WAIFK EA UWA] ;) HJUWBEQMMHÝJ.- DKIJHUQEA] SWBSBWMHBJKFA] ;.;) DKIJHUQEA] EHWAMUKDAJUA SWBSBWMHBJKFA].- Ebs dkijhtueas (prbphaeke `äshmk qua puaea sar daehek) sbj ehramtkdajta prbpbrmhbjkfas (ES) mukjeb kf kudajtkr b ehsdhjuhr af vkfbr ea ujk ea affks, ajtbjmas af vkfbr mbrraspbjehajta k fk btrk kudajtkrë kudajtkrë b ehsdh ehsdhjuhrë raspamthv raspa mthvkda kdajta jta aj fk dh dhsdk prbpb p rbpbrmhýj. rmhýj. Anadpfb> Anadpfb> Qjk parsbjk pars bjk rambrra ramb rra 53 datrbs aj 1 saiujebs. saiujebs. (Aspkmhb vs. Uh Uhadpb) Dkijhtue Dkijhtue K> Aspkmhb Aspkmhb (datrbs) (datrbs) G> Uhadpb Uhadpb (saiujebs (saiujebs))
Pkfbras Pkfbras mbrraspbjehajtas mbrra spbjehajtas 53 <3 ;73 ;03 1 ;3 ;1 73
733 71
Kf ehvhehr sus vkfbras mbrraspbjehajtas af mbmhajta as shadpra mbjstkjta, as eamhr, sh ebs dkijhtueas K y G sbj ehramtkdajta prbpbrmhbjkfas, sa eajbtk> K (ES) G b K G, ajtbjmas K G
O
;.7) DKIJHUQEA] HJPAW]KDAJUA SWBSBWMHBJKFA].- Ebs dkijhtueas sbj hjvarskdajta prbpbrmhbjkfas (HS) mukjeb kf kudajtkr b ehsdhjuhr af vkfbr ea ujk ea affks ajtbjmas af vkfbr vkfbr ea fk fk btrk ehsdhjuhrë ehsdhjuhrë b kudajtkrë kudajtkrë raspamthv rasp amthvkda kdajta jta aj fk dhsdk dhsdk prbpbrmhýj. prbp brmhýj. Anadpfb> Anadpfb> Qjk parsbjk par sbjk rambrra ram brra ;03 od (Pafbmheke (Pafbmheke vs. Uh Uhadpb)
Dkijhtue Dkijhtue K> Pafbmheke Pafbmheke (od/l) (od/l) G> Uhadpb Uhadpb (l)
Pkfbras Pkfbras mbrraspbjehajtas mbrra spbjehajtas 1 ;3 73 53 <3 =7 ;0 < 5 7
Kf dufthpfhmkr sus vkfbras mbrraspbjehajtas af prbeumtb as shadpra mbjstkjta, as eamhr, sh ebs dkijhtueas K y G sbj hjvarskdajta prbpbrmhbjkfas, sa eajbtk> K (HS) G b K
;
G, ajtbjmas
KG4 O
Jbtk> ]h hjtarvhajaj = b dës dkijhtueas tbeks affks fks pbeadbs mbfbmkr aj ujk dhsdk axprashýj. Anadpfb> Anadpfb > ]h ] h K G y K
;
M, sa puaea axpraskr ksä>
K M
G
O
7) WAIFK EA UWA] ]HDSFA EHWAMUK Mbjshsta aj qua ekebs = vkfbras mbrraspbjehajtas k 7 ebs dkijhtueas ehramtkdajta prbpbrmhbjkfas sa eaga ajmbjtrkr uj mukrtb vkfbr. mhmfhstk stk rambrra ramb rra ;7 od aj ;0 dhjutb jutbs. s. ²[ ²[uè uè ehstkjmhk ehstkjmhk rambrrarë ramb rrarë aj a j uj ujkk lbrk2 Anadpfb> Qj mhmfh
]bfumhýj> K dkybr aspkmhb rambrrheb adpfakrë dkybr thadpb, ajtbjmas af aspkmhb y thadpb sbj dkijhtueas ehramtkdajta prbpbrmhbjkfas Dkijhtue K> Aspkmhb (od) G> Uhadpb Uhadpb (dhj) (dhj)
Pkfbras mbrraspbjehajtas ;7 x ;0 03
;7 od
;0 dhj dhj
\
03 dhj dhj
k) Sfkjtab>
Dufthpp fhmk fhmkjeb jeb aj Kspk (Dètbeb> (Dètbe b> Kspk) g) Waifk> Dufth x ∘ ;0 dh dhj 4 ;7 od od ∘ 03 dh dhj x4
;7 od od ∘ 03 dh dhj 4 51 od od ;0 dhj
m) Mbdprbgkmhýj> ]h K G, ajtbjmas> K ;7 4O⇗ 4 4⇗ 4 G ;0 03
Wadpfkz Wadpfkzkk jeb fk raspuastk bgtajh bg tajhek> ek> ;7 51 = = 4 ⇗ 4 ;0 03 5 5
Fb qua sa quaräk mbdprbgkr
=) WAIFK EA = ]HDSFA HJPAW]K Mbjshsta aj qua ekebs = vkfbras mbrraspbjehajtas k 7 dkijhtueas hjvarskdajta prbpbrmhbjkfas sa eagaa ajmbjtrkr uj mukrtb eag mukrtb vkfbr.
Anadpfb> = bgrarbs puaeaj lkmar ujk bgrk aj 75 eäks ²Aj muëjtb thadpb lkrëj fk dhsdk bgrk 7 bgrarbs2
]bfumhýj> Mbj dajbs bgrarbs fk bgrk sa mbjstruhrë aj dës eäks, ajtbjmas af jþdarb ea bgrarbs y af thadpb thadpb sbj dkij kijhtueas htueas hjvarsk jvarskd da jta prbpbrmhbjkf prbp brmhbjkfaa s.
Dkijhtue K> Bgrarbs Bgrarb s G> Uhadpb Uhadpb (eäks) (eäk s)
Pkfbras mbrraspbjehajtas = 7 75 x
k) Sfkjtab> Mbdb ; lbrk 4 03 dhjutbs = bgrarbs
75 eäks
7 bgrarbs
x
Dufthpp fhmk fhmkjeb jeb aj Skrkfafb Skrkfafb (Dètbe ( Dètbeb> b> Skrkf Skrk fafks) g) Waifk> Dufth = bgrarbs 75 eäks 4 7 bgrarbs x x4
= bgrar bgrarbs bs ∘ 75 eäks eäks 4 =0 eäks eäks 7 bgrarbs
m) Mbdprbgkmhýj> ⇗ ∘ 4 ⇗ ∘ 4 ∘ 4 ∘ 4
Wadpfkzk Wadpfkzkjeb jeb af vkfbr mkfmufkeb> mkfmufkeb > = ∘ 75 4 7 ∘ =0 = 0 ⇗ 9 7 4 97
Fb qua sa quaräk mbdprbgkr.
5) WAIFK EA UWA] MBDSQA]UK As kquaff kquaffk aj fk fk qua q ua fks fks dkijhtuea kijhtueass qua sa mbdpkrk mbdpkrkjj sbj = b dës. dës. raifk ea = mbdpuastk thaja thaja = pkrtas> Dètbeb Dètbeb ea fks Wkyks> Fk raifk Wakfhzkebras zkeb ras ea fk fk bgrk b grk b kmmhýj y sus mbjehmhbjas mbjehmhbjas k) Mkusk> Wakfh
g) Mhrmujstkjmhk> Dkijhtueas rafkmhbjkeks mbj af thadpb rakfhzkeb, fk bgrk, bgrk , sus daeheks, daeheks, su eh`h eh`hmuftke muftke m) A`amtb> Fb rakfh
Anadpfb> Skrk lkmar ujk zkjnk ea =3 datrbs ea fkrib pbr ;3 ea kjmlb, ;1 bgrarbs lkj trkgknkeb 0 eäks k rkzýj ea ;7 lbrks ehkrhks. ²Muëjtbs eäks trkgknkrëj ;< bgrarbs k 8 lbrks ehkrhks aj lkmar ujk zkjnk zkjnk ea 51 datrbs ea fkrib pbr 73 7 3 ea kjmlb2
]bfumhýj> K dës bgrarbs (K) fk bgrk sa lkrë aj dajbs eäks (G), pbr fb tkjtb K
;
G. K dajbs
lbrks ehkrhks ea trkgknb (M) fk bgrk sa lkrë aj dës eäks (G), pbr fb tkjtb G
;
M. K dës datrbs ea
fkrib (E) fk bgrk sa lkrë aj dës eäks (G), pbr fb tkjtb G E. X k dës dës datrbs ea kjmlb (A) Fk bgrk sa lkrë aj dës dës thadpb, thadpb, pbr fb fb tkjtb G A. Dkijhtue K> Bgrarbs G> Uhadpb Uhadpb (eäks) (eäk s) M> Uhadpb Uhadpb (lbrks ehkrhks) ehkrhks) E> Datrbs ea fkrib A> Datrbs ea kjmlb
Pkfbras mbrraspbjehajtas ;1 ;< 0 x ;7 8 =3 51 ;3 73
k) Sfkjtab> Mkusk Mhrmuj Mhrmujss tkjmhk Bgrarbs Bg rarbs Eäks Lbrks ehkrhks ehkrhks ;1 0 ;7 ;< x 8
A`amtb Fkrib Kjml Kjmlbb =3 ;3 51 73
g) Waifk> Dètbeb ea fks rkyks ;< ∘ x ∘ 8 ∘ =3 ∘ ;3 ; 3 4 ;1 ∘ 0 ∘ ;7 ∘ 51 5 1 ∘ 73 73 ⇗ x 4
;1∘0∘;7∘51∘73 4 73 eäks eäks ;<∘8∘=3∘;3
e ktbs sa ththaja qua> m) Mbdprbgkmhýj> Ea kmuareb k fbs ektbs ∘ ∘ ∘ ∘ 4 ⇗ 4 4 ∘ ∘ ;1∘0∘;7 ;<∘∘8 ;< ;< 4 ⇗ 4 =3 ∘ ; 3 51 ∘ 73 1 1
Ukf mbdb sa quaräk mbdprbgkr.
1) DH]MAFËJAK EA SWBGFADK] Jbtk> Fbs prbgfadks qua sa prasajtkj k mbjthjukmhýj jb lkj sheb rasuaftbs aj af ;33%, pbr fb qua fk sbfumhýj sbfumhýj mbdpfatk mbdpfatk quaek mbd mb db tkrak tkrak pkrk ustaeas asthdkeb asthdkebss famtbras. amtbra s.
;) Gartlk Hgunès, ujk kdk ea mksk, fkvk pbr mkek =7 dhjutbs ;3 pkjtkfbjas. ²Muëjtbs pkjtkfbjas fkvkrë aj ;l 73 dhjutbs dhjutbs22 Skjtkfbjas ;3
Uhadpb =7 dhj
x
<3 dhj dhj ;3 pkjtkf pkjtkfbjas bjas ∘ <3 dhj dhj 4 71 pkjtk pkjtkfbj fbjas as =7 dhj
x ∘ =7 dh dhj 4 ;3 pk pkjtkfbja bjas ∘ <3 dh dhj ⇗ x 4
7) ;3 bgrarbs puaeaj lkmar ujk bgrk aj 75 eäks ²Aj muëjtb thadpb lkrëj fk dhsdk bgrk < bgrarbs2
Bgrarbs ;3
Eäks 75
<
x
x ∘ < bg bgrarbs 4 ;3 bg bgrar rarbs∘ 75 eä eäks ⇗ x 4
;3 bgrarb bgrarbss ∘ 75 eäks eäks 4 =3 eä eäks < bgrarbs bgrarbs
=) 0 mkgkffbs thajaj rkmhýj pkrk ;1 eäks, sh sa kudajtkj = mkgkffbs dës. ²Skrk muëjtbs eäks kfmkjzkrë fk rkmhýj kjtarhbr2
x ∘ 8 mkg mkgk kffbs 4 0 mk mkgkffbs ∘ ;1 eä eäks ⇗ x 4
Mkgkffbs 0
Eäks ;1
8
x
0 mkgkf mkgkffb fbss ∘ ;1 eäks eäks 4 ;3 eä eäks 8 mkgkffbs
5) Dkrhb ]uëraz trkgknk aj uj mbfaihb ea 933 astuehkjtas, aj af muëf raprbgkrbj af kûb af 1%. ²Muëjtbs astuehkjtas astuehkjtas jb `uarbj `uarbj prbdbvh p rbdbvhebs ebs aj af mbfaihb mbfaihb ea Dkrhb2
Astuehkjtas 933
Sbrmajtkna ;33%
x
x ∘ ;33% 4 933 ast astuehkjtas ∘ 1% ⇗ x 4
1%
933 astueh astuehkjt kjtas as ∘ 1% 4 =1 ;33%
1) ²Eykjhtk Whvark fa praiujtk k su lhnk Adhfy ]uëraz ²Ea què mkjtheke as $ 873 af 73 %2. ²Muëf as fk raspuastk2 Mkjtheke x
Sbrmajtkna ;33%
873
x ∘ 73% 4 $ 873 ∘ ;33% ⇗ x 4
73%
$ 873 873 ∘ ;33% ;33% 4 $ 503 5033 73%
0) Aj uj ajirkjkna, af phûýj dkybr thaja 53 ehajtas y af dajbr thaja 71 ehajtas. ]h af phûýj dkybr ek 733 vuaft vuaftks. ks. ²Muëj ²M uëjtks tks vuaftks vuaftks ek af dajbr2 Ehajtas 53
Puaftks 733
71
x
x ∘ 71 ehajt ajtas 4 53 ehajt ajtas ∘ 733 vuaft aftks ⇗ x 4
53 ehajt ehajtas as ∘ 733 vua vuaftks ftks 4 =73 vuaft vuaftks ks 71 ehajtas
9) Qjk iukrjhmhýj ea ;=33 lbdgras thajaj vävaras pkrk ;73 eäks. ]h sa easak qua fbs vävaras euraj ;3 eäks dës. dës. ²Muëjtbs lbdgras lkgräk qua q ua rathrkr rathrkr ea fk fk iukrjh iukrjhmm hýj2 hý j2
Lbdgras ;=33
Eäks ;73
x
;=3
x ∘ ;=3 ;=3 eä eäks 4 ;=3 ;=33 lbd lbdgras∘ gras∘ ;73 ;73 eäks eäks ⇗ x 4
;=33 lbdg lbdgras ras ∘ ;73 eäk eäkss 4 ;733 lbdgras bdgras ;=3 eäks
;=33 lbdgras lbdgras ” ;733 ;733 lbdgras lbdgras 4 ;33 lbdgras lbdgras <) K uj paýj sa fa b`rama uj suafeb ea $ ;833 kjukfas y uj mkgkffb. Kf mkgb ea < dasas as easpaeheb ramhghajeb ramhghajeb uj tbtkf ea $ ;733 y af mkgkffb. mkgkffb. ²Muëf ²M uëf as af vkfbr vkfbr eaf mkgkff mkg kffb2 b2 m 4 mkgkffb ]uafeb ;833 + m
Dasas ;7
;733 + m
<
< ∘ ($;833+m) 4 ;7 ∘ ($;733 + m) ⇗ $;1733 +
8) ]aiujeb ]uëraz, uj krtaskjb, pajsý lkmar 73 `hiurks ea dkeark aj ;1 eäks, parb tkreý 0 eäks dës pbr trkgknkr 7 lbrks dajbs mkek eäk. ²Muëjtks lbrks trkgkný ehkrhkdajta2
Mkusk Mhrmujstkjm Mhrmujstkjmhk hk Krtaskjb Eäks Lbrks ehkrhks ehkrhks ; ;1 x ; 7; x-7
A`amtb A`amtb @hiurks @hiurks 73 73
; ∘ ; 1 ∘ ∘ 7 3 4 ; ∘ 7 ; ∘ ( ∗ 7 ) ∘ 7 3 ⇗ = 3 3 4 5 7 3 ∗ <5 3 < 53 4 5 73 ∗ = 3 3 ⇗ < 5 3 4 ; 73 ⇗ 4 > ∗ 7 4 9 ∗ 7 4 1
<53 ⇗49 ;73
;3) Dktläks ]uëraz mbjtrktk k 75 bgrarbs, fbs mukfas sa mbdprbdataj k mkvkr ujk zkjnk ea 13 d ea fkrib,
Mkusk Mk usk Mhrmujstkjm Mhrmujstkjmhk hk Bgrarbs Bg rarbs Eäks A`h A`hmhajmhk mhajm hk 75 ;3 ; x 1 ;,1
A`amtb A`amtb Fkrib Kjml Kjmlbb Srb`u Srb `ujeh jeheke eke 13 < 7 91 ;7 =
∘ 1 ∘ ;,1 ∘ 13 13 ∘ < ∘ 7 4 75 ∘ ;3 ;3 ∘ ; ∘ 91 ∘ ;7 ∘ = ⇗ 4
75∘;3∘;∘91∘;7∘= 4 ;3< 1∘;,1∘13∘<∘7
WA@AWAJMHK] GHGFHBIWË@HMK] KXKFK, KXKFK, Brfkj B rfkjeb eb,, (7330), (733 0), Dktadëthmk Dktadëthmk Wamrakth Wamrak thvk, vk, D & P IWË@HM. IWË@HM . Hgkrrk, Amukeb Amukebrr ]QËWA^, Dkrhb ]QËWA^, Dkrhb, (7335), (733 5),
Hjtarkprajehzkna Hjtarkprajehzk na Lbf Lb fästhmb ästhmb ea Dktadëthmk Dktadëthmk,, Ae. Irë`hmks Irë`hmks Sf S fkjatk, Hgkrrk, Amukebr.
]QËWA^, Dkrhb, (7335), (7335 ), Lkmhk uj uj Hjtarkprajehzkna Hjtarkpraje hzkna Lbfästh Lbfästhmb mb ea Ëfiagrk y Iabdaträk, Iabdaträk, Ae. Irë`hmks Sfkjatk, Hgkrrk, Amukebr.
Fhgrbs y krtämufbs sbgra Krhtdèthmk, Ëfiagrk, Iabdaträk, Urhibjbdaträk, Fýihmk Dktadëthmk, Srbgk Srb gkghf ghfhekea hekeas, s, Astkeästhmk Astke ästhmk Easmrhpth Easm rhpthvk, vk, Astkeästhmk Astke ästhmk Hj`arajmhkf Hj`ara jmhkf,, Mëfmuf Më fmufbb Eh`arajmhkf, Eh`arajmhkf, Mëfmuf Më fmufbb Hjtairkf, y Sfkjh`hmkmhbjas Eheëmthmks sa ajmuajtrkj pugfhmkebs aj> lttp>//as.smrhge.mbd/dkrhbsukrazhgunas lttp> ttp>//w //www.dbjbirk ww.dbjbirk`hks `hks.mb .mbd/u d/usukr sukrhhb /par`hf /par`hfas/ as/dkrh dkrhbR bRsukrazR9/dbjbirk sukrazR9/dbjbirk`h `hks ks lttp>//rapbshtbrhb.utj.aeu.am/lkjefa/;7=5109<8/903 lttp> ttp>//w //www.ebmajtas ww.ebmajtashjj hjjbb vkebras.jat/Qsukrhbs/Par/7818; lttp>//krthmufbsdktadkthmk.gfbispbt.mbd