MATHS GRADE 6

PsfO 12 leGg / bzdnj (Fraction and Decimal)

12.1 kg' /fjl[ Q

;dtN' o leGgx¿ (Equivalent fractions)
cl3Nnf sIffx¿df kl9;ss] f leGg;DaGwL ljifoj:ts' f] kg' /fjl[ Q u/f“} M

12 34
24 68

lrqdf bv] fOPsf 5fof kfl/Psf efux¿n] jm| dz 1 , 2 , 3 / 4 hgfp“5g\ . o:tf leGgx¿nfO{
2 4 6 8

;dt'No leGg elgG5 .

2 46 8
3 69 12

lrqdf bv] fOPsf 5fof kfl/Psf efux¿n] jm| dzM 2 , 4 , 6 , 8 hgfp5“ g\ .
3 6 9 12

oL ;a} ;dt'No leGgx¿ x'g\ .
leGgsf] n3T' td kb (Lowest term of a fraction)

1  1 2  1 3  1 4 cyjf 1  2  3  4
2 22 23 24 2 4 6 8

oxf,“ 2 , 3 / 4 ;a} 1 ;“u a/fa/ 5g\ .
4 6 8 2

1 leGgsf] cz+ / x/df sg' } klg ;femf u0' fgv08 gePsfn] o;nfO{ 2,3 / 4 sf] n3Q' d
2 46 8

kb elgG5 .

To;u} /L 4 , 8 , 12 , 16 leGgx¿sf] n3Q' d kb 2 xf] .
6 12 18 24 3

96 ul0ft, sIff ^

leGgx¿sf] tn' gf (Comparison of Fractions) 1
2
1
2

11 1
33 3

1 11 1
4 44 4

11 11 1
55 55 5

11 1 1 11
66 6 6 66

oL leGgx¿sf cz+ slt 5g\ <

s] oL leGgsf x/x¿ Pp6} 5g\ < 5g} g\ eg] x/ a9b\ f] 5 ls 36b\ f] <

leGgsf] cz+ pxL t/ x/ a9b\ } hfb“ f s] c;/ kbf{] /x5] <

leGgsf] cz+ pxL t/ x/ 36b\ } hfG5 eg] leGgdf s] c;/ kbf{] /x5] <

bi| 6Jo M olb bO' { jf bO' e{ Gbf a9L leGgx¿df cz+ Pp6} 5 eg] x/ ;fgf] ePsf] leGg 7n' f] xG' 5 .

tn lbOPsf lrqx¿af6 hgfpg] leGgnfO{ Psk6s x/] f“} M

14 →11111111122222222233333333344444444455555555514666666666777777777888888888999999999000000000111111111222222222333333333
24 →1111111112222222223333333334444444445555555551466666666677777777788888888899999999900000000011111111122222222233333333344444444455555555566666666677777777788888888814999999999000000000111111111222222222333333333444444444555555555
34 →111111111222222222333333333444444444555555555146666666667777777778888888889999999990000000001111111112222222223333333334444444445555555556666666667777777778888888881499999999900000000011111111122222222233333333344444444455555555566666666677777777788888888899999999900000000014111111111222222222111111111222222222333333333444444444555555555

1  2  3
4 4 4

 oL leGgx¿df x/ slt slt 5g\ <
 clg cz+ x¿ a9b\ f] 5 ls 36b\ f] <
 leGgsf] x/ pxL t/ cz+ 36b\ } hfG5 eg] leGgdf s] c;/ k5{ <

bi| 6Jo M bO' { jf bO' e{ Gbf a9L leGgx¿df x/ Pp6} 5 eg] cz+ 7n' f] ePsf] leGg g} 7n' f] xG' 5 .

ul0ft, sIff ^ 97

sg' } bO' { jf bO' e{ Gbf a9L leGgx¿sf] tn' gf ubf,{
 x/ Pp6} 5 eg] cz+ dfq bfH“ bf hg' leGgsf] cz+ 7n' f] xG' 5 ToxL leGg 7n' f] x'G5 .
 x/ km/s km/s 5g\ eg] cz+ / x/nfO{ ;femf u0' fgv08n] ug' /] Pp6} x/ ePsf (like

fractions) leGgx¿df abn/] leGgx¿sf] tn' gf ug{ ;lsG5 .

pbfx/0f 1

3 / 2 tn' gf u/ M
5 3

pQ/

oxf,“ bj' } leGgsf x/ km/s km/s 5g\ . t;y,{ x/ Pp6} agfpg x/x¿ 5 / 3 sf] n=;= lgsfnf“} .

oxf“ 5 / 3 sf] n=;= = 5 x 3 = 15 xG' 5 .

3 nfO{ x/ / cz+ df 3 n] u0' fg ubf,{ 3 = 3 3  9 -x/ 5 nfO{ 15 agfpg x/ /
5 5 5 3 15 cz+ nfO{ 3 n] u0' ff u/s] f] ._

2 2 = 25  10 -x/ 3 nfO{ 15 agfpg x/ /
3 35 15 cz+ nfO{ 5 n] u0' ff u/s] f] ._
3 sf] x/ / cz+ df 5 n] u0' fg ubf,{

ca, bj' } leGgsf] x/ Pp6} 5 / cz+ x¿df 9 < 10

To;sf/0f, 9 10
15  15

To;n} ] 3  2 cyjf 2  3
5 3 3 5

pbfx/0f 2

leGgx¿ 1 , 2 / 3 nfO{ 7n' fb] l] v ;fgf] jm| ddf ldnfP/ nv] .
2 3 4

pQ/

oxf,“ x/x¿ km/s km/s 5g,\ To;n} ] Pp6} x/ ePsf leGgx¿ agfpgk' 5{ .

x/x¿ 2, 3 / 4 sf] n=;= lgsfnf“} hg' 12 xG' 5 .

 1  1 6  6 -x/ 2 nfO{ 12 agfpg x/ / cz+ bj' n} fO{ 6 n] u0' ff u/s] f_]
2 26 12

2  24  8 -x/ 3 nfO{ 12 agfpg x/ / cz+ bj' n} fO{ 4 n] u0' ff u/s] f_]
3 34 12

3  33  9 -x/ 4 nfO{ 12 agfpg x/ / cz+ bj' n} fO{ 3 n] u0' ff u/s] f_]
4 43 12

98 ul0ft, sIff ^

ca, Pp6} x/ ePsf leGgx¿nfO{ 7n' fb] l] v ;fgf] jm| ddf /fVbf,

9 , 8 , 6 cyft{ \ 3 , 2 , 1 x'G5 .
12 12 12 4 3 2

pbfx/0f 3

Pp6f n67\ Lsf] 3 efu sfnf,] 2 efu ;t] f] / afs“ L efu /ftf] nufOPsf] /x5] . sg' /ªsf] efu
7 5

a9L nufPsf] /x5] <

pQ/
klxnf lrqdf ljrf/ u/f}“ M

3 sfnf] 2 ;t] f] < /ftf]
7 5

n67\ Lsf] k/" f efuaf6 sfnf] / ;t] f] nufOPsf] efu 36fPkl5 /ftf] efu slt /x5] kTtf nufpg

;lsG5 .

n67\ Lsf] sfnf] / ;t] f] efu  3  2
7 5

 35  27 -;dfg x/ agfPsf_]
75 57

 15  14  29
35 35 35

To;n} ,] n67\ Lsf] /ftf] efu = -k/" f efu_ – -sfnf] / ;t] f] efu_

 1 29 -k/' f efu egs] f 1 xf] ._
35

 35  29  6 -;dfg x/ agfPsf_]
35 35 35

To;sf/0f sfnf,] ;t] f] / /ftf] efu jm| dzM 3,2 / 6 eof] .
75 35

;dfg x/ agfpb“ f jm| dzM 15 , 14 / 6 ePsf] xg' fn] a9L /ªu\ fOPsf] efu sfnf] xf] .
35 35 35

ul0ft, sIff ^ 99

cEof; 12.1

1. tnsf lrqx¿df 5fof kfl/Psf efunfO{ leGgdf nv] M

(a) 11111111111112222222222222333333333333344444444444445555555555555 11111111111112222222222222333333333333344444444444445555555555555 (b) 11111111111112222222222222333333333333344444444444445555555555555 1111111111111222222222222233333333333334444444444444 1111111111111222222222222233333333333334444444444444 (c) 1111111111111222222222222233333333333334444444444444 11111111111112222222222222333333333333344444444444445555555555555 11111111111112222222222222333333333333344444444444445555555555555 (d) 11111111111112222222222222333333333333344444444444445555555555555 11111111111112222222222222333333333333344444444444445555555555555

11111111111112222222222222333333333333344444444444445555555555555 1111111111111222222222222233333333333334444444444444

2. (a) 3 sf] cz+ / x/nfO{ 2, 3, 4 / 5 n] u'0fg u/]/ ;dt'No leGgx¿ n]v .
5

(b) lgDgfg;' f/ vfnL sf7] fdf e//] 7 sf ;dtN' o leGgx¿ agfpm M
8

×7 7×
8×
×8
7

8

×7 7×
×8 8×

3. tnsf tflnsf x/] / vfnL sf7] fdf ldNg] ;ªV\ of nv] M 1 1
5 5
111
555 11
10 10
11111111
10 10 10 10 10 10 10 10

(a) 1   (b) 5  6 (c) 2  4
5 10 10
10

(d) 5   (e) 3  6 (f) 1  2
5 10 5 10


4. tn lbOPsf sg' sg' leGgx¿ ;dtN' o leGgx¿ xg' \ <

(a) 3 / 12 (b) 6 / 12 (c) 5 / 25 (d) 2 / 18
4 15 7 13 9 45 3 27

5. n3'Qd kbdf ¿kfGt/ u/ M

27 84 126 52 208 150
(a) 108 (b) 96 (c) 396 (d) 76 (e) 312 (f) 250

100 ul0ft, sIff ^

6. tn lbOPsf leGgx¿ bfh“ / larsf] vfnL sf7] fdf <, = jf > lrx\g /fv M

1  2 3  6 (c) 1  1
(a) 3 3 (b) 5 10 3 4

2  2 3  9 1  4
(d) 5 6 (e) 4 12 (f) 5 15

7. tn lbOPsf leGgx¿nfO{ ;fgfb] l] v 7n' f] jm| ddf ldnfP/ nv] M

(a) 1 , 1 / 1 (b) 3 , 4 / 9
2 3 4 4 5 10

(c) 1,2 / 5 (d) 3 , 11 / 7
69 12 10 30 20

8. zLnf / ;dLgfnfO{ dfOhn" ] Ps Pscf6] f /f6] L lbge' of] . zLnfn] 3 efu / ;dLgfn] 5 efu
8 7

/f6] L dfq vfO5g\ eg] s;n] a9L /f6] L vfP <

9. sn} f; 3/df hfb“ f k/" f af6fs] f] 3 efu a;af6, 1 efu 6o\ fS;Laf6 / afs“ L kb} n uP5g\
7 2

eg] pgn] ;ae} Gbf a9L b/' L s;/L kf/ u/] <

12.2 leGgx¿sf] hf8] / 36fp (Addition and subtraction of fractions)

;dfg x/ ePsf leGgx¿sf] hf8] tyf 36fp
-s_ tnsf lrqx¿ x/] / leGgx¿ hf8] g\ ] tl/sf cWoog u/ M

2 + 3 =5
7 77

+

1 2 =3
8+ 8 8

ul0ft, sIff ^ 101

1111111111111111111222222222222222222233333333333333333334444444444444444444555555555555555555566666666666666666667777777777777777777888888888888888888899999999999999999990000000000000000000111111111111111111122222222222222222223333333333333333333444444444444444444455555555555555555556666666666666666666777777777777777777788888888888888888889999999999999999999 + 1111111111222222222233333333334444444444555555555566666666667777777777888888888899999999990000000000 = 1111111111111111111222222222222222222233333333333333333334444444444444444444555555555555555555566666666666666666667777777777777777777888888888888888888899999999999999999990000000000000000000111111111111111111122222222222222222223333333333333333333444444444444444444455555555555555555556666666666666666666777777777777777777788888888888888888889999999999999999999 111111111122222222223333333333444444444455555555556666666666777777777788888888889999999999

;dfg x/ ePsf leGgx¿ hf8] b\ f ;femf x/n] cz+ x¿sf] ofu] kmnnfO{ efu ugk{' b5{ .
pbfx/0f 1
hf8] M

pQ/

pbfx/0f 2
hf8] M
pQ/

-v_ t. ns11111111111f2222222222233333333333l44444444444r55555555555q66666666666x77777777777¿8888888888899999999999x00000000000/]1111111111122222222222/ le=Ggx11111111111¿222222222223333333333344444444444355555555555666666666666f77777777777pg] tl/sf =cWo11111111111o22222222222g3333333333344444444444u/ M

==

o;/L, ;dfg x/ ePsf] leGg 36fpb“ f ;femf x/n] cz+ x¿sf] km/snfO{ efu ugk{' b5{ .

102 ul0ft, sIff ^

pbfx/0f 3 103
pbfx/0f 4
pbfx/0f 5

pbfx/0f 6

ul0ft, sIff ^

104 ul0ft, sIff ^

pbfx/0f 8 105

pbfx/0f 9

cEof; 12.2
1. lx;fa u/ M
2. lx;fa u/ M
3. lx;fa u/ M

ul0ft, sIff ^

4. lx;fa u/ M (ii) (iii)

51
(i) 3  2

(iv) (v) (vi)

(vii) 3 2  2 1 (viii) 10 2  3 1 (ix) 7 2  2 3
5 6 5 6 9 4

5. lx;fa u/ M

123 (ii) 1 1  2 3  4 1
(i) 2  3  4 2 4 2

(iii) 3 3 1 1  2 (iv) 10 2  3 1 1 1
6 4 3 5 10 20

6. /ljg;u“ Pp6f 7n' f] /lh:6/ lyof] . olb p;n] o;sf] 1 efu nl] v;Sof] eg] ca /lh:6/df
2

nV] g slt afs“ L 5 <

7. dL/f hªu\ ndf j:te' fp r/fpg uPsL l14yOge\ fu. ;vfe“ fh]m /] 3/Nokfpmsgb{ e' fopf] g. nc] a;as} ljt:tjs' :tf] e' 12fp
efu 3/ kmsf{Og\ . pgsf a'afn] km]l/

3/df kmsfp{ g afs“ L /x] <

8. Pp6f /fdfo0fsf] lstfa /Lgf, dLgf / /ljgfn] jm| dzM 1 efu, 2 efu / 1 efu k9/]
3 5 6

l;WofP5g\ . ca sNkgfn] ;f] lstfa k9/] l;Wofpgk' bf{ pgn] k:' tssf] slt efu k9g\ k' nf{ <

12.3 leGgx¿sf] u0' fg / efu (Multiplication and division of fractions)

leGg / k0" f{ ;ªV\ ofsf] u0' fg (Product of a fraction and a whole number)

tnsf] pbfx/0f x]/f}“ M 3 310
10

3  2  2 3 x'G5 . -s;/L <_ 111111111111111111222222222222222222333333333333333333444444444444444444555555555555555555666666666666666666777777777777777777888888888888888888999999999999999999000000000000000000111111111111111111222222222222222222333333333333333333444444444444444444555555555555555555666666666666666666777777777777777777888888888888888888999999999999999999000000000000000000111111111111111111
10 10

106 ul0ft, sIff ^

2  3 egs] f] 2 cf6] f 3 hDdf ug]{ xf] . lrqaf6 2 cf6] f 3 hDdf ubf{ 6 cyft{ \ 3
10 10 10 10 5

b]lvof] . To;}n] o:tf] ;d:of ;dfwfg ug]{ k|of; u/f}“ .

3  2  3 2  6  3
10 10 10 5

pbfx/0f 1

u0' fg u/ M 4  3
9

pQ/

4  3  4  3  12  4  1 1
9 9 9 3 3

pbfx/0f 2

4 sf] 2 3 efu slt xG' 5 <
10

pQ/

4 sf] 2 3 efu egs] f] 4  2 3 xf] .
10 10

To;n} ,] 4  2 3  4  23  4  23  92  46  9 1
10 10 10 10 5 5

leGgx¿sf] u0' fg (Product of fractions)

kf;fª;u“ 1 /f]6L 5 . pgn] To;sf] 1
2 2

-cfwf_ efOnfO{ lbOg\ . ca pgn] k'/} cfwf /f6] L cfkm;" u“ ePsf] efOnfO{
/f6] Lsf] slt efu efOnfO{ lbOg\ < leGg nv] . lbPsf]

lrqaf6 yfxf xG' 5 ls efOnfO{ pgn] /f6] Lsf] 1 efu dfq lbOg\ .
4

To;n} ,] 1 sf] 1  1  1  11  1 32 → 111111111112222222222233333333333444444444445555555555566666666666777777777778888888888899999999999000000000001111111111122222222222333333333334444444444455555555555666666666667777777777788888888888999999999990000000000011111111111222222222223333333333311111111112222222222333333333344444444445555555555666666666677777777778888888888999999999900000000001111111111
2 2 2 2 22 4 4 ×3

km]l/, csf]{ pbfx/0f x]/f}“ M

ul0ft, sIff ^ 107

3  2 slt xG' 5 <
4 3

oxf,“ 3  2  3 2  6  1
4 3 43 12 2

leGgnfO{ leGgn] u0' fg ubf{ cz+ cz+ sf] u0' fgkmnnfO{ cz+ df / x/x/sf] u0' fgkmnnfO{ x/df /fvL
gof“ leGg agfOG5 .

pbfx/0f 3

u0' fg u/ M -s_ 1  4 -v_ 3  5
3 15 10 6

pQ/

-s_ 1  4  1 4  4
3 15 3  15 45

-v_ 3  5  35  15  1
10 6 10  6 60 4

pbfx/0f 4

dfg lgsfn M

-s_ 9  5 -v_ 1 kg sf] 3
10 3 2 4

pQ/

-s_ 9  5  95  3  1 1
10 3 10  3 2 2

-v_ 1 kg  3   1  3 kg 3
2 4  2 4   8 kg

 3 1000gm = 375 gm.
8

pbfx/0f 5

/fhn' fO{ cfdfn] lkpg lbge' Psf] 3 lunf; bw' dWo] 2 -bO' { ltxfO_ dfq p;n] lkof] eg,]
4 3

-s_ p;n] Ps lunf;sf] slt efu bw' lkof] xfn] f <

-v_ ca lunf;df slt bw' afs“ L /xG5 <

108 ul0ft, sIff ^

pQ/

-s_ /fhn' ] lkPsf] bw'  3 sf] 2 efu  3  2  1 lunf;
4 3 4 3 2

-v_ lunf;df afs“ L bw'  3 lunf; – /fhn' ] lkPsf] bw'
4

 3  1  3 2  1 lunf; .
4 2 4 4

k0" f{ ;ªV\ ofnfO{ leGgn] efu ug{] (Dividing a whole number by a fraction)

tnsf] pbfx/0f x]/f}“ M

3  1
2

11111111112222222222333333333344444444445555555555n] df sltcf6] f 5g\ eGg] a'emfp“5 .3 13 1 12 125555555555555555555666666666666666666677777777777777777778888888888888888888999999999999999999900000000000000000001111111111111111111222222222222222222233333333333333333334444444444444444444
2 111111111111112222222222222233333333333333124444444444444455555555555555666666666666667777777777777788888888888888 2
12 12111111111111111111222222222222222222333333333333333333444444444444444444555555555555555555666666666666666666777777777777777777888888888888888888999999999999999999000000000000000000
1
2

lrqaf6 :ki6 xG' 5 ls 3 cf6] f l;ªu\ fd] f 6 cf6] f 1 x'G5g\ .
2

o;nfO{ 5f6] s/Ldf,

3  1  3  2
2 1

1 nfO{ 2 agfO{ 3 n] u0' fg u/s] f_]
1
(2

 6  6
1

To:t} lsl;dn] 10  2  10  5  10  5  25
5 2 2

;ªV\ of /v] fdf 3  1 nfO{ o;/L b]vfpg ;lsG5 .
2
11 1 111

22 2 222

012 3 109
ul0ft, sIff ^

leGgnfO{ leGgn] efu ug{] (Dividing a fraction by a fraction)

/fdn] 41 cf6] f la:s6' sf kl' /ofx¿ vfn] /] Ps hgfnfO{ 3 kl' /ofsf b/n] af8“ b\ f slt hgfnfO{
2 4

kU' nf <

ul0ftLo efiffdf 4 1 df sltcf6] f 3 xG' 5g\ klg eGg ;lsG5, cyft{ \ 9  3  6
2 4 2 4

33 3 3 33
44 4 4 44

01 2 3 4 9 5
2

oxf,“ 9  3  6 [ cyft{ ,\ 9  4  36  6 ]
2 4 2 3 6

Pp6f leGgnfO{ csf{] leGgn] efu ubf{ ÷ nfO{ × df abn/] efhs leGgnfO{ pN6fO{ efHo leGg;u“

u'0ff ubf{ cfjZos efukmn lg:sG5 . h:t} M a  c  a  d xG' 5 .
b d b c

pbfx/0f 4

lx;fa u/ M -s_ 6  3 -v_ 3  11
5 5

pQ/

-s_ 6  3  6 5 -v_ 3 1 1  3  6
5 3 5 5

 65  3  5
3 6

= 10  3 5
6

 5  2 1
2 2

pbfx/0f 5

lx;fa u/ M -s_ 2  1 -v_ 3 4  2 1
5 2 5 10

110 ul0ft, sIff ^

pQ/ 2  1  2  2 -v_ 3 4  21  19  21
-s_ 5 2 5 1 5 10 5 10

 2 2  4  19  10 = 19 × 10
5 5 5 21 5 × 21

 38  11271
21

pbfx/0f 6

21 ld= nfdf] sk8faf6 3 ld= nDafO ePsf 6j' m| fx¿ sfl6P eg] hDdf slt 6j' m| f aGnfg\ <
4

pQ/

oxf,“ 21  3  21 4
4 3

 21 4
3

= 28

cyft{ \ 21 ld= nfdf] sk8faf6 3 ld= nDafO ePsf 28 cf]6f 6'j|mfx¿ sf6\g ;lsG5 .
4

cEof; 12.3

1. tn lbOPsf lrqx¿df bfx] f/] f] 5fof kf/s] f] efunfO{ leGgsf] u0' fgkmnsf ¿kdf nv] M

111111111111112222222222222233333333333333444444444444445555555555555566666666666666777777777777778888888888888899999999999999000000000000001111111111111122222222222222 111111111111112222222222222233333333333333444444444444445555555555555566666666666666777777777777778888888888888899999999999999000000000000001111111111111122222222222222333333333333334444444444444455555555555555666666666666667777777777777788888888888888999999999999990000000000000011111111111111222222222222223333333333333344444444444444 222222222333333333444444444555555555666666666777777777888888888999999999000000000111111111222222222333333333444444444

(a) (b) (c)

2. u'0fgkmn lgsfn M

(a) 1  1 (b) 4  1 (c) 1  5
5 3 3 5 10 6

(d) 1 2  2 1 (e) 9  25 (f) 3 2  2 3
3 6 10 30 4 4

ul0ft, sIff ^ 111

3. slt x'G5 M

(a) 1 sf] 3 (b) 5 sf] 12
3 5 6 35

(c) 3 ld= sk8fsf] 1 (d) ?= 2 3 sf] 2
4 3 4 5

4. tn lbOPsf lx;fanfO{ ;ªV\ of/v] fdf bv] fpm M

(a) 2  1 
2

(b) 4 2 
3

5. lx;fa u/ M

(a) 1 1 (b) 12  2 (c) 20  4 (d) 32  2 2 (e) 3  3
2 3 5 7 7 5 8

18 9 (g) 11  3 (h) 35  2 2 (i) 4 4  22
(f) 13  8 2 4 9 3 5 15

6. Zofd;u“ ePsf] ?= 125 sf] 3 efu p;n] ;fyLnfO{ ;fk6 lbP5 / afs“ Lsf] 3 efusf]
5 10

sfkL lsg5] eg] pm;u“ slt ?lkof“ afs“ L xfn] f <

7. sf7df8f}“af6 dl' Unª;Ddsf] af6f] 110 ls=ld= nfdf] 5. ;f] af6fd] f 3 efu an] fotL sDkgLn,]
sDkgLn] sfnfk] q ug{] 5
3
To;kl5 afs“ Lsf] 4 efu lrlgof“ ;xdlt ePcg;' f/ sfd ;DkGg eof] .

ca hDdf slt ls=ld= af6fd] f sfnfk] q xg' afs“ L 5 <

8. Pp6f] ;fgf] v/fof] Psk6sdf 2 ld= plk|mg ;S5 eg] p;nfO{ l;wf af6f]sf]
3

16 ld= b/' L kf/ ug{ slt k6s plkm| gk' nf{ <

9. Pp6f ;fgf] Oofnsf nflu 3 ld= sf] kbf{ rflxG5 eg] 30 ld= sk8fsf] yfgaf6 slt cf6] f
4

‰ofndf kbf{ xfNg ;lsG5 .

10. 9 lSjG6n;Dd afS] g ;Sg] Ps hgf dflg;n] 18 lSjG6n lrgL cf;] fg{ slt k6s afS] gk' nf{ <
11. k/' f rSsfdf Ps ;dsf0] fsf] < -1 rSsfdf 3600 xG' 5 ._
10 2 efux¿ sltcf6] f xfn] fg\
3
Ps

12. 20 ln6/ bw' nfO{ 1 1 ln= c6g\ ] l;;Ldf e/L /fVbf sltcf6] f l;;L eg{ ;lsPnf <
4

112 ul0ft, sIff ^

12.4 leGgsf] ;/nLs/0f (Simplification of fraction)

leGgx¿;u“ +, -, × / ÷ lrxg\ x¿ ldl;P/ cfpb“ f s;/L xn ug{] eGg] ljifodf 5nkmn u/f“} . o;sf

nflu lgDglnlvt pbfx/0fx¿ cWoog u/ M

1 ld= nDafOsf 4 6j' m| f sk8f egs] f] hDdf 1  4  2 ld= sk8f xf] . o;df kml] / 3 ld= hf8] b\ f
2 2 4

hDdf 2 ld=  3 ld=  2 3 ld= nfdf] sk8f xG' 5 . o;nfO{ ul0ftLo efiffdf lgDgfg;' f/ nl] vG5 M
4 4

4 1  3  2  3  2 3
2 4 4 4

To;n} ] sk8fsf] hDdf nDafO 2 3 ld= x'G5 .
4

leGgx¿sf] ;/nLs/0f ubf{ klg k0" f{ ;ªV\ ofx¿sf] ;/nLs/0fsf] em}“ ko| fu] ul/G5 .

tnsf pbfx/0fx¿ x]/f}“ M

pbfx/0f 1

;/n u/ M 2 1  3 1  1  2
2 2 14 3

pQ/

oxf,“ 2 1  3 1  1  2  5  7  1  2
2 2 14 3 2 2 14 3

 5  1  1  2  5  1  2 -klxnf] u0' fg lrxg\ x6fpg_]
2 2 2 3 2 4 3

 30 3  8 -n=;= lnP/ Pp6} leGgdf abNg_]
12

 33  8  25  2 1
12 12 12

pbfx/0f 2

;/n u/ M 3  2 1  1  1
4 3 4 2

pQ/

oxf,“ 3  7  1  1
4 3 4 2

 7  1  1 -u0' fg lrxg\ x6fPsf_]
4 4 2

ul0ft, sIff ^ 113

 7 1 2  7 3  4 =1
4 4 4

ca ÷ / × lrxg\ ePsf ;d:ofx¿ cWoog u/f“} M

pbfx/0f 3

;/n u/ M 4  8  1
5 9 7

o;nfO{ efiffdf 4 nfO{ 8 n] efu u/L cfPsf] efukmnnfO{ 1 n] u0' fg ug{] elgG5 .
5 9 7

pQ/

4 nfO{ 8 n] efu ubf,{
5 9

4  8 154  9  9
5 9 82 10

kml] /, 9  1  9  1  91  9
10 7 10 7 10  7 70

5f6] s/Ldf,

4 × 9 × 1
5 8 7

1 4 × 9 × 1 = 9
5 8 7 70
= 2

pbfx/0f 4

;/n u/ M 1  3 1 1  2 -o;nfO{ efiffdf cg'jfb ug]{ ko| f; u/ ._
2 4 4 3

pQ/

oxf,“ 1  3  1 1  2
2 4 4 3

 1  4  5  2
2 3 4 3

 5  2  5 4  9  3  1 1
6 3 6 6 2 2

114 ul0ft, sIff ^

pbfx/0f 5

;/n u/ M 34 –1 1 1 3 1
5 10 6 10

pQ/

oxf,“ 3 4  1110  1  3 1
5 6 10

 19  11  6  31
5 10 1 10

 19  33  31
5 5 10

 38  66  31
10

 69  66
10
3
 10

pbfx/0f 6

Pp6f 71 ld= nfdf] sk8fnfO{ 5 a/fa/ efu u/L cfPsf] Ps 6j' m| fdf 21 ld= nfdf] sk8f hf8] /]
2 2

l;pb“ f slt nfdf] aG5 <

pQ/

oxf,“ 7 1  5  2 1
2 2

 15  5  5  15  1  5  5  5
2 2 2 5 2 1 

 3  5  8 =4
2 2 2

To;n} ,] hDdf sk8fsf] nDafO 4 ld= x'G5 .

pbfx/0f 7

/fd;u“ 100 kfgf ePsf] Pp6f sfkL lyof] . p;n] 1 efudf gk] fnL, 1 efudf ul0ft / 1 efudf
5 4 10

lj1fg n]v]5 t/ 1 efu RofltP/ uP5 eg] ca pm;u“ slt kfgf sfkL afs“ L /xo\ f] xfn] f <
20

ul0ft, sIff ^ 115

cEof; 12.4 ul0ft, sIff ^
1. ;/n u/ M

2. ;/n u/ M

116

3. ;dfwfg u/ M

(a) Zofdn] 2 cf6] f ;G' tnfdf kT| os] ;G' tnfsf] 1 efu kfp“5 . olb p;n] cf gf] efusf]
3

;G' tnfnfO{ cfkm" ;dt] 2 hgf ;fyLx¿nfO{ a/fa/ u/L af8“ b\ f p;sf] efudf slt
cfp“5 .

(b) 1 / 1 sf] ofu] kmnnfO{ 3 n] u0' ff u/L 2 n] efu u/ .
2 3

(c) 11 / 2 sf] ofu] kmnnfO{ 3 n] efu u/L 1 36fpm .
2 3 3

(d) 31 af6 2 36fP/ cfPsf] dfgnfO{ 1 n] efu u/L 11 n] u0' ff ubf{ slt xG' 5 <
4 5 2 2

(e) xl/;u“ ?= 50 lyof] . cfdfn] pm;“u ePsf] k};fsf] 1 efu ylklbge' of] t/ xl/n]
2

alxgLnfO{ ca pm;u“ ePsf] hDdf k;} fsf] 2 efu lbP/ ;x/ 3'Dg uof] . olb p;n]
5

lnP/ uPsf] k;} fdWo] 1 efudfq vr{ u¥of] eg] hDdf pm;u“ slt afs“ L xfn] f <
2

12.5 leGgnfO{ bzdnjdf / bzdnjnfO{ leGgdf ¿kfGt/

(Conversion of fraction to decimal and vice-versa)

1  0.1 1  0.01 1  0.001
10 100 1000

bzdnj Ps bzdnj zG" o Ps bzdnj zG" o zG" o Ps
-Ps b;fz+ _ -Ps ;tfz+ _ -Ps xhf/fz+ _

ul0ft, sIff ^ 117

tLg bzdnj;Ddsf] ;ªV\ of (2.315) nfO{ cfwf/ b; Ans (Base Ten Block) sf / :yfgdfg
tflnsfdf tn lbOPcg';f/ b]vfpg ;lsG5 .

b; Ps b;f+z ;tfz+ xhf/f+z

23 15

2.315 / of] ;ªV\ ofnfO{ ‘bO' { bzdnj tLg Ps kfr“ ’ eg]/ kl9G5 .
leGgnfO{ bzdnj ;ªV\ ofdf ¿kfGt/
pbfx/0f 1

13 nfO{ bzdnj ;ª\Vofdf ¿kfGt/ u/ .
4

pQ/

1 3  7
4 4

7 nfO{ bzdnjdf abNbf,
4

1.75 (cz+ 7 nfO{ x/ 4 n] efu u/s] f] 7.00 dfg/] efu u/L ;tfz+ df nus] f] ._
4 ) 7.00
4

30

28

20

20

×

bzdnjnfO{ leGgdf ¿kfGt/
tnsf pbfx/0fx¿nfO{ Psk6s x]/f}“ M
pbfx/0f 2

0.2  2  1
10 5

0.2  2 bzdnj laGbk' l5 Pp6f cªs\ 5 eg] x/df 10 x'G5 .
10

118 ul0ft, sIff ^

0.35  35 bzdnj laGbk' l5 bO' c{ f6] f cªs\ x¿ 5g\ eg] of] ;tfz+ xf,] To;n} ] x/df 100 x'G5 .
100

0.675  675 bzdnjkl5 tLgcf6] f cªs\ x¿ 5g\ eg] xhf/fz+ xf,] To;n} ] x/df 1000 x'G5 .
1000

pbfx/0f 3

3.285 nfO{ n3T' td kbdf n}hfpm / leGgdf nv] .
pQ/

3.285  3285 -bzdnj laGb' x6fOPsf_]
1000

 657 -5 n] efu u/s] f_]
200

∴ 3.285  657
200

cEof; 12.5
1. tnsf leGgnfO{ bzdnjdf kl/0ft u/ -bzdnjsf] ltg :yfg;Dd dfq_ M

11 5 22 (d) 12 11 (e) 35
(a) 8 (b) 7 (c) 9 12 16

2. lgDglnlvt bzdnj ;ª\VofnfO{ leGgdf ¿kfGt/ u/ -leGgnfO{ n3'Qd kbdf klg
nh} fpm ._ M

(a) 0.5 (b) 1.3 (c) 2.51 (d) 15.65 (e) 7.509 (f) 12.325

ul0ft, sIff ^ 119

12.6 bzdnjsf] hf8] / 36fp (addition and substraction of decimal)

bzdnjsf] hf8] / 36fpsf nflu lgDglnlvt pbfx/0fsf] cWoog u/f“} M
pbfx/0f 1
hf]8

5.474
 8.450

pQ/ hf8] g\ k' g{] ;ªV\ ofx¿nfO{ :yfgdfg cg;' f/ ldnfP/ /fVg] .
cfjZostfcg;' f/ bzdnj k5fl8 zG" ox¿ yKg] . ca
5.474 ;ª\Vofx¿ hf]8\g]
 8.450
13.924

pbfx/0f 2 cfjZostfcg;' f/ bzdnj k5fl8 zG" o yKg] . :yfgcg;' f/ ldnfP/ ;ªV\ ofx¿
36fpm 36fpg]

32 . 67
12 . 881

pQ/

32 . 670
12 . 881

19.789

bi| 6Jo M hf8] / 36fp ubf{ bzdnjkl5 Tolts} :yfgsf cªs\ x¿ xg' k' 5{ . gku' ] cfjZostfcg;' f/
0 yKg'k5{ .

cEof; 12.6

1. hf8] M (b) 32.69 (c) 1.405
 19.23  0.068
(a) 3.05
 2.79

120 ul0ft, sIff ^

(d) 6.374 (e) 13.54 (f) 21.54
18.966 2.689 23.89
 4.3  9.22
 3.28

(g) 18.00 + 9.099

2. 36fpm M (b) 13.8 (c) 21.081
 6.95  14.069
(a) 5.67
 3.09

(d) 17.704 (e) 14 (f) 52.08
 8.648  12.836  43.68

(g) 2.801 - 1.9 (h) 12 - 8.6 (i) 13.07 – 6.894

3. ;/n u/ M

(a) 6.97 – 13.543 + 8.695

(b) 1.1 – 20.976 + 25.68

4. ?= 192.50 sf] Ps bhg{ sfkL / ?= 40.75 sf] Pp6f snd lsGbf ?= 250 ?lkofa“ f6 slt
lkmtf{ cfp5“ <

5. Pp6f cfotsf] nDafO 14.6 ;=] ld= 5 / nDafOeGbf rf8} fO 1.8 ;=] ld= n] sd 5 eg,] (a)
rf8} fO slt xfn] f < (b) cfotsf] kl/ldlt slt xfn] f <

6. ;dLgfn] ?= 9.75 sf] ;df;] f, ?= 5.50 sf] h/] L / ?= 10.25 sf] Ps sk lrof vfO5g\ eg] (a)
;dLgfn] hDdf slt vr{ ul/5g\ < (b) olb ;dLgfn] k;nn] fO{ ?= 50 sf] gf6] lbPsf] eP slt
?lkof“ lkmtf{ cfp5“ <

7. 30 ls=ld= nfdf] af6f] agfpb“ f 5.75 ls=ld= hg>dbfgaf6 / afs“ L ;/sf/L cgb' fgaf6 vlgof]
eg] ;/sf/L cgb' fgaf6 slt ls=ld= agfpg] sfd eP5 <

8. Pp6f sf7sf] kmns] 3.5 ;]=ld= afSnf] 5 . 4.25 ;=] ld= nfdf] lsnf 7fS] bf lsnfsf] slt efu
kmns] af6 aflx/ lg:sG5 xfn] f <

ul0ft, sIff ^ 121

12.6 bzdnjnfO{ 10 / 10 sf ckjTox{ ¿n] u0' fg / efu ug{]

tnsf pbfx/0fx¿ Ps k6s x]/f}“ M

-s_ 0.1 × 10

1 10  1 1  10  1  1  1
10 10 10 10 100

0.1 × 10 = 1.0 0.1 ÷ 10 = 0.01

-v_ 0.01 × 10

1 10  1 1  10  1  1  1
100 10 100 100 10 1000

0.01 × 10 = 0.1 0.01 ÷ 10 = 0.001

-u_ 0.001 × 10

1 10  1 1  10  1  1  1
1000 100 1000 1000 10 10000

0.001 × 10 = 0.01 0.001 ÷ 10 = 0.0001

oL pbfx/0fx¿af6,
(i) bzdnj ;ªV\ ofnfO{ 10 n] u0' fg ubf{ bzdnj laGb' Ps :yfg bfofl“ t/ ;g{] ub5{ .
(i) bzdnj ;ªV\ ofnfO{ 10 n] efu ubf{ bzdnj laGb' Ps :yfg afofl“ t/ ;g{] ub5{ .

of] kl| jm| ofnfO{ tnsf] lrqaf6 ae' mg\ ] ko| f; u/f“} M

10 × 10 10 10 10 10

1000 100 10 1 0.1 0.01 0.001

÷10 ÷10 ÷10 ÷10 ÷10 ÷10

tnsf sx] L pbfx/0fx¿ klg x/] f“} M

-s_ 12.56 × 10  1256 × 10  1256 = 125.6
100 10

bzdnj ;ªV\ ofnfO{ 10 n] u0' fg ubf{ bzdnj laGb' Ps :yfg bfofl“ t/ ;b5{ .

-v_ 12.567 × 100  12567 × 100  12567 = 1256.7
1000 10

122 ul0ft, sIff ^

bzdnj ;ªV\ ofnfO{ 100 n] u0' fg ubf{ bzdnj laGb' bO' { :yfg bfofl“ t/ ;b5{ .

-u_ 12.5678 ×1000  125678 × 1000 = 12567.8
10000

bzdnj ;ªV\ ofnfO{ 1000 n] u0' fg ubf{ bzdnj laGb' tLg :yfg bfofl“ t/ ;5{ .
-3_ 12.2 ÷ 10 = 1.22

bzdnj ;ªV\ ofnfO{ 10 n] efu ubf{ bzdnj laGb' Ps :yfg afofl“ t/ ;b5{ .
-ª_ 123.4 ÷ 100 = 1.234

bzdnj ;ªV\ ofnfO{ 100 n] efu ubf{ bzdnj laGb' bO' { :yfg afofl“ t/ ;b5{ .
-r_ 1234.5 ÷ 1000 = 1.2345

bzdnj ;ªV\ ofnfO{ 1000 n] efu ubf{ bzdnj laGb' tLg :yfg afofl“ t/ ;b5{ .

pbfx/0f 1

0.0573 nfO{ jm| dzM 10, 100 / 1000 n] u0' fg u/ M

pQ/

0.0573 × 10 = 0.573

0.0573 × 100 = 5.73

0.0573 × 1000 = 57.3

pbfx/0f 2

0.5 nfO{ 10, 100 / 1000 n] efu u/ M

pQ/

0.5 ÷ 10 = 0.05

0.5 ÷ 100 = 0.005

0.5 ÷ 1000 = 0.0005

pbfx/0f 3

vfnL 7fp“ e/ M

1001 100.1

pQ/ ÷10 ÷10 ÷10 ÷10

1001 100.1 10.01 1.001 0.1001

÷10 ÷10 ÷10 ÷10

ul0ft, sIff ^ 123

cEof; 12.7

1. tn lbOPsf kT| os] ;ªV\ ofnfO{ jm| dzM 10, 100 / 1000 n] u'0fg u/ M

(a) 1.2 (b) 10.5 (c) 0.12

(d) 0.025 (e) 0.345 (f) 0.1

2. tn lbOPsf kT| os] ;ªV\ ofnfO{ jm| dzM 10, 100 / 1000 n] efu u/ M

(a) 1234 (b) 360.5 (c) 58.2

(d) 48.5 (e) 0.05 (f) 1.5

3. 9f“rf x]/L vfnL 7fp“df e/ M

(a) ×10 ×10 ×10 ×10

1.001 10.01

(b) 100.1
1001
÷10
÷10 ÷10 ÷10

4. 15 ls=ld= nfdf] af6f] vGg' lyof] . olb lgDgfg;' f/sf JolStx¿n] a/fa/ sfd u/d] f kT| os] n]
slt ls=ld= af6f] vGnfg\ < -ls=ld=df nv] _

(a) 10 hgf (b) 100 hgf (c) 10000 hgf

5. 22 ld= df slt ls=ld= xG' 5 <

6. 675 uf| = egs] f] slt lsnfu] f| d xf] <

124 ul0ft, sIff ^

12.8 bzdnjsf] u0' fg / efu (Multiplication and division of decimal):

bzdnjsf] u0' fg
0.3 × 6

0.3 × 6  3  6  18 = 1.8
10 10
To;}n],
oxf“ 0.3 df bzdnj laGbk' l5 Ps cªs\ ePsf] xg' fn,] pTt/df
0.3 bfofb“ l] v Ps cªs\ 5f8] /] bzdnj laGb' /fVg]
6
1.8

kml] /, 0.56 × 0.2

 56  2 oxf“ 0.56 df bzdnj laGbk' l5 bO' { cªs\ / 0.2 df bzdnj laGbk' l5
100 10 Ps cªs\ ePsf] xg' fn] pTt/df hDdf ltg cªs\ 5f8] /] bzdnj laGb'
/fVg]
 112
1000

= 0.112

bi| 6Jo M 0.01 × 0.002 u0' fg ubf{ klxnf] bzdnj laGb' gePsf] 7fgL 1 × 2 sf] u0' fgkmn lgsfNg]
hg' 2 xG' 5, clg 0.01 / 0.002 df hDdf bzdnj laGbs' f] k5fl8 5 cf6] f cªs\ 5g\ . To;}n] 2 sf]
cufl8 rf/ cf6] f zG" o /fvL bzdnj laGb' /fVbf 0.00002 x'G5 .

tnsf pbfx/0f x]/ M

pbfx/0f 1

lx;fa u/ M 0.02 × 0.03 × 0.3 125

pQ/

0.02
 0.03
0.0006

 0.3
0.00018

csf{] tl/sf,
2 × 3 × 3 = 18

bzdnjkl5 5 cªs\ ePsf] cfjZos u0' fgkmn = 0.00018 x'G5 .

ul0ft, sIff ^

pbfx/0f 2

u0' fg u/ M 0.8 × 2.35
pQ/

2.35
 0.8
1.880

u0' fgkmn 1.880 sf] ;66\ f 1.88 dfq nV] g ;lsG5 . To;sf/0f 2.35 × 0.8 = 1.88 xG' 5 .

bzdnjsf] efu (Division of decimal)
tnsf] pbfx/0f x]/ M

efu ubf{ M 2.4 ÷ 6  24  6  24  1  4 = 0.4
10 10 6 10

To;sf/0f, 2.4 ÷ 6= 0.4

bzdnj ;ªV\ ofnfO{ k0" ffª{ s\ n] efu ubf{ bzdnj ;ªV\ ofnfO{ efu ug{] lalTts} efukmndf klg
bzdnj lrx\g /fVg'k5{ .

pbfx/0f 3

efu u/ M 38.48 ÷ 8

pQ/ klxn] k"0f{ ;ª\Vofn] k"0f{ ;ª\VofnfO{ efu ug]{ . To;kl5
bzdnjkl5sf] 4 nfO{ tn emfg{] lalQs} efukmndf bzdnj
4.81 laGb' /fVg] . 64 nfO{ 8 n] efu ug{] / efukmn bzdnj
laGb'kl5 /fVg] .
8) 38.48
- 32
64
- 64
8
-8

×

To;n} ,] 38.48 ÷ 8 = 4.81

126 ul0ft, sIff ^

cEof; 12.8

1. u'0fg u/ M (b) 8 × 0.6 (c) 9 × 1.5
(a) 2.3 × 6
(e) 8.25
(d) 0.07 1.2
12

(f) 9.34 × 2.5 (g) 5.56 × 1.6 (h) 0.94 × 6.2

2. efu u/ M (b) 1.21 ÷ 11 (c) 14.4 ÷ 6
(a) 6.4 ÷ 8 (e) 7.29 ÷ 9 (f) 0.927 ÷ 3

(d) 1.95 ÷ 5

3. ;/n u/ M (b) 0.2 × (0.7 + 0.07) (c) (5.5 – 3.2) × 1.2
(a) (1.3 + 0.2) × 0.2 (e) (1.1 × 1.5) × 0.7 (f) 10.5 – (1.5 × 0.6)

(d) 3.5 × (1.9 - 0.7)

4. Pp6f cfotsf] Ifq] kmn 14.79 ju{ ld= / rf8} fO 2.9 ld= eP nDafO slt xfn] f <

2.9 m 14.79 m2

5. Pp6f jufs{ f/ vt] sf] jl/kl/sf] 3/] f 9.2 ld= 5 .
e'hfsf] nDafO / If]qkmn kTtf nufpm .

16.9 zG" ofGt (Rounding Off) C D

AB

oxf,“ /v] fv08x¿ AB / CD sf] gfk l7s;u“ lnb“ f,

AB = 5.2 ;=] ld= CD = 3.6 ;=] ld=

AB sf] gfk k0" ffª{ s\ df JoSt ubf{ sl/a 5 ;=] ld= / CD sf] gfk sl/a 4 ;]=ld= 5g\ .

glhssf] PsfOdf JoSt ug{] kl| jm| ofnfO{ zG" ofGt (rounding off) ug]{ elgG5 .

ul0ft, sIff ^ 127

oxf,“ 5.2 df 5 / 6 gfkdWo] 5 sf] glhs kb{5 . 5.2

To;n} ,] 5.2 nfO{ zG" ofGt ubf{ 5 x'G5 . 01 23 4 5 6
To;n} ,] 5.2  5, cyjf 5.2 sl/a 5 xf] .

o;/L tnlt/ -afofl“ t/_ zG" ofGt ugn{] fO{ tNnf] dfgdf zG" ofGt (round down) elgG5 . 3.6

To:t,} 3.6 nfO{ zG" ofGt ubf,{

3.6 4.0 01 23 4

o;/L dflylt/ -bfofl“ t/_ zG" ofGt ugn{] fO{ dflyNnf] dfgdf zG" ofGt (round up) elgG5 . To;}n]

zG" ofGt ubf,{ zG" o kfgk{' g{] ;ªV\ of 0, 1, 2, 3 jf 4 eP Round down / 5, 6, 7, 8 of 9 ePdf round

up ug'{kb{5 .

h:t} M l;w} efu ubf,{

5.024 nfO{ bzdnjsf] t;] f| ] :yfgdf zG" ofGt ubf,{ 3.1428571

7 22)
21
5.024 ≈ 5.02 10

/ 3.6918 nfO{ bzdnjsf] rfy} f] :yfgdf zG" ofGt ubf,{ 7
30
3.6918 ≈3.692 xG' 5 .
28

20

nfO{ bzdnjdf JoSt ubf,{ 14
60

56

22  3.143 -bzdnjsf] ltg :yfg;Dd_ 40
7 35
50

= 3.14 -bzdnjsf] bO' { :yfg;Dd_ 49
= 3.1 -bzdnjsf] Ps :yfg;Dd_ 10 slxNo} lgMz]if x'“b}g . of]
= 3 glhssf] PsfOdf JoSt ubf{ 7
3 k|lj|mof hf/L /xG5 .

o;/L leGgsf] x/n] cz+ nfO{ efu ubf{ bzdnj /flv;sk] l5 klg lgMzi] f xb“' g} eg] bzdnjsf] tLg
jf rf/ :yfg;Dd -jf tfl] sPadfl] hdsf] :yfg;Dd_ dfq /fv] kU' 5 .

pbfx/0f 1

zG" ofGt u/ M 6.02527 nfO{ bzdnjsf]

(i) kfr“ f“} :yfgdf (ii) rfy} f] :yfgdf (iii) t;] f| ] :yfgdf

pQ/

(i) 6.02527

oxf“ zG" o kfgk{' g{] ;ªV\ of 7 xf] . 0 1 2 3 4 5 6 7 8 9 10

128 ul0ft, sIff ^

of] ;ªV\ of 0 / 10 dWo] 10 sf] glhs kb{5 .
To;n} ,] 7 nfO{ zG" o kf/L 2 df 1 hf8] b\ f
6.02527 ≈ 6.0253

(ii) 6.02527≈ 6.025 0 1 2 3 4 5 6 7 8 9 10

lsgeg] ;ªV\ of 2, 0 / 10 dWo] 0 sf] glhs 5 .

(iii) 6.02527 ≈ 6.03

lsgeg] zG" o kfgk{' g{] ;ªV\ of 5, 0 /
10 l7s lardf kb{5 . o:tf] cj:yfdf 0 1 2 3 4 5 6 7 8 9 10
5 nfO{ 7n' f] PsfO 10 lt/ z"GofGt ul/G5 .

cEof; 12.9

1. bzdnjsf] Ps :yfgdf z"GofGt u/ M

(a) 2.62 (b) 3.59 (c) 15.47 (d) 27.63

2. bzdnjsf] b'O{ :yfgdf z"GofGt u/ M

(a) 3.627 (b) 12.592 (c) 17.418 (d) 13.025

3. bzdnjsf] ltg :yfgdf z"GofGt u/ M

(a) 5.3247 (b) 6.5432 (c) 6.4153 (d) 17.343

4. tnsf kT| os] leGgnfO{ bzdnjsf] tLg :yfgdf zG" ofGt u/L JoSt u/ M

1 2 1 (d) 2 12
(a) 3 (b) 3 (c) 6 23

5. glhssf] ;]=ld= df z"GofGt u/ M

(a) 6.3 ;=] ld= (b) 12.5 ;=] ld= (c) 16.8 ;=] ld= (d) 55.5 ;=] ld=

6. glhssf] ?lkof“df z"GofGt u/ M

(a) ?= 5.35 (b) ?= 12.50 (c) ?= 25.73 (d) ?= 24.26

7. glhssf] ls=ld= jf ls=uf| = df zG" ofGt u/ M

(a) 45.6 ls=ld= (b) 147.5 ls=ld= (c) 15.4 ls=uf| = (d) 17.46 ls=uf| =

ul0ft, sIff ^ 129

PsfO 13 cgk' ft, ;dfgk' ft / kl| tzt (Ratio, Proportion and Percentage)

13.1 leGg / kl| tzt (Fraction and percentage)

leGgnfO{ kl| tztdf / kl| tztnfO{ leGgdf ¿kfGt/0f ug{ ;lsG5, h:t} M tnsf sx] L pbfx/0f
x/] f“} M

dflysf] lrqnfO{ ul0ftLo efiffdf nV] bf s;/L nl] vG5 <

leGgdf nV] bf, 1
4

x/ ;o xg' ] u/L lbOPsf] leGgnfO{ ;dtN' o leGgdf ¿kfGt/0f ubf{ M

1 nfO{ x/ / cz+ df 25 n] u0' fg u/f,“}
4

1  1 25  25 o;nfO{ kl| tztdf ¿kfGt/0f ubf{
4 4  25 100

25  25% -% kl| tzt lrxg\ xf] ._
100

cyf{t\

dflysf] lrqdf /ªu\ fPsf] efunfO{ 25 nl] vG5 .
100

= 25% xG' 5 .

ctM sg' } klg leGgnfO{ kl| tztdf abNbf 100 n] u0' fg u/L kl| tzt lrxg\ (%) nV] gk' 5{ . kl| tztnfO{

leGgdf abNbf 100 n] efu ugk{' 5{ .

130 ul0ft, sIff ^

pbfx/0f 1

(a) 75% nfO{ leGgdf ¿kfGt/0f u/ M

75  3
100 4

(b) 3 nfO{ kl| tztdf ¿kfGt/0f u/ M
4

3  3  25  75  75% cyjf, 3  3  100  75%
4 4  25 100 4 4

(c) 100 sf] 10% slt xG' 5 <

100  10  10 xG' 5 .
100

pbfx/0f 2

cfoi' fn] 20 cªs\ sf] k0" ffª{ s\ ul0ftdf 18 cªs\ kfPsf 5g\ eg] pgn] kfPsf] cªs\ slt kl| tzt
xG' 5 <

oxf“, 20 k0" ffª{ s\
dflysf] egfOnfO{ ul0ftLo efiffdf nV] bf,

kfPsf] cªs\ 18 18 kf| Ktfªs\
cfoi' fn] kfPsf] cªs\ = k"0ff{ª\s  20 -;dtN' o leGg jf x/ ;o ePsf_]

 18  5
20  5

 90  90%
100

cfoi' fn] kf| Kt u/s] f] cªs\ = 90% /x5] .

cEof; 13.1

1. tnsf kT| os] kl| tztnfO{ x/ 100 ePsf] leGgdf JoSt u/ M

(a) 20 % (b) 75 % (c) 84 % (d) 68 % (e) 100 %

2. tnsf kT| os] kl| tztnfO{ x/ 100 ePsf] leGgdf JoSt u/L n3T' td ¿kdf ¿kfGt/ u/ M

(a) 12% (b) 16% (c) 25% (d) 42% (e) 85%

ul0ft, sIff ^ 131

(f) 45% (g) 65% (h) 90% (i) 20 % (j) 35%

3. tn lbOPsf kT| os] leGgnfO{ kl| tztdf JoSt u/ M

2 1 12 27
(a) 5 (b) 20 (c) 25 (d) 20

37 1 4 1
(e) 50 (f) 10 (g) 5 (h) 4

4. tnsf ;ªV\ ofsf] dfg lgsfn M

(s) 200 sf] 10% (b) 50 sf] 5% (c) 300 sf] 20%

(d) 100 sf] 20% (e) 300 sf] 4%

5. Pp6f sIffdf 50 hgf ljBfyL{ lyP . tLdWo] 8 hgf cgk' l:yt eP5g\ eg] cgk' l:yt ePsf
ljBfyL{ ;ªV\ ofnfO,{

(a) leGgdf nv]

(b) k|ltztdf n]v .

6. /dz] n] 50 k0" ffª{ s\ ul0ftsf] k/LIffdf 35 cªs\ NofP5g\ eg] pgn] slt kl| tzt NofP5g\ <

7. lznfn] 10 cf6] f ;G' tnf lslgg\ / 3 cf6] f ;G' tnf efO sdnnfO{ lbOg\ eg,]

(a) lznf;u“ slt kl| tzt ;G' tnf afs“ L 5g\ <

(b) sdnn] slt kl| tzt ;G' tnf kfP <

8. ljgon] cfkm\ gf] km6' an l6dn] xfgs] f] 5 ufn] dWo] 3 ufn] u/] eg] ljgon] slt kl| tzt
ufn] xfg5] g\ <

9. Pp6f ljBfnodf cfpg] 20 ljBfyLd{ Wo] 8 hgf ;fOsnaf6 cfpb“ f /x5] g\ eg] afs“ Ln]
:sn' a; ko| fu] ubf{ /x5] g\ . slt kl| tzt ljBfyLn{ ] a; ko| fu] ubf{ /x5] g\ <

10. Pp6f sIffsf 50 ljBfyLd{ Wo] 16 hgf 5fq /x5] g\ eg] slt kl| tzt 5fqf /x5] g\ <

132 ul0ft, sIff ^

13.2 cgk' ft tyf ;dfgk' ft (Ratio and Proportion)

sIff 6 df s6] fsf] ;ªV\ of 40 hgf / s6] Lsf] ;ªV\ of 20 hgf /x5] eg] s6] f / s6] Lsf]
11223344556660 111222333444 s]6f
cgk' ft slt xfn] f < 11223344556650
11223344556640 40 hgf s]6L11111112222222333333344444445555555666666677777778888888
11122233344455566630
111112222233333444445555566666 hgf20
111111111111222222222222333333333333444444444444555555555555666666666666777777777777888888888888999999999999000000000000111111111111222222222222333333333333 20
10

1234560

dflysf] lrqnfO{ leGgdf nV] bf, 40
20

s6] f / s6] Lsf] cgk' ft  40  2 jf 2:1
20 1

ctM s6] fsf] ;ªV\ of s6] Lsf] ;ªV\ ofeGbf bfA] a/ 5 .

cgk' ft egs] f] 2 cf6] f j:ts' f] tn' gfTds kl/df0f yfxf kfpg] kl| jm| of xf] . csf{] zAbdf cgk' ft
egs] f] Pp6f kl/0ffdnfO{ csf{] kl/df0fn] efu ubf{ kf| Kt xg' ] leGgfTds efukmn xf] . a:b nfO{
a/b sf] ¿kdf nl] vG5 .

pbfx/0f 1

;Ltf;u“ 5 cf6] f lstfax¿ 5g\ . To;} u/L pgL;u“ 8 cf6] f sfkL 5g\ eg] lstfa / sfkLsf] cgk' ft
slt xfn] f <

8 cf6] f sfkL 5 cf6] f lstfa
lstfa / sfkLsf] cgk' ft = 5:8

pbfx/0f 2

-s_ 3 : 2 nfO{ n3Q' d kbdf ¿kfGt/0f u/ .
4

3 : 2 nfO{ leGgdf nV] bf,
4

3 1  3  3: 8 -leGgnfO{ cgk' ftdf ¿kfGt/0f ubf_{
4 2 8

ul0ft, sIff ^ 133

-v_ 11 :12
43

11  5
43

11  5
43

55
43

5  3  15 -leGgsf] u0' fg ubf_{
4 5 20

3
4

 3 : 4 -leGgnfO{ cgk' ftdf abNbf_

-u_ 50 k;} f M ?= 1.5

bO' { j:ts' f] kl/df0f tn' gf ubf{ tL j:ts' f] PsfO Pp6} xg' k' 5{ . cgk' ftnfO{ leGgdf nV] bf,

50 -?= 1.5 nfO{ k;} fdf ¿kfGt/0f ubf_{
150

50  1  1: 3
150 3

pbfx/0f 3

-s_ Zofd;u“ / /fd;u“ ePsf] k;} fsf] cgk' ft 3:8 /x5] , olb /fd;u“ 56 ?lkof“ eP Zofd;u“ slt
?lkof“ /x5] <
pQ/
Zofd;u“ ePsf] k;} fnfO{ x dfGbf,
bj' } hgf;u“ ePsf] k;} f tn' gf ubf,{ 3:8 = x:56

cyjf, 3  x
8 56

356  x 8

x  3 56  21
8

Zofd;u“ ePsf] k;} f = ?= 21 /x5] .

-v_
cfot 2cm

4cm ul0ft, sIff ^

dflysf] cfotsf] nDafO tyf rf8} fOsf] cgk' ft kQf nufpm .

134

;dfgk' ft (Proportion)

rf/ hgfn] kfPsf uR' rfx¿sf] ;ªV\ of dfly dfof zªs\ /
bv] fP h:t} dfof / zªs\ /;u“ ePsf] uR' rf, uLtf cfoi' f
uLtf tyf cfo'if;“u ePsf] u'Rrflar t'ngf
ubf{ rf/} hgfn] kfPsf] ;dfgk' ft lgDgfg;' f/
nV] g ;lsG5 M

dfof / zªs\ /;u“ ePsf uR' rf = 4:6 = 2:3

uLtf / cfoi' f;u“ ePsf uR' rf = 6:9 = 2:3

rf/} hgf;u“ ePsf uR' rfsf] cgk' ft =
4:6::6:9 (:: n] ;dfgk' ftnfO{ hgfp5“ ._

dflysf] pbfx/0fdf dfof / zªs\ /;u“ ePsf] uR' rfsf] cgk' ft / uLtf / cfoi' f;u“ ePsf] uR' rfsf]
cgk' ft a/fa/ 5g\ . To;n} ] 4, 6, 6, 9 ;dfgk' ftdf 5 . o;nfO{ 4 : 6 :: 6 : 9 nl] vG5 . :: lrxg\ n]
;dfgk' ft hgfp5“ .

sg' } rf/cf6] f ;ªV\ of sf] cgk' ftdf klxnf] bO' { ;ªV\ ofsf] cgk' ft;u“ clGtd bO' { ;ªV\ ofsf] cgk' ft
a/fa/ xG' 5 eg] To;nfO{ ;dfgk' ft (proportion) elgG5, h:t} M rf/ cf6] f ;ªV\ ofx¿ 15, 20, 30
/ 40 sf] cgk' ft

ul0ft, sIff ^ 135

jf 30  3 jf 3 : 3 klxnf bO' { ;ªV\ ofsf] cgk' ft = bf;] f| ] bO' { ;ªV\ ofsf] cgk' ft xG' 5 ._
40 4 4 4

pbfx/0f 4

olb 3.5m, 14m, 16m / m ;dfgk' flts ;ªV\ of xg' \ eg] x sf] dfg slt xfn] f <

3.5m:14m:16m: xm

3.5m  16m
14m xm

3.5 x  14 16
x  14 16  64m
3.5

xm = 64m

pbfx/0f 5

/dz] / xl/;u“ jm| dzM 15 tyf 16 cf6] f :ofp 5g\ / To;u} /L dLgf;u“ 24 cf6] f :ofp 5g,\ olb
/dz] / xl/;u“ ePsf] :ofpsf] cgk' ft / dLgf / ;l' gtf;u“ ePsf] :ofpsf] cgk' ft klg a/fa/
5g\ eg] ;l' gtf;u“ ePsf] :ofp slt xfn] f <
dfgf,“} ;l' gtf;u“ x :ofp 5g\ .

oxf“, /dz] ;u“ ePsf] :ofpsf] ;ªV\ of = dLgf;u“ ePsf] :ofpsf] ;ªV\ of
xl/;u“ ePsf] :ofpsf] ;ªV\ of ;l' gtf;u“ ePsf] :ofpsf] ;ªV\ of

12  24
15
xx

xx  24 15  30
12

;l' gtf;u“ 30 cf6] f :ofp /x5] g\ .

cEof; 13.2

1. tnsf kT| os] cgk' ftnfO{ n3Q' d ¿kdf JoSt u/ M

(a) 20 : 35 (b) 115 : 60 (c) 25 : 75 (d) 49 : 245

(e) 350 : 400 (f) 1 1 : 3 (g) 2 2 : 4 1 (h) 25 k=} M ?= 1
2 4 8
(l) 500 ml : 5l
(i) 15 cm : 1 m (j) 20 cm : 5 m (k) 250 gm : 1 kg
ul0ft, sIff ^
136

2. Pp6f k/LIffdf ;lDdlnt ePsf ljBfyLd{ Wo] 100 hgf s6] L / 200 hgf s6] f /x5] g\ eg] s6] f
/ s6] Lsf] cgk' ft slt xfn] f <

3. Pp6f cfotsf] nDafO 4 cm / rf8} fO 3 cm /x5] ,
(a) nDafO / rf8} fOsf] cgk' ft slt /x5] <
(b) rf8} fO / nDafOsf] cgk' ft slt /x5] <

4. Pp6f jus{ f] nDafO 4 cm /x5] / csf{] jus{ f] nDafO 6 cm /x5] g\ eg,]
(a) klxnf] / bf;] f| ] jus{ f] nDafOsf] cgk' ft lgsfn .

(b) klxnf] / bf;] f| ] jus{ f] rf8} fOsf] cgk' ft lgsfn .

(c) klxnf] / bf;] f| ] jus{ f] Ifq] kmnsf] cgk' ft slt xG' 5 <

5. tn lbOPsf kT| os] cgk' ft a/fa/ 5g\ eg] vfnL 7fpd“ f sg' cªs\ /fVgk' nf{ <

4  8  ........  ........
6 ........ 36 60

6. sg' } Pp6f uf=lj=;= sf] sn' hg;ªV\ of 35820 dWo] 18900 dlxnf /x5] g\ eg,]

(a) k'?ifsf] ;ª\Vof lgsfn .

(b) k'?if / dlxnf ;ª\Vofsf] cg'kft lgsfn .
(c) dlxnf / sn' hg;ªV\ ofsf] cgk' ft slt slt /x5] <

(d) k?' if / sn' hg;ªV\ ofsf] cgk' ft slt /x5] <

7. Pp6f sf7] fsf] nDafO / rf8} fOsf] cgk' ft 5 : 4 /x]5 . olb nDafO 15 m eP rf8} fO slt
xfn] f <

8. (a) Pp6f sf7] fsf] nDafO / rf8} fOsf] cgk' ft 3:2 /x5] / nDafO 7 m eP sf7] fsf] rf8} fO
slt xfn] f <

(b) bO' { cfotdWo] Pp6f cfotsf] kl/ldlt 150cm / csf{] cfotsf] kl/ldlt 75cm 5 eg]
bj' } cfotsf] kl/ldltsf] cgk' ft slt xfn] f <

9. olb 5 / 8, x / 16 ;dfgk' ftdf eP x sf] dfg kTtf nufpm .

10. x:y = 1:2 / a:b = x:y 5g\ . olb a sf] dfg 6 eP b sf] dfg kQf nufpm .

11. /fd, xl/, Zofd / /dz] ;u“ sx] L ;G' tnfx¿ 5g\ . /fd;u“ 10, xl/;u“ 15 / /dz] ;u“ 30 cf6] f
;'Gtnfx¿ 5g\ . /fd / xl/;“u ePsf ;'Gtnfx¿sf] cg'kft Zofd / /d]z;“u ePsf]
;G' tnfx¿sf] cgk' ft a/fa/ eP Zofd;u“ ePsf ;G' tnfx¿sf] ;ªV\ of kQf nufpm .

12. m sf] dfg kTtf nufpm M

(i) 7  m (ii) 12  3 (iii) 5:6 = m:4 (iv) 11:m =1:13
49 28 m 2

ul0ft, sIff ^ 137

PsfO 14 gfkmf / gfS] ;fg (Profit and Loss)

gfkmf tyf gfS] ;fg zAb Jofkfl/s sf/fa] f/ ubf{ a9L ko| fu] df cfpg] u/s] f] zAb xf] . sg' } JolStn]
Pp6f dN" odf ;fdfg vl/b u/L csf{] dN" odf ;f] ;fdfg lajm| L ubf{ ar] s] f] dN" o lsgs] f] dN" oeGbf
yf/] } jf w/] } xg' hfG5 . o;/L lsgs] f] tyf ar] s] f] dN" osf] cfwf/df gfkmf jf gfS] ;fg xG' 5 . olb lsgs] f]
dN" oeGbf lajm| L u/s] f] dN" o yf/] } ePdf 3f6f jf gfS] ;fg x'G5 . olb lsgs] f] dN" oeGbf lajm| L dN" o a9L ePdf
gfkmf xG' 5 .
jm| o dN" o (Cost Price) : ;fdfg vl/b ubf{ ltl/Psf] dN" onfO{ ;f] ;fdfgsf] jm| o dN" o elgG5 . o;nfO{
5f6] s/Ldf jm| =d=" (C.P.) elgG5 .
ljjm| o dN" o (Selling Price): sg' } klg ;fdfg lajm| L ubf{ kf| Kt xg' ] /sdnfO{ ;f] ;fdfgsf] ljjm| o dN" o
elgG5 . o;nfO{ 5f6] s/Ldf lj=d=" (S.P.) elgG5 . h:t} M /fdn] Pp6f sldh ?= 500 df lsg/]
?= 400 df lajm| L u¥of,] p;nfO{ gfS] ;fg jf gfkmf s] eof] xfn] f ljrf/ u/ t .
oxf,“ lsgs] f] dN" oeGbf lajm| L dN" o yf/] } 5 . To;n} ] p;nfO{ gfS] ;fg eof] .
ctM gfS] ;fg = jm| =d=" — lj=d="

= ?= 500 - ?= 400 = ?= 100
ctM /fdn] 100 ?lkof“ 3f6f vfP/ ar] s] f] /x5] .
olb /fdn] ?= 500 df lsg/] ?= 600 df ar] s] f] eP gfkmf jf gfS] ;fg s] xG' Yof] <
gfkmf xG' Yof] lsgls lsgs] f] dN" oeGbf lajm| L dN" o a9L 5,
gfkmf = lj=d" — jm| =d"

= ?= 600 - ?= 500 = ?= 100

138 ul0ft, sIff ^

pbfx/0f 1

jm| =d=" = ?= 400, lj=d=" = ?= 300 eP gfS] ;fg = <
gfS] ;fg = ?= 400 - ?= 300 = ?= 100

pbfx/0f 2

xl/n] 1 bhg{ sfkL ?= 120 df lsg/] ?= 12 kl| tuf6] fsf b/n] lajm| L ubf{ slt gfkmf xG' 5 <

oxf,“ jm| =d=" = ?= 120
lj=d=" = ?= 12 × 12
= ?= 144

ljjm| o dN" o a9L ePsfn] xl/nfO{ gfkmf eP5 .
To;}n],
gfkmf = lj=d=" — jm| =d="

= ?= 144 – ?= 120 = ?= 24

pbfx/0f 3

2 Kofs6] la:s6' kT| os] nfO{ ?= 6.50 df lsg/] ?= 5.25 sf b/n] lajm| L ubf{ slt gfkmf jf gfS] ;fg
xfn] f <

oxf“, 2 Kofs6] sf] jm| =d=" = 2 × ?= 6.50 = ?= 13
2 Kofs6] sf] lj=d=" = 2 × ?= 5.25 = ?= 10.50

lj=d"= yf]/} ePsfn] gf]S;fg eof] .
To;n} ,] gfS] ;fg = jm| =d=" – lj=d="

= ?= 13 – ?= 10.50

= ?= 2.50

cEof; 14.1

1. tnsf kT| os] cj:yfdf gfkmf lgsfn M

-s_ jm| =d=" = ?= 50 lj=d=\" = ?= 75

-v_ jm| =d=" = ?= 2504 lj=d=\" = ?= 2910

-u_ jm| =d=" = ?= 365 lj=d=\" = ?= 387

-3_ jm| =d=" = ?= 3333 lj=d=\" = ?= 3460

ul0ft, sIff ^ 139

2. tnsf cj:yfdf gfkmf jf gfS] ;fg s] xfn] f, lgsfn M

-s_ lj=d=" = ?= 350 jm| =d=" = ?= 395

-v_ lj=d=\" = ?=3720 jm| =d=" = ?= 3514

-u_ lj=d=" = ?=7590 jm| =d=" = ?= 8350

-3_ lj=d=" = ?= 980 jm| =d=" = ?= 795

3. Pp6f k;nn] ] ?= 20 df lsgs] f] sfkL ?= 22 df lajm| L u/5] . p;nfO{ slt gfkmf jf gfS] ;fg
eof] kTtf nufpm .

4. hdg' fnfO{ Pp6f Jofkf/Ln] ?= 2500 df lsgs] f] 50 ls=uf| = rfdn ?= 2750 df lajm| L u¥of] eg]
Jofkf/Ln] hdg' faf6 slt gfkmf lnof] <

5. 4 ls=uf| = :ofp kl| t ls=uf| = ?= 90 df lsg/] ?= 100 kl| t ls=uf| = df lajm| L ubf{ slt gfkmf
x'G5 <

6. /fdn] 2 cf6] f snd ?= 100 df lsg/] Pp6fnfO{ ?= 30 df / csfn{ fO{ ?= 20 df lajm| L ubf{
p;nfO{ x'g] gfkmf jf gf]S;fg kTtf nufpm .

7. 1 bhg{ ;G' tnf ?= 120 df lsg/] kT| os] sf] ?= 13.30 sf b/n] lajm| L ubf{ gfkmf jf gfS] ;fg slt
xfn] f <

8. ;'hgn] Pp6f elnan ?= 750 df / Pp6f k'm6an ?= 825 df lsg]/ elnannfO{ ?= 650
df / km' 6annfO{ ?= 985 df lajm| L u/5] eg] p;nfO{ gfkmf jf gfS] ;fg s] slt eof] xfn] f <

14.2 gfkmf / gfS] ;fg ldl>t ;fdfGo ;d:ofx¿

tnsf pbfx/0fx¿sf] cWoog u/ M

pbfx/0f 1

Pp6f kmnkm" n k;nn] ] 4 ls=uf| = ;G' tnf ?= 40 kl| t ls=uf| = sf b/n] lsgL ?= 20 kmfObf ug{ rfxG5
eg] p;n] hDdf slt ¿lkofd“ f ;G' tnf aR] gk' nf{ <

oxf,“ hDdf jm| =d=" = ?= 40 × 4 = ?= 160
kmfObf ug{ rfxs] f] dN" o = ?= 20
kmfObf ugk{' bf{ jm| =d=" eGbf lj=d=" a9L xg' k' 5{ / of] lj=d=" jm| =d=" eGbf gfkmf ugk{' g{] /sdn] a9L
x'g cfp“5 . lj=d"= = jm| o dN" o ± gfkmf
∴ hDdf xg' k' g,{] lj=d=" = ?=160 + ?= 20
= ?= 180

140 ul0ft, sIff ^

pbfx/0f 2
Pp6f k;nn] ] ?= 450 df lsgs] f] ;fdfg ?= 125 gfS] ;fg ;x/] aR] gk' ¥of] eg] ljjm| o dN" o slt xfn] f <

oxf“, j|m=d"= = ?= 450
gfS] ;fg = ?= 125
gfS] ;fg ePsf] cj:yfdf jm| o dN" o ljjm| o dN" oeGbf a9L xG' 5 cyjf ljjm| o dN" o jm| o dN" oeGbf
gf]S;fg ePsf] /sdn] 36L x'G5 .
lj=d"= = jm| =d=" – gfS] ;fg

∴ lj=d"= = ?= 450 - ?= 125
= ?= 325

pbfx/0f 3

Pp6f Jofkf/Ln] 4 ld= sk8f ?= 80 kl| tld6/sf b/n] lsg/] hDdf ?= 60 gfkmf ug{ rfxG5 eg] p;n]
4 ld= sk8f hDdf slt dN" odf lajm| L ugk{' nf{ <

oxf“, hDdf jm| =d=" = ?= 80 × 4 = ?= 320
gfkmf
= ?= 60
;q" cg;' f/, lj=d="
= jm| =d=" ± gfkmf

= ?= 320 ± ?= 60
= ?= 380

pbfx/0f 4

xdLbn] 1 bhg{ 86k\ g] ?= 8 kl| tuf6] fsf b/n] lsgs] f] lyof] . p;n] tL 86k\ g] x¿ aR] bf ?= 3 gfS] ;fg
eP5 . hDdf slt ?lkof“df ;a} 86\k]gx¿ a]r]5 kTtf nufpm .

oxf,“ hDdf jm| =d=" = ?=8 × 12
= ?= 96

gfS] ;fg = ?= 3

;q" cg;' f/, lj=d=" = jm| =d=" – gfS] ;fg
= ?= 96 – ?= 3
= ?= 93

ul0ft, sIff ^ 141

pbfx/0f 5

30 cf6] f rSn6] hDdf ?= 50 df lsg/] kl| tuf6] fdf ?= 0.50 gfkmf u/L aR] bf hDdf lj=d=" slt xG' 5 <

oxf,“ hDdf jm| =d=" = ?= 50

gfkmf = 30 × ?= 0.50

= ?= 15

;q" cg;' f/, lj=d=" = jm| =d=" ± gfkmf

= ?=50 + ?= 15

= ?= 65

cEof; 14.2

1. laj|mL d"No lgsfn M gfkmf = ?= 5
-s_ jm| =d=" = ?= 35 gfkmf = ?= 10
-v_ jm| =d=" = ?= 63 gfS] ;fg = ?= 50
-u_ jm| =d=" = ?= 800 gfS] ;fg = ?= 75
-3_ jm| =d=" = ?= 450

2. Pp6f 38L ?= 1000 df lsg/] ?= 30 gfkmf u/L aR] bf ljjm| o dN" o slt xfn] f <

3. sg' } j:t' ?= 310 df lsg/] ?= 125 gfS] ;fg ;x/] aR] gk' bf{ ljjm| o dN" o slt xg' cfp5“ <

4. Ps lSjG6n rfdn ?= 50 kl| t lsnfu] f| dsf b/df lsg/] ?= 150 gfkmf ug{ rfxb“ f hDdf lajm| L
dN" o slt xG' 5 <

5. ?= 40 kl| t lsnfu] f| dsf b/n] Pp6f Jofkf/Ln] 50 lsnf]u|fd Kofh lsGof] t/ p;n] a]Rg]
jn] fdf kl| t lsnfu] f| dsf] ?= 1 n] ahf/ efp 365] eg] p;nfO{ hDdf gfS] ;fg slt eof] <

6. ?= 160 sf b/n] 4 cf6] f emfn] f lsg/] kT| os] emfn] fdf ?= 20 gfkmf u/L aR] bf hDdf ljjm| o dN" o
slt xG' 5 <

7. Pp6f dflg;n] ?= 1580 lt//] lsgs] f] /l] 8of] ?= 175 gfS] ;fg ;x/] aR] of] eg] pSt /l] 8ofs] f]
ljjm| o dN" o slt xfn] f <

142 ul0ft, sIff ^

PsfO 15 Pl] ss lgod (Unitary Method)

15.1 PsfO dN" o / hDdf dN" o lgsfNg] ;d:of

?= 40

4 cf6] f sndsf] dN" o ?= slt xfn] f <

Pp6fsf] dN" oeGbf 4 cf6] fsf] dN" o a9L xg' cfp5“ . o:tf] dN" o lgsfNgsf nflu tn lbOPsf] lgod
nufpg ;lsG5 .

hDdf dN" o = j:tx' ¿sf] ;ªV\ of × Pp6f j:ts' f] dN" o

To;n} ] 4 cf6] f sndsf] hDdf dN" o = 4 × ?= 40
= ?= 160

pbfx/0f 1

1 kfs6] hLjg hnn] 6 lrof lunf; emfn] hLjg hn tof/ kfg{ ;lsG5 eg] 3 kfs6] n] slt lunf;
emfn] hLjg hn tof/ xfn] f <

oxf“, 1 kfs6] = 6 lrof lunf; emfn]
3 kfs6] = 6 x 3 lrof lunf; emfn]
= 18 lrof lunf; emfn]

pbfx/0f 2

Pp6f l;;fsndsf] ?= 3.50 kb5{ eg] 2 bhg{ l;;fsndx¿ lsGg slt ?lkof“ cfjZos knf{ <

oxf,“ Pp6f l;;fsndsf] dN" o = ?= 3.50
lsGgk' g{] l;;fsndsf] ;ªV\ of = 2 bhg{
= 2 × 12
∴ 2 bhg{ l;;fsndsf] dN" o
= 24

= 24 × ?= 3.50

= ?= 84

To;n} ] 2 bhg{ l;;fsnd lsGgsf nflu ?= 84 cfjZos k5{ .

-Pp6f j:ts' f] dN" o yfxf ePdf To:t} vfnsf w/] } j:tx' ¿sf] dN" o lgsfNg ;lsb“ f] /x5] ._

ul0ft, sIff ^ 143

kml] /, ljrf/ u/f“} ls w/] } j:tx' ¿sf] dN" o yfxf ePsf] cj:yfdf Pp6f j:ts' f] dN" o s;/L lgsfNg] <
olb 2 cf6] f k:' tssf] dN" o ?= 200 5 eg] Pp6f k:' tssf] dN" o slt xfn] f, ljrf/ u/f“} .
o:tf] cj:yfdf Pp6f j:ts' f] dN" o lgsfNg tn lbOPsf] lgod nufpg ;lsG5 M

Pp6f j:ts' f] dN" o = j:tx' ¿sf] hDdf dN" o
hDdf j:tx' ¿sf] ;ªV\ of

dflysf pbfx/0fdf Pp6f k:' tssf] dN" o,

= ?= 200 = ?= 100

2

To:t,} olb 8 cf6] f :ofpsf] dN" o ?= 56 xb“' f Pp6fsf] dN" o slt xfn] f, ljrf/ u/ .

Pp6f :ofpsf] dN" o = ?= 56

8

= ?= 7

pbfx/0f 3

olb 20 cf6] f esG' 8fx¿sf] dN" o ?= 12000 eP Pp6fsf] dN" o slt xfn] f <

oxf,“ esG' 8fx¿sf] ;ªV\ of = 20
hDdf dN" o
= ?= 12000

∴ Pp6f esG' 8fsf] dN" o = ?=12000 = ?= 600

20

pbfx/0f 4

15 af/] f l;dG] 6sf] dN" o ?= 8250 eP Ps af/] f l;dG] 6sf] dN" o slt xfn] f <

oxf,“ hDdf l;dG] 6 = 15 af/] f

hDdf dN" o = ?= 8250

∴ Ps af/] f l;dG] 6sf] dN" o = ?=8250

15

= ?=550

144 ul0ft, sIff ^

cEof; 15.1 j:tx' ¿sf] ;ªV\ of

1. tnsf] cj:yfdf hDdf d"No lgsfn M 15
PsfO dN" o -Pp6fsf] dN" o_ 22
65
-s_ ?= 25 57
-v_ ?= 45.50
-u_ ?= 350 hDdf dN" o
-3_ ?= 250.50 ?= 240
?= 576
2. tnsf] cj:yfdf PsfO d"No lgsfn M ?= 2170
j:tx' ¿sf] ;ªV\ of ?= 5800

-s_ 12
-v_ 32
-u_ 70
-3_ 232

3. 1 kfs6] hLjg hnn] 6 lrof lunf; emfn] hLjg hn tof/ kfg{ ;lsG5 eg] 4 kfs6] n] slt
lunf; emfn] hLjg hn tof/ xfn] f <

4. Ps kfs6] la:s6' sf] dN" o ?=17.50 eP 8 kfs6] sf] dN" o slt xfn] f <

5. Ps bhg{ sfkL ?=12 kl| tuf6] fsf b/n] lsGgk' ¥of] eg] hDdf slt ltgk{' 5{ <

6. Ps ls=uf| = rfdnsf] dN" o ?= 49 xb“' f 50 ls=uf| = rfdnsf] dN" o slt xfn] f <

7. 10 bhg{ s/] f lsGbf Ps dflg;n] ?=300 lt/5] eg] 1 bhg{ dfq lsgs] f] eP slt ltgk{' Yof{] <

8. olb 2 bhg{ l;;fsndsf] dN" o ?= 120 eP Pp6f l;;fsndsf] dN" o slt xG' 5 <
9. Zofd;u“ 25 cf]6f sndx¿ 5g\ . p;n] ?= 45 kl| tuf6] fsf b/n] lajm| L ubf{ hDdf slt ?lkof“

kfp5“ <

10. 100 cf6] f rsn6] ePsf] kfs] f lsGbf ?= 125 k5{ eg] Pp6f rsn6] sf] dN" o slt xG' 5 <

11. 12 bhg{ Kofs6] la:s6' sf] dN" o ?= 1152 kb5{ eg] Ps Kofs6] la:s6' sf] dN" o slt xG' 5 <

ul0ft, sIff ^ 145