h;dWo] s6] fx¿sf] ;ª\Vof 862
lyof] eg] s]6Lx¿sf] ;ª\Vof slt lyof] xfn] f <
(ii) ;ndfgn] Pp6f ;fdfg ?= 23,250 df lsg/] ?= 28,580 df aR] bf p;n]
slt gfkmf u¥of] xf]nf <
(iii) Ct'n] Pp6f ;fdfg ?= 18,220 df lsg]/ ?= 14,568 df a]lrg eg] pgn]
pSt ;fdfgdf slt 3f6f a]xfg] {'k¥of] xf]nf <
(iv) Pp6f k;n]n] ?= 25,480 kg]{ 6l] nlehgdf ?= 4,280 5'6 lbP/ a]Rof] eg]
p;n] pSt 6]lnlehg sltdf aR] of] xfn] f <
(v) ;Demgfsf a'afsf] Pp6f aª} s\ vftfdf ?= 68,280 /sd lyof] . pgsf]
aa' fn] pSt a}ªs\ vftfaf6 ?= 12,876 lemSg'eof] eg] ca pSt a}ª\s
vftfdf slt /sd afsF L 5 xfn] f <
(vi) jiff{sf] kl/jf/sf] dfl;s cfDbfgL ?= 32,468 5 / dfl;s vr{ ?= 28,665
eP kl/jf/sf] dfl;s art slt xG' 5 xfn] f <
(vii) bO' c{ f6] f ;ª\Vofx¿sf] of]ukmn 68,250 5 / olb kl5nf] ;ª\Vof 24,380
eP bf;] |f] ;ªV\ of slt xfn] f <
(viii) 4230 df slt hf]8b\ f 6240 x'G5 <

(ix) gfu9'ª\ufdf 6f« lkms kx| /Ln] pkTosfdf k|j]z ug{] ;jf/L ;fwgx¿sf] btf{
u/L /s] 8{ /fVg] ub{5g\ . cluNnf] lbgsf] ;jf/L btf{ gDa/ 25,314 ;Dd
k'u]sf] lyof] . kl5Nnf] lbg pSt ;jf/L btf{ gDa/ a9/] 46,987 ;Dd kU' of]
eg] kl5Nnf] lbg slt ;jf/L ;fwg pkTosf kj| z] u/s] f /x]5g\, kQf
nufpgx' f];\ .

(x) huGgfysf] tna u0fz] sf] tnaeGbf ?= 8,760 n] sd 5 . olb u0fz] sf]
tna ?= 52,316 eP huGgfysf] tna slt xf]nf, kQf nufpgx' f;] \ .

(xi) cfwf/et" ljBfnosf] aª} s\ vftfdf ?= 5,32,460 5 . pSt ljBfnosf]
Ps dlxgfsf] lzIfs tna ljt/0fkZrft\ ljBfno vftfdf ?= 2,95,205
afsF L /x\of] eg] slt ?lkofF lzIfs tnadf vr{ eP5, kQf nufpg'xf;] \ .

48 ul0ft M sIff $

2.2.4 u0' fg (Multiplication)

lj|mofsnfk 1

sIff $ df cWoog ug]{ ldgfsL cfdfn] ldgfsf] nflu 4 bh{g l;;fsnd lsg/]
Nofpge' of] . cfdfn] ldgfnfO{ hDdf sltcf]6f l;;fsnd 5g\, egL ;f]Wg'eof] <
hDdf l;;fsndsf] ;ª\Vof lgsfNbf ldgfn] tnsf] h:t} ul/g\ M

hDdf l;;fsnd = 12 + 12 + 12 + 12
= 48 cf6] f
ldgfsf 7fpdF f tkfO‘{ x'ge' Psf] eP s;/L ugx{' 'GYof] xfn] f <
dflysf] lx;fanfO{ 5f]6s/Ldf o;/L klg ug{ ;lsG5 M

0 12 24 36 48
oxfF 4 k6s 12 ePsfn] o;nfO{ u0' fgsf ¿kdf JoSt ubf,{ 12 × 4 = 48
t;y{ hDdff l;;snd = 48 cf]6f x'G5 .

12 nfO{ 4 k6s hf8] \g' / 4 /
12 u'0fg ug{' Pp6} kf] /x5] .

ljm| ofsnfk 2 4 x 5 = 20
tnsf] u'0fg tflnsf x/] L 5nkmn ug{x' f;] \ . 49

0 4 8 12 16 20 24
ul0ft M sIff $

0 4 8 12 16 20 24 4 x 4 = 16
0 4 8 12 16 20 24 4 n] 36\of] .
4 x 3 = 12
0 4 8 12 16 20 24
4 n] 36o\ f] .
0 4 8 12 16 20 24 4x2=8
0 4 8 12 16 20 24
4 nx] 136=\o4f] .
dflysf] ;ªV\ of /]vf cWoog ubf,{ 4 x 0 = 0 xG' 5 . 4

44 nx] 036=o\ 0f] .

4 n] 36\bf 0 xG' 5 .

z'Go (0) n] hg' ;'s} k"0f{ ;ª\VofnfO{ u'0fg ubf{ u0' fgkmn z'Go (0) x'G5 .

pbfx/0f 1

k/fk] sf/L ;+:yfn] af9L klx/f] kLl8t ;db' fodf kl| tJolSt ?= 935 sf b/n] 142
hgfnfO{ /fxt ljt/0f u¥of] . pSt sfo{j|mddf hDdf slt /sd /fxt :j¿k ljt/0f
ul/of] xf]nf <

;dfwfg
k/f]ksf/L Ps hgf JolStnfO{ ?= 935 sf] b/n] hDdf 142 hgf JolStnfO{ /fxt ljt/0f
u/s] f] 5 . hDdf /xt /sd kQf nufpg ?= 935 nfO{ 142 k6s hf]8g\ k' 5{ . t/ To;/L
hf]8\g' / 142 nfO{ 935 n] u0' fg ug'{ Pp6} xf] . To;n} ] u0' fg u/L x/] f}F .

50 ul0ft M sIff $

9 3 5 935 nfO{ 2 Psn] u0' fg ubf{
× 1 4 2 1 1

1 8 7 0 9 3 5
3 7 4 0 0 × 2
+ 9 3 5 0 0 1 8 7 0

1 3 2 7 7 0 935 nfO{ 4 bzn] u0' fg ubf{
1 2
;a} u0' fgkmnnfO{ hf]8s] f]
9 3 5
× 4 0
3 7 4 0 0

935 nfO{ 1 ;on] u0' fg ubf{

9 3 5
× 1 0 0
9 3 5 0 0

∴ pSt kmfpG8;] gn] /fxt :j¿k ?= 1,32,770 /sd ljt/0f u/5] .

dflysf] u0' fgnfO{ Nofl6; ljlwaf6 u/L x]/f}F M

9 3 5 1 kT| os] sf7] fnfO{ bO' { efudf ljefhg ul/Psf]
1 1 4 5 . tNnf] efun] Ps / dflyNnf] efun]
1 2 bzsf] :yfgnfO{ hgfp5F . h:t} M 2 / 5
0 0 u0' fg ubf{ 10 x'G5 . 1 nfO{ dflyNnf] efu /
0 0 nfO{ tNnf] efudf nl] vG5 . 2 / 3 u'0fg
ubf{ 6 xG' 5 . dflyNnf] efudf 0 /fvL tNnf]
1 9 3 5 efudf 6 n]lvG5 . sf]7fleq ePsf] jf0f
3 1 2 ;ªs\ ]tn] bv] fP h:t} u/L hf8] g\ 'k5{ . hf8]
ubf{ xftnfuL cfpg] ;ª\VofnfO{ dflyNnf]
3 6 2 0 sf7] fdf /fVg] . cGtdf sf]7f aflx/sf] jf0f
1 0 1 lrxg\ n] bv] fP h:t} u/L nV] g'k5{ .

2 8 6 0

7 7 0

u'0fgkmn

∴ 142 x 935 = 1,32,770

t;y{ ?= 1,32,770 /fxt :j¿k

ljt/0f ul/of] .

ul0ft M sIff $ 51

pbfx/0f 2
Pp6f lstfasf] dN" o ?= 220 k5{ eg] To:t} 37 cf6] f lstfasf] dN" o slt k5{ xf]nf <

;dfwfg

ul0ftLo jfSodf nV] bf ca Nofl6; ljlwaf6 x]bf,{

220 × 37 = 2 2 0
1
u'0ffg u//] ubf{
2 2 0 0 0 0 3
× 3 7 7
0 6 6 0
1 5 4 0
+ 6 6 0 1 1 0
8 1 4 0
8 4 4 0

1 4 0

∴ p:t} vfnsf 37 cf6] f lstfanfO{ ?= 8,140 k5{ .

cEof;

1. vfnL 7fpF eg{'xf];\ M

(i) Pp6f slksf] dN" o ?= 20 eP 5 cf6] f slksf] d"No Ö ?=

(ii) Pp6f Kofs6] df 12 cf6] f snd eP 6 cf6] f Kofs]6df cf]6f snd

xG' 5g\ .

(iii) Ps ls=u|f= cªu\ '/sf] d"No ?= 320 eP 8 ls=u|f= sf] dN" o ?= x'G5 .

(iv) Ps ldg6] df 60 ;]s]G8 eP 10 ldg]6df ;s] ]G8 x'G5 .

(v) Ps xKtfdf 7 lbg eP 12 xKtfdf lbg x'G5 .

2. u0' fg ug{'xf];\ M

(i) 1 2 2 (ii) 2 2 0 (iii) 9 6 5 (iv) 8 6 8
× 2 4 3 x 1 1 2 x 4 2 0 x 5 6 0

(v) 5 2 0 (vi) 6 6 8 (vii) 2 8 0 (viii) 7 3 5
x 4 0 0 x 4 9 5 x 5 7 0 x 6 4 8

52 ul0ft M sIff $

3. tnsf k|Zgx¿sf] ;dfwfg ugx'{ f];\ M

(i) Pp6f ld7fOs{ f] a66\ fdf 135 cf]6f ld7fO{ c6fpF5g\ eg] To:t} 115 cf6] f
a6\6fdf sltcf6] f ld7fO{ c6fp5F g\ <

(ii) Pp6f sl[ if kmf/ddf 165 cf6] f nx/df ufn] eF8] f v]tL ul/Psf] /x]5 . olb
Pp6f nx/df 125 cf6] f uf]ne]8F fsf af6] x¿ eP hDdf ufn] e8]F fsf af6] sf]
;ª\Vof kQf nufpg'xf];\ .

(iii) Pp6f ljBfnon] sIff $ / % sf] u/L hDdf 135 hgf ljBfyLn{ fO{ jgefh]
nh} fg] lg0f{o u/]5 . ;fx] Lcg';f/ k|ltljBfyL{ ?= 650 p7fpg] lg0f{o u/5]
eg] hDdf slt /sd ;ª\sng xG' 5 xfn] f < ;f] /sd ?= 1 nfveGbf sltn]
sd xfn] f <

(iv) kmnk"mn ahf/df 125 cf6] f :ofpsf jm| 6] x¿ 5g\ . olb Pp6f jm| ]6df
110 cf]6f :ofpx¿ eP hDdf :ofpsf] ;ª\Vof slt xfn] f < pSt ;ªV\ of
10,000 eGbf sltn] a9L xfn] f <

(v) Pp6f lstfadf 220 k[i7x¿ 5g\ eg] pltg} ki[ 7 ;ª\Vof ePsf 312 cf]6f
lstfadf sltcf]6f k[i7x¿ xG' 5g\ <

kl/of]hgf sfo{ (Project work)

tnsf] tflnsfdf /dz] sf] kl/jf/sf] dfl;s vfBfGgsf] ljj/0f kl/df0f / PsfOdf
lbOPsf] 5 . pSt kl/jf/sf] hDdf vr{ kQf nufpgx' f];\ M

zLifs{ kl/df0f kl| t PsfO dN" o ?= hDdf d"No
rfdn 120 ls=uf| =
tn] 4 ln= 75
lrgL 4 ls=u|f=
gg' 1 kfs]6 160
bw' 40 ln=
90

20
90

ul0ft M sIff $ 53

2.2.5 efu (Division)

lj|mofsnfk 1
/Ltf, dLgf / ;/Ltf tLg hgf ldNg] ;fyLx¿ ljBfnosf] vfhf ;dodf vfhf vfg
a;]5g\ . dLgfn] cfˆgf] l6lkmg aS;df 12 cf]6f n8\8' NofO5g\ / tLg} hgfn] a/fa/
efu nufP/ vfP5g\ .

ca Ps hgfsf efudf sltcf6] f n88\ ' k¥of] xf]nf <
dLgfn] cfˆgf] l6lkmg aS;af6 Ps Pscf6] fsf b/n] ;a} n8\8' g;lspGh]n afFl8g\ .
Ps hgfsf efudf 4 cf6] f n88\ ' k/] . 4 cf]6f n88\ ' tLg 7fpFdf To;}n] 4 × 3 = 12
hDdf n8\8' ;lsof] .

Ps Pscf]6fsf b/n] af8F b\ f Ps k6sdf hDdf sltcf]6f n88\ ' aflF 8of] < sltk6s

afF8b\ f n8\8' ;lsof] xf]nf <

Ps k6sdf 3 cf6] f n88\ ' aflF 8of] . 1 cf]6fsf b/n] tLg hgfnfO{ af8F \bf kT| o]s k6s

3 cf6] fsf b/n] n8\8' 36\b} hfG5 .

12 - 3 = 9 klxnf] k6s
bf;] |f] k6s
9 - 3 = 6 t;] f| ] k6s
6 - 3 = 3 rf}yf] k6s
3 - 3 = 0

hDdf rf/ k6s 36fOof] .

54 ul0ft M sIff $

o;/L 3 cf6] fsf b/n] 36\b} hfbF f 4 k6sdf ;a} n8\8' af8F /] ;lsof] . t;y{ Ps hgfsf

efudf 4 cf6] f n88\ ' k/] .

o;n} fO{ 5f6] s/Ldf o;/L klg ug{ ;lsG5 M
4
3 12 12 af6 3 nuftf/ 4 k6s
-12 36fpg' / 12 nfO{ 3 n] efu ug{'

0 Pp6} kf] /x5] .

o;/L 4 cf6] f n8\8' 3 hgfnfO{ a/fa/ efu nufpFbf Ps hgfsf] efudf 4 cf]6f n88\ '
k/] . oxfF 3 efhs, 4 efukmn / 12 efHo xf] . 3 n] 12 nfO{ lgMz]if efu uof] . efu
lj|mof ul/;sk] l5 ldn] gldns] f] tnsf] ;DaGwaf6 hfrF u/L xg] { ;lsG5 .
efHo Ö efhs × efukmn + zi] f

or, 12 = 3 × 4 + 0

or, 12 = 12 -a/fa/ cfof]_
efunfO{ bfx] fl] /Psf] 36fpsf] 5f6] f] ¿k klg elgG5 .

ljm| ofsnfk 2

si[ 0f, /fhg / ;/fh] tLg bfhe' fO lyP . a'afn] ahf/af6 b'O{ bhg{ sndx¿ lsg]/
Nofpge' of] . ca pSt snd tLg} hgfnfO{ a/fa/ efu nufpbF f Pp6fsf] efudf
sltcf6] f k5{ xfn] f eg]/ pgLx¿ cfkm}F 5nkmn ug{ yfn5] g\ .

? ??

;/f]h M xfdL tLg hgfnfO{ a/fa/ efu nufpbF f 8/8 cf6] f snd efudf k5{ <

si[ 0f M s;/L <

;/fh] M 8 x 3 = 24 xG' 5 lg .

/fhg M 24 cf6] f snd 3 a/fa/ efu nufpg' egs] f] t 3 n] sltnfO{ u'0fg ubf{

24 cfpg] /x]5 eGg] kf] /x5] × 3 = 24 xlu Û

8 × 3 = 24

ul0ft M sIff $ 55

lj|mofsnfk 3

378 cf]6f sfkLx¿ 25 hgf JolStnfO{ a/fa/ efu nufpFbf Ps hgfsf efudf

sltcf]6f sfkL k5g{ \ / clGtddf sltcf]6f sfkLx¿ afsF L /xG5g\ <

dflysf k|Zgdf 25 efhs xf] / 378 efHo xf] . ;jk{ y| d efhs / efHonfO{ tnsf]

h:t} u/L /fvf}F M u0[' fg tflnsf
25 378
efu ug'{eGbf klxn] efhssf] u'0fg tflnsf lgdf{0f 25 × 1 = 25
ug{k' 5{ / To;}sf cfwf/df slt k6s efu nfU5 egL 25 × 2 = 50
x]g{ ;lsG5 . 25 sf] u'0fg tflnsf bfofF lbOPsf] 5 . 25 × 3 = 75
25 × 4 = 100
oxfF efhs 2 cª\ssf] 5 To;}n] efHosf] 7n' f] :yfgsf] 2 25 × 5 = 125
cªs\ sf] ;ª\Vof lnO{ slt k6s efu nfU5 egL x]gk'{ 5{ . 25 × 6 = 150
ca 37 nfO{ 25 n] slt k6s efu nfU5 egL u'0fg 25 × 7 = 175
tflnsfdf x]/fF} . 25 × 8 = 200
25 × 9 = 225
25 × 10 = 250

2 k6s 25 ubf{ 50 x'G5 h'g 37 eGbf 7'nf] eof] . To;}n] 15
37 nfO{ 25 n] 1 k6s dfq efu nfU5 . ca 1 nfO{ 25 378
efukmn /fVg] 7fpdF f /fvL 37 af6 25 36fpFbf 12 xG' 5 . -25
ca efHodf /xs] f] cª\s 8 nfO{ tn emf/f}F . o;f] ubf{ 128
128 x'G5 . ca 25 n] 128 nfO{ slt k6s efu nfU5 -125
egL u'0fg tflnsf x]/fF} .
3

5 k6sdf 25 × 5 = 125 h'g 128 sf] w]/} glhs x'g] ePsfn] 128 nfO{ 5 k6s efu
nfU5 .
ca 5 nfO{ efukmn /fVg] 7fpFdf /fvL 128 af6 125 36fpbF f 3 afFsL /xG5 .
o;/L afFsL ePsf] 3, 25 eGbf ;fgf] ePsfn] 25 n] efu nfUb}g .
o;/L efu ubf{ clGtddf efu gnfUg] u/L afFsL /x]sf] ;ª\VofnfO{ zi] f elgG5 . dflysf]
efudf 3 nfO{ z]if elgG5 eg] 15 nfO{ efukmn elgG5 . t;y{ 378 cf]6f sfkL 25
hgfnfO{ a/fa/ efu nufpFbf Ps hgfsf efudf 15 cf]6f sfkL k5g{ \ / 3 cf6] f slk
afFsL /xG5g\ .

56 ul0ft M sIff $

xfdLn] dfly u/s] f] efu ldNof] jf ldn]g egL s;/L hfrF ug{ ;lsPnf < efu u/s] f]
ldNof] jf ldn]g egL hfFr ug{ tnsf] ;DaGw k|of]u ul/G5 M
efHo Ö efhs × efukmn + z]if

ca, hfFr]/ x/] fF} M
378 = 25 x 15 + 3
or, 378 = 375 + 3
or, 378 = 378 -∴ b'j}lt/ a/fa/ cfPsfn] xfdLn] u/]sf] efu ldNof]_

pbfx/0f 1
efu ug'{xf;] \ / ldn] gldns] f] hfrF ugx{' f];\ .

978 ÷ 36 efhs u'0[ fg tflnsf
zi] f
ca, efu ubf{ 36 × 1 = 36
36 × 2 = 72
27 36 × 3 = 108
36 × 4 = 144
36 978 36 × 5 = 180
-72 36 × 6 = 216
258 36 × 7 = 252
-252 36 × 8 = 288
36 × 9 = 324
6 36 × 10 = 360

hfFr]/ x/] fF} M

efHo = efhs x efukmn + zi] f
or, 978 = 36 × 27 + 6
or, 978 = 972 + 6
or, 978 = 978 -bj' }lt/ a/fa/ eof] To;n} ] lx;fa ldNof_]

ul0ft M sIff $ 57

pbfx/0f 2

efu ug'x{ f;] \ / pQ/ ldn] gldns] f] hfFr ug{x' f;] \ M
162 ÷ 18

efu ubf{, efhs u['0fg tflnsf
z]if
9 18 × 1 = 18
18 × 2 = 36
18 162 18 × 3 = 54
-162 18 × 4 = 72
18 × 5 = 90
6 18 × 6 = 108
18 × 7 = 126
hfrF ]/ xb] f,{ 18 × 8 = 144
18 × 9 = 162
efHo = efhs × efukmn 18 × 10 = 180
or, 162 = 18 x 9
or, 162 = 162

a/fa/ cfof] . lx;fa ldNof] .
∴ t;y{ 9 k6s efu nfU5 .

pbfx/0f 3
olb 15 cf6] f sndsf] dN" o ?= 375 eP Pp6f sndsf] d"No slt k5{ xfn] f <

;dfwfg

oxfF 1 cf6] f sndsf] dN" o lgsfNg'kg{] ePsfn] ?= 375 nfO{ 15 n] efu ug{k' g]{ x'G5,

To;}n] efu ubf,{

25
15 375
-30
75
-75

0

∴ Pp6f sndsf] d"No ?= 25 k5{ . ul0ft M sIff $

58

cEof;

1. vfnL 7fpF egx{' f;] \ M

(i) efhs × efukmn + = efHo x'G5 .

(ii) efHo = × efukmn + z]if xG' 5 .

(iii) olb 18 ÷ 6 = 3 df efHo = , efhs = , efukmn = ,/

xG' 5 . xG' 5 .
xG' 5 .
(iv) olb efhs 4, efukmn 5 / zi] f 3 eP efHo =

(v) olb efhs 3, efukmn 8 / zi] f 0 eP efHo =

2. efu ug'x{ f];\ / pQ/ ldn] gldn]sf] hfrF ug{'xf];\ M

(i) 24 480 (ii) 38 204 (iii) 68 894

(iv) 72 868 (v) 55 995 (vi) 568 ÷ 22

(vii) 654 ÷ 42 (viii) 735 ÷ 23 (ix) 943 ÷ 32

3. tnsf kZ| gx¿sf] pQ/ lbg'xf];\ M

(i) olb 1 ls=u|f= lrgLsf] dN" o ?= 85 k5{ eg] ?= 765 df slt s]=h= lrgL

lsGg ;lsPnf <

(ii) Pp6f afs;df 65 cf]6f ld7fO{ c6\5g\ eg] 530 cf]6f ld7fO{n] To:t} slt

cf]6f afs; el/G5g\ < sltcf]6f ld7fO{ afFsL /xG5g\, kQf nufpgx' f;] \ .

(iii) olb Pp6f a;n] Ps 306fdf 45 ls=ld= ofqf u5{ eg] pxL ultdf 675

ls=ld= ofqf ug{ slt ;do nfUnf <

(iv) sIff $ df ePsf 45 hgf ljBfyL{nfO{ 565 cf]6f snd a/fa/ efu

nufpbF f Ps hgfsf efudf slt k5{ < sltcf]6f snd afsF L /xG5 < pSt

afFsL snddf sltcf6] f snd yKbf 45 hgfnfO{ km]l/ 1/1 cf]6f k'Ug] u/L

efu nufpg ;lsG5 xf]nf <

(v) 1 ls=u|f= rfdnsf] d"No ?= 75 5 . ;z' fGtn] ?= 975 df slt ls=uf| =
rfdn lsGg ;S5 < olb p;n] k;n]nfO{ ?= 1000 lbPsf] eP slt /sd

lkmtf{ cfpF5 <

(vi) Ps j|m6] df 30 cf]6f cG8f xG' 5g\ . ca 975 cf]6f To:t} cG8fn] sltcf]6f

jm| 6] el/G5 / sltcf6] f afFsL /xG5 < afFsL /xs] f] c08fdf sltcf]6f cG8f

yKbf Pp6f j|m]6 el/G5 xf]nf, lx;fa ug'{xf];\ . 59
ul0ft M sIff $

2.2.5 ;/nLs/0f (Simplification)

ljm| ofsnfk 1
cfs[ltn] k;} f hDdf ugs{ f nflu Pp6f vq' 's] lsg/] NofO5g\ . pSt vq' 'sd] f pgn]
;'?df ?= 10 hDdf ul/g\ / pgsf] aa' fn] ?= 20 ylklbg'eof] . sx] L lbgkl5 cfs[ltn]
pSt vq' 's]af6 ?= 5 lems]/ vr{ ul/g\ . ca pSt v'qs' d] f slt /sd afsF L xf]nf <

?=5 lemlsg

?=10 ?=10 ?=5
?=20 ?=20

;dfwfg

dflysf] ;d:ofnfO{ ;ªV\ of /v] fdf /fv/] x/] f}F M
?= 20
?= 10

0 5 10 15 20 25 30 35 40

ca pgL;uF ?= 25 afFsL 5 .
dflysf] lx;fanfO{ ul0ftLo jfSodf n]Vbf,

10 + 20 - 5

= 30 - 5 -klxnf] hf]8 u/s] f_]

= 25

t;y{ pSt v'q'sd] f ?= 25 afFsL 5 .

pbfx/0f 1
;'dg;uF 7 cf]6f l;;fsnd lyP . 3 cf6] f l;;fsnd p;n] ljBfnodf ;fyLx¿nfO{
lbof] . sx] L lbgkl5 pgsL cfdfn] 5 cf]6f l;;fsnd lslglbge' of] . ca ;d' g;Fu
hDdf sltcf6] f l;;fsnd afFsL eof,] kQf nufpg'xf];\ .

60 ul0ft M sIff $

;dfwfg
dflysf] ;d:ofnfO{ ;ªV\ of /v] fdf /fv]/ x]/fF} M

1 2 3 4 5 6 7 8 9 10

hDdf l;;fsnd = 7 - 3 + 5
= 4 + 5 -klxnf 36fpsf] sfd u/s] f_]
=9
t;y,{ p;uF hDdf 9 cf6] f l;;fsnd 5g\ .

cEof;
1. tnsf aS;df lbOPsf jfSox¿ l7s eP (√) lrxg\ / al] 7s eP (×) lrxg\ lbgx' f;] \ M

(i) 8 - 2 - 5 = 11
(ii) 12 - 5 + 3 = 10
(iii) 15 + 4 - 5 = 12
(iv) 11 + 12 + 3 = 26
(v) 8 - 4 + 1 = 5

2. tnsf k|Zgx¿nfO{ ul0ftLo jfSodf n]vL ;/n ug'{xf];\ M

(i) ;'lgtf;Fu 10 cf6] f rsn]6 lyP . pgn] 6 cf]6f rsn6] ;fyLx¿nfO{
aflF 8g\ / kml] / 5 cf6] f lslgg\ eg] ca pgL;uF hDdf slt cf]6f rsn6] 5g\
xfn] f, kQf nufpgx' f];\ .

(ii) /flwsf;uF ?= 30 lyof] . pgnfO{ a'afn] ?= 45 ylklbge' of] / s]xL lbgkl5
pgn] ?= 60 sf] sfkL lslgg\ . ca pgL;Fu hDdf slt /sd afsF L
5, kQf nufpg'xf;] \ .

(iii) 12 / 6 sf] of]ukmnaf6 5 36fpbF f slt x'G5 <

(iv) 8 / 5 sf] km/sdf 3 hf8] \bf slt x'G5 <

(v) 15 cf6] f :ofp /flvPsf] afs;af6 5 cf6] f s'lxPsf] :ofp lgsfnL km]l/
4 cf]6f /fdf| ] :ofp /fVbf pSt afs;df hDdf sltcf]6f :ofp xG' 5g\, kQf
nufpg'xf];\ .

ul0ft M sIff $ 61

2.3 leGg, bzdnj / k|ltzt (Fraction, Decimal and Percentage)

2.3.1 leGg (Fraction)
2.3.1.1 k'g/fjnfs] g (Review)

lj|mofsnfk 1
;ndfgn] Pp6f kfk8nfO{ rf/ a/fa/ efu nufO{ Ps rfy} fO efu cfkm" n] vfP 5g\ .
ca p;n] vfPsf] efunfO{ leGgsf ¿kdf s;/L n]Vg] xf]nf < afsF L /x]sf] kfk8nfO{
leGgdf s;/L n]Vg] xf]nf <

hDdf 4 a/fa/ efu kfk8dWo] 1 efu ;ndfgn] vfof] . To;n} ] p;n] vfPsf] efunfO{
1
leGgdf 4 nl] vG5 .

kml] /, ;ndfgn] vfP h:t} 3 cf]6f efu kfk8 afsF L /xs] f] 5 .
1
o;nfO{ leGgdf 3 cf]6f 4

= 3
4

pbfx/0f 1

cflz/fn] Pp6f /:] 6/' G] 6af6 8 cf6] f a/fa/ cflz/fn] vfPsf]
:nfO; ePsf] lkhf lsg/] NofOg\ . pSt lkhf
lkhfaf6 cfkm" n] 1 cf]6f / bfh'nfO{ 2
cf]6f :nfO; lbOg\ . afsF L /x]sf] lkhfnfO{ cflz/fsf] bfOn]
a'afcfdfnfO{ cfPkl5 vfg] u/L /flvg\ . ca vfPsf] lkhf
cflz/fn] vfPsf] lkhfnfO{ leGgdf s;/L
n]Vg] xf]nf < pgsf] bfh'n] vfPsf] / afFsL ul0ft M sIff $
/xs] f] lkhfnfO{ leGgdf s;/L n]Vg] xfn] f <

62

;dfwfg

oxfF cflz/fn] vfPsf] lkhf 8 efudWo] 1 efu xf]
1
To;}n] 8 nl] vG5 .

cflz/sf] bfh'n] vfPsf] lkhf cflz/fn] vfPh:t}

2 cf]6f :nfO; xf] .
1 2
To;}n] 2 cf]6f 8 = 8 n]lvG5 .

afsF L lkhf 5 cf6] f 1 = 5 xG' 5 .
8 8

cz+ df 1 ePsf] leGgnfO{ PsfO leGg elgG5 . PsfO leGgsf cfwf/df c¿ efunfO{

JoSt ug{ ;lhnf] x'G5 .

cEof;

1. 5fof kfl/Psf] efunfO{ leGgsf ¿kdf nV] gx' f;] \ M

(i) = (ii) =

(iii) = (iv) =

2. tnsf kT| o]s zAbnfO{ leGgsf ¿kdf n]Vg'xf;] \ M

(i) cfwf = (ii) Ps ltxfO =
(iv) Ps rfy} fO =
(iii) b'O{ ltxfO = (vi) tLg rfy} fO =

(v) bO' { rf}yfO =

(vii) 7 efudf 3 efu = (viii) 15 efudf 9 efu =

(ix) 100 efudf 30 efu =

3. k'?iffQ] dn] 3/df cfPsf kfx'gfsf] ;Tsf/sf nflu Pp6f :ofpnfO{ 4 a/fa/ 6'jm| f

xg' ] u/L 3 cf]6f :ofp sf6]5g\ . kfxg' fdWos] f Psn] 2 cf]6f 6j' m| f :ofp vfP5g\

eg] pgn] Pp6f :ofpsf] slt efu vfP5g,\ leGgdf n]Vg'xf];\ .

4. lvngf/fo0fn] cfˆgf] hGdlbgdf NofPsf] s]snfO{ 12 a/fa/ efu nufP5g\ .

pSt 12 efu ss] dWo] >LdtLnfO{ 2 efu, 5f/] fn] 1 efu, 5f/] Ln] 2 efu / cfk"mn]

2 efu vfP5g\ . afFsL ss] ] /l] k|mh]/]6/df /fv5] g\ eg] ;an} ] vfPsf] / /]lk|mh]/6] /df

/fv]sf] ss] nfO{ leGgdf n]Vgx' f];\ .

ul0ft M sIff $ 63

2.3.1.2 ;dfg x/ ePsf leGgx¿sf] tn' gf (Comparison of fraction having

equal denominator)

pbfx/0f 1

;'lznf, ;d' g / /fhg tLg hgf ljBfnosf] rd]gf ux[ df uP5g\ . rd]gf ux[ df ePsf
4 2 5
n88\ 'sf] 11 efu ;'lznfn], 11 efu ;d' gn] / 11 efu /fhgn] vfP5g\ . ;ae} Gbf a9L

/ ;ae} Gbf sd n8\8' s;n] vfP xf]nf, 5nkmn ug'{xf];\ M

;dfwfg 4
11
oxfF ;'lznfn] vfPsf] n8\8' =
= 4 cf6] f
1
2 11

;d' gn] vfPsf] n88\ ' = 11 1
= 11
2 cf]6f

/fhgn] vfPsf] n8\8' = 5
=
11 1
11
5 cf6] f

= 5 > 4 > 2 xg' ] x'Fbf 5 > 4 > 2 eof]
11 11 11

/fhgn] ;ae} Gbf a9L / ;d' gn] ;ae} Gbf sd n8\8' vfPsf /x]5g\ .

dflysf leGgx¿nfO{ ;ªV\ of /]vfdf k|:tt' ubf,{

;d' g ;'lzn /fhg

0 1 2 3 4 5 6 7 8 9 10 11 = 1
11 11 11 11 11 11 11 11 11 11 11

64 ul0ft M sIff $

;ªV\ of /]vfdf bfofFlt/sf leGgx¿ jm| dzM 7n' f] xbF' } hfg] ePsfn] /fhgn] ;ae} Gbf w/] }
/ ;'dgn] ;ae} Gbf sd n88\ ' vfP5g\ .

o;}nfO{ lrqaf6 xb] f{ ;l' zn
;'dg
4 /fhg
11
2
11
5
11

oxfF ;a} leGgdf x/ ;dfg 5 . /fhgn] vfPsf] leGg hgfpg] lrqdf a9L efudf 5fof

k/s] f] / ;d' gn] vfPsf] leGg hgfpg] lrqdf sd efudf 5fof k/]sf] 5 . To;n} ] /fhgn]

a9L n8\8' vfP5g\ / ;d' gn] sd n88\ ' vfP5g\ .

2.3.1.3 ;dfg x/ ePsf leGgx¿sf] hf]8 / 36fp (Addition and Subtraction of

Fraction having same Denominator)

pbfx/0f 1

;l/tfsf a'afn] af/Lsf] 3 efu / ;l/tfn] 2 efu vlg5g\ eg] pgn] / pgsf] a'afn]
6 6
ldn]/ hDdf slt efu af/L vg]5g\ M

;dfwfg 3
6
aa' fn] vGg'ePsf] efu =

;l/tfn] vg]sf] efu = 2
6
3 2
b'j} ldn]/ vgs] f] efu = 6 + 6

=3 cf]6f 1 +2 cf6] f 1
6 6

ul0ft M sIff $ 65

= 5 cf]6f 1
6
5
=6

0 1 2 3 4 5 6 =1
6 6 6 6 6 6

hDdf 5 efu sfd eP5 .
6

b'j} leGgsf] hf8] nfO{ lrqaf6 x/] fF,}

+=

3 2 5
6 6 6
5
t;y{ b'j} ldnL 6 efu af/L vg]5g\ . ;dfg x/ ePsfn] Pp6f dfq
x/ lnO{ cz+ x¿ hfl] 8G5 .
dflysf lj|mofsnfknfO{ ;fdfGoLs/0f ubf{
3 2 3 + 2
6 + 6 = 6

= 5
6
5
∴ b'j} hgf ldnL = 6 efu

cyf{t\ 6 efudf 5 efu af/L vg]5g\ .

;dfg x/ ePsf leGgx¿sf] hf8] ubf{ pxL x/ /fvL c+zx¿ dfq hf]8 ul/G5 .

pbfx/0f 2

ljlkgn] ljBfnoaf6 cfˆgf] 3/ k'Ug 7 ls=ld= lxF8\gk' 5{ . olb p;n] 3 ls=ld=
8 8
lxlF 8;ss] f] eP ca 3/ kU' g slt ls=ld= lxF8g\ 'knf{ M

;dfwfg

ul0ftLo jfSodf nV] bf
7 3
= 8 - 8

=7 cf6] f 1 -3 cf6] f 1
8 8

66 ul0ft M sIff $

= 4 cf6] f 1 = 4
8 8
1
= 2

;ª\Vof /]vfdf /fv]/ x]bf,{

afFsL b'/L

0 1 28 3 84 5 6 7 8 =1
8 8 8 8 8 8

ca p;n] 3/ kU' g 4 ls=ld= lxF8g\ 'k5{ .
8

o;}nfO{ csf{] tl/sfaf6 x]/f},F

– Ö

78 4 83 4
8 8

ca ljlkgnfO{ 3/ k'Ug ls=ld= lx8F \g'knf{ .

dflysf] 36fp lj|mofnfO{ ;dfGoLs/0f ubf,{ 7 efu 5fof kf/]sf]af6, 3 efu 5fof
kf/s] f] efu x6fpFbf 4 efu 5fof
7 – 3 Ö 7-3 kf/s] f] afFsL /xG5 .
8 8 8

Ö 4
8

Ö 1 ;dfg x/ ePsfn] Pp6f dfq
2 x/ lnO{ c+zx¿ 36fPsf]

t;y{ ljlkgnfO{ 3/ kU' g 1 ls=ld= lxF8g\ 'knf{ .
2

;dfg x/ ePsf leGgx¿sf] 36fp ubf{ pxL x/ /fvL c+zx¿sf] dfq 36fp ul/G5 .

ul0ft M sIff $ 67

cEof;

1. tnsf sf7] fdf '>', '<' jf '=' lrxg\ x¿ /fVgx' f];\ M

(i) 3 78 (ii) 2 1
8 5 5

(iii) 8 4 (iv) 2 3
11 11 5 5

(v) 1 56
6

2. tnsf leGgx¿nfO{ 7n' f]b]lv ;fgf] j|md ldnfP/ n]Vgx' f];\ M

(i) 3 , 8 , 5 (ii) 5 , 1 , 4 (iii) 1 , 4 , 3
11 11 11 9 9 9 5 5 5

(iv) 5 , 1 , 7 (v) 1 , 6 , 4
13 13 13 7 7 7

3. tnsf leGgnfO{ ;fgf]bl] v 7'nf] jm| d ldnfP/ n]Vg'xf;] \ M

(i) 6 , 4 , 5 (ii) 1 , 5 , 3 (iii) 3 , 7 , 5
7 7 7 6 6 6 8 8 8

(iv) 9 , 5 , 10 (v) 4 , 1 , 2
11 11 11 5 5 5

4. tnsf] lx;fa ug'{xf];\ M

(i) 3 + 1 (ii) 5 + 2 (iii) 5 + 3 + 2
4 4 7 7 11 11 11

(iv) 5 - 3 (v) 7 - 3 (vi) 8 - 1 - 2
9 9 8 8 13 13 13

5. tnsf kZ| gx¿sf] ;dfwfg ug{x' f;] \ M

(i) l;dg| n] Pp6f s]ssf] 2 efu laxfg / 3 efu a]n'sf vfOl5g eg] sg'
5 5
;dodf pgn] yf]/} s]s vfOl5g\ <
5
(ii) Pp6f k]G6/n] klxnf] lbg Pp6f 3/sf] 9 efu /ª nufP5g\ / bf;] |f] lbg
4 /ª nufP5g\ / slt efu a9L
9 efu /ª nufP5g\ eg] sg' lbg a9L

nufP5g\ <

68 ul0ft M sIff $

((iivii)) Pvkh"lfnOfn;g]gP/np] 84695f sfj|mfn] fO{ a/fa/ 8 efudf sfl65g\ / tLdWo] 3 efu cfkm" n]
8
efu efOnfO{ lbOg\ eg] bj' } ldnL slt efu sfj|mf] vfP5g\ <
4
efu :ofpaf6 9 efu clgtfnfO{ lbOg\ eg] ca pgL;uF

slt :ofp afFsL 5g\ < 5
8 10
(v) Pp6f 6fs] /Ldf /xs] f 10 :ofpdWo] efu sl' xP5g\ eg] gsl' xPsf :ofp

slt /x]5g\ < 8 11
15 15
(vi) jG[ bfn] cfOtaf/ ls=ld= bfl} 8l5g\ / ;f]daf/ ls=ld= bf}l85g\ eg]

s'g lbg slt ls=ld= a9L bfl} 8l5g\ <

kl/ofh] gf sfo{ (Project work)

l/ofn] 100 k"0ff{ªs\ sf] ul0ft ljifosf] k/LIffdf 95 cªs\ / ;/f]hn] 93 cª\s k|fKt
u/5] g\ . pgLx¿n] k|fKt u/]sf] cª\snfO{ leGgdf k:| tt' u/L s;n] a9L cª\s k|fKt
u/]5g\ < slt efun] a9L kf| Kt u/5] g\ < lzIfssf] ;xfotfdf bj' }n] s'g sg' u|8] kfP5g\,
kQf nufpgx' f;] \ .

ul0ft M sIff $ 69

2.3.2 bzdnj (Decimal) : bzf+z / ;ofz+ (Tenths and Hundredths)

lj|mofsnfk 1

bfofsf] klxnf] lrqdf Pp6f cfotnfO{ 10 a/fa/

efudf afF8L 1 efudf xl/of] /ª nufOPsf] 5 .
1
/ªu\ fOPsf] efunfO{ leGgdf 10 nl] vG5 . o;nfO{

Ps bzf+z elgG5 . bzdnjdf 0.1 nl] vG5 . k9b\ f

zG" o bzdnj Ps elgG5 . 1
10
bf;] f| ] lrqdf Pp6f cfotnfO{ 10 a/fa/ efudf af8F L = 0.1

3 efudf lgnf] /ª nufOPsf] 5 . /ªu\ fOPsf] efunfO{
3
leGgdf 10 nl] vG5 . o;nfO{ tLg bzfz+ elgG5 .

bzdnjdf 0.3 nl] vG5 .

o;nfO{ o;/L klg k|:tt' ug{ ;lsG5 .

1 = 1 cf6] f 1 = 1 cf]6f 0.1 = 0.1 3 = 0.3
10 10 10
3 1
10 = 3 cf6] f 10 = 3 cf6] f 0.1 = 0.3

o;u} /L,

4 = 4 cf]6f 1 = 4 cf6] f 0.1 = 0.4
10 10

o;nfO{ ;ªV\ of /v] fdf b]vfpFbf,

0 1 2 3 4 5 6 7 8 9 1
10 10 10 10 10 10 10 10 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

leGgx¿ 1 2 4 5 cflbsf] x/df 10 5 t;y{ logLx¿nfO{ bzf+z elgG5 .
10, 10, 10, 10
bzdnjdf 0.1, 0.2, 0.4, 0.5 nl] vG5 .

70 ul0ft M sIff $

ljm| ofsnfk 2

bfof lbOPsf] lrqdf Pp6f cfotnfO{ 100 a/fa/

efudf aflF 8Psf] 5 . 5 efudf /ftf] /ª nufOPsf]

5 . /ª\ufOPsf] efunfO{ leGgdf 5 n]lvG5 /
100

5 ;ofz+ elgG5 . o;nfO{ bzdnjdf nV] bf 0.05

n]lvG5 . k9b\ f z"Go bzdnj z"Go kfFr egL kl9G5 .

To;}u/L 12 efudf kxnF] f] /ª nufOPsf] 5 . /ª

nufOPsf] efunfO{ leGgdf 12 nl] vG5 / o;nfO{
100

12 ;of+z elgG5 / bzdnjdf 0.12 n]lvG5 . k9\bf

zG" o bzdnj Ps b'O{ egL kl9G5 . o;/L leGgx¿

5 , 9 , 18 , 45 cflbsf] x/df 100 5 .
100 100 100 100

To;}n] logLx¿nfO{ ;ofz+ elgG5 / bzdnjdf jm| dzM

0.05, 0.09, 0.18, 0.45 nl] vG5 .

bzfz+ / ;of+znfO{ tnsf] ;ª\Vof/]vfdf k|:tt' ul/Psf] 5 . pSt ;ª\Vof /v] f cWoog
ug'x{ f];\ .

0 1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 .2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

leGgsf] x/df 10 5 eg] To;nfO{ bzf+z / x/df 100 5 eg] To;nfO{ ;of+z elgG5 .

ul0ft M sIff $ 71

pbfx/0f 1

tnsf lrqx¿df 5fof kfl/Psf] efunfO{ leGg / bzdnjdf nV] gx' f;] \ M

(i) (ii)

leGg Ö 4 leGg Ö 6
10 10

bzdnj Ö 0.4 bzdnj Ö 0.6

(iii)

leGg Ö 2
10

bzdnj Ö 0.2

(iv)

leGg Ö 8
100

bzdnj Ö 0.08

cEof;

2. tnsf leGgx¿nfO{ bzdnjdf n]Vg'xf];\ M

(i) 3 (ii) 8 (iii) 7 (iv) 1 (v) 4
10 10 10 10 10

(vi) 3 (vii) 5 (viii) 1900 (ix) 12050 (x) 48
100 100 100

(xi) 65 (xii) 95
100 100

3. tnsf bzdnj ;ªV\ ofx¿nfO{ leGgdf n]Vg'xf;] \ M (v) 3.5
(i) 0.5 (ii) 0.9 (iii) 0.8 (iv) 2.4 (x) 0.85
(vi) 0.37 (vii) 0.48 (viii) 0.32 (ix) 0.68
ul0ft M sIff $
72

2.3.3 kl| tzt (Percentage)

ljm| ofsnfk 1

bfof lbOPsf] lrqdf Pp6f jufs{ f/ sfuhnfO{ 100

a/fa/ efudf afFl8Psf] 5 . h;sf] 15 efudf 5fof

kfl/Psf] 5 . 5fof kfl/Psf] efunfO{ leGgdf nV] bf
15
100 nl] vG5 / o;nfO{ 15 ;ofz+ elgG5 . bzdnjdf

n]Vbf 0.15 nl] vG5 . ca o;}nfO{ kl| tztdf n]Vbf

15% nl] vG5 . 15% n] k|lt ;odf 15 eGg] hgfpF5 .

To;}u/L 9 efudf lgnf] /ªn] 5fof kfl/Psf] 5 h;nfO{
leGg, bzdnj / k|ltztdf jm| dzM 1090, 0.09 / 9%
nl] vG5 .

olb leGgsf] x/ 100 5 eg] pSt leGgsf] cz+ nfO{ k|ltzt elgG5 . kl| tzt

hgfpgnfO{ (%) lrxg\ sf] k|ofu] ul/G5 .

ljm| ofsnfk 1

bfof lrqdf bv] fP h:t} Pp6f jufs{ f/ sfuh lncfF} /

To;nfO{ 10 a/fa/ efu nufO{ tLg efudf 5fofF kf/f}F .

ca 5fofF kfl/Psf] efunfO{ leGgdf / bzdnjdf slt
3
slt nl] vG5 xfn] f < cjZo klg 10 / 0.3 n]lvG5 .

ca o;nfO{ k|tztdf s;/L n]Vg] xfn] f <

k|ltztdf nV] g leGgsf] x/df 100 x'gk' 5{ . To;sf 3
10
nflu dflysf] juf{sf/ sfuhnfO{ t];f]{k6\l6af6 bz = 0.3

a/fa/ efudf afF8f,}F

ca hDdf sf7] f ;ª\Vof 100 / 30 cf6] fdf 5fof

kf/s] f] b]lvg] 5 . o;nfO{ leGg / b;dnjdf nV] bf
30
jm| dzM 100 / 0.30 nl] vG5 . h;nfO{ 30% nV] g

;lsG5 .

ul0ft M sIff $ 73

ul0ftLo jfSodf o;nfO{ o;/L klg ug{ ;lsG5 M
3 3 x 10
10 = 10 x 10 x/nfO{ 100 agfpg c+z / x/
b'j}nfO{ 10 n] u0' fg u/]sf]
= 30
100

= 30%

pbfx/0f 1

;/:jtL cfwf/e"t ljBfnosf] sIff $ df /x]sf 100 hgf ljBfyLd{ Wo] 5fqf 45 hgf
/x5] g\ . pSt ljBfnodf ePsf 5fqf / 5fqsf] ;ª\VofnfO{ leGg / kl| tztdf n]Vgx' f;] \ .

;dfwfg 5fofF kf//] b]vfpFbf

hDdf ljBfyL{ ;ªV\ of = 100

5fqfsf] ;ª\Vof = 45

5fqsf] ;ª\Vof = 100 - 45 = 55 45
100
5fqfsf] ;ª\VofnfO{ leGgdf nV] bf,

5fqsf] ;ª\VofnfO{ leGgdf n]Vbf, 55
100

k|ltztdf nV] bf,
5fqf = 45% / 5fq = 55%

cEof;

1. tnsf] lrqdf 5fof kfl/Psf] efun] hgfpg] leGg, bzdnj / kl| tzt nV] gx' f;] \ M
(i) (ii)

74 ul0ft M sIff $

2. tnsf k|To]s jfSox¿nfO{ leGg, bzdnj / kl| tztdf nV] gx' f];\ M

(i) x//] fdn] 100 k"0ff{ª\ssf] ul0ftsf] k/LIffdf 80 cªs\ kf| Kt u/5] g\ .

kf| Kt cªs\ sf] leGg Ö

k|fKt cªs\ sf] bzdnj Ö

k|fKt cª\ssf] k|ltzt Ö

(ii) sIff $ df hDdf 100 hgf ljBfyL{df hDdf 45 hgf s]6fx¿ /x5] g\ .

s]6fsf] leGg Ö s]6fx¿sf] bzdnj Ö

s6] fx¿sf] kl| tzt Ö

(iii) sIff $ df 100 hgf ljBfyL{dWo] 8 hgf cgk' l:yt eP5g\ .

cg'kl:yt leGg Ö cgk' l:yt bzdnj Ö

cg'kl:yt kl| tzt Ö

(iv) Pp6f k;n]n] ?= 100 sf] ;fdfg aR] bf ?= 10 gfkmf u/]5 .

gfkmfsf] leGg Ö gfkmfsf] bzdnj Ö

gfkmfsf] kl| tzt Ö

(v) Pp6f k;nn] ] ?= 100 kg]{ ;fdfgdf ?= 1 5'6 lbP5 .

5'6sf] leGg Ö 5'6sf] bzdnj Ö

56' sf] k|ltzt Ö

3. tnsf kT| o]s leGg / bzdnjnfO{ k|ltztdf nV] gx' f];\ M

(a) 20 Ö Ü (b) 45 Ö Ü
100 100

(c) 4 Ö Ü (d) 5 Ö Ü
10 10

(e) 0.03 Ö Ü (f) 0.15 Ö Ü

(g) 0.65 Ö Ü

4. tnsf kT| o]s k|ltztnfO{ leGg / bzdnjdf n]Vg'xf];\ M

(a) 25Ü (b) 20Ü (c) 35Ü

(d) 75Ü (e) 5Ü (f) 3Ü

ul0ft M sIff $ 75

kf7 # gfk (Measurement)

3.1 ;do (Time)
3.1.1 k'g/jnf]sg (Review)
tnsf kZ| gx¿df cfwfl/t /xL 5nkmn ug{x' f];\ M
(i) gk] fnL kfqf]cg';f/ t];|f] dlxgf c;f/ xf] eg] bzf}F dlxgf s'g xfn] f <
(ii) Ps xKtfdf 7 lbg xG' 5 eg] 4 xKtfdf slt lbg xG' 5 <
(iii) 19 lbgdf slt xKtf / slt lbg xG' 5 <
(iv) 5 jif{df slt dlxgf x'G5 <
(v) 16 dlxgfdf slt jif{ / slt dlxgf x'G5 <

lj|mofsnfk 1 M

/Ltfsf] 38Ldf slt
ahs] f] xf]nf <

of] /Ltfsf] 38L xf] . o;df 1 bl] v 12 ;Dd nl] vPsf] 5 . 38Lsf] ;a}eGbf 5f6] f] ;'O{
306f ;'O{ xf] . 306f ;'O{n] Ps kmGsf] nufpbF f 12 306f xG' 5, k'gM Ps kmGsf] nufpFbf12
306f xb'F f bO' { kmGsf] nufpbF f 24 306f x'G5 / of] Ps lbg a/fa/ x'G5 . 38Ldf 306f
;O' e{ Gbf nfdf] ;fy} df6] f] ;'O{nfO{ ldg]6 ;'O{ elgG5 . nfdf] t/ kftnf] ;O' { ;]s]G8 ;'O{
xf] . ldg]6 ;'On{ ] Ps kmGsf] nufpFbf 306f ;O' { Ps cª\sb]lv csf]{ cª\s;Dd kU' b5 .
ldg6] ;O' n{ ] Ps kmGsf] nufpbF f nfUg] ;do 60 ldg6] xf] . ;s] G] 8 ;O' n{ ] Ps kmGsf]
nufpg 60 ;s] G] 8 nfU5 .
o;}n ] 1 lbg = 24 306f
1 306f = 60 ldg]6
1 ldg6] = 60 ;s] G] 8

76 ul0ft M sIff $

ca /Ltfsf] 38Ldf 306f ;'O{ 8 / 9 sf] lardf t/ 8 sf] glhs 5 eg] ldg]6 ;'O{ 3 df
5 . o;}n] of] 38Ldf 8 ah/] 15 ldg6] uPsf] 5 . o;nfO{ 8:15 nl] vG5 . o;/L g} cGo
38Lsf] ;do xg] {x' f];\ / 5nkmn ug'{xf;] \ .

3.1.2 ;dosf PsfOx¿sf] Pscsf{df ¿kfGt/0f (Conversion of units of

time each other)

lj|mofsnfk 1
/dfnfO{ 3/af6 ljBfno k}bn hfg 15 ldg6] ;do nfU5 . o;nfO{ ;s] G] 8df abNg s]
ug{'knf{ < 5nkmn ug{x' f];\ / sIffsf7] fdf k:| t't ugx'{ f;] \ .
1 ldg6] df 60 ;]sG] 8 x'G5 .
2 ldg]6df 60 + 60 = 120 ;]s]G8 xG' 5 .
3 ldg]6df 60 + 60 + 60 = 180 ;s] G] 8 x'G5 .
4 ldg6] df 60 + 60 + 60 + 60 = 240 ;s] ]G8 x'G5 .
5 ldg]6df slt xfn] f, 5nkmn ugx{' f];\ .
o;/L hf]8\b} hfFbf nfdf] x'bF } hfg] ePsfn]
1 ldg]6df 60 ;]sG] 8 xG' 5 .
2 ldg]6df 60 + 60 = 60 × 2 = 120 ;]s]G8 x'G5 .
3 ldg]6df 60 + 60 + 60 = 60 × 3 = 180 ;]s]G8 xG' 5 .
15 ldg]6df 60 × 15 = 900 ;s] ]G8

ldg6] nfO{ ;]s]G8df n}hfFbf 60 nfO{ u'0fg ug'{kb5{ .
/dfnfO{ ljBfnoaf6 3/ kmsg{ ;fOsndf 300 ;]sG] 8 nfU5 eg] slt ldg]6 nfus] f]
/x]5 < o;u} /L ;]sG] 8nfO{ ldg6] df abNgsf] nfu s] ug{] xfn] f <
;donfO{ ;ªV\ of /v] fdf bv] fpFbf,

0 60 120 180 240 300
ljBfno 3/

/dfnfO{ ljBfnoaf6 3/ kmsg{ 5 ldg6] nfU5 .

ul0ft M sIff $ 77

;]sG] 8nfO{ ldg6] df abNgsf nflu,
60 ;s] ]G8 a/fa/ 1 ldg]6 xf] .
120 ;s] G] 8 a/fa/ 120 ÷ 60 = 2 ldg]6 xf] .
180 ;s] G] 8 a/fa/ 180 ÷ 60 = 3 ldg6] xf] .
240 ;]s]G8 a/fa/ 240 ÷ 60 = 4 ldg6] xf] .
300 ;]sG] 8 a/fa/ 300 ÷ 60 = 5 ldg]6 xf] .

;]s]G8nfO{ ldg6] df abNgsf nflu 60 n] efu ugk{' b{5 .
;s] G] 8 ÷ 60 ldg]6

60 ×

lj|mofsnfk 2

tnsf k|Zgx¿df cfwfl/t /xL 5nkmn ugx'{ f];\ M
(i) tkfO‘{nfO{ 3/af6 glhss} f] k;ndf hfg 5 ldg6] nfU5 eg] slt ;]sG] 8 eof] <

(ii) tkfOn‘{ fO{ vfgf vfg 25 ldg]6 nfU5 eg] slt ;s] ]G8 nfU5 <

(iii) /fx] g / p;sf ;fyLx¿n] ljBfnosf] vn] d}bfgdf 540 ;]s]G8;Dd an vn] ] eg]
slt ldg6] v]n] <

dflysf kZ| gx¿sf cfwf/df x]bf{,

(i) k;ndf hfg nfu]sf] ;do 5 ldg6] 5 1 ldg6] = 60 ;s] ]G8
1 ldg6] = 60 ;]s]G8
5 ldg]6 = (60 × 5) ;]sG] 8 = 300 ;s] ]G8

(ii) dflysf k|Zgdf h:t} oxfF vfgf vfg nfu]sf] ;do = 25 ldg]6
1 ldg]6 = 60 ;]sG] 8
25 ldg]6 = (60 × 25) ;]sG] 8 = 150 ;s] ]G8

(iii) /fx] g / p;sf ;fyLx¿nfO{ an v]Ng nfu]sf] ;do = 540 ;s] G] 8
60 ;s] ]G8 = 1 ldg6]
540 ;]sG] 8 = 540 ÷ 60 = 9 ldg6]

78 ul0ft M sIff $

lj|mofsnfk 3
lj=;=+ 2078 sf] j}zfvbl] v rt} ;Ddsf] Sofn]G8/ lbOPsf] 5 . Sofn]G8/df cfwfl/t /xL
tnsf kZ| gdf 5nkmn ugx{' f];\ M

(i) sflQs dlxgfdf slt lbg / slt xKtf 5g\ <
(ii) Ps jif{df hDdf slt dlxgf x'G5g\ <
(iii) k|Tos] dlxgfsf] lbgnfO{ hf8] \bf slt lbg xG' 5 < hf]8]/ xg] {'xf;] \ .

;fdfGotof 1 jif{ = 365 lbg xG' 5 . t/ clwsdf; k/]sf] jif{ 1 jif{ = 366 lbg x'G5 .
gf]6 M g]kfnL kfqf]cg;' f/ sg' } dlxgf 29, 30, 31, 32 lbgsf xG' 5g\ . ;fdfGotof 1
dlxgf = 30 lbg /fvL lx;fa ul/G5 .

pbfx/0f 1

;'lgtfnfO{ bf}8/] 500 ld6/ b'/L kf/ ug{ 3 ldg]6 20 ;]s]G8 nfU5 . pgnfO{ hDdf slt
;]sG] 8 ;do nfu]sf] /x5] <

;dfwfg 1 ldg]6 = 60 ;]s]G8
3 ldg6] 20 ;s] G] 8 = 3 ldg6] + 20 ;s] G] 8
= (60 × 3) ;]s]G8 + 20 ;s] ]G8
= 180 ;s] ]G8 + 20 ;s] G] 8
= 200 ;]s]G8

ul0ft M sIff $ 79

pbfx/0f 2 60 ;s] ]G8 = 1 ldg6]
180 ;]sG] 8nfO{ ldg6] df ¿kfGt/0f ug{'xf];\ M
;dfwfg 180 ;s] ]G8 = 180 ÷ 60
180 ;s] ]G8 = (60 + 60 + 60) ;s] G] 8 60 ) 180 ( 3
= 3 ldg]6 - 180

o;sf/0f 180 ;s] G] 8 = 3 ldg]6 0

pbfx/0f 3

200 ;]s]G8nfO{ ldg]6df ¿kfGt/0f ugx{' f;] \ M 200 ;s] G] 8 = 200 ÷ 60
60 ) 200 ( 3
;dfwfg - 180
200 ;s] ]G8 = (60 + 60 + 60 + 20) ;s] G] 8
= 3 ldg]6 20 ;s] G] 8 20

pbfx/0f 4

Pp6f ;xsf/L ejg lgdf{0f x'g 3 jif{ 6 lbg nfUof] eg] hDdf slt lbg nfu5] , kQf
nufpg'xf];\ M

;dfwfg 1 jif{ = 365 lbg
3 jif{ = (3 x 365) lbg = 1095 lbg
3 jif{ 6 lbg = 1095 lbg + 6 lbg = 1101 lbg

pbfx/0f 5

730 lbgnfO{ jifd{ f ¿kfGt/0f ugx{' f;] \ M 365 lbg = 1 jif{
ul0ft M sIff $
;dfwfg

365 lbg = 1 jif{
730
730 lbg = 365 jif{ = 2 jif{

80

cEof;

1. tnsf kZ| gx¿sf] pQ/ lbg'xf];\ .

(i) 6 ldg6] = ;s] G] 8

(ii) 120 ;]sG] 8 = ldg]6

(iii) 1095 lbg = jif{

(iv) 8 jif{ = lbg

2. ;s] G] 8df ¿kfGt/0f ugx{' f;] \ . (ii) 21 ldg]6
(i) 15 ldg6] (iv) 14 ldg]6 30 ;]sG] 8
(iii) 3 ldg]6 11 ;s] G] 8

3. ;s] G] 8nfO{ ldg6] df ¿kfGt/0f ug{x' f];\ . (ii) 320 ;s] G] 8
(i) 110 ;s] ]G8 (iv) 170 ;]s]G8
(iii) 190 ;]s]G8

4. lbgnfO{ jif{ / lbgdf / jifn{ fO{ lbgdf ¿kfGt/0f ug{x' f];\ .

(i) 730 lbg (ii) 550 lbg (iii) 20 jif{

(iv) 1095 lbg (v) 590 lbg (vi) 25 jif{

(vii) 18 jif{

5. wg]ifnfO{ 3/af6 ljBfno kU' g 21 ldg]6 nfU5 eg] p;nfO{ slt ;s] G] 8 nfu]sf]
/x]5, kQf nufpg'xf;] \ .

6. Pp6f ;f+:sl[ ts sfo{j|mddf cfxfgfn] 260 ;]sG] 8 g[To ul/g\ eg] pgn] slt
ldg6] / slt ;s] ]G8 gT[ o ul/g,\ lgsfNg'xf;] \ .

7. 5 jif{ 20 lbgnfO{ lbgdf ¿kfGt/0f ugx'{ f;] \ .

8. 308 lbg eg]sf] slt jif{ / lbg xf]nf <

kl/ofh] gf sfo{ (Project work)

tkfO{‘ laxfg p7]kZrft\ lbge/df ug{] ljm| ofsnfkx¿sf] ;r" L agfpg'xf];\ . oL ;a}
lj|mofsnfk ug{ nfUg] ;do ldg6] df nV] g'xf;] \ . o;nfO{ ;s] G] 8df ¿kfGt/0f ug'{xf;] \ .

ul0ft M sIff $ 81

3.1.3 ;dosf] hf]8 (Addition of time)

lj|mofsnfk 1

tnsf] 38Lsf] cjnfs] g u/L lgDglnlvt k|Zgx¿df 5nkmn ug{'xf;] \ M
(i) 38Ldf slt ah]sf] 5 <
(ii) 2 306f 30 ldg6] kl5 38Lsf] ;'O{ sxfF sxfF xfn] fg\ <
(iii) 2 306f 30 ldg]6kl5sf] ;dodf 1 306f 15 ldg6] hf]8\bf 38Lsf] ;'O{ sxfF sxfF

kU' nf <
38Ldf 10 ah]/ 10 ldg6] uPsf] 5 . 2 306f 30 ldg6] kl5 38Lsf] ;do kQf nufpgsf
nflu 38Lsf] ;'O{nfO{ 2 306f 30 ldg]6 k/ 3d' fP/ sltdf k'Uof] x]g'{xf];\ < ;f] ;dodf
kml] / 38Lsf] ;'O{nfO{ 1 306f 15 ldg]6 k/ 3'dfpg'xf;] \ . 38Lsf] 306f / ldg]6 ;'O{ sxfF
sxfF xfn] fg\ <

ljm| ofsnfk 2 M

ljBfnon] cfˆgf] ;Dk0" f{ zI} fl0fs ultljlw pNnv] u//] lgdf0{ f u/s] f] SofnG] 8/ xg] 'x{ f];\
/ tnsf k|Zgx¿sf] 5nkmn ug{x' f;] \ M
(i) o; z}lIfs jif{df cfh;Dd hDdf

slt dlxgf ljBfno vN' of] <
(ii) z}lIfs ;qsf] cGt xg' of] jif{df

slt dlxgf afFsL 5 <
(iii) tL afsF L dlxgfx¿df slt xKtf

/x5] g\, nV] gx' f];\ .
(iv) olb Ps lbgdf 6 306f k9fO xG' 5

eg] 5 lbgdf slt 306f k9fO x'G5, kQf nufpgx' f;] \ .

pbfx/0f 1

pQdn] zlgaf/sf lbg 1 306f 25 ldg]6 u[xsfo{ u¥of] . 2 306f 12 ldg6] km'6;ndf
km'6an v]Nof] . o:t} 3 306f 10 ldg]6 6]lnlehg x]¥of] eg] oL tLgcf]6f sfo{ ug{ hDdf
slt ;do nfUof,] kQf nufpg'xf;] \ .

82 ul0ft M sIff $

;dfwfg

ux[ sfo{ ug{ nfu]sf] ;do = 1 306f 25 ldg]6

km'6;ndf km6' jn v]Ng nfu]sf] ;do = 2 306f 12 ldg6]

6]lnlehgdf x/] s] f] ;do = 3 306f 10 ldg6]

ca p;nfO{ nfu]sf] hDdf ;do

306f ldg]6
1 25

2 12

+ 3 10

6 306f 47 ldg]6

pQdnfO{ oL ;Dk"0f{ sfd ug{ 6 306f 47 ldg]6 nfu]sf] /x5] .

;dosf] hf8] df ;dfg PsfOx¿ dfq hfl] 8G5, h:t} M 306f;uF 306f / ldg]6;Fu ldg]6

pbfx/0f 2

cfof{ / pgsf] ;fyLx¿n] an] s' f 5:30 af6 ux[ sfo{ ug{ ;?' u/] . pgLx¿n] 40
ldg6] df u[xsfo{ u//] l;WofP eg] l;WofPkl5 ;do lgsfNg'xf;] \ / Tof] ;donfO{ 38Ldf
bv] fpg'xf];\ M

;dfwfg

cfof{ / pgsf] ;fyLn] u[xsfo{ ug{ ;?' u/s] f] ;do = 5 306f 30 ldg]6

ux[ sfo{ ug{ nfus] f] ;do = 40 ldg]6

l;WofPkl5sf] ;do lgsfNg oL bO' { ;do hf]8g\ 'k5,{

5 306f 30 ldg]6

40 ldg6]

5 70 60 ldg6] = 1 306f
+ 1 - 60
6 10

o;sf/0f pgLx¿n] 6 ah/] 10 ldg6] hfbF f l;WofPsf /x]5g\ .

ul0ft M sIff $ 83

pbfx/0f 3

;l/gfn] 2075 ;fn j}zfvdf cfFksf la?jf /f]lkg\ . 2 jif{ 10 dlxgfkl5 pSt la?jfn]
kmn lbg yfNof] eg] slxnb] ]lv kmNg yfNof] kQf nufpg'xf];\ M

;dfwfg
;l/gfn] la?jf /f]ks] f] ;do = 2075 jif{ 1 dlxgf
cfkF kmNg yfns] f] ;do = 2 jif{ 10 dlxgf

2075 2076 2077

jif{ dlxgf

2075 1

+ 2 10

2077 jif { 11 dlxgf

∴ pSt la?jf 2077 ;fn kmfu'gaf6 kmNg yfns] f] /x]5 .

pbfx/0f 4
tnsf] vfnL sf7] fdf pQ/ nV] gx' f;] \ M

3 lbg 10 306f 2 lbg 5 306f
kl5

;dfwfg

lbg 306f
3 10

+ 2 5

5 lbg 15 306f

o;sf/0f pSt vfnL sf]7fdf 5 lbg 15 306f n]Vg'kb{5 .

84 ul0ft M sIff $

pbfx/0f 5

/Tg/fHo df=lj= df lj=;+= 2076 df3 dlxgfdf 2 xKtf 4 lbg v]nsb' sfoj{ |md ;~rfng
eof] . 1 xKtf 3 lbg cGo cltl/St ljm| ofsnfk ;~rfng eof] eg] hDdf slt ;do
sfoj{ m| d ;~rfng eof,] lgsfNg'xf];\ M

;dfwfg

v]ns'b sfo{j|md ;~rfng ePsf] ;do = 2 xKtf 4 lbg

cltl/St lj|mofsnfk ;~rfng ePsf] ;do = 1 xKtf 3 lbg

ca hf8] ]/ x/] f},F

xKtf lbg
2 4

+ 1 3

3 7 7 lbg = 1 xKtf
+ 1

4 0

∴pSt sfo{jm| d 4 xKtf;Dd ;~rfng eof] .

cEof;

1. tn lbOPsf cfwf/df ;dosf] hf8] ug'{xf;] \ M

(a) jif { dlxgf (b) jif { dlxgf (c) lbg 306f
3 4 6 7 3 12

+ 5 2 + 9 8 + 5 13

(d) lbg 306f (e) 306f ldg6] (f) 306f ldg6]

1 14 7 8 2 37
+ 2 20 + 4 12 + 1 48

(g) xKtf lbg (h) xKtf lbg

2 2 3 5

+ 5 4 + 2 4

ul0ft M sIff $ 85

2. ;Fus} f] 38L xg] {x' f];\ . o;sf] 45 ldg]6 kl5sf] ;do
slt xfn] f, kQf nufpg'xf];\ .

3. vfnL sf7] fdf pkoS' t ;do nV] g'xf];\ M

2:35




4. ;fg] fdsf] clxn]sf] pd]/ 11 jif{ 5 dlxgf 5 . 12 jif{ 2 dlxgf kl5 ;fg] fdsf] pd]/

slt x'G5, kQf nufpg'xf;] \ .
5. lglndfnfO{ udL{ labfsf] u[xsfo{ / kl/of]hgf sfo{ ug{ 4 306f 5 ldg6] nfUof] .

g]kfnLsf] ux[ sfo{ / kl/of]hgf sfo{ ug{ 3 306f 7 ldg6] nfUof] eg] b'j} ljifosf]
sfd l;Wofpg slt ;do nfUnf, kQf nufpg'xf];\ .
6. af:s6] jn v]nf8Ln] df3 dlxgfdf 1 xKtf 4 lbg / kmfu'g dlxgfdf 2 xKtf 5 lbg
cEof; u/] eg] hDdf slt ;do cEof; u/] <
7. ls;fgnfO{ t/sf/L v]tL ugs{ f nflu 6g]n agfpg 2 xKtf 2 lbg nfUof] . df6f]
tof/ u//] lap /f]Kg 1 xKtf 5 lbg nfUof] eg] ;a} sfd l;Wofpg hDdf slt ;do
nfUof,] kQf nufpg'xf];\ .

kl/of]hgf sfo{ (Project work)
tkfOn{‘ ] ca cfpg] zlgaf/ ug{] ;Dk0" f{ sfdsf] l6kf]6 ug'x{ f;] \ . tL sfdx¿ ug{ slt
;do nfUnf < ;Dk"0f{ sfo{ / nfUg] ;do af/] kl| tjb] g tof/ u/L sIffsf]7fdf k:| tt'
ugx'{ f];\ .

86 ul0ft M sIff $

3.1.4 ;dosf] 36fp (Subtraction of time)

lj|mofsnfk 1

lbOPsf] lrqdf 38Ln] hgfPsf] ;do x]/]/ 5nkmn ugx{' f];\ M

(i) ljBfno 5'6\6L xg' ] ;do eGgx' f;] \ .

(ii) ul0ftsf] sIff slt ah] ;?' ePsf] /x]5 < -s_ -v_

ul0ftsf] sIff ljBfno 5'6\6L
(iii) ul0ftsf] sIff 1 306f;Dd ;~rfng x'G5 eg] slt ;?' xg' ] ;do x'g] ;do

ah] ;lsG5 <

(iv) olb 2:35 af6 zf/Ll/s sIff ;~rfng xG' 5 eg] ul0ftsf] sIffsf] ;dflKt zf/Ll/ssf]
sIffsf] ;?' xg' ] ;dolar slt km/s /x5] <

(v) ul0ftsf] sIff ;'? xg' ] ;dob]lv ljBfno 5'66\ L x'g slt ;do afFsL xf]nf <

lbOPsf] 38Lx¿df ljBfno 56' 6\ L x'g] ;do 4:00 ah] / ul0ft sIff ;'? 1:15 ah] x'g]

b]vfPsf] 5 . ul0ftsf] sIff 2:15 ah] ;lsPsf] 5 .

zf/Ll/s sIff / ul0ftsf] sIffsf] larsf] ;do lgsfNg zf/Ll/ssf] sIffsf] ;do 2:35

af6 ul0ftsf] sIff ;dfKt ePsf] ;doaf6 36fpgk' b5{ .

o;sf] nflu 306f / ldg]6sf] 5'6\6f56' \6} nx/df ldnfP/ /fvf}F .

306f ldg]6
2 35

- 2 15

0 20

ul0ftsf] sIff ;dflKt / zf/Ll/s sIffsf] ;?' xg' ] ;do 20 ldg6] sf] km/s /x]5 .

ul0ft ;?' ePsf] ;doaf6 ljBfno 5'6\6L x'g afFsL /x]sf] ;do lgsfNgsf nflu 4:00

ah]af6 1:15 36fpgk' b{5 . ljBfno 56' 6\ L xg' ] ;do 4:00 306fdf dfq 5 . t/ ul0ftsf]

sIffsf] ;do 1:15 df 306f / ldg6] 5 . o:tf] cj:yfdf 36fpg s] ug]{ xfn] f <

306f ldg]6 1 306f = 60 ldg6] x'g] ePsfn]
4 306faf6, 1 306f (60 ldg6] )
3 60 ;fk6L lnP/ 15 36fpg' kb{5 .

4 0

- 1 15

2 45

ul0ft M sIff $ 87

ljBfno 5'6\6L xg' 2 306f 45 ldg6] afsF L /x]5 . ;dosf] hf]8df h:t} 36fpdf klg
;dfg PsfOx¿ dfq 36fpg] ul/G5 .

pbfx/0f 1

lxs]; a]n'sf 8:45 bl] v 6]lnlehg x]g{ a:of] . p;nfO{ dg kg{] sfo{jm| d cfPsfn] p;n]
10:15 ;Dd 6l] nlehg x¥] of] eg] p;n] slt ;do;Dd 6]lnlehg x]¥of] <

;dfwfg

lxs];nfO{ 6]lnlehg x]g{ ;'? u/s] f] ;do = 8:45

6]lnlehg xg] { cGTo u/s] f] ;do = 10:15

of] larsf] ;dofjlw kQf nufpg 36fp ug{k' b5{ .

306f ldg6] 306f ldg]6

10 15 9 60+15 = 75

- 8 45 ⇒ - 8 45

1 30

p;n] 1 306f 30 ldg6] 6]lnlehg x]¥of] .

pbfx/0f 2

lj=;+= 2072 ;fnsf] e'sDkn] eTsfPsf] ljBfno ejg 2074 ;fn c;f]h 1 uta] f6
sfo{ ;?' eO{ lj=;=+ 2077 ;fn r}t d;fGt ;DkGg eof] eg] slt ;dodf lgdf0{ f ;DkGg
eof] <

;dfwfg

ljBfno lgdf{0f ug{ ;?' ul/Psf] ;do = 2074 ;fn c;f]h 1 ut]
ljBfno lgdf0{ f ;DkGg ePsf] ;do = 2077 ;fn r}t d;fGt

36fP/ x]/f,F}

jif{ dlxgf

2077 12

- 2074 5

3 jif{ 7 dlxgf

ejg lgdf{0f ug{ 3 jif{ 7 dlxgf nfu]sf] /x]5 .

88 ul0ft M sIff $

pbfx/0f 3

5 lbgsf nflu uf/] vfaf6 dxG] b| gu/;Ddsf] z}lIfs ed| 0fsf] tof/L ul/Psf]
lyof] . e|d0f ;'? u/s] f] 70 306fkl5 ca slt 306fsf] ed| 0f afFsL /xG5, kQf
nufpg'xf;] \ .

;dfwfg lsg xfn] f <
5 lbg = (24 × 5) 306f = 120 306f

ca, afFsL ;do = 120 - 70 = 50 306f

cEof;

1. tn lbOPsf cfwf/df ;dosf] 36fp ugx{' f;] \ M

(i) lbg 306f (ii) 306f ldg]6 (iii) lbg 306f

6 10 9 48 12 20

- 4 15 - 6 50 - 8 15

(iv) lbg 306f (v) xKtf lbg (vi) xKtf lbg

22 4 13 8 19 6

- 14 10 - 9 5 - 11 5

(vii) jif { dlxgf (viii) jif { dlxgf

15 20 8 12

- 10 11 - 4 11

2. cflzif laxfg 7:30 df lsgdn] ug{ ahf/ uP . 10:45 df 3/ kmls{P eg] slt
;do 3/af6 aflx/ /x], kQf nufpgx' f;] \ .

3. 3 jif{ 6 dlxgfdf slt ;do yk] 8 jif{ 7 dlxgf x'G5 xf]nf <

4. ljdnnfO{ ldgL a;df sf7df8f}aF f6 kf]v/f;Ddsf] ofqf ug{ 6 306f 30 ldg6]
nfUof] . laxfg sf7df8fFa} f6 lx8F s] f] a; 3 306f 10 ldg6] df d'lUng k'u]sf] /x5] .
ca afFsL ofqf slt ;dodf k/' f u/]sf] /x]5, kQf nufpg'xf;] \ .

ul0ft M sIff $ 89

5. s'g} sfdbf/n] 3 xKtf 6 lbgsf nflu kvf{n nufpg] sfdsf] lhDdf lnP5 . p;n]
2 xKtf dfq sfd u//] sfd 5f8] 5] eg] ca slt ;dosf] sfd afsF L 5, kQf
nufpgx' f;] \ .

6. bLlksfn] laxfg 6:45 af6 u[xsfo{ ug{ ;?' ul/g\ . 8:30 ;Dd ;a} ux[ sfo{ k/" f
ul/g\ eg] pgnfO{ ux[ sfo{ ug{ slt ;do nfUof] <
kl/of]hgf sfo{ (Project work)

tkfO‘s{ f] 3/df bl} gs k|ofu] ug]{ 1 xKtfsf pkefU] o ;fdu|Lx¿df n]lvPsf] pkef]Uo
ldlt ;dfKt x'g slt ;do afFsL /x]sf] 5 < l6kf6] u/L sIffsf]7fdf k|:tt' ug'{xf;] \ .

90 ul0ft M sIff $

3.2 d'bf| (Money)
3.2.1 kg' /jnfs] g
tn lbOPsf db' f| x¿sf] cjnfs] g ugx'{ f;] \ M

d'bf| x¿ l;Ssf jf gf6] sf ¿kdf xG' 5g\ . db' |fsf] ;fgf] PsfO k};f xf] . ?= 1 df 100 k;} f
x'G5 . ?= 5, ?= 10, ?=50 x¿df slt k};f x'G5g\ xf]nf, 5nkmn ug{x' f];\ .

3.2.2 db' f| ;DaGwL u0' fg (Multiplication related to money)

ljm| ofsnfk 1

?= 40 ?= 15 ?= 55 10 ?lkofF 50 k;} f

kl| tuf6] f k|ltuf]6f k|ltuf6] f k|ltuf]6f -s]/f_

-:ofp_ -;G' tnf_ -cgf/_

dflysf] lrqsf] cjnf]sg ug{'xf];\ / tnsf kZ| gx¿df 5nkmn ug{'xf];\ M

(i) 3 cf6] f :ofpsf] dN" o slt xfn] f <
(ii) 8 cf6] f sf;] f s]/fsf] d"No slt xf]nf <
(iii) 5 cf]6f ;'Gtnf / 3 cf6] f cgf/sf] hDdf dN" o slt slt xf]nf <
(iv) tkfO‘{;Fu ePsf] ?= 150 af6 kmnk"mn lsGg' k/] sg' s'g kmnk"mn slt dfqfdf

lsGgx' G' 5 <

hDdf d"No Ö PsfO j:t'sf] dN" o × j:t'sf] ;ªV\ of

ul0ft M sIff $ 91

pbfx/0f 1

tn b]vfOPsf vfgs] '/fsf] d"No;r" L x]/L ;fl] wPsf k|Zgx¿sf] hjfkm lbgx' f];\ M
j:tx' ¿ cfhsf] dN" o ;r" L d"No
lrof ?= 20 k|lt sk
b'w ?= 25 k|lt sk

;df;] f 16 ?lkofF 50 k};f k|lt uf]6f

rfpldg ?= 60 kl| t Kn6]

kZ| gx¿
(i) 5 sk lrofsf] dN" o slt knf{ <
(ii) b'O{ sk bw' / b'O{ Kn]6 rfpldgsf] hDdf slt knf{ <
(iii) 3 cf6] f ;df;] fnfO{ slt knf{ <
(iv) 6 cf]6f ;df];f, Ps sk lrof / Ps sk b'wsf] slt knf{ <

;dfwfg

(i) 1 sk lrofsf] dN" o ?= 20 kb{5 .

5 sk lrofsf] dN" o 20 × 5 = ?= 100 kb{5 .
(ii) 1 sk bw' sf] d"No ?= 25 kb{5
2 sk bw' sf] dN" o 25 × 2 = ?= 50 kb{5 .
km]l/,

1 Kn]6 rfpldgsf] dN" o Ö ?= 60
2 Kn]6 rfpldgsf] d"No Ö 60 × 2 = ?= 120 kb5{ .

hDdf = 2 sk b'wsf] dN" o + 2 Kn]6 rfpldg d"No
= ?= 50 + ?= 120
= ?= 170
(iii) Pp6f ;df];fsf] dN" o Ö 16 ?lkofF 50 k};f

3 cf]6f ;df;] fsf] d"No Ö 16 ?lkofF 50 k};f × 3

Ö 48 ?lkofF 150 k;} f 150 k};f = 1 ?lkofF 50 k;} f

92 ul0ft M sIff $

Ö (48 + 1) ?lkofF 50 k};f

Ö 49 ?lkofF 50 k};f

(iv) 6 cf]6f ;df;] fsf] dN" o Ö 16 ?lkofF 50 k;} f × 6

Ö 96 ?lkofF 300 k};f 300 k;} f = 3 ?lkofF

Ö (96 + 3) ?lkofF

Ö 99 ?lkofF

1 sk lrof Ö ?= 20

1 sk b'w Ö ?= 25

hDdf d"No Ö 99 + 20 + 25 = ?= 144

pbfx/0f 2

lx/fn] 5 cf]6f snd lslgg\ . Pp6f sndsf] d"No ?= 25 kb5{ eg] 5 cf]6f sndsf]
d"No slt knf{, kQf nufpg'xf;] \ .

;dfwfg Ö ?= 25
Ö ?= 25 + ?= 25 + ?= 25 + ?= 25 + ?= 25
Pp6f sndsf] dN" o Ö ?= 25 × 5
5 cf6] f sndsf] dN" o Ö ?= 125




pbfx/0f 3

e[s6' L afnSnjsf afnaflnsfx¿n] km'6an k|ltof]lutf cfofh] gf ug{ km6' an lsGgsf
nflu 11 hgfn] kl| tJolSt ?= 100 p7fpg] lgwf] u/] . km6' an k;ndf ku' /] dN" o ;f]Wbf
Pp6f km'6ansf] dN" o ?= 350 /x5] . pgLx¿n] tLgcf]6f anx¿ lsGg] lg0f{o u/] .
hDdf ansf] d"No slt k¥of] xf]nf < pgLx¿;Fu ePsf] /sdn] kU' of] ls ku' ]g, kQf
nufpgx' f;] \ .

ul0ft M sIff $ 93