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cÖv_wgK MwYZ 1

RvZxq wk¶vµg I cvV¨cy¯ZK †evW© KZ…©K 2004 wk¶veQi †_‡K

Z…Zxq †kªYxi cvV¨cy¯ZKiƒ‡c wbav©wiZ

cªv_wgK MwYZ Z…Zxq †kªYx

iPbv mv‡jn& gwZb

G. Gg. Gg. Avn&mvb Dj−vn&

nvwg`v evby †eMg

†gvt iwdKzj Bmjvg

m¤úv`bv W. gywbeyi ingvb †PŠayix

RvZxq wk¶vµg I cvV¨cy —K †evW©, XvKv

2 cÖv_wgK MwYZ

RvZxq wk¶vµg I cvV¨cy ZK †evW© 69-70, gwZwSj evwYwR¨K GjvKv, XvKv-1000

KZ©„K cÖKvwkZ|

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cix¶vg ~jK ms¯‹iY

cÖ_g gy ªY : A‡±vei, 2003

cybg©y ªY : †m‡Þ¤i- 2009

cÖPQ` I wPÎv¼‡b KvRx mvBdzÏxb AveŸvm

Kw¤cDUvi K‡¤cvR †jRvi ¯‹¨vb wjt

wWRvBb RvZxq wk¶vµg I cvV¨cy ZK †evW©

me©Rbxb cÖv_wgK wk¶v Kg©m ~wPi AvIZvq miKvi KZ…©K webvg~‡j¨ weZi‡Yi Rb¨

gy ª‡Y :

cÖv_wgK MwYZ 3

f~wgKv cÖv_wgK wk¶vi ¸YMZ gvb Dbœq‡bi j‡¶¨ 1992 mv‡j evsjv‡`‡k cÖv_wgK wk¶v ¯Z‡i †hvM¨ZvwfwËK wk¶vµg cÖeZ©b Kiv nq| 2002 mvj ch ©š— cÖvq GK hyM a‡i G wk¶vµg Abyhvqx cÖYxZ cVb cvVb mvgMÖx eëüZ n‡q Avm‡Q|

B‡Zvg‡a¨ ¯vfvweKfv‡e Áv‡bi RM‡Z N‡U‡Q A‡bK cwieZ©b| GB cwieZ©‡bi mv‡_ mvgÄm¨ ivLv Ges mgvR I †`‡ki Pvwn`v †gUv‡bvi j‡¶¨ cÖv_wgK ch©v‡q cÖPwjZ wk¶vµg g~j¨vqb, cwigvR©b I bevqb Kivi Rb¨ PÆMÖvg, wm‡jU I ewikvj wefv‡Mi cÖv_wgK wk¶vi Dbœqb cÖKí (2q ch©vq), GbwmwUwe BDwbU, XvKv, GKwU we¯Z…Z Kg©cwiKíbv MÖnY K‡i| G †cÖw¶‡Z cÖv_wgK ¯Z‡ii wk¶vµg Ges cÖ_g I wØZxq †kªYxi cVb cvVb mvgMÖx g~j¨vqb I cwigvR©b Kvh©µg (2000-2002) cwiPvjbv Kiv nq| G Kvh©µ‡gi Aax‡b cÖv_wgK ¯Z‡ii wk¶vi j¶¨ I D‡Ïk¨, cÖvwš—K †hvM¨Zv Ges cÖ_g I wØZxq †kªYxi Rb¨ welqwfwËK AR©b Dc‡hvMx †hvM¨Zv cybtwba©viY Kiv nq| cieZ©x‡Z (2002-2003) D³ BDwbU Z…Zxq †_‡K cÂg †kªYx ch©š— welqwfwËK AR©b Dc‡hvMx †hvM¨Zvmg~n cybtwba©vi‡Yi Kvh©µgI cwiPvjbv K‡i| Gi wfwË‡Z Z…Zxq †kªYxi wewfbœ wel‡qi bZzb cVb cvVb mvgMÖx cÖYqb Kiv nq| ÔwØZxq cÖv_wgK wk¶v †m±i cÖKíÕ G Kvh©µg cwiPvjbvq Avw_©K mnvqZv `vb K‡i| MwYZ wel‡qi Rb¨ wPwýZ †kªYxwfwËK AR©b Dc‡hvMx †hvM¨Zvi wfwË‡Z PÆMÖvg, wm‡jU I ewikvj wefv‡Mi wk¶vi Dbœqb cÖKí (2q ch©vq), GbwmwUwe BDwbU, XvKv KZ©„K wbe©vwPZ Ges MYcÖRvZš¿x evsjv‡`k miKv‡ii cÖv_wgK I MYwk¶v gš¿Yvjq KZ©„K Aby‡gvw`Z †jLK I m¤úv`K Z…Zxq †kªYxi Rb¨ bZzb GB MwYZ cvV¨cy ZKwU iPbv I m¤úv`bv K‡ib| †kªYx wk¶K, wk¶K cÖwk¶K, welq we‡klÁ I wk¶vµg we‡klÁ‡ì mnvqZvq GKwU Kg©kvjvi gva¨‡g eBwUi †hŠw³K g~j¨vqb Kiv nq|

MwYZ wel‡qi cwigvwR©Z wkLbµ‡g †kªYx ci¤úivq †hvM¨Zv¸‡jv wPwýZ Kiv n‡q‡Q| wkïiv hv‡Z Zv cy‡ivcywi AR©b Ki‡Z cv‡i †mw`‡K we‡kl j¶ ivLv n‡q‡Q| MvwYwZK a¨vb aviYv †ek RwUj| ZvB MvwYwZK aviYv ¯cóZi Kivi Rb¨ welqe¯‘ Dc¯’vc‡b cÖ‡qvRbxq e¨vL¨v I Qwe eënvi Kiv n‡q‡Q| wkL‡b wkï‡ì AvMÖn m„wó I cvV MÖnY mnR Kivi Rb¨ Abykxj‡bi eë¯’v ivLv n‡q‡Q| eBwU‡Z MvwYwZK mgm¨v¸‡jv ÔmnR †_‡K KwVbÕ ixwZ AbymiY K‡i mvRv‡bv n‡q‡Q Ges G¸‡jv mgvav‡b wkïiv hv‡Z D‡`¨vMx n‡Z cv‡i †mw`‡KI we‡kl `„wó †Ìqv n‡q‡Q| Avkv Kiv hvq, G‡Z wkï‡ì MvwYwZK aviYvi myôz weKvk NU‡e Ges MwY‡Zi cÖwZ A‡nZzK fxwZ ` ~i n‡e|

mswk−ó e¨w³e‡M©i mhZœ cÖqvm I mZK©Zv m‡Ë¡I eBwU‡Z wKQy ÎywU wePz¨wZ †_‡K †h‡Z cv‡i| G wel‡q Avgv‡ì AewnZ Kiv n‡j Avgiv Aek¨B cÖ‡qvRbxq ms‡kva‡bi eë ’v MÖnY Kie|

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cÖ‡dmi †gv: †gv¯Zdv KvgvjDwÏb †Pqvig¨vb RvZxq wk¶vµg I cvV¨cy¯ZK †evW©

4 cÖv_wgK MwYZ

m~wPcÎ welqe¯‘ c„ôv

MYbv Ki, co I †jL 1 − 11 ¯’vbxq gvb 12 − 16 †Rvo-we‡Rvo msL¨v 17 msL¨vi Zzjbv 18−22 Abykxjbx−1 23−25 †hvM 26−29 we‡qvM 30−32 †hvM we‡qv‡Mi m¤úK© 33 Abykxjbx-2 34−35 †hvM I we‡qvM msµvšÍ mgm¨v 36 Abykxjbx-3 37−38 ¸Y 39−47 Abykxjbx-4 48−49 fvM 50−54 Abykxjbx-5 55 mgm¨vi mgvavb 56−58 (†hvM, we‡qvM, ¸Y I fvM msµvšÍ) Abykxjbx-6 59−60 MvwYwZK cÖZxK 61−62 Abykxjbx-7 63−64 evsjv‡`kx gy ªv I †bvU 65−71 Abykxjbx-8 72−73 fMœvsk 74−94 Abykxjbx-9 95−96 cwigvc 97−103 Abykxjbx-10 104 mgq cwigvc 105−107 Abykxjbx-11 108 w`bcwÄi e¨envi 109−110 R¨vwgwZ 111−118 Abykxjbx-12 119

cÖv_wgK MwYZ 1

msL¨v MYbv Ki, co I †jL (GKwU K‡i †`Lv‡bv nj)

Qwe `‡ki

mvnv‡h¨ †jLA‡¼ †jL

K_vq †jL

1 `k 8

18 AvVvi

− `k −

− `k −

− `k −

− `k −

− `k −

− `k −

− `k −

− `k −

− `k −

2 cÖv_wgK MwYZ

`k I k‡Zi `j ˆZwi Ki ( GKwU K‡i †`Lv‡bv nj)

cÖv_wgK MwYZ 3

MYbv Ki I co

9 `k 9 Avi 1 mgvb 10 `k mgvb 1 kZ

99 + 1 = 100 GKk

1 kZ 1

101

GKk GK

1 kZ 2

102

GKk yB

1 kZ 3

103

GKk wZb

1 kZ 4

104

GKk Pvi

1 kZ 5

105

GKk cuvP

•

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4 cÖv_wgK MwYZ

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MYbv Ki I co

1 kZ 6

106

GKk Qq

1 kZ 7

107

GKk mvZ

1 kZ 8

108

GKk AvU

1 kZ 9

109

GKk bq

1 kZ 1 `k

110

GKk `k

1 kZ 1 `k 1

111

GKk GMvi

1 kZ 9 `k 9

199

GKk wbivbeŸB

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cÖv_wgK MwYZ 5

MYbv Ki I co

2 kZ

200 yBk

2 kZ 5 k 9

259 yBk EblvU

5 kZ 6 k 8

568

cuvPk AvUlwÆ

7 kZ 9 k 4

794 mvZk

PzivbeŸB

8 kZ 0 k 9

809 AvUk bq

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6 cÖv_wgK MwYZ

MYbv Ki I co

7 kZ 9 `k 4

794 mvZk

PzivbeŸB

7 kZ 9 `k 4

794 mvZk

PzivbeŸB

7 kZ 9 `k 4

794 mvZk

PzivbeŸB

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••••••••••

•••••

cÖv_wgK MwYZ 7

`k-`k K‡i MYbv MYbv Ki I co

1 `k

10 `k

2 `k 20 wek

3 `k 30 wÎk

4 `k 40

Pwj−k

5 `k 50

cÂvk

6 `k 60 lvU

7 `k 70 mËi

8 `k 80 Avwk

9 `k

90 beŸB

10 `k

100 GKk

8 cÖv_wgK MwYZ

kZ−kZ K‡i MYbv MYbv Ki I co

1 kZ 100 GKk

2 kZ 200 `yBk

3 kZ 300wZbk

4 kZ 400Pvik

5 kZ 500cuvPk

6 kZ

600 Qqk

7 kZ

700 mvZk

8 kZ 800

AvUk

9 kZ 900

bqk

10 kZ 1000

GK nvRvi

cÖv_wgK MwYZ 9

nvRvi-nvRvi K‡i MYbv MYbv Ki I co

1 nvRvi 1000

GK nvRvi

2 nvRvi 2000 nvRvi

3 nvRvi 3000nvRvi

4 nvRvi

4000

nvRvi

5 nvRvi

5000

nvRvi

6 nvRvi

6000

nvRvi

7 nvRvi 7000

nvRvi

8 nvRvi 8000

nvRvi

9 nvRvi 9000

nvRvi

10 nvRvi

10000 nvRvi

10 cÖv_wgK MwYZ

co

msL¨v A‡¼

nvRvi kZK `kK GKK msL¨v K_vq

AhyZ mnmª

101 1 0 1 GKk GK

109 1 0 9 GKk bq

115 1 1 5 GKk c‡bi

519 5 1 9 cuvPk Ewbk

1000 1 0 0 0 GK nvRvi

1928 1 9 2 8 GK nvRvi bqk AvUvk

2379 2 3 7 9 yB nvRvi wZbkZ EbAvwk

4830 4 8 3 0 Pvi nvRvi AvUk wÎk

7075 7 0 7 5 mvZ nvRvi cuPvËi

8609 8 6 0 9 AvU nvRvi QqkZ bq

9999 9 9 9 9 bq nvRvi bqk wbivbeŸB

10000 1 0 0 0 0 `k nvRvi

1 AhyZ = 10 nvRvi

cÖv_wgK MwYZ 11

co, A‡¼ I K_vq †jL (GKwU K‡i †`Lv‡bv nj) msL¨v A‡¼

nvRvi kZK `kK GKK msL¨v K_vq

AhyZ mnmª

5739 5 7 3 9 cuvP nvRvi mvZk EbPwj−k

425

1640

3333

5009

8270

10000

6085

9601

co I A‡¼ †jL (GKwU K‡i †`Lv‡bv nj)

msL¨v K_vq nvRvi

kZK `kK GKK msL¨v A‡¼ AhyZ mnmª

`yB nvRvi mvZk EbbeŸB 2 7 8 9 2789

wZbk mvZvwk

AvUk cuvP

GK nvRvi bqk cÂvk

Pvi nvRvi cuPvËi

mvZ nvRvi wZbk weqvwj −k

AvU nvRvi Avwk

bq nvRvi bqk beŸB

`k nvRvi

12 cÖv_wgK MwYZ

¯’vbxq gvb Qwe †`‡L msL¨v †jL

(GKwU K‡i †`Lv‡bv nj) ¯’vbxq gvb †jL

(GKwU K‡i †`Lv‡bv nj)

6 5

` G6 9 9 GKK = 9 6 `kK = 60

` G 8 1 ` G 7 5

` G 4 9

` G 9 9

wb‡Pi msL¨v¸‡jv †`‡L GK‡Ki I `k‡Ki N‡ii A¼ †jL| (GKwU K‡i †`Lv‡bv nj)

34 GKK `kK

52 GKK `kK

68 `kK GKK

91 `kK GKK

87 GKK `kK

78 `kK GKK

4 3

cÖv_wgK MwYZ 13

¯’vbxq gvb k ` G5 6 3

3 GKK = 3 6 `kK = 60 5 kZK = 500

m k ` G1 0 0 0

0 GKK = 0 0 kK = 0 0 kZK = 0

1 mnmª = 1000m k ` G1 1 1 1

1 GKK = 1 1 `kK = 10 1 kZK = 100

1 mnmª = 1000m k ` G9 0 5 7

7 GKK = 7 5 `kK = 50 0 kZK = 0

9 mnmª = 9000A m k ` G1 0 0 0 0

0 GKK = 0 0 GKK = 0 0 `kK = 0 0 mnmª = 0

1 AhyZ = 10000

14 cÖv_wgK MwYZ

¯’vbxq gvb Qwe †`‡L msL¨v †jL

(GKwU K‡i †`Lv‡bv nj)¯’vbxq gvb †jL

(GKwU K‡i †`Lv‡bv nj)

m k ` G5 0 9 2 2 GKK = 2 9 `kK = 90 0 kZK = 0

5 mnmª = 5000

m k ` G3 8 5 1

m k ` G6 2 0 9

m k ` G7 1 6 5

m k ` G9 9 8 9

cÖv_wgK MwYZ 15

¯’vbxq gvb AhyZ mnmª kZK `kK GKK msL¨v K_vq

1 GK

1 0 `k

1 0 0 GK kZ

1 0 0 0 GK nvRvi

1 0 0 0 0 `k nvRvi

D`vniY 1| 7 9 4 3 msL¨vwUi cÖwZwU A‡¼i ¯’vbxq gvb wbY©q Ki| 7 9 4 3 3 Gi ¯’vbxq gvb = 3 GKK = 3 4 Gi ¯’vbxq gvb = 4 `kK = 4 0 9 Gi ¯’vbxq gvb = 9 kZK = 9 0 0 7 Gi ¯’vbxq gvb = 7 mnmª = 7 0 0 0

D`vniY 2| 8 0 2 5 msL¨vwUi cÖwZwU A‡¼i ¯’vbxq gvb wbY©q Ki| 8 0 2 5 5 Gi ¯’vbxq gvb = 5 GKK = 3 2 Gi ¯’vbxq gvb = 2 `kK = 4 0 0 Gi ¯’vbxq gvb = 0 kZK = 9 0 0 8 Gi ¯’vbxq gvb = 8 mnmª = 7 0 0 0

16 cÖv_wgK MwYZ

¯’vbxq gvb wbY©q Ki (`yBwU K‡i †`Lv‡bv nj)

8529

9 Gi ¯’vbxq gvb = 9 2 Gi ¯’vbxq gvb = 20 5 Gi ¯’vbxq gvb = 500 8 Gi ¯’vbxq gvb = 8000

5947

7 Gi ¯’vbxq gvb = 4 Gi ¯’vbxq gvb = 9 Gi ¯’vbxq gvb = 5 Gi ¯’vbxq gvb =

694

9 Gi ¯’vbxq gvb = 90 4 Gi ¯’vbxq gvb = 4 6 Gi ¯’vbxq gvb = 600

4806


9705


8092


9864


2987


cÖv_wgK MwYZ 17

†Rvo we‡Rvo msL¨v †Rvo msL¨v we‡Rvo msL¨v

10, 32, 54, 76, 98 msL¨v¸‡jvi GK‡Ki N‡i 0, 2, 4, 6, 8 Av‡Q|

GZGe, †h‡Kvb msL¨vi GK‡Ki N‡i 0, 2, 4, 6 ev 8 _vK‡j †m¸‡jv †Rvo msL¨v|

21, 43, 55, 67, 89 msL¨v¸‡jvi GK‡Ki N‡i 1, 3, 5, 7, 9 Av‡Q|

AZGe, †h‡Kvb msL¨vi GK‡Ki N‡i 1, 3, 5, 7 ev 9 _vK‡j †m¸‡jv we‡Rvo msL¨v

wb‡Pi msL¨v¸‡jv †_‡K †Rvo I we‡Rvo msL¨v¸‡jv Avjv`v K‡i LvwjN‡i †jL (GKwU K‡i †`Lv‡bv nj)

msL¨v †Rvo msL¨v we‡Rvo msL¨v

89, 314, 336, 423, 667, 134, 245, 98

314, 336, 134, 98

89, 423, 667, 245

116, 223, 425, 356, 237, 430, 278, 409

609, 708, 534, 463, 612, 615, 727, 790

1020, 1042, 1045, 1057, 2131, 3134, 3336, 2223

809, 970, 4336, 5127, 6241, 6565, 5798

7001, 7203, 7310, 5436, 905, 938, 6292, 10000

3572, 5718, 4679, 791, 8843, 9992, 4445, 97

18 cÖv_wgK MwYZ

msL¨vi Zzjbv eo †QvU wbY©q I cÖZxK e¨envi K‡i †jLv 78 = 7 `k 8

61 = 6 `k 1

∴ 78 eo, 61 †QvU

ev, 78 > 61; > eo wPý

covi wbqg : 78, 61 †_‡K eo|

Avevi, Gfv‡eI †jLv hvq

61 < 78; < †QvU wPý

covi wbqg : 61, 78 †_‡K †QvU|

yB A‡¼i yBwU msL¨vq †h msL¨vwUi `k‡Ki A¼ eo, †mB msL¨vwUB eo|

94 = 9 `k 4

98 = 9 `k 8

∴ 94 †QvU, 98 eo

ev, 94 < 98

yB A‡¼i yBwU msL¨v `k‡Ki A¼ mgvb n‡j, †h msL¨vwUi GK‡Ki A¼ †QvU, †mB msL¨vwU †QvU|

100 = 10 `k

99 = 9 `k 9

∴ 100 eo, 99 †QvU

ev, 100 > 99

wZb A‡¼i msL¨v `yB A‡¼i †h †Kvb msL¨v †_‡K eo|

475 = 4 kZ 7 `k 5

459 = 4 kZ 5 `k 9

∴ 475 eo, 459 †QvU

ev, 475 > 459

wZb A‡¼i yBwU msL¨v kZ‡Ki A¼ GKB n‡j, †h msL¨vwUi `k‡Ki A¼ eo, †mB msL¨vwU eo|

cÖv_wgK MwYZ 19

msL¨vi Zzjbv 147 = 1 kZ 4 `k 7

148 = 1 kZ 4 `k 8

∴ 147 †QvU, 148 eo

ev, 147 < 148

wZb A‡¼i yBwU msL¨vq kZ‡Ki I `k‡Ki A¼ mgvb n‡j, †h msL¨vwUi GK‡Ki A¼ eo, †mB msL¨vwU eo|

801 = 8 kZ 0 `k 1

699 = 6 kZ 9 `k 9

∴ 801 eo, 699 †QvU

ev, 801 > 699

wZb A‡¼i yBwU msL¨vi g‡a¨ †h msL¨vwU‡Z kZ‡Ki A¼ eo, †mB msL¨vwU eo|

8725 = 8 nvRvi 7 kZ 2 `k 5

9878 = 9 nvRvi 8 kZ 7 `k 8

∴ 8725 †QvU, 9878 eo

ev, 8725 < 9878

Pvi A‡¼i yBwU msL¨vi g‡a¨ †h msL¨vwU‡Z nvRv‡ii A¼ eo, †mB msL¨vwU eo|

6789 = 6 nvRvi 7 kZ 8 `k 9

6776 = 6 nvRvi 7 kZ 7 `k 6

∴ 6789 eo, 6776 †QvU

ev, 6789 > 6776

Pvi A‡¼i `yBwU msL¨vi g‡a¨ nvRv‡ii A¼ GKB n‡j, kZK, `kK I GK‡Ki A¼ †`‡L eo †QvU wbY©q Ki‡Z nq|

10000 = 10 nvRvi

9999 = 9 nvRvi 9 kZ 9 `k 9

∴ 10000 eo, 9999 †QvU

ev, 10000 > 9999

yBwU msL¨vi g‡a¨ GKwU cuvP A‡¼i I GKwU Pvi A‡¼i n‡j, cuvP A‡¼i msL¨vwU me mgqB eo|

20 cÖv_wgK MwYZ

eo †QvU wbY©q Ki I cÖZxK e¨envi K‡i †jL (wZbwU K‡i †`Lv‡bv nj) 207, 211

207 †QvU, 211 eo 207 < 211

289, 283

289 eo, 283 †QvU 289 > 283

790, 892

892 eo, 790 †QvU 892 > 790

301, 299

⎯⎯ eo, ⎯⎯ †QvU

3452, 3500

⎯⎯ †QvU ⎯⎯ eo

4571, 4671

⎯⎯ eo, ⎯⎯ †QvU

6789, 7790

⎯⎯ eo, ⎯⎯ †QvU

7028, 7038

⎯⎯ †QvU ⎯⎯ eo

8109, 8099

⎯⎯ †QvU ⎯⎯ eo

9909, 9999

⎯⎯ eo, ⎯⎯ †QvU

6006, 6077

⎯⎯ †QvU ⎯⎯ eo

cÖv_wgK MwYZ 21

msL¨v Abyµ‡g mvRv‡bv eo †_‡K †QvU I †QvU †_‡K eo µ‡g mvRv‡bv

eo †_‡K †QvU †QvU †_‡K eo

88, 59, 76, 91

GLv‡b, 91 eo, 88 †QvU

88 eo, 76 †QvU

76 eo, 59 †QvU

∴ 91, 88, 76, 59

ev, 91 > 88 > 76 > 59

88, 59, 76, 91

GLv‡b, 59 †QvU, 76 eo

76 †QvU, 88 eo

88 †QvU, 91 eo

∴ 59, 76, 88, 91

ev, 59 < 76 < 88 < 91

340, 356, 389, 346

GLv‡b, 389 eo, 356 †QvU

356 eo, 346 †QvU

346 eo, 340 †QvU

∴ 389, 356, 346, 340

ev, 389 > 356 > 346 > 340

340, 356, 389, 346

GLv‡b, 340 †QvU, 346 eo

346 †QvU, 356 eo

356 †QvU, 389 eo

∴ 340, 346, 356, 389

ev, 340 < 346 < 356 < 389

781, 692, 835, 901, 792

GLv‡b, 901 eo, 835 †QvU

835 eo, 792 †QvU

792 eo, 781 †QvU

781 eo, 692 †QvU

∴ 901, 835, 792, 781, 692

ev, 901 > 835 > 792 > 781 > 692

781, 692, 835, 901, 792

GLv‡b, 692 †QvU, 781 eo

781 †QvU, 792 eo

792 †QvU, 835 eo

835 †QvU, 901 eo

∴ 692, 781, 792, 835, 901

ev, 692 < 781 < 792 < 835 < 901

22 cÖv_wgK MwYZ

msL¨v Abyµ‡g mvRv‡bv eo †_‡K †QvU I †QvU †_‡K eo µ‡g mvRvI (GKwU K‡i †`Lv‡bv nj)

msL¨v eo †_‡K †QvU †QvU †_‡K eo

91, 78, 99, 66

99, 91, 78, 70,

66 99 > 91 > 78

> 70 > 66

66, 70, 78, 91, 99

66 < 70 < 78 < 91

< 99

677, 777, 690, 791, 760

409, 496, 460, 480, 491

699, 791, 889, 691, 776

99, 81, 77, 100, 107, 103

875, 760, 870, 799, 800

499, 509, 480, 600, 700

600, 802, 714, 799, 607

999, 899, 890, 799, 900

599, 590, 600, 670, 672

779, 680, 791, 699, 709

cÖv_wgK MwYZ 23

AbykxjbxÑ1 1| Qwe †`‡L †duvUv MYbv Ki I msL¨v †jL : A‡¼ K_vq

2| 105 †_‡K 160 ch©šÍ msL¨v¸‡jv ci ci A‡¼ †jL|

3| wb‡Pi msL¨v¸‡jv co I LvwjN‡i A‡¼ †jL :

2 kZ 5 `k 9 7 nvRvi 7 kZ 8

6 kZ 3 `k 5 nvRvi 5 kZ 5 k 5

9 kZ 8 8 nvRvi

1 nvRvi 4 kZ 7 k 1 9 nvRvi 3 kZ 4

3 nvRvi 9 `k 9 nvRvi 9

4| A‡¼ †jL :

GKk yB, AvUk beŸB, bqk Qq, wZb nvRvi cuvPk ewÎk, mvZ nvRvi Pzivwk, cuvP nvRvi

Pvik Dwbk, AvU nvRvi AvUk AvUvwk, Qq nvRvi GK, bq nvRvi beŸB, `k nvRvi|

24 cÖv_wgK MwYZ

5| K_vq †jL :

202, 543, 960, 1369, 3095, 5807, 6666, 7950, 8009, 9070

6| LvwjNi c~iY Ki :

(K) 10, 20, , , , 60 , 80, , 100

(L) 100, , 300, 400, , , 700, , , 1000

(M) 1000, , , 4000, 5000, , ,

, 9000,

7| wb‡Pi msL¨v¸‡jvi cÖwZwU A‡¼i ¯’vbxq gvb wj‡L LvwjNi c~iY Ki : 5367−G 6753−G 7536−G 3675−G

7 Gi ¯’vbxq gvb = 7 700 7000 70

6 Gi ¯’vbxq gvb =



8| mwVK msL¨v‡Z †Mvj `vM `vI :

(K) †Kvb& msL¨vwUi `k‡Ki N‡i 3 Av‡Q? 453, 391, 732

(L) †Kvb& msL¨vwU nvRv‡ii N‡i 5 Av‡Q? 9250, 7536, 5129

(M) †Kvb& msL¨vwUi GK‡Ki N‡i 9 Av‡Q? 6219, 9126, 2691

(N) †Kvb& msL¨vwUi kZ‡Ki N‡i 1 Av‡Q? 8601, 6190, 1432

(O) †Kvb& msL¨vwUi `k‡Ki N‡i 7 Av‡Q? 4670 7203 6715

cÖv_wgK MwYZ 25

9| LvwjN‡i Av‡Mi I c‡ii msL¨v †jL :

499 ; 578 ;

705 ; 900 ;

10| wb‡Pi msL¨vi †Rvov¸‡jv †_‡K eoÑ†QvU wbY©q Ki :

365, 356 834, 921; 2485, 2396

5298, 6189 7009, 9007; 8000, 7999|

11| wb‡Pi msL¨v¸‡jv‡K eo †_‡K †QvU I †QvU †_‡K eo µ‡g mvwR‡q †jL : (K) 432, 428, 565, 605, 342

(L) 702, 720, 699, 996, 969

(M) 583, 848, 584, 398, 839

(N) 3670, 3706, 4021, 4210, 3999|

12| wb‡Pi msL¨v¸‡jv †_‡K †Rvo I we‡Rvo msL¨v Avjv`v Ki Ges †QvU †_‡K eo µ‡g mvwR‡q †jL :

384, 495, 999, 2530, 4268, 3597,

6092, 7381, 8643, 9776|

13| wb‡Pi LvwjN‡i eoÑ†QvU cÖZxK †jL :

(K) 463 436 (L) 800 809

(L) 2532 2352 (N) 5899 6000

(O) 3609 3906 4000

(P) 8990 9899 9099

26 cÖv_wgK MwYZ

†hvM †hvM Ki (nv‡Z bv †i‡L)

34 3 `k 4

+ 23 + 2 `k 3

57 5 `k 7

14 1 `k 4

32 3 `k 2

+ 53 + 5 `k 3

99 9 `k 9

100

602 6 kZ 0 `k 2

154 1 kZ 5 `k 4

+ 230 + 2 kZ 3 `k 0

986 9 kZ 8 `k 6

1021 1 nvRvi 0 kZ 2 `k 1

3202 3 nvRvi 2 kZ 0 `k 2

143 1 kZ 4 `k 3

+ 4430 + 4 nvRvi 4 kZ 3 `k 0

8796 8 nvRvi 7 kZ 9 `k 6

365 + 34

= 3 kZ 6 `k 5 + 3 `k 4

= 3 kZ 9 `k 9

= 399

4031 + 41 + 1906

= 4 nvRvi 0 kZ 3 `k 1 + 4 `k 1

+ 1 nvRvi 9 kZ 0 `k 6

= 5 nvRvi 9 kZ 7 `k 8

= 5978

cÖv_wgK MwYZ 27

†hvM Ki 54 402 42 1007

+ 45 + 397 613 3450

99 799 + 200 2021

855 + 3520

9998

63 + 36 = 99

721 + 12 + 100 = 833

4021 + 305 + 1452 = 5778

42 50 60

+ 24 + 24 + 39

641 310 413

+ 256 + 421 + 264

31 260 429

27 107 360

+ 41 + 321 + 100

10 3021 2104

21 2405 1540

45 + 3272 3202

+ 13 + 3153

24 + 43 =

50 + 39 =

402 + 396 =

529 + 40 =

24 + 42 + 13 =

340 + 103 + 41 =

501 + 48 + 350 =

32 + 21 + 32 + 14 =

2041 + 3425 + 2130 =

4102 + 1534 + 2051 =

†hvM Ki (nv‡Z †i‡L)

45 4 `k 5

+ 27 + 2 `k 7

72 6 `k 12

= 7 `k 2

= 72

28 cÖv_wgK MwYZ

+1 78 7 `k 8 + 15 + 1 `k 5 93 8 `k 13 = 9 `k 3 = 93

+1 +1

4 3 7 4 kZ 3 `k 7

2 6 5 2 kZ 6 `k 5

+ 1 7 2 + 1 kZ 7 `k 2

8 7 4 7 kZ 16 `k 14

= 7 kZ 17 `k 4

= 8 kZ 7 `k 4

= 874

96 + 89 = 9 `k 6 + 8 `k 9 = 17 `k 15 = 18 `k 5 = 185

65 484 167 + 37 + 879 452 102 1363 + 374 993 88 + 77 = 165 148 + 561 + 67 = 776

†hvM Ki (nv‡Z †i‡L) 49 76 89 + 63 + 67 + 98

450 387 572 + 368 + 508 + 489

147 350 508 282 407 490 + 306 + 284 + 373

35 + 46 =

57 + 44 =

68 + 52 =

124 + 238 =

480 + 367 =

204 + 386 + 42 =

382 + 268 + 100 =

cÖv_wgK MwYZ 29

+1 +1 +1 1 nvRvi 2 kZ 3 `k 5 1 2 3 5 2 nvRvi 3 kZ 1 `k 4 2 3 1 4 6 kZ 8 `k 2 + 6 8 2 = 3 nvRvi 11 kZ 12 `k 11 4 2 3 1 = 3 nvRvi 11 kZ 13 `k 1 = 3 nvRvi 12 kZ 3 `k 1 = 4 nvRvi 2 kZ 3 `k 1 = 4231

+1 +2 +2 3 nvRvi 4 kZ 2 `k 5 3 4 2 5 3 nvRvi 3 kZ 6 `k 7 3 3 6 7 1 nvRvi 3 kZ 8 `k 2 1 3 8 2 1 nvRvi 6 kZ 4 `k 6 + 1 6 4 6 8 nvRvi 16 kZ 20 `k 20 9 8 2 0 = 8 nvRvi 16 kZ 22 `k 0 = 8 nvRvi 18 kZ 2 `k 0 = 9 nvRvi 8 kZ 2 `k 0 = 9820

4523 1064 1768 2507 + 3254 4875 9545 + 1439 9885

1426 + 2035 + 4782 = 8243 2407 + 885 + 1646 = 4938 2046 + 3807 + 11682 = 7535 275 + 86 + 1000 + 5608 = 6969

367 402 684 742 396 176 + 176 + 427 + 453 461 4321 832 689 2687 5426 247 1754 1374 + 21 + 476

203 + 456 + 387 = 515 + 277 + 190 = 4560 + 3809 = 4523 + 4280 + 307 = 1460 + 3287 + 4789 = 245 + 278 + 469 + 3871 =

30 cÖv_wgK MwYZ

we‡qvM

65 6 `k 5

− 41 − 4 `k 1 24 2 `k 4

766 7 kZ 6 `k 6 GLv‡b, 766 we‡qvRb

− 423 − 4 kZ 2 `k 3 423 we‡qvR¨ 343 3 kZ 4 `k 3 343 we‡qvMdj = 343

876 we‡qvRb

− 514 we‡qvR¨ 362 we‡qvMdj

508 − 304 GLv‡b, we‡qvRb 508

= 5 kZ 0 `k 8 − 3 kZ 0 `k 4 we‡qvR¨ 304

= 2 kZ 0 `k 4 we‡qvMdj 304

= 204

857 735 1649 8765

− 613 − 214 − 317 − 4321 244 521 1332 4444

563 − 241 = 322

795 − 412 = 383

962 − 530 = 432

684 we‡qvRb

− 341 we‡qvR¨ we‡qvMdj

897 GLv‡b, 897

− 435 435 we‡qvMdj

789 − 406 = ⎯⎯ , GLv‡b, 789 we‡qvRb, 406 we‡qvR¨,

we‡qvMdj ⎯⎯|

968 − 510 = ⎯⎯ , GLv‡b, we‡qvRb ⎯⎯ , we‡qvR¨ ⎯⎯ ,

we‡qvMdj ⎯⎯|

cÖv_wgK MwYZ 31

we‡qvM Ki (nv‡Z †i‡L) 53 5 `k 3

− 24 − 2 `k 4 29 2 `k 9 = 29 e¨vL¨v : we‡qvR‡bi GK‡Ki N‡ii 3 †_‡K we‡qv‡R¨i GK‡Ki N‡ii 4 eo| GRb¨ 3 †_‡K 4 we‡qvM Kiv hvq bv, we‡qvR‡bi 1wU `‡ki evwÛj †Lvjv n‡q‡Q| G‡Z `‡ki evwÛj n‡q‡Q 4wU Ges †Lvjv KvwV 13wU| Gi †_‡K 2wU `‡ki evwÊj I 4wU †Lvjv KvwV ev` †`Iqv n‡q‡Q| Aewkó _v‡K 2wU `‡ki evwÊj I 9wU †Lvjv KvwV| A_©vr, 2 `k 9| 86 8 `k 6 7 `k 16

− 48 − 4 `k 8 − 4 `k 8 38 3 `k 8 = 38

867 8 kZ 6 `k 7 7 kZ 15 `k 17

− 498 − 4 kZ 9 `k 8 − 4 kZ 9 `k 8 369 3 kZ 6 `k 9 = 369

536 5 kZ 3 `k 6 4 kZ 12 `k 16 635 − 478

−287 − 2kZ 8 `k 7 − 2 kZ 8 `k 7 = 6 kZ 3 `k 5 − 4 kZ 2 kZ 4 `k 9 7 `k 8

= 249 = 5 kZ 12 `k 15 − 4 kZ 7 `k 8 = 1 kZ 5 `k 7 = 157

`kK GKK 8 = 7 + 1

↓ 7 − 4

6 ↓ 16 8

3 8

kZK `kK GKK8=7+1

↓ 7 − 4

6=5+1 ↓ 15 9

7 ↓ 17 8

3 6 9

32 cÖv_wgK MwYZ

we‡qvM Ki 845 751 − 267 = 484

− 486 804 − 619 = 185

359 7165 − 3076 = 4089 9 5 4 8

− 6 8 7 9 2 6 6 9

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

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

850 2610 4050

− 451 − 702 − 3152

635 − 246 =

702 − 403 =

1242 − 553 =

6104 − 5213 =

8426 − 7439 =

7120 8240 9032

− 4923 − 4546 − 8136

kZK `kK GKK 8

−4+1 4+10

8+1 5+10

6

3 5 9

nvRvi kZK `kK GKK 9

− 6+1 5+10

8+1 4+10

7+1 8+10

9

2 6 6 9

cÖv_wgK MwYZ 33

†hvM we‡qv‡Mi m¤úK©

30 + 15 = 45 45 − 30 = 15

45 − 15 = 30 15 + 30 = 45

we‡qvM †hv‡Mi wecixZ

36 + 10 = 46 36 − 9 = 27

46 − 10 = 36 36 − 27 = 9

46 − 36 = 10 27 + 9 = 36 45 + 15 = 60 65 + 22 = 87

60 − = 45 − 65 = 22

60 − = 15 87 − 22 =

77 − = 54 88 + 12 =

54 + 23 = 100 − 88 =

− 54 = 23 − 12 = 88

+15

−15

4530

−30

+30

1545

34 cÖv_wgK MwYZ

AbykxjbxÑ2 1| †hvM Ki : (K) 44 (L) 302 (M) 210 (N) 1504 + 22 + 546 467 3021 + 322 4200 + 164 (O) 367 (P) 478 (Q) 210 (R) 1504 + 476 305 657 1876 + 530 756 3718 + 675 + 2680 2| †hvM Ki :

(K) 61 + 16 = (L) 326 + 602 =

(M) 456 + 546 = (N) 376 + 673 + 763 =

(O) 2561 + 1562 + 3578 =

(P) 4678 + 467 + 2784 + 1648 =

3| we‡qvM Ki : (K) 968 (L) 7562 (M) 6894 − 426 − 340 − 4102 (N) 702 (O) 9120 (P) 6147 − 304 − 571 − 5658 4| we‡qvM Ki :

(K) 675 − 41 = (L) 9786 − 6120 =

(M) 714 − 316 = (N) 8001 − 7016 =

(O) 8342 − 506 = (P) 7124 − 6125 =

cÖv_wgK MwYZ 35

5| we‡qvM Ki Ges cv‡ki LvwjNi c ~iY Ki : (K) 820 GLv‡b, we‡qvRb

− 731 we‡qvR¨

we‡qvMdj

(L) 1604 GLv‡b, we‡qvMdj

− 708 we‡qvRb

we‡qvR¨

(M) 482 − 183 = GLv‡b, we‡qvMdj

we‡qvRb

we‡qvR¨ 6| LvwjN‡i mwVKfv‡e we‡qvRb, we‡qvR¨, we‡qvMdj †jL : (K) 840 − 651 = 189 GLv‡b, 189

840

651

(L) 2041 GLv‡b, 2041

− 1832 209

209 1832 7| LvwjNi c~iY Ki : (K) 53 + 27 = (L) 88 − 19 =

80 − 27 = + 69 = 88

80 − 53 = 88 − = 19

(M) 64 + = 92 (N) 125 + 75 =

− 64 = 28 − 75 = 125

92 − = 64 200 − = 75

36 cÖv_wgK MwYZ

†hvM we‡qvM msµvšÍ mgm¨v D`vniY 1| wgVvcyKzi ¯‹z‡j wk¶v_©x wQj 365 Rb| AviI 125 Rb wk¶v_©x fwZ© nj| GLb †gvU KZRb wk¶v_©x nj? mgvavb : wk¶v_©x wQj 365 Rb fwZ© nj + 125 Rb

∴ †gvU wk¶v_©x 490 Rb DËi : 490 Rb| D`vniY 2| wRwbqv †eM‡gi gvwmK Avq 7650 UvKv| Zuvi gvwmK e¨q 6700 UvKv| Zuvi gvwmK Rgv KZ? mgvavb : gvwmK Avq 7650 UvKv gvwmK e¨q − 6700 UvKv

∴ gvwmK Rgv 950 UvKv DËi : gvwmK Rgv 950 UvKv| D`vniY 3| mxgvi Av‡Q 575 UvKv| mxgvi †_‡K ixgvi 190 UvKv Kg Av‡Q| Zv‡`i yB R‡bi UvKv GKÎ Ki‡j bxcvi UvKvi mgvb nq| bxcvi KZ UvKv Av‡Q? mgvavb : mxgvi Av‡Q 575 UvKv| mxgvi †P‡q ixgvi Av‡Q 190 UvKv Kg| ∴ ixgvi Av‡Q (575 − 190) UvKv = 385 UvKv| ∴ bxcvi Av‡Q (575 + 385) UvKv = 960 UvKv| DËi : 960 UvKv| D`vniY 4| †mwjg mv‡ne 525 UvKv wb‡q evRv‡i †M‡jb| wZwb 150 UvKvi gvQ, 90 UvKvi †Zj Ges 75 UvKvi kvK mewR wKb‡jb| Zuvi wbKU Avi KZ UvKv _vKj? mgvavb : gvQ 150 UvKv wQj 525 UvKv †Zj 90 UvKv LiP − 315 UvKv kvK mewR 75 UvKv _vKj 210 UvKv †gvU 315 UvKv DËi : †mwjg mv‡n‡ei 210 UvKv _vKj|

cÖv_wgK MwYZ 39

¸Y

4 4 4 12

4 + 4 + 4 = 12 4, 3 evi mgvb 12 4 × 3 = 12

3 3 3 3 12

3 + 3 + 3 + 3 = 12 3 × 4 = 12

3, 4 evi mgvb 12

∴ 4 × 3 = 3 × 4 = 12

4 × 5 = 20

5 × 4 = 20 LvwjNi c~iY Ki (GKwU K‡i †`Lv‡bv nj)

7 × 5 = 5 × 7 = 35 7 × 8 = 8 × 7 =

4 × 9 = 9 × 4 = 6 × 8 = 8 × 6 =

5 × 8 = 8 × 5 = 7 × 9 = 9 × 7 =

9 × 6 = 6 × 9 = 5 × 6 = 6 × 5 =

40 cÖv_wgK MwYZ

LvwjNi c~iY Ki

× 1 2 3 4 5 6 7 8 9 10

1 5 9

2 6 14

3 6 18 30

4 12 36

5 5 35

6 12 60

7 28 42

8 8 64

9 36 81

10 50 100

cÖv_wgK MwYZ 41

¸‡Yi bvgZv (11 †_‡K 20)

× 1 2 3 4 5 6 7 8 9 10

11 11 22 33 44 55 66 77 88 99 110

12 12 24 36 48 60 72 84 96 108 120

13 13 26 39 52 65 78 91 104 117 130

14 14 28 42 56 70 84 98 112 126 140

15 15 30 45 60 75 90 105 120 135 150

16 16 32 48 64 80 96 112 128 144 160

17 17 34 51 68 85 102 119 136 153 170

18 18 36 54 72 90 108 126 144 162 180

19 19 38 57 76 95 114 133 152 171 190

20 20 40 60 80 100 120 140 160 180 200

42 cÖv_wgK MwYZ

LvwjNi c~iY Ki 13

×1

13

×2

13

×3

13

×4

13

×5

13

×6

13

×7

13

×8

13

×9

13

×10

15

×1

15

×2

15

×3

15

×4

15

×5

15

×6

15

×7

15

×8

15

×9

15

×10

17

×1

17

×2

17

×3

17

×4

17

×5

17

×6

17

×7

17

×8

17

×9

17

×10

18

×1

18

×2

18

×3

18

×4

18

×5

18

×6

18

×7

18

×8

18

×9

18

×10

bvgZvi mvnv‡h¨ LvwjNi c ~iY Ki

× 1 2 3 4 5 6 7 8 9 10

11 55 88

14 42 98

16 16 96 160

19 38 171

20 60 160

cÖv_wgK MwYZ 43

¸Y D`vniY 1| 43 e¨vL¨v :

×3 4 3 × 3

129 3 × 3 = 9 = 9

4 `k × 3 = 12 `k = 120 129 DËi : 129

D`vniY 2| 28 e¨vL¨v :

×7 2 8 × 7

196 8 × 7 = 56 = 56

2 `k × 7 = 14 `k = 140 196 DËi : 196

D`vniY 3| 213 e¨vL¨v :

×3 2 1 3 × 3

639 3 × 3 = 9 = 9

1 `k × 3 = 3 `k = 30

2 kZ × 3 = 6 kZ = 600 639 DËi : 639 D`vniY 4| 683 e¨vL¨v : ×9 6 8 3 × 9 6147 3 × 9 = 27 = 27 8 `k × 9 = 72 `k = 720 6 kZ × 9 = 54 kZ = 5400 6147 DËi : 6147

44 cÖv_wgK MwYZ

msw¶ß wbq‡g ¸Y

D`vniY 5|

13 × 10 = 13 × 1 `k

= 13 `k

= 130

∴ 13 × 10 = 130

DËi : 130

D`vniY 6|

57 × 20 = 57 × 2 `k

= 114 `k

= 1140

∴ 57 × 20 = 1140

DËi : 1140

D`vniY 7|

135 × 30 = 135 × 3 `k

= 405 `k

= 4050

∴ 135 × 30 = 4050

DËi : 4050

D`vniY 8|

340 × 50 = 340 × 5 `k

= 1700 `k

= 17000

∴ 340 × 50 = 17000

DËi : 17000

D`vniY 9|

82 × 100 = 82 × 1 kZ

= 82 kZ

= 8200

∴ 82 × 100 = 8200

DËi : 8200

D`vniY 10|

100 × 100 = 100 × 1 kZ

= 100 kZ

= 10000

∴ 100 × 100 = 10000

DËi : 10000

cÖv_wgK MwYZ 45

D`vniY 11| ¸Y Ki : e¨vL¨v : 34 3 4 1 2 × 12 4 = 4 2 = 2 mgvavb : 3 `k = 30 1 `k = 10 34 GLb, 3 4 × 2 × 12 4 × 2 = 8 68 30 × 2 = 60 340 68 408 3 4 × 10 ∴ 34 × 2 = 68 4 × 10 = 40 34 × 10 = 340 30 × 10 = 300 34 × 12 = 408 340 DËi : 408 D`vniY 11| ¸Y Ki : e¨vL¨v : 273 2 7 3 2 9 × 29 3 GKK = 3 9 GKK = 9mgvavb : 7 `kK = 70 2 `kK = 20 273 2 kZK = 200 × 29 GLb, 2 7 3 × 9 2457 3 × 9 = 27 5460 70 × 9 = 630 7917 200 × 9 = 1800 2457 Ges 2 7 3 × 20 ∴ 273 × 9 = 2457 3 × 20 = 60 273 × 20 = 5460 70 × 20 = 1400 273 × 29 = 7917 200 × 20 = 4000 DËi : 408 5460gšÍe¨ : wk¶v_©xiv cÖPwjZ c×wZ‡Z ¸Y A¼ Ki‡e, Gi evB‡i Zv‡`i wbKU †Kvb e¨vL¨v PvIqv n‡e bv|

46 cÖv_wgK MwYZ

D`vniY 13| ¸Y Ki : 370

× 18 mgvavb : 370

× 18 2960 3700 6660 DËi : 6660

D`vniY 14| ¸Y Ki : 209

× 46 mgvavb : 209

× 46 1254 8360 9614 DËi : 9614

D`vniY 15| GKwU ev‡· 25wU PK Av‡Q| Giƒc 15wU ev‡· KqwU PK Av‡Q? mgvavb : 25

× 15 125 250 375 DËi : 375wU|

D`vniY 16| GKwU eB‡q 35wU c„ôv Av‡Q| Gi~c 16wU eB‡q KZwU c„ôv n‡e?mgvavb : 35

× 16 210 350 560 DËi : 560wU|

cÖv_wgK MwYZ 47

¸Y¨, ¸YK I ¸Ydj 12 × 8 = 96| GLv‡b, 12 ¸Y¨, 8 ¸YK, 96 ¸Ydj|

24 GLv‡b,

× 15 ¸Y¨ 24

120

240 ¸YK 15

360 ¸Ydj 360

15 GLv‡b,

× 24 24 ¸YK

60

300 15 ¸Y¨

360 360 ¸Ydj

yBwU msL¨vi ¸‡Yi †¶‡Î :

• †h msL¨v‡K ¸Y Kiv nq Zv nj ¸Y¨|

• †h msL¨v w`‡q ¸Y Kiv nq Zv nj ¸YK|

• ¸Y K‡i †h msL¨v cvIqv hvq Zv nj ¸Ydj|

LvwjNi c~iY Ki :

7 × 9 = 63 GLv‡b, ¸Y¨ ; ¸YK ; ¸Ydj

13 × 6 = 78 GLv‡b, ¸Y¨ ; ¸YK ; ¸Ydj

59 GLv‡b, 472 65 GLv‡b, ¸Y¨

× 8 59 × 9 ¸Ydj

472 8 585 ¸YK

3 × 8 = 24 ; 8 × 3 = 24 ∴ 3 × 8 = 8 × 3

6 × 7 = 42 ; 7 × 6 = 42 ∴ 6 × 7 = 7 × 6

13 × 9 = 117 ; 9 × 13 = 117 ∴ 13 × 9 = 9 × 13 j¶ Kwi : ¸Y¨ I ¸Y‡Ki ¯’vb wewbg‡q ¸Ydj GKB _v‡K|

48 cÖv_wgK MwYZ

AbykxjbxÑ4 1| LvwjNi c ~iY Ki :

(K) 6 × 9 = (L) 8 × = 72

(M) 15 × 7 = (N) 12 × 5 =

(O) 18 × = 108 (P) × 9 = 171

(Q) 53 × 16 = 848 GLv‡b, ¸YK ; ¸Y¨ ; ¸Ydj |

2| ¸Ydj wbY©q Ki :

(K) 65 × 4 (L) 46 × 7 (M) 71 × 9

(N) 97 × 8 (O) 88 × 6 (P) 234 × 4

(Q) 532 × 8 (R) 406 × 9 (S) 728 × 9

(T) 876 × 5 (U) 936 × 7 (V) 999 × 8

3| msw¶ß wbq‡g ¸Ydj wbY©q Ki :

(K) 61 × 10 (L) 43 × 20 (M) 36 × 50

(N) 24 × 100 (O) 387 × 10 (P) 250 × 20

(Q) 187 × 30 (R) 67 × 50 (S) 100 × 100

4| ¸Ydj wbY©q Ki :

(K) 327 (L) 910 (M) 849 (N) 65

× 9 × 5 × 9 × 25

(O) 48 (P) 98 (Q) 99 (R) 325

× 27 × 65 × 87 × 14

(S) 148 (T) 309 (U) 230 (V) 408

× 47 ×32 × 28 × 19

50 cÖv_wgK MwYZ

fvM fv‡Mi aviYv

j¶ Kwi : 15wU †cbwmj †_‡K cÖwZ ev‡i 5wU K‡i wb‡j 3 evi †bIqv hvq| fvM nj cybt cybt we‡qvM|

63 ÷ 9 = KZ?

9 × KZ? = 63

9 × 7 = 63

∴ 63 ÷ 9 = 7

9 ) 63 ( 7 63 0

35 ÷ 5 =

54 ÷ 6 =

48 ÷ 8 =

81 ÷ 9 =

7 ) 49 ( 8 ) 72 ( 4 ) 32 (

15 − 5 = 10

10 − 5 = 5

5 − 5 = 0

cÖv_wgK MwYZ 51

fvR¨, fvRK, fvMdj I fvM‡kl 8) 149 (18 GLv‡b, 149 fvR¨ 8 8 fvRK 69 18 fvMdj 64 5 fvM‡kl 5

∴ 149 = 8 × 18 + 5 A_©vr, fvR¨ = fvRK × fvMcj + fvM‡kl LvwjNi c ~iY Ki :

35 ÷ 5 = 7

GLv‡b, fvRK

fvR¨

fvMdj

6) 80 ( 13 6 20 18 2

j¶ Kwi : †h msL¨v w`‡q fvM Kiv nq Zv fvRK| †h msL¨v‡K fvM Kiv nq Zv fvR¨| fvM K‡i †h msL¨v cvIqv hvq Zv fvMdj| fv‡Mi †k‡l †h msL¨v †_‡K hvq Zv fvM‡kl; fvM‡kl Aek¨B fvRK †_‡K

†QvU| fvM‡kl k~b¨ n‡j fvR¨, fvRK Øviv wbt‡k‡l wefvR¨|

28 ÷ 4 = 7 ↓ ↓ ↓ fvR¨ fvRK fvMdj

fvR¨ ÷ fvRK = fvMdj

fvR¨ = fvRK × fvMdj 28 = 4 × 7

wbt‡k‡l fvM n‡”Q ¸‡Yi wecixZ cÖwµqv

fvRK

fvR¨

fvMdj

fvM‡kl

52 cÖv_wgK MwYZ

5 ÷ 5 = KZ?

5 × KZ? = 5 5 ) 5 ( 1

5 × 1 = 5 5

∴ 5 ÷ 5 = 1 0

9 ÷ 1 = KZ?

KZ? × 9 = 9 1) 9 ( 9

1 × 9 = 9 9

∴9 ÷1 = 9 0

0 ÷ 2 = 0 2 ) 0 ( 0 0 0

0 ÷ 5 = 0 5 ) 0 ( 0 0 0

5 ÷ 0 = †Kvb msL¨v bvB|

†Kvb msL¨v‡K k ~b¨ (0) Øviv fvM Kiv hvq bv| GRb¨ fvRK †Kvb Ae¯’vq k~b¨ (0) n‡Z cv‡i bv|

j¶ Kwi :

fvRK I fvR¨ mgvb n‡j fvMdj 1 nq|

fvRK 1 n‡j fvMdj fv‡R¨i mgvb nq|

fvR¨ 0 n‡j fvMdjI 0 nq|

fvRK 0 n‡j fvM Kiv hvq bv, GRb¨ fvRK †Kvb Ae¯’vq 0 n‡Z cv‡i bv|

LvwjNi c ~iY Ki ( yBwU K‡i †`Lv‡bv nj)

fvR¨ 42 55 25 78 85 61 0 95 17 0

fvRK 6 9 4 5 8 7 7 9 3 3

fvMdj 7 8

fvM‡kl 0 5

cÖv_wgK MwYZ 53

fvM D`vniY 1| 336 †K 3 Øviv fvM Ki|

mgvavb : 3) 336 (112 e¨vL¨v : 336 = 3 kZ 3 `k 6

3 3) 3 kZ 3 `k 6 ( 1 kZ 1 `k 2

3 3 kZ

3 3 `k

6 3 `k

6 6

0 6

0

DËi : fvMdj 112|

D`vniY 2| 371 †K 7 w`‡q fvM Ki|

mgvavb : 7 ) 371 (53 e¨vL¨v : 371 = 3 kZ 7 `k 1

35 = 37 `k 1

21 7) 37 `k 1 ( 5 `k 3

21 35 `k

0 21

21

0

DËi : fvMdj 53|

j¶ Kwi : fv‡R¨i kZ‡Ki N‡ii 3, fvRK 7 Gi †P‡q †QvU| ZvB kZK I `k‡Ki N‡ii A¼ GKÎ K‡i 37 `kK †bIqv n‡q‡Q|

56 cÖv_wgK MwYZ

mgm¨vi mgvavb (†hvM, we‡qvM, ¸Y I fvM msµvšÍ)

D`vniY 1| GKwU ¯‹z‡j 15wU P‡Ki ev· Av‡Q| cÖwZ ev‡· 50wU K‡i PK Av‡Q| GQvov AviI 25wU PK i‡q‡Q| H ¯‹z‡j †gvU KZwU PK i‡q‡Q? mgvavb : 1wU ev‡· PK Av‡Q 50wU

∴ 15wU ev‡· PK Av‡Q (50 × 15) wU GLb, 50

× 15 750wU 250 + 25wU 500 †gvU PK Av‡Q 775 wU 750 DËi : 775wU| D`vniY 2| ivwKe cÖwZ gv‡m 60 UvKv Rgv iv‡L| 12 gv‡m †h UvKv Rgv nq Zv †_‡K †m iwY‡K 85 UvKv †`q| Zvi wbKU KZ UvKv Aewkó _v‡K? mgvavb : 1 gv‡m Rgv K‡i 60 UvKv

∴ 12 gv‡m Rgv K‡i (60 × 12) UvKv GLb, 60

× 12 120 720 UvKv

600 −85 UvKv 720 ivwK‡ei _v‡K 635 UvKv DËi : 635 UvKv|

cÖv_wgK MwYZ 57

D`vniY 3| GKwU c¨v‡K‡U 50wU j‡RÝ Av‡Q| Gi †_‡K 8wU j‡RÝ †i‡L †Ìqv nj| Aewkó j‡RÝ 7 R‡bi g‡a¨ mgvb fv‡M fvM K‡i †Ìqv nq| cÖ‡Z¨‡K KqwU K‡i j‡RÝ cvq?

mgvavb : 8wU ev‡` j‡R‡Ýi msL¨v : 50 − 8 = 42

GLb, 42wU j‡RÝ 7 R‡bi g‡a¨ fvM Kiv nq|

∴ 42 ÷ 7 = 6

∴ cÖ‡Z¨‡K 6wU K‡i j‡RÝ cvq|

DËi : 6wU|

D`vniY 4| GKwU Zv‡K 86wU eB Av‡Q| Ab¨ GKwU Zv‡K 94wU eB Av‡Q| eB¸‡jv †_‡K cÖ‡Z¨K‡K 6wU K‡i eB w`‡j KZ Rb‡K †Ìqv hv‡e?

mgvavb : GKwU Zv‡K eB Av‡Q 86wU

Aci Zv‡K eB Av‡Q 94wU

†gvU eB Av‡Q 180 wU

GLb, 6) 180 ( 30

18 0 0 0

∴ 30 Rb‡K †Ìqv hv‡e| DËi : 30 Rb‡K|

cÖv_wgK MwYZ 61

MvwYwZK cÖZxK MvwYwZK cÖwµqv cÖZxK

ÒwZb †hvM `yBÓ †K †jLv nq, 3 + 2| GLv‡b, 3 I 2 nj msL¨v cÖZxK Ges + wPýwU nj †hvM cÖwµqvi cÖZxK, hv‡K ms‡¶‡c ejv nq †hvM wPý|

MvwYwZK cÖwµqv cÖZxK¸‡jv nj :

cÖZxK e¨envi K‡i cÖKvk Ki (GKwU K‡i †`Lv‡bv nj) : GKk wek †hvM cuwPk 120 + 25

wZbk we‡qvM GKk wZb

Pvik ¸Y evi

`yBk cÂvk fvM cuwPk

m¤úK© cÖZxK yBwU msL¨vi g‡a¨ m¤úK© cÖZxK¸‡jv nj :

D`vniY : (K) 3 + 2 = 5 (L) 9 Ñ 3 > 4

(M) 10 ÷ 5 < 3 MvwYwZK m¤úK© cÖZxK e¨envi K‡i cÖKvk Ki :

+ †hvM Ñ we‡qvM × ¸Y ÷ fvM

= mgvb > eo < †QvU

Pvik wbivbeŸB, cuvPkZ †_‡K †QvU 499<500 cuvP nvRvi †_‡K Qq nvRvi evi eo 6012 > 5000

62 cÖv_wgK MwYZ

N‡i h_vh_ MvwYwZK cÖwµqv cÖZxK (+, Ñ, ×, ÷) emvI ( yBwU K‡i †`Lv‡bv nj) : 145 15 = 160 145 + 15 = 160

655 500 = 155

250 3 = 750 250 × 3 = 750

715 25 = 740

830 5 = 166

125 12 = 1500

873 6 = 867

D`vniY : N‡i h_vh_ m¤úK© cÖZxK ( >, = , < ) emvI : (K) 250 + 30 30 + 250

(L) 250 × 3 3 × 250

(M) 250 Ñ 30 230

(N) 250 × 3 650

(O) 250 ÷ 5 55 mgvavb : (K) 250 + 30 = 280 (L) 250 × 3 = 750 Avevi, 30 + 250 = 280 Avevi, 3 × 250 = 750 myZivs, 250 + 30 = 30 + 250 myZivs, 250 × 3 = 3 × 250 DËi : 250 + 30 = 30+250 DËi : 250 × 3 = 3 × 3 × 250

(M) 250 − 30 = 220 (N) 250 × 3 = 750 GLv‡b, 220 < 230 GLv‡b, 750 > 650 DËi : 250 Ñ 30 < 230 DËi : 250 × 3 > 650

(O) 250 ÷ 5 = 50 GLv‡b, 50 < 55 DËi : 250 ÷ 5 < 55

cÖv_wgK MwYZ 63

AbykxjbxÑ7 1| cÖZx‡Ki mvnv‡h¨ cÖKvk Ki : (K) `yBk cÂvk †hvM mËi| (L) GK nvRvi GKk GK †hvM GKk wZb| (M) Pvik cuPvËi we‡qvM wZbk mvZ| (N) AvUk beŸB we‡qvM beŸB| (O) mvZk cuqZvwj−k ¸Y c‡bi| (P) yB nvRvi bqk cuPvËi †_‡K yB nvRvi cuPvËi we‡qvM| (Q) bqk weivbeŸB‡K †lvj w`‡q fvM| (R) AvU Gi wZb¸‡Yi mv‡_ PweŸk †hvM| 2| MvwYwZK cÖZxK e¨envi K‡i cÖKvk Ki : (K) AvUk beŸB †_‡K AvUk Avwk †QvU| (L) AvU nvRvi cuvPk †_‡K AvU nvRvi cuqZvwj−k †QvU| (M) wZb nvRvi mvZk wek †_‡K Pvi nvRvi eo| (N) bq nvRvi bqk wbivbeŸB †_‡K `k nvRvi eo|

3| N‡i cÖ‡hvR¨ cÖZxK (+, Ñ, ×, ÷ ) emvI:

(K) 250 60 = 310

(L) 150 6 = 900

(M) 350 40 = 310

(N) 840 2 = 838

(O) 999 11 = 1010

(P) 660 0 = 0

(Q) 783 3 = 261

(R) 270 5 = 1350

(S) 830 0 = 830

64 cÖv_wgK MwYZ

4 | N‡i cÖ‡hvR¨ cÖZxK (>, = , <) emvI :

(K) 275 + 85 400

(L) 275 + 75 300

(M) 807 + 700 1000

(N) 999 + 111 10000

(O) 1209 − 1200 10

(P) 875 − 0 875

(Q) 720 − 700 20

(R) 920 × 3 2700

(S) 920 × 3 2760

(T) 920 × 3 2600

(U) 265 ÷ 5 53

(V) 265 ÷ 5 50

(W) 265 ÷ 5 61

(X) 126 ÷ 9 11

5| MvwYwZK cÖZx‡Ki mvnv‡h¨ cÖKvk Ki : (K) 64 †K 8 w`‡q fvM Ki‡j fvMdj 8 Gi mgvb| (L) 50 †K 0 w`‡q ¸Y Ki‡j ¸Ydj 0 Gi mgvb| (M) 17 Gi mv‡_ 43 †hvM Ki‡j †hvMdj 45 Gi eo| (N) 17 Gi mv‡_ 43 †hvM Ki‡j †hvMdj 60 Gi mgvb| (O) 35 Gi 3 ¸Y 100 †_‡K eo| (P) 120 †K 8 w`‡q fvM Ki‡j fvMdj 16 Gi †QvU|

cÖv_wgK MwYZ 65

evsjv‡`kx gy`ªv I †bvU

5 cqmv 10 cqmv 25 cqmv 50 cqmv 50 cqmv 1 UvKv

1 UvKv 1 UvKv 2 UvKv 2 UvKv 2 UvKv

5 UvKv 5 UvKv 5 UvKv 10 UvKv 10 UvKv

66 cÖv_wgK MwYZ

evsjv‡`kx †bvU

20 UvKv 20 UvKv

50 UvKv 50 UvKv

100 UvKv 100 UvKv

500 UvKv 500 UvKv

1000 UvKv 1000 UvKv

cÖv_wgK MwYZ 67

evsjv‡`kx gy ªv I †bvU

68 cÖv_wgK MwYZ

evsjv‡`kx †bvU 5 cqmvi 5wU gy ªv mgvb 25 cqmv| 10 cqmvi 5wU gy ªv mgvb 50 cqmv| 25 cqmvi 4wU gy ªv mgvb 1 UvKv| 50 cqmvi 2wU gy ªv mgvb 1 UvKv| 5wU 10 cqmvi I 1wU 25 cqmvi gy ªv mgvb 75 cqmv| 2wU 25 cqmvi I 1wU 50 cqmvi gy ªv mgvb 1 UvKv| 5 UvKvi 4wU †bvU mgvb 20 UvKv| 10 UvKvi 5wU †bvU mgvb 50 UvKv| 50 UvKvi 10wU †bvU mgvb 500 UvKv| 3wU 10 UvKvi †bvU I 1wU 20 UvKvi †bvU mgvb 50 UvKv| 20 UvKvi 5wU †bvU mgvb 100 UvKv| 50 UvKvi 8wU †bvU I 100 UvKvi 1wU †bvU mgvb 500 UvKv| 500 UvKvi 1wU †bvU I 100 UvKvi 5wU †bvU mgvb 1000 UvKv

LvwjNi c~iY Ki :

10 cqmvi wU gy ªv mgvb 1 UvKv|

5 cqmvi wU gy ªv mgvb 1 UvKv|

4wU cqmvi gy ªv I wU 10 cqmvi gy ªv mgvb 50 cqmv|

5 UvKvi 20wU †bvU mgvb UvKv|

10 UvKvi wU †bvU mgvb 100 UvKv|




cÖv_wgK MwYZ 69

UvKv-cqmv †jLvi c×wZ 5 cqmv = 0.05 UvKv

8 cqmv = 0.08 UvKv

10 cqmv = 0.10 UvKv

15 cqmv = 0.15 UvKv

20 cqmv = 0.20 UvKv

30 cqmv = 0.30 UvKv

50 cqmv = 0.50 UvKv

75 cqmv = 0.75 UvKv

80 cqmv = 0.80 UvKv

99 cqmv = 0.99 UvKv

100 cqmv = 1.00 UvKv

1 UvKv 25 cqmv = 1.25 UvKv



100 UvKv = 100.00 UvKv


A‡¼ †jL (GKwU K‡i †`Lv‡bv nj) :

evBk UvKv bq cqmv = 22.09 UvKv AvVvi UvKv beŸB cqmv = UvKv

cuvP UvKv cuvP cqmv = UvKv cuwPk UvKv cuPvËi cqmv = UvKv

`k UvKv AvU cqmv = UvKv yBk cuqlwÆ UvKv k cqmv = UvKv

Dwbk UvKv = UvKv cuvPk UvKv cuPvËi cqmv = UvKv

K_vq †jL (GKwU K‡i †`Lv‡bv nj) :

3.23 UvKv = wZb UvKv †ZBk cqmv

5.02 UvKv =

75.09 UvKv =

501. 90 UvKv =

70 cÖv_wgK MwYZ

D`vniY 1| 60 cqmv, 25 cqmv Ges 15 cqmvq †gvU KZ nq? mgvavb : 60 cqmv 25 cqmv

15 cqmv ∴ 100 cqmv = 1.00 UvKv †gvU : 100 cqmv = 1 UvKv DËi : 1 UvKv| D`vniY 2| 2wU 25 cqmvi gy ªv, 4wU 10 cqmvi gy ªv I 3wU 5 cqmvi gy ªvq †gvU KZ UvKv nq?

mgvavb : 2wU 25 cqmvi gy ªv = 25 cqmv × 2 = 50 cqmv

4wU 10 cqmvi gy ªv = 10 cqmv × 4 = 40 cqmv

3wU 5 cqmvi gy ªv = 5 cqmv × 3 = 15 cqmv †gvU = 105 cqmv DËi : 1.05 UvKv| = 1.05 UvKv|

D`vniY 3| †hvM Ki : (K) 47 UvKv 20 cqmv (L) 85.25 UvKv (M) 8.40 UvKv + 28 UvKv 50 cqmv + 63.10 UvKv 25.75 UvKv 75 UvKv 70 cqmv 148.35 UvKv + 47.15 UvKv 81.30 UvKv DËi : 75 UvKv 70 cqmv| DËi : 148.35 UvKv| DËi : 81.30 UvKv|

D`vniY 4| we‡qvM Ki : (K) 97 UvKv 50 cqmv (L) 38.75 UvKv (M) 60.00 UvKv

−25 UvKv 25 cqmv −26.20 UvKv −35.40 UvKv 72 UvKv 25 cqmv 12.55 UvKv 24.60 UvKv DËi : 72 UvKv 25 cqmv| DËi : 12.55 UvKv| DËi : 24.60 UvKv|

cÖv_wgK MwYZ 71

D`vniY 5| i“wei 50.75 UvKv wQj| Zvi AveŸv Zv‡K eB †Kbvi Rb¨ 80 UvKv w`‡jb| i“wei GLb KZ UvKv nj?

mgvavb : i“wei wQj 50.75 UvKv

AveŸv w`‡jb + 80.00 UvKv

†gvU 130.75 UvKv

DËi : 130.75 UvKv|

D`vniY 6| ARq evRv‡i wM‡q 57.50 UvKvi gvQ wKbj| †m †`vKvb`vi‡K 100 UvKvi GKwU †bvU w`j| †`vKvb`vi Zv‡K KZ UvKv †diZ †`‡e?

mgvavb : 100.00 UvKv

Ñ57.50 UvKv

42.50 UvKv

DËi : 42.50 UvKv|

D`vniY 7| GKwU nuv‡mi `vg 100 UvKv| GKwU gyiwMi `vg 90.75 UvKv| ivRy GKwU nuvm I GKwU gyiwM wKbj| †m †`vKvb`vi‡K 500 UvKvi GKwU †bvU w`j| †`vKvb`vi KZ UvKv †diZ †`‡e?

mgvavb : nuv‡mi `vg 100.00 UvKv 500.00 UvKv

gyiwMi `vg 90.75 UvKv − 190.75 UvKv

†gvU 190.75 UvKv 309.25 UvKv

DËi : 309.25 UvKv|

72 cÖv_wgK MwYZ

AbykxjbxÑ8 1| A‡¼ †jL :

mËi cqmv, bq cqmv, mvZ UvKv wZb cqmv, Avwk UvKv|

GKk mvZvwk UvKv beŸB cqmv, weivbeŸB UvKv mvZ cqmv|

2| K_vq †jL :

0.06 UvKv, 3.09 UvKv, 7.19 UvKv, 299.99 UvKv, 547.50 UvKv|

3| (K) 55 cqmv, 15 cqmv I 20 cqmv GK‡Î KZ cqmv nq?

(L) 4wU 10 cqmvi gy ªv I 6wU 5 cqmvi gy ªv GK‡Î KZ cqmv nq?

(M) 5wU 25 cqmvi gy ªv I 6wU 5 cqmvi gy ªvi GK‡Î KZ cqmv nq?

4| †hvM Ki : (K) 45 UvKv 50 cqmv (L) 57.08 UvKv (M) 526.27 UvKv + 60 UvKv 25 cqmv + 25.27 UvKv + 127.28 UvKv (N) 205.75 UvKv (O) 28.03 UvKv (P) 460.75 UvKv + 25.08 UvKv 17.78 UvKv 234.25 UvKv + 2.25 UvKv + 83.50 UvKv 5| we‡qvM Ki : (K) 85 UvKv 75 cqmv (L) 31.87 UvKv (M) 100.70 UvKv

− 16 UvKv 50 cqmv − 15.25 UvKv −71.38 UvKv

cÖv_wgK MwYZ 73

6| Kwig 17.50 UvKvi Pvj I 37.25 UvKvi Wvj wKbj| †m †gvU KZ UvKv

LiP Kij? 7| wUUz 35.75 UvKv w`‡q yBwU LvZv wK‡b †`vKvb`vi‡K 50 UvKvi GKwU

†bvU w`j| †`vKvb`vi Zv‡K KZ UvKv †diZ †`‡e? 8| eveyj GK Szwo Avg wK‡b †`vKvb`vi‡K Av‡gi g~j¨ wn‡m‡e 50 UvKvi

1wU †bvU Ges 20 UvKvi 3wU †bvU w`j| H GK Szwo Av‡gi g~j¨ KZ? 9| 100 UvKvi 1wU †bvU 20 UvKvi 1wU †bvU †_‡K KZ UvKv †ewk? 10| 50 UvKvi 1wU †bvU 500 UvKvi 1wU †bvU †_‡K KZ UvKv Kg? 11| GKwU LvZvi g~j¨ 15.50 UvKv Ges GKwU Kj‡gi g~j¨ 10.00 UvKv|

†`vKvb`vi‡K GKwU 50.00 UvKvi †bvU w`‡j KZ UvKv †diZ cvIqv hv‡e?

12| 100 UvKvi 2wU †bvU, 50 UvKvi 3wU †bvU A‡c¶v KZ †ewk?

74 cÖv_wgK MwYZ

fMœvsk fMœvs‡ki aviYv

12 ev, A‡a©K ev, Aa©vsk

ev, `yB fv‡Mi GK


ev, `yB fv‡Mi GK


ev, `yB fv‡Mi GK

13 ev, GK Z…Zxqvsk ev,

wZb fv‡Mi GK


wZb fv‡Mi GK


wZb fv‡Mi GK

14 ev, GK PZz_©vsk ev,

Pvi fv‡Mi GK


Pvi fv‡Mi GK


Pvi fv‡Mi GK

23 ev, yB Z…Zxqvsk ev,

wZb fv‡Mi yB


wZb fv‡Mi yB


wZb fv‡Mi yB

cÖv_wgK MwYZ 75

fMœvsk fMœvs‡ki aviYv

34 ev, wZb PZz_©vsk ev,

Pvi fv‡Mi wZb


Pvi fv‡Mi wZb


Pvi fv‡Mi wZb

45 ev, Pvi cÂgvsk ev,

cuvP fv‡Mi Pvi


cuvP fv‡Mi Pvi


cuvP fv‡Mi Pvi

16 ev, GK lôvsk ev, Qq fv‡Mi GK

27 ev, `yB mßgvsk ev, mvZ fv‡Mi `yB

38 ev, wZb Aógvsk ev, AvU fv‡Mi wZb

29 ev, yB begvsk ev, bq fv‡Mi yB

j¶ Kwi : mvaviY fMœvsk †jLvi Rb¨ GKwU †jLv (ÒÑÓ) UvKv n‡q‡Q| hZ fvM Kiv n‡q‡Q Zv‡K wb‡P Ges hZ fvM †bIqv n‡q‡Q Zv‡K Dc‡i †jLv n‡q‡Q|

76 cÖv_wgK MwYZ

wb‡Pi Qwe¸‡jv‡Z 12 Ask †cbwmj w`‡q Kvj Ki :




cÖv_wgK MwYZ 77

wb‡Pi Qwe¸‡jvi Kvj Ask A‡¼ I K_vq †jL : (GKwU K‡i †`Lv‡bv nj)

Qwe A‡¼ †jL K_vq †jL

12

A‡a©K

78 cÖv_wgK MwYZ

GKvs‡ki aviYv

6 6 12 Gi

12 = 6

12

12 Gi 13 = 4

12

12 Gi 14 = 3

12

12 Gi 16 = 2

12

4 4 4

3 3 3 3

2 2 2 2 2 2

16 Gi 12 = 8

16

16 Gi 14 = 4

16

16 Gi 18 = 2

16

8 8

4 4 4 4

2 2 2 2 2 2 2 2

cÖv_wgK MwYZ 79

6 = 2 × 3 = 3 × 2

∴ 6 ÷ 2 = 3 Avevi 6 ÷ 3 = 2

∴ 6 Gi 12 = 3 Ges 6 Gi

13 = 2

8 = 2 × 4 = 4 × 2

∴ 8 ÷ 2 = 4 Avevi 8 ÷ 4 = 2

∴ 8 Gi 12 = 4 Ges 8 Gi

14 = 2

10 = 2 × 5 = 5 × 2

∴ 10 ÷ 2 = 5 Avevi 10 ÷ 5 = 2

∴ 10 Gi 12 = 5 Ges 10 Gi

15 = 2

15 = 3 × 5 = 5 × 3

∴ 15 ÷ 3 = 5 Avevi 15 ÷ 5 = 3

∴ 15 Gi 13 = 5 Ges 15 Gi

15 = 3

20 = 4 × 5 = 5 × 4

∴ 20 ÷ 4 = 5 Avevi 20 ÷ 5 = 4

∴ 20 Gi 14 = 5 Ges 20 Gi

15 = 4

30 = 5 × 6 = 6 × 5

∴ 30 ÷ 5 = 6 Avevi 30 ÷ 6 = 5

∴ 30 Gi 15 = 6 Ges 30 Gi

16 = 5

42 = 6 × 7 = 7 × 6

∴ 42 ÷ 6 = 7 Avevi 42 ÷ 7 = 6

∴ 42 Gi 16 = 7 Ges 42 Gi

17 = 6

54 = 6 × 9 = 9 × 6

∴ 54 ÷ 6 = 9 Avevi 54 ÷ 9 = 6

∴ 54 Gi 16 = 9 Ges 54 Gi

19 = 6

80 cÖv_wgK MwYZ

LvwjNi c~iY Ki :

56 = 7 × = 8 × 72 = × 9 = × 8

56 Gi 17 = 72 Gi

19 =

1 eQi ev 12 gv‡mi 12 Ask = 6 gvm

1 eQi ev 12 gv‡mi 13 Ask = gvm



1 gvm ev 30 w`‡bi 12 Ask = 15 w`b

1 gvm ev 30 w`‡bi 13 Ask = w`b



1 w`b ev 24 N›Uvi 12 Ask = 12 NÈv

1 w`b ev 24 N›Uvi 13 Ask = NÈv

cÖv_wgK MwYZ 81




1 NÈv ev 60 wgwb‡Ui 12 Ask = 30 wgwbU

1 NÈv ev 60 wgwb‡Ui 13 Ask = wgwbU





1 UvKv ev 100 cqmvi 12 Ask = 50 cqmv

1 UvKv ev 100 cqmvi 14 Ask = cqmv



82 cÖv_wgK MwYZ

1 wgUvi ev 100 †mw›UwgUv‡ii 12 Ask = 50 †mw›UwgUvi

1 wgUvi ev 100 †mw›UwgUv‡ii 14 Ask = †mw›UwgUvi



18 Gi 13 Ask = 6 24 Gi Ask = 8

25 Gi 15 Ask = 5 30 Gi

16 Ask =

32 Gi Ask = 8 36 Gi Ask = 9

1 eQ‡ii Ask = 4 gvm 1 gv‡mi 13 Ask = w`b

1 N›Uvi 16 Ask = wgwbU 1 UvKvi Ask = 25 cqmv

1 wgUv‡ii Ask = 20 †m. wg. 1 w`‡bi Ask = N›Uv

cÖv_wgK MwYZ 83

fMœvs‡ki ni I je

e„‡Ëi wfZ‡ii Kvj 14 Ask |

34 je (hZ fvM †bIqv n‡q‡Q)

ni (hZ fvM Kiv n‡q‡Q)

j¶ Kwi : `v‡Mi Dc‡ii msL¨vwU je Ges wb‡Pi msL¨vwU ni|

25 fMœvs‡k 5 ni Ges 2 je

wb‡Pi LvwjNi c~iY Ki (GKwU K‡i †`Lv‡bv nj) :

38 fMœvs‡k 8 ni Ges 3 je

35 fMœvs‡k 3 Ges 5

78 fMœvs‡k 8 Ges 7

7 je Ges 9 ni n‡j fMœvskwU n‡e

6 ni Ges 5 je n‡j fMœvskwU n‡e

5 je Ges 7 ni n‡j fMœvskwU n‡e



84 cÖv_wgK MwYZ

mgZzj fMœvsk

12

24

36

48

510

12,

24,

36,

48,

510 fMœvsk¸‡jv mgZzj|

†Kbbv, †h‡Kvb e¯‘i 12 Ask,

24 Ask,

36 Ask,

48 Ask,

510 Ask ci¯úi mgvb|

12243648510

12,

24,

36,

48,

510 mgZzj fMœvsk|

13

26

39

13 ,

26,

39 mgZzj fMœvsk|

cÖv_wgK MwYZ 85

mgZzj fMœvsk †Pbv

13,

26 mgZzj fMœvsk|

13

26

GLv‡b, 13

26

24,

36 mgZzj fMœvsk|

24 ∴

24 =

36

36 GLv‡b,

24

36

12,

34 mgZzj fMœvsk bq|

12 ∴

12 =

34 mgvb bq|

34 GLv‡b,

12

34

3 × 2 = 6 1 × 6 = 6

4×3 = 12 2×6 = 12

2 × 3 = 6 1 × 4 = 4

∴ 13 =

26

86 cÖv_wgK MwYZ

8 Gi 24 = 4 8 Gi

48 = 4

∴ 24,

48 mgZzj fMœvsk| ∴

24 =

48

GLv‡b, 24

48

12 Gi 34 = 9

12 Gi 46 = 8 GLv‡b,

34

46

∴ 34,

46 mgZzj fMœvsk bq| ∴

34,

46 mgvb bq|

mgZzj fMœvsk MVb12 Gi mgZzj fMœvsk :

12 =

1 × 22 × 2

= 24 ;

12 =

1 × 32 × 3

= 36 ;

12 =

1 × 42 × 4

= 48 ;

12 =

1 × 52 × 5

= 510 ;

24,

36,

48,

510 BZ¨vw`

12 Gi mgZzj fMœvsk|

2 × 3 = 6 1 × 4 = 4

4 × 4= 163 × 6= 18

cÖv_wgK MwYZ 87

13 Gi mgZzj fMœvsk :

13 =

1 × 23 × 2

= 26 ;

13 =

1 × 33 × 3

= 39 ;

13 =

1 × 43 × 4

= 412 ;

13 =

1 × 53 × 5

= 515 ;

26,

39,

412,

515 BZ¨vw`



23 =

2 × 23 × 2

= 46 ;

23 =

2 × 33 × 3

= 69 ;

23 =

2 × 43 × 4

= 812 ;

23 =

2 × 53 × 5

= 1015 ;

46,

69,

812,

1015 BZ¨vw`



14 =

1 × 24 × 2

= 28 ;

14 =

1 × 34 × 3

= 312 ;

14 =

1 × 44 × 4

= 416 ;

14 =

1 × 54 × 5

= 520 ;

28,

312,

416,

520 BZ¨vw`


88 cÖv_wgK MwYZ

1| wb‡Pi cÖ‡Z¨K fMœvs‡ki PviwU mgZzj fMœvsk wbY©q Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

(K) 23,

46 ,

69 ,

812 ,

1015

(L) 34, , , , ,

(M) 35, , , , ,

(N) 56, , , , ,

(O) 27, , , , ,

2| LvwjNi c ~iY Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

(K) 12 =

4

8 ;

12 =

6 ; 12 =

7

;

12 =

3

(L) 13 =

5

;

15 =

2

;

25 =

10 ; 34 =

8

(M) 16 =

3

;

17 =

3

;

17 =

14 ; 45 =

15

cÖv_wgK MwYZ 89

3| wb‡Pi cÖ‡Z¨K fMœvsk †Rvov mgZzj wKbv Zv wbY©q K‡i duvKv N‡i DËi †jL (cÖ_g yBwU K‡i †`Lv‡bv nj) :

(K) 34,

68

(L) 46,

13

(M) 36,

48

(N) 17,

214

(O) 36,

612

(P) 23,

36

1 Gi mgZzj fMœvsk

1 = 11

22

33

44

1 × 2 = 2 1 × 3 = 3 1 × 4 = 4

2 × 1 = 2 3 × 1 = 3 4 × 1 = 4

∴ 22,

33 ,

44 BZ¨vw` cÖ‡Z¨KwU fMœvsk 1 Gi mgZzj|

A_©vr, 1 = 11 =

22 =

33 =

44 BZ¨vw`|

34

68

4 × 6 = 24 3 × 8 = 24

mgZzj

46

13

6 × 1 = 6 4 × 3 = 12

mgZzj bq

90 cÖv_wgK MwYZ

fMœvs‡ki Zzjbv

34

24

34 eo

24 †_‡K; A_©vr,

34 >

24

Avevi, 24 †QvU

34 †_‡K;

24 <

34

38

58

38 †QvU

58 †_‡K; A_©vr,

38 <

58

Avevi, 58 eo

38 †_‡K;

58 >

38

58 = 8 fv‡Mi 5 fvM

38 = 8 fv‡Mi 3 fvM

49 = 9 fv‡Mi 4 fvM

79 = 9 fv‡Mi 7 fvM

715 = 15 fv‡Mi 7 fvM

615 = 15 fv‡Mi 6 fvM

5 > 3 ∴ 58 >

38

4 > 7 ∴ 49 <

79

7 > 6 ∴ 715 >

615

cÖv_wgK MwYZ 91

fMœvs‡ki †hvM

12

12

1

12 +

12 = 1

wKš‘, 1 = 22 =

1 + 12

∴ 12 +

12 =

1 + 12

13

13

23

13 +

13 =

23

wKš‘, 23 =

1 + 13

∴ 13 +

13 =

1 + 13

25

15

35

25 +

15 =

35

wKš‘, 35 =

2 + 15

∴ 25 +

15 =

2 + 15

58

28

78

58 +

28 =

78

wKš‘, 78 =

5 + 28

∴ 58 +

28 =

5 + 28

26 +

36 =

2 + 36

= 56

49 +

39 =

4 + 39

= 79

92 cÖv_wgK MwYZ

38 +

18 =

3 + 18

= 48

= 12

KviY, 48 I

12 mgZzj fMœvsk|

512 +

412 =

5 + 412

= 912

= 34

KviY, 912 I

34 mgZzj fMœvsk|

LvwjNi c ~iY Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

1| 411 +

511 =

912 2|

14 +

24 =

3| 27 +

37 = 4|

58 +

38 =

5| 712 +

312 = 6|

914 +

414 =

7| 715 +

415 = 8|

516 +

916 =

9| 919 +

519 = 10|

720 +

920 =

11| 825 +

925 = 12|

712 +

412 =

cÖv_wgK MwYZ 93

fMœvs‡ki we‡qvM

45

25

25

45 −

25 =

25 Avevi,

25 =

4 − 25

∴ 45 −

25 =

4 − 25 =

25

1 = 33

23

13

1 − 23 =

13 Avevi,

13 =

3 − 23

∴ 1 − 23 =

33 −

23 =

3 − 23 =

13

94 cÖv_wgK MwYZ

79 −

59 =

7 − 59

= 29

1 − 25 =

55 −

25

= 5 − 2

5 = 35

58 −

38 =

5 − 38

= 28

= 14

KviY, 28 I

14 mgZzj fMœvsk|

712 −

512 =

7 − 512

= 212

= 16

KviY, 212 I

16 mgZzj fMœvsk|

LvwjNi c ~iY Ki (cÖ_gwU K‡i †`Lv‡bv nj) :

1| 812 −

312 =

512 2|

713 −

513 =

3| 910 −

510 = 4| 1 −

49 =

5| 1 − 78 = 6|

815 −

115 =

7| 920 −

720 = 8|

725 −

225 =

9| 931 −

831 = 10|

912 −

712 =

cÖv_wgK MwYZ 95

AbykxjbxÑ9 1| wb‡Pi fMœvsk¸‡jv A‡¼ †jL :

(K) Pvi fv‡Mi wZb fvM (L) mvZ fv‡Mi cuvP fvM

(M) AvU fv‡Mi wZb fvM (N) GMvi fv‡Mi bq fvM

(O) c‡bi fv‡Mi mvZ fvM (P) EbwÎk fv‡Mi AvU fvM

2| wb‡Pi fMœvsk¸‡jv K_vq †jL :

(K) 35 (L)

78 (M)

415 (N)

522 (O)

733 (P)

939

3| (K) 8 Gi 34 = KZ? (L) 9 Gi

23 = KZ?

(M) 12 Gi 13 = KZ? (N) 12 Gi

26 = KZ?

(O) 20 Gi 14 = KZ? (P) 25 Gi

35 = KZ?

4| (K) 1 eQ‡ii 12 Ask = KZ gvm? (L) 1 gv‡mi

13 Ask = KZ w`b?

(M) 1 w`‡bi 14 Ask = KZ N›Uv? (N) 1 UvKvi

14 Ask = KZ cqmv?

(O) 1 wgUv‡ii 110 Ask = KZ †mw›UwgUvi?

(P) 1 N›Uvi 15 Ask = KZ wgwbU?

96 cÖv_wgK MwYZ

5| wb‡Pi fMœvsk¸‡jvi †Kvb †Rvov mgZzj Ges †Kvb †Rvov mgZzj bq, wbY©q Ki :

(K) 12,

24 (L)

35,

610 (M)

27,

37 (N)

34,

38

(O) 45,

23 (P)

46,

812 (Q)

26,

312

6| wb‡Pi fMœvsk¸‡jv Zzjbv K‡i K_vq cÖKvk Ki :

(K) 34 >

14 (L)

25 <

35 (M)

710 >

510 (M)

710 <

910

7| wb‡Pi fMœvsk¸‡jv Zzjbv K‡i eo ev †QvU Gi cÖZxK LvwjN‡i †jL :

(K) 56

36 (L)

58

78 (M)

712

912

(N) 716

516 (O)

625

925 (P)

821

521

8| †hvM Ki :

(K) 34 +

14 (L)

35 +

15 (M)

26 +

36

(N) 57 +

17 (O)

511 +

311 (P)

913 +

213

(Q) 820 +

520 (R)

725 +

525 (S)

231 +

831

9| we‡qvM Ki :

(K) 37 −

27 (L)

79 −

59 (M) 1 −

13

(N) 1 − 56 (O)

818 −

518 (P)

521 −

421

cÖv_wgK MwYZ 97

cwigvc ˆ`N©¨ cwigvc

†¯‹j †`‡L LvwjNi c ~iY Ki :

ˆ`N© cwigv‡ci GKK : wgUvi 1 wgUvi = 100 †mw›UwgUvi ev †m. wg.

1 †mw›UwgUvi ev 1 †m. wg.

†m. wg.

†m. wg.

†m. wg.

†m. wg.

†m. wg.

†m. wg.

†m. wg.

†m. wg.

cÖv_wgK MwYZ 99

ˆ`N©¨ cwigv‡ci GKKvewj

LvwjNi c ~iY Ki ( yBwU K‡i †`Lv‡bv nj) :

4 wgUvi 20 †mw›UwgUvi = 420 †mw›UwgUvi

8 wgUvi = †mw›UwgUvi

5 wgUvi 50 †mw›UwgUvi = †mw›UwgUvi

2 wK‡jvwgUvi 500 wgUvi = 2500 wgUvi

4 wK‡jvwgUvi = wgUvi

3 wK‡jvwgUvi 200 wgUvi = wgUvi

8 wK‡jvwgUvi 300 wgUvi = wgUvi

D`vniY 1| 25 wgUvi‡K †mw›UwgUv‡i cÖKvk Ki| mgvavb : 1 wgUvi = 100 †mw›UwgUvi

∴ 25 wgUvi = 25 × 100 †mw›UwgUvi = 2500 †mw›UwgUvi DËi : 2500 †mw›UwgUvi| D`vniY 2| 4 wK‡jvwgUvi‡K wgUv‡i cÖKvk Ki| mgvavb : 1 wK‡jvwgUvi = 1000 wgUvi|

∴ 4 wK‡jvwgUvi = 4 × 1000 wgUvi = 4000 wgUvi DËi : 4000 wgUvi| D`vniY 3| 22 wgUvi 25 †mw›UwgUvi‡K †mw›UwgUv‡i cÖKvk Ki| mgvavb : 22 wgUvi 25 †mw›UwgUvi

= 22 × 100 †mw›UwgUvi + 25 †mw›UwgUvi = 2200 †mw›UwgUvi + 25 †mw›UwgUvi = 2225 †mw›UwgUvi DËi : 2225 †mw›UwgUvi|

100 †mw›UwgUvi = 1 wgUvi ev 1 wg. 1000 wgUvi = 1 wK‡jvwgUvi ev 1 wK. wg.

100 cÖv_wgK MwYZ

IRb cwigvc 5 MÖvg 10 MÖvg 20 MÖvg 50 MÖvg 100 MÖvg 200 MÖvg 500 MÖvg 1 wK‡jvMÖvg ev 1 †KwR

IRb cwigv‡ci GKK : MÖvg

1000 MÖvg = 1 wK‡jvMÖvg ev 1 †KwR

cÖv_wgK MwYZ 101

`vM †U‡b wgj Ki (GKwU K‡i †`Lv‡bv nj) : 950 MÖvg

3 wK‡jvMÖvg 2500 MÖvg

2wU 200 MÖvg I 1wU 50 MÖvg evULviv 3000 MÖvg

1wU 500 MÖvg, 2wU 200 MÖvg I 1wU 50 MÖvg evULviv 3050 MÖvg

1wU 1 †KwR, 1wU 500 MÖvg I 1wU 100 MÖvg evULviv 250 MÖvg

7 †KwR 1 †KwR 600 MÖvg

2 †KwR 500 MÖvg 450 MÖvg



102 cÖv_wgK MwYZ

D`vniY 1| 6 †KwR‡K MÖv‡g cÖKvk Ki| mgvavb : 1 †KwR = 1000 MÖvg

∴ 6 †KwR = 6 × 1000 MÖvg = 6000 MÖvg DËi : 6000 MÖvg| D`vniY 2| 3 †KwR 750 MÖvg‡K MÖv‡g cÖKvk Ki| mgvavb : 1 †KwR = 1000 MÖvg 3000 MÖvg

∴ 3 †KwR = 3 × 1000 MÖvg + 750 MÖvg = 3000 MÖvg 3750 MÖvg DËi : 3750 MÖvg| D`vniY 3| GKwU 1 †KwR, GKwU 500 MÖvg I GKwU 200 MÖvg evULvivq KZ MÖvg nq? mgvavb : GKwU 1 †KwR evULviv = 1000 MÖvg GKwU 500 MÖvg evULviv = 500 MÖvg GKwU 200 MÖvg evULviv = 200 MÖvg †gvU = 1700 MÖvg DËi : 1700 MÖvg| D`vniY 4| GKRb †`vKvb`vi 1 e¨vM Avjy gvc‡Z 2wU 1 †KwR, 1wU 200 MÖvg evULviv e¨envi Ki‡jb| H 1 e¨vM Avjyi IRb KZ MÖvg?

mgvavb : 2wU 1 †KwRi evULviv = 2 × 1000 MÖvg = 2000 MÖvg 2wU 1 †KwRi evULviv = 2000 MÖvg 1wU 200 MÖvg evULviv = 200 MÖvg

∴ 1 e¨vM Avjyi IRb = 2200 MÖvg DËi : 2200 MÖvg|

cÖv_wgK MwYZ 103

Zij c`v‡_©i AvqZb cwigvc wjUvi cwigvc

Zij c`v‡_©i AvqZb cwigv‡ci GKK : wjUvi

1 wjUvi = 1000 wgwjwjUvi

Zij c`v‡_©i AvqZb cwigv‡ci GKK: wjUvi

104 cÖv_wgK MwYZ

AbykxjbxÑ10 1| `vM †U‡b mwVK GK‡Ki m‡½ wgj Ki : XvKv †_‡K dwi`cy‡ii ~iZ¡ wgUvi

GKwU †cbwm‡ji ˆ`N© wjUvi

N‡ii ˆ`N© wgwjwjUvi

GK e¯Zv Pv‡ji IRb MÖvg

GK †evZj †mvqvweb †Zj wK‡jvwgUvi

yBwU †QvU U‡g‡Uvi IRb wK‡jvMÖvg

GK Pv PvgP cvwbi AvqZb †mw›UwgUvi

2| LvwjNi c~iY Ki : (K) 1 †KwR = MÖvg (N) 1 wgUvi = †m. wg.

(L) 1 wK‡jvwgUvi = wgUvi (O) 1 wg. 70 †m. wg. = †m. wg.

(M) wgwjwjUvi = 1 wjUvi (P) 2 †KwR 500 MÖvg = MÖvg

3| †mw›UwgUv‡i cÖKvk Ki : (K) 25 wgUvi (L) 18 wgUvi 72 †m. wg. (M) 47 wgUvi 52 †m. wg. (N) 39 wgUvi 20 †m. wg.

4| wgUv‡i cÖKvk Ki : (K) 3 wK‡jvwgUvi (L) 10 wK‡jvwgUvi (M) 5 wK‡jvwgUvi 750 wgUvi (N) 8 wK‡jvwgUvi 500 wgUvi

5| MÖv‡g cÖKvk Ki : (K) 7 wK‡jvMÖvg (L) 3 wK‡jvMÖvg 300 MÖvg (M) 6 wK‡jvMÖvg 500 MÖvg (N) 8 wK‡jvMÖvg 900 MÖvg

6| wgUvi wdZv w`‡q †Zvgvi †kªYxK‡¶i ˆ`N© gvc Ges LvZvq †jL| 7| wgUvi †¯‹j w`‡q †Zvgvi emvi †eÂwUi ˆ`N© I cÖ ’ gvc Ges LvZvq

†jL| 8| 1 e¨vM je‡Yi IRb 1 †KwR 50 MÖvg| GwU gvc‡Z Kx Kx evULviv jvM‡e?

cÖv_wgK MwYZ 105

mgq cwigvc Nwo †`Lv

106 cÖv_wgK MwYZ

Nwo †`Lv

cÖv_wgK MwYZ 107

mgq cwigv‡ci †hvM I we‡qvM mgq cwigv‡ci GKKvewj

60 †m‡KÛ = 1 wgwbU 60 wgwbU = 1 N›Uv 24 N›Uv = 1 w`b 7 w`b = 1 mßvn

30 w`b = 1 gvm 12 gvm = 1 eQi 365 w`b = 1 eQi

D`vniY 1| †hvM Ki : N›Uv wgwbU †m‡KÛ 4 25 21 9 20 28 mgvavb : N›Uv wgwbU †m‡KÛ 4 25 21 9 30 28 13 55 49 DËi : 13 N›Uv 55 wgwbU 49 †m‡KÛ

D`vniY 2| we‡qvM Ki : N›Uv wgwbU †m‡KÛ 24 35 45 12 20 35 mgvavb : N›Uv wgwbU †m‡KÛ 24 35 45 12 20 35 12 15 10 DËi : 12 N›Uv 15 wgwbU 10 †m‡KÛ

D`vniY 3| wibv mKv‡j 2 N›Uv 15 wgwbU Ges iv‡Î 3 N›Uv 30 wgwbU c‡o| †m †gvU KZ mgq c‡o? mgvavb : NÈv wgwbU 2 15 + 3 30 5 45 DËi : 5 NÈv 45 wgwbU| D`vniY 4| Aby mKvj 7 Uv 20 wgwb‡U gvgv evwoi c‡_ iIqvbv nj| 9 Uv 50 wgwb‡U †m gvgv evwo †cuŠQj| Zvi gvgv evwo †h‡Z KZ mgq jvMj? mgvavb : NÈv wgwbU 9 50 − 7 20 2 30 DËi : 2 NÈv 30 wgwbU|

108 cÖv_wgK MwYZ

AbykxjbxÑ11 1| LvwjNi c ~iY Ki : (K) 3 w`b = N›Uv (L) 2 N›Uv = wgwbU

(M) w`b = 2 mßvn (N) 1 eQi = w`b

(O) gvm = 1 eQi (P) †m‡KÛ = 1 wgwbU

2| †hvM Ki : (K) wgwbU †m‡KÛ (L) wgwbU †m‡KÛ 4 23 7 15 2 12 5 28 (M) N›Uv wgwbU †m‡KÛ N›Uv wgwbU †m‡KÛ 5 12 23 9 19 12 3 25 26 7 22 28 3| we‡qvM Ki : (K) wgwbU †m‡KÛ (L) wgwbU †m‡KÛ 9 7 15 27 6 5 4 18 (M) N›Uv wgwbU †m‡KÛ N›Uv wgwbU †m‡KÛ 18 52 25 15 17 30 7 16 17 8 0 5 4| wgRvb mv‡ne mKv‡j 1 N›Uv 20 wgwbU nuv‡Ub| wZwb we‡K‡j 2 N›Uv 15

wgwbU nuv‡Ub| wZwb †gvU KZ mgq nuv‡Ub? 5| Kwji ¯‹zj QywU nq 11 Uv 30 wgwb‡U| evmvq †cuŠQv‡Z Zvi mgq jv‡M 25

wgwbU| †m KqUvi mgq evmvq †cuŠ‡Q? 6| iwY 7Uv 15 wgwb‡U cov ïi“ K‡i| 10 Uv 50 wgwb‡U †m cov †kl K‡i|

†m †gvU KZ mgq c‡o? 7| GK w`b mKvj 8 Uv 25 wgwb‡U e„wó ïi“ nq| e„wó †kl nq 10 Uv 20

wgwb‡U| KZ¶Y e„wó n‡qwQj?

cÖv_wgK MwYZ 109

w`bcwÄi eënvi w`bcwÄ GLv‡b, 1410 evsjv mv‡ji êkvL gv‡mi 1 ZvwiL †mvgevi| êkvL gv‡mi 10 ZvwiL eyaevi| êkvL gv‡mi 26 ZvwiL ïµevi| êkvL gv‡mi 7, 14, 21 I 28 ZvwiL iweevi| êkvL gv‡mi e„n¯úwZev‡ii ZvwiL¸‡jv n‡”Q 4, 11, 18 I 25| êkvL gv‡m †mvgevi Av‡Q 5wU|

GLv‡b, 2003 Bs‡iwR mv‡ji b‡f¤î gv‡mi 1 ZvwiL kwbevi| b‡f¤î gv‡mi 14 ZvwiL ïµevi| b‡f¤î gv‡mi 25 ZvwiL g½jevi| b‡f¤î gv‡mi 2, 9, 16, 23 I 30 ZvwiL n‡”Q iweevi| b‡f¤î gv‡mi kwbev‡ii ZvwiL¸‡jv n‡”Q, 1, 8, 15, 22 I 29| b‡f¤î gv‡m e„n¯úwZevi Av‡Q 4wU|

w`bcwÄ‡Z _v‡K : (K) wbw ©ó mvj (L) gv‡mi bvg (M) ev‡ii bvg (N) evi Abyhvqx wewfbœ ZvwiL|

êkvL 1410kwb iwe †mvg g½j eya e„n: ïµ 1 2 3 4 56 7 8 9 10 11 1213 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30 31

b‡f¤î 2003kwb iwe †mvg g½j eya e„n: ïµ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

110 cÖv_wgK MwYZ

evsjv gv‡mi w`‡bi msL¨v gvm w`‡bi msL¨v gvm w`‡bi msL¨v

êkvL 31 KvwZ©K 30 ˆR¨ô 31 AMÖnvqY 30 Avlvp 31 †cŠl 30 kªveY 31 gvN 30 fv ª 31 dvêyb 30

Avwk¦b 30 ˆPÎ 30 2003 mv‡ji wewfbœ gv‡mi w`‡bi msL¨v

gvm w`‡bi msL¨v gvm w`‡bi msL¨vRvbyqvwi 31 RyjvB 31 †deª“qvwi 28 AvM÷ 31

gvP© 31 †m‡Þ¤î 30 GwcÖj 30 A‡±vei 31

†g 31 b‡f¤i 30 Ryb 30 wW‡m¤i 31

LvwjNi c~iY Ki (GKwU K‡i †`Lv‡bv nj) :1410 mv‡ji êkvL gv‡mi 4 ZvwiL e„n¯úwZ evi

1410 mv‡ji êkvL gv‡mi 23 ZvwiL evi

1410 mv‡ji êkvL gv‡mi cÖ_g †mvgevi ZvwiL

2003 mv‡ji b‡f¤i gv‡mi †kl eyaevi ZvwiL

2003 mv‡ji b‡f¤i gv‡m ïµevi Av‡Q wU

2003 mv‡ji b‡f¤i gv‡mi 20 ZvwiL evi

evsjv eQ‡ii 5g gvm

Bs‡iwR eQ‡ii 7g gvm

evsjv eQ‡ii 1g gvm

dvêyY gv‡mi c‡ii gvm

cÖv_wgK MwYZ 111

R¨vwgwZK AvK…wZ

wewfbœ AvK…wZi Nbe¯‘

112 cÖv_wgK MwYZ

`vM †U‡b bv‡gi mwVK wP‡Îi wgj Ki (GKwU K‡i †`Lv‡bv nj) :

cÖv_wgK MwYZ 113

e¯‘i Qwe †`‡L AvK…wZi bvg †jL (GKwU K‡i †`Lv‡bv nj) :

114 cÖv_wgK MwYZ

wÎfyR

K L M GKwU wÎfyR| KL, LM Ges MK wÎfyRwUi wZbwU evû|

evû‡f‡` wÎfyR K L M GKwU mgevû wÎfyR| KviY, evû K L =evû L M = evû M K

K L M GKwU mgwØevû wÎfyR| KviY, evû K L = evû K M

K L M GKwU welgevû wÎfyR KviY, †Kvb yBwU evûB mgvb bq|

cÖv_wgK MwYZ 115

evû‡f‡` wÎfyR wÎfyR¸‡jvi Wv‡b Zv‡`i bvg †jL (cÖ_gwU K‡i †`Lv‡bv nj) :

welgevû wÎfyR

116 cÖv_wgK MwYZ

evû‡f‡` wÎfyR

GKwU KvUv Ask KvM‡Ri Dc‡i †i‡L wZb w`‡K `vM Uvbv n‡q‡Q| G‡Z GKwU welgevû wÎfyR n‡q‡Q|

†cv÷ KvW©wU†KvYvKzwb

KvUv n‡q‡Q| GKwU †cv÷ KvW©

K L M Ask KvM‡Ri Dci †i‡L GKwU wÎfyR Uvbv nj| GwU GKwU mgwØevû wÎfyR|

yBwU `vM eivei †cv÷ KvW©wU KvUv n‡q‡Q|

†cv÷ KvW©wUi L K I L M cÖv‡šÍ mgvb ` ~i‡Z¡ yBwU `vM †Ìqv

n‡q‡Q|

KvwVi wZb cÖv‡š— †duvUv †Ìqv n‡q‡Q| c‡i †duvUv¸‡jv †¯‹j eënvi K‡i †hvM Kiv n‡q‡Q| GwU GKwU mgevû wÎfyR|

KvwV wZbwUi cÖvš—wgwj‡q mgevû wÎfy‡Ri AvKvi ˆZwi n‡q‡Q|

3 †m. wg. j¤^v wZbwU KvwV

cÖv_wgK MwYZ 117

PZzf~©R wewfbœ AvK…wZi PZzfy©R

†h¸‡jv PZzfy©R Zvi Wv‡b LvwjN‡i (√ ) wPý `vI| †h¸‡jv PZzfy©R bq Zvi Wv‡b LvwjN‡i ( × ) wPý `vI yBwU K‡i †`Lv‡bv nj :

118 cÖv_wgK MwYZ

PZzfy©R AvqZvKvi I eM©vKvi PZzfy©R KLMN GKwU AvqZ ev AvqZvKvi PZzfy©R| GLv‡b, KL = NM, KN = LM Ges KM = LN AvqZ ev AvqZvKvi PZzfy©R

KLMN GKwU eM© ev eM©vKvi PZzfy©R| GLv‡b, KL = LM = MN = NK Ges KM = LN

Dc‡ii Qwe¸‡jvi g‡a¨ Pzwo, AvswU I PvKvi AvK…wZ †Mvj| M−v‡mi gyL †MvjvKvi|

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