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![2 cÖv_wgK MwYZ
RvZxq wk¶vµg I cvV¨cy¯ZK †evW©
69-70, gwZwSj evwYwR¨K GjvKv, XvKv-1000
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cÖ_g gy`ªY : A‡±vei, 2003
cybg©y`ªY : †m‡Þ¤^i- 2009
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Uploaded bySARA MEHER
প্রাথমিক গনিত ৩
The document introduces a new textbook for third grade students in Bangladesh. It was developed by the Primary Education Development Project of Rajshahi, Rangpur and Dinajpur Divisions to evaluate and revise the existing primary education curriculum and textbooks. The textbook was prepared by experienced teachers and curriculum experts based on a detailed analysis of the learning outcomes and standards for third grade. It aims to improve students' conceptual understanding and problem-solving abilities through clear explanations and examples. Feedback from field testing was used to finalise the content and presentation. The textbook is expected to enhance students' interest in learning and performance in mathematics.
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প্রাথমিক গনিত ৩
- 2. cÖv_wgK MwYZ 1 RvZxqwk¶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
- 3. 2 cÖv_wgK MwYZ RvZxqwk¶vµg I cvV¨cy¯ZK †evW© 69-70, gwZwSj evwYwR¨K GjvKv, XvKv-1000 KZ©„K cÖKvwkZ| [ cÖKvkK KZ„©©K me©¯^Z¡ msiw¶Z ] 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 :
- 4. cÖv_wgK MwYZ 3 f~wgKv cÖv_wgKwk¶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 e¨eüZ 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 wfwˇZ 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 wfwˇZ 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‡`i 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 e¨envi Kiv n‡q‡Q| wkL‡b wkï‡`i AvMÖn m„wó I cvV MÖnY mnR Kivi Rb¨ Abykxj‡bi e¨e¯’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 †`Iqv n‡q‡Q| Avkv Kiv hvq, G‡Z wkï‡`i 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‡`i AewnZ Kiv n‡j Avgiv Aek¨B cÖ‡qvRbxq ms‡kva‡bi e¨e¯’v MÖnY Kie| huv‡`i cÖ‡Póvq eBwU cÖKvwkZ nj, Zuv‡`i mevB‡K RvbvB Avš—wiK ab¨ev`| cÖ‡dmi †gv: †gv¯Zdv KvgvjDwÏb †Pqvig¨vb RvZxq wk¶vµg I cvV¨cy¯ZK †evW©
- 5. 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
- 6. cÖv_wgK MwYZ 1 msL¨v MYbvKi, co I †jL (GKwU K‡i †`Lv‡bv nj) Qwe `‡ki mvnv‡h¨ †jL A‡¼ †jL K_vq †jL 1 `k 8 18 AvVvi − `k − − `k − − `k − − `k − − `k − − `k − − `k − − `k − − `k −
- 7. 2 cÖv_wgK MwYZ `kI k‡Zi `j ˆZwi Ki ( GKwU K‡i †`Lv‡bv nj)
- 8. cÖv_wgK MwYZ 3 MYbvKi 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 • • • • • • • • • • • • • • • •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• ••••••••••
- 9. 4 cÖv_wgK MwYZ •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• ••••••••• ••••••••• ••••••••• ••••••••• ••••••••• ••••••••• ••••••••• ••••••••• ••••••••• ••••••••• • • • • • • • • • MYbvKi 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 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• •••••••••• ••••••••••
- 10. cÖv_wgK MwYZ 5 MYbvKi 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 •••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
- 11. 6 cÖv_wgK MwYZ MYbvKi 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 •••••••••• •••••••••• •••••
- 12. cÖv_wgK MwYZ 7 `k-`kK‡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
- 13. 8 cÖv_wgK MwYZ kZ−kZK‡i MYbv MYbv Ki I co 1 kZ 100 GKk 2 kZ 200 `yBk 3 kZ 300 wZbk 4 kZ 400 Pvik 5 kZ 500 cuvPk 6 kZ 600 Qqk 7 kZ 700 mvZk 8 kZ 800 AvUk 9 kZ 900 bqk 10 kZ 1000 GK nvRvi
- 14. cÖv_wgK MwYZ 9 nvRvi-nvRviK‡i MYbv MYbv Ki I co 1 nvRvi 1000 GK nvRvi 2 nvRvi 2000 nvRvi 3 nvRvi 3000 nvRvi 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
- 15. 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
- 16. 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‡¼AhyZ 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
- 17. 12 cÖv_wgK MwYZ ¯’vbxqgvb Qwe †`‡L msL¨v †jL (GKwU K‡i †`Lv‡bv nj) ¯’vbxq gvb †jL (GKwU K‡i †`Lv‡bv nj) 6 5 ` G 6 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
- 18. cÖv_wgK MwYZ 13 ¯’vbxqgvb k ` G 5 6 3 3 GKK = 3 6 `kK = 60 5 kZK = 500 m k ` G 1 0 0 0 0 GKK = 0 0 `kK = 0 0 kZK = 0 1 mnmª = 1000 m k ` G 1 1 1 1 1 GKK = 1 1 `kK = 10 1 kZK = 100 1 mnmª = 1000 m k ` G 9 0 5 7 7 GKK = 7 5 `kK = 50 0 kZK = 0 9mnmª = 9000 A m k ` G 1 0 0 0 0 0 GKK = 0 0 GKK = 0 0 `kK = 0 0 mnmª = 0 1 AhyZ = 10000
- 19. 14 cÖv_wgK MwYZ ¯’vbxqgvb Qwe †`‡L msL¨v †jL (GKwU K‡i †`Lv‡bv nj) ¯’vbxq gvb †jL (GKwU K‡i †`Lv‡bv nj) m k ` G 5 0 9 2 2 GKK = 2 9 `kK = 90 0 kZK = 0 5 mnmª = 5000 m k ` G 3 8 5 1 m k ` G 6 2 0 9 m k ` G 7 1 6 5 m k ` G 9 9 8 9
- 20. cÖv_wgK MwYZ 15 ¯’vbxqgvb 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
- 21. 16 cÖv_wgK MwYZ ¯’vbxqgvb 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 8 Gi ¯’vbxq gvb = 6 Gi ¯’vbxq gvb = 0 Gi ¯’vbxq gvb = 4 Gi ¯’vbxq gvb = 9705 7 Gi ¯’vbxq gvb = 5 Gi ¯’vbxq gvb = 0 Gi ¯’vbxq gvb = 9 Gi ¯’vbxq gvb = 8092 9 Gi ¯’vbxq gvb = 0 Gi ¯’vbxq gvb = 2 Gi ¯’vbxq gvb = 8 Gi ¯’vbxq gvb = 9864 9 Gi ¯’vbxq gvb = 4 Gi ¯’vbxq gvb = 8 Gi ¯’vbxq gvb = 6 Gi ¯’vbxq gvb = 2987 2 Gi ¯’vbxq gvb = 7 Gi ¯’vbxq gvb = 9 Gi ¯’vbxq gvb = 8 Gi ¯’vbxq gvb =
- 22. cÖv_wgK MwYZ 17 †Rvowe‡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
- 23. 18 cÖv_wgK MwYZ msL¨viZzjbv 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|
- 24. cÖv_wgK MwYZ 19 msL¨viZzjbv 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|
- 25. 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
- 26. cÖv_wgK MwYZ 21 msL¨vAbyµ‡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
- 27. 22 cÖv_wgK MwYZ msL¨vAbyµ‡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
- 28. 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|
- 29. 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 = 5 Gi ¯’vbxq gvb = 3 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
- 30. 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 eoцQvU 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 eoцQvU cÖZxK †jL : (K) 463 436 (L) 800 809 (L) 2532 2352 (N) 5899 6000 (O) 3609 3906 4000 (P) 8990 9899 9099
- 31. 26 cÖv_wgK MwYZ †hvM †hvMKi (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
- 32. cÖv_wgK MwYZ 27 †hvMKi 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
- 33. 28 cÖv_wgK MwYZ +1 787 `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 =
- 34. 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 =
- 35. 30 cÖv_wgK MwYZ we‡qvM 656 `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 ⎯⎯|
- 36. cÖv_wgK MwYZ 31 we‡qvMKi (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 GKK 8=7+1 ↓ 7 − 4 6=5+1 ↓ 15 9 7 ↓ 17 8 3 6 9
- 37. 32 cÖv_wgK MwYZ we‡qvMKi 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
- 38. cÖv_wgK MwYZ 33 †hvMwe‡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
- 39. 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 =
- 40. 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
- 41. 36 cÖv_wgK MwYZ †hvMwe‡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|
- 42. cÖv_wgK MwYZ 37 Abykxjbx-3 1|`xev evwl©K cix¶vq evsjvq 85, MwY‡Z 90 Ges Bs‡iwR‡Z 80 b¤^i †c‡q‡Q| †m wZb wel‡q †gvU KZ b¤^i †c‡q‡Q? 2| Ave`yj nvwjg mv‡ne 375 UvKvi Pvj, 175 UvKvi gvQ I 90 UvKvi †Zj wKb‡jb| wZwb †gvU KZ UvKv LiP Ki‡jb? 3| wgZvi Av‡Q 175 UvKv| †m 75 UvKvi eB I 35 UvKvi LvZv wKbj| Zvi Kv‡Q KZ UvKv iBj? 4| mye‡ji ˆ`wbK Avq 275 UvKv| Zvi ˆ`wbK e¨q 180 UvKv| †m cÖwZw`b KZ UvKv Rgv K‡i? 5| wRKzi 271 UvKv I wcKzi 108 UvKv Av‡Q| Zv‡`i `yB R‡bi UvKv GKÎ Ki‡j UzKzi UvKvi mgvb nq| wZb R‡bi †gvU KZ UvKv Av‡Q? 6| Z…Zxq, PZz_© I cÂg †kªYxi wk¶v_©x‡`i eb‡fvR‡bi Rb¨ 3000 UvKvi cÖ‡qvRb| PZz_© I cÂg †kªYx †_‡K h_vµ‡g 1055 UvKv I 1200 UvKv DVv‡bv n‡q‡Q| Z…Zxq †kªYx †_‡K KZ UvKv DVv‡Z n‡e? 7| 476 Gi mv‡_ KZ †hvM Ki‡j †hvMdj 900 n‡e? 8| Kvgvj mv‡n‡ei Ni †givg‡Zi Rb¨ 9175 UvKvi cÖ‡qvRb| wZwb 1225 UvKvi cvU, 5025 UvKvi Mi“ I 1900 UvKvi QvMj wewµ K‡ib| Ni †givg‡Zi Rb¨ Avi KZ UvKv †RvMvo Ki‡Z n‡e? 9| myR‡bi Av‡Q 975 UvKv| myg‡bi Av‡Q 688 UvKv| mygb‡K Avi KZ UvKv w`‡j Zvi UvKv myR‡bi UvKvi mgvb n‡e?
- 43. 38 cÖv_wgK MwYZ 10|†Kvb bvm©vwi‡Z 805wU Pviv MvQ wQj| 617wU Pviv MvQ wewµ n‡q †Mj| H bvm©vwi‡Z AviI 575wU Pviv MvQ Avbv nj| GLb KZwU Pviv MvQ nj? 11| 917 †_‡K KZ we‡qvM Ki‡j we‡qvMdj 409 n‡e? 12| `yBwU msL¨vi †hvMdj 1475| GKwU msL¨v 986, AciwU KZ? 13| †g‡qi eqm 16 eQi Ges wcZvi eqm 47 eQi| 12 eQi ci Zv‡`i †gvU eqm KZ n‡e? 14| GKwU we`¨vj‡q 1275 R‡bi Avmb e¨e¯’v Av‡Q| 1326 R‡bi emvi Rb¨ Avi KZwU Avm‡bi e¨e¯’v Ki‡Z n‡e? 15| GK e¨w³ 2025 UvKv wb‡q C‡`i evRvi Ki‡Z †M‡jb| evwo wd‡i †`‡Lb Zuvi Kv‡Q 627 UvKv Av‡Q| wZwb KZ UvKv LiP K‡i‡Qb? 16| †Zvgv‡`i ¯‹z‡j 321 Rb wk¶v_©x wQj| 75 Rb ¯‹zj Z¨vM K‡i‡Q| bZzb fwZ© n‡q‡Q 109 Rb| GLb ¯‹z‡j KZ Rb wk¶v_©x Av‡Q? 17| imyjcyi MÖv‡gi RbmsL¨v 5781 Rb| cyi“‡li msL¨v 2985 Rb| bvixi msL¨v KZ? 18| `yBwU msL¨vi we‡qvMdj 1013| eo msL¨vwU 2104; †QvU msL¨vwU KZ? 19| GKwU †Uª‡bi Avmb msL¨v 1575wU| `yB w`‡b Avm‡bi wUwKU †ePv n‡q‡Q 509wU I 425wU| KZwU Avm‡bi wUwKU †ePv nqwb? 20| †Kvb evMv‡b 179wU Avg MvQ Av‡Q| H evMv‡b Avg Mv‡Qi †P‡q 30wU KuvVvj MvQ †ewk Av‡Q| evMv‡b †gvU KZwU MvQ Av‡Q?
- 44. cÖv_wgK MwYZ 39 ¸Y 44 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 =
- 45. 40 cÖv_wgK MwYZ LvwjNic~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
- 46. cÖv_wgK MwYZ 41 ¸‡YibvgZv (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
- 47. 42 cÖv_wgK MwYZ LvwjNic~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
- 48. cÖv_wgK MwYZ 43 ¸Y D`vniY1| 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
- 49. 44 cÖv_wgK MwYZ msw¶ßwbq‡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
- 50. cÖv_wgK MwYZ 45 D`vniY11| ¸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 = 9 mgvavb : 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 5460 gšÍe¨ : 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|
- 51. 46 cÖv_wgK MwYZ D`vniY13| ¸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|
- 52. 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|
- 53. 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
- 54. cÖv_wgK MwYZ 49 5|GKwU c¨v‡K‡U 45wU j‡RÝ Av‡Q| Giƒc 12wU c¨v‡K‡U KZ¸‡jv j‡RÝ Av‡Q? 6| GKwU Mvwo NÈvq 54 wK‡jvwgUvi hvq| GKB MwZ‡Z 15 NÈvq MvwowU KZ wK‡jvwgUvi hv‡e? 7| nvwjgv LvZzb †mjvB K‡i cÖwZw`b 125 UvKv Avq K‡ib| wZwb 15 w`‡b KZ UvKv Avq K‡ib? 8| GKwU eB‡q 104 c„ôv Av‡Q| Giƒc 25wU eB‡q †gvU KZ c„ôv Av‡Q? 9| Acyi `v`vi eqm Acyi eq‡mi 9 ¸Y| Acyi eqm 9 eQi n‡j, Zvi `v`vi eqm KZ? 10| Zzwj ˆ`wbK 7 N›Uv Nygvq| 1 gv‡m †m KZ NÈv Nygv‡e? (1 gvm = 30 w`b) 11| GKwU Uªv‡K 40 e¯Ív Pvj a‡i| Giƒc 6wU Uªv‡K KZ e¯Ív Pvj ai‡e? 12| Abyi †h UvKv Av‡Q Zv wgbyi UvKvi 9 ¸Y| wgbyi 88 UvKv _vK‡j Abyi KZ UvKv Av‡Q? 13| GKwU nuv‡mi `vg 88 UvKv| Giƒc 25wU nuv‡mi `vg KZ? 14| †Kvb evMv‡bi GKwU mvwi‡Z 28wU MvQ Av‡Q| H evMv‡b 17wU mvwi‡Z KZwU MvQ Av‡Q? 15| GKwU Szwo‡Z 112wU †cqviv Av‡Q| Hiƒc 19wU Szwo‡Z †gvU KZwU †cqviv Av‡Q?
- 55. 50 cÖv_wgK MwYZ fvM fv‡MiaviYv 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
- 56. 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 × fvMdj28 = 4 × 7 wbt‡k‡l fvM n‡”Q ¸‡Yi wecixZ cÖwµqv fvRK fvR¨ fvMdj fvM‡kl
- 57. 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
- 58. cÖv_wgK MwYZ 53 fvM D`vniY1| 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|
- 59. 54 cÖv_wgK MwYZ D`vniY3| 279 †K 6 w`‡q fvM Ki| mgvavb : 6 ) 279 ( 46 e¨vL¨v : 279 = 27 `k 9 24 6 ) 27 `k 9 ( 4 `k 6 39 24 `k 36 39 3 36 3 DËi : fvMdj 46, fvM‡kl 3| gšÍe¨ : wk¶v_©xiv cÖPwjZ wbq‡g fvM A¼ Ki‡e, Gi evB‡i Zv‡`i wbKU †Kvb e¨vL¨v PvIqv n‡e bv| D`vniY 4| 509 †K 7 w`‡q fvM Ki| mgvavb : 7 ) 509 ( 72 49 19 14 5 DËi : fvMdj 72, fvM‡kl 5| D`vniY 5| 660 †K 6 w`‡q fvM Ki| mgvavb : 6 ) 660 ( 110 6 6 6 0 DËi : fvMdj 110| D`vniY 6| 8wU LvZvi `vg 88 UvKv| cÖwZwU LvZvi `vg KZ? mgvavb : 8 ) 88 ( 11 8 8 8 0 DËi : 11 UvKv| D`vniY 7| 4wU‡Z GK nvwj nq| 60wU Kjvq KZ nvwj n‡e? mgvavb : 4 ) 60 ( 15 4 20 20 0 DËi : 15 nvwj|
- 60. cÖv_wgK MwYZ 55 AbykxjbxÑ 5 1| fvM Ki : (K) 63 ÷ 9 (L) 84 ÷ 4 (M) 330 ÷ 3 (N) 242 ÷ 2 (O) 126 ÷ 6 (P) 550 ÷ 5 (Q) 404 ÷ 4 (R) 567 ÷ 9 (S) 800 ÷ 9 (T) 428 ÷ 4 (U) 931 ÷ 8 (V) 948 ÷ 6 2| LvwjNi c~iY Ki : (K) 28 ÷ 4 = 7 GLv‡b, 4 , 28 , 7 | (L) 7 ÷ 1 = 7 GLv‡b, fvR¨ I mgvb| (M) 5 ÷ 5 = 1 GLv‡b, fvMdj Ges I mgvb| (N) 8 ) 325 ( 40 GLv‡b, fvMdj , fvM‡kl | 32 5 3| 36 wgUvi j¤^v KvwV †_‡K 2 wgUvi K‡i KZ LÊ †K‡U †bIqv hv‡e? 4| GKwU evm 4 NÈvq 92 wK‡jvwgUvi hvq| evmwU cÖwZ NÈvq KZ wK‡jvwgUvi hvq? 5| †Kvb †kªYx‡Z 40 Rb QvÎ Av‡Q| cÖwZ †e‡Â 5 Rb K‡i em‡j KqwU †e jvM‡e? 6| GKwU †cbwm‡ji `vg 4 UvKv| 96 UvKvq KqwU †cbwmj cvIqv hv‡e? 7| 9wU Kj‡gi `vg 99 UvKv| cÖwZwU Kj‡gi `vg KZ?
- 61. 56 cÖv_wgK MwYZ mgm¨vimgvavb (†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|
- 62. cÖv_wgK MwYZ 57 D`vniY3| GKwU c¨v‡K‡U 50wU j‡RÝ Av‡Q| Gi †_‡K 8wU j‡RÝ †i‡L †`Iqv nj| Aewkó j‡RÝ 7 R‡bi g‡a¨ mgvb fv‡M fvM K‡i †`Iqv 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 †`Iqv 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 †`Iqv hv‡e| DËi : 30 Rb‡K|
- 63. 58 cÖv_wgK MwYZ D`vniY5| 4 Rb Qv‡Îi cÖ‡Z¨‡Ki 25wU K‡i j‡RÝ Av‡Q| mK‡ji j‡RÝ GKÎ Kiv nj| Zvici j‡Rݸ‡jv 5 R‡bi g‡a¨ mgvb fv‡e fvM K‡i †`Iqv nj| GLb cÖ‡Z¨‡K KqwU K‡i j‡RÝ †cj? mgvavb : 25 GLb, 5 ) 100 ( 20 ×4 10 100 0 0 0 ∴ †gvU j‡RÝ Av‡Q 100wU| ∴ cÖ‡Z¨‡K 20wU K‡i j‡RÝ †cj| DËi : 20wU| D`vniY 6| gwbi AveŸv 99wU wjPz mgvb wZb fvM Ki‡jb| `yB fvM wZwb gwb‡K w`‡jb| gwb KqwU wjPz †cj? mgvavb : 3) 99 (33 ∴ GK fv‡M wjQy Av‡Q 33wU| 9 gwb †cj `yB fvM| 9 9 GLb, 33 0 × 2 66 ∴ gwb wjPz †cj 66wU| DËi : 66wU|
- 64. cÖv_wgK MwYZ 59 AbykxjbxÑ6 1|GKwU †kªYx‡Z 42 Rb QvÎ Av‡Q| cÖ‡Z¨K 7 UvKv K‡i Puv`v w`j| †gvU KZ UvKv Puv`v nj? 2| GKwU Szwo‡Z 45wU Avg Av‡Q| Giƒc 12wU Szwo‡Z KZwU Avg Av‡Q? 3| †Kvb †kªYx‡Z 45 Rb QvÎ Av‡Q| cÖwZ †e‡Â 5 Rb K‡i QvÎ e‡m| H †kªYxi Rb¨ KZwU †e‡Âi cÖ‡qvRb n‡e? 4| 36 wgUvi j¤^v GKwU wdZv‡K mgvb 9 fv‡M fvM Kiv nj| cÖwZ fv‡M KZ wgUvi wdZv nj? 5| †Kvb †kªYx‡Z 12wU †e Av‡Q| 8wU †e‡Â 5 Rb K‡i QvÎx em‡Z cv‡i| evwK 4wU‡Z 6 Rb K‡i em‡Z cv‡i| 12wU †e‡Â †gvU KZ Rb QvÎx em‡Z cv‡i? 6| GKwU kv‡U©i g~j¨ 75 UvKv| GKwU c¨v‡›Ui g~j¨ 120 UvKv| 25wU kvU© I 6wU c¨v›U wKb‡Z KZ UvKv jvM‡e? 7| GKwU ej‡cb I 6wU LvZvi g~j¨ GK‡Î 77 UvKv| GKwU LvZvi g~j¨ 12 UvKv| GKwU ej‡c‡bi g~j¨ KZ? 8| cÖwZ Kuvw`‡Z 82wU Kjv _vKvi K_v| Giƒc 8wU Kuvw`i Kjv MYbv Kiv nj| ¸‡Y †`Lv †Mj †gvU 15wU Kjv Kg Av‡Q| Kuvw`¸‡jv‡Z †gvU KZ¸‡jv Kjv wQj?
- 65. 60 cÖv_wgK MwYZ 9|GKwU Szwo‡Z 35wU Avg Av‡Q| Aci GKwU Szwo‡Z 37wU Avg Av‡Q| `yB Szwo Avg GKÎ K‡i 9 R‡bi g‡a¨ fvM K‡i †`Iqv nj| cª‡Z¨‡K KqwU K‡i Avg †cj ? 10| mywg 40 wU j‡R‡Ýi GKwU c¨v‡KU wKbj| †m c¨v‡KU †_‡K 5wU j‡RÝ mwi‡q ivL‡jv| Aewkó j‡RÝ 5 Rb eÜzi gv‡S mgvb fv‡e fvM K‡i w`j| mywgi eÜziv cÖ‡Z¨‡K KqwU K‡i j‡RÝ †cj? 11| ivbvi 99 UvKv Av‡Q| †m H UvKvi bq fv‡Mi GK fvM Qwe‡K w`j| GLb ivbvi KZ UvKv iBj? 12| 88 wgUvi j¤^v wdZvi 8 fv‡Mi 6 fvM kvnvbv‡K †`Iqv nj| kvnvbv KZ wgUvi wdZv †cj? 13| 25 Rb gwnjvi cÖ‡Z¨‡K 75 UvKv K‡i Puv`v w`‡jb| GB Puv`v GKÎ K‡i 15 Rb eb¨v `yM©Z †jv‡Ki gv‡S mgvb fv‡e weZiY Kiv nj| cÖ‡Z¨‡K KZ UvKv K‡i †c‡jb? 14| wcZvi eZ©gvb eqm cy‡Îi eZ©gvb eq‡mi 3 ¸Y| 5 eQi c~‡e© cy‡Îi eqm 8 eQi wQj| wcZvi eZ©gvb eqm KZ? 15| Kvjvg mv‡ne cÖwZ Szwo 75 UvKv `‡i 7 Szwo Avg weµq Ki‡jb| wZwb 98 UvKv w`‡q GKwU Rvgv wKb‡jb| Aewkó UvKv e¨vs‡K Rgv ivL‡jb| wZwb KZ UvKv e¨vs‡K Rgv ivL‡jb? 16| GKwU Mvwo g‡bvni`x †_‡K mKvj 6Uvq hvÎv ïi“ K‡i| †Kv_vI bv †_‡g `ycyi 12Uvq nweM‡Ä †cŠ‡Q| hw` MvwowU NÈvq 45 wK‡jvwgUvi hvq, Z‡e g‡bvni`x †_‡K nweM‡Äi `~iZ¡ KZ?
- 66. cÖv_wgK MwYZ 61 MvwYwZKcÖZxK 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
- 67. 62 cÖv_wgK MwYZ N‡ih_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
- 68. 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
- 69. 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|
- 70. cÖv_wgK MwYZ 65 evsjv‡`kxgy`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
- 71. 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
- 72. cÖv_wgK MwYZ 67 evsjv‡`kxgy`ªv I †bvU
- 73. 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| 100 UvKvi wU †bvU mgvb 500 UvKv| 20 UvKvi wU †bvU mgvb 500 UvKv| 100 UvKvi wU †bvU mgvb 1000 UvKv|
- 74. 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 1 UvKv 80 cqmv = 1.80 UvKv 90 UvKv 9 cqmv = 90.09 UvKv 100 UvKv = 100.00 UvKv 100 UvKv 5 cqmv = 100.05 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 `yBkcuqlwÆUvKv`kcqmv= 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 =
- 75. 70 cÖv_wgK MwYZ D`vniY1| 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|
- 76. cÖv_wgK MwYZ 71 D`vniY5| 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|
- 77. 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
- 78. 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?
- 79. 74 cÖv_wgK MwYZ fMœvsk fMœvs‡kiaviYv 1 2 ev, A‡a©K ev, Aa©vsk ev, `yB fv‡Mi GK 1 2 ev, A‡a©K ev, Aa©vsk ev, `yB fv‡Mi GK 1 2 ev, A‡a©K ev, Aa©vsk ev, `yB fv‡Mi GK 1 3 ev, GK Z…Zxqvsk ev, wZb fv‡Mi GK 1 3 ev, GK Z…Zxqvsk ev, wZb fv‡Mi GK 1 3 ev, GK Z…Zxqvsk ev, wZb fv‡Mi GK 1 4 ev, GK PZz_©vsk ev, Pvi fv‡Mi GK 1 4 ev, GK PZz_©vsk ev, Pvi fv‡Mi GK 1 4 ev, GK PZz_©vsk ev, Pvi fv‡Mi GK 2 3 ev, `yB Z…Zxqvsk ev, wZb fv‡Mi `yB 2 3 ev, `yB Z…Zxqvsk ev, wZb fv‡Mi `yB 2 3 ev, `yB Z…Zxqvsk ev, wZb fv‡Mi `yB
- 80. cÖv_wgK MwYZ 75 fMœvsk fMœvs‡kiaviYv 3 4 ev, wZb PZz_©vsk ev, Pvi fv‡Mi wZb 3 4 ev, wZb PZz_©vsk ev, Pvi fv‡Mi wZb 3 4 ev, wZb PZz_©vsk ev, Pvi fv‡Mi wZb 4 5 ev, Pvi cÂgvsk ev, cuvP fv‡Mi Pvi 4 5 ev, Pvi cÂgvsk ev, cuvP fv‡Mi Pvi 4 5 ev, Pvi cÂgvsk ev, cuvP fv‡Mi Pvi 1 6 ev, GK lôvsk ev, Qq fv‡Mi GK 2 7 ev, `yB mßgvsk ev, mvZ fv‡Mi `yB 3 8 ev, wZb Aógvsk ev, AvU fv‡Mi wZb 2 9 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|
- 81. 76 cÖv_wgK MwYZ wb‡PiQwe¸‡jv‡Z 1 2 Ask †cbwmj w`‡q Kvj Ki : wb‡Pi Qwe¸‡jv‡Z 1 3 Ask †cbwmj w`‡q Kvj Ki : wb‡Pi Qwe¸‡jv‡Z 1 4 Ask †cbwmj w`‡q Kvj Ki : wb‡Pi Qwe¸‡jv‡Z 2 3 Ask †cbwmj w`‡q Kvj Ki :
- 82. cÖv_wgK MwYZ 77 wb‡PiQwe¸‡jvi Kvj Ask A‡¼ I K_vq †jL : (GKwU K‡i †`Lv‡bv nj) Qwe A‡¼ †jL K_vq †jL 1 2 A‡a©K
- 83. 78 cÖv_wgK MwYZ GKvs‡kiaviYv 6 6 12 Gi 1 2 = 6 12 12 Gi 1 3 = 4 12 12 Gi 1 4 = 3 12 12 Gi 1 6 = 2 12 4 4 4 3 3 3 3 2 2 2 2 2 2 16 Gi 1 2 = 8 16 16 Gi 1 4 = 4 16 16 Gi 1 8 = 2 16 8 8 4 4 4 4 2 2 2 2 2 2 2 2
- 84. cÖv_wgK MwYZ 79 6= 2 × 3 = 3 × 2 ∴ 6 ÷ 2 = 3 Avevi 6 ÷ 3 = 2 ∴ 6 Gi 1 2 = 3 Ges 6 Gi 1 3 = 2 8 = 2 × 4 = 4 × 2 ∴ 8 ÷ 2 = 4 Avevi 8 ÷ 4 = 2 ∴ 8 Gi 1 2 = 4 Ges 8 Gi 1 4 = 2 10 = 2 × 5 = 5 × 2 ∴ 10 ÷ 2 = 5 Avevi 10 ÷ 5 = 2 ∴ 10 Gi 1 2 = 5 Ges 10 Gi 1 5 = 2 15 = 3 × 5 = 5 × 3 ∴ 15 ÷ 3 = 5 Avevi 15 ÷ 5 = 3 ∴ 15 Gi 1 3 = 5 Ges 15 Gi 1 5 = 3 20 = 4 × 5 = 5 × 4 ∴ 20 ÷ 4 = 5 Avevi 20 ÷ 5 = 4 ∴ 20 Gi 1 4 = 5 Ges 20 Gi 1 5 = 4 30 = 5 × 6 = 6 × 5 ∴ 30 ÷ 5 = 6 Avevi 30 ÷ 6 = 5 ∴ 30 Gi 1 5 = 6 Ges 30 Gi 1 6 = 5 42 = 6 × 7 = 7 × 6 ∴ 42 ÷ 6 = 7 Avevi 42 ÷ 7 = 6 ∴ 42 Gi 1 6 = 7 Ges 42 Gi 1 7 = 6 54 = 6 × 9 = 9 × 6 ∴ 54 ÷ 6 = 9 Avevi 54 ÷ 9 = 6 ∴ 54 Gi 1 6 = 9 Ges 54 Gi 1 9 = 6
- 85. 80 cÖv_wgK MwYZ LvwjNic~iY Ki : 56 = 7 × = 8 × 72 = × 9 = × 8 56 Gi 1 7 = 72 Gi 1 9 = 1 eQi ev 12 gv‡mi 1 2 Ask = 6 gvm 1 eQi ev 12 gv‡mi 1 3 Ask = gvm 1 eQi ev 12 gv‡mi 1 4 Ask = gvm 1 eQi ev 12 gv‡mi 1 6 Ask = gvm 1 gvm ev 30 w`‡bi 1 2 Ask = 15 w`b 1 gvm ev 30 w`‡bi 1 3 Ask = w`b 1 gvm ev 30 w`‡bi 1 5 Ask = w`b 1 gvm ev 30 w`‡bi 1 6 Ask = w`b 1 w`b ev 24 N›Uvi 1 2 Ask = 12 NÈv 1 w`b ev 24 N›Uvi 1 3 Ask = NÈv
- 86. cÖv_wgK MwYZ 81 1w`b ev 24 N›Uvi 1 4 Ask = NÈv 1 w`b ev 24 N›Uvi 1 6 Ask = NÈv 1 w`b ev 24 N›Uvi 1 8 Ask = NÈv 1 NÈv ev 60 wgwb‡Ui 1 2 Ask = 30 wgwbU 1 NÈv ev 60 wgwb‡Ui 1 3 Ask = wgwbU 1 NÈv ev 60 wgwb‡Ui 1 4 Ask = wgwbU 1 NÈv ev 60 wgwb‡Ui 1 5 Ask = wgwbU 1 NÈv ev 60 wgwb‡Ui 1 6 Ask = wgwbU 1 NÈv ev 60 wgwb‡Ui 1 10 Ask = wgwbU 1 UvKv ev 100 cqmvi 1 2 Ask = 50 cqmv 1 UvKv ev 100 cqmvi 1 4 Ask = cqmv 1 UvKv ev 100 cqmvi 1 5 Ask = cqmv 1 UvKv ev 100 cqmvi 1 10 Ask = cqmv
- 87. 82 cÖv_wgK MwYZ 1wgUvi ev 100 †mw›UwgUv‡ii 1 2 Ask = 50 †mw›UwgUvi 1 wgUvi ev 100 †mw›UwgUv‡ii 1 4 Ask = †mw›UwgUvi 1 wgUvi ev 100 †mw›UwgUv‡ii 1 5 Ask = †mw›UwgUvi 1 wgUvi ev 100 †mw›UwgUv‡ii 1 10 Ask = †mw›UwgUvi 18 Gi 1 3 Ask = 6 24 Gi Ask = 8 25 Gi 1 5 Ask = 5 30 Gi 1 6 Ask = 32 Gi Ask = 8 36 Gi Ask = 9 1 eQ‡ii Ask = 4 gvm 1 gv‡mi 1 3 Ask = w`b 1 N›Uvi 1 6 Ask = wgwbU 1 UvKvi Ask = 25 cqmv 1 wgUv‡ii Ask = 20 †m. wg. 1 w`‡bi Ask = N›Uv
- 88. cÖv_wgK MwYZ 83 fMœvs‡kini I je e„‡Ëi wfZ‡ii Kvj 1 4 Ask | 3 4 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| 2 5 fMœvs‡k 5 ni Ges 2 je wb‡Pi LvwjNi c~iY Ki (GKwU K‡i †`Lv‡bv nj) : 3 8 fMœvs‡k 8 ni Ges 3 je 3 5 fMœvs‡k 3 Ges 5 7 8 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 8 ni Ges 7 je n‡j fMœvskwU n‡e 9 ni Ges 8 je n‡j fMœvskwU n‡e
- 89. 84 cÖv_wgK MwYZ mgZzjfMœvsk 1 2 2 4 3 6 4 8 5 10 1 2, 2 4, 3 6, 4 8, 5 10 fMœvsk¸‡jv mgZzj| †Kbbv, †h‡Kvb e¯‘i 1 2 Ask, 2 4 Ask, 3 6 Ask, 4 8 Ask, 5 10 Ask ci¯úi mgvb| 1 2 2 4 3 6 4 8 5 10 1 2, 2 4, 3 6, 4 8, 5 10 mgZzj fMœvsk| 1 3 2 6 3 9 1 3 , 2 6, 3 9 mgZzj fMœvsk|
- 90. cÖv_wgK MwYZ 85 mgZzjfMœvsk †Pbv 1 3, 2 6 mgZzj fMœvsk| 1 3 2 6 GLv‡b, 1 3 2 6 2 4, 3 6 mgZzj fMœvsk| 2 4 ∴ 2 4 = 3 6 3 6 GLv‡b, 2 4 3 6 1 2, 3 4 mgZzj fMœvsk bq| 1 2 ∴ 1 2 = 3 4 mgvb bq| 3 4 GLv‡b, 1 2 3 4 3 × 2 = 6 1 × 6 = 6 4×3 = 12 2×6 = 12 2 × 3 = 6 1 × 4 = 4 ∴ 1 3 = 2 6
- 91. 86 cÖv_wgK MwYZ 8Gi 2 4 = 4 8 Gi 4 8 = 4 ∴ 2 4, 4 8 mgZzj fMœvsk| ∴ 2 4 = 4 8 GLv‡b, 2 4 4 8 12 Gi 3 4 = 9 12 Gi 4 6 = 8 GLv‡b, 3 4 4 6 ∴ 3 4, 4 6 mgZzj fMœvsk bq| ∴ 3 4, 4 6 mgvb bq| mgZzj fMœvsk MVb 1 2 Gi mgZzj fMœvsk : 1 2 = 1 × 2 2 × 2 = 2 4 ; 1 2 = 1 × 3 2 × 3 = 3 6 ; 1 2 = 1 × 4 2 × 4 = 4 8 ; 1 2 = 1 × 5 2 × 5 = 5 10 ; 2 4, 3 6, 4 8, 5 10 BZ¨vw` 1 2 Gi mgZzj fMœvsk| 2 × 3 = 6 1 × 4 = 4 4 × 4= 16 3 × 6= 18
- 92. cÖv_wgK MwYZ 87 1 3Gi mgZzj fMœvsk : 1 3 = 1 × 2 3 × 2 = 2 6 ; 1 3 = 1 × 3 3 × 3 = 3 9 ; 1 3 = 1 × 4 3 × 4 = 4 12 ; 1 3 = 1 × 5 3 × 5 = 5 15 ; 2 6, 3 9, 4 12, 5 15 BZ¨vw` 1 3 Gi mgZzj fMœvsk| 2 3 Gi mgZzj fMœvsk : 2 3 = 2 × 2 3 × 2 = 4 6 ; 2 3 = 2 × 3 3 × 3 = 6 9 ; 2 3 = 2 × 4 3 × 4 = 8 12 ; 2 3 = 2 × 5 3 × 5 = 10 15 ; 4 6, 6 9, 8 12, 10 15 BZ¨vw` 2 3 Gi mgZzj fMœvsk| 1 4 Gi mgZzj fMœvsk : 1 4 = 1 × 2 4 × 2 = 2 8 ; 1 4 = 1 × 3 4 × 3 = 3 12 ; 1 4 = 1 × 4 4 × 4 = 4 16 ; 1 4 = 1 × 5 4 × 5 = 5 20 ; 2 8, 3 12, 4 16, 5 20 BZ¨vw` 1 4 Gi mgZzj fMœvsk|
- 93. 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) 2 3, 4 6 , 6 9 , 8 12 , 10 15 (L) 3 4, , , , , (M) 3 5, , , , , (N) 5 6, , , , , (O) 2 7, , , , , 2| LvwjNi c~iY Ki (cÖ_gwU K‡i †`Lv‡bv nj) : (K) 1 2 = 4 8 ; 1 2 = 6 ; 1 2 = 7 ; 1 2 = 3 (L) 1 3 = 5 ; 1 5 = 2 ; 2 5 = 10 ; 3 4 = 8 (M) 1 6 = 3 ; 1 7 = 3 ; 1 7 = 14 ; 4 5 = 15
- 94. 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) 3 4, 6 8 (L) 4 6, 1 3 (M) 3 6, 4 8 (N) 1 7, 2 14 (O) 3 6, 6 12 (P) 2 3, 3 6 1 Gi mgZzj fMœvsk 1 = 1 1 2 2 3 3 4 4 1 × 2 = 2 1 × 3 = 3 1 × 4 = 4 2 × 1 = 2 3 × 1 = 3 4 × 1 = 4 ∴ 2 2, 3 3 , 4 4 BZ¨vw` cÖ‡Z¨KwU fMœvsk 1 Gi mgZzj| A_©vr, 1 = 1 1 = 2 2 = 3 3 = 4 4 BZ¨vw`| 3 4 6 8 4 × 6 = 24 3 × 8 = 24 mgZzj 4 6 1 3 6 × 1 = 6 4 × 3 = 12 mgZzj bq
- 95. 90 cÖv_wgK MwYZ fMœvs‡kiZzjbv 3 4 2 4 3 4 eo 2 4 †_‡K; A_©vr, 3 4 > 2 4 Avevi, 2 4 †QvU 3 4 †_‡K; 2 4 < 3 4 3 8 5 8 3 8 †QvU 5 8 †_‡K; A_©vr, 3 8 < 5 8 Avevi, 5 8 eo 3 8 †_‡K; 5 8 > 3 8 5 8 = 8 fv‡Mi 5 fvM 3 8 = 8 fv‡Mi 3 fvM 4 9 = 9 fv‡Mi 4 fvM 7 9 = 9 fv‡Mi 7 fvM 7 15 = 15 fv‡Mi 7 fvM 6 15 = 15 fv‡Mi 6 fvM 5 > 3 ∴ 5 8 > 3 8 4 > 7 ∴ 4 9 < 7 9 7 > 6 ∴ 7 15 > 6 15
- 96. cÖv_wgK MwYZ 91 fMœvs‡ki†hvM 1 2 1 2 1 1 2 + 1 2 = 1 wKš‘, 1 = 2 2 = 1 + 1 2 ∴ 1 2 + 1 2 = 1 + 1 2 1 3 1 3 2 3 1 3 + 1 3 = 2 3 wKš‘, 2 3 = 1 + 1 3 ∴ 1 3 + 1 3 = 1 + 1 3 2 5 1 5 3 5 2 5 + 1 5 = 3 5 wKš‘, 3 5 = 2 + 1 5 ∴ 2 5 + 1 5 = 2 + 1 5 5 8 2 8 7 8 5 8 + 2 8 = 7 8 wKš‘, 7 8 = 5 + 2 8 ∴ 5 8 + 2 8 = 5 + 2 8 2 6 + 3 6 = 2 + 3 6 = 5 6 4 9 + 3 9 = 4 + 3 9 = 7 9
- 97. 92 cÖv_wgK MwYZ 3 8 + 1 8= 3 + 1 8 = 4 8 = 1 2 KviY, 4 8 I 1 2 mgZzj fMœvsk| 5 12 + 4 12 = 5 + 4 12 = 9 12 = 3 4 KviY, 9 12 I 3 4 mgZzj fMœvsk| LvwjNi c~iY Ki (cÖ_gwU K‡i †`Lv‡bv nj) : 1| 4 11 + 5 11 = 9 12 2| 1 4 + 2 4 = 3| 2 7 + 3 7 = 4| 5 8 + 3 8 = 5| 7 12 + 3 12 = 6| 9 14 + 4 14 = 7| 7 15 + 4 15 = 8| 5 16 + 9 16 = 9| 9 19 + 5 19 = 10| 7 20 + 9 20 = 11| 8 25 + 9 25 = 12| 7 12 + 4 12 =
- 98. cÖv_wgK MwYZ 93 fMœvs‡kiwe‡qvM 4 5 2 5 2 5 4 5 − 2 5 = 2 5 Avevi, 2 5 = 4 − 2 5 ∴ 4 5 − 2 5 = 4 − 2 5 = 2 5 1 = 3 3 2 3 1 3 1 − 2 3 = 1 3 Avevi, 1 3 = 3 − 2 3 ∴ 1 − 2 3 = 3 3 − 2 3 = 3 − 2 3 = 1 3
- 99. 94 cÖv_wgK MwYZ 7 9 − 5 9= 7 − 5 9 = 2 9 1 − 2 5 = 5 5 − 2 5 = 5 − 2 5 = 3 5 5 8 − 3 8 = 5 − 3 8 = 2 8 = 1 4 KviY, 2 8 I 1 4 mgZzj fMœvsk| 7 12 − 5 12 = 7 − 5 12 = 2 12 = 1 6 KviY, 2 12 I 1 6 mgZzj fMœvsk| LvwjNi c~iY Ki (cÖ_gwU K‡i †`Lv‡bv nj) : 1| 8 12 − 3 12 = 5 12 2| 7 13 − 5 13 = 3| 9 10 − 5 10 = 4| 1 − 4 9 = 5| 1 − 7 8 = 6| 8 15 − 1 15 = 7| 9 20 − 7 20 = 8| 7 25 − 2 25 = 9| 9 31 − 8 31 = 10| 9 12 − 7 12 =
- 100. 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) 3 5 (L) 7 8 (M) 4 15 (N) 5 22 (O) 7 33 (P) 9 39 3| (K) 8 Gi 3 4 = KZ? (L) 9 Gi 2 3 = KZ? (M) 12 Gi 1 3 = KZ? (N) 12 Gi 2 6 = KZ? (O) 20 Gi 1 4 = KZ? (P) 25 Gi 3 5 = KZ? 4| (K) 1 eQ‡ii 1 2 Ask = KZ gvm? (L) 1 gv‡mi 1 3 Ask = KZ w`b? (M) 1 w`‡bi 1 4 Ask = KZ N›Uv? (N) 1 UvKvi 1 4 Ask = KZ cqmv? (O) 1 wgUv‡ii 1 10 Ask = KZ †mw›UwgUvi? (P) 1 N›Uvi 1 5 Ask = KZ wgwbU?
- 101. 96 cÖv_wgK MwYZ 5|wb‡Pi fMœvsk¸‡jvi †Kvb †Rvov mgZzj Ges †Kvb †Rvov mgZzj bq, wbY©q Ki : (K) 1 2, 2 4 (L) 3 5, 6 10 (M) 2 7, 3 7 (N) 3 4, 3 8 (O) 4 5, 2 3 (P) 4 6, 8 12 (Q) 2 6, 3 12 6| wb‡Pi fMœvsk¸‡jv Zzjbv K‡i K_vq cÖKvk Ki : (K) 3 4 > 1 4 (L) 2 5 < 3 5 (M) 7 10 > 5 10 (M) 7 10 < 9 10 7| wb‡Pi fMœvsk¸‡jv Zzjbv K‡i eo ev †QvU Gi cÖZxK LvwjN‡i †jL : (K) 5 6 3 6 (L) 5 8 7 8 (M) 7 12 9 12 (N) 7 16 5 16 (O) 6 25 9 25 (P) 8 21 5 21 8| †hvM Ki : (K) 3 4 + 1 4 (L) 3 5 + 1 5 (M) 2 6 + 3 6 (N) 5 7 + 1 7 (O) 5 11 + 3 11 (P) 9 13 + 2 13 (Q) 8 20 + 5 20 (R) 7 25 + 5 25 (S) 2 31 + 8 31 9| we‡qvM Ki : (K) 3 7 − 2 7 (L) 7 9 − 5 9 (M) 1 − 1 3 (N) 1 − 5 6 (O) 8 18 − 5 18 (P) 5 21 − 4 21
- 102. 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.
- 103. 98 cÖv_wgK MwYZ wdZvev †Uc wdZv w`‡q †g‡c LvwjNi c~iY Ki : †m. wg. †m. wg. †m. wg. ˆ`N©¨ cÖ¯’ D”PZv
- 104. 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.
- 105. 100 cÖv_wgK MwYZ IRbcwigvc 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
- 106. 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 3 †KwR 50 MÖvg 1950 MÖvg 1 †KwR 950 MÖvg 7000 MÖvg
- 107. 102 cÖv_wgK MwYZ D`vniY1| 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|
- 108. cÖv_wgK MwYZ 103 Zijc`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
- 109. 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?
- 110. cÖv_wgK MwYZ 105 mgqcwigvc Nwo †`Lv
- 111.
- 112. cÖv_wgK MwYZ 107 mgqcwigv‡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|
- 113. 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?
- 114. cÖv_wgK MwYZ 109 w`bcwÄie¨envi w`bcwÄ GLv‡b, 1410 evsjv mv‡ji ˆekvL gv‡mi 1 ZvwiL †mvgevi| ˆekvL gv‡mi 10 ZvwiL eyaevi| ˆekvL gv‡mi 26 ZvwiL ïµevi| ˆekvL gv‡mi 7, 14, 21 I 28 ZvwiL iweevi| ˆekvL gv‡mi e„n¯úwZev‡ii ZvwiL¸‡jv n‡”Q 4, 11, 18 I 25| ˆekvL gv‡m †mvgevi Av‡Q 5wU| GLv‡b, 2003 Bs‡iwR mv‡ji b‡f¤^i gv‡mi 1 ZvwiL kwbevi| b‡f¤^i gv‡mi 14 ZvwiL ïµevi| b‡f¤^i gv‡mi 25 ZvwiL g½jevi| b‡f¤^i gv‡mi 2, 9, 16, 23 I 30 ZvwiL n‡”Q iweevi| b‡f¤^i gv‡mi kwbev‡ii ZvwiL¸‡jv n‡”Q, 1, 8, 15, 22 I 29| b‡f¤^i gv‡m e„n¯úwZevi Av‡Q 4wU| w`bcwćZ _v‡K : (K) wbw`©ó mvj (L) gv‡mi bvg (M) ev‡ii bvg (N) evi Abyhvqx wewfbœ ZvwiL| ˆekvL 1410 kwb 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 31 b‡f¤^i 2003 kwb 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
- 115. 110 cÖv_wgK MwYZ evsjvgv‡mi w`‡bi msL¨v gvm w`‡bi msL¨v gvm w`‡bi msL¨v ˆekvL 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¨v Rvbyqvwi 31 RyjvB 31 †deª“qvwi 28 AvM÷ 31 gvP© 31 †m‡Þ¤^i 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 ˆekvL gv‡mi 4 ZvwiL e„n¯úwZ evi 1410 mv‡ji ˆekvL gv‡mi 23 ZvwiL evi 1410 mv‡ji ˆekvL 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
- 116. cÖv_wgK MwYZ 111 R¨vwgwZKAvK…wZ wewfbœ AvK…wZi Nbe¯‘
- 117. 112 cÖv_wgK MwYZ `vM†U‡b bv‡gi mwVK wP‡Îi wgj Ki (GKwU K‡i †`Lv‡bv nj) :
- 118. cÖv_wgK MwYZ 113 e¯‘iQwe †`‡L AvK…wZi bvg †jL (GKwU K‡i †`Lv‡bv nj) :
- 119. 114 cÖv_wgK MwYZ wÎfyR KL 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|
- 120. cÖv_wgK MwYZ 115 evû‡f‡`wÎfyR wÎfyR¸‡jvi Wv‡b Zv‡`i bvg †jL (cÖ_gwU K‡i †`Lv‡bv nj) : welgevû wÎfyR
- 121. 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 †`Iqv n‡q‡Q| KvwVi wZb cÖv‡š— †duvUv †`Iqv n‡q‡Q| c‡i †duvUv¸‡jv †¯‹j e¨envi 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
- 122. cÖv_wgK MwYZ 117 PZzf~©R wewfbœAvK…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 :
- 123. 118 cÖv_wgK MwYZ PZzfy©R AvqZvKviI 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|
- 124. cÖv_wgK MwYZ 119 AbykxjbxÑ12 1|`vM †U‡b bv‡gi mv‡_ wP‡Îi wgj Ki : AvqZ mgevû wÎfyR e„Ë welgevû wÎfyR eM© mgwØevû wÎfyR 2| GKwU welgevû wÎfyR AvuK| 3| GKwU AvqZ ev AvqZvKvi PZzfy©R AvuK| 4| GKwU eM© ev eM©vKvi PZzfy©R AvuK| 5| cv‡ki Qwe‡Z KqwU wÎfyR i‡q‡Q|