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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

8 Tavi 1 • racionaluri ricxvebi

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;

Tavi 1 • racionaluri ricxvebi 9


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

10 Tavi 1 • racionaluri ricxvebi

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

Tavi 1 • racionaluri ricxvebi 11


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

12 Tavi 1 • racionaluri ricxvebi

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


upasuxe kiTxvebs

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%

Tavi 3 • monacemebi 157


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.

158 Tavi 3 • monacemebi

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

Tavi 3 • monacemebi 159


3.3 rogor vipovoT ricxvi misi procentis mixedviT

mizani:

160 Tavi 3 • monacemebi

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.

Tavi 3 • monacemebi 161


upasuxe kiTxvebs

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)

162 Tavi 3 • monacemebi

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?

Tavi 3 • monacemebi 163


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).

164 Tavi 3 • monacemebi

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.

Tavi 3 • monacemebi 165


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.

upasuxe kiTxvebs

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.

Tavi 3 • monacemebi 167


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

Tavi 3 • monacemebi 169


aodenobas.

170 Tavi 3 • monacemebi

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?

Tavi 3 • monacemebi 171


3.5 monacemTa ricxviTi maxasiaTeblebi

mizani:

172 Tavi 3 • monacemebi

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?).

upasuxe kiTxvebs

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 .

Tavi 3 • monacemebi 173


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;

174 Tavi 3 • monacemebi

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?

.

Tavi 3 • monacemebi 175


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.

upasuxe kiTxvebs

176 Tavi 3 • monacemebi

• 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).

Tavi 3 • monacemebi 177


178 Tavi 3 • monacemebi

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?

Tavi 3 • monacemebi 179


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.

180 Tavi 3 • monacemebi

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.

Tavi 3 • monacemebi 181


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:

182 Tavi 3 • monacemebi

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

+ =

2. ucnobis Semotana da monacemebTan dakavSireba:

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.

1. gantolebis Sedgena:

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


2. ucnobis Semotana da monacemebTan dakavSireba:

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.

3. gantolebis Sedgena:

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


upasuxe kiTxvebs

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?

Tavi 4 • erTwevri da mravalwevri 235


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.

Tavi 4 • erTwevri da mravalwevri 237

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