cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 1 2025 gƒj eBGqi AwZwiÚ Ask cÉ^g AaÅvq: Õ¼vfvweK msLÅv I f™²vsk Abykxjbx-1.1 Abykxjbxi cÉGk²i cvkvcvwk ‰ AsGk ˆ`Iqv cÉk²àGjv meÆvwaK ài‚Z½ w`Gq cÉÅvKwUm KGiv| ZvnGj ‰ AaÅvGqi mKj wkLbdGji Ici cÉÕ§wZ mÁ·®² nGe| cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb cÉk² 1 ‘bq jÞ mvZ nvRvi ‰Kk cuPvwk’ K^vq ˆjLv ‰KwU msLÅv| K. Õ¼vfvweK msLÅv KvGK eGj? 2 L. K^vq ˆjLv msLÅvwUGK AGâ wjGL wecixZfvGe mvRvGj ˆh msLÅv cvIqv hvq Zv Av ̄¦RÆvwZK c«¬wZGZ I ˆ`kxq c«¬wZGZ K^vq ˆjL| 4 M. DóxcGKi msLÅvwUGK wecixZfvGe mvRvGj ˆh msLÅv cvIqv hvq ˆmB msLÅvwUi mv^ÆK AâàGjvi Õ©vbxq gvb wbYÆq Ki| 4 1 bs cÉGk²i mgvavb K kƒbÅ eÅZxZ cƒYÆ msLÅvàGjvGK Õ¼vfvweK msLÅv eGj| 1, 2, 3, 4, 5, 6, 7, 8, 9 msLÅvàGjv nGjv Õ¼vfvweK msLÅv| L AGâ wjGL msLÅvwU nGe 9,07,185| msLÅvwUi AâàGjvGK wecixZfvGe mvRvGj cvIqv hvq 581709| msLÅvwUGK Wvb w`K ˆ^GK wZb Aâ cici Kgv ewmGq cvB, 581,709 myZivs, Av ̄¦RÆvwZK c«¬wZGZ K^vq cÉKvk KiGj nq: cuvPk ‰Kvwk nvRvi mvZk bq| (Dîi) Avevi, msLÅvwUGZ ˆ`kxq c«¬wZGZ Kgv ewmGq cvB, 5,81,709 ‰Lb jGÞi NGi ‰KwU Aâ 5, AhyZ I nvRvGii NGi `yBwU Aâ wgwjGq 81, kZGKi NGi 7, `kGKi NGi 0 ‰es ‰KGKi NGi 9| myZivs, msLÅvwUGK K^vq cÉKvk KiGj nq : cuvP jÞ ‰Kvwk nvRvi mvZk bq| (Dîi) M ‘L’ ‰ cÉvμ¦ AGâ ˆjLv msLÅvwU nGjv 5,81,709 msLÅvwUGZ mv^ÆK AâàGjv nGjv 9, 7, 1, 8 ‰es 5| 9 ‰i Õ©vbxq gvb 9 ‰KK ev 9 1 = 9 ev bq 7 ‰i Õ©vbxq gvb 7 kZK ev 7 100 = 700 ev mvZk 1 ‰i Õ©vbxq gvb 1 nvRvi ev 1 1000 = 1000 ev ‰K nvRvi 8 ‰i Õ©vbxq gvb 8 AhyZ ev 8 10000 = 80000 ev Avwk nvRvi 5 ‰i Õ©vbxq gvb 5 jÞ ev 5 100000 = 500000 ev cuvP jÞ 5, 81, 709, msLÅvwUGZ mv^ÆK AâàGjv 5, 8, 1, 7, 9 ‰i Õ©vbxq gvb h^vKÌGg cuvP jÞ, Avwk nvRvi, ‰K nvRvi, mvZk ‰es bq| (Dîi) cÉk² 2 5609070 ‰KwU msLÅv| K. msLÅvwUGK ˆ`kxq ixwZGZ Kgv ewmGq wjL| 2 L. cÉ`î msLÅvwUi mv^ÆK AâàGjvi Õ©vbxq gvb wbYÆq Ki| 4 M. msLÅvwUi AâàGjv ‰Kevi eÅenvi KGi mvZ Aâ wewkÓ¡ e†nîg msLÅv MVb Ki| 4 2 bs cÉGk²i mgvavb K 56,09,070 L 5 6 0 9 0 7 0 M msLÅvwUi AâàGjv nGjv 5, 6, 0, 9, 0, 7, 0 e†nîg msLÅvwU 9765000 cÉk² 3 `yBk cuqZvwÍÏk ˆKvwU cçvk jÞ cqwòk nvRvi Qqk cçvk ‰KwU msLÅv| K. msLÅvwUGK AGâ wjL| 2 L. cÉ`î msLÅvwUGK Av ̄¦RÆvwZK c«¬wZGZ K^vq wjL| 4 M. cÉ`î msLÅvwUi meàGjv Aâ ‰Kevi eÅenvi KGi Þz`ËZg msLÅv MVb Ki| 4 3 bs cÉGk²i mgvavb K 2,45,50,35,650 L Av ̄¦RÆvwZK c«¬wZGZ Kgv ewmGq cvB, 2,455,035,650 K^vq = `yB wewjqb Pvik cçv®² wgwjqb cuqwòk nvRvi QqkZ cçvk| M cÉ`î msLÅvwUi meàGjv Aâ nGjv 2, 4, 5, 5, 0, 3, 5, 6, 5, 0 0 cÉ^Gg ^vKGj meàGjv Aâ wbGq msLÅv MwVZ nGZ cvGi bv| Þz`ËZg msLÅvwU 2003455556 7 `kK = 70 9 nvRvi = 9000 6 jÞ 5 wbhyZ 56 jÞ 600000 5000000 2 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ cÉk² 4 wkÞK ˆevGWÆ 8 AâwewkÓ¡ e†nîg I Þz`ËZg msLÅv wjLGjb| K. 4 wewjqb = KZ ˆKvwU? 2 L. 8 Aâ wewkÓ¡ Þz`ËZg msLÅv = KZ wgwjqb? 4 M. 9 AâwewkÓ¡ Þz`ËZg msLÅv ‰es 8 AâwewkÓ¡ e†nîg msLÅvi cv^ÆKÅ KZ? 4 4 bs cÉGk²i mgvavb K Avgiv Rvwb, 1 wewjqb = 100 ˆKvwU 4 ,, = (100 4) ˆKvwU = 400 ˆKvwU L 8 AâwewkÓ¡ Þz`ËZg msLÅvwU nGjv : 10000000 ˆhGnZz wgwjqb nGjv Av ̄¦RÆvwZK c«¬wZ Kgv eÅenvi KGi cvB, 10,000,000 8 AâwewkÓ¡ Þz`ËZg msLÅv = 10 wgwjqb M 9 AâwewkÓ¡ Þz`ËZg msLÅv = 100000000 ‰es 8 AâwewkÓ¡ e†nîg msLÅv = 99999999 cv^ÆKÅ = (100000000 99999999) = 1 cÉk² 5 9 AâwewkÓ¡ ‰KwU e†nîg I ‰KwU Þz`ËZg msLÅv hvG`i cÉ^Gg 2 ‰es ˆkGl 1 AvGQ| K. ˆ`kxq msLÅv cVb ixwZGZ cçg, lÓ¤, mμ¦g I AÓ¡g Õ©vbGK Kx ejv nq? 2 L. Þz`ËZg msLÅvwU wbYÆq Ki| 4 M. e†nîg msLÅvwUi AâàGjvGK wecixZKÌGg mvRvGj ˆh msLÅv nq Zv K^vq cÉKvk Ki| 4 5 bs cÉGk²i mgvavb K cçg Õ©vb = AhyZ lÓ¤ Õ©vb = jÞ mμ¦g Õ©vb = wbhyZ AÓ¡g Õ©vb = ˆKvwU L 9 AâwewkÓ¡ ‰KwU Þz`ËZg msLÅv hvi cÉ^Gg 2 ‰es ˆkGl 1 AvGQ| Þz`ËZg msLÅvwU 200000001 M 9 AâwewkÓ¡ ‰KwU e†nîg msLÅv hvi cÉ^Gg 2 ‰es ˆkGl 1 AvGQ| e†nîg msLÅvwU 299999991 e†nîg msLÅvwUi AâàGjvGK wecixZKÌGg mvRvGj ˆh msLÅv nq Zv nGjv 19,99,99,992 K^vq wjLGj nGe@ Ewbk ˆKvwU wbivbe»B jÞ wbivbe»B nvRvi bqk weivbe»B| cÉk²eÅvsK ‰LvGb gvÕ¡vi ˆUÇBbvi cÉYxZ Dîimn AviI wKQy iPbvgƒjK I mswÞμ¦-Dîi cÉk² ˆ`Iqv nGqGQ| ‰ cÉk²àGjv ˆZvgvGK AwaK AbykxjGb mnvqZv KiGe| cÉk²àGjvi mgvavb Zzwg wbGR wbGRB KiGe| cÉk² 6 6, 3, 8, 5, 0, 2 AâàGjv jÞ Ki| A K. ‰K wgwjqGb KZ jÞ? 2 L. DóxcGKi AâàGjv gvò ‰Kevi eÅenvi KGi Qq AGâi e†nîg I Þz`ËZg msLÅv MVb Ki| 4 M. e†nîg I Þz`ËZg msLÅvwUGK Av ̄¦RÆvwZK c«¬wZGZ K^vq ˆjL| 4 Dîi: K. `k jÞ; L. 8,65,320 ‰es 2,03,568; M. AvUk cuqlwë nvRvi wZbk wek ‰es `yBk wZb nvRvi cuvPk AvUlwë| cÉk² 7 ‘`yB ˆKvwU ‰Kvwk jÞ wòk nvRvi bqk AvU’ K^vq ˆjLv ‰KwU msLÅv| A K. msLÅvwUGK AGâ ˆjL| 2 L. msLÅvwUi AâàGjv gvò ‰Kevi eÅenvi KGi Qq AGâi e†nîg I Þz`ËZg msLÅv MVb Ki| 4 M. e†nîg msLÅvwUi ˆhmKj mv^ÆK Aâ AvGQ ZvG`i Õ©vbxq gvb wbYÆq Ki| 4 Dîi: K. 2, 81, 30, 908; L. 9, 83, 210; 1, 02, 389; M. bq jÞ, Avwk nvRvi, wZb nvRvi, `yBk ‰es `k| cÉk² 8 ‘AvU wewjqb wZbk PwÍÏk wgwjqb Pvik ewòk nvRvi cuvP’ Av ̄¦RÆvwZK c«¬wZGZ K^vq ˆjLv ‰KwU msLÅv| B K. 1 wewjqGb KZ ˆKvwU? 2 L. ˆ`kxq ixwZGZ msLÅvwUGK K^vq ˆjL| 4 M. msLÅvwUGK wecixZKÌGg mvwRGq Av ̄¦RÆvwZK c«¬wZGZ K^vq ˆjL| 4 Dîi: K. 100 ˆKvwU; L. AvUk ˆPŒwòk ˆKvwU Pvi jÞ ewòk nvRvi cuvP; M. cuvP wewjqb `yB wgwjqb wZbk PwÍÏk nvRvi Pvik AvUwòk| cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 3 Abykxjbx-1.2 Abykxjbxi cÉGk²i cvkvcvwk ‰ AsGk ˆ`Iqv cÉk²àGjv meÆvwaK ài‚Z½ w`Gq cÉÅvKwUm KGiv| ZvnGj ‰ AaÅvGqi mKj wkLbdGji Ici cÉÕ§wZ mÁ·®² nGe| cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb cÉk² 1 0, 5, 9, 7, 2, 6, 3, 4, 1 A K. mv^ÆK AâàGjv ‰Kevi eÅenvi KGi mvZ AGâi e†nîg I Þz`ËZg msLÅv wjL hvi cÉ^Gg 7 ‰es ˆkGl 4 AvGQ| 2 L. weGRvo ˆgŒwjK msLÅvàGjv «¼viv MwVZ e†nîg msLÅv 3 I 5 «¼viv wefvRÅZv cixÞv KGiv| 4 M. mv^ÆK AâàGjv eÅenvi KGi e†nîg I Þz`ËZg msLÅvi ˆhvMdj wbYÆq KGiv| ˆhvMdjwU wK 9 «¼viv wefvRÅ? 4 1 bs cÉGk²i mgvavb K cÉ`î mv^ÆK AâàGjv w`Gq MwVZ cÉ^Gg 7 I ˆkGl 4 ^vKGe ‰gb e†nîg msLÅvwU 7965324| (Dîi) Avevi, cÉ`î mv^ÆK AâàGjv w`Gq MwVZ cÉ^Gg 7 I ˆkGl 4 ^vKGe ‰gb Þz`ËZg msLÅvwU 7123564| (Dîi) L cÉ`î msLÅvàGjvi gGaÅ weGRvo ˆgŒwjK msLÅvàGjv nGjv : 3, 5, 7 msLÅvàGjv «¼viv MwVZ e†nîg msLÅvwU 753 753 msLÅvwUi AâàGjv ˆhvMdj = 7 + 5 + 3 = 15; hv 3 «¼viv wefvRÅ 753 msLÅvwU 3 «¼viv wefvRÅ| (Dîi) Avevi, 753 msLÅvwUi Aâ Õ©vbxq Aâ 3| 753 msLÅvwU 5 «¼viv wefvRÅ bq| (Dîi) M cÉ`î AâàGjv eÅenvi KGi MwVZ e†nîg msLÅvwU 97654321| ‰es Þz`ËZg msLÅvwU 12345679 e†nîg I Þz`ËZg msLÅvi ˆhvMdj = 97654321 + 12345679 = 110000000 msLÅvwUi AâàGjvi ˆhvMdj = 1 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 0 = 2; hv 9 «¼viv wefvRÅ bq| 110000000 msLÅvwU 9 «¼viv wefvRÅ bq| (Dîi) cÉk² 2 6, 0, 4, 2, 8 KGqKwU Aâ ˆ`qv nj| K. 3 ˆ^GK 30 chÆ ̄¦ ˆgŒwjK msLÅvàGjv wjL| 2 L. Aâàwj «¼viv MwVZ e†nîg msLÅv 4 I 6 «¼viv wefvRÅ wKbv wbYÆq Ki| 4 M. DóxcGK cÉ`î AâàGjvi 1g wZbwU Aâ wbGq MwVZ `yB AGâi mv^ÆK msLÅvàGjvi ˆKvbàGjv ˆRvov mnGgŒwjK wbYÆq Ki| 4 2 bs cÉGk²i mgvavb K 3 ˆ^GK 30 chÆ ̄¦ ˆgŒwjK msLÅvàGjv nGjv : 3, 5, 7, 11, 13, 17, 19, 23, 29 L cÉ`î AâàGjv nGjv : 6, 0, 4, 2, 8 e†nîg msLÅv = 86420 Avgiv Rvwb, ˆKvGbv msLÅvi ‰KK I `kK Õ©vGbi Aâ `yBwU «¼viv MwVZ msLÅv 4 «¼viv wefvRÅ nGj, H msLÅvwU 4 «¼viv wefvRÅ nGe| ‰LvGb, 20 4 = 5 e†nîg msLÅvwU 4 «¼viv wefvRÅ| Avevi, ˆKvGbv msLÅv 2 ‰es 3 «¼viv wefvRÅ nGj msLÅvwU 6 «¼vivI wefvRÅ nGe| ˆhGnZz cÉ`î msLÅvwU 2 «¼viv wefvRÅ wK ̄§ 3 «¼viv wefvRÅ bq| A^Ævr e†nîg msLÅvwU 6 «¼viv wefvRÅ bq| M cÉ`î DóxcGKi 1g wZbwU Aâ 6, 0, 4 cÉ^g wZbwU Aâ wbGq MwVZ `yB AGâi mv^ÆK msLÅvàGjv nGjv : 64, 46, 60, 40 64 ‰i àYbxqKàGjv nGjv : 1, 2, 4, 8, 16, 32, 64 46 ‰i àYYxqKàGjv nGjv : 1, 2, 23, 46 60 ‰i àYbxqKàGjv nGjv : 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 40 ‰i àYbxqKàGjv nGjv : 1, 2, 4, 5, 8, 10, 20, 40 cÉ`î msLÅvàGjvi gGaÅ ‰KvwaK mvaviY àYbxqK AvGQ| ˆRvov mnGgŒwjK ˆKvbwUB bq| cÉk² 3 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 `kwU msLÅv| K. mnGgŒwjK msLÅv KvGK eGj? 2 L. ˆgŒwjK msLÅv eÅZxZ weGRvo msLÅvàGjv ˆjL ‰es ‰G`i ˆhvMdj 3 «¼viv wefvRÅ wK bv Zv wbaÆviY Ki? 4 M. DóxcGKi weGRvo ˆhŒwMK msLÅvàGjvi gGaÅ ˆKvb msLÅv ˆRvov mnGgŒwjK wbYÆq Ki| 4 3 bs cÉGk²i mgvavb K `yB ev ZGZvwaK msLÅvi mvaviY àYbxqK ÷ay 1 nGj msLÅvàGjv ciÕ·i mnGgŒwjK| ˆhgb, 2 ‰es 13 ciÕ·i mnGgŒwjK| L weGRvo msLÅvàGjv nGjv : 31, 33, 35, 37, 39 ˆgŒwjK msLÅv eÅZxZ weGRvo msLÅvàGjv nj: 33, 35, 39 msLÅvàGjvi ˆhvMdj = 33 + 35 + 39 = 107 Avgiv Rvwb, ˆKvGbv msLÅvi AâàGjvi ˆhvMdj 3 «¼viv wefvRÅ nGj, H msLÅvwU 3 «¼viv wefvRÅ nGe| ‰LvGb, 1 + 0 + 7 = 8 hv 3 «¼viv wefvRÅ bq| ˆhvMdj 3 «¼viv wefvRÅ bq| M weGRvo msLÅvàGjv nGjv : 31, 33, 35, 37, 39 weGRvo ˆhŒwMK msLÅvàGjv nGjv : 33, 35, 39 33 ‰i àYbxqKàGjv nGjv : 1, 3, 11 35 ‰i àYbxqKàGjv nGjv : 1, 5, 7, 35 4 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ 39 ‰i àYbxqKàGjv nGjv : 1, 3, 13, 39 33 ‰es 35 ‰i gGaÅ 1 Qvov AbÅ ˆKvGbv mvaviY àYbxqK ˆbB| 33 ‰es 35 ciÕ·i mnGgŒwjK| Avevi, 35 ‰es 39 ‰i gGaÅ 1 Qvov AbÅ ˆKvGbv mvaviY àYbxqK ˆbB| 35 ‰es 39 ciÕ·i mnGgŒwjK| cÉk² 4 wg÷, wgbvGK ‰K AGâi QqwU msLÅv wjLGZ ejvq ˆm 2, 0, 8, 7, 3 I 4 wjLGjv| K. Aâ cvZb ejGZ Kx ˆevSvq? msLÅv cÉZxKàGjvi gGaÅ mv^ÆK Aâ KqwU? 2 L. ˆ`LvI ˆh, wgbvi wjLv AâàGjv «¼viv MwVZ e†nîg I Þz`ËZg msLÅvi weGqvMdj 9 «¼viv wefvRÅ| 4 M. 475 2- ‰ wPwn×Z Õ©vGb ˆKvb PviwU AGâi ˆh ˆKvGbvwU emvGj cÉwZGÞGò MwVZ msLÅv 3 «¼viv wefvRÅ nGe? 4 4 bs cÉGk²i mgvavb K ˆKvGbv msLÅv Aâ «¼viv ˆjLvGK AâcvZb eGj| msLÅv cÉZxKàGjv nGjv : 0, 1, 2, 3, 4, 5, 6, 7, 8 ‰es 9 mv^ÆK Aâ 9wU| L wgbvi wjLv AâàGjv w`Gq MwVZ e†nîg msLÅv 874320 ‰es Þz`ËZg msLÅv 203478 weGqvMdj = (874320 203478) = 670842 Avevi, ˆKvGbv msLÅvi AâàGjvi ˆhvMdj 9 «¼viv wefvRÅ nGj, cÉ`î msLÅvwU 9 «¼viv wefvRÅ nGe| ‰LvGb, 6 + 7 + 0 + 8 + 4 + 2 = 27 hv 9 «¼viv wefvRÅ| weGqvMdj 9 «¼viv wefvRÅ| (ˆ`LvGbv nGjv) M Avgiv Rvwb, ˆKvGbv msLÅvi AâàGjvi ˆhvMdj 3 «¼viv wefvRÅ nGj, H msLÅvwU 3 «¼viv wefvRÅ nGe| 0 ‰i RbÅ 4 + 7 + 5 + 0 + 2 = 18 hv 3 «¼viv wefvRÅ 1 ‰i RbÅ 4 + 7 + 5 + 1 + 2 = 19 hv 3 «¼viv wefvRÅ bq 2 ‰i RbÅ 4 + 7 + 5 + 2 + 2 = 20 hv 3 «¼viv wefvRÅ bq 3 ‰i RbÅ 4 + 7 + 5 + 3 + 2 = 21 hv 3 «¼viv wefvRÅ 4 ‰i RbÅ 4 + 7 + 5 + 4 + 2 = 22 hv 3 «¼viv wefvRÅ bq 5 ‰i RbÅ 4 + 7 + 5 + 5 + 2 = 23 hv 3 «¼viv wefvRÅ bq 6 ‰i RbÅ 4 + 7 + 5 + 6 + 2 = 24 hv 3 «¼viv wefvRÅ 7 ‰i RbÅ 4 + 7 + 5 + 7 + 2 = 25 hv 3 «¼viv wefvRÅ bq 8 ‰i RbÅ 4 + 7 + 5 + 8 + 2 = 26 hv 3 «¼viv wefvRÅ bq 9 ‰i RbÅ 4 + 7 + 5 + 9 + 2 = 27 hv 3 «¼viv wefvRÅ wPwn×Z Õ©vGb emGe, 0, 3, 6, 9 cÉk²eÅvsK ‰LvGb gvÕ¡vi ˆUÇBbvi cÉYxZ Dîimn AviI wKQy iPbvgƒjK I mswÞμ¦-Dîi cÉk² ˆ`Iqv nGqGQ| ‰ cÉk²àGjv ˆZvgvGK AwaK AbykxjGb mnvqZv KiGe| cÉk²àGjvi mgvavb Zzwg wbGR wbGRB KiGe| cÉk² 6 1 ˆ^GK 50 ‰i gGaÅ wKQy ˆgŒwjK I ˆhŒwMK msLÅv we`Ågvb| A K. cÉ^g 6wU ˆgŒwjK msLÅv ˆjL| 2 L. ˆhŒwMK msLÅv KqwU ‰es ˆhàGjv 5 «¼viv wefvRÅ ˆmàGjv ˆjL| 4 M. DóxcK nGZ cÉvμ¦ ˆgŒwjK msLÅvàGjvi mgwÓ¡ 9 «¼viv wefvRÅ wKbv hvPvB Ki| 4 Dîi: K. 2, 3, 5, 7, 11, 13; L. 34wU; 10, 15, 20, 25, 30, 35, 40, 45, 50; M. 9 «¼viv wefvRÅ bq| cÉk² 7 3, 0, 5, 2, 7 KGqKwU AsK ˆ`Iqv nGjv| K. 3 ˆ^GK 20 chÆ ̄¦ ˆgŒwjK msLÅvàGjv wjL| 2 L. Aâàwj «¼viv MwVZ e†nîg msLÅv 4 ‰es 6 w`Gq wefvRÅ wKbv Zv wbYÆq Ki| 4 M. e†nîg msLÅvwUi mv^ÆK AâàGjvi cÉGZÅKwUi Õ©vbxq gvb wbYÆq KGiv| 4 Dîi: K. 3, 5, 7, 11, 13, 17, 19; L. 4 «¼viv wefvRÅ ‰es 6 «¼viv wefvRÅ bq; M. wek, wZbkZ, cuvP nvRvi, mîi nvRvi cÉk² 8 3052 ‰KwU msLÅv| B K. msLÅvwUi Õ©vGb KZ emvGj msLÅvwU 9 «¼viv wefvRÅ nGe? 2 L. msLÅvwUi AâàGjv eÅenvi KGi cuvP AGâi e†nîg I Þz`ËZg msLÅv MVb Ki| 4 M. e†nîg I Þz`ËZg msLÅvwU 4 ‰es 5 «¼viv wefvRÅ wK bv Zv wbYÆq Ki| 4 Dîi: K. -‰i Õ©vGb 8 emvGj msLÅvwU 9 «¼viv wefvRÅ nGe; L. e†nîg msLÅv 85320 ‰es Þz`ËZg msLÅv 20,358; M. e†nîg msLÅvwU 4 I 5 «¼viv wefvRÅ| Þz`ËZg msLÅvwU 4 I 5 «¼viv wefvRÅ bq| cÉk² 9 31, 32, 33, 34, 35, 36, 37, 38, 39, 40 `kwU msLÅv| A K. mnGgŒwjK msLÅv KvGK eGj? 2 L. ˆgŒwjK msLÅv eÅZxZ weGRvo msLÅvàGjv ˆjL ‰es ‰G`i ˆhvMdj 3 «¼viv wefvRÅ wK bv Zv wbaÆviY Ki| 4 M. ‘L’ nGZ cÉvμ¦ msLÅvàGjvi gGaÅ ˆKvb msLÅv ˆRvov mnGgŒwjK wbYÆq Ki| 4 Dîi: L. 3 «¼viv wefvRÅ bq; M. 33 I 35 msLÅv«¼q ciÕ·i mnGgŒwjK; 35 I 39 msLÅv«¼q ciÕ·i mnGgŒwjK| cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 5 Abykxjbx-1.3 Abykxjbxi cÉGk²i cvkvcvwk ‰ AsGk ˆ`Iqv cÉk²àGjv meÆvwaK ài‚Z½ w`Gq cÉÅvKwUm KGiv| ZvnGj ‰ AaÅvGqi mKj wkLbdGji Ici cÉÕ§wZ mÁ·®² nGe| cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb cÉk² 1 PviwU N¥Ÿv cÉ^Gg ‰KGò ˆeGR cGi h^ vKÌGg 16, 24, 30 I 36 ˆmGK ́£ A ̄¦i A ̄¦i evRGZ jvMj| B K. w«¼Zxq I PZz^Æ ˆmGK ́£«¼Gqi mvaviY àYbxqKàGjv ˆjL|2 L. KZÞY cGi N¥ŸvàGjv cybivq ‰KGò evRGe? 4 M. Qq AGâi ˆKvb Þz`ËZg msLÅvi mvG^ 5 ˆhvM KiGj ˆhvMdj ˆmGK ́£àGjv «¼viv wb:GkGl wefvRÅ nGe? 4 1 bs cÉGk²i mgvavb K ‰LvGb, 24 = 1 24 36 = 1 36 = 2 12 = 2 18 = 3 8 = 3 12 = 4 6 = 4 9 = 6 6 24 ‰i àYbxqKàGjv 1, 2, 3, 4, 6, 8, 12, 24 36 ‰i àYbxqKàGjv 1, 2, 3, 4, 6, 9, 12, 18, 36 24 I 36 ‰i mvaviY àYbxqKàGjv 1, 2, 3, 4, 6, 12 (Dîi) L ‰LvGb, 16, 24, 30 I 36 ‰i j.mv.à. nGe N¥ŸvàGjv cybivq ‰KGò evRvi mgq| (ˆmGKG ́£) 2 , 16, 24, 30, 36 2 , 8, 12, 15, 18 2 , 4, 6, 15, 9 3 , 2, 3, 15, 9 , 2, 1, 5, 3 wbGYÆq j.mv.à. = 2 2 2 2 3 3 5 = 720 720 ˆmGK ́£ ev 12 wgwbU ci N¥ŸvàGjv cybivq ‰KGò evRGe| (Dîi) M Qq AGâi Þz`ËZg msLÅv 100000 ZvwjKvi msLÅvàGjvi j.mv.à w`Gq fvM KGi msLÅvwU wbYÆq KiGZ nGe| ‘L’ nGZ cvB, j.mv.à = 720 ‰Lb, 720 ) 100000 ( 138 720 2800 2160 6400 5760 640 100000 msLÅvwU 720 «¼viv wefvRÅ bq| AZ‰e wefvRÅ msLÅvwU nGe (100000 640) = 99360 wK ̄§ ‰wU cuvP AGâi msLÅv| Qq AGâi Þz`ËZg msLÅvwU = 100000 + (720 640) = 100000 + 80 = 100080 myZivs Qq AGâi ˆh Þz`ËZg msLÅvi mvG^ 5 ˆhvM KiGj ˆhvMdj 16, 24, 30 I 36 «¼viv wb:Gkl wefvRÅ nGe Zv nGjv (100080 5) ev, 100075 (Dîi) cÉk² 2 `ywU msLÅvi àYdj 8670 I M.mv.à. 17| K. cÉ`î M.mv.à. ‰es 18-‰i ˆgŒwjK àYbxqKàGjv wbYÆq Ki| 2 L. msLÅv `ywUi j.mv.à. KZ nGe wbYÆq Ki| 4 M. ‰KwU msLÅv j.mv.à. ‰i 1 3 àY nGj Aci msLÅvwU KZ? 4 2 bs cÉGk²i mgvavb K cÉ`î M.mv.à = 17 17 ‰i ˆgŒwjK àYbxqK = 17 18 ‰i ˆgŒwjK àYbxqK = 2, 3 L Avgiv Rvwb, M.mv.à j.mv.à = msLÅv `ywUi àYdj ev, 17 j.mv.à = 8670 j.mv.à = 8670 17 = 510 17 ‰es 510 ‰i j.mv.à wbYÆq KiGZ nGe| 17 17, 510 1, 30 j.mv.à = 17 30 = 510 M ‘L’ ˆ^GK cvB, j.mv.à = 510 ‰KwU msLÅv = j.mv.à ‰i 1 3 = 510 ‰i 1 3 = 170 Avgiv Rvwb, `yBwU msLÅvi àYdj msLÅv `yBwUi M.mv.à I j.mvà ‰i àYdGji mgvb| j.mv.à M.mv.à = `ywU msLÅvi àYdj ev, ‰KwU msLÅv Aci msLÅv = 8670 ev, 170 Aci msLÅv = 8670 ev, Aci msLÅv = 8670 170 Aci msLÅv = 51 cÉk² 3 wbGPi wZbwU msLÅvi `j ˆ`Iqv nGjv: A (16, 24, 30, 36), B (189, 210), C (109, 144) K. B `Gji msLÅv«¼q mnGgŒwjK wKbv wbYÆq Ki| 2 L. C `Gji 1g msLÅvGK ˆKvb ˆKvb msLÅv «¼viv fvM KiGj cÉwZGÞGò 4 AewkÓ¡ ^vKGe? 4 M. cuvP AGâi ˆKvb e†nîg msLÅvGK A `Gji msLÅvàGjv «¼viv fvM KiGj cÉwZevi fvMGkl 12 ^vKGe? 4 6 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ 3 bs cÉGk²i mgvavb K B `jwU nGjv B (189, 210) 189 ‰i àYbxqKàGjv nGjv : 1, 3, 7, 9, 21, 27, 63, 189 210 ‰i àYbxqKàGjv nGjv : 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210 ˆhGnZz msLÅv«¼Gqi mvaviY àYbxqK ‰KvwaK B (189, 210) mnGgŒwjK bq| L C `Gji 1g msLÅvwU nGjv 109 109 ˆ^GK 4 weGqvM KiGj cvB, 109 4 = 105 105 ‰i àYbxqKàGjv nGjv 1, 3, 5, 7, 15, 21, 35, 105 ˆhGnZz 4 msLÅvwU 1 I 3 ˆ^GK eo| wbGYÆq msLÅvàGjv nGjv : 5, 7, 15, 21, 35 I 105 M cuvP AGâi e†nîg msLÅv = 99999 A `Gji msLÅvàGjv 16, 24, 30, 36 cÉ`î fvRK 16, 24, 30, 36 ‰i j.mv.à. wbYÆq Kwi: 2 16, 24, 30, 36 2 8, 12, 15, 18 2 4, 6, 15, 9 3 2, 3, 15, 9 2, 1, 5, 3 wbGYÆq j.mv.à. = 2 2 2 3 2 5 3 = 720 Avgiv Rvwb, cuvP AGâi e†nîg msLÅv = 99999 720 ) 99999 ( 138 720 2799 2160 6399 5760 639 DcGivÚ fvM cÉwKÌqvq ˆ`Lv hvq ˆh, 99999 msLÅvwU 720 «¼viv wefvRÅ bq| 720 w`Gq fvM KiGj 639 AewkÓ¡ ^vGK| fvRÅ 99999 ˆ^GK 639 Kg nGj cÉvμ¦ msLÅvwU 720 «¼viv wefvRÅ nGe| Avevi, fvRK (720 @ 639) = 81 ˆewk nGjI 720 «¼viv wefvRÅ nGe| wK ̄§ (99999 + 81) = 100080 msLÅvwU 6 Aâ wewkÓ¡| 720 «¼viv wb:GkGl wefvRÅ 5 Aâ wewkÓ¡ e†nîg msLÅv = (99999 – 639) = 99360 wK ̄§ cÉk²vbymvGi, fvMGkl 12 ^vKGZ nGe| wbGYÆq cuvP AGâi e†nîg msLÅvwU = 99360 + 12 = 99372 (Dîi) cÉk² 4 wZbwU MÉn 360 w`Gb, 660 w`Gb I 1650 w`Gb ‰KwU bÞòGK cÉ`wÞY KGi| K. `ywU msLÅvi àYdj Zvi j.mv.à. ‰i PviàY| msLÅv `ywUi M.mv.à. wbYÆq Ki| 2 L. DóxcGKi MÉnàGjv ‰KB ˆiLv eivei AeÕ©vb KGi NƒYÆb ÷i‚ Kivi KZw`b ci cybivq ‰KB ˆiLv eivei AeÕ©vb KiGe? 4 M. 120 eQGi MÉnàGjv KZevi ‰KB ˆiLv eivei AeÕ©vb KiGe? 4 4 bs cÉGk²i mgvavb K gGb Kwi, `ywU msLÅvi j.mv.à = K `ywU msLÅvi àYdj = K 4 ev, `ywU msLÅvi àYdj = 4K Avevi, M.mv.à j.mv.à = `ywU msLÅvi àYdj ev, M.mv.à K = 4K M.mv.à = 4 L ‰LvGb, 360, 660, 1650 ‰i j.mv.à nGe MÉnàGjvi cybivq ‰KB ˆiLv eivei AeÕ©vb Kivi mgq| 2 360, 660, 1650 2 180, 330, 825 3 90, 165, 825 5 30, 55, 275 11 6, 11, 55 6, 1, 5 j.mv.à = 2 2 3 5 11 6 5 = 19800 cybivq 19800 w`b ci ‰KB ˆiLv eivei AeÕ©vb KiGe| M ‘L’ ˆ^GK cvB, cybivq ‰KB ˆiLv eivei AeÕ©vb KiGe 19800 w`b ci 1 eQi = 365 w`b 120 eQi = 365 120 = 43800 w`b 19800 w`Gb wgwjZ nq 1 evi 1 ,, ,, ,, 1 19800 evi 43800 ,, ,, ,, 43800 19800 evi = 2.12 evi A^Ævr 3 evi| cÉk² 5 159wU Avg, 227wU Rvg I 401wU wjPz KGqKRb evjGKi gGaÅ fvM KGi ˆ`qv nj| ‰GZ 3wU Avg, 6wU Rvg I 11wU wjPz AewkÓ¡ iBj| K. ˆKvb cÉKvGii KqwU dj fvM KGi ˆ`Iqv nGjv? 2 L. meGPGq ˆewk KZRb evjGKi gGaÅ, djàGjv mgvbfvGe fvM KGi ˆ`Iqv hvGe? 4 M. cÉGZÅK evjK KqwU KGi cÉGZÅK cÉKvGii dj cvGe? 4 cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 7 5 bs cÉGk²i mgvavb K Avg ˆ`qv nGjv (159 3) = 156wU Rvg ˆ`qv nGjv (227 6) = 221wU wjPz ˆ`qv nGjv (401 11) = 390wU L meGPGq ˆewk evjGKi msLÅvi cwigvY nGjv 156, 221, 390 ‰i M.mv.à ‰i mgvb| 156) 221 (1 156 65) 156 (2 130 26) 65 (2 52 13) 26 (2 26 0 13) 390 (30 39 0 0 0 13 Rb evjGKi gGaÅ djàGjv mgvbfvGe fvM KGi ˆ`qv hvGe| M cÉGZÅK evjK Avg cvGe (156 13)wU = 12wU cÉGZÅK evjK Rvg cvGe (221 13)wU = 17wU cÉGZÅK evjK wjPz cvGe (390 13)wU = 30wU cÉk²eÅvsK ‰LvGb gvÕ¡vi ˆUÇBbvi cÉYxZ Dîimn AviI wKQy iPbvgƒjK I mswÞμ¦-Dîi cÉk² ˆ`Iqv nGqGQ| ‰ cÉk²àGjv ˆZvgvGK AwaK AbykxjGb mnvqZv KiGe| cÉk²àGjvi mgvavb Zzwg wbGR wbGRB KiGe| cÉk² 6 ‰KwU ˆjvnvi cvZ I ‰KwU Zvgvi cvGZi Š`NÆÅ h^vKÌGg 672 ˆm.wg I 960 ˆm.wg.| K. 672 ‰i ˆgŒwjK àYbxqKàGjv ˆjL| 2 L. cvZ `yBwU ˆ^GK ˆKGU ˆbIqv ‰KB gvGci meGPGq eo UzKivi Š`NÆÅ KZ nGe? 4 M. cÉGZÅK cvGZi UzKivi msLÅv wbYÆq Ki| 4 Dîi: K. 2, 2, 2, 2, 2, 3, 7 L. 96 ˆm.wg. M. 7wU ‰es 10 wU cÉk² 7 16, 24, 30, 36 KGqKwU msLÅv| K. 30 I 36 ‰i àYbxqKàGjv ˆjL| 2 L. ˆgŒwjK àYbxqGKi mvnvGhÅ 16 I 24 ‰i j.mv.à wbYÆq Ki ‰es ˆ`LvI ˆh, msLÅv `yBwUi àYdj = BnvG`i j.mv.à M.mv.à| 4 M. cuvP AGâi e†nîg msLÅv wbYÆq Ki hvGK DóxcGKi msLÅvàGjv «¼viv fvM KiGj cÉwZGÞGò 5 AewkÓ¡ ^vGK| 4 Dîi: K. 30 ‰i àYbxqKàGjv 1, 2, 3, 5, 6, 10, 15, 30; 36 ‰i àYbxqKàGjv 1, 2, 3, 4, 6, 9, 12, 18, 36 L. 48; M. 99365 cÉk² 8 8, 12 `yBwU msLÅv| K. 1g msLÅvwUi mvG^ 10 ˆhvM KGi bZyb ‰KwU msLÅv MVb Ki ‰es bZzb msLÅvwUi 3wU àwYZK ˆjL| 2 L. bZyb msLÅvwU ‰es cÉ`î msLÅv `yBwUi fvM cÉwKÌqvq M.mv.à. ˆei Ki| 4 M. ˆKvb Þz`ËZg msLÅvGK ‘L’ ˆ^GK cÉvμ¦ M.mv.à., bZyb msLÅv ‰es cÉ`î msLÅv `yBwU «¼viv fvM KiGj cÉGZÅK eviB fvMGkl 1 ^vGK| 4 Dîi: K. 18, 36, 54; L. 2; M. 73 cÉk² 9 28, 36, 54, 72 ‰es 144 cuvPwU msLÅv| K. 1g msLÅvwUi ˆgŒwjK àYbxqKàGjv ˆjL| 2 L. ˆgŒwjK àYbxqGKi mvnvGhÅ msLÅvàGjvi j.mv.à. wbYÆq Ki| 4 M. ˆ`LvI ˆh, BDwKÑWxq c«¬wZGZ cÉvμ¦ j.mv.à. ˆgŒwjK àYbxqK c«¬wZGZ cÉvμ¦ j.mv.à. ‰i mgvb| 4 Dîi: K. 2, 2, 7; L. 3024 ; M. 3024 8 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ Abykxjbx-1.4 Abykxjbxi cÉGk²i cvkvcvwk ‰ AsGk ˆ`Iqv cÉk²àGjv meÆvwaK ài‚Z½ w`Gq cÉÅvKwUm KGiv| ZvnGj ‰ AaÅvGqi mKj wkLbdGji Ici cÉÕ§wZ mÁ·®² nGe| cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb cÉk² 1 AvGbvqvi mvGne Zuvi Rwg ˆ^ GK eQGi 20 1 10 KzB ́Ÿvj Avgb, 30 1 20 KzB ́Ÿvj Bwi ‰es 10 1 50 KzB ́Ÿvj AvDk avb ˆcGjb| A K. cÉ`î f™²vskàGjvGK AcÉK‡Z f™²vsGk cÉKvk Ki| 2 L. f™²vskàGjvGK gvGbi AatKÌg AbymvGi mvRvI| 4 M. AvGbvqvi mvGne Zuvi Rwg ˆ^GK `yB eQGi KZ KzB ́Ÿvj avb ˆcGqGQb? 4 1 bs cÉGk²i mgvavb K ‰LvGb, f™²vskàGjv AcÉK‡Z f™²vsGk cÉKvk Kiv nGjv: 20 1 10 = 10 20 + 1 10 = 201 10 30 1 20 = 20 30 + 1 20 = 601 20 10 1 50 = 50 10 + 1 50 = 501 50 L ‘K’ ˆ^GK cvB, f™²vskàGjv nGjv 201 10 , 601 20 , 501 50 f™²vskàGjvi ni 10, 20, 50 ‰i j.mv.à. = 100 1g f™²vsk = 201 10 = 201 10 10 10 = 2010 100 [ ã 100 10 = 10] 2q f™²vsk = 601 20 = 601 5 20 5 = 3005 100 [ ã 100 20 = 5] 3q f™²vsk = 501 50 = 501 2 50 2 = 1002 100 [ ã 100 50 = 2] mgni wewkÓ¡ f™²vsk 2010 100 , 3005 100 , 1002 100 ‰i jeàGjvi gGaÅ Zzjbv KGi cvB, 3005 > 2010 > 1002 ev, 3005 100 > 2010 100 > 1002 100 A^Ævr, 601 20 > 201 10 > 501 50 myZivs gvGbi AatKÌg AbymvGi mvwRGq cvB, 30 1 20 > 20 1 10 > 10 1 50 (Dîi) M cÉwZ eQi AvGbvqvi mvGne Rwg ˆ^GK Avgb avb ˆcGqGQb = 20 1 10 KzB ́Ÿvj ev 201 10 KzB ́Ÿvj Bwi ” ” = 30 1 20 ” ev, 601 20 ” AvDk ” ” = 10 1 50 ” ev, 501 50 ” cÉwZ eQi avb ˆcGqGQb = 201 10 + 601 20 + 501 50 KzB ́Ÿvj = 2010 + 3005 + 1002 100 ” = 6017 100 ” 2 eQGi ˆgvU avb ˆcGqGQb = 1 § 2 6017 § 100 50 KzB ́Ÿvj = 6017 50 ” = 12017 50 ” (Dîi) cÉk² 2 4 1 4 , 10 5 14 , 38 11 21 wZbwU mvaviY f™²vsk| K. cÉK‡Z I AcÉK‡Z f™²vsk KvGK eGj? 2 L. cÉ`î f™²vskàwjGK gvGbi Ea»ÆKÌg AbymvGi mvRvI| 4 M. ˆkl `ywU f™²vsGki ˆhvMdGji mGã KZ ˆhvM KiGj msLÅvwU 100 nGe? 4 2 bs cÉGk²i mgvavb K ˆh f™²vsGki ni AGcÞv je ˆQvU ZvGK cÉK‡Z f™²vsk eGj| ˆhgb 3 5 ‰KwU mvaviY f™²vsk| ‰B f™²vsGk je 3 I ni 5| ‰LvGb je, ni ˆ^GK ˆQvU| ‰wU ‰KwU cÉK‡Z f™²vsk| ˆh f™²vsGki jGei gvb nGii gvb AGcÞv eo, ˆmB f™²vskGK AcÉK‡Z f™²vsk eGj| ˆhgb 8 5 mvaviY f™²vsGk je, ni ˆ^GK eo| ‰wU ‰KwU AcÉK‡Z f™²vsk| L cÉ`î f™²vsk 4 1 4 , 10 5 14 , 38 11 21 f™²vskàGjvGK AcÉK‡Z f™²vsGk cÉKvk KGi cvB, 17 4 , 145 14 , 809 21 f™²vsGki ni 4, 14, 21 ‰i j.mv.à = 84 cÉ^g f™²vsk = 17 4 = 17 21 4 21 = 357 84 [ˆhGnZz 84 6 = 21] 2q f™²vsk = 145 14 = 145 6 14 6 = 870 84 [ 84 14 = 6] 3q f™²vsk = 809 21 = 809 4 21 4 = 3236 84 [ 84 21 = 4] cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 9 mgni wewkÓ¡ f™²vsk 357 84 , 870 84 , 3236 84 ‰i jeàGjvi gGaÅ Zzjbv KGi cvB, 357 < 870 < 3236 357 84 < 870 84 < 3236 84 A^Ævr 17 4 < 145 14 < 809 21 gvGbi Da»ÆKÌg AbymvGi mvwRGq cvB, 4 1 4 < 10 5 14 < 38 11 21 (Dîi) M gGb Kwi, ˆkl `yBwU f™²vsGki mGã K ˆhvM KiGj msLÅvwU 100 nGe| cÉk²gGZ, 10 5 14 + 38 11 21 + K = 100 ev, 145 14 + 809 21 + K = 100 ev, 435 + 1618 42 + K = 100 ev, 2053 42 + K = 100 ev, K = 100 2053 42 = 4200 2053 42 = 2147 42 = 51 5 42 ˆkl `yBwU f™²vsGki A^Ævr 10 5 14 I 38 11 21 ‰i ˆhvMdGji mvG^ 51 5 42 ˆhvM KiGj msLÅvwU 100 nGe| cÉk² 3 iwng mvGne Zuvi 250 kZvsk Rwgi 6 16 Ask RwgGZ cvU, 10 24 Ask RwgGZ avb Pvl KiGjb ‰es evwK RwgGZ e†ÞGivcY KiGjb| K. iwng mvGne KZ kZvsk RwgGZ avb Pvl KiGjb? 2 L. 2 kZvsk RwgGZ Drcvw`Z cvGUi gƒjÅ 1200 UvKv nGj, wZwb ˆgvU KZ UvKvi cvU wewKÌ KiGZ cviGeb? 4 M. iwng mvGne KZ kZvsk RwgGZ e†ÞGivcY KiGjb? 4 3 bs cÉGk²i mgvavb K avb Pvl KiGjb 250 kZvsk Rwgi 10 24 Ask = 250 10 24 kZvsk = 625 6 kZvsk (Dîi) L cvGUi Rwgi cwigvY = 250 kZvsk Rwgi 6 16 Ask = 250 6 16 kZvsk = 375 4 kZvsk 2 kZvsk RwgGZ Drcvw`Z cvGUi gƒjÅ = 1200 UvKv 1 ,, ,, ,, ,, ,, = 1200 2 UvKv 375 4 ,, ,, ,, ,, ,, = 1200 2 375 4 UvKv = 56250 UvKv wZwb ˆgvU 56250 UvKvi cvU wewKÌ KiGZ cviGeb| M ‘K’ I ‘L’ nGZ cvB, avGbi Rwgi cwigvY = 625 6 kZvsk cvGUi Rwgi cwigvY = 375 4 kZvsk ˆgvU avb I cvGUi Rwgi cwigvY = 625 6 + 375 4 kZvsk = 1250 + 1125 12 kZvsk = 2375 12 kZvsk ˆgvU Rwgi cwigvY 250 kZvsk iwng mvGne e†ÞGivcY KiGjb Zvi cwigvY = 250 2375 12 kZvsk = 3000 2375 12 kZvsk = 625 12 kZvsk = 52 1 12 kZvsk cÉk² 4 32 wgUvi Š`GNÆÅi ‰KwU wdZvi 83 4 wgUvi mv`v is, 101 8 wgUvi jvj is ‰es evwK Ask KvGjv| K. mv`v I jvj AsGki cwigvYGK AcÉK‡Z f™²vsGk i...cv ̄¦i Ki| 2 L. wdZvwUi jvj ‰es mv`v AsGki ˆgvU cwigvY wbYÆq Ki| 4 M. wdZvq KvGjv AsGki cwigvY ˆei Ki| 4 4 bs cÉGk²i mgvavb K wdZvwUi mv`v AsGki cwigvY = 8 3 4 wgUvi = 35 4 wgUvi (Dîi) wdZvwUi jvj AsGki cwigvY = 10 1 8 wgUvi 10 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ = 81 8 wgUvi (Dîi) L ‘K’ nGZ cvB, wdZvwUi mv`v AsGki cwigvY 35 4 wgUvi ‰es wdZvwUi jvj AsGki cwigvY 81 8 wgUvi wdZvwUi jvj I mv`v AsGki ˆgvU cwigvY = 35 4 + 81 8 wgUvi = 70 + 81 8 wgUvi = 151 8 wgUvi M ˆ`Iqv AvGQ, wdZvwUi Š`NÆÅ = 32 wgUvi ‘L’ nGZ cvB, jvj I mv`v AsGki cwigvY = 151 8 wgUvi KvGjv AsGki cwigvY = 32 151 8 wgUvi = 256 151 8 wgUvi = 105 8 wgUvi = 13 1 8 wgUvi cÉk² 5 Dîiv ˆ^GK gwZwSGji `ƒiZ½ 16 wKGjvwgUvi| ˆKvGbv eÅwÚ 45 8 wKGjvwgUvi c^ wiÝvq , 9 3 20 wKGjvwgUvi c^ evGm ‰es evwK c^ ˆnuGU ˆMGjb| K. AcÉK‡Z f™²vsk KvGK eGj? 2 L. wiÝv ‰es evGm ˆgvU AwZKÌv ̄¦ `ƒiZ½ wbYÆq Ki| 4 M. H eÅwÚ KZ wKGjvwgUvi c^ ˆnuGU ˆMGjb? 4 5 bs cÉGk²i mgvavb K ˆh f™²vsGki jGei gvb nGii gvb AGcÞv eo, ˆmB f™²vskGK AcÉK‡Z f™²vsk eGj| ˆhgb 8 5 , 13 12 BZÅvw`| L wiÝv I evGm ˆgvU AwZKÌv ̄¦ `ƒiZ½ = 4 5 8 + 9 3 20 wKGjvwgUvi = 185 + 366 40 ,, = 551 40 ,, = 13 31 40 wKGjvwgUvi (Dîi) M Dîiv ˆ^GK gwZwSGji `ƒiZ½ 16 wKGjvwgUvi ‘L’ ˆ^GK cvB wiÝv I evGm ˆgvU AwZKÌv ̄¦ `ƒiZ½ = 551 40 wgUvi H eÅwÚ ˆnuGU ˆMGjb = 16 551 40 wKGjvwgUvi = 640 551 40 ,, = 89 40 ,, = 2 9 40 wKGjvwgUvi (Dîi) cÉk²eÅvsK ‰LvGb gvÕ¡vi ˆUÇBbvi cÉYxZ Dîimn AviI wKQy iPbvgƒjK I mswÞμ¦-Dîi cÉk² ˆ`Iqv nGqGQ| ‰ cÉk²àGjv ˆZvgvGK AwaK AbykxjGb mnvqZv KiGe| cÉk²àGjvi mgvavb Zzwg wbGR wbGRB KiGe| cÉk² 6 25 wgUvi jÁ ¼v ‰KwU euvGki 5 4 25 wgUvi KvGjv, 7 1 4 wgUvi jvj, 4 3 10 wgUvi njy` ‰es evKx Ask mv`v is Kiv nGjv| B K. jvj I njy` AsGki cwigvYGK AcÉK‡Z f™²vsGk cÉKvk Ki| 2 L. euvkwUi KZ wgUvi mv`v is Kiv nGqGQ wbYÆq Ki| 4 M. euvkwUGZ ˆKvb is ˆewk RvqMv RyGo iGqGQ Zv wbYÆq Ki| 4 Dîi: K. 29 4 wgUvi; 43 10 wgUvi L. 8 29 100 wgUvi M. mv`v Ask cÉk² 7 25 wgUvi jÁ¼v ‰KwU euvGki 5 4 25 wgUvi KvGjv, 71 4 wgUvi jvj ‰es 4 3 10 wgUvi njy` is Kiv nGjv| C K. 63 I 91 mnGgŒwjK wKbv hvPvB KGiv| 2 L. euvkwUi KZUzKz is Kiv nqwb? 4 M. ˆKvb is meGPGq ˆewk Kiv nGqGQ? 4 Dîi: K. mnGgŒwjK bq; L. 8 29 100 wgUvi; M. jvj is cÉk² 8 Kwjg mvGne Zuvi Rwg ˆ^GK eQGi 20 1 10 KzB ́Ÿvj Avgb, 30 1 20 KzB ́Ÿvj Bwi ‰es 10 1 50 KzB ́Ÿvj AvDk avb ˆcGjb| K. cÉ`î f™²vskàGjvGK AcÉK‡Z f™²vsGk cÉKvk Ki| 2 L. f™²vskàGjvGK gvGbi AatKÌg AbymvGi mvRvI| 4 M. Kwjg mvGne Zuvi Rwg ˆ^GK `yB eQGi KZ KzB ́Ÿvj avb ˆcGqGQb? 4 Dîi: K. 20 1 10 = 10 20 + 1 10 = 201 10 cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 11 30 1 20 = 20 30 + 1 20 = 601 20 10 1 50 = 50 10 + 1 50 = 501 50 L. 30 1 20 > 20 1 10 > 10 1 50 M. 12017 50 KzB ́Ÿvj cÉk² 9 20 ˆKwR 97 8 MÉvg, 65 ˆKwR 17 3 16 MÉvg, 80 ˆKwR 23 1 24 MÉvg| K. 17 3 16 ˆK AcÉK‡Z f™²vsGk i...cv ̄¦i Ki| 2 L. cÉ^g `yBwU cwigvGYi ˆhvMdj ˆei Ki| 4 M. ‘L’ ‰ cÉvμ¦ ˆhvMdj ˆ^GK ˆkGli cwigvY weGqvM Ki| 4 Dîi: K. 275 16 ; hv AcÉK‡Z f™²vsk| L. 85 ˆKwR 27 1 16 MÉvg; M. 5 ˆKwR 4 1 48 MÉvg 12 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ Abykxjbx-1.5 Abykxjbxi cÉGk²i cvkvcvwk ‰ AsGk ˆ`Iqv cÉk²àGjv meÆvwaK ài‚Z½ w`Gq cÉÅvKwUm KGiv| ZvnGj ‰ AaÅvGqi mKj wkLbdGji Ici cÉÕ§wZ mÁ·®² nGe| cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb cÉk² 1 Rvgvj mvGne Zuvi evevi mÁ·wîi 7 18 AsGki gvwjK| Zuvi evevi mÁ·wîi cwigvY 36 ‰Ki| wZwb Zuvi mÁ·wîi 1 3 Ask ˆgGqGK ‰es 1 2 Ask ˆQGjGK w`Gjb| A K. Rvgvj mvGnGei mÁ·wîi cwigvY KZ? 2 L. wZwb Zuvi ˆQGj I ˆgGqGK KZ ‰Ki KGi mÁ·wî w`Gjb? 4 M. ˆQGj ˆgGqi mÁ·wîi cwigvY ZvG`i evevi mÁ·wîi KZ Ask wbYÆq Ki| 4 1 bs cÉGk²i mgvavb K ˆ`Iqv AvGQ, Zuvi evevi mÁ·wîi cwigvY 36 ‰Ki Rvgvj mvGnGei mÁ·wîi cwigvY 36 2 ‰i 7 18 ‰Ki = 14 ‰Ki (Dîi) L ‘K’ nGZ cvB, Rvgvj mvGnGei mÁ·wîi cwigvY = 14 ‰Ki wZwb Zuvi ˆQGjGK w`Gjb 14 7 ‰i 1 2 ‰Ki = 7 ‰Ki (Dîi) wZwb Zuvi ˆgGqGK w`Gjb 14 ‰i 1 3 ‰Ki = 14 3 ‰Ki (Dîi) M ‘L’ nGZ cvB, ˆQGj I ˆgGqi mÁ·wîi cwigvY h^vKÌGg 7 ‰Ki I 14 3 ‰Ki| wZwb Zuvi ˆQGj I ˆgGqGK ˆgvU mÁ·wî w`Gjb 7 + 14 3 ‰Ki = 21 + 14 3 ,, = 35 3 ‰Ki awi, Rvgvj mvGnGei ˆgvU mÁ·wî 1 Ask 14 ‰Ki = 1 Ask 1 ,, = 1 14 ,, 35 3 ,, = 1 35 5 14 2 3 ,, = 5 6 Ask (Dîi) cÉk² 2 ‰KwU cvwb fwZÆ evjwZi IRb 30 1 4 ˆKwR| evjwZwUi 1 4 Ask cvwbcƒYÆ ^vKGj Zvi IRb 81 2 ˆKwR nq| A K. wgkÉ f™²vsk `ywUGK mij AbycvGZ cÉKvk Ki| 2 L. mÁ·ƒYÆ evjwZi cvwbi IRb KZ? 4 M. Lvwj evjwZi IRb KZ? 4 2 bs cÉGk²i mgvavb K wgkÉ f™²vsk `yBwU 30 1 4 I 8 1 2 wgkÉ f™²vsk `yBwUi mij AbycvZ = 30 1 4 : 8 1 2 = 121 4 : 17 2 = 121 : 34 L evjwZi IRb + evjwZi 1 Ask ev mÁ·ƒYÆ Ask cvwbi IRb = 121 4 ˆKwR evjwZi IRb + evjwZi 1 4 Ask cvwbi IRb = 17 2 ˆKwR evjwZi 1 1 4 Ask cvwbi IRb = 121 4 17 2 ˆKwR ev, ,, 4 1 4 ,, ,, ,, = 121 34 4 ˆKwR ev, ,, 3 4 ,, ,, ,, = 87 4 ˆKwR ,, 1 ev mÁ·ƒYÆ ,, ,, ,, = 87 4 4 3 ˆKwR = 29 ˆKwR mÁ·ƒYÆ evjwZi cvwbi IRb = 29 ˆKwR (Dîi) M Lvwj evjwZi IRb = 121 4 29 ˆKwR = 121 116 4 ˆKwR = 5 4 ˆKwR = 1 1 4 ˆKwR cÉk² 3 ingvb mvGne Zuvi gvwmK ˆeZGbi 1 7 Ask evRvi m`vB ‰i KvGR, 3 14 Ask ˆQGjGgGqG`i ˆjLvcovi RbÅ, 1 6 Ask evwofvovq LiP KGib| evwK UvKvi 3 4 Ask wZwb AbÅ KvGR LiP KGib ‰es AewkÓ¡ 5000 UvKv Rgv KGib| A K. mij Ki: 5 6 ‰i 12 25 1 3 4 1 13 23 5 + 3 20 2 L. DóxcGKi f™²vskàGjvi M.mv.à. wbYÆq Ki| 4 M. ingvb mvGnGei gvwmK ˆeZb KZ? 4 (–) cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 13 3 bs cÉGk²i mgvavb K 5 6 ‰i 12 25 1 3 4 1 13 2 3 5 + 3 20 = 5 6 ‰i 12 25 1 3 1 4 1 13 13 5 + 3 20 = 5 6 ‰i 12 25 1 12 1 13 13 5 + 3 20 = 5 6 ‰i 12 25 13 12 156 13 5 + 3 20 = 5 6 ‰i 12 25 1 156 13 5 + 3 20 = 5 6 ‰i 12 25 1 60 + 3 20 = 5 6 ‰i 12 25 1 + 9 60 = 5 6 ‰i 12 25 10 60 = 1 15 L DóxcGKi f™²vskàGjv 1 7 , 3 14 , 1 6 , 3 4 Avgiv Rvwb, f™²vskàGjvi M.mv.à = f™²vskàGjvi jGei M.mv.à f™²vskàGjvi nGii j.mv.à 1, 3, 1, 3 ‰i M.mv.à = 1 2 7, 14, 6, 4 7 7, 7, 3, 2 1, 1, 3, 2 7, 14, 6, 4 ‰i j.mv.à = 2 7 3 2 = 84 f™²vskàGjvi M.mv.à = 1 84 (Dîi) M gGb Kwi, mÁ·ƒYÆ mÁ·wî 1 Ask ingvb mvGne LiP KGib ˆeZGbi 1 7 + 3 14 + 1 6 Ask = 12 + 18 + 14 84 Ask = 44 84 Ask = 11 21 Ask evwK Ask = 1 11 21 Ask = 21 11 21 = 10 21 AbÅvbÅ LiGPi Ask = 10 21 ‰i 3 4 = 5 14 ˆgvU LiGPi Ask = 11 21 + 5 14 = 22 + 15 42 = 37 42 AewkÓ¡ Ask = 1 37 42 = 42 37 42 = 5 42 5 42 Ask = 5000 UvKv 1 ev mÁ·ƒYÆ Ask = 5000 42 5 UvKv = 42000 UvKv (Dîi) cÉk² 4 ‰KwU cvIqvi ˆRbvGikb ˆKvÁ·vbx ˆgvU Drcv`Gbi 2 5 Ask cvIqvi evmv evwoGZ ‰es 7 20 Ask cvIqvi ‰KwU gvGKÆGU mieivn KGi| AewkÓ¡ Drcv`Gbi 8 25 Ask ivÕ¦vi jÅvÁ·GcvÕ¡ R¾vjvGbv ‰es AbÅvbÅ KvGR mieivn Kiv nq| evwK 68 ˆgMvIqvU cvIqvi nvmcvZvGj mieivn Kiv nq| A K. DóxcGKi f™²vskàGjvGK ˆKvb e†nîg msLÅv «¼viv fvM KiGj fvMdj cƒYÆmsLÅv nGe? 2 L. cvIqvi ˆRbvGikb ˆKvÁ·bxwU ˆgvU KZ cwigvY cvIqvi Drc®² KGi? 4 M. hw` nvmcvZvj I gvGKÆGU cvIqvi mieivGni cÉGqvRb cƒGeÆi Zzjbvq h^vKÌGg 1 7 17 àY ‰es 23 7 àY nq, ZGe ˆKvÁ·vbxwUGK Avi KZ cvIqvi Drcv`b KiGZ nGe? 4 4 bs cÉGk²i mgvavb K DóxcGKi f™²vskàGjv 2 5 , 7 20 , 8 25 f™²vskàGjvi M.mv.à = f™²vskàGjvi jGei M.mv.à f™²vskàGjvi nGii j.mv.à 2, 7, 8 ‰i M.mv.à = 1 5, 20, 25 ‰i j.mv.à = 100 f™²vskàGjvi M.mv.à = 1 100 L cvIqvi ˆRbvGiUi ˆKvÁ·vbx mieivn KGi = 2 5 + 7 20 = 8 + 7 20 = 15 20 Ask AewkÓ¡ ^vGK = 1 15 20 = 5 20 = 1 4 Ask 14 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ ivÕ¦vq I AbÅvbÅ KvGR mieivn KGi = 1 4 ‰i 8 25 = 2 25 Ask ˆgvU mieivn KGi = 15 20 + 2 25 Ask = 75 + 8 100 ,, = 83 100 Ask AewkÓ¡ ^vGK = 1 83 100 Ask = 17 100 Ask 17 100 Ask = 68 ˆgMvIqvU 1 ev mÁ·ƒYÆ Ask = 68 100 17 ˆgMvIqvU = 400 ˆgMvIqvU M cƒGeÆi Zzjbvq nvmcvZvGj cvIqvi cÉGqvRb = 68 ‰i 24 17 ˆgMvIqvU = 96 ˆgMvIqvU nvmcvZvGj cvIqvi e†w«¬ cvq = (96 68) ˆgMvIqvU = 28 ˆgMvIqvU Avevi, gvGKÆGU cvIqvi mieivn KGi = 400 7 20 ˆgMvIqvU = 140 ˆgMvIqvU cƒGeÆi Zzjbvq gvGKÆGU cvIqvi mieivn KGi = 140 ‰i 17 7 ˆgMvIqvU = 340 ˆgMvIqvU gvGKÆGU cvIqvi e†w«¬ cvq = (340 140) ˆgMvIqvU = 200 ˆgMvIqvU ˆKvÁ·vwb AviI cvIqvi Drcv`b KiGe = (28 + 200) ˆgMvIqvU = 228 ˆgMvIqvU (Dîi) cÉk² 5 K = 1 20 23 4 5 16 1 3 8 ‰i 51 2 + 5 7 3 14 L = 1 13 32, M = 48 1 8 K. 8, 22 5, 8 10 ‰i M.mv.à. KZ? 2 L. DóxcGKi ‘K’ ‰i gvGbi mij Ki| 4 M. `yBwU f™²vsGki àYdj M| 1wU f™²vsk L nGj Aci f™²vskwU wbYÆq Ki| 4 5 bs cÉGk²i mgvavb K DóxcGKi f™²vskàGjv 8, 2 2 5 , 8 10 ev 8, 12 5 , 8 10 Avgiv Rvwb, f™²vskàGjvi M.mv.à = f™²vskàGjvi jGei M.mv.à f™²vskàGjvi nGii j.mv.à 8, 12, 8 ‰i M.mv.à = 4 1, 5, 10 ‰i j.mv.à = 10 f™²vskàGjvi M.mv.à = 4 10 = 2 5 (Dîi) L K = 1 20 23 4 5 16 1 3 8 ‰i 5 1 2 + 5 7 3 14 = 43 23 69 16 11 8 ‰i 11 2 + 10 3 14 = 43 23 69 16 121 16 + 7 14 = 43 23 69 16 121 16 + 1 2 = 43 23 69 16 121 + 8 16 = 43 23 69 16 16 129 = 43 23 23 43 = 1 (Dîi) M cÉk²gGZ, `yBwU f™²vsGki àYdj = 48 1 8 = 385 8 ‰KwU f™²vsk = 1 13 32 = 45 32 Aci f™²vskwU = 385 8 45 32 = 385 8 32 45 = 308 9 = 34 2 9 (Dîi) cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 15 cÉk²eÅvsK ‰LvGb gvÕ¡vi ˆUÇBbvi cÉYxZ Dîimn AviI wKQy iPbvgƒjK I mswÞμ¦-Dîi cÉk² ˆ`Iqv nGqGQ| ‰ cÉk²àGjv ˆZvgvGK AwaK AbykxjGb mnvqZv KiGe| cÉk²àGjvi mgvavb Zzwg wbGR wbGRB KiGe| cÉk² 6 41 6 ‰ KwU w gkÉ f™²vsk ‰es 45 8 ‰KwU AcÉK‡Z f™²vsk| A K. 2 3 I 5 6 ‰i M.mv.à. KZ? 2 L. f™²vskàGjvi M.mv.à. ˆei Ki| 4 M. ˆ`LvI ˆh, f™²vskàGjvi àYdj ZvG`i M.mv.à. I j.mv.à.- ‰i àYdGji mgvb| 4 Dîi: K. 1 6 L. 5 24 cÉk² 7 Kvgvj Zvi evevi mÁ·wîi 7 18 Ask cvq| Zvi `yB ˆevb hvG`i cÉGZÅGK Zvi AGaÆK mÁ·wî cvq| Kvgvj Zvi mÁ·wîi 5 6 Ask wZb m ̄¦vGbi gGaÅ mgvbfvGe fvM KGi ˆ`q| A K. KvgvGji cÉGZÅK ˆevb Zvi evevi mÁ·wîi KZ Ask cvq? 2 L. KvgvGji cÉGZÅK m ̄¦vb KvgvGji evevi mÁ·wîi KZ Ask cvq? 4 M. KvgvGji cÉGZÅK m ̄¦vGbi mÁ·wîi gƒjÅ 35 jvL UvKv nGj KvgvGji evevi ˆgvU mÁ·wîi gƒjÅ KZ? 4 Dîi: K. 7 36 Ask; L. 35 324 Ask; M. 3 ˆKvwU 24 jvL UvKv| cÉk² 8 9 1 24 , 35 32 , 7 80 KGqKwU f™²vsk| A K. 21 2 5 2 = KZ? 2 L. f™²vskàGjvi je ‰i M.mv.à. ˆei Ki| 4 M. ˆKvb e†nîg msLÅv w`Gq f™²vskàGjvGK fvM KiGj fvMdj cÉwZ ˆÞGò cƒYÆmsLÅv nGe| 4 Dîi: K. 1 L. 7; M. 7 480 16 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ Abykxjbx-1.6 Abykxjbxi cÉGk²i cvkvcvwk ‰ AsGk ˆ`Iqv cÉk²àGjv meÆvwaK ài‚Z½ w`Gq cÉÅvKwUm KGiv| ZvnGj ‰ AaÅvGqi mKj wkLbdGji Ici cÉÕ§wZ mÁ·®² nGe| cixÞvq Kgb ˆcGZ AviI cÉk² I mgvavb cÉk² 1 kixd mvGne evRvGi wMGq ˆgvU 3 ˆKwR evRvi KiGjb| wZwb 1.5 ˆKwR ˆcuqvR I ˆcuqvGRi 2 3 àY gwiP ‰es wKQy cwigvY imyb KÌq KiGjb| A K. wZwb KZ ˆKwR gwiP KÌq KiGjb? 2 L. wZwb KZ ˆKwR imyb KÌq KiGjb Zv mvaviY f™²vsGk ˆjL| 4 M. 1 ˆKwR ˆcuqvGRi gƒjÅ 100 UvKv, 1 ˆKwR gwiGPi gƒjÅ 80 UvKv ‰es 1 ˆKwR imyGbi gƒjÅ 90 UvKv nGj wZwb ˆgvU KZ UvKvi evRvi KiGjb? 4 1 bs cÉGk²i mgvavb K gwiGPi cwigvY = 2 3 ˆcuqvGRi cwigvY = 2 3 1.5 ˆKwR = / 2 / 3 / 15 5 / 1 10 / 5 / ˆKwR = 1 ˆKwR (Dîi) L ˆcuqvR I gwiGPi cwigvY = (1.5 + 1) ˆKwR = 2.5 ˆKwR imyb KÌq KiGjb = (3 2.5) ˆKwR = 0.5 ˆKwR = 5 10 ˆKwR = 1 2 ˆKwR (Dîi) M 1 ˆKwR ˆcuqvGRi gƒjÅ = 100 UvKv 1.5 ˆKwR ˆcuqvGRi gƒjÅ = (100 1.5) UvKv = 150 UvKv 1 ˆKwR gwiGPi gƒjÅ = 80 UvKv 1 ˆKwR imyGbi gƒjÅ = 90 UvKv 1 2 ev 0.5 ˆKwR imyGbi gƒjÅ = 90 1 2 UvKv = 45 UvKv wZwb ˆgvU evRvi KiGjb (150 + 80 + 45) UvKv = 275 UvKv| (Dîi) cÉk² 2 cuvPwU N¥Ÿv ‰KGò ˆeGR h^vKÌGg 20, 25, 30, 36 I 48 ˆmGKí ci ci evRGZ jvMj| K. DóxcGKi mLÅvàGjvi ˆKvbàGjv ˆRvov mnGgŒwjK wbYÆq Ki|2 L. N¥ŸvàGjv 5 N¥Ÿv ci meÆGgvU KZevi ‰KGò evRGe? 4 M. cçg N ́ŸvwU mwiGq wbGj evwK PviwU N ́Ÿv KZÞY cici ‰KmvG^ evRGe? 4 2 bs cÉGk²i mgvavb K DóxcGKi msLÅvàGjv nGjv : 20, 25, 30, 36, 48 20 ‰i àYbxqKàGjv nGjv : 1,2, 4, 5, 10, 20 25 ‰i àYbxqKàGjv nGjv : 1, 5, 25 30 ‰i àYbxqKàGjv nGjv : 1, 2, 3, 5, 6, 10, 15, 30 36 ‰i àYbxqKàGjv nGjv : 1, 2, 3, 4, 6, 9, 12, 18, 36 48 ‰i àYbxqKàGjv nGjv : 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 25 ‰es 36 ‰i mvaviY àYbxqK ÷ay 1 myZivs Bnv ciÕ·i mnGgŒwjK| 25 ‰es 48 ‰i mvaviY àYbxqK ÷ay 1 myZivs Bnv ciÕ·i mnGgŒwjK| L ‰LvGb, 20, 25, 30, 36, 48 ‰i j.mv.à. hZ nGe ZGZv ˆmGK ́£ cGi N¥ŸvàGjv ‰KGò evRGe| 2 20, 25, 30, 36, 48 2 10, 25, 15, 18, 24 3 5, 25, 15, 9, 12 5 5, 25, 5, 3, 4 1, 5, 1, 3, 4 j.mv.à. = 2 2 3 5 5 3 4 = 3600 ˆmGK ́£ = (3600 60) wgwbU = 60 wgwbU = 1 N¥Ÿv ˆhGnZz cÉ^g evi ‰KGò evRGe 1 N ́Ÿv ci ZvnGj 5 N ́Ÿv ci wgwjZ nGe (1 + 5) evi = 6 evi M cçg N¥ŸvwU mwiGq wbGj evwK PviwU N¥Ÿv nGjv : 20, 25, 30, 36 ‰LvGb, 20, 25, 30, 36 ‰i j.mv.à hZ nGe ZGZv ˆmGK ́£ cGi N¥ŸvàGjv ‰KGò evRGe| 2 20, 25, 30, 36 2 10, 25, 15, 18 3 5, 25, 15, 9 5 5, 25, 5, 3 1, 5, 1, 3 j.mv.à. = 2 2 3 5 5 3 = 900 ˆmGK ́£ = 900 60 wgwbU = 15 wgwbU cÉ^g AaÅvq Õ¼vfvweK msLÅv I f™²vsk 17 cÉk² 3 Kwei mvGne 62580 UvKvi 2 7 fvM Zuvi Õ ̈xGK, 3 14 fvM Zuvi ‰K cyòGK ‰es evwK UvKv Zuvi wZb ˆgGqi gvGS mgvb wZb fvM KGi w`Gjb| K. ˆKvb Þz`ËZg msLÅvGK DóxcGKi f™²vskàGjv «¼viv fvM KiGj fvMdj cƒYÆmsLÅv nGe? 2 L. Kwei mvGnGei cÉGZÅK ˆgGq Zuvi cyGòi ˆ^GK KZ UvKv Kg ˆcj? 4 M. Kwei mvGne Zuvi cyòGK KZ UvKv Kg w`Gj cyò ‰es wZb KbÅvi cÉGZÅKGK mgvb UvKv w`GZ cviGeb? cÉGZÅK KbÅv cƒGeÆi Zzjbvq KZ UvKv ˆewk cvGe? 4 3 bs cÉGk²i mgvavb K DóxcGKi f™²vskàGjv nGjv 2 7 ‰es 3 14 Avgiv Rvwb, f™²vskàGjvi j.mv.à. = f™²vskàGjvi jGei j.mv.à. f™²vskàGjvi nGii M.mv.à. cÉ`î f™²vskàGjvi je 2 ‰es 3 ‰i j.mv.à. = 6 cÉ`î f™²vskàGjvi ni 7 ‰es 14 ‰i j.mv.à. = 7 wbGYÆq j.mv.à. = 6 7 wbGYÆq Þz`ËZg msLÅvwU 6 7 L cyò ˆcj = 62580 UvKvi 3 14 Ask = 13410 UvKv Õ ̈x ˆcj 62580 ‰i 2 7 Ask = 17880 UvKv cyò ‰es Õ ̈x ˆcj (13410 + 17880) UvKv = 31290 UvKv wZb ˆgGq ˆcj (62580 31290) UvKv = 31290 UvKv cÉGZÅK ˆgGq ˆcj = (31290 3) UvKv = 10430 UvKv cÉGZÅK ˆgGq Kg ˆcj = (13410 10430) UvKv = 2980 UvKv M ‘L’ ˆ^GK cvB, cÉGZÅK ˆgGq ˆcj 10430 UvKv cyò ˆcj 13410 UvKv wZb KbÅv ˆgvU ˆcj 31290 UvKv cyò ˆcj 13410 UvKv wZb KbÅv ‰es cyò ˆgvU ˆcj 44700 UvKv wZb KbÅv ‰es cyò cÉGZÅK mgvb UvKv ˆcGj ˆgvU UvKvGK 4 w`Gq fvM KiGZ nGe| cÉGZÅGK cvq = (44700 4) UvKv = 11175 UvKv cyò Kg ˆcj (13410 11175) UvKv = 2235 UvKv cÉGZÅK ˆgGq cƒGeÆi Zzjbvq ˆewk ˆcj = (11175 10430) UvKv = 745 UvKv cÉk² 4 wbGPi wPòwU jÞ Ki ‰es cÉk²àGjvi Dîi `vI: K. mij Ki: 7 8 ‰i 4 5 3 4 ‰i 9 10 – 1 2 5 9 2 L. cÉwZ wgUvi euvk is KiGZ 12.50 UvKv LiP nGj Kv`vq I cvwbGZ ˆh Ask AvGQ, ZvGZ is KiGZ ˆgvU KZ LiP nGe? 4 M. euvkwUGK Zvi Kuv`vq ^vKv AsGki 0.25 Ask DcGi DVvGbv nGj cvwbi Dci euvkwUi KZ Ask we`Ågvb ^vKGe? 4 4 bs cÉGk²i mgvavb K 7 8 ‰i 4 5 3 4 ‰i 9 10 1 2 5 9 = 7 8 ‰i 4 5 3 4 ‰i 9 10 1 2 5 9 = 7 10 27 40 1 2 5 9 = 7 10 40 27 1 2 5 9 = 28 27 5 18 = 56 15 54 = 41 54 (Dîi) L euvkwUi Kv`vq I cvwbGZ AvGQ = (0.15 + 0.65) Ask = 0.8 Ask = 8 10 Ask = 4 5 Ask gGb Kwi, mÁ·ƒYÆ euvkwU 1 Ask euvkwUi cvwbi DcGi AvGQ = 1 4 5 Ask = 5 4 5 Ask = 1 5 Ask cÉk²vbymvGi, euvkwUi 1 5 Ask = 4 wgUvi euvkwUi 1 ev mÁ·ƒYÆ Ask = 4 5 wgUvi = 20 wgUvi Kuv`vq I cvwbGZ AvGQ 20 ‰i 4 5 Ask = 16 wgUvi 1 wgUvi euvk is KiGZ LiP nq 12.50 UvKv 16 ,, ,, ,, ,, ,, ,, (12.50 16) UvKv = 200 UvKv Kv`v cvwbi DcGi 4 wgUvi cvwbGZ 0.65 Ask Kv`vq 0.15 Ask 18 cvGéix ‰GKi wfZi me: lÓ¤ ˆkÉwY MwYZ M Kuv`vq AvGQ 0.15 Ask DcGi DVvGbv nGj Kuv`vq AvGQ (0.15 0.15 ‰i 0.25) Ask = (0.15 0.0375) Ask = 0.1125 Ask cvwbGZ ‰es Kuv`vq AvGQ (0.65 + 0.1125) Ask = 0.7625 Ask ˆhGnZz mÁ·ƒYÆ euvk 1 Ask cvwbi DcGi ^vKGe (1 0.7625) Ask = 0.2375 Ask = 19 80 Ask cÉk² 5 ‰K eÅwÚ 59wU ˆcqviv, 87wU Avg ‰es 259wU wjPz Zvi wKQy e®¬yi gvGS fvM KGi ˆ`qvi ci 8wU ˆcqviv, 2wU Avg ‰es 4wU wjPz AewkÓ¡ ^vGK| A K. 15 23 ‰es 21 5 f™²vsk«¼qGK mgni wewkÓ¡ f™²vsGk cÉKvk Ki| 2 L. meÆvwaK KZ Rb e®¬yGK wZwb DóxcGKi bÅvq djàGjv fvM KGi w`GqwQGjb? 4 M. cÉwZwU wjPzi gƒjÅ I cÉwZwU ˆcqvivi gƒjÅ h^vKÌGg cÉwZwU AvGgi gƒGjÅi 1 8 fvM ‰es 3 4 fvM| hw` meàGjv dGji ˆgvU gƒjÅ 2618 UvKv nq, ZGe ‰K nvwj Avg, ‰K nvwj ˆcqviv I ‰K nvwj wjPzi gƒjÅ wbYÆq Ki| 4 5 bs cÉGk²i mgvavb K cÉ^g f™²vsk = 15 23 w«¼Zxq f™²vsk, 2 1 5 = 11 5 15 23 ‰es 11 5 f™²vsk«¼qGK mgni wewkÓ¡ f™²vsGk cÉKvk KiGZ nGe| cÉ`î f™²vskàGjvi ni 23 ‰es 5 ‰i j.mv.à. = 115 cÉ^g f™²vsk = 15 23 = 15 5 23 5 = 75 115 [ 115 23 = 5] w«¼Zxq f™²vsk = 11 5 = 11 23 5 23 = 253 115 [ 115 5 = 23] L fvM KGi ˆ`qv ˆcqvivi msLÅv (59 8)wU = 51wU fvM KGi ˆ`qv AvGgi msLÅv (87 2)wU = 85wU fvM KGi ˆ`qv wjPzi msLÅv (259 4)wU = 255wU meÆvwaK msLÅK e®¬yi msLÅv nGjv 51, 85, 255 ‰i M.mv.à. ‰i mgvb| 51) 85 51 (1 34) 51 34 (1 17) 34(2 34 0 17) 255 17 (15 85 85 0 wbGYÆq e®¬yi msLÅv = 17 Rb M gGb Kwi, 1wU AvGgi gƒjÅ K UvKv 87wU AvGgi gƒjÅ 87 K UvKv = 87K UvKv cÉwZwU wjPzi gƒjÅ = K ‰i 1 8 UvKv = K 8 UvKv 259wU wjPzi gƒjÅ = 259 K 8 UvKv = 259K 8 UvKv cÉwZwU ˆcqvivi gƒjÅ = K ‰i 3 4 UvKv = 3K 4 UvKv 59wU ˆcqvivi gƒjÅ = 59 3K 4 UvKv = 177K 4 UvKv cÉk²gGZ, 177K 4 + 87K + 259K 8 = 2618 ev, 354K + 696K + 259K 8 = 2618 ev, 1309K 8 = 2618 ev, K = 2618 8 1309 K = 16 1wU AvGgi gƒjÅ 16 UvKv 1 nvwj = 4wU AvGgi gƒjÅ (16 4) UvKv = 64 UvKv 1wU ˆcqvivi gƒjÅ = 3 16 4 UvKv = 12 UvKv 1 nvwj ev 4wU ˆcqvivi gƒjÅ (12 4) UvKv = 48 UvKv 1wU wjPzi gƒjÅ 16 8 = 2 UvKv 1 nvw