Com. Mathematics (VII)

kf7 3 k0" ffª{ \s

(Integers)

3.0 k'g/jnfs] g (Review)

tn lbOPsf cj:yfsf] cWoog u/L ;d"xdf 5nkmn ug'x{ f];\ M
-s_ A B C D E F G H I

–2 0 2

;ª\Vof /v] fdf –2, 0 / 2 nfO{ B, D / F n] hgfPsf] 5 . A, C, G / I n] s'g sg'
k"0ff{ªs\ nfO{ hgfp5F <

-v_ k"0ffª{ s\ x¿ 8, –2, 3, 0 / –6 nfO{ ;ªV\ of /v] fdf s;/L b]vfOG5 <
-u_ s'g} Pp6f laGb'nfO{ pbu\ d laGb' dfg/] pSt laGb'b]lv 5 PsfO bfofsF f] :yfg

A / 5 PsfO afofFsf] :yfg B nfO{ ;ª\Vof /]vfdf s;/L bv] fpg ;lsG5 <

3.1 k0" ff{ªs\ sf] hf8] (Addition of Integer)

ljm| ofsnfk 1

● bO' { km/s km/s /ªsf cfotfsf/ sfuhsf 6j' m| fx¿ lngx' f];\ . Pp6f /ªsf
sfuhsf 6j\ |mfx¿df ʻ+ʼ lrxg\ n]Vg'xf;] \ . bf];|f] /ªsf sfuhsf 6'jm| fx¿df ʻ–ʼ

lrxg\ n]Vgx' f];\ .

● k0" ff{ªs\ sf] ;d"xaf6 sg' } bO' c{ f]6f ;ªV\ of lngx' f;] \ / of]ukmn slt xG' 5 <
sfuhsf 6'j|mfx¿af6 x]g'x{ f];\ .

h:t} M (-5) + (+3) 5 cf6] f ʻ−ʼ
3 cf6] f ʻ+ʼ
– – – – –
+ + +

● ca ʻ+ʼ nl] vPsf] / ʻ−ʼ nl] vPsf] sfuhsf 6'jm| fx¿nfO{ hf]8L agfP/
x6fpg'xf;] \ .

– – – – –
+ + +

2 cf]6f ʻ−ʼ lrx\g ePsf sfuhsf 6j' |mfx¿ afsF L /x]sf 5g\ .
To;}n], (–5) + (+3) = –2 eof] .

ul0ft, sIff & 45

ljm| ofsnfk 2
;ªV\ of /v] fsf] dfWodaf6 k"0ff{ªs\ sf] hf]8
tn lbOPsf ;ªVof /v] fsf cjnfs] g u/L ;dx" df 5nkmn ugx{' f];\ M
-s_ (+2) + (+4) = ?

-c_ ;ª\Vof /]vf agfpgx' f];\ .

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

-cf_ pbu\ d laGb' -zG" o_ af6 @ PsfO bfofF hfg'xf;] \ .

+2

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

-O_ ca pSt laGba' f6 k'gM 4 PsfO bfofFtkm{ hfg'xf;] \ .

–4 –3 –2 –1 0 1 2 3 4 5 6 7

-O{_ s'g ;ª\Vofdf k'Uge' of] < l6kf6] ug'x{ f;] \ .
ctM (+2) + (+4) = (6) xG' 5 .
-v_ (+6) + (–2) = ?
pb\ud laGb'af6 6 PsfO bfofF / ;f] laGba' f6 2 PsfO afofF kms{bf s'g laGb'df
k'luG5 < ;ªV\ of /]vf agfP/ l6kf]6 ug'{xf];\ .
-u_ (+2) + (–7) = ?
pbu\ d laGb'af6 2 PsfO bfofF / ;f] laGb'af6 7 PsfO afofF kmsb{ f sg' laGb'df
k'luG5 < ;ªV\ of /]vf agfP/ l6kf6] ugx'{ f;] \ .
-3_ (–3) + (–5) = ?
pbu\ d laGba' f6 3 PsfO afofF / ;f] laGba' f6 k'gM 5 PsfO afofF g} hfbF f sg'
laGbd' f k'luG5 < ;ª\Vof /]vfdf bv] fpgx' f];\ .

46 ul0ft, sIff &

k"0ffª{ s\ sf] hf]8sf lgodx¿ (Properties of Addition of Integers)
1. aGbL lgod (Closure Property)
k"0ffª{ \ssf] ;dx" Z = {..., –3, –2, –1, 0, 1, 2, 3, ...} af6 sg' } b'O{cf6] f ;ª\Vofx¿
lngx' f;] \ . pSt ;ª\Vofx¿sf] ofu] kmn lgsfNgx' f;] \ . glthf s] cfp5F < 5nkmn
ugx{' f];\ .

h:t} M 0 + 1 = 1

–2 + 1 = –1
–2 – 3 = –5

s'g} klg bO' {cf6] f k"0ff{ª\ssf] ofu] kmn k0" ff{ª\s g} x'G5 .

olb a / b s'g} bO' {cf6] f k"0ff{ª\s eP a + b klg k"0ff{ª\s g} xG' 5 . o;nfO{
hf8] sf] aGbL lgod elgG5 .

2. jm| d ljlgdo lgod (Commutative Law)
k0" ff{ª\ssf] ;dx" af6 s'g} b'O{cf]6f ;ªV\ of lngx' f;] \ . pSt ;ªV\ ofsf] j|md kl/jt{g u/L
of]ukmn lgsfNg'xf];\ . glthf s] cfp5F , 5nkmn ug'x{ f;] \ .

h:t} M 2 + 3 = 3 + 2 = 5

–1 + 1 = 1 – 1 = 0
–2 + (–3) = (–3) + (–2) = –5

k"0ff{ª\sx¿nfO{ h'g;'s} jm| ddf /fv/] ofu] kmn lgsfNbf klg kl/0ffd Pp6} k"0ffª{ s\
cfp5F .

s'g} b'Oc{ f]6f k0" ff{ªs\ x¿ a / b eP a + b = b + a x'G5 . o;nfO{ hf8] sf]
j|md ljlgdo lgod elgG5 .

ul0ft, sIff & 47

3. ;ª3\ Lo lgod (Associative Property)
k"0ffª{ s\ sf] ;d"xaf6 sg' } klg tLgcf6] f ;ª\Vof lngx' f;] \ . tL ;ªV\ ofx¿nfO{ h'g;'s}
j|mddf /fv/] klxnf b'O{cf6] f k"0ffª{ \snfO{ hf]8/] cfPsf] hf]8kmndf t;] f| ] k"0ff{ª\s
hf8] g\ x' f];\ . glthf s] cfp5F , 5nkmn ugx{' f];\ .

h:t}M – 3, –2 / –5 df

[(–3) + (–2)] + (–5) = (–3) + [(–2) + (–5)]
– 5 – 5 = – 3 – 7
⸫ –10 = –10

olb a, b / c s'g} k0" ff{ªs\ eP (a + b) + c = a + (b + c) x'G5, o;nfO{
hf8] sf] ;ª3\ Lo lgod elgG5 .

4. hf8] sf] ljk/Lt (Additive Inverse)

–5 + (+5) = ?

(+2) + (–2) = ? 5nkmn ug'{xf;] \ .
sg' } b'O{cf6] f k"0ffª{ \sx¿ hf]8b\ f of]ukmn zG" o (0) xG' 5 eg] ltgLx¿ Ps csfs{ f]
hf]8sf] ljk/Lt x'G5g\ .

sg' } klg k0" ffª{ \s a sf nflu C0ffTds k"0ff{ªs\ (–a) xG' 5, hxfF, a + (–a)
= (–a) + a = 0 xG' 5 . a / –a Ps csfs{ f hf8] sf] ljk/Lt xg' \ .

pbfx/0f 1

–3 sf] 4 PsfO afofF kg{] k0" ff{ª\s n]Vgx' f;] \ .
;dfwfg
oxf,F

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

–3 af6 4 PsfO afofF kg{] k"0ffª{ s\ (–7) xf] .

48 ul0ft, sIff &

pbfx/0f 2

;ªV\ of /v] fsf] k|of]u u/L hf]8 ug{x' f];\ M

-s_ (+4) + (+6)

;dfwfg +10
oxf,F
+4 +6

–2 –1 0 1 2 3 4 5 6 7 8 9 10 11

ctM (+4) + (+6) = +10 xG' 5 .

-v_ (–3) + (–4)

–7

–4 –3

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4

ctM (–3) + (–4)

= –3 –4

= –7 x'G5 .

pbfx/0f 3

k"0ffª{ \sx¿ (–7), (–2) / (+6) af6 ;ª3\ Lo lgodsf] k/LIf0f ugx'{ f;] \ M
;dfwfg

oxfF,

(–7) + (–2) + (+6)
= [(–7) + (–2)] + (+6)
= –9 + 6
= –3

km]l/,

(–7) + [(–2) + (+6)]
= (–7) + (+4)
= –7 + 4
= –3

ctM [(–7) + (–2)] + (+6) = (–7) + [(–2) + (+6)] = –3

ul0ft, sIff & 49

3.2 k"0ffª{ s\ sf] 36fp (Subtraction of Integers)

lj|mofsnfk 3

/fd a;kfs{af6 20 km kj" {df /x]sf] :yfg C df uP . kml] / pxL af6f] x'Fb} :yfg C
af6 13 km klZrddf kg]{ :yfg B df kmls{P .

olb a;kfs{nfO{ pbu\ dlaGb,' k"j{lt/sf] b/' LnfO{ wgfTds dfg lng] xf] eg] /fd

a;kfs{af6 slt lsnfl] d6/ 6f9f 5g\ < lbOPsf] ;ªV\ of /]vfsf] cjnfs] g u/L

5nkmn ug'x{ f;] \ . a;kfs{ kj" {
klZrd
B C
lj|mofsnfk 4

tn lbOPsf ;ª\Vof /]vfaf6 k"0ff{ªs\ sf 36fpsf pbfx/0fsf] cWoog u/L 5nkmn

ugx{' f];\ M

1. (+5) – (+2) = ?

zG" o (0) af6 5 PsfO bfofF uP/ kg' M 2 PsfO afofF kms{bf s'g laGbd' f
k'luG5 < ;ª\Vof /v] faf6 l6kf6] ug'x{ f];\ .

+5
–2

–4 –3 –2 –1 0 1 2 3 4 5 6 7

+3

3 PsfO bfofF cyf{t\ (+3) df k'luG5 .
ctM (+5) – (+2) = (+3) x'G5 .

2. (+3) – (+5) = ?

pbu\ d laGb' (0) af6 3 PsfO bfofF hfg'xf];\ . k'gM pSt laGba' f6 5 PsfO afofFtkm{
kmsg{ x' f];\ . sg' laGbd' f kl' uG5 < ;ª\Vof /]vf agfP/ 5nkmn ug'x{ f;] \ .

–5
–2 +3

–5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

50 ul0ft, sIff &

2 PsfO afofF cyft{ \ (–2) df k'luG5 .
ctM (+3) – (+5) = (–2) xG' 5 .

pbfx/0f 4
-s_ (–4) – (–5) nfO{ ;ª\Vof /v] fdf bv] fpg'xf;] \ M
;dfwfg
oxf,F (–4) – (–5)

= (–4) +5
=1

+5
–4 +1
–6 –5 –4 –3 –2 –1 0 1 2 3 4

ctM (–4) – (–5) = +1 xG' 5 .

-v_ (–2) – (+5) nfO{ ;ªV\ of /]vfdf bv] fpg'xf;] \ M

;dfwfg

oxf, (–2) – (+5)

= –2 – 5

= –7

–7 –2
–5

–8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3

ctM (–2) – (+5) = –7 xG' 5 .

ul0ft, sIff & 51

pbfx/0f 5
;/n ug'x{ f];\ M
(+250) + (–275) – (+148) + (+207) – (175)
;dfwfg

oxfF (+250) + (–275) – (+148) + (+207) – (175)

= 250 – 275 – 148 + 207 – 175
= (–25) + 59 – 175
= –25 + 59 – 175
= 34 – 175
= –141

3.3 k0" ffª{ s\ sf] lg/kI] fdfg (Absolute Value of Integers)

tn lbOPsf] ;ªV\ of /]vfsf] cjnf]sg u/L 5nkmn ugx{' f];\ M

B a;kfs{ A

–4km –3km –2km –1km 0 1km 2km 3km 4km

oxfF pb\ud laGb' a;kfs{ xf] . a;kfsa{ f6 4 km bfofF :yfg A 5 eg] 4 km afofF
:yfg B 5 .

:yfg A bl] v B ;Ddsf] b'/L slt 5 <

s] 4km + (–4) km = 0 x'G5, 5nkmn ug'{xf;] \ .

:yfg A b]lv B ;Ddsf] b/' L 4 km + 4 km = 8 km x'G5 . b/' L slxNo} klg C0ffTds
x'bF }g . To;n} ] –4 / +4 bj' s} f] lg/kI] f dfg 4 xG' 5 . –4 / 4 ljd'v k0" ffª{ s\ xg' \ .

s'g} klg k"0ffª{ \ssf] wgfTds ;fª\lVos dfgnfO{ lg/kI] fdfg elgG5 .
To;n} ,] |+a| = |–a| = a xG' 5 .

sg' } k"0ffª{ s\ ;ª\Vof /]vfsf] pb\ud laGb' z"Goaf6 hlt b'/Ldf 5 l7s Tolt g} b/' Ldf
/xs] f] csf]{ ljk/Lt lbzfsf] k"0ff{ªs\ nfO{ Tof] k0" ffª{ \ssf] ljd'v elgG5 .

52 ul0ft, sIff &

cEof; 3.1

1. tn lbOPsf k"0ff{ª\sx¿sf] 8 PsfO bfofF kg{] k0" ffª{ \s n]Vg'xf;] \ M

-s_ (–2) -v_ (–6) -u_ 0 -3_ (+3) -ª_ (+5)
2. tn lbOPsf k0" ffª{ \sx¿sf] 8 PsfO afofF kg]{ k0" ffª{ s\ nV] gx' f;] \ M

-s_ (–3) -v_ (–4) -u_ 0 -3_ (+2) -ª_ (+7)
3. tn lbOPsf k"0ff{ª\sx¿sf] ljdv' k"0ffª{ s\ n]Vgx' f;] \ M

-s_ (+3) -v_ (+4) -u_ (–8) -3_ 0 -ª_ (–2)
4. tn lbOPsf k0" ff{ª\sx¿sf] lg/kI] f dfg n]Vgx' f;] \ M

-s_ |+10| -v_ |–6| -u_ |–5| -3_ |+4| -ª_ |–9|
5. ;ªV\ of /v] fsf] ko| f]u u/L of]ukmn lgsfNgx' f;] \ M

-s_ (+3) + (+4) -v_ (–4) + (–3)

-u_ (+5) + (–2) -3_ (–5) + (+2)

6. ;ª\Vof /v] fsf] k|ofu] u/L 36fpg'xf];\ M

-s_ (–8) – (–3) -v_ (+9) – (–4)

-u_ (+4) – (+5) -3_ (+7) – (+2)

7. jm| d ljlgod lgodsf] ko| f]u u/L bj' } tl/sfn] tn lbOPsf k"0ff{ª\sx¿sf]
of]ukmn lgsfNg'xf];\ M

-s_ (+3) / (+6) -v_ (+4) / (–3)

-u_ (+5) / (–3) -3_ (–3) / (–1)

8. tn lbOPsf k0" ffª{ \sx¿sf] ;ª3\ Lo lgod k|of]u u/L bj' } tl/sfn] ofu] kmn
lgsfNg'xf];\ M

-s_ (+2), (–3) / (–5) -v_ (+4), (–3) / (+6)

-u_ (–5), (+4) / (0) -3_ (–2), (–5) / (+8)

9. (+9) / o;sf] ljdv' k0" ffª{ \ssf] ofu] kmn slt xG' 5, nV] g'xf;] \ .

10. (+30) df slt hf8] \bf (–30) x'G5, nV] g'xf;] \ .

ul0ft, sIff & 53

11. (–25) af6 slt 36fpbF f (–20) x'G5, nV] gx' f;] \ .

12. s'g} b'Oc{ f]6f k0" ffª{ s\ x¿sf] ofu] kmn (–115) 5 . tLdWo] Pp6f k0" ffª{ s\ 175
eP csf]{ k0" ff{ªs\ kQf nufpgx' f;] \ .

13. bO'\ {cf6] f a;x¿ Ps} :yfgaf6 Ps} ;dodf 5'65] g\ . Pp6f a;n] 125 km kj" {
ofqf u¥of] / csf{] a;n] 120 km klZrd ofqf u¥of] . tL b'O{ a;n] kf/ u/s] f]
hDdf b/' L kQf nufpg'xf];\ .

14. ;/n ugx{' f;] \ M

-s_ (–30) – (–40) – (–20) + (+2)

-v_ (+75) – (–14) – (–10) + (+1)

-u_ (–40) + (–25) + (+60) + (–5)

-3_ (–30) – (–40) – (–20) – (–10)

15. tn lbOPsf b'j} tflnsfx¿af6 k|To]s kª\lSt, kT| os] nx/ / ljs0f{x¿sf]
ofu] kmn lgsfNg'xf];\ . s'g tflnsfsf] glthfdf kªl\ St, nx/ / ljs0f{sf]
of]ukmn Pp6} cfpF5, n]Vg'xf];\ .

tflnsf 1 –1 tflnsf 2
–2
–5 3 –4 1 –10 0
–5 7 –4 –3 –2
0 –3 –6 4 –7

pQ/

1. -s_ +6 -v_ +2 -u_ +8 -3_ +11 -ª_ +13

2. -s_ –11 -v_ –12 -u_ –8 -3_ –6 -ª_ –1

3. -s_ –3 -v_ –4 -u_ +8 -3_ ljd'v xF'b}g -ª_ +2

4. -s_ 10 -v_ 6 -u_ 5 -3_ 4 -ª_ 9

5-8. lzIfsnfO{ bv] fpgx' f];\ . 9. 0 10. –60 11. –5

12. –290 13. 245 km 14. -s_ +32 -v_ +100 -u_ –10

-3_ +40 15. lzIfsnfO{ bv] fpg'xf;] \ .

54 ul0ft, sIff &

3.4 k"0ffª{ \ssf] u'0fg (Multiplication of Integers)

tnsf] cj:yfsf] cjnf]sg u/L 5nkmn ugx{' f;] \ M

kj" {

klZrdaf6 k"j{tkm{ 40 km k|lt 306fsf b/n] ul' 8/x]sf] a; 8 306fdf slt b/' L
kf/ unf{ <

8 306fdf a;n] kf/ u/]sf] b/' L = 8 × 40 km

= 320 km

lj|mofsnfk 1

tn lbOPsf] u'0fg tflnsf k"/f ugx'{ f;] \ / ;f]lwPsf ;ªVofx¿sf] u'0fgkmn slt
x'G5 < 5nkmn u/L kQf nufpg'xf;] \ .

×3 2 1 0 –1 –2 –3
39 6 –3 0 3 6 9
26 4 –2 0 2 4 6
13 2 –1 0 1 2 3
0
–1
–2
–3

-s_ (–3) × (–2) =
-v_ (+2) × (–1) =
-u_ (+1) × 0 =

;dfg lrx\g wgfTds k"0ff{ª\s × wgfTds k0" ff{ªs\ = wgfTds k"0ffª{ \s
ePsf] C0ffTds k0" ff{ª\s × C0ffTds k"0ff{ª\s = wgfTds k"0ff{ªs\

ljk/Lt lrxg\ wgfTds k"0ff{ªs\ × C0ffTds k0" ffª{ \s = C0ffTds k"0ff{ªs\
ePsf] C0ffTds k0" ffª{ s\ × wgfTds k"0ffª{ s\ = C0ffTds k"0ffª{ \s

ul0ft, sIff & 55

ljm| ofsnfk 2 -;ª\Vof /]vfsf] k|ofu] af6 u'0fg_

tnsf ;ª\Vof /]vfsf] ko| f]uaf6 ul/Psf ;ª\Vofx¿sf] u0' fglj|mof cjnfs] g u/L
5nkmn ug'{xf];\ M

-s_ (–2) × 3 = ? –6
–2 –2
–2

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4

pbu\ d laGb'af6 afofFtkm{ 2 PsfO 3 k6s hfFbf sg' laGb'df kl' uG5, 5nkmn
ug'{xf;] \ M

(–2) × 3
= –6

ctM (–2) × 3 = –6

-v_ 2 × (–3) = ?

pb\ud laGba' f6 afoftF km{ 3 PsfO 2 k6s hfgx' f];\ . s'g laGbd' f k'luG5 < ;ªVof

/v] f agfO{ l6kf]6 ug'x{ f;] \ . –6

–3 –3

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4

oxfF,

2 × (–3)
= –6

ctM 2 × (–3) = (–6) x'G5 .

56 ul0ft, sIff &

-u_ (–2) × (–3) = ?

(–2) × 3 n] lj|mofsnfk 3 -s_ df h:t} pb\ud laGb'af6 afoftF km{ 2 PsfO 3 k6s
hfFbf k'lug] laGb' hgfpF5 . 3 sf] cufl8sf] '–' lrx\gn] klxns] f] lbzfsf] ljk/Lt
lbzfdf hfg] eGg] ae' mfpF5 . To;}n] pb\ud laGb'af6 (–2) × 3 n] lbg] laGb;' Dd
slt PsfO x'G5, uGgx' f;] \ /pb\ud laGba' f6 Tolt g} PsfO bfoflF t/ hfg'xf];\ . sg'
laGb'df k'luG5 < ;ªVof /v] fdf b]vfpgx' f];\ .

–7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7

oxfF, (–2) × (–3)

=6

ctM (–2) × (–3) = 6 xG' 5 .

k"0ffª{ \ssf] u'0fgsf lgodx¿ (Properties of Multiplication of Integers)

1. aGbL lgod (Closure Property)
k0" ff{ªs\ sf] ;dx" af6 sg' } bO' { ;ª\Vofx¿ lngx' f];\ . pSt ;ª\Vofx¿sf] u0' fgkmn
lgsfNg'xf;] \ . glthf s] cfpF5, 5nkmn ug{x' f];\ .

h:t} M (–5) × (+4) = (–20)

(–4) × (–2) = (+8)
1×0=0

sg' } klg b'Oc{ f6] f k"0ffª{ \ssf] u'0fgkmn klg k0" ffª{ s\ g} xG' 5 . o;nfO{ u'0fgsf]
aGbL lgod elgG5 .
olb a × b b'O{ k0" ff{ªs\ x¿ eP a × b klg k"0ff{ªs\ xG' 5 .

2. jm| d ljlgdo lgod (Commutative Property)
k0" ffª{ \ssf] ;dx" af6 s'g} b'O{ ;ªV\ ofx¿ lng'xf];\ . tL ;ªV\ ofnfO{ jm| d kl/jt{g u/L
u'0fgkmn lgsfNgx' f];\ . glthf s] cfpF5, 5nkmn ug{'xf;] \ .

h:t} M 3 × 2 = 2 × 3 = 6

(–8) × (+3) = (+3) × (–8) = –24
1×0=0×1=0

ul0ft, sIff & 57

s'g} klg b'Oc{ f]6f k"0ffª{ s\ sf] u'0fgkmn ltgLx¿sf] :yfg abNbf xg' ] u0' fgkmn;uF
a/fa/ xG' 5, o;nfO{ u'0fgsf] jm| d ljlgdo lgod elgG5 .

olb a / b bO' c{ f6] f k0" ff{ª\sx¿ 5g\ eg] a × b = b × a xG' 5 .

3. ;ª3\ Lo lgod (Associative Property)
s'g} tLgcf6] f k0" ffª{ snfO{ km/s km/s j|mddf /fv]/ klxnf b'Oc{ f]6fsf] u'0fgkmndf
t;] f| n] ] u'0fg ubf{ u0' fgkmn s] cfpF5, 5nkmn ug{x' f;] \ .

h:t} M tLgcf6] f k0" ffª{ s\ x¿ 2, 3 / –4 df,

[2 × 3] × (–4) = 2 × [3 × (–4)]
6 × (–4) = 2 × (–12)

⸫ –24 = –24

sg' } tLgcf]6f k"0ff{ªsnfO{ hg' ;'s} j|mddf /fv]/ klxnf bO' c{ f6] fsf] u'0fgkmndf
t;] |fn] ] u0' fg ubf{ u0' fgkmn a/fa/ xG' 5 .

olb a, b / c tLgcf6] f k0" ffª{ x\ ¿ x'g\ eg],
(a × b) × c = a × (b × c) x'G5 .

4. kb ljR5b] g lgod (Distributive Property)

h:t} M (+6), (+3) / (–2) df,

+6 [(+3) + (–2)] = (+6) × (+3) + (+6) × (–2)

cyjf +6 (+1) = 18 – 12

cyjf +6 = +6 olb a, b / c tLgcf]6f k"0ffª{ \sx¿ x'g\ eg]
a(b + c) = a × b + a × c x'G5 .

5. 1 sf] u0' fg lgod (Multiplicative Property of 1)

(–5) × 1 = (–5)

1 × (+6) = +6 olb a Pp6f k"0ff{ª\s xf] eg] a × (+1) = (+1) × a = a x'G5 .

6. 0 sf] u0' fg lgod (Multiplicative Property of Zero)

2×0=0 olb a Pp6f k"0ffª{ \s xf] eg] a × 0 = 0 × a = 0 xG' 5 .
0 × 2 = 0

58 ul0ft, sIff &

pbfx/0f 1

;ªV\ of /v] fsf] k|ofu] u/L (–4) × 3 sf] u0' fg ugx{' f];\ M
;dfwfg

oxfF, –12
–4 –4
–4

–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3

ctM (–4) × 3 = (–12)

pbfx/0f 2
u'0fg ug{'xf];\ M (+12) × (–8) × (+2)
;dfwfg

oxfF, (+12) × (–8) × (+2)

= (–96) × (+2)

= –192

pbfx/0f 3
u0' fgsf] ;ª\3Lo lgod k|of]u u/L (+5), (+6) / (–7) nfO{ bj' } tl/sfn] u'0fg ug'x{ f;] \ M
;dfwfg

oxf,F (+5) × (+6) × (–7)

= [(+5) × (+6)] × (–7)
= (+30) × (–7)
= (–210)

km]l/, (+5) × [(+6) × (–7)]

= (+5) × (–42)
= (–210)
⸫ [(+5) × (6)] × (–7) = (+5) × [(6) × (–7)] = (–210)

ul0ft, sIff & 59

pbfx/0f 4
u'0fgsf] kb ljR5]bg lgod ko| f]u u/L ;/n ug{'xf];\ M
-s_ (–5) × [(+24) – (–6)]
;dfwfg
oxf,F (–5) × [(+24) – (–6)]

= (–5) × (+24) – (–5) × (–6)
= (–120) – (+30)
= –120 – 30
= –150
⸫ [(–5) + [(+24) – (–6)] = –150

3.5 k0" ff{ªs\ sf] efu (Division of Integers)

tn lbOPsf pbfx/0f cjnfs] g u/L 5nkmn ugx{' f;] \ M

(–8) × (–4) = 32
32 ÷ (–8) = ?
32 ÷ (–4) = ?

32 ÷ (–8) = (–4) x'G5 / 32 ÷ (–4) = (–8) x'G5 .

;dfg lrx\g wgfTds k"0ff{ª\s ÷ wgfTds k"0ff{ª\s = wgfTds k"0ff{ª\s
ePsf] C0ffTds k"0ffª{ \s ÷ C0ffTds k0" ff{ªs\ = wgfTds k"0ffª{ s\
ljk/Lt lrx\g wgfTds k0" ff{ª\s ÷ C0ffTds k"0ffª{ \s = C0ffTds k"0ffª{ \s
ePsf] C0ffTds k0" ff{ª\s ÷ wgfTds k0" ff{ªs\ = C0ffTds k"0ffª{ \s

60 ul0ft, sIff &

cEof; 3.2
1. vfnL 7fpdF f pko'St ;ª\Vof eg'x{ f];\ M

-s_ ÷ (–6) = 4
-v_ 81 ÷ = (–9)
-u_ 19 × = 0
-3_ –20 × = –20
-ª_ + (–45) = 1

2. ;ªV\ of /v] fsf] ko| fu] u/L u'0fg ugx'{ f];\ M -ª_ (+5) × (–4)
-s_ (+3) × (+2) -v_ (–5) × (+3)
-u_ (+2) × (–6) -3_ (–4) × (–3)

3. u'0fgsf] ;ª3\ Lo lgod ko| fu] u/L bj' } tl/sfn] u0' fgkmn lgsfNg'xf];\ M

-s_ (+3) × (+4) × (+5) -v_ (+7) × (–5) × (–3)

-u_ (–2) × (–2) × (–2) -3_ (+4) × (+8) × (–5)

4. u'0fgsf] kb ljR5]bg lgod ko| fu] u/L ;/n ugx'{ f];\ M
-s_ (+6) × [(–8) + (+30)]
-v_ (–9) × [(+24) – (–6)]
-u_ (+7) × [(–12) – (+8)]
-3_ (–8) × [(–3) + (–5)]
5. efukmn lgsfNg'xf];\ M
-s_ (+36) ÷ (+6) -v_ (–45) ÷ (+5)
-u_ (+54) ÷ (–6) -3_ (–95) ÷ (–1)

6. lbOPsf ;ª\Vofx¿sf] ;/n ugx'{ f;] \ M
-s_ [(+7) × (+8) × (–6)]÷ (–3)
-v_ [(+12) × (–8)] ÷ [(+2) × (–1)]

ul0ft, sIff & 61

-u_ [(+6) × (+4)] ÷ [(–3) × (–2)]
-3_ (+5) × (–4) × (–8) × (–3)
7. b'Oc{ f]6f k0" ff{ªs\ sf] u'0fgkmn (+63) 5 . Pp6f k0" ffª{ s\ (+7) eP csf]{ k"0ff{ªs

kQf nufpg'xf];\ .
8. (–5) nfO{ sltn] u'0fg ubf{ u0' fgkmn (+90) xG' 5, kQf nufpg'xf];\ .
9. (+56) nfO{ sltn] efu ubf{ efukmn (+7) xG' 5, kQf nufpg'xf;] \ .
10. u'0fgkmn (–144) agfpg (–12) nfO{ sltn] u0' fg ugk{' b{5 <
11. u0' fgkmn (–169) agfpg (–13) nfO{ sltn] u'0fg ug{k' b5{ <

12. Pp6f xflh/Lhjfkm kl| tofl] utfdf k|To]s ;xL pQ/sf nflu (+5), k|Tos] unt
pQ/sf nflu (–2) / pQ/ eGg g;sd] f (0) lbg] lgod agfOPsf] /x]5 .
-s_ ;d"x A n] 4 cf]6f ;xL hjfkm / 5 cf]6f unt hjfkm lbP5 eg] hDdf
slt cªs\ k|fKt u/]5 <
-v_ ;d"x B n] 5 cf]6f ;xL hjfkm / 5 cf]6f g} unt hjfkm lbP5 . hDdf
slt cªs\ kf| Kt u/]5 <
-u_ s'g ;dx" n] a9L cª\s kf| Kt u/]5g\ < sltn] k|fKt u/]5g,\ kQf nufpgx' f;] \ .

13. Pp6f xflh/Lhjfkm kl| tofl] utfdf kT| o]s ;xL pQ/sf nflu (+3) / kT| o]s
unt pQ/sf nflu (–2) lbg] lgod agfOPsf] 5 .
-s_ ;d"x A n] hDdf 18 cªs\ k|fKt u/5] . h;df 12 cf]6f kZ| gsf] unt
pQ/ lbP5 eg] sltcf6] f kZ| gsf] ;xL hjfkm lbP5, kQf nufpgx' f;] \ .
-v_ ;d"x B n] hDdf (–5) cª\s k|fKt u/5] . h;df 7 cf]6f k|Zgsf] unt
pQ/ lbP5 eg] sltcf6] f kZ| gsf] ;xL hjfkm lbP5, kQf nufpg'xf];\ .
-u_ s'g ;d"xn] w]/}cf6] f kZ| gx¿sf] ;xL hjfkm lbP5 < sltn] lbP5, kQf
nufpg'xf];\ .

62 ul0ft, sIff &

kl/of]hgf sfo{

tn lbOPsf k0" ffª{ s\ x¿larsf] ul0ftLo lj|mofnfO{ ;ªV\ of/v] fdf bv] fpgx' f];\
/ sIffdf k:| tt' ug{x' f;] \ M

1. (+3) × (+2)

2. (+3) × (–2)

3. (–3) × (+2)
4. (–3) × (–2)

pQ/

1. -s_ –24 -v_ –9 -u_ 0 -3_ +1 -ª_ +46

2-4. lzIfsnfO{ bv] fpgx' f;] \ .

5. -s_ +6 -v_ –9 -u_ –9 -3_ +95

6. -s_ +112 -v_ +48 -u_ +4 -3_ –480

7. +9 8. –18 9. +8 10. 12 11. 13

12. -s_ 10 -v_ 15 -u_ ;d"x B n] 5 cª\s a9L kf| Kt u/]5 .

13. -s_ 14 -v_ 3 -u_ ;dx" s, 11 cf6] f

ul0ft, sIff & 63

3.6 k0" ffª{ \ssf] ;/nLs/0f (Simplification of Integers)

k"0ff{ªs\ sf] hf]8, 36fp, u0' fg / efu;DaGwL ;d:ofx¿ cluNnf kf7x¿df 5nkmn
ul/;ss] f 5f}F . ca xfdL k0" ffª{ \ssf] ;/nLs/0fsf af/]df 5nkmn ug{] 5fF} .

ljm| ofsnfk 1

tn lbOPsf] ul0ftLo ;d:ofsf] cjnfs] gaf6 ;f]lwPsf kZ| gx¿df 5nkmn ugx{' f];\ .

Pp6f xflh/Lhjfkm k|ltofl] utfdf k|To]s ;xL pQ/sf nflu (+5) / k|Tos] unt pQ/
sf nflu (–3) lbg] lgod /x]5 .

-s_ ;d"x cGgk0" f{n] hDdf 104 cªs\ kf| Kt u/L ljhoL eP5 . h;df 25 cf]6f
kZ| gsf] ;xL hjfkm lbPsf /x]5g\ eg] sltcf6] f k|Zgsf] pQ/ unt u/5] g\ <

-v_ of] ;d:of ;dfwfg ug{ s'g sg' ul0ftLo lj|mofx¿ ugk{' nf{ <

oxf F lbOPsf] ;d:ofnfO{ ul0ftLo jfSodf nV] bf,

25 × (+5) + × (–3) = 104

unt pQ/ lbOPsf k|Zg ;ª\Vof = x -dfgf_}F

ca 25 × (+5) + x × (–3) = 104

cyjf 125 – 3x = 104

cyjf –3x = 104 – 125

cyjf –3x = -21

x = 7

7 cf]6f k|Zgsf] hjfkm unt u/5] .

pbfx/0f 1 [(+25) ÷ (–5) u/s] f]]
;/n ugx'{ f;] \ M

(+12) + (–5) + (+25) ÷ (–5) – (–6) × (+7)

= (+12) + (–5) + (–5) – (–6) × (+7)
= (+12) + (–5) + (–5) – (–42)
= (+7) + (–5) + 42
= +2 + 42
= 44

64 ul0ft, sIff &

● hf]8, 36fp tyf u'0fg ldl>t ;d:ofdf klxnf u0' fgsf] sfd ugk'{ b5{ .

● hf8] , 36fp tyf efu ljm| of ;dfjz] ePdf ;d:ofsf] ;dfwfg ubf{
;ae} Gbf klxnf efu ljm| of ug{k' b{5 .

● u0' fg / efu ;dfjz] ePsf ;d:ofdf klxnf efu ljm| of ug{] jf afofFaf6
bfoflF t/ ;/n ub{} hfFbf hg' lrx\g klxnf cfp5F , ToxL lj|mof klxnf
ugk'{ b5{ .

cEof; 3.3

1. ;/n ug{x' f;] \ M
-s_ (–6) × (–4) ÷ (+4) + (–5) – (–1)

-v_ (–15) ÷ (+5) × (–4) + (–10) – (+7)

-u_ (–12) + (+16) × (–27) ÷ (–9)

-3_ (–3) × (+16) – {(+12) ÷ (+6) + (–10)}

2. Pp6f ljBfnodf ePsf] xflh/Lhjfkm kl| tof]lutfdf k|To]s ;xL pQ/sf nflu
(+10) / kT| os] unt pQ/sf nflu (–5) lbg] lgod /x5] .

-s_ lgnf] ;bgn] 60 cª\s k|fKt u/]5 . h;df 2 cf]6f kZ| gsf] pQ/ unt
lbP5 eg] sltcf]6f kZ| gsf] ;xL hjfkm lbP5, kQf nufpg'xf;] \ .

-v_ kxn]F f] ;bgn] 20 cªs\ kf| Kt u/5] . h;df 4 cf6] f k|Zgsf] pQ/ unt
eg5] eg] sltcf6] f k|Zgsf] ;xL hjfkm lbP5, kQf nufpg'xf];\ .

-u_ s'g ;dx" n] w]/}cf6] f kZ| gsf] ;xL hjfkm lbP5 / pSt ;dx" n] yf/] }
k|Zgsf] ;xL hjfkm lbg] ;d"xn]eGbf sltcf6] f a9L kZ| gsf] ;xL hjfkm
lbP5, nV] g'xf];\ .

3. /fdsf] a}ªs\ vftfdf jz} fv dlxgfsf] ;'?df ?= 5000 lyof] . j}zfv 7 ut]
?= 30,000 tna p;sf] vftfdf hDdf eof] . jz} fv 9 ut] lah'nLsf] dx;'n
?= 945 pSt vftfaf6 ae' mfP . jz} fv 11 ut] kmf]gdf ?= 500 pSt vftfaf6
l/rfh{ u/]5g\ . jz} fv 15 ut] ABC vfB :6f/] nfO{ pSt vftfaf6 ?= 10,000
lt/5] g\ eg] ca pgsf] vftfdf slt /sd afsF L 5, kQf nufpgx' f;] \ .

ul0ft, sIff & 65

4. klxnf] k0" ff{ª\s bf;] |f] k"0ffª{ s\ sf] kfFr u0' ffeGbf 2 n] a9L 5 . olb klxnf]
k0" ffª{ s\ 77 5 eg] bf];f| ] k"0ffª{ s\ kQf nufpgx' f];\ .

5. Pp6f 1000 l Ifdtfsf] 6\ofª\sLdf 500 l kfgL 5 . o;df b'O{cf]6f wf/fx¿ h8fg
ul/Psf 5g\ . 1 ldg6] df wf/f A n] 25 l kfgL e5{ / wf/f B n] 15 l kfgL aflx/
kmfN5 . olb bj' } wf/fnfO{ 5 ldg6] ;Dd vf]lnof] eg] 6\ofªs\ Ldf slt ln6/ kfgL
xG' 5 <

kl/ofh] gf sfo{

xfd|f] b}lgs hLjgdf k"0ff{ªs\ sf] hf8] / 36fpsf] k|of]u ePsf sg' }
kfFrcf]6f pbfx/0fx¿ n]vL sIffdf k|:tt' \ ug'{xf;] \ .

pQ/ -v_ –5 -u_ +36 -3_ –40
-v_ 4 -u_ lgnf] ;bgn], 3 cf6] f
1. -s_ +2
2. -s_ 7 4. 15 5. 560 l
3. ?= 23555

66 ul0ft, sIff &

kf7 4 cfg'kflts ;ªV\ of

(Rational Number)

4.0 kg' /jnf]sg (Review)

k"0ffª{ \sx¿sf] ;dx" Z = {..., –3, –2, –1, 0, 1, 2, 3, ...} af6 s'g} b'O{ ;ª\Vofx¿
lng'xf;] \ . tL ;ªV\ ofx¿sf] hf8] , 36fp, u'0fg / efudWo] sg' s'g ljm| of ;Dej
xfn] f, 5nkmn ug{x' f];\ .

4.1 cfg'kflts ;ª\Vofsf] kl/ro (Introduction to Rational Number)

lj|mofsnfk 1

lbOPsf] cj:yfsf] cWoog ugx{' f];\ / ;ª\Vof /v] fsf cfwf/df ;f]lwPsf kZ| gdf
5nkmn ugx{' f];\ .

k"0ffª{ \ssf] ;d"xaf6 b'O{cf6] f ;ª\Vofx¿ 3 / –4 lnOPsf] 5 . logLx¿lar hf]8,
36fp, u'0fg / efu lj|mof ul/Psf] 5 .

3 + (–4) = –1

3 – (–4) = 3 + 4 = 7

3 × (–4) = –132 –4 –3 –2 –1 0 1 2 3 4 5 6
3 ÷ (–4) =
–4

-s_ s] bO' c{ f6] f k"0ffª{ s\ nfO{ hf]8b\ f, 36fpFbf / u0' fg ubf{ ;w}F k"0ff{ª\s g} x'G5 <

-v_ k0" ff{ªs\ ;ª\Vofx¿larsf] efukmn s] xf]nf <
3
-u_ s] –4 k"0ffª{ s\ sf] ;ªV\ of /v] fdf 5 <

s'g} klg bO' {cf6] f k0" ff{ª\s hf]8\bf, 36fpFbf, u0' fg ubf{ k0" ffª{ \sg} cfpF5 . t/
Pp6f k"0ffª{ \snfO{ csf]{ k"0ff{ªs\ n] efu ubf{ ;wF} k0" ff{ªs\ gxg' klg ;S5 .

hcf:gtk' } Mflt23s, 21;, 54ª\Vko"0fxff¿ª{ \sxxg' ¿\ xf]Ogg\ . logLx¿ a sf ¿kdf cfpF5g\ . o:tf ;ªVofx¿
b hgfOG5
. cfg'kflts ;ª\Vofsf] ;dx" nfO{ Q n]

Q = {..., –4, –3, –52, –2, –32, –1, –12, 0, 2 , 1 , 1, ... }
3 2

ul0ft, sIff & 67

sg' } klg ;ªV\ ofnfO{ a sf ¿kdf JoSt ug{ ;lsG5 eg] To:tf]
b
;ªV\ ofnfO{ cfg'kflts ;ª\Vof (Rational Number) elgG5 . hxfF a / b bj' }

k0" ffª{ s\ x¿ x'g\ / b ≠ 0 5 .

lj|mofsnfk 2
tn lbOPsf k|ZgnfO{ ;fyLx¿lar ;d"xdf 5nkmn u/L lgisif{ sIffdf k|:tt'
ug'x{ f];\ .

-s_ s] ;a} kf| s[lts ;ª\Vof cfgk' flts ;ª\Vof x'g\ <

-v_ s] ;a} k0" ff{ªs\ cfgk' flts ;ª\Vof xG' 5g\ <

;a} k0" ffª{ \sx¿ cfgk' flts ;ª\Vofx¿sf] ;d"xdf kg]{ ePsfn]
k0" ffª{ \sx¿sf] ;d"x cfgk' flts ;ª\Vofx¿sf] pko'St pk;d"x xf] .
To;n} ] Z ⸦ Q nV] g ;lsG5 .

cfgk' flts ;ªV\ ofsf ljzi] ftfx¿ (Properties of Rational Numbers)
tnsf cfg'kflts ;ª\Vofsf] ljzi] ftfsf] cjnfs] g u/L 5nkmn ug{x' f];\ .
1. PsfTds lgod (Identity Property)

hf8] sf] PsfTds lgod u'0fgsf] PsfTds lgod

1 + 0 = 0 + 1 = 1 1 × 1 = 1 × 1 = 1
2 2 2 2 2 2

–2 +0=0+ –2 = –2 –2 × 1 = 1 × –2 = –2
3 3 3 3 3 3

sg' } klg cfg'kflts ;ª\Vofdf zG" o sg' } klg cfg'kflts ;ªV\ ofnfO{ 1 n]
(0) hf8] b\ f cfpg] ;ª\Vof ToxL ;ªV\ of u0' fg ubf{ ToxL ;ª\Vof cfp5F . o;nfO{
xG' 5 . o;nfO{ hf]8sf] PsfTds lgod
elgG5 . u'0fgsf] PsfTds lgod elgG5 .

68 ul0ft, sIff &

2. ljk/Lt u0' f (Inverse Property)

hf]8sf] ljk/Lt u'0f u0' fgsf] ljk/Lt u'0f

–1 + 1 = 0 2 × 1 = 1
2

–1 + 1 = 0 2 × 3 =1
2 2 3 2

s'g} klg cfgk' flts ;ª\Vof a df –a sg' } klg cfg'kflts ;ªV\ of a nfO{
b b ab cfp5F b
nfO{ hf8] \bf zG" o cfpF5 eg] o;nfO{ n] u0' fg ubf{ 1 csfs{ f . To;}
a –a n] b
hf8] sf] ljk/Lt u0' f elgG5 . b / b a / a nfO{ Ps u0' fgsf]
Ps csfs{ f hf]8sf ljk/Lt . b
x'g\ ljk/Lt dflgG5 .

3. jm| d ljlgdo u'0f (Commutative Property)

hf8] sf] jm| d ljlgdo u'0f u0' fgsf] j|md ljlgdo u'0f

1 + 2 = 2 + 1 1 × 2 = 2 × 1
2 3 3 2 2 3 3 2

olb a / c cfgk' flts ;ª\Vofx¿ xg' \ eg,] olb a / c cfgk' flts ;ª\Vofx¿ x'g\ eg,]
b d b d
a c c a
b + d = d + b nfO{ hf]8sf] j|md lbajl×gddc o= cdel×gGab5n.fO{ u'0fgsf] j|md

ljlgdo elgG5 .

4. ;ª\3Lo lgod (Associative Property)

hf]8sf] ;ª3\ Lo lgod u0' fgsf] ;ª3\ Lo lgod

1 + 23+ 3 = 1 + 2 + 3 1 × 2 × 3 = 1 × 2 × 3
2 5 2 3 5 2 3 5 2 3 5

lgod, lgod,

olb a , c = e cfgk' flts ;ª\Vofx¿ xg' \ olb a = c = e cfgk' flts ;ª\Vofx¿
b d f b d f
eg,] xg' \ eg,]
e
a + c + e = a + c + e a × c × e = a × c × f
b d f b d f b d f b d

nfO{ hf]8sf] ;ª\3Lo lgod elgG5 . nfO{ u'0fgsf] ;ª3\ Lo lgod elgG5 .

ul0ft, sIff & 69

5. aGbL lgod (Closure Property)

hf8] sf] aGbL lgod u'0fgsf] aGbL lgod

;1122 ª+/\V32o23 fc=xfgf3] +6k' . 4flt=s76 ;ªV\ ofx¿ xg' \ . 1 / 2 cfgk' flts ;ª\Vofx¿ x'g\ .
klg cfgk' flts 2 3
1 2 1
2 × 3 = 3 klg cfg'kflts ;ªV\ of xf] .

b'\Oc{ f]6f cfg'kflts ;ª\Vofsf] b\'Oc{ f6] f cfgk' flts ;ª\Vofx¿sf]
of]ukmn klg cfg'kflts ;ªV\ of x'G5 . u0' fgkmn klg cfg'kflts ;ª\Vof g} x'G5 .
o;nfO{ hf8] sf] aGbL lgod elgG5 . o;nfO{ u0' fgsf] aGbL lgod elgG5 .

4.2 bzdnj / cfgk' flts ;ª\Vof (Decimal and Rational Number)

ljm| ofsnfk 3

tn tflnsfdf lbOPsf leGgnfO{ bzdnj ;ªV\ ofdf ¿kfGt/0f u/L vfnL 7fpF
eg'x{ f;] \ / lgisif{nfO{ 5nkmn ugx'{ f];\ .

jm| =;= leGg bzdnj ;ªV\ of

1 1 0.5
2

2 1 0.333
3

3 2 0.285714285714
7

4 2 ...
3

5 3 ...
10

6 5 ...
13

leGgnfO{ bzdnjdf ¿kfGt/0f ubf{ cGTo xg' ,] cGToxLg / k'g/fj[Q bzdnjdf
JoSt ug{ ;lsG5 .

70 ul0ft, sIff &

1. cGTo xg' ] bzdnj ;ª\Vof (Terminating Decimal)

tn lbOPsf pbfx/0fsf] cjnfs] g u/L 5nkmn ug{x' f;] \ M

1 = 0.25, 1 = 0.125, 1 = 1.5
4 8 3
1 3 3
4 , 8 , 2 nfO{ bzdnj ;ªs\ t] df ¿kfGt/0f ubf{ bzdnj k5fl8sf ;ª\Vofx¿

lglZrt :yfgdf cGTo ePsf 5g\ .

cfg'kflts ;ªV\ ofsf] x/n] cz+ nfO{ efu ubf{ efukmndf bzdnj k5fl8sf ;ª\Vofx¿sf]
cGTo xG' 5 eg] To:tf] ;ªV\ ofnfO{ cGTo xg' ] bzdnj ;ªV\ of elgG5 .

2. cGToxLg k'g/fjQ[ bzdnj ;ª\Vof (Non Terminating Recurring Decimal)

leGgx¿ 1 , 2 , 4 nfO{ bzdnj ;ª\Vofdf ¿kfGt/0f ugx{' f;] \ . glthfsf cfwf/df
3 9 11
s] lgisif{ lgl:sG5, 5nkmn ugx{' f;] \ .
1
3 = 0.3333...

2 = 0.2222...
9
4
11 = 0.363636...

2 = 0.285714285714...
7

0.333... nfO{ 0.3 klg nV] g ;lsG5 .

dflysf leGgnfO{ bzdnj ;ªst] df ¿kfGt/0f ubf{ bzdnjkl5sf ;ª\Vofx¿
cGTo ePsf 5g} g\ . Pp6} ;ª\Vof bf]xf]l//x]sf 5g\ . o:tf ;ª\VofnfO{ cGToxLg
kg' /fj[Q bzdnj ;ª\Vof elgG5 .

cGTo xg' ] / cGToxLg kg' /fjQ[ bzdnj ;ªV\ ofnfO{ cfg'kflts ;ª\Vof elgG5 .

gf]6 M olb cfg'kflts ;ª\Vofsf] x/df 2 cyjf 5 sf ckjTo{ /xs] f 5g\ eg]

Tof] ;ª\Vof cGTo xg' ] bzdnj ;ª\Vof x'G5 .
7
h:t} M 1 , 1 , 10 , 275, ...
2 5
olb cfgk' flts ;ªV\ ofsf] x/df 2 / 5 afxs] c¿ ;ªV\ of ePdf To:tf bzdnj
1 2 5
;ª\Vof cGToxLg kg' /fj[Q xG' 5g,\ h:t} M 3 , 3 , 7 , ...

ul0ft, sIff & 71

4.3 bzdnjnfO{ leGgdf ¿kfGt/0f
(Conversion of Decimal into Fraction)

I. cGTo x'g] bzdnj ;ªV\ ofnfO{ leGgdf ¿kfGt/0f

0.75 nfO{ leGgdf s;/L ¿kfGt/0f ug{ ;lsG5, 5nkmn ugx{' f;] \ .

0.75 eg]sf] ;of+zsf] 75 c+z eGg] a'lemG5 .
753
To;n} ] 100 4 xG' 5 . 0.75 df bzdnjkl5 b'Oc{ f]6f ;ª\Vofx¿ 5g\ .

cyf{t\ 3 xG' 5 . To;n} ] 0.75 nfO{ 100 n] u'0fg / efu ug{k' 5{ .
4

csf]{ tl/sf M 0.75 nfO{ leGgdf ¿kfGt/0f ubf{,

0.75 × 100
100
75
= 100

= 3
4

II. cGToxLg kg' /fj[Q bzdnjnfO{ leGgdf ¿kfGt/0f

0.3 nfO{ leGgdf s;/L ¿kfGt/0f ug{ ;lsG5 < 5nkmn ugx{' f];\ .

dfgfF}, x = 0.3

x = 0.33 ... (i) 0.3 df bzdnjkl5 Pp6}
;ªV\ of 3 dfq kg' /fjQ[ 5,
;dLs/0f (i) nfO{ 10 n] u0' ff ubf,{ To;n} ] 10 n] u0' ff ug'{kb{5 .

10x = 3.33 ... (ii)

ca ;dLs/0f (ii) af6 ;dLs/0f (i) 36fpFbf,

10x – x = 3.33... – 0.33...

cyjf 9x = 3
3
cyjf x = 19
3
ctM cyjf x =
1 .
0.3 = 3 xG' 5

72 ul0ft, sIff &

pbfx/0f 1

tn lbOPsf leGgx¿nfO{ bzdnjdf ¿kfGt/0f u/L cGTo x'g] / cGToxLg kg' /fjQ[ bzdnj
;ª\Vofx¿ 5'6\ofpg'xf;] \ .

-s_ 5 -v_ 2
8 3

;dfwfg

-s_ 5 0.625
8
8 50
= 0.625 – 48
20
0.625 cGTo x'g] bzdnj ;ª\Vof xf] . – 16
40
– 40
×

-v_ 2 0.666
3
3 20
= 0.666... – 18
20
= 0.6 ... – 18
20
0.6 cGToxLg kg' /fj[Q bzdnj ;ªV\ of xf] . – 18
2

pbfx/0f 2

tn lbOPsf bzdnj ;ªV\ ofnfO{ leGgdf ¿kfGt/0f ug{x' f;] \ M
-s_ 0.35 -v_ 0.41
;dfwfg

-s_ 0.35

= 0.35 × 100 = 35 = 7
100 100 20

ul0ft, sIff & 73

-v_ 0.41 (i)
dfgf}F, x = 0.41
x = 0.4141...

;dLs/0f (i) nfO{ 100 n] u0' fg ubf{,

100x = 41.4141... (ii)

ca ;dLs/0f (ii) af6 ;dLs/0f (i) 36fpbF f,

100x – x = 41.4141... – 0.4141...

cyjf 99x = 41

cyjf x = 41
99

ctM 0.41 = 41 x'G5 .
99

pbfx/0f 3

1 / 5 sf larsf s'g} b'O{cf]6f cfg'kflts ;ª\Vof lgsfNg'xf;] \ M
2 6

;dfwfg 5
1 6
2 / sf] larsf] cfgk' flts ;ªV\ of lgsfNg,

15 1 5
2+ 6 2 6
Pp6f ;ªV\ of = 2 / sf] cf;} t dfg lgsfNg] .

= 3 + 5 × 1 5 × 1 4 × 1 4 2
6 2 = 6 2 = 3 2 = 6 = 3

csf]{ ;ª\Vof = 1 / 2 sf] cf};t dfg
2 3

12
2+
= 2 3

= 3 + 4 × 1 = 7
6 2 12

1 / 5 sf lardf kg]{ s'g} b'O{ cfg'kflts ;ª\Vofx¿ 2 / 7 xg' \ .
2 6 3 12

74 ul0ft, sIff &

bzdnj ;ª\VofnfO{ tnsf] h:tf] rf6a{ f6 b]vfpg ;lsG5 M
bzdnj ;ªV\ of

cGTo xg' ] bzdnj ;ª\Vof cGTo gxg' ] bzdnj ;ª\Vof

cfg'kflts ;ªV\ of kg' /fj[Q xg' ] k'g/fj[Q gx'g]
bzdnj ;ªV\ of bzdnj ;ª\Vof

cEof; 4.1

1. tn lbOPsf leGgx¿nfO{ bzdnjdf ¿kfGt/0f u/L cGTo x'g] / cGToxLg kg' /fjQ[
bzdnj ;ª\Vofx¿ 56' o\ fpgx' f;] \ .

-s_ 12 -v_ 3 -u_ 2 -3_ 15 -ª_ 17
5 7 2 13

-r_ 55 -5_ 37 -h_ 25 -em_ 12
10 20 17 25

2. tn lbOPsf ;ª\Vofx¿sf] hf8] sf] ljk/Lt ;ªV\ of n]Vg'xf;] \ M

-s_ 25 -v_ –5 -u_ 22 -3_ 12 -ª_ –11
7 12 7 8

3. tn lbOPsf ;ªV\ ofx¿sf] u'0fgsf] ljk/Lt ;ªV\ of n]Vg'xf];\ M

-s_ 43 -v_ 25 -u_ –2 -3_ 22 -ª_ 1
10 3 12 8

4. tn lbOPsf bzdnj ;ª\VofnfO{ leGgdf ¿kfGt/0f ug'{xf];\ M

-s_ 0.5 -v_ 0.7 -u_ 0.24 -3_ 0.27

-ª_ 1.57 -r_ 2.35 -5_ 7.025

ul0ft, sIff & 75

5. 1 / 3 sf larsf sg' } b'O{cf6] f cfg'kflts ;ª\Vof n]Vgx' f;] \ .
2 4

6. 1 / 3 sf] larsf s'g} b'O{cf6] f cfg'kflts ;ªV\ of kQf nufpg'xf];\ .
3 4
7. tnsf k|Zgsf] pQ/ sf/0f;lxt n]Vg'xf];\ M

-s_ s] ;a} k|fsl[ ts ;ªV\ of cfg'kflts ;ª\Vof x'g\ <

-v_ s] ;a} k0" f{ ;ª\Vof cfgk' flts ;ª\Vof x'g\ <

-u_ s] ;a} k0" ffª{ \s cfgk' flts ;ª\Vof x'g\ <

-3_ s] z"Go (0) cfg'kflts ;ªV\ of xf] <
-ª_ s] ;a} cfgk' flts ;ªV\ of k"0ffª{ \s x'g\ <

kl/of]hgf sfo{

k|s[lts ;ªV\ of, k0" f{ ;ªV\ of, k0" ffª{ \s / cfgk' flts ;ª\Vofsf] ;DaGwnfO{
lrqaf6 bv] fpgx' f;] \ / sIffdf k:| t't ug{x' f;] \ .

pQ/
lzIfsnfO{ b]vfpgx' f;] \ .

76 ul0ft, sIff &

kf7 5 leGg / bzdnj

(Fraction and Decimal)

5.1 leGg (Fraction)

5.1.0 kg' /jnfs] g (Review)

tn lbOPsf leGg;DaGwL ;d:ofsf af/d] f ;dx" df 5nkmn u/L lx;fa ug'x{ f;] \ M

-s_ 51 + 1 -v_ 556 – 1 2
7 3

-u_ 6× 2 -3_ 12 ÷ 2
5 5

5.1.1 leGgsf] ;/nLs/0f (Simplification of Fractions)

ljm| ofsnfk 1

tn lbOPsf ul0ftLo ;d:ofnfO{ cWoog ug{'xf;] \ / ;fl] wPsf kZ| gdf 5nkmn
ug{'xf;] \ M

l;h{gfsf] dfl;s cfDbfgL ?= 24000 5 . pgn] Ps dlxgfdf cfDbfgLsf]
tLg efusf] Ps efu lzIffdf vr{ ul/g\ . To:t} cfwf efu vfgfdf vr{ ul/g\ eg],

-s_ pgn] hDdf cfDbfgLsf] slt efu vr{ ul/5g\ <

-v_ hDdf slt /sd vr{ ul/5g\ < 1
3
oxfF lzIffdf vr{ ePsf] cfDbfgLsf] efu = tLg efusf] Ps efu =
vfgfdf vr{ ePsf] cfDbfgLsf] efu
= cfwf efu = 1
2
hDdf vr{ = <

hDdf vr{ lgsfNg b'j} vr{ hf8] \g'kb{5 .

To;}n] hDdf vr{ = 1 + 1
3 2

= 1 × 2 + 1 × 3 c;dfg x/nfO{
3 2 2 3 ;dfg agfPsf]

= 2 + 3
6 6

ul0ft, sIff & 77

hDdf vr{ = 5 -n3'Qd kbdf n]Vbf_
6
ca hDdf vr{ /sd = cfDbfgLsf] 5 efu
6
= ?= 24000 × 5
6

= ?= 4000 × 5

= ?= 20,000

csf]{ tl/sf -gdg' f lrq0f ljlwaf6_

lzIffdf cfDbfgLsf] tLg efusf] Ps efu vr{ ePsfn] l;h{gfsf] cfDbfgLnfO{ tLg
a/fa/ efu nufpbF f,

l;h{gfsf] cfDbfgL = ?= 24,000

xxx

3x = 24000

cyjf x = 24000
cyjf x = 3

8000

ctM lzIffdf vr{ = ?= 8,000

km]l/ vfgfdf cfDbfgLsf] cfwf efu vr{ ePsfn,]

l;h{gfsf] cfDbfgL = ?= 24,000

yy

2y = 24000

cyjf y = 24000
cyjf y = 2

12000

ctM vfgfdf vr{ = ¿= 12,000

ctM hDdf vr{ = lzIffdf vr{ + vfgfdf vr{

= 8000 + 12000

= ?= 20,000
hDdf vr{ ?=20000
hDdf vrs{ f] efu = hDdf cfDbfgL = ?=24000 = 5
6

78 ul0ft, sIff &

pbfx/0f 1

(z23flGetful-gs_s_5 ]tf5gqfflqj/fBsaffnf] Fs;oLsª5\Vf]ofsqfI5sffglt7\ ed/gxf],]548<hgf ljBfyL{ 5g\ . tLdWo] tLg efusf] b'O{ efu

-v_ 5fqsf] ;ªV\ of slt /x5] , kQf nufpg'xf];\ .
;dfwfg

oxf F hDdf ljBfyL{ ;ª\Vof = 48 2
3
5fqfsf] ;ªV\ of = hDdf ljBfyL{sf] efu
= 48 ×
2
3
= 32

5fqsf] ;ª\Vof = hDdf ljBfyL{ ;ªV\ of − 5fqfsf] ;ªV\ of
= 48 − 32

= 16

ctM 5fqf 32 hgf / 5fq 16 hgf /x]5g\ .

csf{] tl/sf -gdg' f lrq0f ljlwaf6_

5fqfsf] ;ª\Vof tLg efusf] b'O{ efu,

48

5fqf 5fqf 5fqf

xxx

3x = 48

x = 48 5fqf = 2 efu
⸫ x = 3 5fq = 1 efu
16

5fqfsf] ;ª\Vof = 2 efu

= 2 × 16

= 32

5fqsf] ;ªV\ of = 48 – 32

= 16

ul0ft, sIff & 79

pbfx/0f 2

Pp6f cfwf/et" ljBfnodf ePsf hDdf ljBfyL{sf] 2 efu 5fq 5g\ . olb 5fqfx¿ 90
5
hgf 5g\ eg] 5fq slt hgf /x5] g\ < hDdf ljBfyL{ slt hgf /x]5g\ <

;dfwfg

5fq 5fqf

xxxxx

oxfF 3x = 90

x = 90
3

= 30

5fqsf] ;ª\Vof 2x = 30 × 2

= 60 hgf

hDdf ljBfyL{ ;ªV\ of = 5x = 5 × 30 = 150 hgf

pbfx/0f 3

k13l| j/0f;nG]bcLkfˆng] f16] heGdfulbvgdfPf5Pgp\ 6. f;sa}e]sGblsf gw5]]/g} s\ .s] pgss;fn;] vfyfLPx5¿gd,\ WkoQ] sf [knfnuf] p12gex' ff];u\,Mcdgn]
;dfwfg

oxfF s[kfn] vfPsf] ss] sf] efu = 1
2
1
cdgn] vfPsf] s]ssf] efu = 3

;GbLkn] vfPsf] s]ssf] efu = 1
6

ca tLgc} f]6f leGgsf] x/df ePsf cªs\ sf] n=;= lgsfNgx' f;] \ .

n=;= = 6

t'ngf ugs{ f nflu ;a} leGgsf
x/x¿ ;dfg xg' k' 5{ .

80 ul0ft, sIff &

ca ;a} leGgsf] x/ 6 agfpgx' f];\ .

1 = 1 × 3 = 3 2 sf ckjTo{x¿ 2, 4, 6, 8, ...
2 2 3 6 3 sf ckjTo{x¿ 3, 6, 9, ...
6 sf ckjTox{ ¿ 6, 12, ...
1 = 1 × 2 = 2 ;a}eGbf ;fgf] ;femf ckjTo{ = 6
3 3 2 6

1 = 1 × 1 = 1
6 6 1 6

oxf F 3 egs] f] 3 cf6] f 1 xf] . 2 egs] f] 2 cf6] f 1 xf] / 1 egs] f] 1 cf6] f 1 xf] .
6 6 6 6 6 6

To;}n] oL 3 cf]6f leGgdWo] 3 7'nf] leGg xf] .
6
1
⸫ 2 efu vfg] sk[ fn] ;a}eGbf w]/} s]s vfO5g\ .

cEof; 5.1

1. ;/n ugx{' f;] \ M

-s_ 11113 – 183 + 287 -v_ 1319 – 4 8 – 6151
9

-u_ 1 + 1 × 6 – 4 ÷ 21 -3_ 1 + 2 ÷ 3 × 4 + 5 – 2
2 3 2 5 2 3 5 5 9

2. 618 – 8 nfO{ 30 af6 36fP/ 141 hf8] g\ x' f];\ .

3. ;GbLksf] dfl;s cfDbfgL ?= 27,000 5 . pgn] 15u/e5] fgu\ vfhfdf vr{ u/5] g\ .
110efu sk8fdf 2 eg],
vr{ u/5] g\ / 5 oftfoftdf vr{

-s_ hDdf slt efu vr{ u/]5g\ <

-v_ slt efu art u/]5g\ <

-u_ slt ?lkofF art u/]5g,\ kQf nufpgx' f];\ .

4. /dfOnf] dn] fdf /d] fn] cfkm;" Fu ePsf] ?lkofsF f] 1 efu dgf]/~hgdf / 1
5 2
efu vfgfsf nflu vr{ ul/g\ . sg' zLif{sdf w]/} vr{ u/]sL /lx5g\, kQf

nufpg'xf;] \ .

ul0ft, sIff & 81

5. sIff 7 df ePsf 40 hgf ljBfyL{dWo] 1 efu ljBfyL{n] cª\u]|hL ljifo
dg k/fpF5g\ 2 5 ljBfyL{n] lj1fg
dg k/fpF5g\ . 5 ljBfyL{n] ul0ft ljifo dg k/fpF5g\ . afsF L
.

-s_ cª\uh|] L dg k/fpg] ljBfyL{sf] ;ª\Vof kQf nufpgx' f];\ .

-v_ ul0ft dg k/fpg] ljBfyL{sf] ;ª\Vof kQf nufpgx' f;] \ .

-u_ lj1fg dg k/fpg] ljBfyLn{ fO{ leGgsf ¿kdf nV] gx' f];\ .

6. /~h'nfO{ pgsf aa' fn] ?= 6,000 lbge' of] . pgn] tLg efusf] Ps efu
?lkofFn] lstfa lslgg\ . rf/ efusf] Ps efu ?lkofnF ] sk8f lslgg\ . kfrF efusf]

Ps efu ?lkofF ed| 0fdf vr{ ul/5g\ eg],

-s_ hDdf slt efu vr{ ul/5g\ <

-v_ hDdf slt ?lkofF vr{ ul/5g\ <

-u_ slt /sd art ul/5g,\ kQf nufpgx' f];\ .

7. Ct'n] Pp6f :ofpsf] 3 efu vfO5g\ / afFsL efu pgsf] efO ;d' gn]
5
vfP5g\ .

-s_ ;'dgn] vfPsf] :ofpsf] efunfO{ leGgdf n]Vg'xf];\ .

-v_ s;n] slt efu :ofp a9L vfP5, kQf nufpgx' f;] \ .

8. dfx] gn] cfkm;" Fu ePsf] /sddWo] 2 efu >LdtLnfO{ lbP5g\ . afsF L /sdaf6
/51x5] egf\ue5g],f/] fnfO{ / 1 3 60,000 lbPsf
3 efu 5f/] LnfO{ lbP5g\ . >LdtLnfO{ ?=

-s_ 5f/] fnfO{ slt /sd lbPsf /x]5g\ <

-v_ 5f/] LnfO{ slt /sd lbPsf /x5] g\ <

-u_ 5f]/f / 5f]/LdWo] s;nfO{ sltn] w/] } /sd lbPsf /x]5g\, kQf nufpgx' f;] \ .

9. ks| [ltn] cfkm;" uF ePsf] /sdsf] 4190e0fuxf] vr{ u/]5g\ . afsF L /ssltda/fs6d52 efu
x/fP5g\ . olb x/fPsf] /sd ?= eg] pgL;Fu klxn] lyof],

kQf nufpgx' f;] \ .

82 ul0ft, sIff &

10. Pespffuo6j{ f5m| dlfjqdBfff/nuPo31sdefffluy45P5efefqugx]¿l5jfB;qfxdn¿fojs]z5fgeh\ PD.5dlfgj\lBj. fBnofloybsL{ 1;f]9ªj0\Vgohefgfkh]f QljfsBnffouyj{ fLp{m| djggd'xeff];fh]41\ .

kl/of]hgf sfo{

tkfOF{ cfkmn" ] Ps lbgdf ljBfnodf latfpg] ;dodWo] tn lbOPsf
ljm| ofsnfkx¿sf nflu slt slt ;do vr{gx' 'G5 < leGgsf ¿kdf n]vL
sIffdf k|:tt' ug'{xf;] \ M
-s_ laxfgLsf] k|fyg{ f
-v_ vfhf vfg] ;do
-u_ cWoog ;do
-3_ cGo ljm| ofsnfksf] ;do

pQ/

1. -s_ 12 1265 -v_ 149679 -u_ –25 -3_ 1
9

2. 3381 3. -s_ 7 efu -v_ 3 efu -u_ ?= 8,100
10 10

4. lzIfsnfO{ b]vfpg'xf;] \ . 5. -s_ 8 hgf -v_ 16 hgf

-u_ 2 efu 6. -s_ 47 efu -v_ ?= 4,700 -u_ ?= 1300
5 60


7. -s_ 2 -v_ Ctn' ] 1 8. -s_ ?= 6000
5 5

-v_ ?= 10,000 -u_ 5f/] LnfO{ ?= 4000 9. ?= 3000 10. 600

ul0ft, sIff & 83

5.2 bzdnj (Decimal)

5.2.0 kg' /jnfs] g (Review)
tn lbOPsf bzdnj ;ª\Vofsf] lx;fa ugx{' f];\ M

-s_ 8.97 + 23.2 -v_ 3.6 × 5.8

-u_ 7.7 – 2.8 -3_ 17.40 ÷ 4

bzdnj ;ªV\ ofsf] hf]8, 36fp, u'0fg / efu ubf{ s] s] s/' fdf Wofg lbgk' b5{ <
;fyL ;dx" df 5nkmn u/L lgisif{ lgsfNgx' f;] \ .

5.2.1 bzdnjsf] ;/nLs/0f (Simplification of Decimal)
ljm| ofsnfk 1

Oz{ fn] sIffsf7] f ;hfpgsf nflu sx] L 9frF fx¿ agfpbF } lyOg\ . ;f] 9frF f agfpgsf

nflu nDafO 4.5 cm ePsf jufs{ f/ sfuhsf 6j' m| fx¿ rflxG5 . pgL;uF 54 cm
nDafO / 4.5 cm rf}8fO ePsf] cfotfsf/ sfuh 5 . To; sfuhaf6 4.5 cm
nDafOsf sltcf]6f 6'jm| fx¿ aGnfg < pgn] sfuhsf 6'jm| fx¿sf] ;ªV\ of kQf

nufpg 54 nfO{ 4.5 n] efu ul/g\ . s] of] ;xL 5, 5nkmn u/L lgisif{ lgsfNg'xf];\ .

ljm| ofsnfk 2

lrqdf bv] fOPsf] 6a] 'nsf] cfotfsf/ ;txsf] 3.5 ft 5.25 ft
nDafO 5.25 ft / rf8} fO 3.5 ft /x]5 .

-s_ 6]an' sf] kl/ldlt slt xfn] f <

-v_ kl/ldlt lgsfNg sg' ul0ftLo lj|mof
k|ofu] ugk{' nf,{ 5nkmn ug'{xf;] \ .

oxfF 6a] 'nsf] kl/ldlt lgsfNg,

pSt 6]a'nsf] nDafO / rf}8fO bj' }nfO{ b'O{ bO' { k6s hf]8\g'kb{5 .

cyft{ \ 6]a'nsf] kl/ldlt = 2(l + b) x'G5 .

ctM 6a] n' sf] kl/ldlt = 2(5.25 ft + 3.5 ft)

= 2 × 8.75 ft

= 17.5 ft

84 ul0ft, sIff &

pbfx/0f 1

tn lbOPsf] ul0ftLo ;d:ofnfO{ s;/L ;/n ug{ ;lsG5 < ;f]lwPsf k|Zgx¿df 5nkmn
ub}{ lgisif{ lgsfNg'xf];\ M
;/n ug'x{ f];\ M

7.5 + {6.72 ÷ 2.8 (3.59 – 1.49)}

-s_ s] 3.59 af6 1.49 g36fO 2.8 n] 6.72 nfO{ efu ug{ ldN5 <
-v_ s'g sfi] 7leqsf] sfd klxn] ug'{knf{ <
-u_ ;/n ubf{ sfi] 7leqsf] j|md s] xfn] f <
;/n ubf{ jm| dz M efu, u0' fg, hf8] / 36fpsf] sfd ul/G5 t/ oxfF ;j{k|yd 3.59
af6 1.49 nfO{ 36fpg' kb{5 . cfPsf] glthfnfO{ 2.8 ;uF u'0fg ug{'kb5{ . To;kl5
dfq 6.72 nfO{ efu ug{ ;lsG5 . To;n} ] (3.59 – 1.49) nfO{ ;fgf] sfi] 7df / {6.72
÷ 2.8 (3.59 – 1.49)} nfO{ demfn} f sfi] 7df /flvPsf] 5 .
;dfwfg

oxfF 7.5 + {6.72 ÷ 2.8 (3.59 – 1.49)}

= 7.5 + {6.72 ÷ 2.8 (2.1)}
= 7.5 + {6.72 ÷ 5.88}
= 7.5 + 1.14
= 8.64

rf/ ;fwf/0f lj|mofx¿ (+, –, ×,÷) / sf]i7x¿;lxtsf] bzdnjsf]
;/nLs/0f ubf{ k"0f{ ;ª\Vofsf] ;/nLs/0fdf h:t} sfi] 7leq
;dfjz] ePsf lj|mofnfO{ klxnf ul/;s]kl5 afFsL ljm| ofx¿
ub{} hfgk' b{5 . ;/nLs/0fdf ko| fu] ePsf sfi] 7x¿ j|md;} Fu
;fgf] sfi] 7 ( ), demfn} f sfi] 7 { } / 7n' f] sf]i7 [ ] leq ;dfj]z
ePsf lj|mofx¿ ug{k' b{5 .

ul0ft, sIff & 85

cEof; 5.2

1. ;/n ug{'xf];\ M

-s_ 1.44 ÷ 1.2 + 6.2

-v_ 12.75 – {4.38 – (2.4 × 4.32 ÷ 3.6 – 0.85)}

-u_ 1.2 × 1.2 – 0.4 × 0.4
2.4 – 1.6

-3_ 4.5 × 4.5 – 2.1 × 2.1
4.5 + 2.1
1
2. lqe'hsf] Ifq] kmn = 2 × cfwf/ × prfO xG' 5 . olb Pp6f lqeh' sf] cfwf/

25.75 cm / prfO 30.15 cm 5 eg] Ifq] kmn kQf nufpg'xf];\ .

3. Pp6f cfotfsf/ au}Frfsf] nDafO 22.66 m / rf}8fO 15.65 m 5 .

-s_ ;f] auFr} fsf] If]qkmn slt x'G5 <
-v_ pSt au}Frfsf] kl/ldlt slt x'G5, kQf nufpg'xf;] \ .

4. Pp6f juf{sf/ 6a] n' sf] kl/ldlt 24.4 ft 5 . 6]a'nsf] nDafO kQf nufpg'xf;] \ .

5. Pp6f cfotfsf/ hUufsf] If]qkmn 215.66 m2 5 . olb ;f] hUuf 67.35 m
nfdf] eP slt km/flsnf] xf]nf, kQf nufpg'xf;] \ .

6. Pp6f juf{sf/ vt] sf] nDafO 8.45 m 5 eg] pSt vt] sf] kl/ldlt kQf
nufpg'xf];\ .

kl/ofh] gf sfo{

klqsf jf OG6/g6] af6 cfhsf] db' f| ljlgdo b/ tflnsf vf]Hg'xf;] \ . sg' }
klg kfFrcf6] f b]zsf] vl/b b/ / laj|mL b/ l6Kgx' f;] \ / ltgLx¿larsf]
km/s lgsfn/] sIffdf k|:t't ug'{xf;] \ .

pQ/ -v_ 10.4 -u_ 1.6 -3_ 2.4
1. -s_ 7.4 3. -s_ 354.62 m2 -v_ 76.62 m

2. 388.18 cm2 5. 3.20 m 6. 33.8 m
4. 6.1 ft

86 ul0ft, sIff &

kf7 6 cg'kft / ;dfgk' ft

(Ratio and Proportion)

6.1 cgk' ft (Ratio)

6.1.0 k'g/jnf]sg (Review)

cg'hfsf] tf}n 30 kg / /d]zsf] tfn} 60 kg 5 . cgh' f / /d]zsf] tfn} nfO{ s;/L
t'ngf ug{ ;lsG5, 5nkmn ug'{xf];\ M

cGt/sf ¿kdf,

cgh' f / /dz] sf] tfn} sf] cGt/ = (60 – 30) kg = 30 kg
efukmnsf ¿kdf,

cg'hfsf] tfn} = 30 kg = 1
/dz] sf] tfn} 60 kg 2

cGt/n] v'b slt cª\sn] ;fgf] jf 7'nf] eGg] ae' mfpF5 eg] efukmnn] slt u'0ffn]
;fgf] jf 7'nf] eGg] ae' mfp5F .

6.1.1 cgk' ftsf] kl/ro (Introduction of Ratio)
ljm| ofsnfk 1

lbOPsf] tflnsf cjnfs] g u/L ;flwPsf k|Zgx¿df 5nkmn ug'x{ f];\ M
j:t' d"No

?= 10

?= 70

?= 5

ul0ft, sIff & 87

-s_ sfkLsf] dN" o l;;fsndsf] d"Nosf] slt u0' ff 5 <
-v_ O/h] /sf] d"No l;;fsndsf] dN" osf] slt u0' ff 5, tn' gf ugx'{ f];\ .

sfkLsf] dN" o ?=70 = 7 sfkLsf] d"No l;;fsndsf]
l;;fsndsf] dN" o = ?=10 1 d"Nosf] 7 u0' ff 5 .

O/h] /sf] d"No ?=5 = 1 O/h] /sf] dN" o l;;fsndsf]
l;;fsndsf] d"No = ?=10 2 d"Nosf] cfwf 5 .

Pp6f kl/df0fsf cfwf/df csf{] kl/df0fnfO{ t'ngf ug{' g} cgk' ft xf] .

bO' c{ f6] f Pp6} PsfOdf ePsf kl/df0fx¿ a / b sf] cg'kft a jf a:b
b
x'G5, hxfF b ≠ 0 . a:b nfO{ a is to b eg/] kl9G5 . hxfF a / b cg'kftsf

kbx¿ x'g\ .

cg'kftsf] ;/n ¿k (Simplest form of a Ratio)

lj|mofsnfk 2

tn lrqdf bv] fOPsf] l;;fsndsf] nDafO 18 cm / Jof; 6 mm 5 . l;;fsndsf]
Jof; / o;sf] nDafOsf] cg'kft slt xG' 5 <

18 cm 6 mm

s] 18 cm = 3 n]Vg ;lsG5 <
6 mm 1

s] l;;fsndsf] nDafO o;sf] Aof;sf] tLg u'0ff dfq 5, 5nkmn ugx{' f;] \ .

l;;fsnsf] nDafO olb lbOPsf kl/df0fx¿sf
Jof;sf] tLg u0' ff dfq PsfOx¿ km/s 5g\ eg] Pp6}
PsfOdf kl/jtg{ u//] t'ngf
5g} , s;/L ug{] <
ug'{kb5{ .

88 ul0ft, sIff &

l;;fsndsf] nDafO ldlnld6/df kl/jtg{ ubf,{

18 cm = 18 × 10 mm = 180 mm x'G5 .

To;n} ],

l;;fsndsf] Jof; / o;sf] nDafOsf] cg'kft = 10 mm = 1 nV] g ;lsG5 .
180 mm 18

● cgk' ftn] ;dfg PsfO ePsf kl/df0fnfO{ dfq tn' gf ub{5 .
● cgk' ft n]Vbf PsfO nV] gx' b'F }g .

cgk' ft / leGgsf] ;DaGw (Relation of Ratio and Fraction)

lj|mofsnfk 3
Pp6f j[Q lvRgx' f;] \ . o;nfO{ kfFr a/fa/ efudf af8g\ 'xf;] \ .

-s_ 5fof kfl/Psf] efunfO{ leGgdf nV] g'xf];\ .
-v_ 5fof kfl/Psf] efu / 5fof gkfl/Psf] efusf] cgk' ft lgsfNgx' f];\ .
-u_ leGg / cgk' ftdf s] km/s 5, 5nkmn ug'x{ f];\ .

oxfF, 3
5
5fof kfl/Psf] efusf] leGg =
5fof kfl/Psf]
/ gkfl/Psf] efusf] cgk' ft = 2 = 3:2
3

leGgdf 5 n] 5 efudWo] 3 efu a'emfpF5 .
3
cgk' ftdf 3:2 n] hDdf 3 + 2 = 5 efu eGg] ae' mfp5F .

pbfx/0f 1
lrqdf bv] fOPsf bO' {cf]6f aflN6gsf] IfdtfnfO{ cgk' ftdf n]Vgx' f];\ M

9 l Ifdtf 15 l Ifdtf

ul0ft, sIff & 89

;dfwfg

oxfF klxnf] aflN6gsf] Ifdtf = 9l = 3×3l = 3 = 3:5
bf];|f] aflN6gsf] Ifdtf 15 l 3×5l 5

ctM klxnf] aflN6gsf] Ifdtf M bf];f| ] aflN6gsf] Ifdtf = 3:5 5 .

pbfx/0f 2

tn lbOPsf] ;ª\Vof /v] fdf k|Tos] efun] 1cm hgfp5F .

0 1 2 3 4 5 6 7 8 9

A B C D E F G H I J

AC / AF sf] cgk' ft slt xG' 5 <
;dfwfg

oxfF AC : AF
AC
= AF

= 2 cm = 2 = 2:5
5 cm 5

pbfx/0f 3

lji0f'n] ul0ftdf 100 k0" ff{ª\sdf 75 cª\s kf| Kt u/5] g\ . lj1fgdf 75 k"0ffª{ s\ df 50

cªs\ k|fKt u/]5g\ < s'g ljifodf /fd|f] cªs\ kf| Kt u/]5g\, kQf nufpg'xf];\ M

;dfwfg

oxf F ul0ftdf k|fKt u/]sf] cª\ssf] leGg = 17050 = 3
4

lj1fgdf k|fKt u/]sf] cª\ssf] leGg = 50 = 2
75 3

;dfg x/ agfpFbf,

ul0ftdf kf| Kt u/s] f] cªs\ sf] leGg = 3 × 3 =192
4 3

lj1fgdf k|fKt u/s] f] cªs\ sf] leGg = 2 × 4 =182
3 4

⸫ 9 > 8 To;n} ] lji0fn' ] ul0ftdf /fd|f] u/s] f /x5] g\ .
12 12

90 ul0ft, sIff &

pbfx/0f 4

c~h' / cdgsf] prfO jm| dz M 145 cm / 165 cm 5 . pgLx¿sf] prfOsf] cgk' ft
lgsfNg'xf];\ .

;dfwfg

oxfF c~h' / cdgsf] prfOsf] cgk' ft = 145 cm
165 cm

= 5 × 29 = 39 = 29 : 33
5 × 33 33

cEof; 6.1

1. tnsf kT| os] sf] cgk' ft n]vL n3'Qd kbdf ¿kfGt/0f ug{'xf];\ M
-s_ 10 cm / 100 cm -v_ ?= 180 / ?= 240 -u_ 10 kg / 2 kg
-3_ 8 306f / 24 306f -ª_ 250 ml / 1000 ml -ª_ 2.5 kg / 7.5 kg

2. tn lbOPsf] ;ª\Vof /]vfdf kT| os] efun] 1 cm hgfpF5 . ;f]lwPsf b'/Lsf]
cgk' ft kQf nufpg'xf;] \ M

0 1 2 3 4 5 6 7 8

A B C D E F G H I

-s_ AB : AG -v_ AC : DI -u_ CF : CH
-3_ BG : BI -ª_ AF : AI

3. lbOPsf] lrqaf6 5fof kfl/Psf] / 5fof gkfl/Psf] efusf] cgk' ft lgsfNgx' f;] \ M

-s_ -v_ -u_

ul0ft, sIff & 91

4. Pp6f ljBfnodf 450 ljBfyL{ 5g,\ h;dWo] 180 5fqfx¿ 5g\ eg,]

-s_ 5fqf / hDdf ljBfyL{sf] cg'kft lgsfNg'xf];\ .
-v_ 5fq / hDdf ljBfyLs{ f] cgk' ft lgsfNg'xf;] \ .
-u_ 5fqf / 5fqsf] cgk' ft lgsfNgx' f];\ .
5. Pp6f ljBfnodf 25 hgf lzIfs / 500 hgf ljBfyL{ 5g\ eg] lzIfs /
ljBfyLs{ f] cgk' ft kQf nufpgx' f];\ .

6. kd| ]zsf] prfO 165 cm / kl| dnfsf] prfO 150 cm 5 eg],

-s_ kd| ]z / kl| dnfsf] prfOsf] cg'kft lgsfNg'xf;] \ .
-v_ kl| dnf / kd| ]zsf] prfOsf] cgk' ft lgsfNg'xf];\ .
7. /fxn' n] cª\uh]| L ljifodf 50 k"0ffª{ \sdf 40, gk] fnLdf 30 k0" ff{ªs\ df 20 /
ul0ftdf 20 k"0ff{ªs\ df 13 k|fKt u/5] g\ .

-s_ cª\u|]hL / g]kfnL b'O{ ljifodf s'gsf] glthf /fd|f] 5 <
-v_ g]kfnL / ul0ftdWo] s'g ljifosf] glthf /fdf| ] 5 <
-u_ ;ae} Gbf /fdf| ] glthf s'g ljifodf 5, cg'kft lgsfn]/ tn' gf ug{'xf;] \ .

8. Pln;f / lbk]zn] ?= 6000 nfO{ 5:7 sf] cgk' ftdf afªF \bf bj' n} ] slt slt
?lkofF kfpnfg, kQf nufpg'xf];\ .

kl/of]hgf sfo{

tkfOsF f] ljBfnosf] sIff 5 bl] v 10 ;Ddsf 5fq / 5fqf ljBfyL{sf]
;ª\Vof l6kf6] ugx{' f;] \ . sIffut ¿kdf k|To]s sIffsf,

-s_ 5fqf / 5fqsf] cg'kft
-v_ 5fqf / hDdf ljBfyLs{ f] cgk' ft
-u_ 5fq / hDdf ljBfyL{sf] cg'kft lgsfNgx' f];\ / lgisif{ sIffdf k:| t't

ug'x{ f];\ .

pQ/

1 bl] v 3 ;Dd lzIfsnfO{ b]vfpg'xf];\ .

4. -s_ 2 -v_ 3 -u_ 2 5. 1:20 6. -s_ 11:10
5 5 3

-v_ 10:11 7. lzIfsnfO{ bv] fpg'xf];\ . 8. ?= 2500, ?= 3500

92 ul0ft, sIff &

6.2 ;dfgk' ft (Proportion)

6.2.0 kg' /jnf]sg (Review)

2 sf ;dtN' o leGgx¿ x'g] u/L tn lbOPsf] tflnsf k/" f ugx{' f;] \ M
5

2 = 4 2 = 10
5 10 5

2 = 6 2 = 30
5 5

2 = 2 = 14
5 5
20

s] 4 , 6 , 8 ;a}sf cg'kftx¿ Pp6} 5g\ < 5nkmn u/L lgisif{ lgsfNg'xf];\ .
10 15 10

6.2.1 ;dfgk' ftsf] kl/ro (Introduction of Proportion)
ljm| ofsnfk 1

bO' {cf6] f a/fa/ gfksf jufs{ f/ sfuh lngx' f];\ . klxnf]nfO{ a/fa/ rf/ efudf
afF8g\ 'xf;] \ . bf;] f| n] fO{ a/fa/ 16 efudf afF8\gx' f];\ .

lrq I lrq II

klxnf] lrqaf6 5fof kfl/Psf] / gkfl/Psf] efusf] cg'kft lgsfNg'xf];\ .

5fof kfl/Psf] efu = 1
5fof gkfl/Psf] efu 3

ca, bf];|f] lrqaf6 5fof kfl/Psf] / gkfl/Psf] efusf] cg'kft lgsfNgx' f];\ .

5fof kfl/Psf] efu = 4 = 1
5fof gkfl/Psf] efu 12 3

ctM 1 / 4 a/fa/ cg'kftx¿ x'g\ .
3 12

a/fa/ cg'kftnfO{ ;dfg'kft elgG5 .

ul0ft, sIff & 93

lj|mofsnfk 2

bLkG] bn| ] ul0ft ljifodf 30 k"0ffª{ s\ sf] k/LIffdf 25 cª\s k|fKt u/]5g\ . gdb{ fn] 24
k0" ff{ª\ssf] k/LIffdf 20 cª\s kf| Kt ul/5g\ . s;sf] glthf /fd|f] 5 < logLx¿sf]
kf| Ktfª\ssf] cgk' ft slt xfn] f < s;/L tn' gf ug{ ;lsPnf, 5nkmn ug{x' f;] \ .

bLkG] bs| f] kf| Ktfªs\ / k"0ff{ªs\ sf] cgk' ft = 25 = 5
30 6

gdb{ fsf] kf| Ktfª\s / k0" ffª{ \ssf] cg'kft = 20 = 5
24 6

25 / 20 sf] cgk' ft a/fa/ ePsfn] b'j}sf] glthf a/fa/ bl] vG5 .
30 24

ctM 25 = 20 x'G5 .
30 24

b'O{cf]6f cgk' ftx¿ a/fa/ ePsfn] oL cgk' ftnfO{ ;dfgk' ft elgG5 . oxfF 25 nfO{

klxnf] kb, 30 nfO{ bf;] f| ] kb, 20 nfO{ t;] f| ] kb / 24 nfO{ rfy} f] kb elgG5 .

rf/cf]6f ;ª\Vofx¿ a, b, c / d df a / b sf] cg'kft c / d sf] :c: cg:'kdftj;f baFu

=a/dcfan/]lv5Gg5\ e. gc] ya,jbf ,ac / d ;dfg'kftdf 5g\ elgG5 . o;nfO{ a:b
× d = b × c n]lvG5 .

dflysf pbfx/0fdf 25 / 20 ;dfg'kflts 5g\ .
30 24

Extremes

o;nfO{ 25:30 :: 20:24 klg nl] vG5 .

Means

aflx/sf b'Oc{ f]6f kbnfO{ 5p] 5p] sf kbx¿ (Extremes) elgG5 . leqsf b'Oc{ f]6f
kbnfO{ larsf kbx¿ (Means) elgG5 . oxfF 25 / 24 Extremes xg' \ eg] 30 /
20 Means x'g .

25 × 24 = 30 × 20 = 600 eof] .

5]p 5p] sf kbx¿ (Extremes) / larsf kbx¿ (Means) sf]
56' 6\ f 5'6\6} u'0fgkmn a/fa/ xG' 5 .

94 ul0ft, sIff &