magaliTad, Tu sawyisi fasia : a: lari da moxda

sarCevi
1. Sesavali ----------------------------------------------------------- 5
2. miznebi mimarTulebebis mixedviT ----------------------------- 5
3. sakiTxTa CamonaTvali mimarTulebebis mixedviT------------- 8
4. meTodika ---------------------------------------------------------- 10
5. VII klasis savaraudo Tematuri gegma --------------------------- 13
6. miznebi meToduri rekomendaciebi, pasuxebi
Semajamebeli samuSaoebis nimuSebi ---------------------------- 15
7. maTematikuri TamaSebi ----------------------------------------- 68
4

III Tavi
monacemebi
Tavis ZiriTadi mizani: SevuqmnaT moswavles warmodgena monacemebze, maT saxeebze
da mopovebis xerxebze; vaswavloT Tvisebriv da raodenobriv monacemTa erTobliobis
mowesrigeba; vaswavloT cxriliT sargebloba da cxrilis Sedgena; CamovuyaliboT
diagramebis agebisa da wakiTxvis unar-Cvevebi; vaswavloT monacemTa
ricxviTi maxasiaTeblebis (sixSire, moda, gabnevis diapazoni, saSualo) gamoTvla.
gavufarTovoT moswavles warmodgena ricxvebis sxvadasxva ekvivalenturi formiT
Caweris SesaZleblobebis Sesaxeb. SevaswavloT ricxvis procentuli Cawera,
sididis procentis gamoTvla, monacemTa Cawera procentebSi da gamovumuSaoT
Sesabamisi gamoTvlebisa da Sefasebis unarebi.
am Tavis erT-erTi ZiriTadi Temaa `procenti~, romelic saxelmZRvaneloSi
warmodgenilia sami paragrafiT. TiToeul maTgans eTmoba ori gakveTili.
3.2 ricxvis procentuli Cawera
3.3 rogor vipovoT ricxvi misi procentis mixedviT?
3.4 ori sididis fardobis procentuli warmodgena.
nawilebze amocanebi da maT Soris sami ZiriTadi amocana (ricxvis nawilis povna,
ricxvis povna misi nawilis mixedviT, erTi ricxvi meore ricxvis ra nawilia), VI
klasis saxelmZRvaneloSi dawvrilebiT iyo gadmocemuli. am TavSi amocanebi
nawilebze gasameorebel savarjiSoebSia mocemuli. nawilebze amocanebis
analogiurad, ZiriTadi amocanebi procentebze Semdegi saxisaa:
1) mocemuli ricxvis procentis povna.
2) ricxvis povna misi procentis mixedviT.
3) ori ricxvis procentuli Tanafardobis gamoTvla.
3.1 monacemTa saxeebi da maTi mopovebis saSualebebi
mizani: SevuqmnaT moswavleebs warmodgena monacemebis raobisa da maTi ZiriTadi
saxeebis Sesaxeb. gavacnoT moswavleebs monacemTa mopovebis xerxebi (gamokiTxva,
dakvirveba, gazomva, eqsperimenti) da SevZinoT maT am xerxebis praqtikuli
gamoyenebis Cvevebi.
codna da unarebi: paragrafSi gadmocemuli masalis Seswavlis Sedegad moswavle
SeZlebs gansazRvros, Tu ra monacemebis flobaa saWiro dasmuli amocanis
amosaxsnelad da ganasxvaos, romelia maT Soris raodenobrivi da romeli –
Tvisebrivi. SeZlebs misi gadasawyveti amocanisaTvis monacemTa Segrovebis
Sesaferisi saSualebis SerCevasa da gamoyenebas.
meToduri komentarebi: paragrafis teqstSi Setanil gakveTilze gansaxilvel sam
magaliTs saWiroebis SemTxvevaSi SeiZleba davumatoT sxva magaliTebi.kerZod,
#4. ris mixedviT Rebulobs diasaxlisi gadawyvetilebas namcxvris gamosacxobad
saWiro produqtebis SeZenis Sesaxeb? (receptis mixedviT).
# 5. zogi qarxana yoveldRiurad produqciis aTasobiT erTeuls awarmoebs. xarisxis
gakontrolebis mizniT drodadro xdeba gamoSvebuli produqciis SedarebiT
mcire nawilis SemTxveviT amorCeva da amorCeulis vargisianobis Seswavla.
amis Sedegad miRebuli monacemebis safuZvelze keTdeba daskvnebi mTeli produqciis
xarisxze.
magaliTi 6. moqalaqis ra monacemebis gagebas SevZlebT misi piradobis mowmobidan?
pasportidan?
sasurvelia, rom am paragrafis Seswavlis droisTvis moswavleebs sabunebismetyvelo
sagnebidan (fizika, qimia, biologia) SeeZloT bunebis raime kanonis
dasaxeleba, romlis aRmoCenac mecnieruli dakvirvebis an eqsperimentis Sedegad
moxda. Tu es ase araa, mouyvaneT moswavleebs TqvenTvis cnobili magaliTebi.
komentarebi savarjiSoebis Sesaxeb da pasuxebi. #1. raodenobrivi maxasiaTeblebi:
42

mricxveli, mniSvneli. Tvisebrivi: wesieri, arawesieri, ukveci, kvecadi.
#3. samkuTxedis raodenobrivi maxasiaTeblebi: gverdebis sigrZeebi, kuTxeebis
sidideebi, simaRleebis sigrZeebi da a. S. Tvisebrivi: tolgverda, tolferda,
maxvilkuTxa, marTkuTxa, blagvkuTxa. analogiuri kiTxva SeiZleba davsvaT
kuTxisaTvis (monacemebi: gradusuli zoma, maxvili, marTi, blagvi, gaSlili).
#7. amocanis mizania, moswavles intuiciis doneze (sakiTxSi CarRmavebis gareSe)
SevuqmnaT warmodgena respondentTa `warmomadgenlobiTi~ jgufis Sesaxeb.
sasurvelia sxva, analogiuri magaliTis ganxilva (vTqvaT, fexburTis nacvlad
saopero xelovneba).
#10. a) raodenobrivi: ubnebis raodenoba, sigrZe, kuTxeebi mezobel ubnebs Soris.
Tvisebrivi: Sekruli texili. b) raodenobrivi: gverdebis raodenoba, perimetri,
kuTxeebis sidideebi. Tvisebrivi: amozneqili, araamozneqili.
#13. gamocnobis SesaZlo meTodebi:
• `pirdapiri~, rodesac kiTxvas Tanmimdevrulad davusvamT 10-s
(`Cafiqrebuli ricxvi 10-is tolia?~), mere 11-s da a. S. (dagvWirdeba ara
umetes 89 kiTxvisa);
• `poziciuri~ – wina meTodiT jer gamovicnoT Cafiqrebuli ricxvis
pirveli cifri (maqsimum 8 kiTxva), Semdeg – meore (maqsimum 9 kiTxva, sul
– ara umetes 17 kiTxvisa);
• `orobiTi~ – saZiebo aris Tanmimdevruli `ganaxevreba~ kiTxviT
`Cafiqrebuli ricxvi naklebia 55-ze?~ da misi analogiurebiT (dagvWirdeba
sul didi 7 kiTxva). am mniSvnelovani xerxis gaazreba moswavleebs
gauadvildebaT, Tu winaswar ganxiluli iqneba 2, 4, 8, 16 sagnidan erTi
Cafiqrebulis amocnobis amocana.
#20 MK da AC monakveTebi AMCK rombis diagonalebia, xolo O maTi gadakve-
Tis wertili, anu OK = OM . amave dros,
0
AK = AM da ∠ KAO = 30 . maSasadame, AMK samkuTxedi
tolgverda. Tu SemovitanT aRniS-
C
M D
O
vnas: OK = OM = x , maSin CK = AK = 2. x CBK −
0
Si ∠ C = 30 , BK = x.
vinaidan AB = BK + KA = 3x
0
3 sm, miviRebT, rom x = 1sm. AK = 2x
= 2 sm. B
30
K
A
pasuxi: rombis gverdis sigrZea 2 sm.
#21 ormoci boZis SemTxvevaSi TiToeuli gverdis gaswvriv 11 boZi da boZebs
Soris 10 Sualedi iqneba. e. i. kvadratis gverdis sigrZe 15 m iqneba, xolo farTobi
225 kv.m.
n
aba scade! TiToeuli marTkuTxedis ori
t
mezobeli gverdi didi kvadratis perimetris c
Semadgeneli nawilia. am gverdebis sigrZeebis
jami 15 sm-is tolia. maSasadame, marTkuTxedebis
perimetrebis jami iqneba 30 sm.
m
marTlac, Tu didi kvadratis perimetrs P k
asoTi aRvniSnavT, maSin naxazis mixedviT: P =
k+ c+ n+ t+ m+ e+ f + d , rac amocanis pirobiT
15 sm-is tolia. naxazze mocemuli
e
marTkuTxedebis perimetrebis jami iqneba:
f d
2 n+ c + 2 t+ m + 2 d + e + 2 k+ f =
( ) ( ) ( ) ( )
( n c t m d e k f )
2 + + + + + + + = 2⋅ 15 = 30 (sm).
mivaqcioT yuradReba, rom saZiebeli jami ar aris damokidebuli mcire
kvadratis gverdis sigrZeze da ganlagebaze.
SesaZlebelia Tu ara? ara, es SeuZlebelia.
davuSvaT, rom Tavisufal ujredebSi ricxvebis Cawera amocanis pirobis dacviT
aris SesaZlebeli. vTqvaT, zeda striqonis Tavisufal ujredSi Cawerilia ricxvi
43

x, xolo qveda striqonis Tavisufal ujredSi – ricxvi y. marcxena zeda da
marcxena qveda 2x2 kvadrats ori saerTo ujredi aqvs:
2 x 4
2 x 4
7 y 8
7 y 8
am kvadratebSi Cawerili ricxvebis jami toli unda iyos. amitom 2+ x = 7+y.
maSasadame, x ricxvi y ricxvze 5-iT meti unda iyos. marjvena zeda da marjvena
qveda 2x2 kvadratisaTvis Catarebuli analogiuri msjeloba ki gviCvenebs, rom
x ricxvi y ricxvze 4-iT meti unda iyos, rac wina daskvnas ewinaaRmdegeba.
amocanisadmi interesi gaizrdeba, Tu moswavleebs winaswar SeTavazebT amocanis
pirobis dacviT Semdegi kvadratis Sevsebas:
2 4
7 9
(es SesaZlebelia, magaliTad, Tu aviRebT x = 7 , y = 2, xolo SuaTana striqonSi
CavwerT 4-s, 0-s da 2-s ). Tu moswavleebi am amocanas damoukideblad ver daZleven,
mieciT miTiTebebi. amocanis garCevisas moswavleebi miiReben dasabuTebuli
msjelobisa da cxrilis SevsebasTan dakavSirebuli procedurebis Catarebis
garkveul gamocdilebas.
3.2 ricxvis procentuli Cawera
mizani: moswavleebs unda SevaswavloT: ricxvis Cawera procentis saxiT da
procentis saxiT Cawerili ricxvis warmodgena wiladisa da aTwiladis saxiT;
ricxvis (sididis) nawilsa da procents Soris kavSiris damyareba; gamovumuSaoT
am kavSirebis gamoyenebiT amocanebis amoxsnis unar-Cvevebi.
codna da unarebi: moswavlem unda icodes Cveulebrivi wiladis an aTwiladis
saxiT Cawerili ricxvis Cawera procentis saxiT da, piriqiT, procentis
warmodgena Cveulebrivi wiladis an/da aTwiladis saxiT. moswavlem unda icodes,
rom:
1) procenti gviCvenebs ricxvSi measedebis raodenobas.
2) ricxvis procentis saxiT Casawerad igi 100-ze unda gavamravloT.
3) procentis wiladis an aTwiladis saxiT Casawerad procentis ricxvi iyofa
100-ze.
4) ricxvis (sididis) nawilis procentis saxiT da, piriqiT, procentis nawilis
saxiT Casawerad saWiroa analogiuri moqmedebebis Catareba.
5) ricxvis (sididis) p%
-is gamosaTvlelad saWiroa procenti CavweroT nawilis
saxiT da gamovTvaloT ricxvis (sididis) nawili, an, rac igivea, ricxvi
(sidide) gavamravloT p /100 -ze.
moswavles unda SeeZlos.
1) imis gamoTvla, Tu erTi sidide meore sididis ramden procents Seadgens
2) ricxvis povna misi mocemuli procentiT.
3) procentebiT warmodgenili monacemebiT sargebloba;
4) praqtikuli amocanebis amoxsna ricxvis (sididis) procentis gamoTvlaze.
meToduri komentarebi: moswavle kargad unda daeuflos ricxvis sxvadasxva
saxiT Caweras. unda gaiTavisos, rom procenti measedebad Cawerili ricxvia.
moviyvanoT sidideTa measedebad Caweris saWiroebis magaliTebi (magaliTad,
1lari iyofa 100 TeTrad, amitom 0,75 lari = 75 TeTrs, an 1 metri iyofa 100
44

santimetrad, amitom 0,52m=52sm da sxva). davsvaT kiTxvebi: ra moqmedebas
vasrulebT larebis TeTrebSi an metrebis santimetrebSi gadasayvanad? ra
moqmedebas vasrulebT TeTrebis larebSi gadasayvanad? am analogiiT
ganvumartoT ricxvis procentis saxiT da, piriqiT, procentis wiladisa da
aTwiladis saxiT Cawera.
moswavleebs SeiZleba avuxsnaT, rom wiladma 1 miiRo gansakuTrebuli saxelwodeba
`naxevari~, 1 3 -ma – `mesamedi~, 1 1
2
-ma – `meoTxedi~ da -ma – `procenti~,
anu `measedi~, -ma – `promile~, anu `meaTasedi~.
4 100
1
1000
sanam amocanebis amoxsnaze gadavidodeT, unda amovxsnaT magaliTebi ricxvis
procentuli da procentis ricxvis saxiT gamosaxvaze. am mizniT, umjobesia,
Ipirvel gakveTilze ganxilul iqnes teqstis magaliTebi, savarjiSoebi 1, 3, 5 da
me-8 savarjiSos (cxrili) nawili (mag., me-2 da me-3 striqoni).
meore gakveTilze amoixsneba teqstis amocana, me-6 da me-7 savarjiSoebis
nawili da ramdenime amocana. orive dRes davaleba klasSi amoxsnilis msgavsi
miecema. magaliTad, me-6 da me-7 savarjiSoebis darCenili nawili da ramdenime
amocana.
komentarebi savarjiSoebis Sesaxeb da pasuxebi. # 1-5 da #8 savarjiSoebis
Sesrulebis Sedegad moswavles gamoumuSavdeba unari: Caweros ricxvi
procentis saxiT da procenti warmoadginos aTwiladis an wiladis saxiT.
# 6 , # 7, #9, #10, #11. unda gamoiyenon ricxvis procentis gamoTvlis wesebi.
#11. amocanis a) punqtisaTvis erT-erTi monacemi unda moiZion wina amocanaSi.
kerZod, saqarTvelos teritoria moicavs 69700kv.km-s.
#13. amocanis amoxsnis gza airCios TviTon moswavlem. magaliTad, jer daiTvlis,
Tu ramdeni moswavle eswreba da gamoaklebs moswavleTa saerTo raodenobidan
maT raodenobas, an jer procents gamoiTvlis im moswavleebisas, romelnic
acdenen da Semdeg maT raodenobas. pasuxi: 5 #14. gasavleli gza mTeli gzis
60%-ia , anu 5×0,6=3(km). # 15. friadosania 9 gogona.
#18. a) diagramaze naCvenebi bostneulis wili aris 25%+30%+15%+5%, anu 75%. im
bostneulis wili, romelic araa diagramaze naCvenebi, aris 100%-75%, anu 25%.
b) gamoiTvlian 12 tonis Sesabamis procentebs. pasuxi: 3t; 3,6t; 1,8t; 0,6t.
#19. 15× 2 × 15 =1,5(lari). #21. or fila SokoladSi giam daxarja 3 lari, ris
3 100
Semdegac darCa 7 lari. 14 cal saReW rezinaSi giam gadaixada 7 lari=1, 4 lari.
5
gias darCa 10-(3+1,4) lari, anu 5,6 lari. pasuxi: a) erTi fila Sokoladi Rirda 1,5
lari; b) erTi saReWi rezina Rirda 10 TeTri. gias darCa 5,6 lari.
#22. Seadginon amocanis Sesabamisi Seadginon sqematuri naxazi da ise amoxsnan.
10 45 60
vinaidan usg( 10, 45,60)=5, yovel or xes Soris manZili 5 m unda iyos. amitom
saWiroa (10+45+60):5+1-4=20 xis dargva.
#23 meore pirobiT gvanca ar cxovrobs mesame (bolo) sarTulze da araa meSvide
klaseli. igi an pirvel, an meore sarTulze cxovrobs da an meeqvse, an merveklaselia.
mesame pirobiT lana araa arc meeqvse da arc merveklaseli.e.i lana meSvideklaselia
da gvancas binis zemoT, anu mesame sarTulze cxovrobs. gvanca merveklaselia da
meore sarTulze cxovrobs. maSasadame, salome meeqvseklaselia da I sarTulze
cxovrobs.
45

3.3 rogor vipovoT ricxvi misi procentis mixedviT?
mizani: moswavleebs vaswavloT ricxvis(sididis) povna misi procentiT; mocemuli
procentiT gazrdiT an SemcirebiT miRebuli ricxvis gamoTvla.
codna da unari: moswavlem unda icodes:
1) ricxvis povnis wesi misi procentis mixedviT;
2) ricxvis mocemuli procentis gazrda-SemcirebiT miRebuli ricxvis
gamoTvla.
moswavles unda SeeZlos:
1) ricxvis povna misi procentis mixedviT;
2) ricxvis mocemuli procentis gazrda-SemcirebiT miRebuli ricxvis gamo-
Tvla;
3) amocanebis amoxsna sididis (magaliTad, fasis) ramdenime procentiT
momateba-daklebaze.
meToduri komentarebi: moswavleebma meeqvse klasidan ician ricxvis povna misi
nawilis mixedviT. nawilis mixedviT (amocanebi nawilebze gasameorebel
savarjiSoebSi wina paragrafSic hqondaT micemuli). amitom SegviZlia procentis
mixedviT ricxvis povna daviyvanoT nawilebze analogiur amocanaze.
sididis (magaliTad, fasis) procentiT momateba-daklebis amocanebSi
aucilebelia moswavleebs avuxsnaT, rom ` a sidide gaizarda (Semcirda) p%
-iT~
niSnavs ara a+ p%
-s, aramed a sidides daemata (gamoaklda) amave a sididis p % ,
a×
p a×
p
anu ( a + )-s ( ( a − )-s).
100 100
p
p
sasargebloa, agreTve, aRniSnuli formulebis a (1 + ) da a(1 − ) saxiT
100
100
gamoyeneba. magaliTad `fasi gaizarda 15%-iT~ niSnavs sawyisi fasis 1,15-ze
gamravlebas, xolo `fasi Semcirda 15%-iT~ niSnavs sawyisi fasis 0,85-ze
gamravlebas. aseT amocanebSi sawyisi sidide (magaliTad, sawyisi fasi) iTvleba
100%-ad da mas emateba an akldeba mocemuli procenti. aseTi midgoma aadvilebs
iseTi amocanebis amoxsnas, romlebSic xdeba mravaljeradi momateba-dakleba.
magaliTad, Tu sawyisi fasia a lari da moxda orjeradi gaZvireba jer 10%-iT da
Semdeg 20%-iT, maSin miRebuli fasi iqneba a × 1,1× 1, 2 = 1, 32 , anu sabolood
gaZvireba 32%-iT momxdara.
moswavleebs unda davusvaT Semdegi kiTxvebi:
_ ras niSnavs saqonlis 25%-iT gaZvireba? (pasuxi: sawyisi fasis 1,25-ze
gamravlebas).
_ ras niSnavs 90%-iani fasdakleba? (pasuxi: sawyisi fasis 0,1-ze gamravlebas, anu
sawyisi fasis meaTeds).
_ fasis 1,5-jer gazrda ramdeni procentiT gaZvirebaa? (50%-iT).
_ fasis 4-jer Semcireba ramdeni procentiT fasdaklebaa? (75%-iT) da a. S.
komentarebi savarjiSoebis Sesaxeb da pasuxebi. #1---#13 paragrafSi amoxsnili
savarjiSoebis analogiuria.
#15. a)10000× 1,1 = 11000 ; b)10000× 1,1 × 1,1 = 11000× 1,1 = 12100 .
#16. amocana amoixsnas etapebad:
I etapi. gamovTvaloT Rvinis warmoebaSi dabandebuli Tanxis raodenoba:
20000× 0, 4 = 8000 (l).
II etapi. gamovTvaloT sakonservo warmoebaSi dabandebuli Tanxis raodenoba:
20000 − 8000 = 12000 (l)
III etapi. gamovTvaloT TiToeul warmoebaSi miRebuli mogeba:
8000 × 0,6 = 4800 (l), 12000× 0,5 = 6000 (l)
IV etapi. gamovTvaloT saboloo Tanxa:
20000 + 4800 + 6000 = 308000 (l)
#17. vTqvaT, marTkuTxedis sigrZea a , xolo sigane b . maSin misi farTobi
S = a× b. sigrZis 10%-iT gazrdiT miviRebT 1,1a -s, xolo siganis 20%-iT gazrdiT
46

– 1, 2b -s. miRebuli marTkuTxedis farTobi iqneba 1,1a × 1, 2b = 1, 32ab = 1, 32S
. e. i.
farTobi gaizrdeba 32%-iT.
#18. vTqvaT, Tambaqos nawarmi Rirda a lari. 20 da 40 procentiT orjeradi
gaZvirebis Semdeg misi Rirebuleba iqneba a× 1, 2 × 1, 4 = 1, 68a(l). e. i. Tambaqo
gaZvirda 68%-iT.
#19. sasurvelia, amoixsnas jgufur mecadineobaze. moswavleebs avuxsnaT, rom
mTeli wre Seesabameba saqarTvelos mTel mosaxleobas. gamoTvlebis dros
sasurvelia kalkulatoris gamoyeneba.
# 20. nawilebze es savarjiSo aucileblad gavakeTebinoT, raTa moswavleebi
Semdegi paragrafisaTvis SevamzadoT.
3.4 ori sididis fardobis procentuli warmodgena
mizani: moswavleebs vaswavloT, ras gviCvenebs ori sididis fardobis
procentuli Canaweri. gamovumuSaoT maT procentebis saxiT warmodgenili
monacemebis Sedgenisa da gamoyenebis unari.
codna da unari: moswavlem unda icodes:
1) ra SemTxvevaSi SeiZleba ori sididis Sefardebis procentis saxiT
warmodgena;
2) ras gviCvenebs ori sididis procentuli fardoba.
moswavles unda SeeZlos:
1) erTi sididis gamosaxva meore sididis procentiT;
2) ori sididis fardobis procentis gamoyeneba praqtikuli amocanebis
amosaxsnelad;
3) procentis saxiT warmodgenili monacemebis Sedgena da gamoyeneba.
meToduri komentarebi: moswavleebma ician, rogor ipovon, erTi ricxvi (sidide)
meoris ra nawilia. Tu nawilis maCvenebel wilads CavwerT procentis saxiT, anu
gavamravlebT 100-ze, miviRebT fardobis procents, romelic gviCvenebs, erTi
ricxvi (sidide) meoris ramdeni procentia. amitom am paragrafis amocanebi
SegviZlia nawilebze amocanebis analogiurad amovxsnaT. sasurvelia,
davubrundeT wina paragrafis me-20 savarjiSos da amovxsnaT procentebSi.
sabolood moswavleebma unda aiTvison `formula~:
procenti = nawili ×100.
paragrafis amocanebTan dakavSirebiT unda aRvniSnoT, rom rZeSi (amocana1)
cximis procentuli Semadgenloba gviCvenebs, 100g rZeSi ramdeni grami cximia (4,5
g), xolo burTis tyorcnis sizuste gviCvenebs, imave sizustiT as-asjer rom
etyorcnaT, ramden burTs Caagdebdnen
( pirveli kalaTburTeli 75-s, meore 70-s). gansakuTrebuli yuradReba mivaqcioT
xsnarebsa da naerTebSi amaTuim nivTierebis koncentarciis cnebas, romelic
procentuli fardobiT ganimarteba.
16 200
komentarebi savarjiSoebis Sesaxeb da pasuxebi. #3.b) × 100% = % ; d)
72 9
7
× 100% = 35% .#4. TiToeuli monacemi unda gavyoT moswavleTa mTlian
20
raodenobaze (miviRebT, ra nawilia) da gavamravloT 100-ze.
# 5. sasurvelia, mivceT Sebrunebuli amocanebic. magaliTad, ra nawilia 25%,
50%, 1%, 100%. #15. amocana SeiZleba amoixsnas ori gziT:
I gza. jer gavigoT namatis raodenoba: 18– 12 = 6 (TeTri). gamovTvaloT 6-isa da
18-is fardobis procenti: 6 100% 50%.
12 × =
II gza. gamovTvaloT 18 -isa da 12-is fardobis procenti: 18 ( 100)% 150%.
12 × = iyo
100% da gaxda 150%, anu gazrdila 50%-iT.
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100
# 11. 100% 25%.
100 + 300 ⋅ =
6
# 12. 200⋅ 0,03 = 6; ⋅ 100% = 1%.
600
# 13. marilwylis wona 120 g iqneba. aqedan 30g marilia, xolo 90 g wyali.
pasuxi: 90 g wyali unda davumatoT.
# 16. pasuxi: 40%. #17. gamoTvalon da Sedegi Seamowmon kalkulatoriT.
#19-20 amocanebSi moswavleebi gaixseneben wriul da svetovan diagramebs.
#21. arsebobs ori aseTi wertili. monakveTis SigniT da mis gagrZelebaze unda
gadavzomoT B-dan 3,2sm. #22. Tavidan mocemuli ricxvi 100%-ia. 3-jer gazrdiT
iqneba 300%, e. i. gaizarda 200%-iT. #23. Tavidan mocemuli ricxvi 100%-ia.
aTjer Semcirebis Semdeg darCeba 10%, e.i. Semcirda 90%-iT.
#24. aseT amocanebSi unda gavutoloT im nivTierbis masebi (am SemTxvevaSi
marilis), romelic ucvleli rCeba. vTqvaT unda davamatoT x g wyali. wylis
damatebamde marilis masa iyo 200⋅ 0,3=60g, xolo Semdeg( x+200)⋅ 0,2. miviReT
gantoleba: ( x+200)⋅ 0,2 = 60. pasuxi: 100g.
#25. amocanis Sesabamisi gantolebaa 500⋅ 0,8+300⋅ 0,7=800⋅ x. pasuxi: 76,25%.
500⋅ 0,8 + 300⋅0,7 305 1
gamosaxulebiT ki ase amoxsnian: = = 76 .
800 4 4
#26. vTqvaT iyo x g, maSin 0,3x+ 50 = ( x+ 50)
⋅ 0,8 , saidanac x = 20. pasuxi: 20g.
#28. 1-ze patara wiladebi, romelTa mniSvneli 12-ia, aris:
1 2 3 4 5 6 7 8 9 10 11
, , , , , , , , , , .
12 12 12 12 12 12 12 12 12 12 12 maTgan 4 1
= , xolo 10 = 5 .
4 10
-sa da
12 3 12 6 12 12
-s Soris moqceuli wiladebis mricxvelebis jamia: 5+6+7+8+9=35.
Semajamebeli samuSao #5
I varianti
1) gamoTvale: a)15-is 20%; b)180-is 0,6 nawili.
2) ipove ricxvi, romlis: a) 3 5
nawili tolia 1,8-is; b) 25% tolia 16-is.
3) mariams saSinao davalebis Sesrulebaze 3sT da 20 wT dasWirda, anas ki –
am drois 75%. ramdeni wuTiT swrafad Seasrula anam davaleba?
4) marwyvis gamyidvelma gadawyvita, gaekeTebina yoveldRiuri Canaweri
myidvelebisa da maT mier nayidi marwyvis raodenobis Sesaxeb. xuTi dRis
Semdeg mis mier Segrovili informacia aseTi saxis iyo:
I dR.: 7 myidv. gaiyida 7kg marwyvi
II dR.: 10 myidv. gaiyida 12kg marwyvi
III dR.: 13 myidv. gaiyida 20kg marwyvi
IV dR.: 16 myidv. gaiyida 33kg marwyvi
V dR.: 21 myidv. gaiyida 40 kg marwyvi
am monacemebis mixedviT Seadgine cxrili da upasuxe SekiTxvebs:
a) sul ramdeni kg marwyvi gaiyida?
b) marwyvze moTxovnileba izrdeboda Tu mcirdeboda?
48

g) ramdeni procentiT meti marwyvi gayida gamyidvelma IV dRes II dResTan
SedarebiT?
d) gaagrZelos Tu Sewyvitos gamyidvelma marwyvis gayidva? ratom?
e) saSualod ramden kg marwyvs yidulobda erTi myidveli?
II varianti
2) gamoTvale: a) 20-is 15%; b) 120-is 2 3 nawili.
3) ipove ricxvi, romlis: a) 0,8 nawili tolia 8-is; b) 20% tolia 18-is.
3) nikas soflidan qalaqSi Casasvlelad 2 sT da 20 wT dasWirda, ukan
dasabruneblad ki – am drois 80% . ramdeni wuTiT naklebi dro moandoma
nikam sofelSi dabrunebas qalaqSi CasvlasTan SedarebiT?
4) mandarinis gamyidvelma gadawyvita, gaekeTebina yoveldRiuri Canaweri
myidvelebisa da maT mier nayidi mandarinis raodenobis Sesaxeb
(kilogramebSi). xuTi dRis Semdeg mis mier Segrovili informacia aseTi
saxis iyo:
I dR.: 12 myidv. gaiyida 30kg mandarini
II dR.: 9 myidv. – 21kg mandarini
III dR.: 8 myidv. – 13kg mandarini
IVdR.: 6 myidv. – 10kg mandarini
V dR.: 3 myidv. – 6kg mandarini
monacemebis mixedviT Seadgine cxrili da upasuxe SekiTxvebs:
a) sul ramdeni kg mandarini gaiyida?
b) izrdeboda Tu mcirdeboda moTxovnileba?
g) ramdeni procentiT naklebi mandarini gayida gamyidvelma meoTxe dRes
meore dResTan SedarebiT?
d) gaagrZelos Tu Sewyvitos gamyidvelma mandarinis gayidva? ratom?
e) saSualod ramden kg mandarins yidulobda erTi myidveli?
Sefasebis sqema – sul 10qula
1) amoxsna a) 0,5qula, amoxsna b) 0,5qula . sul -----1 qula.
2) amoxsna a) 1qula, amoxsna b) 1qula. sul ---- 2 qula.
3) gadaiyvana saaTebi wuTebSi – 0,5 qula, gamoTvala procenti – 1 qula,
gamoTvala sxvaoba – 0,5 qula. sul-----2 qula.
4) upasuxa SekiTxvebs: sul -----5 qula.
a) gamoTvala, ramdeni kg marwyvi gayida gamyidvelma sul – 1 qula.
b) gaigo, izrdeboda moTxovnileba marwyvze Tu mcirdeboda – 1 qula.
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g) gamoTvala, ramdeni procentiT meti marwyvi gayida gamyidvelma IV dRes II
dResTan SedarebiT – 1 qula.
d) upasuxa SekiTxvebs – 1 qula.
e) gamoTvala saSualo – 1 qula.
3.5 monacemTa ricxviTi maxasiaTeblebi
mizani: moswavleebs SevaxsenoT monacemTa adre naswavli ricxviTi
maxasiaTeblebi (udidesi da umciresi monacemebi, diapazoni, sixSire, saSualo) da
vaswavloT monacemTa moda. vaswavloT monacemTa sxvadasxva saxis diagramebiT
(maT Soris wertilovani da xazovani diagramebiT) saxiT warmodgena.
codna da unari: moswavlem unda icodes:
4) monacemTa ricxviTi maxasiaTeblebis (udidesi da umciresi monacemebi,
diapazoni, sixSire, saSualo) gamoTvla;
5) mocemul monacemebSi modis povna;
6) sxvadasxa saxiT warmodgenili monacemebis wakiTxva da analizi;
7) mocemuli monacemebis sxvadasxva saxis diagramiT warmodgena.
komentarebi savarjiSoebis Sesaxeb da pasuxebi #4. pasuxi: x= 3; #5. pasuxi: a= 3,6;
#8. pasuxi: g); #11. pasuxi: 7,5kg; #12. daiwereba ori wrfivi gantoleba;
#16. a) 5 b) 2,5 #17baRSi msxlis xeebis raodenoba xeebis saerTo raodenobis 1 6
nawilia. e. i. 1 2 12.
6 x = → x = baRSi 12 xea, maTgan atmisaa14, xolo msxlis _ 6.
Tavis mimoxilva.
#7. Camofasda 30%-iT.
#13. razmSi 49 jariskacia.
#15. 4 saaTSi jariskaci gaivlis gzis 6/5 nawils, amitom igi B-dan A-sken
gamobrundeba da gaivlis gzis 1/5 nawils, anu 4 km-s. pasuxi: 16km.
16. pasuxi: 150m. #17. vTqvaT, saWiroa x g xorbali. vwerT gantolebas:
x⋅0,8 ⋅ 1, 2 = 480 ⇒ x = 500 (g) #18. svetovani diagramidan vgebulobT A raionis
mosaxleobis raodenobas, xolo wriuli diagramidan(10000), am raodenobis
Sesabamis procents (20%). maSasaame, mosaxleobis saerTo raodenobaa 50000. C
raionSi _ 22 500, xolo A raionSi _ 17500.
50

Semajamebeli samuSao #6
I varianti
1. gamoTvale 60-is 2 nawilis 20%.
3
2. gamoTvale 2,5; 4; -2; 6,2; - 3,7; monacemebis a) saSualo; b) gabnevis
diapazini.
3. mocemuli sixSiruli cxrilis mixedviT masa a 30 32 25 b
gamoTvale
sixSire 4 1 2 1 2
a) a-s mniSvneloba, Tu cnobilia, rom monacemTa
modaa 29;
b) 2a+b, Tu cnobilia, rom monacemTa saSualo 0-is tolia.
4. ra koncentraciis xsnari miiReba, Tu 3kg Saqars 9kg wyalSi gavxsniT?
1. gamoTvale 45-is 2 nawilis 50%.
5
II varianti
2. gamoTvale 2; 4,5; -2,3; 6,4; - 3,1; monacemebis a) saSualo; b) gabnevis
diapazini.
3. mocemuli sixSiruli cxrilis mixedviT masa a 35 30 20 b
gamoTvale
sixSire 2 3 2 1 4
a) b-s mniSvneloba, Tu cnobilia, rom monacemTa
modaa 25.
b) a+2b, Tu cnobilia, rom monacemTa saSualo 0-is tolia.
4. ra koncentraciis xsnari miiReba, Tu 3kg Saqars 9kg wyalSi gavxsniT?
Sefasebis sqema – sul 10qula
1. gamoTvala nawili ---- 1 qula
gamoTvala procenti ---- 1 qula
2. a) ------- 1 qula
b)--------1 qula
3. a) ----1 qula
b) ---- 3 qula
4. gamoTvala mTliani masa ----1qula
gamoTvala procentuli fardoba ----- 1 qula
testis pasuxebi:
1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
b a a g b b a a a d g a b b g d a g b a d g d a d
51