गणित कक्षा ८

11.4 = k~rcfwf/ k4ltnfO{ bzdnj k4ltdf ¿kfGt/

tl/sf,
- ;jk{ y| d :yfgdfg tflnsfcg;' f/ lj:tfl/t ¿kdf nV] g]
- ;a} 3ftfªs\ x¿ u0' ff u/L ;/n ug{]
- k0" f{ ;ªV\ ofdf JoSt ug{]

pbfx/0f 4

tnsf k~rcfwf/ ;ªV\ ofnfO{ bzdnj ;ªV\ ofdf ¿kfGt/ u/ M

-s_ 43215 -v_ 133205
;dfwfg

43215 nfO{ :yfgdfg tflnsfdf /fVbf,
Ps ;o klRr; klRr; kf“r Ps

53 52 51 50

4 321

ca lj:tfl/t ¿kdf nV] bf,

43215 = 4 × 53 + 3 × 52 +2 × 51 + 1× 50
= 4 ×125 +3 × 25 +2 × 5 +1×1

= 500 + 75+ 10+ 1

= 58610

To;sf/0f 43215 =58610 xG' 5
-v_ 134205

:yfgdfg tflnsfdf /fVbf,

54 53 52 51 50

13 4 2 0

ca tflnsfcg;' f/ lj:tfl/t ¿kdf nV] bf,

134205 = 1×54 + 3×53 +4×52 + 2×51+ 0×50
= 1×625 +3×125 +4×25 +2×5+ 0

= 625+ 375+ 100+ 10

= 111010

t;y,{ 134205 = 111010 xG5 .

96 xfdf| ] ul0ft, sIff *

cEof; 11.1

1. tnsf ;ªV\ ofx¿nfO{ k~rcfwf/ ;ªV\ ofdf ¿kfGt/0f u/ M

-s_ 9 (v_13 -u_ 21 -3_ 26 -ª_ 45

-r_ 86 -5_ 194 -h_ 404 -em_ 497 -`_ 1234

2. tnsf k~rcfwf/ ;ªV\ ofx¿nfO{ bzdnj ;ªV\ ofx¿df ¿kfGt/0f u/ M

-s_ 24 -v_1015 -u_ 300 -3_ 4321
5 -r_ 20235 5 5
-`_ 20145
-ª_ 4415 -5_ 42015 -h_ 33135

-em_ 123045 -6_ 101235 -7_ 214325

11.5 låcfwf/ ;ªV\ of k4lt (Binary Number System)

tn ;u“ s} f] pbfx/0f x/] f“} .

29 = 28 + 1 = 8X 3+ 1 X 4 + 1 = 8 X2 + 8 X 1 + 1 X 4 + 1
∴29 =16 +8 +4 +1 =24 +23+22 +1
= 1 x 24+ 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20

= 1 x 24+ 1 x 23 +1 x 22 + 0 x 21 + 1 x 20 xG' 5 .

To:t,} sg' } ;ªV\ ofnfO{ 0, 1, 2, 3 / 4 dfq ko| fu] u/] em“} sg' } klg ;ªV\ ofnfO{ 0 / 1 dfq ko| fu]
u/L 2 sf] 3ftsf ¿kdf nl] vG5 . :yfgdfg tflnsfdf Ps, b'O,{ rf/, cf7 , ;f/] , alQ;, ………
jf 20, 21, 22, 23, 54, …… xG' 5 eg] To; ;ªV\ ofªs\ g k4ltnfO{ låcfwf/ ;ªV\ ofªs\ g k4lt
(binary numeration system) elgG5 . dflysf] pbfx/0fnfO{ 29 = 111012 nl] vG5 .

pbfx/0f 1
43 nfO{ låcfwf/ k4ltdf lj:tfl/t ¿kdf nv] .

;dfwfg
43 nfO{ 2 sf] 3ftsf] ;dx" df ljefhg ubf,{

43 = 32 + 8 + 2 + 1

= 2 x 2 x 2 x 2 x 2 + 2 x 2 x 2 + 2 +1

∴ 432 = 1 x 25 + 0 x 24 +1 x 23+ 0 x 22 +1 x 21+1 x 20 xG' 5 .

xfdf| ] ul0ft, sIff * 97

11.6 bzdnj ;ªV\ ofnfO{ låcfwf/ k4ltdf ¿kfGt/0f

cfwf/ 10 ePsf] ;ªV\ ofnfO{ cfwf/ 2 ePsf] ;ªV\ ofdf s;/L ¿kfGt/0f ug{ ;lsG5 x/] .

pbfx/0f 2

7510 nfO{ låcfwf/ k4ltdf ¿kfGt/0f u/ .
;dfwfg

2 75 zi] f

2 37 1 – cfwf/ 10 df zi] f 0 bl] v 9 ;Dd xG' 5 .

2 18 1 – cfwf/ 2 df zi] f 0 / 1 dfq xG' 5 .

29 0 – t;y{ sg' } ;ªV\ ofnfO{ låcfwf/df ¿kfGt/0f ug{ 2 n]
efu ug{] / zi] f nV] b} hfg] ugk{' 5{ .
24 1

22 0

21 0

01

ca zi] fx¿nfO{ jm| dzM tnaf6 dflysf] jm| ddf nV] bf 1001011 xG' 5 .

t;y{ 7510 = 10010112 xG' 5 .
låcfwf/ k4ltdf :yfgdfg tflnsfnfO{ lgDgfg;' f/ bv] fpg ;lsG5 M

28 27 26 25 24 23 22 21 20

256… 128 64 32 16 8 4 2 1

k:| tt' tflnsfsf] ko| fu] n] låcfwf/ ;ªV\ ofnfO{ lj:tt[ ¿kdf nV] g ;lsG5 / låcfwf/ ;ªV\ ofnfO{
bzdnj jf cGo k0| ffnLdf ¿kfGt/0f ug{ ;lsG5 .

11.7 låcfwf/ k4ltnfO{ bzdnj k4ltdf ¿kfGt/0f

xfdLnfO{ yfxf 5 ls låcfwf/ k4ltdf sg' } klg ;ªV\ ofnfO{ cfwf/ 2 df / 2 sf] 3ftfªs\ sf
¿kdf JoSt ul/G5 . ca o;nfO{ cfwf/ 10 jf bzdnj k4ltdf s;/L ¿kfGt/0f ug,{] tnsf]
pbfx/0f x/] f“} .

pbfx/0f 4

tnsf låcfwf/ ;ªV\ ofnfO{ bzdnj k4ltdf ¿kfGt/0f u/ M tl/sf M
– cfwf/ 2 sf] 3ftfªs\ sf]
-s_ 1001011 -v_ 1100101
2 2 ¿kdf lj:tfl/t ¿kdf nV] g]
– ;/n ug]{
– pQ/ n]Vg]

98 xfdf| ] ul0ft, sIff *

;dfwfg
-s_ 10010112 = 1 × 26 +0 × 25 + 024 +1 × 23 + 0 × 22 +1 × 21 + 1 × 20

= 26 + 0+ 0+ 23+ + 0+21 + 1
= 64+ 8+ 2+ 1
= 75

10

∴ 10010112 = 7510

ct M 11001012 = 1×26 +1×25 + 0×24 +0×23 + 1×22 +0×21 + 1×20

= 26 + 25+ 0+0+ 22+ 0+21
= 64+ 32+ 4+ 1
= 10110
∴ 11001012 = 10110

cEof; 11.2

1. tnsf ;ªV\ ofx¿ sg' ;ªV\ of k4ltdf 5g,\ nv] M

-s_ 10011 -v_ 350 -u_ 1001 -3_ 42
2 2

-ª_ 555 -r_ 77532 -5_ 100100112 -h_ 257903

2. tnsf bzdnj k4ltsf ;ªV\ ofnfO{ låcfwf/ k4ltdf ¿kfGt/0f u/ M

-s_ 4 -v_ 9 -u_ 12 -3_ 25 -ª_ 35

-r_ 65 -5_ 94 -h_ 135 -em_190 -`_ 275
-6_ 220 -7_ 512 -8_ 530

3. tnsf låcfwf/ ;ªV\ ofnfO{ bzdnj k4ltdf ¿kfGt/0f u/ M

-s_ 11002 -v_ 100102 -u_ 111102 -3_ 1000012

-ª_ 111111 -r_ 1100011 -5_ 1110011 -h_ 1100110011
2 2 2 2

-em_10101011102 -`_ 1000010002 -6_ 1011101112 -7_ 110110110012

4. olb sg' } ;ªV\ ofsf] bzdnj k4ltdf 723 n] hgfOG5 eg] pSt ;ªV\ ofsf] dfg låcfwf/

k4ltdf slt xfn] f <

5. 100000001 nfO{ bzdnj k4ltdf nv] .
2

xfdf| ] ul0ft, sIff * 99

kf7

12 k0" ffª{ s\ x¿ (Integers)

12.1 kg' /jnfs] g ( Review)

k0Ò f{ ;ªV\ ofx¿sf] ;dxÒ W = {0,1,2,3,4,5,……} df tnsf pbfx/0fx¿sf] cWoog u/ .

3 + 4 = ?, 3–3=? 3–4=?

dflysf] t;] f| ] pbfx/0fdf Pp6f ;fgf] kf| sl[ ts ;ªV\ ofaf6 7n' f] kf| sl[ ts ;ªV\ of 36fpb“ f gof“
kf| sl[ ts ;ªV\ of jf k0Ò f{ ;ªV\ of xg' ;Sbg} -lsg <_ . To;sf/0f k0Ò f{ ;ªV\ ofsf] ;dxÒ n] dfq
;a} ;ªV\ ofx¿nfO{ hgfpg ;lsPg . gof“ ;ªV\ ofx¿sf] cfjZostf dx;;' eof] / C0ffTds
kÒ0f{ ;ª\Vof¿sf] cfljisf/ eof] . k|fs[lts ;ª\Vofx¿sf] ;dÒx, zÒGo / C0ffTds kÒ0f{
;ªV\ ofx¿sf] ;dxÒ ldn/] ags] f] ;ªV\ ofx¿sf] ;dxÒ nfO{ k0" ffª{ s\ x¿sf] ;dxÒ elgG5 . o;nfO{
(Z) n] hgfOG5 . Z = {…………-4, -3 ,-2, -1, 0, 1, 2,, 3, 4, ………} xG' 5 . ;fy,} {1, 2, 3, 4,
………} nfO{ wgfTds k0Ò f;{ ªV\ ofx¿sf] ;dxÒ / {-1,- 2, -3, -4, ………} nfO{ C0ffTds k0Ò f{
;ªV\ ofx¿sf] ;dxÒ elgG5 . k0Ò f;{ ªV\ ofx¿sf] ;dxÒ nfO{ ;ªV\ of /v] fåf/f lgDgfg;' f/ bv] fpg
;lsG5 M

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

k0" f{ ;ªV\ ofx¿sf ljm| ofx¿sf lgodx¿ (Laws of Operation of Integers):
dflysf] ;ªV\ of/v] faf6 sg' } tLg ;ªV\ of np] m, h:t} M -2 , 3 / 4

-2 + 4 = ?; 4+ (-2) = ?; 4+0 = ?; 0+4=?; -2+ (4 +3) =?; ( -2+4) +3 =?; -2+2=?

dflysf kZ| gx¿af6 s] yfxf xG' 5 <
dflysf kZ| gx¿sf pQ/x¿af6 lgDglnlvt lgodx¿df ;fdfGoLs/0f ug{ ;lsG5 M
k0" ffª{ s\ x¿sf] hf8] sf lgodx¿ (Laws of addition of Integers)
olb a, b, c, tLgcf6] f k0" ffª{ s\ x¿ eP
-s_ aGbL lgod (closure law) : a+b / a+b+c klg k0" ffª{ s\ g} xG' 5g\ .

100 xfdf| ] ul0ft, sIff *

-v_ ljlgod lgod (cummutative law) : a+b= b+a, a+c=c+a, b+c= c+b xG' 5 .

-u_ ;ª3\ Lo lgod (associative law) : (a+b) +c = a+ (b+c) xG' 5 .

-3_ PsfTds lgod (identity law) : a+0 =0+a = a xG' 5 .

-ª_ ljk/Lt kl/0ffdsf] lgod (inverse law )M ;a} a sf nflu k0" ffª{ s\ sf] ;dx" df -a xG' 5 .

;fy} a + (–a) = (-a) + a = 0 xG' 5 .

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

– + + = – -– 7n' f] cªs\ ePdf_ -3 + 2 = -1
+ + – = + -+7n' f] cªs\ ePdf_ 3 + (-2) = 1
– + – = – xG' 5 . -3 + (-2) = -5

To:t,} tnsf kZ| gx¿sf] pQ/ kQf nufpm / sfkLdf nv] M

-3 X 2 = ? ; 2 X (-3) = ? ; 2 X 1= ?; 1 X 2 =?; -3X (2 X 4) =? ;( -3 X 2) X 4 =?

dflysf kZ| gx¿af6 s] yfxf xG' 5, kQf nufpm .

k0" ffª{ s\ x¿sf] u0' fgnfO{ lgDglnlvt tflnsfaf6 :ki6 kfg{ ;lsG5 M

× -4 -3 -2 -1 0 1 2 3 4

-4 16 12 8 4 0 -4 -8 -12 -16
-3 12 9 6 3 0 -3 -6 -9 -12
-2 8 6 4 2 0 -2 -4 -6 -8
-1 4 3 2 1 0 -1 -2 -3 -4
0 0 0 00 000 0 0
1 -4 -3 -2 -1 0 1 2 3 4
2 -8 --6 --4 -2 0 2 4 6 8
3 -12 -9 -6 -3 0 3 6 9 12
4 -16 -12 -8 -4 0 4 8 12 16

dflysf kZ| gx¿ / tflnsfsf cfwf/df k0" ffª{ s\ x¿sf] u0' fgsf lgDglnlvt lgodx¿ agfpg ;lsG5 M

xfdf| ] ul0ft, sIff * 101

k0" ffª{ s\ sf] u0' fgsf lgodx¿ [Law of Multiplication of Integrs]

olb a, b, c tLgcf6] f k0" ffª{ s\ x¿ eP

-s_ aGbL lgod (closure law) :
a × b, b × c, c × a k0" ffª{ s\ xG' 5 .

-v_ ljlgodsf] lgod (commutative law) :
a × b, = b × a xG' 5 .

-u_ ;ª3\ Lo lgod (associative law) :
(a×b)×c = a × (b×c) xG' 5 .

-3_ kbljR5b] g÷ljt/0ffTds lgod (distributive law) :
a×(b+c)= a × b + a × c xG' 5 . cyjf a ( b+c) = ab+ac

-ª_ PsfO lgod (Identity law) :
a x 1 = 1 x a = a xG' 5 .

[ + x + = + , – x + = – , + x – = –, / – x – = + xG' 5 .
/ – ÷ – = + xG' 5 .
To;u} /L sg' } k0" ffª{ s\ n] csf{] k0" ffª{ s\ nfO{ efu ubf,{

[+ ÷ + = +, + ÷ – = –, – ÷ + = –

o;/L k0" f{ ;ªV\ ofx¿ / ltgLx¿sf ;fwf/0f ljm| ofx¿ Pjd\ ltgLx¿sf] lgodx¿sf af/d] f xfdLx¿n] cl3Nnf
sIffx¿df cWoog ul/;ss] f 5f“} . ca xfdL k0" f{ ;ªV\ ofx¿sf ;/nLs/0fsf af/d] f cWoog ub5{ f“} .

12.1 k0" ffª{ s\ x¿sf] ;/nLs/0f ( Simplification of Integers)

xfdLn] hf8] (+), 36fp (–), u0' fg (×) / efu (÷) ;lDdlnt ;/n ubf{ ;jk{ y| d efusf,] To;kl5
jm| dzM u0' fg, hf8] / 36fpsf] ljm| of ugk{' 5,{ h:t} M

pbfx/0f 1

;/n u/ M 25–24 ÷ 8 + 3×2

;dfwfg

oxf,“ 25 - 24 ÷ 8 + 3 × 2

= 25 – 3 + 3 × 2(÷)

= 25 – 3 + 6 (×)

= 25 + 6 – 3 (+)

= 31 – 3 (–)

= 28

;fgf] sfi] 7af6 jm| dzM demfn} f sfi] 7 / 7n' f] si7sf ljm| ofx¿ ugk{' g{] xG' 5 . To;kl5 sfi] 7leq
jm| dzM dn] aGb - _ efu -÷_ ,u0' fg -X_, hf8] -+_ / 36fp - – _ sf] sfd ul/G5 .

102 xfdf| ] ul0ft, sIff *

pbfx/0f 2
;/n u/ M -19 + [27 - {14 + ( 5-2 ) × 4 ÷ 2}]

;dfwfg

-19 + [27 - {14 + ( 5-2 ) × 4 ÷ 2}] [ ( ) sf] ljm| of ]
= -19 + [27 - {14 + 3 × 4 ÷ 2}] [ ÷ sf] ljm| of ]
= -19 + [27 - {14 + 3 × 2}] [x sf] ljm| of ]
= -19 + [27 - {14 + 6}] [{} sf] ljm| of]
= -19 + [27 - 20 ] ( [ ] sf] ljm| of)
= -19 + 7
= -12

cEof; 12.1

1. ;/n u/ M

-s_ 17-{19-2(1+3)} -v_ 20-{8-(15+2)}

-u_ 25- {16÷(17-9)} -3_ -16 + {8 x(2+4)}

-ª_ 50÷{18 - (4 x 10 ÷ 2)} -r_ [-20÷{40-6(7-2)}] +16

-5_ 5[152 - {7-8(9-2)}] -h_ 11 x 11÷[-11÷{12-(13-12)}]

-em_ 24÷[18-3{5+(6-9)}]+8 -`_ [-2+{11x (8+4)÷3}]+21

-6_ 64÷8-2[3+{7-3(3+4-2)}] -7_ -64 ÷16+[12x{6÷(16÷10-2)}]

-8_ 80÷4[400÷4{7+(19+8-24)}]

2. lbOPsf] tflnsfdf 1 bl] v 9 ;Ddsf cªs\ x¿ gbfx] fl] /g] u/L e/ 6
h;df kT| os] kªS\ ltaf6 / /v] Loaf6 ljs0fx{ ¿sf] ofu] kmn 15 xG' 5 .
51
3. 4 sf] tLg u0' ffaf6 7 36fP/ 5 hf8] b\ f slt xG' 5 < 4

4. 15 sf] 4 u0' ffsf] 6 efusf] 1 efuaf6 3 36fP/ 5 n] u0' ff ubf{ slt xG' 5 <

5. 20 sf] Ps rfy} fOnfO{ 6 n] u0' ff u//] 5 hf8] L 4 36fpb“ f slt xG' 5 <

6. 8 sf] 5 u0' ffnfO{ 4 n] efu u/L 10 hf8] /] 20 36fpb“ f slt xfn] f <

7. 5 / 3 sf] ofu] kmnaf6 6 36fO 9 n] u0' ff ubf{ slt xG' 5 <

8. 64 nfO{ 13 / 9 sf] km/sdf 4 hf8] L efu u//] 8 36fpb“ f slt xG' 5 <

9. 72 df o;s} f] Ps rfy} fO hf8] L cfPsf] ofu] kmndf 72 s} 8 efusf] 1 efu / 1 hf8] b\ f slt xG' 5 <

10. 36 df kml] / ToxL ;ªV\ of, To;sf] cfwf / km] l/ cfwfsf] cfwf hf8] L 1 hf8] b\ f slt xG' 5 <
xfdf| ] ul0ft, sIff *
103

13kf7 cfgk' flts ;ªV\ ofx¿
(Rational Numbers)

104 xfdf| ] ul0ft, sIff *

xfdf| ] ul0ft, sIff * 105

106 xfdf| ] ul0ft, sIff *

;dfwfg
-s_ 6.3 x 103 = 6.3 x 1000 = 6300.0 = 6300
-v_ 4.579 x 106 = 4.579 x 1000000

= 4579000.000 = 4579000

-u_ 7.4 x 10–5 = 7.4  7.4  0.000074 tl/sf M
105 100000
C0ffTds lrxg\ ePsf] 3ftfªs\ nfO{ x/df nh} fg]
-3_ 3.579 x 10–4 = 3.579
104 10 sf] 3ftfªs\ nfO{ lj:tfl/t ¿kdf nV] g]

 0.3579  0.03579 To;kl5 ;ªV\ ofsf] cufl8 x/df ePsf] zG" o
1000 100 a/fa/sf] zG" o ykL bzdnj lrxg\ nfO{ cufl8
a9fpg]
 0.00130579 0.0003579

cEof; 13.1

1. tnsf bzdnj ;ªV\ ofx¿nfO{ j1} flgs ;ªs\ t] df nv] M

-s_ 45 -v_ 3400 -u_ 0.000023 -3_ 101000
-h_ 0.00671
-ª_ 0.010 -r_ 45.01 -5_ 7000000 -7_ 87200
-t_ 3456.78
-em_ 625.6 -`_ 0.07882 -6_ 118000

-8_ 0.00000272 -9_ 0.000037 -0f_ 74171.7

2. tnsf j1} flgs ;ªs\ t] x¿nfO{ bzdnj ;ªV\ ofdf ¿kfGt/0f u/ M

-s_ 2.30 x 104 -v_ 5.40 x 101 -u_ 1.76 x 100 -3_ 1.76 x 10–3

-ª_ 7.4 x 10–5 -r_ 1.901 x 10–7 -5_ 1.525 x 106 -h_ 6.58157 x 107

-em_ 5.256 x 108 -`_ 5.23 x 10–7 -6_ 8.71 x 10–8 -7_ 7.75763 x 10–9

3. Pp6f ;fdfg;lxtsf] 6s« sf] tfn} 12,000 kg 5 eg] pSt tfn} nfO{ j1} flgs ;ªs\ t] nv] .

4. cfug{ sf] k/df0fs' f] cwJ{ of; 0.000,000,000,098 ld6/ eP o;sf] j1} flgs ;ªs\ t] nv] .

5. 3 x 108m/s n] ks| fzsf] xfjfdf ult hgfp5“ eg] To;sf] bzdnj dfg slt xG' 5 <

6. 30 lbg ePsf] dlxgfdf 6480000 ;s] G] 8 xG' 5 eg] o;sf] j1} flgs ;ªs\ t] slt xG' 5 <

xfdf| ] ul0ft, sIff * 107

13.2 j1} flgs ;ªs\ t] df nl] vPsf ;ªV\ ofx¿sf] ;/nLs/0f

(Simplification of Numbers with Scientific Notations)

(I) j1} flgs ;ªs\ t] ePsf ;ªV\ ofx?sf] hf8] / 36fp (Addition and Subtraction)

tnsf pbfx/0f x/] M

pbfx/0f 4

;/n u/ M

-s_ 3.4 x 102 + 4.57 x 103 -v_ 4.54 x 10–3 – 2.4 x 10–3

;dfwfg

-s_ 3.4 x 102 + 4.57 x 103

oxf“ bj' } kbdf 10 sf] 3ftfªs\ a/fa/ 5g} . t;y{ logLx¿nfO{ hf8] g\ ldNbg} / bj' } kbdf 10 sf]
3ftfªs\ a/fa/ agfpgk' g{] xG' 5 .

olb bj' } kbsf] 3ftfªs\ a/fa/ 5g} eg] ;ªV\ ofsf j1} flgs ;ªs\ t] x¿ hf8] g\ / 36fpg ldNbg} .

oxf,“ 3.4 x 102 = 0.34 x 103 xG' 5 . j1} flgs ;ªV\ ofx¿ hf8] b\ f÷36fpb“ f u0' ffªs\
ca, 3.4 x 102 = 4.57 x 103 hfl] 8G5÷36fOG5 / 10 sf] 3ftfªs\ h:tfsf]
t:t} /flvG5 .
= 0.34 x 103 + 4.57 x 103
= (0.34 + 4.57) x 103

= 4.91 x 103

-v_ 4.54 x 10–3 – 2.4 x 10–3

= (4.54 – 2.4) x 10–3 [ bj' d} f ;dfg 3ftfªs\ –3 ePsfn] ]
= 2.14 x 10-3

(II) j1} flgs ;ªs\ t] df nl] vPsf ;ªV\ ofx¿sf] u0' fg / efu

(Multiplication and Division of Numbers with Scientific Notations)

tnsf pbfx/0f x/] f“} M

pbfx/0f 5

;/n u/ M

-s_ (2.00 x 103) x (4.12 x 104) -v_ 9.60  107
1.60  104

108 xfdf| ] ul0ft, sIff *

;dfwfg -bO' { j1} flgs ;ªs\ t] df nl] vPsf ;ªV\ ofnfO{ u0' fg ubf{
-s_ (2.00 x 103) x (4.12 x 104) u0' ffªs\ x¿sf] u0' fg ul/G5 / 3ftfªs\ hfl] 8G5 ._

= 2.00 x 4.12 x 103+4 -sg' } j1} flgs ;ªs\ t] df nl] vPsf] ;ªV\ ofnfO{ csf{]
j1} flgs ;ªs\ t] df nl] vPsf] ;ªV\ ofn] efu ubf{
= 8.24 x 107 u0' ffªs\ n] efu ul/G5 / 3ftfªs\ 36fOG5 ._

-v_ 9.60  107
1.60  104

= 9.60 1074
1.60

= 6.0 x 103

cEof; 13.2

1. ;/n u/ / j1} flgs ;ªs\ t] df nv] M

-s_ (1.2 x 105) + (5.35 x 106) -v_ 6.91 x 10–2 + 2.4 x 10–3

-u_ 9.70 x 106 + 8.3 x 105 -3_ 3.67 x 102 – 1.6 x 101

-ª_ 8.41 x 10–5 – 7.00 x 10–6 -r_ 1.33 x 105 – 4.9 x 104

2. ;/n u/ / j1} flgs ;ªs\ t] df nv] M

-s_ (4.3 x 108) x (2.0 x 106) -v_ (6.0 x 103) x (1.5 x 10–2)

-u_ (1.5 x 10 –2) x (8.0 x 10–1) -3_ (5.23 x 1011) x (3.0 x 10–10)

-ª_ 1.20 108 -r_ 7.8 1012 -5_ 8.4 104
3.0 103 1.3  10 13 1.2  10 3

-h_ 5.6 1018 -em_ 8.1  109 -`_ 3.25  10 10
1.4 108 9.0 108 1.625 10 15

3. ;/n u/ M

-s_ (1.1103)  2.3103 -v_ 9.8 108  4.9 108
1.7  10 6 7.0 107

-u_ (2.1106 )(4.0 103) -3_ 6.48 105
4.2  10 4 (2.4 104 ) (1.8 102)

4. Pp6f 6o\ fªs\ Ldf 3.2 x 104 ln6/ kfgL 5 / bf;] f| ] 6o\ fªs\ Ldf 1.3 x 103 ln6/ kfgL 5 eg] bj' }

6o\ fªs\ Ldf u/L hDdf slt kfgL xfn] f <

5. 2.7 x 109 km kf/ ugk'{ g{] Pp6f /s6] n] 1.35 x 109 b/' L kf/ ul/;Sof] eg] ca slt b/' L kf/
ug{ afs“ L /xo\ f] <

6. 9.6 x 106 ln6/ k6] f« n] nfO{ 1.6 x 103 ln6/sf slt cf6] f a/fa/ 6o\ fªs\ Ldf /fVg ;lsPnf <

xfdf| ] ul0ft, sIff * 109

kf7 jf:tljs ;ªV\ ofx¿

14 (Real Numbers)

14.0. kg' /jnfs] g (Review)

;ªV\ ofx¿sf] ljsf; jm| dsf] nfdf] ;do;Dd sg' } bO' { ;ªV\ ofx¿larsf rf/ ljm| ofx¿ ubf{ cfgk' flts
;ªV\ ofx¿ g} kofK{ t lyP . h:t} M sg' } bO' { ;ªV\ ofx¿ hf8] b\ f, 36fpb“ f, u0' fg ubf{ jf efu ubf{
cfgk' flts ;ªV\ of g} xG' 5 . To;} jm| ddf 2 sf] jud{ n" kQf nufpg, x2 -2 = 0 df x sf] dfg kQf
nufpg cfgk' flts ;ªV\ ofx¿af6 dfq ;Dej ePg / cGTo gxg' ] jf kg' /fjl[ Q gxg' ] bzdnj
;ª\Vofx¿sf] cfjZostf b]lvof] . ;fy} Ps PsfO e'hf ePsf] ju{sf] las0f{sf] nDafO kQf
nufpgsf nflu gof“ ;ªV\ ofx¿sf] cfudg cfjZos bl] vof] / ltgsf] vfh] L eof] . h;nfO{ cgfgk' flts
;ªV\ of (irrational number) elgG5 . h:t} M jQ[ sf] kl/lw / Aof;sf] cgk' ft, 2, 3, cflb . 2
nfO{ ;ªV\ of /v] fdf lgDgfg;' f/ k:| tt' ug{ ;lsG5 M

O nfO{ pbu\ d laGb' dfgL P(1,1) laGb' lnpm / OP hf8] . To;kl5 OP sf] b/' L lgbz{] fªs\
HofldtLåf/f kQf nufpm .

oxf,“ OP  (0 1)2  (0 1)2  1  1  2 xG' 5 .
OP a/fa/sf] cwJ{ of; lnO{ O nfO{ sG] b| dfg/] Pp6f cwj{ Q[ lvr . To; cwj{ Q[ sf] kl/lwn]
;ªV\ of /v] fnfO{ sf6s] f] 7fpd“ f k5{ -s;/L <_ . o;/L Pp6f cgfgk' flts ;ªV\ ofnfO{ klg ;ªV\ of
/v] fdf k:| tt' ug{ ;lsG5 .

14.1. jf:tljs ;ªV\ ofx¿sf] kl/ro (Introduction to Real Numbers)

cfgk' flts ;ªV\ ofx¿sf] ;dx" ( Q ) / cgfgk' flts ;ªV\ ofx¿sf] ;dx" (Ir) sf] ;o+ fh] g ;dx" nfO{
jf:tljs ;ªV\ ofsf] ;dx" elgG5 . o;nfO{ R n] hgfOG5 / R = Q ∪ Ir xG' 5 .
cyft{ , sg' } klg ;ªV\ ofnfO{ ;ªV\ of /v] fdf k:| tt' ug{ ;lsG5 eg] ;f] ;ªV\ ofnfO{ jf:tljs ;ªV\ of
elgG5 .

110 xfdf| ] ul0ft, sIff *

jf:tljs ;ªV\ ofx¿sf] ;dx" nfO{ eg] lrqdf lgDgfg;' f/ bv] fpg ;lsG5 M

QR
Z
W

N

Ir oxf,“ N  W  Z  Q  R,Ir  R xG' 5 .
jf:tljs ;ªV\ ofx¿nfO{ lgDgfg;' f/ kj| fx tflnsf (flow chart) af6 bv] fpg ;lsG5 M

jf:tljs ;ªV\ of (R)

cfgk' flts ;ªV\ ofx¿ (Q) cgfgk' flts ;ªV\ ofx¿ (Ir)

k0" ffª{ s\ x¿ (Z) leGg ;ªV\ ofx¿ (F)

k0" f{ ;ªV\ ofx¿ (W) C0ffTds k0" ffª{ s\ x¿ (Z)

kf| sl[ ts ;ªV\ ofx¿ (N) zG" o (O)

hf/] ;ªV\ ofx¿ lahf/] ;ªV\ ofx¿

14.1.2. bzdnj / cgfgk' flts ;ªV\ ofx¿ (Decimal and Irrational Numbers)
tnsf ;ªV\ ofx¿nfO{ x/] M

23  4.75; 20  3.3333....; = 3.14285....
4 6

lbOPsf bzdnj ;ªV\ ofx¿df sg' sg' cGTo xg' ,] sg' bfx] fl] /g] jf kg' /fjl[ Q xg' ] / sg' cGTo
gxg' ] / kg' /fjl[ Q gxg' ] bzdnj ;ªV\ of xg' ,\ 56' o\ fpm .

xfdf| ] ul0ft, sIff * 111

112 xfdf| ] ul0ft, sIff *

xfdf| ] ul0ft, sIff * 113

114 xfdf| ] ul0ft, sIff *

xfdf| ] ul0ft, sIff * 115

;dfwfg

oxf,“ -s_ -oxf“ x/df 5 / 2 nfO{ x6fpg x/ / cz+ bj' d} f 2 n] u0' ff ug_{]

= 32  6
22 2

-v_ 12
5

 223  2 3  5 -x/ cz+ bj' d} f 5 n] u0' fg ubf_{
55 5

 2 3 5  2 15  2 15
5 5 52 5

-u_ 5 2 52  10 -x/ cz+ bj' d} f 2 n] u0' fg u/s] f_]
2 2 22 2

-3_ 3  3  3 3 3 3 3 3
3 3 3 33 3

-ª_ 2 2 3 1  2( 3  1)  2  3 1 2 6 2
3 1 3 1 3 1 ( 3)2  (1)2 31 2

14.2.2 dn" lrxg\ - _ ;dfjz] ePsf ;/n

Pp6} ;ªV\ ofdf dn" lrxg\ ePsf cleJo~hsx¿nfO{ aLhLo cleJo~hsx¿ h:t} hf8] / 36fp
ug{ ;lsG5, h:t} M

pbfx/0f 3

;/n u/ M

-s_ 3 5  5 -v_ 7 2  5 2 -u_ 9 3  3 2  6 3  5 8
;dfwfg
oxf,“ -s_ 3 5  5 -v_ 7 2  5 2

= (3+1) 5 = (7-5) 2

=4 5 =2 2 xfdf| ] ul0ft, sIff *

116

-u_ 9 3  3 2  6 3  5 8  -v_ 3 5  2 2  5 5

9 3 3 2 6 3 5 222 ;dfwfg
 9 3 3 2 6 3 52 2
 9 3  3 2  6 3  10 2 oxf,“  3 5  2 2  5 5
 (9  6) 3  (3  10) 2
 3 3  13 2 3 52 2 3 55 5
 6 10  15 52
pbfx/0f 6  6 10  15 5
u0' fg u/ M  75  6 10
-s_ 2 3  3 2

;dfwfg
oxf,“

2 33 2
 23 3  2  6 32
6 6

dn" lrxg\ ;lDdlnt u0' fg ubf{ dn" lrxg\ aflx/sf] cªs\ ;u“ dn" lrxg\ aflx/sf] ;ªV\ of / dn"
lrxg\ leqsf] ;ªV\ of;u“ dn" lrxg\ leqsf] ;ªV\ of u0' ff ul/G5 .

pbfx/0f 5

;/n u/M -s_ 125  80 -v_ 2 28  3 49 10 7
;dfwfg

-s_ 125  80 -v_ 2 28  3 49  10 7

 555  22225  2 22 7  3 7 7  10 7
 52  5  22  22  5  22 7  3 7  10 7
5 54 5  4 7  21  10 7
9 5  14 7  21
 7(2 7  3)
xfdf| ] ul0ft, sIff *
117

cEof; 14.2

1. tnsf ;ªV\ ofx¿sf] x/sf] cfgk' ftLs/0f u/ M

-s_ 3 -v_ 4 -u_ 7 -3_ 9 -ª_ 22
2 5 8 3 11

-r_ 10 -5_ 11 -h_ 5 3 -em_ 3 2 -`_ 3
48 44 5 1 3 2

-6_ 3 7 -7_ 62
4 24

2. ;/n u/ M

-s_ 3 5  6 5 -v_ 3 10 3 10 -u_ 7 7  5 7  3 7

-3_ 10 3  3 3 -ª_ 3 20  2 45 -r_ 21 7  3 28  63

-5_ 125 5 3 5 -h_  11  121  44

-em_ 128  50 -`_ 63 2 28  5 7
-6_ 288  72  8 -7_ 3 17  68  153

-8_ 12 24  3 216  5 54  600

3. ;/n u/ M

 -s_ 2 3 3 5  5 15  -v_ 3 7  2 28 4 7

 -u_ 9 125  6 180 3 6  -3_ 8 6 3 2  8 48

-ª_ 5 7 3 54 3  -r_ 9 13  4 52  3 117

4. cfgk' ftLs/0f u/L ;/n u/ M

-s_ 3 5 -v_ 5 3 2 2 -u_ 3 1
2 7 55

-3_ 45  125  3 -ª_ 7  300  3 48
5 75

118 xfdf| ] ul0ft, sIff *

kf7 cgk' ft, ;dfgk' ft / kl| tzt

15 (Ratio, Proportion and Percentage)

15.0 kg' /jnfs] g ( Review)

tnsf jfSox¿ k9 / ldNg] jfSonfO{ Ps 7fpd“ f nv] M

-s_ /fd;u“ ?= 450 5 . -v_ kfv] /f – sf7df8fs“} f] a; ef8f ?= 500 5 .

-u_ kD] afsf] tfn} 50 kg 5 . -3_ ljkgf;u“ ?= 500 5 .

-ª_ sf7df8f“} – w/fgsf] a; ef8f ?= 950 5 . -r_ /ljnfnsf] tfn} 55 kg 5 .

dflysf jfSox¿df -v_ / -ª_ bj' } ef8f b/ xg' \ . h;df sf7df8fa“} f6 kfv] /f / w/fgsf] ef8f b/
lbOPsf] 5 . kfv] /f / w/fgsf] ef8f b/ jm| dzM ?= 500 / ?= 950 5 . sf7df8fa“} f6 kfv] /f /

w/fgsf] ef8f cgk' ft  500  10 5 . o;nfO{ 10:19 nl] vG5 .
950 19

To:t,} cGo Pp6} u0' f ePsf kl/df0fx¿ s] s] xg' ,\ kQf nufO{ cgk' ft lgsfn .

15.1. cgk' ft (Ratio)

bO' { cf6] f ;dfg PsfO ePsf kl/df0fnfO{ tn' gf ug{ ko| fu] ul/g] leGgnfO{ cgk' ft elgG5 . olb a
a
/b sf] Pp6} PsfO 5 eg] ltgLx¿sf] cgk' ftnfO{ b jf a:b nl] vG5 . hxf“ a nfO{ klxnf] kb

(antecedent) / b nfO{ bf;] f| ] kb (consequent) elgG5 .

h:t} M kl| dnfsf] prfO 5 lkm6 5 / /ldnfsf] prfO 4 lkm6 5 eg] pgLx¿sf] prfOsf] cgk' ft 5:4
eof] . cgk' ftnfO{ Gog" td -n3Q' d_ leGgdf nl] vG5 .

kD] af / ;fg] fdsf] prfOsf] cgk' ft 4:5 5 . ca kD] afsf] prfO 40 OGr eP ;fg] fdsf] prfO slt xfn] f <

kD] afsf] prfO M ;fg] fdsf] prfO = 4:5

cyjf, kD] afsf] prfO = 4
;fg] fdsf] prfO 5

cyjf, 40 OGr = 4
;fg] fdsf] prfO 5

;fg] fdsf] prfO  40 5  50 OGr .

4

o;/L sg' } cgk' ft / Pp6f kl/df0f yfxf 5 eg] csf{] kl/df0f klg kQf nufpg ;lsG5 .
xfdf| ] ul0ft, sIff *
119

pbfx/0f 1

tnsf kl/df0fx¿nfO{ cgk' ftdf ¿kfGt/0f u/ M

-s_ 200 k;} f / 200 ?lkof“

-v_ 4 kg / 5000 gm

;dfwfg

-s_ oxf,“ 200 k;} f / 200 ?lkof“ bj' d} f Pp6} PsfO 5g} . t;y,{ 200 k;} f = ?= 2 xG' 5 .

ctM cgk' ft = 2 ?lkof“ 1 = 1:100
200 ?lkof“ 100

-v_ 4 kg / 5000 gm

o;df klxnf] kl/df0f = 4 kg

bf;] f| ] kl/df0f = 5000 gm = 5kg

ctM cgk' ft  4kg  4 : 5
5kg

pbfx/0f 2

Ct' / /ZdLn] Pp6f j:td' f 10:13 sf] cgk' ftdf nufgL u/] . olb Ctn' ] ?= 5000 nufgL ul/g\ eg]
/ZdLn] slt ul/g\ xfn] f <

;dfwfg

oxf,“ Ct' / /ZdLsf] nufgLsf] cgk' ft 10:13

Cts' f] nufgL = ?= 5000

/ZdLsf] nufgL = ?

ca, Cts' f] nufgL  10
/ZdLsf] nufgL 13

cyjf, ?= 5000  10
/ZdLsf] nufgL 13

/ZdLsf] nufgL ?=  13 5000  6500
10

120 xfdf| ] ul0ft, sIff *

pbfx/0f 3
/fx] g, laGb' / /fdljnf;n] Pp6f Joj;fodf 3:4:5 sf] cgk' ftdf nufgL u/] . olb pgLx¿n]
?= 36,000,000 hDdf u/5] g\ eg] kT| os] n] slt slt ?lkof“ nufgL u/s] f /x5] g\ <
;dfwfg
oxf,“ hDdf /sd = ?= 36,000,000

/ cgk' ftnfO{ x dfGbf kT| os] sf] nufgL 3x, 4x / 5x xG' 5 .
ca, kZ| gcg;' f/ 3x, + 4x + 5x = ?= 36,000,000
cyjf, 12x = 36,000,000

cyjf, x = 36000000 = ?= 3,000,000

12

To;sf/0f, /fx] gsf] nufgL = 3x = 3 × ?= 3,000,000 = ?= 9,000,000

laGbs' f] nufgL = 4x = 4× ?= 3,000,000 = ?= 12,000,000

/fdljnf;sf] nufgL = 5x = 5 × ?= 3,000,000 = ?= 15,000,000

cEof; 15.1

1. tnsf kT| os] cj:yfdf klxnf] / bf;] f| ] kl/df0fsf] cgk' ftnfO{ Gog" td leGgsf ¿kdf nv] M

-s_ 5 hrs / 10 hrs -v_ 3 ft / 9 ft

-u_ 750 gram / 1.5 kg -3_ 20 cm / 25 cm

-ª_ 375 ml / 1l -r_ Rs. 75 / 750 paisa

2. gk] fn df= lj= sf] lzIfs / ljBfyL{ cgk' ft 1:32 5 . olb pSt ljBfnodf hDdf 25 hgf
lzIfs eP ljBfyL{ ;ªV\ of slt xfn] f <

3. 1:4000 sf] :sn] df lvlrPsf] gS;fdf bO' { 7fpl“ arsf] b/' L 4 cm 5 eg] pSt :yfgx¿larsf]
jf:tljs b/' L slt xfn] f <

4. bO' { ;ªV\ ofx? 3:4 sf] cgk' ftdf /xs] f 5g\ . olb bj' } ;ªV\ ofdf 3 hf8] b\ f 2:3 sf] cgk' ftdf
xG' 5g\ eg] tL ;ªV\ ofx¿ kQf nufpm .

5. ;z' fGt / PGhnn] ?= 600 nfO{ 5:7 sf] cgk' ftdf af8“ b\ f bj' n} ] slt slt ?lkof“ kfpnfg,\ kQf
nufpm .

6. Pp6f kl/jf/df vfgf / lzIffdf vrs{ f] cgk' ft 4:5 5 . olb lzIffdf dfl;s ?= 6750 vr{ xG' 5
eg] vfgfdf slt vr{ rflxPnf <

xfdf| ] ul0ft, sIff * 121

7. 8, 9 / 10 jifs{ f aflnsfx¿nfO{ ?. 216 pgLx¿sf] pd/] sf] cgk' ftdf af8“ b\ f kT| os] n] slt
slt ?lkof“ kfpnfg\ <

8. ljlkg, cdt[ / cflifzn] 2:5:6 sf] cgk' ftdf nufgL u/L Pp6f Joj;fo ;~rfng u/] . Ps
jifk{ l5 pgLx¿n] ?= 65,000,000 cfDbfgL u/] eg] kT| os] n] slt slt /sd cfDbfgL u/] xfn] fg\ <

9. A n] eGbf B n] bfA] a/ / B n] eGbf C n] tA] a/ /sd hDdf ubf{ ?= 98460 hDdf eof] eg]
kT| os] n] slt slt /sd hDdf u/] xfn] fg\ <

15.2. ;dfgk' ft (Proportion)

sIff 8 df 24 hgf 5fqf / 27 hgf 5fq 5g\ . To:t} sIff 9 df 32 hgf 5fq / 36 hgf 5fqf 5g\
eg] bO' c{ f6] f sIffdf slt slt cgk' ftdf 5fq / 5fqf /x5] g,\ kQf nufpm .

bj' } sIffdf 5fq / 5fqflarsf] cgk' ft s:tf] 5, a/fa/ 5 ls 5g} x/] .

sg' } bO' { cgk' ftnfO{ Gog" td leGgdf nV] bf cgk' ft a/fa/ xG' 5 eg] To:tf cgk' ftx¿nfO{ ;dfgk' ft
elgG5 . olb a:b = c:d 5 eg] a:b / c:d ;dfgk' ft xG' 5g\ / a, b, c / d ;dfgk' flts xG' 5g\ . o;nfO{
a:b::c:d klg nl] vG5 .

dflysf] pbfx/0fdf 24 / 32 ;dfgk' flts 5g\ .
27 36

o;nfO{ 24:27 = 32:36 nl] vG5 . Extremes
Means

o;nfO{ 24:27::32:36 klg nl] vG5 . h;df aflx/sf bO' { kbnfO{ extremes elgG5, h:t} M 24 / 36

leqsf bO' { kbnfO{ means elgG5, h:t} M 27 / 32

extremes / means sf] 56' 6\ f 56' 6\ } u0' fgkmn a/fa/ xG' 5 .

cyft{ , a / c ;dfgk' ftdf 5g\ of] ac xG' 5 .
b d bd

cyjf, a × d = b × c xG' 5 .

o;nfO{ ko| fu] u//] ;dfgk' ftdf /xs] f tLgcf6] f kb lbPdf afs“ L kb kQf nufpg ;lsG5 .

pbfx/0f 1

;dfgk' ftdf /xs] f kbx¿dWo] bf;] f| ,] t;] f| ] / rfy} f] kb jm| dzM 4, 6 / 8 eP klxnf] kb kQf nufpm .

;dfwfg . t;y,{ x6 xG' 5 .
oxf“ klxnf] kb x dfgf“} . 48
xfdf| ] ul0ft, sIff *
x, 4, 6 / 8 ;dfgk' flts 5g\

122

xfdf| ] ul0ft, sIff * 123

7. 7 ldg6] df 21 kg ds} lk:g] 366\ nfO{ 15kg ds} lk:g slt ;do nfU5 xfn] f <
8. clgtfsf] ul0ft / lj1fgsf] kf| Ktfªs\ sf] cgk' ft 10:12 5 . olb p;sf] lj1fgsf] kf| Ktfªs\ 80

eP ul0ftsf] kf| Ktfªs\ slt xfn] f <
9. sfk] Lnfn] gl} ts lzIff / Jofj;flos lzIff tyf cªu\ h]| L / lj1fgdf ;dfgk' flts cªs\ kf| Kt

ul/g\ . olb tL ljifox¿df jm| dzM 25, 30, 75 / x kf| Kt ul/g\ eg] x sf] dfg slt xfn] f <
10. ?=180 df 12 cf6] f sfkL kfOG5 eg] ?= 225 df sltcf6] f sfkL kfOG5 <
11. rGbd| f / kY[ jLsf] u?' Tjfsif0{ fsf] cgk' ft 1:6 5 . kY[ jLdf 90 N tfn} ePsf j:ts' f] tfn}

rGbd| fdf slt xfn] f, kQf nufpm .
13. Pp6f ld7fOd{ f bw' / lrgLsf] cgk' ft 5:3 5 . olb bw' 750 gm 5 eg] lrgLsf] efu slt xfn] f <

15.3 kl| tzt (Percentage)

tnsf leGgnfO{ x/] f“} M

sIffsf ;a} ljBfyLn{ fO{ bO' { ;dx" df ljefhg u/L tflnsfdf lbPsf pbfx/0fx¿af/] 5nkmn u/ M

tflnsf s tflnsf v

oxf,“ 1 egs] f] 100 efudf 50 efu /x5] , o;nfO{ 50 kl| tzt elgG5 .
2

To:t} 3 egs] f] 100 efudf 60 efu /x5] , o;nfO{ slt kl| tzt elgG5 <
5

0.33 egs] f] 33 kl| tzt eof] eg] 0.80 a/fa/ kl| tzt slt xfn] f <

33% = 33 = 0.33 xG' 5 .
100

gf6] M leGg jf bzdnjnfO{ kl| tztdf ¿kfGt/0f ug{ 100 n] u0' ff u/L % lrxg\ /fVg] .

124 xfdf| ] ul0ft, sIff *

kT| os] ljBfyLn{ ] tnsf bO' c{ f6] f tflnsfdf ePsf kZ| gx¿sf] pQ/ kQf nufO{ ;dx" df 5nkmn u/ .

tflnsf -s_ tflnsf -v_

50 sf] 16 = slt 50 sf] 20 = < kl| tztnfO{ leGg jf bzdnjdf
100 sf] 16 = < 125 sf] 8 = < ¿kfGt/0f ug{ 100 n] efu u/L
320 sf] 16 = < 250 sf] 4 = < % lrxg\ x6fpg] .
500 sf] 16 = < 500 sf] 2 = <

1020 sf] 16 = < 1000 sf] 1 = <

dflysf] tflnsfaf6 s] yfxf xG' 5 lgisif{ kQf nufpm .

To;sf/0f kl| tzt Pp6f dfkg xf,] h;df sg' } kl/df0fnfO{ 100 sf] efusf ¿kdf JoSt ul/G5 .

15.3.1 lbOPsf] kl| tzt a/fa/ ;ªV\ of kQf nufpg] (To find the number of given percentage)

pbfx/0f 1

560 hgf ;lDdlnt sIff * sf] clGtd k/LIffdf 40% A+, 30% A, 20% B u8]| xfl;n u/] / afs“ Ln]
C u8]| xfl;n u/] eg] slt hgfn] C u8]| xfl;n u/] kQf nufpm .

;dfwfg

oxf“ hDdf ljBfyL{ = 560

A+ xfl;n ug{] ljBfyL{ = 560 sf] 40%
A u8]| xfl;n ug{] ljBfyL{
 560 40  224 hgf
100

= 560 sf] 30

 560 30  168 hgf
100

B u8]| xfl;n ug{] ljBfyL{ = 560 sf] 20%

 560 20  112 hgf
100

A+, A / B u8]| xfl;n ug{] ljBfyL{ ;ªV\ of = 224+168+112 = 504 hgf

ca C u8]| xfl;n ug{] ljBfyL{ ;ªV\ of = 560 - 504 = 56 hgf

csf{] tl/sf, hDdf A+, A / B xfl;n ug{] kl| tzt = 40% + 30% + 20% = 90%

C u8]| xfl;n ug{] ljBfyL{ kl| tzt = 100% - 90% = 10%

ca, C u8]| xfl;n ug{] ljBfyL{ ;ªV\ of = 560 sf] 10%  560 10  56 hgf
100

xfdf| ] ul0ft, sIff * 125

pbfx/0f 2

lbOPsf] tflnsfdf Pp6f k;ndf ljleGg ;fduL| sf] dN" o;r" L lbOPsf] 5 . Pp6f ;6,{ kfOG6 /
Hofs6] lsGgsf nflu hDdf slt ?lkof“ cfjZos k5{ xfn] f <

;dfwfg d"No;"rL
oxf“ hDdf lsGgk' g{] ;fdfgsf] dN" o j:t' dN" o

;6{ ?= 250 ;6{ ?= 250

kfOG6 ?= 475 kfOG6 ?= 475

Hofs]6 ?= 1200 Hofs]6 ?= 1200

hDdf ?= 1925 x/s] ;fdfgdf 20% 56' .

56' kl| tzt = 20%

ca, 56' /sd = 1975 sf] 20%

= ?= 395
hDdf cfjZos ?lkof“ = ?= 1975 - 56'

= ?= 1975 - ?= 395
= ?= 1580
gf6] M o;nfO{ 56' 6\ f56' 6\ } ;fdfgsf] 56' 36fP/ klg ug{ ;lsG5 .

15.3.2. lbOPsf] ;ªV\ ofsf] kl| tzt lgsfNg] (To find the Percentage of Given Number)

pbfx/0f 3

ut jifs{ f] kl| t af/] f lh/f dl;gf] rfdnsf] dN" o ?= 1200 lyof] . clxn] pSt rfdn a9/] ?= 1500
eof] eg] pSt rfdnsf] dN" o slt kl| tzt a9o\ f] <

;dfwfg

oxf,“ ut jifs{ f] rfdnsf] dN" o = ?= 1200

clxn] rfdnsf] dN" o = ?= 1500

a9s] f] dN" o = ? 1500 - ?= 1200 = ?= 300

a9s] f] kl| tzt = <

126 xfdf| ] ul0ft, sIff *

ca, a9s] f] kl| tzt = x dfGbf
?=1200 sf] x = ?=300

cyjf,

∴

t;y{ pSt rfdnsf] dN" o 25% n] jl[ 4 eof] .
-gf6] M kl| tzt lgsfNbf k/' fgf] kl/df0fsf] ;fkI] fdf lgsflnG5 . h:t} M rfdnsf] dN" o ?= 1200 sf]
lgsflnof] t/ ?= 1500 sf] cfwf/df xfO] g ._

cEof; 15.3

1. tnsf leGg jf bzdnjnfO{ kl| tztdf ¿kfGt/0f u/ M

-s_ -v_ 0.34 -u_ -3_ 0.59 -ª_

2. lbOPsf kl| tztnfO{ leGgdf ¿kfGt/0f u/ M

-s_ 45 -v_ 70% -u_ 25 % -3_ 91% -ª_ 53%
4

3. dfg kQf nufpm M

-s_ 250 sf] 10% -v_ 150 sf] 90% -u_ 180 sf] 12.5% -3_ 220 sf] 20%

4. slt kl/df0fsf]

-s_ 15% n] ? 225 xG' 5 < -v_ 21% n] 42 ld6/ xG' 5 <

-u_ 25% n] 12.5 lbg xG' 5 < -3_ 12% n] 72 hgf ljBfyL{ xG' 5 <

5. b;}“ ahf/df Pp6f Rofªu\ f| sf] dN" o ?= 12,000 lyof,] h;df 12% 56' lyof] eg] slt ¿ko} f“ 56'
/x5] , 56' kl5 pSt Rofªu\ f| ] lsGg slt ltg{' knf{ <

6. sIff 7 sf 75 ljBfyLx{ ¿df 8% cgQ' L0f{ eP eg] slt hgf pQL0f{ eP <

7. ?= 18,500 tna ePsf] Ps hgf sdr{ f/Ln] 13% /sd s/ ltgk{' 5{ eg] slt /sd s/ ltgk{' nf{ <

8. sIff 8 sf 80 ljBfyLx{ ¿dWo] 5 hgf cgk' l:yt eP eg] slt kl| tzt ljBfyL{ pkl:yt eP <

xfdf| ] ul0ft, sIff * 127

9. Pp6f ;x/sf] hg;ªV\ of 2,666,200 5 / jl[ 4 b/ 1.50% 5 eg] Ps jif{kl5 pSt
hg;ªV\ of sltn] a9n\ f, kQf nufpm .

10. ?= 17,000 cfDbfgL ePsf] Pp6f lzIfsn] 15% cfos/ ltgk{' 5{ eg] s/ lt/k] l5 slt /sd kf| Kt
u5g{ \ xfn] f, kQf nufpm .

11. Pp6f ;x/sf] hg;ªV\ of hDdf 3,40,000 5 . ltgLx¿dWo] 25,500 hgfn] sDKo6' / ko| fu] u5g{ \
eg] slt kl| tztn] sDKo6' /sf] ko| fu] ubf{ /x5] g\ <

12. tnsf] tflnsfdf ljleGg j:tx' ¿sf] dN" o / 56' kl5sf] dN" o lbOPsf] 5 M

j:t' dN" o -?=_ 56' kl5sf] dN" o -?=_

6f]kL 350 315

;6{ 500 420

h'Qf 950 900

kfOG6 800 720

Hofs]6 1250 1100

emf]nf 600 500

dflysf] tflnsf ko| fu] u/L tnsf j:tx' ¿sf] 56' kl| tzt kQf nufpm .

-s_ 6fk] L -v_ ;6{ -u_ hQ' f

-3_ kfOG6 -ª_ Hofs6] -r_ emfn] f

13. cfOtaf/ lrl8of3/ 3D' gs] f] ;ªV\ of 840 lyof] . ;fd] af/ pSt ;ªV\ of 36/] 420 eof] eg] slt
kl| tztn] 36o\ f] xfn] f, kQf nufpm .

14. ?= 4500 dN" o ePsf] dfa] fOn ;6] nfO{ ?= 4200 df lsGg ;:tf] k5{ ls 7% 56' df lsGbf ;:tf]
knf,{ kQf nufpm .

15. ljsf;n] Pp6f /ªu\ Lg TV ;6] nfO{ 13% 56' df lsGbf ?= 30,00 56' kfof] eg] pSt TV ;6] sf]
cªl\ st dN" o slt /x5] , kQf nufpm .

16. /ljgn] 900 k0" ffª{ s\ df 780 cªs\ kf| Kt u/] / ljkgfn] 800 k0" ffª{ s\ df 700 kf| Kt ul/g\ eg]
s;n] w/] } kl| tzt cªs\ kf| Kt u¥of] xfn] f, kQf nufpm .

128 xfdf| ] ul0ft, sIff *

kf7

16 gfkmf / gfS] ;fg (Profit and Loss)

16.0. kg' /jnfs] g ( Review)

tnsf kZ| gx¿df 5nkmn u/f“} M

-s_ Pp6f 38LnfO{ ?= 450 df lsg/] ?= 500 df aR] bf slt gfkmf jf 3f6f xG' 5 <

-v_ Pp6f sndnfO{ ?= 50 df lsg/] ?= 40 df aR] bf slt gfkmf jf 3f6f xG' 5 <

kZ| g g=+ -s_ df gfkmf eof] lsgls o;df ljjm| o dN" o w/] } 5 . o:tf] cj:yfdf gfkmf egs] f] ljjm| o
dN" o / jm| o dN" osf] km/s xf] .

cyft{ gfkmf (profit) = ljjm| o dN" o (selling price) - jm| o dN" o (cost price) xG' 5 .

To:t} bf;] f| d] f jm| o dN" oeGbf ljjm| o dN" o sd 5 .
t;y{ gfS] ;fg eof] / gfS] ;fg = jm| o dN" o – ljjm| o dN" o xG' 5 .

klxnf]df, bf];|f]df

gfkmf = ?= 50 gfS] ;fg = ?= 10

jm| o dN" o = ?= 450 5 jm| o dN" o = ?= 50

ca, 50  100 % gfkmf kl| tzt xf] . ca, gfS] ;fg kl| tzt  10 100%  20% eof] .
450 50

∴ gfkmf kl| tzt = jf:tljs gfkmf x 100% ∴ gfS] ;fg kl| tzt =jf:tljjs|m=dg'f]S;fgx 100%
j|m=d'

pbfx/0f 1

?= 3450 df lsg]sf] Pp6f afvf| nfO{ 2 dlxgfkl5 a]Rbf ?= 1450 gfS] ;fg eof] eg] pSt afvf| sf]
ljj|mo d"No kQf nufpm .

;dfwfg

oxf“ jm| o dN" o (C.P.) = ?= 3450
gfS] ;fg (L) = ?= 1450
ljjm| o dN" o (S.P.) = ?

xfdf| ] ul0ft, sIff * 129

xfdLnfO{ yfxf 5, gfS] ;fg (L) = C.P. - S.P.

?= 1450 = ?= 3450 - S.P.

cyjf, S.P. = ?= (3450 - 1450) = ?= 2,000

∴ ljjm| o dN" o (S.P.) = ? 2,000

pbfx/0f 2

:dLtfn] ?= 1500 df 50 cf6] f aNa NofOg\ . h;df 4 cf6] f km\ oh' uO;ss] f /x5] g\ . afs“ L aNax¿nfO{
pgn] kl| t aNa ?= 35 sf b/n] aR] bf pgnfO{ slt kl| tzt gfkmf jf gfS] ;fg eof] xfn] f <

;dfwfg

oxf“ 50 cf6] f aNasf] jm| o dN" o (C.P.) = ?= 1500

km\ oh' uPsf aNa ;ªV\ of = 4

l7s cj:yfdf ePsf aNa = 50 - 4 = 46

Pp6f aNasf] ljjm| o dN" o = ?= 35

46 cf6] f aNasf] ljjm| o dN" o (S.P.) = 35 × 46 = ?= 1610

oxf,“ jm| o dN" oeGbf ljjm| o dN" o w/] } eof] . o; sf/0f pgnfO{ gfkmf eof] .

ctM gfkmf = S.P. - C.P.

= ?= 1610 - ?=1500 = ?= 110

ca, gfkmf kl| tzt = jf:tljs gfkmf x 100%
jm| o dN" o

= 110 100%
1500

 22 %
3

= 71%
3

= 7.33%

ctM pgnfO{ 7.33% gfkmf eof] .

130 xfdf| ] ul0ft, sIff *

cEof; 16.1

1. tnsf cfs“ 8fx¿ ko| fu] u//] gfkmf jf gfS] ;fg kQf nufpm M

jm| o dN" o -?=_ ljjm| o dN" o -?=_

-s_ 300 330

-v_ 5000 4500

-u_ 7000 7700

-3_ 10,000 9,990

2. kZ| g g= 1 sf cfs“ 8fx¿af6 gfkmf jf gfS] ;fg kl| tzt kQf nufpm .

3. cdt[ fn] Pp6f ;f8L ? 1350 df aR] bf ? 150 gfkmf eof] eg] pSt ;f8Lsf] jm| o dN" o slt xfn] f <

4. ?= 760 df lsgs] f] Pp6f SofNsn' 6] / aR] bf ?= 50 gfS] ;fg eof] eg] pSt SofNsn' 6] /sf] ljjm| o
dN" o slt /x5] , kQf nufpm .

5. ?= 1450 df lsgs] f] Pp6f afvf| nfO{ ?= 1740 df aR] bf slt kl| tzt gfkmf jf gfS] ;fg eof] <

6. ?= 15000 sf] ;fOsnaf6 10% gfkmf lng sltdf aR] g' knf{ <

7. ?= 13000 df lsgs] f] uf?] nfO{ ?= 14300 df aR] bf slt kl| tzt gfkmf jf gfS] ;fg xG' 5 <

8. ?= 58,500 df lsgs] f] Pp6f df6] /;fOsn aR] bf 8% 3f6f eof] eg] pSt df6] /;fOsnsf] ljjm| o
dN" o kQf nufpm .

9. /fh] Ln] 100 cf6] f cG8f ? 900 df lslgg\ h;df 8 cf6] f cG8f km6' s] f /x5] g\ . afs“ L cG8fnfO{
pgn] kl| t uf6] f ? 10.50 df aR] bf slt kl| tzt gfkmf jf gfS] ;fg eof] xfn] f <

10. ke| n' ] 500 cf6] f sv' /' f lsgs] fdf 75 cf6] f lr;fn] ] d/] . afs“ L sv' /' f kl| t Pssf] ?= 120 df
aR] bf p;n] 2% gfkmf eof] eg] sv' /' fsf] hDdf jm| o dN" o slt /x5] <

11. Ps hgf vfBfGg k;nn] ] ?= 40 kl| t sh] Lsf] 50 sh] L / ?= 50 kl| t sh] Lsf] 50 sh] L rfdn
ldnfP/ kl| t sh] L ?= 48 df aR] of] eg] p;nfO{ slt kl| tzt gfkmf jf gfS] ;fg eof] <

12. Pp6f sDKo6' / ;6] ?= 40,000 df lajm| L ubf{ 25% gfkmf eof] eg] o;sf] jm| o dN" o slt xfn] f <

13. PGhnn] bO' c{ f6] f lstfa kT| os] sf] ?= 500 sf b/df lsGof] . pSt lstfa lajm| L ubf{ p;nfO{
jm| dzM Pp6f lstfadf 25 % gfkmf / csf{] lstfadf 25 % gfS] ;fg eof] eg] p;nfO{ slt
kl| tzt gfkmf jf gfS] ;fg eof] xfn] f <

xfdf| ] ul0ft, sIff * 131

16.1 56' ( /Discount ) dN" o clejl[ 4 s/ (Value Added Tax)

16.1.1 56' (Discount) cfrfo{ k:' ts k;n

lbOPsf] lansf] cWoog u/L tnsf kZ| gsf] au/, kfv] /f

pQ/ vfh] L u/ M gfd M ljkgf e08f/L

-s_ lstfasf] cªl\ st dN" o slt 5 < jm| =;=+ lstfasf] gfd dN" o -?_ kl/df0f /sd-?=_

-v_ 56' slt kl| tzt /x5] < 1. zAbsfz] 350 1 350

-u_ slt /sd 56' kfOof] <

-3_ ljkgfn] pSt zAbsfz] nfO{ slt 56' M 12% cfpg] /sd 42
/sd ltl/g\ <

dflysf kZ| gx¿sf pQ/x¿af/d] f hDdf /sd -?_ 308
;dx" df 5nkmn u/L lgisif{ kQf nufpm .
cIf?kL ?= tLg ;o cf7 dfq . lajm| t] f

Jofkf/Ln] ;fdfgsf] dN" o lgwf/{ 0f u/L uf| xsnfO{ atfpg] dN" onfO{ cªl\ st dN" o (marked price)
elgG5 . sg' } j:ts' f] cªl\ st dN" o df sx] L /sd sd u/L lajm| L ul/Psf] 5 eg] pSt sd ul/Psf]
/sdnfO{ 56' (discount) elgG5 . 56' cªl\ st dN" osf] ;fkI] fdf xG' 5 .
cyft{ , 56' /sd = cªl\ st dN" o (M.P.) sf] 56' kl| tzt

= M.P. × 56' kl| tzt xG' 5 .

cªl\ st dN" odf sx] L 56' u//] ;fdfg lslgG5 eg] 56' kl5sf] dN" onfO{ jf:tljs dN" o elgG5 .

jf:tljs dN" o = M.P. - 56' /sd xG' 5 .

pbfx/0f 1

Pp6f emfn] fsf] cªl\ st dN" o ?= 750 5 . olb pSt emfn] f lsGbf 8% 56' kfOG5 eg] emfn] fsf]
jf:tljs dN" o slt xfn] f, kQf nufpm .

;dfwfg

oxf“ cªl\ st dN" o (M.P.) = ?= 750

56' kl| tzt = 8%

ca, 56' /sd = M.P. sf] 8%

= ?= 750 sf] 8

132 xfdf| ] ul0ft, sIff *

 750  8 = ?= 60
100

To; sf/0f, ljjm| o dN" o = cªl\ st dN" o — 56'

= ?= 750 - 60

= ?= 690

16.1.2 dN" o clejl[ 4 s/ (Value Added Tax)

lbOPsf] lansf] cWoog u/L / s] s] kfp5“ f,}

;fyLx¿lar 5nkmn u/ . tn' ;L lahn' L k;n

oxf“, hgsk'/wfd
gfd M sNkgf /fo ofbj
lx6/sf] cªl\ st dN" o ?= 1700 5 .
jm| =;=+ ;fdfgsf] gfd dN" o -?=_ kl/df0f /sd -?=_
d"=c=s = 13%
1. lx6/
ltgk{' g{] /sd = ?= 1921 1700 1 1700

d"=c=s= = ?= 1921 - ?= 1700

= ?= 221

a9s] f] /sd kl| tzt  221 100%  13% d=' c=s= 13% n] cfpg] /sd 221
1700 hDdf /sd -?=_
?= 1921

cIf?kL PM=s===x=h==f/===g==f}==;=o===P==S=s=f=O=;===d==fq . ljjm]| tf

j:t' tyf ;j] f lajm| L ubf{ kT| os] r/0fdf jl[ 4 xg' ] dN" odf nfUg] s/nfO{ dN" o clejl[ 4 s/ (VAT)
elgG5 . cfkmn" ] lsgs] f] j:td' f 9j' fgL, ladf, sld;g cflb hf8] /] ;j] f zN' s / 56' 36fP/ dN" o

clejl[ 4 s/ (VAT) nfUg] dN" o sfod ul/G5 . ;fy} 56' lbPsf] j:td' f 56' 36fP/ cfPsf] dN" odf dN" o
clejl[ 4 s/ (VAT) nfUg] ub5{ . dN" o clejl[ 4 s/ j:ts' f] ljjm| o dN" odf hfl] 8G5 . dN" o clejl[ 4
s/ hf8] k] l5sf] dN" onfO{ jf:tljs dN" o elgG5 .

VAT% = VAT /sd X 100 VAT /sd = jf:tljs dN" o - ljjm| o dN" o
ljjm| o dN" o

pbfx/0f 2

?= 1500 ahf/ dN" o ePsf] Pp6f /l] 8of] ;6] lsGbf 10% 56' kfOG5 / 13% dN" o clejl[ 4 s/ (VAT)
ltgk{' 5{ eg] pSt /l] 8of] ;6] nfO{ slt ?lkof“ ltgk{' nf{ <

xfdf| ] ul0ft, sIff * 133

;dfwfg
oxf,“ cªl\ st dN" o (M.P) = ?= 1500

56' = 10%

56' /sd = ?= 1500 sf] 10%  1500  10 = ?= 150
100

To; sf/0f, 56' kl5sf] /sd = ?= 1500 - ?= 150 = ?= 1350

VAT = 13%

ca, VAT /sd = ?=1350 sf] 13%

 1350  13 = ?= 175.50
100

ca /l] 8ofs] f] ljjm| o dN" o = ?= 1350 ± ?= 175.50 = ?= 1525.50
To;sf/0f pSt /l] 8of] ;6] lsGg ?= 1525.50 ltgk'{ 5{ .
gf6] M 56' nfO{ cªl\ st dN" o (M.P) af6 36fOG5 eg,] VAT nfO{ ljjm| o dN" o (S.P.) df hfl] 8G5 .

cEof; 16.2

1. ?= 210 cªl\ st dN" o ePsf] lstfadf 12% 56' 5 eg] ;f] lstfanfO{ slt ltgk{' nf{ <

2. Pp6f Hofs6] sf] cªl\ st dN" o ?= 2250 5 . olb k;nn] ] pSt Hofs6] df 8% 56' df lajm| L u5{ eg]
pSt Hofs6] lsGgsf nflu slt ?lkof“ ltgk{' nf{ <

3. tnsf j:tx' ¿sf] jf:tljs dN" o kQf nufpm M

j:t' cªl\ st dN" o (MP) 5'6
b/fh ?= 9950
sDKo'6/ ?= 25,500 12%
38L ?= 1250 8%
SofNs'n]6/ ?= 1500 5%
7%

4. olb 10% 56' df lsGbf Pp6f /ªl\ ug l6eL ;6] nfO{ ?= 13950 k¥of] eg] ;f] TV ;6] sf] cªl\ st
dN" o slt xfn] f, kQf nufpm .

5. Pp6f cfO/gsf] cªl\ st dN" o ?= 500 5 . k;nn] ] pSt cfO/gnfO{ ?= 460 df lajm| L ubf{ p;n]
slt kl| tzt 56' lbof,] kQf nufpm .

134 xfdf| ] ul0ft, sIff *

6. tnsf j:tx' ¿sf] cªl\ st dN" o kQf nufpm M

j:t' 5'6 56' kl5sf] dN" o jf jf:tljs dN" o
-s_ :ofp 3% ?= 116.40 kl| t ls=uf| =
-v_ cfn' 4% ?= 144 kl| t wfgL{
-u_ bfn 7% ?= 186 kl| t 2kg
-3_ Rofp 9% ?= 409.50 kl| t kg

7. tnsf j:tx' ¿sf] 56' kl| tzt kQf nufpm M

j:t' cªl\ st dN" o 56' kl5sf] dN" o

-s_ df]afOn ?= 7,000 ?= 6440

-v_ /]l8of] ?= 1160 ?= 1044

-u_ l6=eL= ?= 6400 ?= 6080

-3_ lx6/ ?= 5950 ?= 5355

8. 14% 56' df lsGbf Pp6f l:j6/nfO{ ?= 1075 k¥of] eg] ;f] l:j6/sf] cªl\ st dN" o slt xfn] f <

9. Pp6f 6rn{ fO6sf] jm| o dN" o ?= 1400 5 . To; 6rs{ f] cªl\ st dN" o jm| o dN" osf] 40% n] a9L
5 . olb k;nn] ] pSt 6rn{ fO{ 20% 56' df aR] of] eg,]

-s_ pSt 6rs{ f] cªl\ st dN" o slt xfn] f <

-v_ jm]| tfn] slt ?lkof“ 56' kfof] <

-u_ jm]| tfn] slt ?lkofd“ f pSt 6r{ lsGof] <

-3_ k;nn] ] pSt 6ra{ f6 slt ?lkof“ gfkmf u¥of] < kQf nufpm .

10. tnsf j:tx' ¿ lsGbf ltgk{' g{] lan /sd kQf nufpm M

j:t' cªl\ st dN" o 5'6 d"=c=s=

-s_ ljBt' Lo hu ?= 980 5% 13%

-v_ l6=eL= ;6] ?= 22,500 11% 13%

-u_ dfa] fOn kmfg] ?= 6,800 14% 13%

-3_ sDKo'6/ ?= 10,500 13% 13%

11. ?= 1600 sf] ljBt' sf] landf 3% 56' lnO{ 13% d=" c=s= hf8] b\ f hDdf slt ltgk{' nf{ <

12. kl| t JolSt 200 sf] 6 hgfsf] hDdf landf 8% 56' kl5 13% VAT hf8] b\ f slt /sd ltgk{' 5{ <

13. ?= 4500 cªl\ st dN" o ePsf] ;fOsnnfO{ 13% 56' kl5 13% d=" c=s= ltbf{ slt ?lkof“ knf{ <

xfdf| ] ul0ft, sIff * 135

17kf7 Pl] ss lgod
(Unitary Method)

17.0. kg' /jnfs] g (Review)

tnsf] tflnsf x/] f“} / lbOPsf kZ| gx¿sf af/d] f 5nkmn u/f“} M

tflnsf 1

l6s6 ;ªV\ of 12 8 4 6 1

hDdf dN" o -?=_ 60 40 20 30 ?

k|Zgx¿
-s_ 12 cf6] f l6s6sf] dN" o slt ?lkof“ 5 <
-v_ 6 cf6] f l6s6sf] dN" o slt 5 <
-u_ 1 cf6] f l6s6sf] dN" o slt xfn] f <
-3_ l6s6 ;ªV\ of / dN" olar s:tf] ;DaGw /xs] f] 5 <

tflnsf 2

sfd k/' f ug{ nfUg] lbg 24681

hDdf sfdbf/ ;ªV\ of 12 6 4 3 ?

k|Zgx¿
-s_ 2 hgfnfO{ sfd k/" f ug{ slt lbg nfUnf <
-v_ 6 hgfnfO{ sfd k/" f ug{ slt lbg nfUnf <
-u_ 1 hgfnfO{ slt lbg nfUnf <
-3_ sfdbf/ ;ªV\ of / sfd k/" f ug{ nfUg] lbglar s:tf] ;DaGw /xs] f] 5 <

17.1 kT| oIf / ckT| oIf ljr/0f (Direct and Indirect Variation)

dflysf] tflnsf 1 af6 l6s6sf] ;ªV\ of 36b\ } hfb“ f hDdf dN" o klg 36b\ } uPsf] / l6s6 ;ªV\ of a9b\ f
hDdf dN" o klg a9s] f] yfxf kfpg ;lsG5 . To;nfO{ kT| oIf ljr/0f ePsf] dflgG5 .

bO' c{ f6] f r/x¿dWo] Pp6f r/df ePsf] sdL jf jl[ 4n] csf{] r/df klg ToxL cgk' ftdf sdL jf jl[ 4
bl] vG5 eg] tL r/x¿larsf] ;DaGwnfO{ kT| oIf ljr/0f (direct variation) elgG5 .

136 xfdf| ] ul0ft, sIff *

To:t,} tflnsf g= 2 df sfd ug{] lbg a9fpb“ } hfb“ f hDdf sfdbf/ ;ªV\ of 36b\ } uPsf] kfOG5 . t;y{
sfd ug{] lbg / sfdbf/larsf] ;DaGw ljk/Lt xG' 5 . To;n} ] sfd ug{] lbg / sfdbf/larsf] ;DaGw
ckT| oIf ljr/0f ePsf] dflgG5 .

bO' c{ f6] f r/x¿dWo] Pp6f r/df sdL jf jl[ 4 xb“' f csf{] r/df ToxL cgk' ftdf jl[ 4 jf sdL cfp5“
eg] tL r/x¿larsf] ;DaGwnfO{ ckT| oIf ljr/0f (Indirect Variation) elgG5 .

sg' } Ps PsfO j:ts' f] dfg kQf nufP/ w/] } jf yf/] } j:ts' f] dfg kQf nufpg] ul0ftLo ljlwnfO{
Pl] ss lgod elgG5 .

pbfx/0f 1

10 kg :ofpsf] dN" o ?= 750 k5{ eg] 6kg :ofpsf] dN" o slt knf{ <

;dfwfg

:ofpsf] kl/df0f / :ofpsf] dN" odf xb] f,{

a9L :ofp eP a9L dN" o, sd :ofp eP sd dN" o

10 kg :ofpsf] dN" o ?= 750 k5{ .

1 kg :ofpsf] dN" o ?= 750 k5{ . ( kT| oIf ljr/0f ePsfn] kl/df0f 36b\ f dN" o klg 365\ .
10
 To; sf/0f 750 nfO{ 10 n] efu ug{] ._

= ?= 75

6 kg :ofpsf] dN" o ?= 75 × 6

= ?= 450 k5{ ( kT| oIf ljr/0fdf kl/df0f a9b\ f dN" o klg a95\ .
To; sf/0f 75 nfO{ 6 n] u0' ff ug{] ._

To;n} ,] 6 kg :ofpsf] dN" o ?= 450 k5{ .

pbfx/0f 2

18 lbgdf sg' } sfd k/" f ug{ 10 hgf sfdbf/ rflxG5 . ToxL sfd 15 lbgdf k/" f ug{ slt hgf yk
sfdbf/sf] cfjZostf knf{ <

;dfwfg

oxf,“ sfd ug{] lbg / sfdbf/ ;ªV\ of xb] f,{

sd lbg eP a9L sfdbf/ rflxG5 .

a9L lbg eP sd sfdbf/ rflxG5 .

xfdf| ] ul0ft, sIff * 137

138 xfdf| ] ul0ft, sIff *

xfdf| ] ul0ft, sIff * 139

cEof; 17.1

1. 12 cf6] f sIffsf7] f ePsf] ljBfnodf hDdf 300 hgf ljBfyL{ Ifdtf lyof] . olb 375 hgf
ljBfyL{ egf{ eP eg] yk sltcf6] f sIffsf7] f rflxPnf <

2. olb 4 bhg{ sndsf] dN" o ?= 576 k5{ eg] ?= 228 df slt cf6] f snd kfOPnf <
3. Ps hgf wfjsn] 45 ldg6] df 18 km bf8} k/' f ug{ ;S5 eg] 30 km b/' L kf/ ug{ slt ;do

nfUnf < kQf nufpm .
4. Pp6f dfnafxs 6s« 48 km kl| t 306fn] u8' b\ f sg' } b/' L 6 306fdf k/" f ub5{ . olb pSt 6s« sf]

ult 36/] 36 km kl| t 306f eof] eg] pSt b/' L slt 306fdf kf/ unf{ <
5. sg' } Pp6f sfd k/" f ug{ 20 hgf sfdbf/nfO{ 15 lbg nfU5 . pSt sfd 12 lbgdf l;Wofpg

slt hgf sfdbf/ yKgk' nf{ <
6. Ps lsnf] lk7fs] f] dN" o ?= 28 xb“' f Pp6f /f6] Lsf] tfn} 496 uf| d lyof] . olb lk7fs] f] dN" o ?= 32

kl| t s=] hL= xb“' f /f6] Lsf] tfn} slt xfn] f < -dfgf,“} /f6] Lsf] dN" o oyfjt\ /xG5 ._
7. sg' } sfd k/" f ug{ 12 hgfnfO{ 14 lbg nfU5 . olb sfdbf/ yk/] 21 hgf agfOof] eg] pSt sfd

slt lbgdf ;lsPnf <
8. sg' } Pp6f Jof/s] df 200 hgf l;kfxLnfO{ 30 lbg kU' g] /f;g 5 . pSt /f;g 40 lbgnfO{

k¥' ofpg slt hgf l;kfxLnfO{ cGoq ;fgk{' nf{ <
9. Pp6f df6] /;fOsn 50 km kl| t 306fsf b/n] u8' b\ f sg' } b/' L kf/ ug{ 7 306f nfU5 . olb

p;nfO{ 5 306fdf pSt b/' L kf/ ugk{' ¥of] eg] pSt df6] /;fOsnsf] ult sltn] a9fpg'
knf{ <
10. 3 cf6] f s;' L{ / 4 cf6] f 6a] nsf] hDdf dN" o ?= 7,540 k5{ . olb Pp6f s;' Ls{ f] dN" o ?= 220
k5{ eg] Pp6f 6a] nsf] dN" o kQf nufpm .
11. 5 cf6] f ufO{ / 2 cf6] f uf?] sf] hDdf dN" o ?= 1,35,000 5 . olb Pp6f uf?] sf] dN" o ?= 17,500
eP Pp6f ufOs{ f] dN" o slt xfn] f <

140 xfdf| ] ul0ft, sIff *

kf7 ;fwf/0f Aofh

18 (Simple Interest)

18.0 kg' /jnfs] g (Review)
tnsf jfSosf] cWoog u/L lbOPsf kZ| gx¿sf] pQ/ vfh] M

xfdf| ] ul0ft, sIff * 141

142 xfdf| ] ul0ft, sIff *

cEof; 18.1

1. ;fwf/0f Aofh (I) kQf nufpm M

-s_ ;fj“ f = ?= 500 Aofh b/ (R)= 3% ;do = 3 jif{
;do = 2 jif{
-v_ ;fj“ f = ?= 9500 Aofh b/ (R) = 11 % ;do = 4 dlxgf
;do = 1 dlxgf
2
Aofh = ?= 378
-u_ ;fj“ f = ?= 12600 Aofh b/ (R) = 15 % Aofh = ?= 650
Aofh = ?= 900
2 Aofh = ?= 350

-3_ ;fj“ f = ?= 9990 Aofh b/ (R) = 24% Aofh = ?= 292
5 Aofh = ?= 1080
2. ;do (T) kQf nufpm M Aofh = ?= 1155
-s_ ;fj“ f = ?= 1260 Aofh b/ (R) = 5% Aofh = ?= 648
-v_ ;fj“ f = ?= 1250 Aofh b/ (R) = 13%
-u_ ;fj“ f = ?= 4500 Aofh b/ (R) = 4% Aofh = ?= 39.96
Aofh = ?= 400
-3_ ;fj“ f = ?= 2400 Aofh b/ (R) = 25 % Aofh = ?= 2062.50
3
143
3. Aofh b/ (R) kQf nufpm M

-s_ ;fj“ f = ?= 1460 ;do (T)= 30 dlxgf

-v_ ;fj“ f = ?= 7,200 ;do (T)= 5 jif{

-u_ ;fj“ f = ?= 6,000 ;do (T)= 3 jif{ 6 dlxgf

-3_ ;fj“ f = ?= 2,160 ;do (T)= 4 jif{

4. ;fj“ f kQf nufpm M

-s_ Aofh b/ (R) 24 % ;do (T)= 1 dlxgf
5

-v_ Aofh b/ (R) 62% ;do (T)= 5 jif{
3 ;do (T)= 15 jif{

-u_ Aofh b/ (R) 41%
6

xfdf| ] ul0ft, sIff *

-3_ Aofh b/ (R) = 9% ;do (T)= 9 jif{ Aofh = ?= 810

5. dGhn' ] jflifs{ 7% sf b/n] Aofh kfpg] u/L ?= 3500 gk] fn aª} s\ lnld68] df hDdf ul/g\
eg] 4 jifk{ l5 pgn] slt Aofh kfpl“ 5g,\ kQf nufpm .

6. dfOnf bfOn] jflifs{ 6.6% Aofh b/df aª} s\ af6 ?= 18000 C0f lnP eg] 30 dlxgfkl5 pgn]
aª} s\ df slt Aofh ae' mfpgk' nf,{ kQf nufpm .

7. sfhLn] /fli6o« jfl0fHo aª} s\ af6 4 jifk{ l5 ?= 550 Aofh kf| Kt ug{ rfxG5 . p;n] clxn] 5.5%
Aofh b/df slt /sd hDdf ugk{' nf,{ kQf nufpm .

8. lagfnfO{ ?= 7600 aª} s\ df /fv] afkt aª} s\ n] 3 jifk{ l5 ?= 1254 Aofh lbof] eg] Aofh b/ slt
/x5] , kQf nufpm .

9. ?= 12, 000 nfO{ aª} s\ df 25 % Aofhb/n] /fVbf slt jifd{ f ;fj“ f a/fa/ Aofh xG' 5 <
2

10. 7 jifk{ l5 Aofh ?= 4200 kfpgsf nflu 6% Aofh b/df clxn] slt /sd hDdf ugk{' nf{ <

11. 10% Aofh b/n] ?= 1080 sf] 4 jifd{ f slt Aofh cfpnf / slt jifd{ f ?= 900 sf] 12% sf
b/n] plQg} Aofh cfp5“ <

18.2. ld>wg (Amount)

/fl] hgfn] aª} s\ df ?= 10000 hDdf ubf{ 3 jifk{ l5 hDdf ?= 12100 kf| Kt ul/g\ . o;df hDdf /sd
eGgfn] s] al' emG5, ;fyLx¿lar 5nkmn u/L nv] .

lglZrt ;dokZrft\ sg' } klg ;fj“ f /sddf Aofh yk u/L Psdi' 6 kb| fg ul/g] /sdnfO{ ld>wg
elgG5 . o;nfO{ A n] hgfOG5 .

ld>wg [Amount (A)] = ;fj“ f [Principal (P)] + Aofh [Interest (I)] xG' 5 .

cyft{ , ld>wg = ;fj“ f + Aofh

A = P + I …………..(i)

xfdLnfO{ yfxf 5 PTR ……..(ii)
I = 100

(i) / (ii) nfO{ ldnfpb“ f,

PTR xfdf| ] ul0ft, sIff *
A = P + 100

144

xfdf| ] ul0ft, sIff * 145