matematika_VII.pdf

sarCevi 

Tavi1 

racionaluri ricxvebi 

1.1 simravle 6 

1.2 simravleTa gaerTianeba da TanakveTa 13 

1.3 dadebiTi da uaryofiTi ricxvebi 19 

1.4 ricxviTi wrfe 23 

1.5 mopirdapire ricxvebi. ricxvebis Sedareba 26 

1.6 ricxvis moduli 31 

1.7 ricxvebis Sekreba 34 

1.8 ricxvebis gamokleba 40 

1.9 ricxvebis gamravleba da gayofa 45 

1.10 ricxviTi gamosaxulebis mniSvnelobis gamoTvla 51 

1.11 kalkulatori 55 

I Tavis mimoxilva 59 

testi 64 

Tavi 2 

mravalkuTxedebi 

2.1 geometriuli figurebi 67 

2.2 sxivi, monakveTi, texili 72 

2.3 mosazRvre da vertikaluri kuTxeebi 75 

2.4 mravalkuTxedi 83 

2.5 samkuTxedi 85 

2.6 agebis amocanebi 91 

2.7 paralelur wrfeTa Tvisebebi 98 

2.8 samkuTxedis kuTxeTa Tvisebebi 101 

2.9 tolferda samkuTxedi 105 

2.10 samkuTxedebis toloba 110 

2.11 samkuTxedebis ageba 116 

2.12 paralelogrami 121 

2.13 wrewiris qorda da mxebi 127 

2.14 praqtikuli amocanebi 131 

II Tavis mimoxilva 135 

testi 143 

Tavi 3. 

monacemebi 

3.1 monacemTa saxeebi da maTi mopovebis saSualebebi 150 

3.2 ricxvis procentuli Cawera 155 

3.3 rogor vipovoT ricxvi misi procentis mixedviT? 160 

3.4 ori sididis fardobis procentuli Cawera 165 

3.5 monacemTa ricxviTi maxasiaTeblebi 172 

III Tavis mimoxilva 176 

4 Tavi 1 • racionaluri ricxvebi

testi 180 

Tavi 4 

erTwevri da mravalwevri 

4.1 cvladiani gamosaxuleba 183 

4.2 toli gamosaxulebebi 187 

4.3 naturalurmaCvenebliani xarisxi 192 

4.4 xarisxis Tvisebebi 197 

4.5 erTwevri da mravalwevri 201 

4.6 moqmedebebi mravalwevrebze 208 

4.7 mravawevris mravalwevrze gamravleba 213 

4.8 mravalwevris mamravlebad daSla 218 

4.9 gantoleba 224 

4.10 wrfivi gantoleba 228 

4.11 amocanis amoxsnis algebruli meTodi 232 

4.12 mimdevrobebi 238 

4.13 ariTmetikuli progresia 243 

IV Tavis mimoxilva 248 

testi 254 

Tavi 5 

koordinatTa sistema 

5.1 wertilis koordinatebi 257 

5.2 wertilis ageba misi koordinatebiT 261 

5.3 grafiki 264 

5.4 pirdapirproporciulobis grafiki 270 

5.5 figuraTa paraleluri gadatana 275 

V Tavis mimoxilva 279 

testi 283 

pasuxebi 286 

5

Tavi 1 racionaluri ricxvebi 

1.1 simravle 

mizani: 

6 Tavi 1 • racionaluri ricxvebi 

gavecnoT simravlisa da qvesimravlis cnebebs. 

xSir SemTxvevaSi Cveni msjelobis sagani aris ara calkeuli obieqti, 

aramed maTi garkveuli jgufi, krebuli, erToblioba. magaliTad, 

SesaZloa vmsjelobdeT: 

� ara calkeul ferwerul tiloebze, aramed maT koleqciaze; 

� ara calkeul fexburTelebze, aramed safexburTo gundze; 

� ara sibrtyis calkeul wertilebze, aramed maT iseT 

erTobliobaze, rogoricaa monakveTi. 

sxvadasxva saxis erTobliobis, jgufisa Tu krebulis aRsaniSnavad 

maTematikaSi gamoiyeneba erTiani termini - simravle. swori warmodgena 

simravleze, rogorc maTematikur cnebaze, SeiZleba konkretuli 

magaliTebisa da simravleTa zogadi Tvisebebis gacnobiT CamoviyaliboT. 

magaliTad, ganxilvis sagani SeiZleba iyos: 

� erovnuli biblioTekis wignTsacavSi daculi wignebis simravle; 

� saklaso oTaxis merxebis simravle; 

� naturaluri ricxvebis simravle; 

� mocemuli samkuTxedis wertilTa simravle; 

� oTxkuTxedebis simravle da a.S. 

yoveli simravle elementebisgan Sedgeba. naturaluri ricxvebis 

simravlis elementebia ricxvebi 1, 2, 3 da a.S. ricxvi 9 aris naturaluri 

ricxvebis simravlis elementi, xolo ricxvi 2,5 ar aris naturaluri 

ricxvebis simravlis elementi. 

simravlis elementebis miTiTeba moxerxebulia maTi sruli 

CamonaTvalis moyvaniT. simravle, romelic Sedgeba a, b, c da d 

elementisagan, ase CavweroT: {a, b, c, d}. kerZod, erTniSna naturaluri 

ricxvebis simravle ase Caiwereba: {1, 2, 3, 4, 5, 6, 7, 8, 9}. 

elementebis CamonaTvalis saxiT SesaZlebelia mocemul iqnas mxolod 

is simravleebi, romelTa elementebis raodenoba sasrulia, anu sasruli 

simravleebi. 

usasrulo simravlis mosacemad uTiTeben iseT Tvisebebs, romelic 

gaaCnia am simravlis yovel elements da ar gaaCnia arcerT sxvas. 

magaliTad, 5-is jeradi kenti naturaluri ricxvebis simravle aris 

usasrulo simravle, romlis elementebia ricxvebi 5, 15, 25, 35, 45 ... . 

simravle gansazRvrulia, Tu garkveviTaa cnobili, romeli elementi 

ekuTvnis am simravles da romeli - ara. 

simravleebs, rogorc wesi, didi laTinuri asoebiT aRvniSnavT: A, B, C, ... , 

simravleTa elementebs ki patara laTinuri asoebiT: a, b, c, ... . is faqti, 

rom x aris A simravlis elementi, Semdegnairad Caiwereba: x∈A. ikiTxeba:

“x ekuTvnis A-s”. y∉B Canaweri ikiTxeba: “y ar ekuTvnis B-s” da niSnavs, 

rom y ar aris B simravlis elementi. 

ganvixiloT 3-is jeradi luwi naturaluri ricxvebis simravle. es 

simravle A-Ti aRvniSnoT. A simravle usasruloa. misi elementebia 

ricxvebi: 6, 12, 18, 24 ... . B-Ti aRvniSnoT 6-is jeradi naturaluri 

ricxvebis simravle. B simravlis elementebia: 6, 12, 18, 24 ... . rogorc 

vxedavT, A da B simravle erTi da imave elementebisgan Sedgeba _ es 

simravleebi tolia, A = B. 

simravleebs, romlebic erTi da imave elementebisgan Sedgeba, 

toli simravleebi ewodeba. 

vTqvaT, A romelime skolis meSvideklaselTa simravlea, xolo B amave 

skolis moswavleTa simravle. yoveli meSvideklaseli, cxadia, skolis 

moswavlec aris. amitom, A simravle B simravlis nawili, anu B_s 

qvesimravlea. 

A simravles ewodeba B simravlis qvesimravle, Tu A simravlis 

yoveli elementi amave dros B simravlis elementicaa. 

is faqti, rom A simravle B simravlis qvesimravlea, ase Caiwereba: A⊂B. 

simravleebs Soris mimarTebis TvalsaCinod warmosadgenad venis 

diagramas iyeneben. 

magaliTad, A ⊂ B mimarTebis SemTxvevaSi venis 

diagrama marjvniv moTavsebul naxazzea 

warmodgenili. am naxazze A simravlis elementebs 

pirobiTad mcire wris wertilebi, xolo B simravlis 

elementebs – didi (momcveli) wris wertilebi 

Seesabameba. 

simravles, romelic arcerT elements ar Seicavs carieli simravle 

ewodeba. carieli simravlis magaliTebia: 2-is jeradi kenti ricxvebis 

simravle, mTvareze mcxovreb adamianTa simravle da a.S. cariel simravles 

∅ simboloTi aRniSnaven. carieli simravle nebismieri simravlis 

qvesimravles warmoadgens. 

sasruli A simravlisTvis n(A)-Ti A simravlis elementebis raodenobas 

aRniSnaven. magaliTad, Tu A = {a, b, g}, maSin n(A) = 3, xolo Tu A = {a, b, 

g, 5, 7,}, maSin n(A) = 5. cxadia, rom n(∅) = 0. 

upasuxe kiTxvebs 

1. aris Tu ara ricxvi 42 Svidis jeradi orniSna naturaluri ricxvebis 

simravlis elementi? miekuTvneba Tu ara am simravles ricxvi 7? ricxvi 54? 

2. rogor ikiTxeba da ras niSnavs Canaweri x∈E ? Canaweri c∉F ? 

3. rogor simravleebs ewodeba sasruli simravleebi? 

4. rogor simravleebs ewodeba toli simravleebi? 

5. rogor ganimarteba simravlis qvesimravle ? 

6. rogor ikiTxeba da ras niSnavs Canaweri D⊂E ? 

B 

A 

Tavi 1 • racionaluri ricxvebi 7

savarjiSoebi 

1 a) daasaxele saqarTvelos qalaqebis simravlis ramdenime 

elementi; 

b) daasaxele zRvispira qalaqebis simravlis ramdenime elementi; 

g) daasaxele qalaqi, romelic ekuTvnis saqarTvelos qalaqebis 

simravles, magram ar ekuTvnis zRvispira 

qalaqebis simravles; 

d) daasaxele elementi, romelic ekuTvnis 

zRvispira qalaqebis simravles da 

ar ekuTvnis saqarTvelos qalaqebis 

simravles. 

2 ganvixiloT S = {a, b, g, d, e, 2, 5, 3, 1, 0} 

simravle. 

a) ramdeni elementia S simravleSi? 

b) Cawere S simravleSi Semavali laTinuri asoebis simravle. 

ramdeni elementia am simravleSi? 

g) S simravlis elementebidan ramdenia qarTuli anbanis aso? 

d) S simravlis elementebidan ramdeni ar aris naturaluri 

ricxvi? 

e) S simravlis elementebis saerTo raodenobis ra nawilia 

laTinuri asoebi? naturaluri ricxvebi? 

3 daasaxele ocze naklebi martivi ricxvebis simravlis yvela 

elementi. ramdeni elementia am simravleSi? Cawere es simravle 

elementebis CamonaTvalis saxiT. 

4 daasaxele evropis saxelmwifoTa simravlis ramdenime elementi. 

ekuTvnis Tu ara am simravles germania? brazilia? espaneTi? 

CineTi? 

5 

vTqvaT, F aris ukveci wesieri wiladebis simravle. 

sasrulia Tu ara es simravle? 

Semdegi Canawerebidan romelia marTebuli? 

a) 1 – 2 ∈ F ; b) 2 – 5 ∉F ; g) 5 – 3 ∈F ; d) 3 – 24 ∈F . 

6 vTqvaT, A qarTuli xmovani asoebis simravlea. Semdegi 

Canawerebidan romlebia marTebuli? 

7 


a) b ∈ A; b) u ∈A; g) d ∉ A. d) u ∈A; 

qvemoT CamoTvlil simravleTagan romelia sasruli da romeli - 

usasrulo? 

a) orniSna naturaluri ricxvebis simravle; 

b) kenti naturaluri ricxvebis simravle; 

g) 100-is gamyofebis simravle;

d) 3-is jeradi naturaluri ricxvebis simravle; 

e) monakveTis wertilTa simravle; 

v) samkuTxedis wveroebis simravle. 

8 qvemoCamoTvlil simravleTagan romelia sasruli da romeli - 

usasrulo? 

a) mTeli ricxvebis simravle; 

b) luwi naturaluri ricxvebis simravle; 

g) 1000-is gamyofebis simravle; 

d) 10-is jeradi naturaluri ricxvebis simravle; 

e) mocemuli sxivis wertilTa simravle; 

v) mocemuli oTxkuTxedis gverdebis simravle. 

9 daasaxele 5-is jeradi naturaluri ricxvebis simravlis ramdenime 

elementi. aris Tu ara sasruli es simravle? 

10 gansazRvre sityvierad usasrulo simravle, romlis 

elementebia: 

a) 1, 3, 5, 7, ... ; 

b) 1, 2, 3, 4, ... ; 

g) 1, 4, 9, 16, ... ; 

d) 10, 100, 1000, 10000, ... 

11 gansazRvre sityvierad simravle, romlis elementebia: 

a) 2, 3, 5, 7, 11, 13 ... ; 

b) 9, 18, 27, 36, ... ; 

g) a, b, g, d, ... ; 

d) a, b, c, d, ... 

12 mocemulia oTxi simravle: 

A={b, r, a, z, i}, B={b, z, a, r, i} 

C={a, z, r, i} D={z, a, r, i} 

Semdegi Canawerebidan romeli Canawerebia marTebuli? 

a) A=B; b) A= C; g) B⊂ C; d) D⊂ A; e) b ∈A; v) b∉ B. 

13 ganixile A ={a, b, c, d, e} simravle da Cawere: 

a) A simravlis xmovnebis qvesimravle; 

b) A simravlis Tanxmovnebis qvesimravle; 

g) A simravlis ramdenime erTelementiani qvesimravle; 

d) A simravlis ramdenime oTxelementiani qvesimravle. 

14 Cawere A ={20, 3, 7 – , 0, 4, 5, 9 

8 1 – } simravlis 

3 

a) mTeli ricxvebis qvesimravle; 

b) naturaluri ricxvebis qvesimravle; 

g) kenti ricxvebis qvesimravle; 

d) 3-is jeradi ricxvebis qvesimravle; 


e) ramdenime orelementiani qvesimravle; 

v) ramdenime samelementiani qvesimravle. 

15 mocemuli diagramis mixedviT gaarkvie, romeli Canaweria 

marTebuli: 

16 vTqvaT, A skolis moswavleTa simravlea, B skolis warCinebul 

moswavleTa simravle. mocemuli diagramebidan romeli gamosaxavs 

A da B simravleebs Soris mimarTebas? 

a) b) g) d) 

A 

17 mocemuli diagramebidan romeli gamosaxavs A da B simravleebs 

Soris mimarTebas, Tu: 

a) A naturalur ricxvTa simravle, xolo B luw naturalur 

ricxvTa simravlea? 

b) A luw ricxvTa simravle, xolo B kent ricxvTa simravlea? 

g) A Svidis jeradi ricxvebis simravle, xolo B rvis jeradi 

ricxvebis simravlea 

a) b) g) d) 

18 ras udris n(A), Tu A 

a) qarTuli anbanis Tanxmovani asoebis simravlea; 

b) 9-is jeradi orniSna ricxvebis simravlea; 

g) 100-is gamyofebis simravlea; 

d) 30-is martivi gamyofebis simravlea. 

19 vTqvaT M M marTkuTxedebis simravlea,K K kvadratebis simravle, 

xolo O O oTxkuTxedebis simravle. Semdegi diagramebidan romeli 

gamosaxavs M M, K K da O O simravleebs Soris mimarTebas? 

a) b) g) 

M 


O 

C 

A 

B 

K 

B 

A 

M 

a) A⊂B; b) A⊂C; 

g) C⊂A; d) B⊂A. 

B 

A 

B 

A 

B B A 

A 

B 

K 

O 

B 

A 

O 

A 

K 

M 

B

20 vTqvaT, A 3-is jeradi naturaluri ricxvebis simravlea, B 6-is 

jeradi naturaluri ricxvebis simravle, xolo C 15-is jeradi 

naturaluri ricxvebis simravle. Semdegi naxazebidan romeli 

gamosaxavs A, B da C simravleebs Soris mimarTebas? 

a) b) g) 

21 ra simravleebi SeiZleba iyos gamosaxuli mocemul naxazze? 

a) A - adamianebi, B - skolis moswavleebi, 

C – studentebi; 

b) A- cxovelebi, B- lomebi, C - vefxvebi; 

g) A - ricxvebi, B - martivi ricxvebi, 

C – luwi ricxvebi. 

B 

A 

d) A - ricxvebi, B - mTeli ricxvebi, 

C - naturaluri ricxvebi. 

C 

22 vTqvaT M M 2-is jeradi, F 5-is jeradi, xolo K K 10-is jeradi 

ricxvebis simravlea. gamosaxe am simravleebs Soris mimarTeba 

diagramis saSualebiT. 

23 

Cawere Sereuli ricxvis saxiT: 

a) 7 – 2 b) 1 

–9 

e) 17 – v) 

16 

13 – 

9 

24 

Sekvece wiladi: 

a) 7 – 14 b) 8 – 16 

e) 18 – 54 

C 

A 

B 

v) 27 – 63 

C 

g) 23 – 13 

z) 23 – 11 

g) 25 – 35 

z) T) 

d) 69 – 25 

T) 68 – 24 

d) 69 – 46 

25 

ricxvebis: 1, 2, 3, gamoyenebiT Seadgine yvela SesaZlo 

a) wesieri wiladi; b) arawesieri wiladi. 

26 

rogor Seicvleba wiladis mniSvneloba, Tu mis mricxvels: 

a) gavzrdiT samjer? b) SevamcirebT orjer? 

27 

rogor Seicvleba wiladis mniSvneloba, Tu mis mniSvnels: 

a) gavzrdiT samjer? b) SevamcirebT orjer? 

A 

28 mocemuli wiladis mricxveli gaamravles 15-ze. ra ricxvze unda 

gaamravlon wiladis mniSvneli, rom miiRon mocemul wiladze 

B 

A 

B 

C 


a) 5-jer didi ricxvi? 

b) 3-jer patara ricxvi? 

29 Seadare ricxvebi: 

a) 1 – 2 da 3 – 4 b) 3 – 10 da 2 – 15 g) 9 – 16 da 9 – 11 d)1 1 – 2 da 1 3 – 4 

e) 11 – da 

21 3 – v) 

14 2 – da 

5 13 – z) da T) 7 da 

40 35 – 

6 

30 

eka, Tamuna da naTia biblioTekars wignebis aRdgenaSi miexmarnen. 

ekam wignebis 1 – aRadgina, Tamunam _ 

80 5 – , xolo naTiam _ 

16 3 – . 

8 

romelma gogonam aRadgina meti wigni? 

31 

gigim 20 amocanidan sworad 15 amocana amoxsna, nikam ki 10 

amocanidan mxolod 3 amocana amoxsna SecdomiT. romeli 

moswavlis Sedegia ukeTesi? 

32 

cxrilSi mocemulia giorgis saskolo weris Sedegebi sam saganSi: 

1 gamoTvaleT 

2 gamoTvaleT 

3 wrewirze 

sagani 


qulaTa 

maqsimumi 

dagrovili 

qulebi 

qarTuli ena 100 83 

qimia 50 42 

maTematika 25 20 

gaarkvie, romeli sagani icis giorgim ukeTesad. 

jgufuri samuSao 

n(A), Tu A 

a) 12-is gamyofebis simravlea; 

b) 10-is jeradi orniSna naturaluri ricxvebis simravlea; 

g) 35-is martiv gamyofTa simravlea. 

n(A) - n(B), sadac 

a) A samniSna naturaluri ricxvebis, xolo B orniSna 

naturaluri ricxvebis simravlea; 

b) A im orniSna ricxvebis simravlea, romelTa CanawerSi 

mxolod kenti cifrebia gamoyenebuli, xolo B im orniSna 

ricxvebis simravle, romelTa CanawerSi mxolod luwi cifrebia 

gamoyenebuli. 

moniSnulia 5 wiTeli da 5 lurji wertili. A im 

monakveTebis simravlea, romelTa boloebidan erTi wiTeli, 

meore ki lurji wertilia, xolo B im monakveTebis simravle, 

romelTa boloebi lurji wertilebia. romelia meti n(A) Tu 

n(B)? ramdeniT?

3.2 ricxvis procentuli Cawera 

mizani: 

aTwiladis 100-ze 

gamravlebisas mZime 

ori cifriT marjvniv 

gadagvaqvs. 

viswavloT: 

1. ricxvis procentuli Cawera; 

2. procentis Cawera wiladis saxiT; 

3. ricxvis procentis gamoTvla. 

monacemebis gadmosacemad xSirad ricxvis procentul 

Canawers iyeneben. 

ricxvis procentuli Canaweri gviCvenebs ricxvSi 

measedebis raodenobas, magaliTad: 1 = 50 

2 100 =50%; 

0,75=75%. 

procentis saxiT Casawerad ricxvs vamravlebT 100-ze da vuwerT 

procentis niSans: `%„. 

aTwiladis 100-ze 

gayofisas mZime ori 

cifriT marcxniv 

gadagvaqvs. 

magaliTi 1. CavweroT procentis saxiT ricxvi: a) 3 

1 ; 

4 

b) 0,432. 

3 3 

amoxsna: a) 1 = (1 × 100)% = 175% ; 

4 4 

b) 0,432 = (0,432× 100)% = 43,2% . 

aseve unda moviqceT nawilis procentuli CawerisaTvis. magaliTad, 

ricxvis 3 

5 nawili igivea, rac am ricxvis 3 × 100 procenti, anu 60%; 0,36 

5 

nawili igivea, rac 0,36× 100 procenti, anu 36%. piriqiT, 

ricxvis 1% am 

ricxvis measedi 

nawilia. 

procentis saxiT Cawerili ricxvis wiladis saxiT Casawerad 

sakmarisia, procentis ricxvi gavyoT 100-ze. 

magaliTi 2. CavweroT wiladis saxiT: a) 48%; b) 

325,8%. 

12 

amoxsna: a) 48% = 48:100 = ; 

25 

b) 325,8%=325,8:100 = 3,258. 

amocana 1. levanis klasSi 30 moswavlea, maTgan 60% gogonaa. ramdeni 

gogonaa levanis klasSi? 

ricxvis 100% TviT 

am ricxvis tolia. 

60 3 

amoxsna: CavweroT 60% wiladis saxiT: 60% = = . 

100 5 

e. i. leva nis TanaklaselTa 3 

nawili, anu 

3 

30× = 18 

5 

moswavle gogonaa. 

5 

pasuxi: levanis klasSi 18 gogonaa. 

a ricxvis p procenti a ricxvis p - 100 nawilis, anu 

magaliTi 3. vipovoT 120-is 2,5%. 

120× 2,5 

amoxsna: = 3 . 

100 

a x p 

- 100 -is tolia. 

Tavi 3 • monacemebi 155


156 Tavi 3 • monacemebi 

1. rogor CavweroT ricxvi procentis saxiT? 

2. rogor CavweroT procentulad Cawerili ricxvi wiladad? 

3. ra kavSiria ricxvis nawilsa da procents Soris? 

4. rogor gamovTvaloT mocemuli ricxvis procenti? 

5. Camoayalibe aTwiladis procentuli Caweris wesi. 

6. Camoayalibe procentis aTwiladad Caweris wesi. 

savarjiSoebi 

1 Cawere mocemuli ricxvi procentis saxiT: 

a) 1; b) 4 1 

7 

13 16 

; g) ; d) ; e) ; v) ; 

5 10 10 20 25 

z) 19 

; T) 

1 

1 ; i) 

50 2 3 

2 ; k) 0,5; l) 0,26; m) 0,256; 

4 

n) 1,01; o) 1,15; p) 2,2356. 

2 procentulad mocemuli ricxvi Cawere wiladis saxiT: 

a) 3%; b) 17%; g) 10%; d) 1%; e) 25%; v) 5%; z) 50%; T) 85%; 

i) 115%; k) 27,5%; l) 32,215%; m) 1 

3 % ; n) 

3 

1 

33 % ; o) 0,01%. 

3 

3 ricxvis ramdeni procentia am ricxvis 

a) 1 

1 

11 

nawili? b) nawili? g) nawili? d) 0,2 nawili? 

2 4 20 

e) 0,75 nawili? v) 0,99 nawili? z) 0,125 nawili? T) 1 

3 nawili? 

4 ricxvis ra nawilia am ricxvis 

a) 25%? b) 50%? g) 10%? d) 100%? e) 67%? v) 2,5%? z) 0,1%? 

5 gamoTvale: 

a) 20-is 25%; b) 45-is 10%; g) 200-is 5%; d) 7,2-is 50%; 

e) 3,5-is 1%; v) 1,7-is 80%; z) 1 

1 

-is 3%; T) 3 -is 33%. 

3 33 

6 naxazis mixedviT daadgine, wris ramdeni procentia 

gaferadebuli. 

a) b) g) d)

7 romelia meti: 

a) 12-is 25% Tu 20-is 10% ? 

b) 10-is 5% Tu 15-is 3%? 

g) 3,6-is 50% Tu 50-is 3,6%? 

d) 1 

1 

-is 100% Tu 100-is 

10 10 %? 

8 Seavse cxrili 

procenti aTwiladi wiladi 

10% 

0,25 

3/8 

15,6% 

0,268 

2/25 

9 Saqris Warxlisagan miRebuli Saqris wona Warxlis wonis 13%-ia. 

ramdeni kg Saqari miiReba 200kg Saqris Warxlisagan? 

10 Saqris lerwmisagan miRebuli Saqris wona lerwmis wonis 15%-ia. 

ramdeni kg Saqari miiReba 1tona Saqris lerwmisagan? 

11 saqarTvelos farTobi 69700 kvadratuli kilometria. misi 

teritoriis 54% mTebs ukavia. ramdeni kvadratuli kilometri 

ukavia mTebs? 

12 klasis 25 moswavlidan gakveTils eswreboda 

80%. ramdeni moswavle ar eswreboda 

gakveTils? 

13 ramdeni TeTria 2 laris 20%? 

14 ramdeni kilometri darCeba gasavleli 

mgzavrs gzis 40%-is gavlis Semdeg, Tu 

mTeli gzis sigrZe 5 kilometria? 

15 klasis 30 moswavlidan 60% gogonaa. gogonaTa 50% friado sania. 

ramdeni gogonaa friadosani? 

16 cxrilSi mocemulia, saqarTvelos teritoriis ramdeni procenti 

ukavia mis xuT regions: 

qarTli 19,3% 

kaxeTi 15,6% 

afxazeTi 12,7% 

imereTi 8,7% 

samegrelo 6,5% 


a) gadaixaze cxrili, monacemebis mixedviT gamoTvale regionebis 

farTobebi da miRebuli Sedegebi Seitane cxrilis mesame svetSi 

(gamoTvlebis gasamartiveblad gamoiyene kalkulatori). 

b) saqarTvelos teritoriis ramden procents Seadgens danarCeni 

regionebi? 

g) miRebuli Sedegebis mixedviT Seadgine svetovani diagrama. 

17 yuTSi 20 burTia. burTebis 20% TeTria, 25% _ Savi, danarCeni ki 

_ wiTeli. burTebis ramdeni procentia: a) wiTeli? b) TeTri an 

Savi? g) ramdeni burTia TeTri an wiTeli? 

18 maRaziaSi erTi Tvis ganmavlobaSi 

12 t bostneuli 

gaiyida. svetovan 

diagramaze naCvenebia 

gayiduli zogierTi 

saxeobis bostneulis wili 

procentebSi. 

a) ramdeni procentia im 

bostneulis wili, romelic 

araa diagramaze 

naCvenebi? 

b) TiToeuli saxeobis ramdeni tona bostneuli gaiyida? 

19 ramdeni laria 15 laris 2 

nawilis 15%? 

3 

20 daasabuTe, rom a ricxvis b procenti da b ricxvis a procenti 

erTmaneTis tolia. 

21 gias 10 lari hqonda. am Tanxis 30%-iT man ori fila Sokoladi 

iyida, darCenili Tanxis 20%-iT ki _ 14 cali saReWi rezina. 


a) ra Rirda erTi fila Sokoladi? 

b) ra Rirda erTi saReWi rezina? 

g) ra Tanxa darCa gias? 

gayiduli bostneuli procentebSi 

35 

30 

25 

20 

15 

10 

5 

0 

kartofili 

kombosto 

stafilo 

22 gzis sworxazovani monakveTis gaswvriv dargulia oTxi xe. manZili 

mezobel xeebs Soris 10, 45 da 60 metria. sul cota, ramdeni 

xis dargva mogviwevs damatebiT, nebismier or mezobel xes Soris 

manZili toli rom aRmoCndes? 

23 gvanca, lana da salome erTi saxlis pirvel, meore da mesame 

sarTulze cxovroben (ara aucileblad am Tan mimdevrobiT). maTgan 

erTi meeqvse, meore meSvide da mesame ki – merveklaselia. 

• meeqvseklaseli gogona cxovrobs pirvel sarTulze. 

xaxvi

• meSvideklaseli gogona cxovrobs gvancas binis zemoT. 

• merveklaseli gogona cxovrobs lanas binis qvemoT. 

romeli gogona romelklaselia da meramdene sarTulze 

cxovrobs? 

24 nikam daxarjuli bunebrivi gazis safasuri 125 lari gadaixada. 

gadaxdilma Tanxam misi xelfasis 0,12 nawili Seadgina. ra xelfasi 

aqvs nikas? 

25 salomem 135 lari eleqtroenergiis, 38lari dasufTavebis da 12 

lari sasmeli wylis moxmarebis safasuri gadaixada. gadaxdilma 

Tanxam misi xelfasis 0,2 nawili Seadgina. ra xelfasi aqvs 

salomes? 

26 ipove proporciis ucnobi wevri: 

2 4 

2 4 

x 2 7 14 5 y 

a) = ; b) = ; g)1, = ; d) 3 = 5 ; e) 

9 3 t 9 4 1, 2 1 

1 

z 

2 

27 ricxvi 512 dayavi 

a) 5-is, 11-is da 16-is proporciul nawilebad 

b) 4 : 5 : 7 SefardebiT. 

28 gadaixaze da Seavse cxrili 

a) Tu cnobilia, rom x da y sidideebs Soris 

pirdapirproporciuli damo kidebulebaa: 

4 × 1, 5 1, 4 : 0, 7 

= 

. 

x 0,75 + 1,25 

x 1 2 0,5 5 30 3 0,2 

y 3 15 36 0,6 2 0,3 

b) Tu cnobilia, rom x da y sidideebs Soris ukuproporciuli 

damo kidebulebaa: 

x 2 1 0,5 5 11 3 0,2 

y 10 15 100 0,5 2 0,9 


3.3 rogor vipovoT ricxvi misi procentis mixedviT 

mizani: 


viswavloT: 

1. ricxvis gamoTvla misi procentiT; 

2. mocemuli procentiT gazrdiT an SemcirebiT 

miRebuli ricxvis gamoTvla; 

3. praqtikuli amocanebis amoxsna fasis momatebadaklebasa 

da xarjTaRricxvaze. 

ganvixiloT amocana, romlis amosaxsnelad dagvWirdeba ricxvis povna 

misi mocemuli procentiT. 

amocana 1. 20% saSemosavlos daqviTvis Semdeg maswavleblis xelfasma 

360 lari Seadgina. ramdenia maswavleblis xelfasi daqviTis gareSe? 

amoxsna: maswavleblis xelfasi daqviTvis gareSe CavTvaloT 100%-ad. 

maSin 20%-is daqviTvis Semdeg darCeboda xelfasis 80%, rac amocanis 

pirobis Tanaxmad 360 laris tolia. 

amocanis amosaxsnelad saWiroa vipovoT ricxvi, romlis 80% 360-is 

tolia. 

vTqvaT, es ricxvia x, maSin misi 80% iqneba 0,8x. amocanis pirobis 

Tanaxmad vwerT: 

0,8x = 360 

Tu am tolobis orive mxares gavyofT 0,8-ze, miviRebT: 

x = 360 : 0,8 = 450. 

pasuxi: maswavleblis xelfasi daqviTvis gareSe 450 laria. 

imisaTvis, rom vipovoT x ricxvi, romlis p procenti a ricxvis 

tolia, saWiroa: 

p 

I. p% CavweroT wiladis saxiT (miviRebT -s). 

100 

p 

II. a ricxvi gavyoT wiladis saxiT Caweril p% -ze (anu -ze). 

100 

magaliTi 1. vipovoT x ricxvi, romlis 

a) 25% udris 20-s; b) 3,125% udris 1,5-s. 

I. 25%=0,25 I. 3,125%=0,03125 

I I. x = 20 : 0,25 = 80 I I. x = 1,5:0,03125 =48 

praqtikaSi xSirad gvxvdeba amocana, romelSic 

mocemuli ricxvi (sidide) izrdeba an mcirdeba mocemuli 

procentiT. magaliTad: 

amocana 2. benzinis fasma 10%-iT moimata. ra Rirs litri 

benzini amJamad, Tu momatebamde misi fasi 2,1 lari iyo? 

amoxsna: 2,1 laris 10%-iT gazrdaSi igulisxmeba 2,1 laris- 

Tvis misive 10%-is damateba, anu 2,1l + 2,1l × 0,1=2,31l. 

ricxvis 10%iT 

gazrda niSnavs 

am ricxvis 1,1-ze 

gamravlebas.

imave amocanis amoxsna SegviZlia Semdegi msjelobiT: 

Tu momatebamde benzinis fass, anu 2,1 lars, CavTvliT 100%-ad, maSin 

10%-is momatebis Semdeg fasi gaxdeba 110%. 2,1-is 110% ki udris: 

2,1× 1,10 = 2,31 . 

pasuxi: amJamad benzini 2lari da 31 TeTri Rirs. 

a ricxvis p% -iT gazrdiT miRebuli ricxvis gamosaTvlelad 

saWiroa: 

I. gamovTvaloT a ricxvis p % , anu . 

I I. a ricxvs davumatoT misi p % , anu , miviRebT 

magaliTi 2. ra ricxvi miiReba, Tu12-s 5% -iT gavzrdiT? 

12× 5 

I. 12-is 5% udris =0,6. 

100 

I I. 12 + 0,6 = 12,6. 

am magaliTSi imave Sedegis misaRebad sakmarisia gamoviTvaloT 12-is 

(100 + 5)%, anu 12 gavamravloT 1,05-ze: 

12× 1,05 = 12,6. 

amocana 3. maRaziam saSobaod 60%-iani fasdakleba gamoacxada. ra fasSi 

gaiyideba fexsacmeli, romelic fasdaklebamde 120 lari Rirda? 

amoxsna: 60% -iani fasdakleba niSnavs Zveli fasis 

Semcirebas am fasis 60%-iT. gamovTvaloT 120-is 

60%: 120× 0,60 = 72 . e. i. fasi 72 lariT Semcirebula. 

amitom axali fasi iqneba 

120- 72 = 48 (lari). 

SevniSnoT, rom es amocana SeiZleba Semdegi 

msjelobiTac amovxsnaT: sawyisi fasi 120 lari 

CavTvaloT 100%-ad. maSin 60%-iT fasdaklebis Semdeg 

miviRebT 40%-s. 120-is 40% ki udris: 120× 0,40 = 48 . 

ricxvis 60%-iT 

Semcireba niSnavs pasuxi: fexsacmeli 48 larad gaiyideba. 

am ricxvis 0,4-ze 

a 

gamravlebas. 

ricxvis p% -iT SemcirebiT miRebuli ricxvis gamosaTvlelad 

saWiroa: 

a× p 

I. gamovTvaloT a ricxvis p % , anu . 

100 

a× p 

I I. a ricxvs gamovakloT misi p % , anu a − . 

100 

magaliTi 3. ra ricxvi miiReba, Tu 80-s SevamcirebT 25%-iT? 

I. 80-is 25% udris 80× 0,25=20. 

I I. 80 - 20 =60. 

am magaliTSi imave Sedegis misaRebad sakmarisia gamoviTvaloT 80-is 

(100 – 25)%, anu 80 gavamravloT 0,75-ze: 80× 0,75 = 60. 



1. rogor gamoviTvaloT ricxvi misi procentis mixedviT? 

2. ras niSnavs ricxvis gazrda mocemuli procentiT? 

3. ras niSnavs ricxvis Semcireba mocemuli procentiT? 

4. ra ricxvi miiReba a ricxvis p% -iT gazrdiT? 

5. ra ricxvi miiReba a ricxvis p% -iT SemcirebiT? 

6. rogor gamoiTvli umoklesi gziT 6-is 40%-iT gazrdiT miRebul ricxvs? 

7. rogor gamoiTvli umoklesi gziT 6-is 40%-iT SemcirebiT miRebul 

ricxvs? 

8. miiReba Tu ara igive a ricxvi, Tu a -s gavzrdiT 10%-iT, Semdeg miRebul 

ricxvs SevamcirebT 10%-iT? 

9. ramdeni procentiT gaizrdeba sabolood ricxvi, romel sac gavzrdiT 

jer 10%-iT, xolo Semdeg kidev 20%-iT? 

savarjiSoebi 

1 ipove ricxvi, romlis 

a) 10% udris 5-s; b) 5% udris 10-s; g) 25% udris 25-s; 

d) 


1 

33 % 

3 udris 100-s; e) 0,1% udris 1-s; v) 120% udris 12-s. 

2 ra ricxvi miiReba 

a) 10-is 10%-iT gazrdiT? b) 36-is 50%-iT 

gazrdiT? 

g) 30-is 200%-iT gazrdiT? d) 0,48-is 1%-iT 

gazrdiT? 

e) 10-is 10%-iT SemcirebiT? v) 36-is 50%-iT 

SemcirebiT? 

z) 30-is 20%-iT SemcirebiT? T) 200-is 13%-iT 

SemcirebiT? 

3 vanos klaselTa 25% qeraTmiania. ramdeni moswavlea vanos klasSi, 

Tu maT Soris 8 moswavlea qeraTmiani? 

4 ninos klaselTa 20% friadosania. ramdeni moswavlea ninos 

klasSi, Tu maT Soris 6 moswavlea friadosani? 

ricxvis p%-iT 

gazrda igivea, rac 

am ricxvis 100 + p 

-ze 

100 

gamravleba, xolo 

ricxvis p%-iT 

Semcireba igivea, rac 

am ricxvis 100 − p 

-ze 

100 

gamravleba. 

5 medeam kviras Tavisufali drois 40% kompiuterul TamaSebs 

dauTmo. ramdeni saaTi hqonda Tavisufali im dRes medeas, Tu is 

2 saaTis ganmalobaSi kompiuteriT TamaSobda? 

6 Tanaklaselebma eqskursiisaTvis Segrovili Tanxis 30% samgzavro 

xarjebisaTvis gadades, darCenili 140 lariT ki sursaTi SeiZines. 

ra Tanxa Seagroves Tanaklaselebma eqskur siisaTvis?

7 moswavlis CanTa 40%-iani fasdaklebis 

Semdeg 18 larad gaiyida. ra Rirda 

CanTa fasdaklebamde? 

8 50%-iani saSobao fasdaklebiT 

fexsacmlis fasma 45 lari Seadgina. ra 

Rirda fexsacmeli fasdaklebamde? 

9 policielis xelfasma 40% -iT gazrdis 

Semdeg 1260 lari Seadgina. ra xelfasi 

hqonda policiels gazrdamde? 

10 bankSi Setanil Tanxas yovelwliurad 

9% emateba. ra Tanxa daugrovdeba 

laSas erT welwadSi, Tu man bankSi 500 lari Seitana? 

11 wignis maRazia gamomcemlobidan miRebul wignebs gamomcemlobis 

fasze 25%-is damatebiT yidis. ra mogeba darCeba maRazias 1000 

wignis gayidvis Semdeg, Tu gamom cemlobis mier dadgenili erTi 

wignis fasi 8 laria? 

12 yava moxalvisas wonis 12%-s kargavs. ramdeni grami moxaluli yava 

miiReba 200 grami nedli yavisagan? 

13 xorblisagan misi masis 75% fqvili miiReba. ramdeni kg xorbalia 

saWiro 3kg fqvilis misaRebad? 

14 puris comi gamocxobisas wonis 20%-s 

iklebs. 

a) ramdeni 400 -gramiani puri 

gamocxveba 10 kg comisagan? 

b) ramdeni kg comia saWiro 5 kg puris 

gamosacxobad? 

15 qalaqSi amJamad 10 000 mosaxlea. mosaxleobis ricxvi 

yovelwliurad 10%-iT izrdeba. ramdeni mosaxle iqneba am qalaqSi 

a) erTi wlis Semdeg? 

b) ori wlis Semdeg? 

16 biznesmenma Tavisi fulis 40% Rvinis warmoebaSi daabanda, 

saidanac 60% mogeba miiRo, danarCeni Tanxa sakonservo 

warmoebaSi Cado, saidanac 50% mogeba miiRo. ramdeni laris 

mflobeli gaxda biznesmeni, Tu misi sawyisi Tanxa 20 000 lars 

Seadgenda? 

17 ramdeni procentiT gaizrdeba marTkuTxedis formis miwis nakveTis 

farTobi, Tu mis sigrZes 10, xolo siganes 20 procentiT gavzrdiT? 


18 Tambaqos nawarmi jer 20, Semdeg 40 procentiT gaZvirda. sul 

ramdeni procentiT gaZvirda Tambaqos nawarmi? 

19 wriul diagramaze mocemulia 

saqarTvelos mosaxleobis ricxovneba 

procentebSi religiuri aRmsareblobis 

mixedviT 2002 wlis aRweris 

monacemebiT. 

a) diagramis saSualebiT gamo Tvale 

sxvadasxva aRmsareb lobis mqone 

mosaxleobis rao denoba (saqarTvelos 

mTeli mosaxleoba 2002 wlis aRweriT 

4 356 000 kacs Seadgenda). 


b) mocemuli diagramis mixedviT Seadgine 

cxrili, romelic gvi Cvenebs mosaxleobis 

rao deno bas aRmsareblobis mixedviT 

rogorc procentebSi, ise ricxobrivad. 

20 gamoTvale: 

a) 1sT-is ra nawilia 3wT; b) 1m-is ra nawilia 20 sm; 

g) 1kg-is ra nawilia 300g; d) 1t-is ra nawilia 250kg; 

e) 1m2 –is ra nawilia 25dm2 ; v) 1ha-is ra nawilia 4000m2 ; 

z) 1dm3-is ra nawilia 10sm3 ; T) 1m3 i) 1sT-is ra nawilia 1wm; 

–is ra nawilia 100l; 

aba, scade! 

daexmare ganawilebaSi 

1. maranSi 7 qvevri savsea RviniT, 7 ki – naxevradsavse, xolo 

danarCeni carielia. Rvino da qvevrebi samma Zmam Tanabrad 

unda gainawilos ise, rom TiToeuls toli raodenobis qvevrebi 

da Rvino unda ergos. ganawilebisas qvevridan qvevrSi Rvinis 

gadasxma ar SeiZleba. rogor u nda gainawilon? 

2. rvalitriani qila rZiTaa savse. rZe orma qalma Tanabrad unda 

gainawilos. erTs mxolod 5-litriani, xolo meores 3-litriani 

carieli bidoni aqvs. rogor gainawilon? 

4% 

1% 

10% 1% 

84% 

marTlmadidebluri 

kaTolikuri 

grigorianuli 

mahmadianuri 

sxva

3.4 ori sididis fardobis procentuli warmodgena 

mizani: 

viswavloT: 

1. ori sididis fardobis procentuli Cawera; 

2. monacemTa gamosaxva fardobis procentiT. 

praqtikuli amocanebis amoxsnisas sidideebis Sefardebas xSirad 

procentebiT gamosaxaven. 

amocana 1. rZe maRalcximianad iTvleba, Tu masSi cximis Semcve loba 

aranakleb 4,5%-ia. qarxanaSi Semosuli rZis cximianobis dasadge nad 

Seamowmes 1,5kg rZis sinji. aRmoCnda, rom igi 63g cxims Seicavda. 

SeiZleba Tu ara, es rZe maRalcximianad CaiTvalos? 

amoxsna. gamovsaxoT rZisa da masSi Semavali cximis wona erTi da imave 

erTeulebSi: 

1,5kg=1500g. 

gamoTvlebi 

kalkulatoriT 

vipovoT, rZis wonis ra nawils warmoadgens cximis 

wona da gamovsaxoT igi procentebSi: 

63 ÷ 1500 x 100 = 4,2 

. 

daskvna: am rZeSi cximis Semcveloba 4,2%-ia. maSasadame, rZe maRalcximiani 

ar aris. 

amocana 2. erTma kalaTburTelma matCis ganmavlobaSi 

20 tyorcna ganaxor ciela, romelTagan 15 tyorcna 

Sedegiani iyo, xolo meorem 40-jer tyorcna, aqedan 28 

Caagdo kalaTSi. romeli kalaTburTeli ufro zustad 

tyorcnis? 

amoxsna. imis gasarkvevad, Tu ra sizustiT tyorcnidnen 

kalaT burTe lebi, daviTvaloT maT mier natyorcni 

burTebis ramdeni procenti moxvda kalaTSi. 

gamoTvlebi CavataroT or etapad: 

I etapi: gavigoT, TiToeuli kalaTburTelis mier natyorcni 

burT ebis raodenobis ra nawils Seadgens Cagdebuli 

burTebis raodenoba: 

pirveli kalaTburTeli: 15 : 20 = 0, 75 

meore kalaTburTeli: 28 : 40 = 0, 7 

II etapi: miRebuli ricxvebi CavweroT procentis saxiT: 

pirveli kalaTburTeli: 0, 75 = ( 0, 75 × 100)% = 75% 

meore kalaTburTeli: 0, 7 = (0, 7 × 100)% = 70% 

daskvna: pirveli kalaTburTelis tyorcnis sizustea 75%, meoresi ki 

_ 70%. maSasadame, pirveli kalaTburTeli ufro zustad tyorcnis 

burTs. 


imisaTvis, rom gavigoT, a sidide b sididis ramdeni procentia, 

saWiroa: 

I. Tu es sidideebi zomis gansxvavebul erTeulebSia mocemuli, 

gadaviyvanoT isini erTsa da imave erTeulebSi. 

II. gavyoT a sidide b -ze da ganayofi warmovadginoT procentis saxiT, 

a 

e. p% 

i. = ×gavamravloT 100 100-ze: 

b 

a 

p% 

= × 100 

b 

ama Tu im narevSi (xsnarSi, naerTSi) raime nivTierebis procentis saxiT 

Caweril wils koncentracia ewodeba. magaliTad, ganxilul amocana1-Si 

gamoTvlili cximis koncentracia rZeSi 4,2%-is toli iyo. 

amocana 3. 90%-iani koncentraciis spirtis 3 litr xsnars 2 litri 

sufTa wyali daamates. ra koncentraciis xsnari miiRes? 

amoxsna. 3 litri xsnari 90%, anu 3⋅ 0,9 =2,7litr spirts Seicavda. 

wylis damatebis Semdeg xsnaris moculoba gaxda 5 litri, spirtis 

raodenoba ki ar Secvlila. amitom miRebuli xsnaris koncentracia 

iqneba 

2,7 

× 100 = 54% . 

5 

pasuxi: miRebuli xsnaris koncentraciaa 54%. 

gaxsovdes! 

procentebiT SeiZleba gamoisaxos mxolod erTi da imave erTeulebiT 

gazomili sidideebis Sefardeba. 


1. ra SemTxvevaSi SeiZleba warmovadginoT ori sididis Sefardeba 

procentis saxiT? 

2. rogor gamovsaxoT erTi sidide meore sididis procentebiT? 

3. Seicvleba Tu ara pirveli amocanis pasuxi, Tu cximisa da rZis wonas 

gamovsaxavT wonis sxva erTeulebiT (magaliTad, kilogramebiT)? 

4. ras ewodeba narevis koncentracia? 

5. Semcirdeba Tu gaizrdeba spirtis xsnaris koncentracia da ramdenjer, 

Tu xsnars imave moculobis wyals SevurevT? 

166 Tavi 3 • monacemebi

savarjiSoebi 

1 gamoTvale, a ricxvi b ricxvis ramdeni procentia, Tu: 

a) a = 9, b = 36 b) a = 15, b = 75 

g) a = 2, 4, b = 4,8 d) a = 10, b = 5 

2 gamoTvale, a sidide b sididis ramdeni procentia, Tu: 

a) a = 12 wT, b = 60wT b) a = 1, 5 kg, b = 30kg 

5 

g) a = dm, b = 2m d) a = 0, 3 lari, b = 40TeTri 

3 qvemoT CamoTvlilTagan romeli sidideebis fardoba 

SeiZleba gamoisaxos procentebiT? SesaZleblobis 

SemTxvevaSi gamoTvale: 

a) 3,5 kg da 4 lari; b) 16 wuTi da 1,2 saaTi; 

g) 2m 2 da 3,2m; d) 7sm da 2dm. 

4 cxrilSi moyvanilia VII klasis 25 moswavlis gamokiTxvis 

Sedegebi, Tu vin sportis romel saxeobas misdevs: 

sportis moswavleTa moswavleTa 

saxeoba raodenoba saerTo 

raodenobis 

procenti 

fexburTi 8 

kalaTburTi 3 

Widaoba 2 

Wadraki 4 

curva 1 

arc erTi 7 

a) gadaixaze cxrili da Seavse misi bolo sveti; 

b) monacemebis mixedviT aage svetovani diagrama; 

g) Seadgine analogiuri cxrili Seni klasis mixedviT. 

5 gamoTvale a da b ricxvebis procentuli fardoba, Tu: 

a) a ricxvi b ricxvis 2/5 nawilia; 

b) a ricxvi b ricxvis 3/10 nawilia; 

g) a ricxvi b ricxvis 13/14 nawilia; 

d) a ricxvi b ricxvis 0,9 nawilia; 

e) a ricxvi b ricxvis meoTxedia; 

v) a ricxvi orjer metia b ricxvze. 


6 klasSi 12 biWi da 15 gogonaa. klasis moswavleTa ramdeni 

procentia gogona? 

7 benzingasamarT sadgurs 3 tona benzinis maragi hqonda. 

sadgurma 0,54 tona benzini gayida. gamoTvale, mTeli 

maragis ramdeni procenti darCa gasayidi. 

8 cxrilSi mocemulia dedamiwis kontinentTa farTobi 

kvadratul kilometrebSi. 

kontinentis 

kontinentis 

farTobi 

kontinenti xmeleTis saerTo 

farTobi farTobis 

procenti 

evrazia 55119000 

CrdiloeT amerika 242300000 

samxreT amerika 17814000 

afrika 30218000 

avstralia 8503000 

antarqtida 

xmeleTis saerTo 

farTobi 

14245000 

gamoTvale xmeleTis saerTo farTobi, gadaixaze da Seavse 

cxrili. 

9 gamoTvale, ( a+ b)-s 

ramdeni procentia ( a− b), 

Tu a = 1, 9 , 

b = 0,6 . 

10 Senadnobi 60g vercxlis da 40 g oqrosa da 100g spilenZisagan 

Sedgeba. gamoTvale am SenadnobSi Semavali TiToeuli 

metalis koncentracia. 

11 ra koncentraciis xsnars miviRebT, Tu 100g marils 300g 

wyalSi gavxsniT? 

12 200g zRvis wyals, romelic 3% marils Seicavda, 400g 

sufTa wyali Seuries. ra koncentraciis xsnari miiRes? 

13 ramdeni grami wyali unda davumatoT 30g marils, rom 

miviRoT 25%-iani koncentraciis marilwyali? 

168 Tavi 3 • monacemebi

14 figuris farTobis ramden procents Seadgens 

gaferadebuli nawilis farTobi? 

a 

d 

b g 

e v 

15 ramdeni procentiT gaZvirda eleqtroenergia (anu 

sawyisi fasis ramdeni procentia namati), Tu adre 

1 kilo vati 12 TeTri Rirda, xolo amJamad erTi 

kilovatis fasia 18 TeTri? 

16 ramdeni procentia fasdakleba maRaziaSi, Tu labada, 

romelic 200 lari Rirda, 120 larad gayides? 

17 cxrilSi mocemulia saqarTvelos mosaxleobis 

raodenoba 1913, 1970 da 2002 wlebis aRweriT. 

wlebi mosaxleobis raodenoba cvlilebebi 

procentebSi 

1913 2 601 000 

1970 4 686 000 moimata 

2002 4 356 000 daiklo 

gamoTvale procentebSi mosaxleobis raodenobis cvlileba 

momdevno aRwerisas wina aRwerasTan SedarebiT. 

18 aleqsande erTi wlis gamavlobaSi 4 santimetriT gaizarda da 

misma simaRlem 164 sm-s miaRwia. ramdeni procentiT moimata 

simaRleSi aleqsandrem erTi wlis gamavlobaSi? 

19 wriul diagramaze mocemulia fermeris mier miRebuli sxvadasxva 

jiSis yurZnis mosavlis raodenobebi tonebSi. 

gamoTvale: 

daimaxsovre 

1 

1% 

100 = 

1 

10% 

10 = 

1 

20% 

5 = 

1 

25% 

4 = 

1 1 

= 33 % 

3 3 

1 

50% 

2 = 

1 = 

100% 

a) mTliani mosavlis ramden procents Seadgens saferavis jiSis 

yurZnis mosavali; 

b) ra gardusuli zomis seqtori Seesabameba rqawiTelis yurZnis 


aodenobas. 


wriul diagramaze aRniSnuli monacemebis mixedviT aage 

Sesabamisi svetovani diagrama. 

20 svetovan diagramaze mocemulia firmis TanamSromeli qalebisa da 

mamakacebis raodenobebi. 

20 

15 

10 

5 

0 

qalebi 

mamakacebi 

diagramis mixedviT upasuxe kiTxvebs: 

a) ramdeni TanamSromelia sul firmaSi? 

b) TanamSromelTa ramdeni procentia qali? 

g) ramdengradusiani seqtori Seesabameba Sen mier agebul wriul 

diagramaze mamakacebis raodenobas? 

aage mocemuli diagramis Sesabamisi wriuli diagrama da miuTiTe 

masze firmis TanamSromeli qalebisa da mamakacebis raodenobebi 

procentebSi. 

21 gaavle wrfe da masze aRniSne A da B wertilebi, romelTa Soris 

manZili 8 sm-ia. amave wrfeze moniSne iseTi C wertili, rom BC 

mona kve Tis sigrZe AB monakveTis sigrZis 40% iyos. ramdeni aseTi 

wertili arse bobs? ras udris AC monakveTis sigrZe? 

22 ramdeni procentiT gaizrdeba ricxvi, Tu mas samjer gavzrdiT? 

23 ramdeni procentiT Semcirdeba ricxvi, Tu mas aTjer 

SevamcirebT? 

24 ramdeni g wyali unda davasxaT 30% koncentraciis mqone 200g 

marilwyals, rom miviRoT 20% koncentraciis xsnari? 

25 Seuries 500grami 80% koncentraciis da 300grami 70% 

koncentraciis spirits xsnarebi. ra koncentraciis xsnari miiRes? 

26 mas Semdeg, rac oqrosa da vercxlis Senadnobs 50g oqro 

daumates, SenadnobSi oqros koncentracia 30%-dan 80 %-mde 

gaizarda. ramdeni grami iyo Senadnobi oqros damatebamde? 

27 ipoveT udidesi samniSna ricxvi, romelic iyofa 3-zec da 5-zec. 

28 risi tolia iseTi wiladebis mricxvelebis jami, romelTa 

mniSvnelia 12, xolo maTi mniSvnelobebi 1/3-sa da 5/6-s Sorisaa?

29 gamoTvale 2-is, 2-is da 3-is saSualo ariTmetikuli. 

30 ipove x, Tu x-sa da 4,2-is saSualo ariTmetikuli 5-is tolia. 

31 1, x, 5, y, z …ricxvTa mimdevrobis meore wevridan dawyebuli 

yoveli wevri misi mezoblebis saSualo ariTmetikulis tolia. 

ipove x, y da z. 

wyvilebSi samuSao 

TiToeulma Tqvenganma damoukideblad SearCieT ori aranulovani 

ricxvi. gamoTvaleT, meore ricxvis ramdeni procentia pirveli 

ricxvi. Semdeg gamo TvaleT, pirveli ricxvis ramdeni procentia meore 

ricxvi. miRebuli ori ric xvi gadaamravleT, pasuxi daamrgvaleT 

mTel ricxvamde da CaiwereT. Sea dare Sen mier miRebuli Sedegi Seni 

megobrebis mier miRebul Sedegebs. ra kanonzomierebas amCnevT? axseniT 

es kanonzomiereba. 

gamoTvlebisas SegiZliaT isargebloT kalkulatoriT. 

qarTuli xuroTmoZRvrebisa da 

arqiteqturis nimuSebi 

gelaTis samonastro 

kompleqsi 

quTaisis maxloblad, wyalwiTelas 

xeobaSi aago mefe 

daviT aRmaSenebelma gelaTis 

monasteri da akademia, romelsac 

Tanamedroveni “axal ierusalimsa 

da meore aTens” 

uwodebdnen. 

daarsda 1106 wels. 

yaurmis xidi 

Zv. qarTuli xuroTmoZ- 

Rvrebis erT-erTi SesaniSnavi 

Zeglia qvis xidi mdinare 

faravanze. igi XII - XIV saukuneebiT 

TariRdeba. 

a) romel geometriul figurebs xedav nagebobebSi? 

b) ramdeni saukunisa gelaTis kompleqsi? yaurmis xidi? 


3.5 monacemTa ricxviTi maxasiaTeblebi 

mizani: 


davadginoT sxvadasxva saxiT warmodgenili 

monacemebis ricxviTi maxasiaTeblebi: sixSire, 

diapazoni, saSualo, moda. 

dedis SekiTxvaze, Tu ra niSnebze swavlobs, nikam upasuxa: — cxrianze, 

radgan Cems mier miRebul niSnebSi 9 qula meti raodenobiTaa. 

dedam dRiuridan amowera sxvadasxva saganSi semestris ganmavlobaSi 

nikas mier miRebuli niSnebi: 

7, 8, 9, 6, 9, 7, 9, 6, 7, 6, 9, 6, 8, 9, 7, 9, 6, 9, 10, 8. 

Semdeg daalaga miRebuli niSnebi zrdis mixedviT da Seadgina sixSireTa 

cxrili: 

niSani 6 7 8 9 10 

sixSire 5 4 3 7 1 

saSualo qulis gamosaTvlelad dedam qulebis maT sixSirebze 

namravlebi Sekriba da miRebuli Sedegi sixSireTa jamze gayo: 

6 ⋅ 5 + 7 ⋅ 4 + 8 ⋅ 3 + 9 ⋅ 7 + 10 ⋅ 1 

saSualo = = 7,75 

5+ 4+ 3+ 7+ 1 

miuxedavad imisa, rom nikas SefasebebSi 9 qulis sixSire yvela sxva 

qulis sixSires sWarbobs, semestris manZilze nikas mier miRebuli 

qulaTa saSualo 8 qulaze naklebi aRmoCnda. udidesi sixSiris monacems 

moda ewodeba. ganxululi monacemebis modaa 9. 

nikas mier miRebuli qulebidan udidesi qulaa 10, umciresi ki 6. maTi 

sxavoba 4 gviCvenebs, Tu ramdenadaa gabneuli monacemebi. sazogadod, 

udidesi da umciresi monacemis sxvaobas ricxviTi monacemebis gabnevis 

diapazoni ewodeba. 

T(CO ) 

amocana1. Tiniko sami dRis 

ganmavlobaSi 9 00 saaTidan 

saRamos 21 00 saaTamde 

yovel 3 saaTSi erTxel 

iwerda haeris temperturas 

da miRebuli monacemebis 

safuZvelze Seadgina xazovani 

diagrama. diagramis mixedviT 

gamovTvaloT monacemTa 

ricxviTi maxasiaTeblebi. 

amoxsna. 

8 

7 

6 

5 

4 

3 

2 

1 

0 

9 saaTi 12 saaTi 15 saaTi 18saaTi 21saaTi 

I dRes umciresi temperatura iyo +20 dilis 900 saaTze, udidesi +70 _ 

1500 saaTze. am dRes gabnevis diapazoni 50-is tolia. monacemTa modaa 50 , 

xolo saSualo temperatura dRis ganmavlobaSi 

(20 +50 +70 +50 +30 ): 5 = 4,40 . 

I dRe 

I I dRe 

I I I dRe

I I dRes umciresi temperatura iyo +1,5 0 dilis 9 00 saaTze, udidesi +6 0 

_15 00 saaTze. am dRes gabnevis diapazoni 4,5 0 -is tolia. monacemTa modaa 

4 0 , xolo saSualo temperatura dRis ganmavlobaSi (1,5 0 +4 0 +6 0 +4 0 +1 0 

):5=3,3 0 . 

mesame dRis ricxviTi maxasiaTeblebi gamoTvale damoukideblad. 

daviTvaloT gabnevis diapazoni samive dRis ganmavlobaSi: am 

SemTxvevaSi umciresia III dRis monacemi+ 0,5 0 , xolo udidesi _ I dRis 

monacemi 7 0 . amitom gabnevis diapazoni iqneba 6,5 0 . 

rogorc vxedavT, gabnevis diapazoni samive dRis ganmavlobaSi 

aRemateba calkeuli dReebis gabnevis diapazonebs. SevniSnoT, rom 

sazogadod, ricxviTi simravlis gabnevis diapazoni yovelTvis 

aranaklebia, vidre misi nebismieri qvesimravlis gabnevis diapazoni 

(axseni, ratom?). 


1. ra monacems hqvia moda? 

2. rogor daiTvli monacemTa saSualos? 

3. rogor daiTvli monacemTa gabnevis diapazons? 

4. eTanxmebi Tu ara nikas mosazrebas, rom is cxrianze swavlobs? 

5. SeiZleba Tu ara ricxviT monacemTa gabnevis diapazoni naklebi iyos 

vidre am monacemTa nawilis gabnevis diapazoni? 

savarjiSoebi 

1 paragrafSi mocemuli diagramis mixedviT ipove samive dRis 

monacemTa moda. 

2 paragrafSi mocemuli diagramis mixedviT gaarkvie romeli 

winadadebaa marTebuli: 

a) samive dRis SemTxvevaSi udidesi temperatura 15 saaTze iyo; 

b) samive dRis SemTxvevaSi umciresi temperatura 9 saaTze iyo; 

g) 15-dan 21 saaTamde SualedSi samive dRes temperatura 

klebulobda; 

d) 9-dan 15 saaTamde SualedSi samive dRes temperatura 

matulobda. 

3 mocemuli sixSireTa cxrilis mixedviT ipove monacemTa saSualo, 

moda da diapazoni; 

monacemebi -1,4 +2,3 -4 +1,4 +6 -4,4 +3,3 4 

sixSire 1 3 2 5 2 3 4 1 

4 ras udris x, Tu x, 5, 3, 8, 4, 5, 3, 10 ricxviTi monacemebis 

modaa 3. 

5 ras udris a, Tu zrdis mixedviT dalagebuli a, 5, 6,5, 8,4, 11 

ricxviTi monacemebis gabnevis diapazonia 7,4 . 


6 ras udris b, Tu klebis mixedviT dalagebuli b, 5,1, 4,7, 3,4, 

-1 ricxviTi monacemebis gabnevis diapazonia 8. 

7 ras udris x, Tu 2,5, -3, 7, x, -4,2, -5, 3, 1,7 ricxviTi 

monacemebis saSualoa 1,25? 

8 dasavleT saqarTvelos ianvris Tvis haeris temperaturis gabnevis 

diapazoni 8 gradusi, xolo aRmosavleT saqarTveloSi imave 

Tvis gabnevis diapazoni 12 gradusi iyo. SeiZleba Tu ara mTel 

saqarTveloSi igive maCvenebeli iyos a) 7 gradusi? b) 11 gradusi? 

g) 13 gradusi? 

9 cxrilSi mocemulia eqvsi Tanaklaselis 

Sedegi 100 m-ze sirbilSi. 

a) daalage es monacemebi klebis mixedviT; 

b) gamoTvale monacemTa saSualo da 

gabnevis diapazoni; 


g) romel moswavles aqvs saukeTeso 

Sedegi? 

d) romeli moswavlis Sedegia 

saSualosTan yvelaze axlos? 

10 xazovan diagramaze mocemulia wisqvil-kombinatis mier sami 

wlis ganmavlobaSi gadamuSavebuli xorblis raodenoba (tonebSi) 

kvartlebis mixedviT. 

diagramis mixedviT 

daadgine TiToeuli 

wlis da mTlianad sami 

wlis ganmavlobaSi 

monacemTa ricxviTi 

maxasiaTeblebi: 

a) udidesi da umciresi 

monacemebi; 

b) monacemTa saSualo; 

g) moda; 

d) gabnevis diapazoni. 

60 

50 

40 

30 

20 

10 

0 

I kvartali 

I I kvartali 

I I I kvartali 

11 sami sazamTros saSualo masaa 7kg, meoTxes masa 9kg. ra saSualo 

masa aqvs oTxive sazamTros? 

12 mocemulia erTi klasis moswavleebis mier maTematikis 

sakontroloSi miRebuli, zrdis mixedviT dalagebuli, qulaTa 

sixSiruli cxrili. 

qula a 4 5 6 7 8 b 

sixSire 1 2 3 5 8 5 1 

salome 16,2 wm 

giga 14,7 wm 

naTia 15,9 wm 

luka 15,3 wm 

kaxa 17,4 wm 

lela 19,2 wm 

I V kvartali 

2009w. 

2010w. 

2011w.

gamoTvale a da b qulebis mniSvnelobebi, Tu cnobilia, rom 

qulaTa saSualoa 6,4, xolo gabnevis diapazoni — 7. 

13 16 sm 3 moculobis brinjaos zodi 162 grams iwonis. ramdens iwonis 

brinjaos birTvi, romlis moculoba 24 sm 3 -ia? 

14 or qalaqs Soris manZili 55 km-ia, xolo am qalaqebs Soris 

manZili rukaze 2,2 sm-is tolia. rogoria am rukis masStabi? 

15 ABC samkuTxedSi ∠ A = 68 0 , ∠ B = 42 0 . qvemoT CamoTvlili 

utolobebidan romelia WeSmariti? 

a) AC < AB < BC; b) AB < AC < BC; g) BC < AB < AC; d) AC < BC < AB. 

16 gamoTvale: 

a) 

; b) 

16 baRSi msxlis, alublisa da atmis xeebia. xeebis saerTo 

raodenobis nawili msxlisa da alublis xeebia, xolo atmis 

xeebis raodenoba xeebis raodenobis -s Seadgens. gamoTvale 

ramdenia baRSi atmis da ramdeni msxlis xe, Tu alublis xe sul 

oria. 

SeiZleba Tu ara? 

mocemulia zrdis mixedviT dalagebuli monacemebi: 

x; 3; 5; 6; 8; y. 

SeiZleba Tu ara, rom am monacemTa gabnevis diapazoni iyos 

a) 4? b) 7? 

. 


I I I Tavis mimoxilva 

ra viswavleT am TavSi? 

1. raodenobrivi da Tvisobrivi monacemebis mopovebis saSualebebi; 

2. ricxvis Cawera procentis saxiT da procentis saxiT Cawerili ricxvis 

Cawera wiladis saxiT; 

3. ricxvis procentis povna da ricxvis povna misi procentiT an 

nawiliT ; 

4. fardobis procentuli Cawera da gamoyeneba; 

5. sididis procentuli zrda-kleba da martivi xarjTaRricxva; 

6. monacemTa mowesrigeba da warmodgena cxrilisa da diagramis saxiT; 

7. monacemTa ricxviTi maxasiaTeblebis (saSualo, moda, gabnevis 

diapazoni) gamoTvla. 



• ra saxis monacemebs icnob? 

• monacemTa mopovebis ra saSualebebi arsebobs? 

• rogor Cawer ricxvs procentis saxiT? 

• rogor Cawer procentis saxiT Caweril ricxvs wiladis saxiT? 

• ra kavSiria ricxvis procentsa da nawils Soris? 

• rogor ipovi a ricxvis p procents? 

• rogor ipovi ricxvs, romlis p procenti a ricxvis tolia? 

• rogor ipovi a ricxvze p procentiT met (nakleb) ricxvs? 

• rogor ipovi a ricxvi b ricxvis ramdeni procentia? 

• ra aris monacemTa moda? 

• ra aris monacemTa sixSiruli cxrili? 

• rogor daiTvli monacemis sixSires? 

• rogor daiTvli ricxviT monacemTa saSualos? 

• ra aris ricxviT monacemTa gabnevis diapazoni? 

• ra saxis diagramebi ici? 

• rogor aageb xazovan (svetovan) diagramas?

SeurCie marcxena svetSi mocemul winadadebas sityva marjvena 

svetidan 

1. ricxviT moncemTa udidesi da umciresi monacemis 

sxvaobas gabnevis ------- ewodeba 

2. udidesi sixSiris monacems ----- ewodeba 

3. ricxvis p/100 nawils am ricxvis p ------- ewodeba 

4. monacemis gameorebaTa raodenobas am monacemis 

------- ewodeba 

5. ricxviTi monacemebis jamis monacemTa raodenobaze 

ganayofs monacemTa ------ ewodeba 

6. ricxvis erTi procenti misi -------- nawilia 

savarjiSoebi 

moda 

sixSire 

diapazoni 

saSualo 

procenti 

measedi 

1 gamoTvale zepirad: 

a) 5-is ra nawilia 2,5; b) 27-is ra nawilia 3; 

g) 55-is 2 

nawili; d) 20-is 0,7 nawili ; 

11 

e) ricxvi, romlis 3 

4 

udris 6-s ; v) ricxvi, romlis udris 8-s; 

7 9 

z) 50-is 10% ; T) ricxvi, romlis 50% udris 60-s; 

i) 1kg-is ramdeni procentia 300g; k) 1sm-is ramdeni procentia 6 mm; 

l) 1sT-s ramdeni procentia 12 wT; m) 1t-s ramdeni procentia 200kg. 

2 gamoTvale: 

a) 50 ramdeni procentiT metia 40-ze; 

b) 40 ramdeni procentiT naklebia 50-ze. 

3 gazarde 20%-iT 

a) 3 lari; b) 5 tona; g) 1,2 metri; d) 25 kg; e) 3,6 

4 Seamcire 40%-iT 

a) 5 lari; b) 3 tona; g) 1,5 metri; d) 2,5 kg; e) 36sm. 

5 TaTias klasSi 25 moswavlea. maTi 60% gogonaa. gogonebis 1/5 

nawili friadosania. a) ramdeni friadosania TaTias klasSi? b) ra 

sididis (gradusebSi) seqtori Seesabameba vaJebis raodenobas im 

wriul diagrmaze, romelzec TaTias Tanaklaseli gogonebisa da 

vaJebis raodenobebi iqneba naCvenebi? aage es diagrama. 

6 ornitologi (frinvelTa yofis 

Semswavleli mecnieri) akvirdeba, 

dReSi ramdenjer tovebs SaSvi budes 

bartyebisTvis saWmlis misatanad. aRricxva 

rom gaudvildes, dakvirvebaTa dRiurSi 

budidan SaSvis yovel gamofrenas igi xazis 

CamosmiT aRniSnavs (ix. qvemoT). 



00 00 

06 −12 

00 00 

12 −18 

am monacemebis mixedviT upasuxe kiTxvebs: 

a) monacemTa mopovebis romel xerxs mimarTavs ornitologi? 

b) ramdenjer datova bude Savma dRis pirvel naxevarSi? 

g) dRis romel naxevarSia SaSvi ufro aqtiuri? 

d) saaTSi saSualod ramdenjer tovebs SaSvi budes dRis 

ganmavlobaSi? 

7 ramdeni procentiT Camofasda qurqi, romlis sawyisi fasi 500 

lari iyo, xolo Camofasebis Semdeg 350 larad gaiyida? 

8 ramdeni procentiT gaZvirda fqvili, Tu adre 1kilogrami 80 

TeTri Rirda, axla ki misi Rirebuleba 1lari da 20 TeTria? 

9 tolferda samkuTxedis fuZis sigrZe perimetris 30%-ia. 

perimetris ramdeni procentia ferdis sigrZe? 

10 qalaqis mcxovrebTa erTi nawili laparakobs mxolod qarTulad, 

nawili mxolod rusulad, danarCeni qarTuladac da rusuladac. 

qarTulad laparakobs am qalaqis mcxovrebTa 80%, rusulad_ 

60%. qalaqis mcxovrebTa ramdeni procenti laparakobs orive 

enaze? 

11 cxrilSi mocemulia kamaTelze mosuli qulebis sixSireTa cxrili 

qula 1 2 3 4 5 6 

sixSire 4 3 5 6 4 3 

cxrilis mixedviT upasuxe kiTxvebs: 

a)sul ramdenjer gaagores kamaTeli? b) ras udris monacemTa 

diapazoni? 

g) ras udris monacemTa moda? 

d) ras udris monacemTa saSualo? 

12 wriul diagramaze mocemulia erTi klasis 25 moswavlis moswrebis 

Sedegebi pirveli trimestris mixedviT procentebSi. diagramis 

mixedviT daadgine: 

a) ramdeni warCinebulia klasSi?

) ramdeni moswavlis Sedegia 

damakmayofilebeli an kargi? 

g) ramden gradusiani 

seqtori Seesabameba 

aradamakmayofilebeli 

Sedegis mqone moswavleebs? 

d) rogori Sefasebis moswavleebi 

Warboben klasSi? 

aradamakmayofilebeli 

12% 

damakmayofilebeli 

36% 

13 SekiTxvaze, ramdeni jariskacia mis 

razmSi, leitenantma upasuxa, rom misi 

razmis 2 

1 

nawili sadarajozea, 

7 7 nawili 

hospitalSi, xolo danarCeni 28 jariskaci 

mwyobrSia. ramdeni jariskacia razmSi? 

14 SekiTxvaze, ramdeni moswavlea mis 

klasSi, levanma upasuxa, rom misi klasis 

moswavleTa 20% friadosania, 40% 

oTxosani, 25% samosani, xolo 3 moswavle 

orosania. ramdeni moswavlea levanis 

klasSi? 

Sedegebi 

15 samxedroTa or A da B banaks Soris manZili 20 

kilometria. A dajgufebis meTaurma jariskacs 

gadasca amanaTi, ubrZana, mietana igi B dajgufebis 

meTaurisaTvis da maSinve ukan gamobrunebuliyo. 

jariskacma 2 saaTSi gaiara banakebs Soris manZilis 

60%. A banakidan ra manZilze iqneba jariskaci 

gamgzavrebidan 4 saaTis Semdeg, Tu is imave 

siCqariT imoZravebs? 

warCinebuli 

20% 

kargi 

32% 

16 dedam Tamriko wamlis sayidlad gagzavna. manZili saxlidan 

afTiaqamde 1,5 km-ia. am manZilis 55% Tamrikom 10 wuTSi gaiara. 

afTiaqidan ra manZilze iqneba Tamriko gagzavnidan 20 wuTis 

Semdeg, Tu imave siCqariT ivlis? 

17 xorbali iZleva 80% fqvils, xolo SoTis puri mis gamosacxobad 

saWiro fqvilze 20%-iT mZimea. ramdeni grami xorbalia saWiro 

erTi 480-gramiani SoTis puris gamosacxobad? 

18 mas Semdeg, rac fexburTelTa gundi erTma Tavdamsxmelma datova, 

gundSi TavdamsxmelTa raodenoba 25%-dan 20%-mde Semcirda. 

ramdeni fexburTeli iyo gundSi Tavdamsxmelis wasvlamde? 

19 mas Semdeg, rac xsnaridan 2kg wyali aaorTqles, xsnaris 

koncentracia 80%-dan 90%-mde gaizarda. ramdeni kilogrami iyo 

xsnari wylis aorTqlebamde? 


20 qalaqi sami A, B da C 

raionisagan Sedgeba. 

wriul diagramaze 

naCvenebia raionebis 

mosaxleobis wili 

procentebSi, xolo 

4000 

svetovan diagramaze 

1000 

mosaxleobis raodenoba 

A raionSi. gamoTvale 

mosaxleobis raodenoba B da C raionebSi. 


aba scade! 

A raioni 

C raioni 

45% 

B raioni 

35% 

ipove naturaluri ricxvi, romlis Tavisive 32%-ze gayofis Sedegad 

arasrul ganayofad amave ricxvis 12% miiReba. 

testi 

cxrilSi Setanilia im qveynebis monacemebi, romlebic 2004 wels 

Sediodnen pirvel aTeulSi erTi maCvenebliT mainc: farTobiT, 

mosaxleobiT, erToblivi Sida produqtiT. 

qveyana dedaqalaqi farTobi mosaxleoba 

erToblivi 

Sida produqti 

(aTasi km2 ) (mln) (mlrd $) 

avstralia kanbera 7 6 8 7 2 0 , 1 692,4 

argentina buenos-airesi 2 7 6 7 3 9 , 5 172,1 

aSS vaSingtoni 9 6 2 9 2 9 5 , 7 12438,9 

bangladeSi daka 1 4 4 1 4 4 , 3 61,9 

brazilia brazilia 8 5 1 3 1 8 6 , 1 732,1 

germania berlini 3 5 7 8 2 , 4 2906,7 

didi 

britaneTi 

londoni 2 4 4 6 0 , 4 2295,1 

espaneTi madridi 5 0 4 4 0 , 3 1120,3 

iaponia tokio 3 7 7 1 2 7 , 4 4799,1 

indoeTi deli 3 2 8 8 1 0 8 0 , 3 749,4 

indonezia jakarta 1 9 1 9 2 4 1 , 9 284,1 

italia romi 3 0 1 5 8 ,1 1836,4 

kanada otava 9 9 8 4 3 2 , 8 1098,4 

nigeria abuja 9 2 3 1 2 8 , 8 91,6 

pakistani islamabadi 8 0 3 1 6 2 , 4 90,3 

ruseTi moskovi 1 7 0 7 5 1 4 3 , 4 755,4 

safrangeTi parizi 5 4 7 6 0 , 7 2216,3 

sudani xartumi 2 5 0 6 4 0 , 2 25,4 

yazaxeTi astana 2 7 1 7 1 5 ,2 51,9 

CineTi pekini 9 5 9 7 1 3 0 6 , 3 1843,1

cxrilis mixedviT upasuxe 1 _ 5 kiTxvebs: 

1. msoflios romel qveyanaSia mosaxleoba yvelaze meti? 

a) ruseTSi; b) CineTSi; g) iaponiaSi; d) indoeTSi. 

2. romeli qveyanaa teritoriiT yvelaze didi? 

a) ruseTi; b) CineTi; g) iaponia; d) aSS; 

3. romeli qveynis erToblivi Sida produqtia yvelaze meti? 

a) aSS b) CineTi g) iaponia d) germania 

4. ramdeniT naklebia mosaxleoba safrangeTSi, vidre nigeriaSi? 

a) 58 mln; b) 59,7 mln; g) 68,1 mln; d) 45 mln. 

5. ramdeni dolariT metia erToblivi Sida produqtis warmoeba 

germaniaSi, vidre ruseTSi? 

a) 1258,3 mlrd. $; b) 2151,3 mlrd. $; g) 1580,4 mlrd. $; d) 980,8 

mlrd. $. 

6. Tu x, 1, 7, 5, 4, 5, 3, 7 ricxviTi monacemebis modaa 7, maSin 

ras udris x =, 

a) 1; b) 7; g) 5; d)3. 

7. Tu zrdis mixedviT dalagebuli a, 3, 5, 8, 10 ricxviTi 

monacemebis gabnevis diapazonia 8 , maSin a = 

a) 2; b) 5; g)3; d) 8. 

8. ras udris x, Tu -3, 5, x, -5, 3, 7 ricxviTi monacemebis 

saSualoa 1? 

a)- 5; b) 5; g) -1; d) 1. 

9. VII klasis moswavle gogonebis simaRleTa gabnevis diapazoni 13smia, 

vaJebisa ki 10 sm. ramdeni sm SeiZleba iyos simaRleTa gabnevis 

diapazoni am klasSi? 

a) 14 sm; b) 11 sm; g) 10 sm; d) 9 sm. 

10. 15 laris 40% tolia: 

a) 1l 40 T; b) 60 l; g) 200 T; d) 6 l. 

11. erTi saaTis 20% aris: 

a) 1 

20 

sT; b) 1 

5 

wT; g) 12 wT; d) 20 wT. 

12. ricxvi, romlis 5% 6-is tolia, aris: 

a) 120; b) 56; g) 360; d) 30. 

13. 20 aris 80 -is 

a) 4%; b) 25%; g) 30%; d) 5%. 

14. ra ricxvis 10% -ia 24-is 3/8 nawili? 

a) 160-is; b) 120-is; g) 72-is; d) 90-is. 

15. ramdeni procentiT gaizrdeba ricxvi, Tu mas gavadidebT 5-jer? 

a) 10%-iT; b) 900%-iT; g) 400%-iT; d) 90%-iT. 

16. ramdeni procentiT Semcirdeba ricxvi, Tu mas SevamcirebT 5-jer? 

a) 5%-iT; b) 80%-iT; g) 500%-iT; d) 50%-iT. 


17. 10% -iT gaZvirebis Semdeg litri benzinis fasma 2,2 lari 

Seadgina. ra Rirda litri benzini gaZvirebamde? 

a) 2 lari; b)1l 99 TeTri; g) 1l 10T; d) 1l90 TeTri . 

18. puris fasi 40 TeTridan 50 TeTramde gaizarda. ramdeni 

procentiT gaZvirda puri? 

a) 35%-iT; b) 10%-iT; g) 25%-iT; d) 50%-iT; 

19. ramdeni kilometria realuri manZili qalaqebs Soris, Tu am 

qalaqebs Soris manZili 1: 500 000 masStabis rukaze 7 sm-ia? 

a) 12 km; b) 35 km; g) 350 km; d) 60km. 

20. 200 mililitri xsnari 40 miligram wamals Seicavs. ramden wamals 

Seicavs 300 mililitri igive xsnari? 

a) 60 mg; b) 500 mg; g) 400 mg; d) 80 mg. 

21. rogori iyo qalaqSi mosaxleobis raodenoba erTi wlis win, 

Tu axla qalaqSi 220 000 mcxovrebia, xolo wliuri namati 10%-s 

Seadgens? 

a) 198000; b) 210000; g) 120000; d) 200000. 

22. marTkuTxedis sigane perimetris 20%-ia. perimetris ramdeni 

procentia am marTkuTxedis sigrZe? 

a) 15%; b) 25%; g) 30%; d) 40%. 

diagramze mocemulia erT-erT maRaziaSi gayiduli kompiuterebis 

raodenoba kvartalebis mixedviT. diagramis mixedviT upasuxe # 23, 

# 24, 25 kiTxvebs: 


60 

50 

40 

30 

20 

10 

I kvartali 

II kvartali 

III kvartali 

IV kvartali 

23. ramdeni kompiuteri gaiyida am maRaziaSi wlis ganmavlobaSi? 

a) 160; b) 165; g) 170; d) 180. 

24. ramdeni procentiT naklebi kompiuteri gaiyida pirvel kvartalSi, 

vidre meoreSi? 

a) 20%-iT; b) 25%-iT; g) 10%-iT; d) 40%-iT. 

25. wlis ganmavlobaSi gayiduli kompiuterebis raodenobis ra nawili 

gaiyida III kvartalSi? 

a) 1 ; b) 2 ; g) 

4 

5 

1 ; d) 1 . 

5 

6

4.11 amocanis amoxsnis algebruli meTodi 

mizani: 

232 Tavi 4 • erTwevri da mravalwevri 

viswavloT: 

1. algebruli meTodiT amocanis amoxsnis safexurebi; 

2. cxrilis gamoyeneba. 

amocanis amoxsnis algebruli meTodi gulisxmobs amocanis pirobebis 

Sesabamisi gantolebis Sedgenas da am gantolebis amoxsnis safuZvelze 

amocanis gadawyvetas. es meTodi Sedgeba Semdegi safexurebisagan: 

1. sityvieri modelis Sedgena; 

2. ucnobis Semotana da monacemebTan dakavSireba; 

3. algebruli modelis (gantolebis) Sedgena; 

4. gantolebis amoxsna; 

5. amocanis SekiTxvaze pasuxis gacema. 

amovxsnaT ramdenime amocana algebruli meTodis gamoyenebiT. 

amocana1. antilopas wamSi 20 metris garbena 

SeuZlia. jiqi masze swrafi cxovelia. mas wamSi 25 

metris garbena ZaluZs, magram am siCqariT jiqs 

mxolod 20 wamis ganmavlobaSi SeuZlia sirbili. ra 

manZilze ar unda miuSvas antilopam jiqi, rom Tavi 

usafrTxod igrZnos? 

amoxsna: 

1. sityvieri modeli 

antilopas 

ganarbeni 

usafrTxo 

manZili 

+ = 

2. ucnobis Semotana da monacemebTan dakavSireba: 

vTqvaT, usafrTxo manZili jiqsa da antilopas Soris 

x metria. 20 wamSi antilopa gairbens 20× 20 = 400 

metrs, xolo jiqi _ 25× 20 = 500 metrs. 

3. gantolebis Sedgena: 

400 + x = 500 

4. gantolebis amoxsna: 

x = 100 . 

5. pasuxi: 

jiqis 

ganarbeni 

manZili, romel ze axlos antilopam jiqi ar unda miuSvas, aris 100 m.

amocana 2. moswavleTa sazafxulo banakma SeiZina 1490 laris Rirebulebis 

oTxadgiliani da eqvsadgiliani 16 karavi. erTi oTxadgiliani karvis fasia 

80 lari, xolo eqvsadgilianisa 110 lari. ramdeni oTxadgiliani da ramdeni 

eqvsadgiliani karavi SeiZina banakma? 

amoxsna: 

1. sityvieri modeli: 

oTxadgilian 

karvebSi 

gadaxdili 

Tanxa 

eqvsadgilian 

karvebSi 

gadaxdili 

Tanxa 

+ = 


mTliani 

Tanxa 

vTqvaT, oTxadgiliani karavi x cali iyides. maSin eqvs adgi lians 

(16- x ) cals iyididnen. oTxadgilian karvebSi gadaixdid nen 80x lars, 

eqvsadgilianebSi _ 110(16- x ) lars. yvela karavSi erTad gadaixdidnen 

(80 x + 110(16- x )) lars. 


80 x + 110 (16- x ) = 1490 

2. gantolebis amoxsna; 80x+ 1760 − 110x= 1490 

− 30x =−270 

x = 9 

5. pasuxi: banakma SeiZina 9 oTxadgi li ani da 7 eqvsadgiliani karavi. 

ganxiluli amocanis amoxsnisas ganto lebis Sedgena gagiadvildeba, Tu 

Seadgen cxrils. 

oTxadgiliani 

karvebi 

eqvsadgiliani 

karvebi 

karvebis erTi karvis yvela karvis 

raodenoba Rirebuleba Rirebuleba 

x 80 80x 

16–x 110 110(16–x) 

amocana 3. erT benzingasamarT sadgurs 2 300 litri benzini hqonda, 

meores _ 2800 litri. dRis ganmavlobaSi meore sadguridan gaiyida 

xuTjer meti benzini, vidre pirvelidan, ris Semdeg orive benzingasamarT 

sadgurSi Tanabari raodenobis benzini aRmoCnda. ramdeni litri benzini 

gaiyida TiToeuli sadguridan? 

amoxsna: 

1. sityvieri modeli 

pirvel sadgurSi 

darCenili benzini 

= 

meore sadgurSi 

darCenili benzini 

Tavi 4 • erTwevri da mravalwevri 233


vTqvaT, pirveli sadguridan gaiyida x litri benzini, maSin meore 

sadguridan gaiyideboda 5x litri. pirvelSi darCenili benzinis 

raodenobaa (2 3000- x ) litri, xolo meoreSi (2 800-5x ) litri. 


4. gantolebis amoxsna: 

234 Tavi 4 • erTwevri da mravalwevri 

2300 − x= 2800 − 5x. 

4x = 500 , x = 125. 

5. pasuxi: pirveli saduridan gaiyida 125 l benzini, meoridan — 125× 

5 l=625 l. 

gantolebis Sedgena am amocanaSic cxrilis gamoyenebiT SeiZleba: 

benzingasamarTi 

I sadguri 

benzingasamarTi 

I I sadguri 

iyo gaiyida darCa 

2 300 l xl (2 300–x)l 

2 800 l 5xl (2 800–5x)l 

algebruli meTodi warmatebiT SegviZlia gamoviyenoT geometriuli 

amocanebis amosaxsnelad. 

amocana 4. AC fuZis mqone tolferda ABC samkuTxedSi gavlebulia 

AD biseqtrisa. C wverosTan mdebare gare kuTxe 1,5-jer metia ADC 

kuTxeze. vipovoT ABC samkuT xedis kuTxeebi. 

B 

moc: ΔABC, AB = BC , 

AD biseqtrisaa, 

∠ DCK = 1, 5∠ADC 

amoxsna: 

u.g. ∠A, ∠B, ∠C 

I. sityvieri modeli: 

mocemulobis Tanaxmad ∠ DCK = 1, 5∠ADC 

II. ucnobis Semotana da monacemebTan dakav Sireba: 

A 

C K 

0 

vTqvaT ∠ ACD = x . maSin, 

0 

∠ BAC = x , 

0 

x 

∠ DAC = , 

2 

0 

0 0 x 

∠ ADC = 180 − ( x + ), 

2 

0 0 

∠ DCK = 180 −x. 

III. gantolebis Sedgena: 

0 

0 0 0 0 x 

180 − x = 1,5(180 − ( x + )). 

2 

IV. gantolebis amoxsna: 

0 0 0 9 0 

180 − x = 270 − x , 

4 

9 0 0 0 0 

x − x = 270 −180 

4 

5 0 0 0 0 

x = 90 , x = 72 . 

0 0 4 

0 

V. pasuxi: ∠ A=∠ C = 72 , ∠ B= 180 −( ∠ A+∠ C) 

= 36 . 

D


1. ra safexurebs gulisxmobs amocanis amoxsnis algebruli meTodi? 

2. ra kavSiria amocanis amoxsnis pirvel da mesame safexurebs Soris? 

3. ris safuZvelze SevadgineT amocana 1-is amoxsnis sityvieri modeli? 

4. amocana 1-is mixedviT SeZlebs Tu ara jiqi antilopas daWeras, Tu 

manZili maT Soris iqneba 95 metri? 120 metri? 

savarjiSoebi 

1 erTi ricxvi meoreze 3-jer mcirea. ipove es ricxvebi, Tu maTi 

sxvaobaa 16 

2 erTi ricxvi meoreze 4-jer didia. ipove es ricxvebi, Tu maTi 

jamia 85. 

3 erTi mosazRvre kuTxe meoreze 360-iT didia. ipove am kuTxeebis 

gradusuli zomebi. 

4 erTi mosazRvre kuTxe meoreze 4-jer didia. ipove am kuTxeebis 

gradusuli zomebi. 

5 erT bidonSi orjer meti rZea, vidre meoreSi. ramdeni litri 

rZea TiToeul bidonSi, Tu oriveSi 30 litri rZea? 

6 erTi kilogrami pirveli xarisxis brinji 0,2 lariT iafia 

umaRlesi xarisxis brinjze. ra Rirs TiToeuli xarisxis erTi 

kilogrami brinji, Tu Tinikom TiTo kilogram orive xarisxis 

brinjSi erTad 3,6 lari gadaixada? 

7 saaxalwlod erT maRaziaSi ori yuTiT meti maSxala gaiyida, 

vidre meoreSi. ramdeni yuTi maSxala gaiyida TiToeul maRaziaSi, 

Tu oriveSi erTad 22 yuTi gaiyida? 

8 or yuTSi 72 kilogrami vaSlia. erT maTganSi 3-jer meti vaSlia, 

vidre meoreSi. ramdeni kilogrami vaSlia TiToeul yuTSi? 

9 dilidan Sesvenebamde maRaziam 64 kilogrami Saqris fxvnili 

gayida. Sesvenebis Semdeg kidev Semoitanes 100 kilogrami da 

gayides 38 kilogrami. saRamos maRaziaSi darCenili iyo 136 

kilogrami Saqris fxvnili. ramdeni kilogrami Saqris fxvnili iyo 

maRaziaSi diliT? 

10 imis Semdeg, rac avtobusSi avida 10 mgzavri da Camovida 6, 

avtobusSi 28 mgzavri aRmoCnda. ramdeni mgzavri iyo avtobusSi 

Tavdapirvelad? 


11 marTkuTxedis perimetri 48 santimetria. misi erTi gverdis 

sigrZe 5-jer metia meoreze. gamoTvale marTkuTxedis gverdebis 

sigrZeebi. 

12 marTkuTxedis formis samosaxlo nakveTis sigrZe 20 metriT metia 

mis siganeze. nakveTze Semovlebuli Robis sigrZe 160 metria. 

gamoTvale nakveTis farTobi. 

13 navma dinebis mimarTulebiT sami, xolo sawinaaRmdego 

mimarTulebiT ori saaTi imoZrava. gamoTvale mdinaris dinebis 

siCqare, Tu navis sakuTari siCqarea 8,5 km/sT da navma orive 

mimarTulebiT 44,5 kilometri gaiara. 

14 navma dinebis mimarTulebiT ori, xolo sawinaaRmdego 

mimarTulebiT erTi saaTis ganmavlobaSi 43 kilometri gaiara. 

gamoTvale navis sakuTari siCqare, Tu dinebis siCqarea 2,5 km/sT. 

15 vaSlis Ciris miRebisas vaSli Tavisi wonis 85%-s kargavs. 

a) ramdeni kilogrami nedli vaSlia saWiro 20 kilogrami Ciris 

misaRebad? 

b) ramdeni kilogrami Ciri miiReba 200 kilogrami nedli 

vaSlisagan? 

16 yava moxalvisas Tavisi wonis 12%-s kargavs. 

a) ramdeni kilogrami nedli yavaa saWiro 5 kilogrami moxaluli 

yavis misaRebad? 

b) ramdeni kilogrami moxaluli yava miiReba 20 kilogrami nedli 

yavisagan? 

17 nato 15 wliT ufrosia nanaze. aTi wlis Semdeg nato 2-jer 

ufrosi iqneba nanaze. ramdeni wlisaa TiToeuli gogona? 

18 mama 50 wlisaa, Svili 28-is. ramdeni wlis win iyo Svili mamaze 

2-jer umcrosi? 

19 samkervalo firmas SekveTa 10 dReSi unda Seesrulebina. firma 

yoveldRiurad imaze 27 nawarmiT mets amzadebda, vidre 

SemkveTTan iyo SeTanxmebuli da SekveTa 7 dReSi Seasrula. 

amasTanave sxva SemkveTsac gaumzada 54 aseTi nawarmi. dReSi 

ramden nawarms amzadebda firma. 

20 giam erT dRes romanis 3 

5 

nawili waikiTxa, meore dRes _ darCenilis 

naxevari da ukanaskneli 20 gverdi. ramden gverdiani romani 

waikiTxa giam? 

21 ori qalaqidan, romelTa Soris manZili 330 kilometria, 

erTmaneTis Sesaxvedrad erTdroulad ori avtomobili gamovida. 

erTi avtomobilis siCqare 10km/sT-iT metia meore avtomobilis 

236 Tavi 4 • erTwevri da mravalwevri

siCqareze. 2 saaTis Semdg avtomobilebs Soris manZili 30 

kilometrs udrida. ra siCqariT moZraobdnen avtomobilebi? 

22 336 mm sisqis Zeli 30 mm sisqis ficrebad unda daixerxos. ramdeni 

ficari miiReba daxerxvis Semdeg, Tu TiToeuli gaxerxvisas 

naxerxis saxiT 4 mm sisqis masala ixarjeba? 

23 fexburTis gundis 16 moTamaSis saSualo asaki 23 weli iyo. mas 

Semdeg, rac gundidan veterani fexburTeli wavida, saSualo asaki 

22 weli gaxda. ramdeni wlis iyo veterani fexburTeli? 

24 daamtkice, rom gemis sakuTari siCqare dinebis mimarTulebiT 

siCqarisa da sawinaaRmdego mimarTulebiT siCqaris saSualo 

ariTmetikulis tolia. 

25 gemi A da B navsadgurebs Soris manZils dinebis mimarTulebiT 8 

saaTs, xolo sawinaaRmdego mimarTulebiT 10 saaTs andomebs. ra 

dros moandomebs gemi amave manZilis gavlas mdgar wyalSi? 

26 marTkuTxedi, romlis sigrZe 80sm, xolo sogane 60sm-ia, 48 tol 

kvadratad aris dayofili. ipove TiToeuli kvadratis gverdis 

sigrZe. 

27 ori kubidan erTis moculoba 27-jer metia meore kubis 

moculobaze. ramdenjer metia pirveli kubis waxnagis farTobi 

meore kubis waxnagis farTobze? 

aba scade! 

daamtkice, rom toli perimetrebis mqone marTkuTxedebs 

Soris udidesi farTobi kvadrats aqvs.

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