2

4

6

12

8

10

88

1010

nawili

moswavlis wigni

maswavleblis wigni moswavlis wigni

dafinansebulia

`moswavleebis

saxelmZRvaneloebiT

uzrunvelyofis programis~

farglebSi

maTematika

��maswavleblis wigni

inteleqti

3

rCevebi wigniT sargeblobisTvis .................................................................................... 4

1. gamravlebis Tvisebebi. gavixsenoT gamravlebis cxrili .............................. 5

2. gamravleba da gayofa. gayofis Tvisebebi ......................................................... 14

3. vamravlebT nulebiT daboloebul ricxvebs .................................................... 22

4. vamrgvalebT Tanamamravlebs da vpoulobT namravls miaxloebiT ......... 29

proeqti ............................................................................................................................. 34

5. erTniSna ricxvze gamravleba ................................................................................. 35

6. orniSna ricxvze gamravleba .................................................................................... 42

7. vamravlebT samniSna ricxvze .................................................................................. 47

8. moqmedebaTa Tanamimdevroba .................................................................................... 53

9. gayofa naSTiT ............................................................................................................... 58

10. erTniSna ricxvze gayofa .......................................................................................... 64

jgufuri proeqti ........................................................................................................ 68

11. Sesabamisoba ..................................................................................................................... 71

proeqti ............................................................................................................................. 75

12. gamravlebisa da gayofis ucnobi komponentis povna ................................... 78

13. amocanebis amoxsna (amocanebi zedmeti mocemulobiT) ................................ 81

SeamowmeT Tqveni codna ............................................................................................ 86

damatebiTi amocanebi .................................................................................................. 87

V Tavis Semajamebeli amocanebi............................................................................. 92

pasuxebi ............................................................................................................................. 95

maTematikuri simboloebi da maTi gamoyenebis magaliTebi ..................... 106

ariTmetikuli moqmedebebis Sesrulebis Tanamimdevroba ......................... 106

ariTmetikuli moqmedebebis Tvisebebi .............................................................. 106

zomis erTeulebi ........................................................................................................ 107

sivrculi figurebi ................................................................................................... 107

maTematikuri terminebis leqsikoni ................................................................... 108

sarCevi

naturaluri ricxvebis gamravleba da gayofa5

4

r C e v e b i w i g n i T s a r g e b l o b i s T v i s

waikiTxeT es rCevebi, gaecaniT specialuri simboloebis daniSnulebas.

es simbolo klasSi Sesasrulebeli samuSaoebis dawyeba­

ze migviTiTebs. am amocanebis nomrebi wreebSia Cas­

muli, feradicaa, magaliTad, 4 .

es simbolo Sin muSaobisTvis gankuTvnil amocanebs

warmogvidgens. am amocanebis nomrebi samkuTxedebSia

Casmuli, magaliTad, 5 .

`sxvadasxva~ _ saintereso faqtebsa da istoriebs

warmogvidgens.

am simbolos mosdevs amocanebi, romelTa amosaxsnelad

sakmao gamWriaxobis gamovlenac mogiwevT. am amo canebis

nomrebi kvadratebSia Casmuli, magaliTad, 1 .

am simboloTi jgufuri muSaobis proeqtebia gamo­

yofili _ aseTi davalebebi TanaklaselebTan erTo­

blivad muSaobis gamocdilebas SegmatebT.

am simboloTi TamaSebia aRniSnuli – maTi saSuale­

biT maTematikuri codnis gamoyenebis gamocdilebas

SeiZenT.

vin iqneba pirveli. am simboloTi warmodgenili amo­

canebis amoxsnisas ecadeT, gamWriaxobasTan erTad

sxa rti azrovnebac gamoavlinoT.

teqstis ukeT gagebis mizniT, saWirod CavTvaleT

zogierTi sityvis warmomavlobis, misi msgavsi mniSvn­

elobebis miTiTeba. aseTi Canawerebi teqstSi moniSnu­

lia aq gamosaxuli simboloTi.

am simboloTi gamoyofili yvela amocana unda gadaita­

noT rveulSi da DSemdeg SeasruloT.

Cveni

leqsikoni

jgufuri

saSinao

moisazre

sxvadasxva

TamaSi

klasSi

gaufrTxildiT wigns!

masSi Tqvenma Canawerebma (amoxsnebma, miTiTebebma) wigni SeiZleba sxvebisTvis

gamousadegari gaxados!

5

• ramdeni wrea TiToeul striqonSi?

• sul ramdeni striqonia wreebiT Sedgenili?

• ipoveT wreebis odenoba gamravlebis gamoyenebiT da

Se a dareT Tqven mier miRebuli Sedegi aq warmo dgenils _

6⋅4=24.

axla amave wreebiT Sedgenil 4 svets davxedoT.

TiToeul svetSi 6 wrea.

• amjerad rogor vipovoT wreebis odenoba ricxvebis gamravlebiT?

am msjelobas mivyavarT gamravlebis I Tvisebamde:

gadanacvlebadobis Tviseba magaliTi

TanamamravlTa gadanacvle­

biT namravli ar icvleba.

6 ⋅ 4 = 24

4 ⋅ 6 = 24

viTvliT kvadratebis odenobas, gvaqvs:

4 ⋅ 8 =32

axla kvadratebis es erToblioba swori xa­

ziT gamoyofil or nawilad warmovadginoT:

4 ⋅ 3 =12 4 ⋅ 5 = 20

• sul oTxi striqoni gvaqvs, TiToeuli striqonSi kvadratebis

toli raodenobaa: 3 + 5. sul – 4 ⋅ (3 + 5). maSasadame,

4 ⋅ (3 + 5) = (4 ⋅3 ) + (4 ⋅ 5 ).

I Tanamamravli II Tanamamravli namravli

I Tanamamravli II Tanamamravli namravli

naturaluri ricxvebis gamravlebada gayofa

1 gamravlebis Tvisebebi. gavixsenoT gamravlebis cxrili

gakveTilebi 93-95

5ricxvebze moqmedebebis codnis gareSe SeuZlebelia swavlaSi war matebis imedi gqondeT. am TavSi Tqven mniSvnelovnad gaiR­rma vebT cod nas am sakiTxebSi, SeiswavliT araerTi saintereso amo canis amox snis gzebs. gisurvebT warmatebas.

6

am magaliTiT gamravlebis kidev erT Tvisebas warmovadgenT:

jamze gamravlebis Tviseba(ganrigebadobis Tviseba)

magaliTebi

ricxvis jamze gamravlebisas es ricxvi

unda gavamravloT TiToeul Sesakrebze,

Semdeg ki miRebuli namravlebi SevkriboT

5⋅(3+4)=5⋅3+5⋅45⋅7=15+20

35=35

4⋅(3+4+2)=4⋅3+4⋅4+4⋅24⋅9=12+16+8

36=36

axla gamravlebis kidev erTi Tvisebis _ jufTebadobis Tvisebis _ magaliTebi warmovadginoT: vamravlebT ricxvebs: 3⋅2⋅4. gamravleba SeiZleba sxvadasxva xerxiT SevasruloT: erTi da imave raodenobis

wreebi jer 4 jgufad davyoT, Semdeg – sam jgufad.

maSasadame, (3⋅2)⋅4=3⋅(2⋅4)

ricxvebis gamravlebisas Tanamamravlebi SeiZleba nebismieri

TanamimdevrobiT davajgufoT.

gavixsenoT gamravlebis kidev ori Tviseba:

• SeecadeT sityvierad CamoayaliboT 1­isa da 0­is es Tvisebebi.

axla gamravlebis cxrili gavixsenoT.

× 0 1 2 3 4 5 6 7 8 9

0 0

1 0 1

2 0 2 4

3 0 3 6 9

4 0 4 8 12 16

5 0 5 10 15 20 25

6 0 6 12 18 24 30 36

7 0 7 14 21 28 35 42 49

8 0 8 16 24 32 40 48 56 64

9 0 9 18 27 36 45 54 63 72 81

8⋅1=8, 1⋅8=8. 9⋅0=0, 0⋅9=0.

magaliTi 1 magaliTi 2

4 jgufidan TiToeulSi

3⋅2 wrea, sul 4⋅(3⋅2), anu

(3⋅2)⋅4 wre

aq sami jgufidan

TiToeulSi 2⋅4 wrea,

sul 3⋅(2⋅4) wre

7

klasSi

Sevsebulia mxolod `diagonali~ (ricxvebi wiTladaa gaferadebu­

li) da `diagonalis~ qveda ujrebi.

moifiqreT raime wesi (magaliTad, gamravlebis Tviseba an striqo­

nebis Sevseba erTi da imave bijiT gadaTvlili ricxvebiT), romlis

gamoyenebiTac SeiZleba Seivsos danarCeni ujrebi.

• ra ricxvebia warmodgenili diagonalze?

• ra Tvisebaze migviTiTebs meore svetSi dawerili nulebi?

• ra Tvisebaze migviTiTebs pirveli da mesame svetebis Sedareba?

SearCieT swori pasuxi (imuSaveT rveulSi):

1 ricxvis nulze gamravlebiT aucileblad miiReba

a) erTi b) ori

g) igive ricxvi d) nuli.

2 ricxvis erTze gamravlebiT aucileblad miiReba

a) nuli b) ori

g) igive ricxvi d) sxva ricxvi.

3 ra ricxvs aRniSnavs , Tu ⋅ 7=7 ⋅ 8?

a) 7 b) 1 g) 8 d) 5.

4 ra ricxvia aRniSnuli n asoTi, Tu n⋅(6+7)=5⋅6+5⋅7?

a) 6 b) 7 g) 1 d) 5.

5 CawereT gamosaxuleba namravlis saxiT: 6+6+6+6+6.

a) 6⋅6 b) 4⋅6 g) 5⋅6 d) 3⋅6.

6 ramdeniTaa 3⋅9 meti 2⋅9­ze?

a) 8­iT b) 9­iT g) 3­iT d) 2­iT.

7 (2⋅3)⋅8=

a) 36 b) 48 g) 24 d) 16.

8 CawereT jamis saxiT gamosaxuleba: 3⋅4.

a) 4+4+4 b) 3+3+4 g) 4+4 d) 4+3+4+3.

8

nimuSi

9 Tu ninos kargad axsovs, rom 7⋅7=49, maSin 8⋅7­is sapovnelad

sakmarisia man 49­s

a) gamoaklos 7 b) miumatos 8

g) miumatos 7 d) gamoaklos 8.

10 mocemulia gamosaxulebebi:

I. 2⋅9+5⋅9 II. 3⋅9+4⋅9 III. 4⋅9+3⋅9 IV. 7⋅9+4⋅9 V. 5⋅9+3⋅9.

romeli gamosaxulebis mniSvnelobis tolia 7⋅9?

a) I da IV b) II da V

g) mxolod II da III d) I, II da III.

11 romeli suraTi dagexmarebaT, rom ipovoT 3­jer 4?

a)     b)      g)     d)    

                                                                  

                     

12 TiToeul suraTze gamosaxuli sagnebis odenoba warmoadgineT

gamravlebisa da Sekrebis Semcveli gamosaxulebebiT:

a) b)

g) d)

ipoveT namravli:

13 3⋅2 14 3⋅7 15 4⋅6 16 4⋅5

17 6⋅6 18 8⋅7 19 9⋅7 20 9⋅9

21 4⋅4 22 6⋅9 23 (2⋅3)⋅8 24 7⋅(3⋅2)

25 2⋅(2⋅2)⋅2.

3⋅4=4+4+4

9

gamoiyeneT sqema:

am sqemis mixedviT SeadgenT tolobas: 9 ⋅ 4 = 6 ⋅

miTiTeba

ipoveT namravli ori sxvadasxva xerxiT:

26 3⋅(2+5) 27 4⋅(2+3) 28 5⋅(2+6)

ipoveT ucnobi ricxvi

29 8 ⋅ =8 30 ⋅ 5=0 31 6 ⋅ (4+3)= ⋅ 4+ ⋅ 3

32 2 ⋅ (4 ⋅ 3)=( ⋅ 4) ⋅ 3 33 8 ⋅ =0 34 6 ⋅ =9 ⋅ 6

35 ⋅ 1=13 36 ⋅ (4+2+1)=3 ⋅ 4+3 ⋅ 2+ ⋅ 1.

ipoveT ucnobi ricxvebi:

37 4 ⋅ ( ⋅ 9)=( ⋅ 6) ⋅

38 4 ⋅ (5 ⋅ )=( ⋅ 5) ⋅ 7

39 ⋅ (4+ +1)=5 ⋅ 4+ ⋅ 3+ ⋅ 1.

amoxseniT amocanebi:

40 levanma akaki wereTlis, vaJa­fSavelasa da galaktion tabiZis oTx­

oTxi leqsi iswavla. ramdeni leqsi iswavla levanma?

41 baTumis portSi Semosul satvirTo ge­

mebs 4 kviris ganmavlobaSi clidnen. yo­

vel kviraSi icleboda 9 gemi. ramdeni

kvi ra dasWirde boda amave gemebis daclas,

Tu yo vel kviraSi daicleboda 6 gemi?

42 ra kanonzomierebiT SeiZleba iyos warmodgenili ricxvebi?

1, 4, 9, 16.

dawereT Semdegi 5 ricxvi imave kanonzomierebis mixedviT.

4 kvira

9 gemi

? kvira

6 gemi

10

moisazre

ecadeT warmoadginoT amocana

sqemis saxiT, magaliTad, ase:

miTiTeba

43 kalaTburTelTa matCis saukeTeso moTamaSe

4 periodidan TiToeulSi TamaSidan 6 qulas

agrovebda, jarimebidan _ 3 qulas. sul ramdeni

qula daagrova am moTamaSem? ipoveT qulebis

odenoba ori xerxiT.

amocanis pirobis mixedviT SeadgineT ricxviTi gamosa-

xuleba, romelic Seicavs gamravlebisa da Sekrebis moqme-

debebs. upasuxeT amocanis kiTxvas:

44 qarTvelma kalaTburTelma zaza faCuliam erT­erT SexvedraSi 6

samquliani da amdenive orquliani burTi Caagdo. manve 7 erTquliani

jarima gamoiyena. sul ramdeni qula daagrova am SexvedraSi zaza

faCuliam?

45 ori soflidan erTmaneTis

Se saxvedrad ori turisti gae­

marTa. pirveli yovel saaTSi 5

km­s gadioda. meore _ 3 km­s. 3

saaTis Semdeg maT Soris manZili 2 km­ia. ra manZilia soflebs So ris?

ucnobi ricxvi x-iTaa aRniSnuli. ipoveT x:

1 6 ⋅ 8 = (6⋅x) – (6 ⋅ 1) 2 (6 ⋅ 3) – (6⋅2) = (6 ⋅ x)

3 6 ⋅ 0 = (6 ⋅ 8) – (6 ⋅ x) 4 (7 ⋅ 4) = (7 ⋅ 2) + 7 ⋅ x

amoxseniT amocanebi (isargebleT gamravlebis cxriliT):

5 nikam Cafiqrebuli ricxvi jer 2­ze gaamravla, Semdeg 9­ze.

waSala bolo (erTeulebis) cifri da miiRo 7. ipoveT Cafiqrebuli

ricxvi.

4 periodi

TamaSidan

sajarimoebidan

11

SearCieT swori pasuxi (imuSaveT rveulSi):

1 7 ⋅ 1=

a) 1 b) 8 g) 9 d) 7.

2 8 ⋅ 0=

a) 0 b) 8 g) 1 d) 9.

3 ra ricxvia aRniSnuli n­iT, Tu n ⋅ 8=8 ⋅ 9?

a) 8 b) 9 g) 10 d) 17.

4 ra ricxvia aRniSnuliT x­iT, Tu 7 ⋅ (5+x)=7 ⋅ 5+7 ⋅ 3?

a) 5 b) 7 g) 3 d) 4.

5 ramdeniT aris 5 ⋅ 9 meti 2 ⋅ 9­ze?

a) 27­iT b) 18­iT g) 9­iT d) 36­iT.

6 4 ⋅ 2 ⋅ 8=

a) 36 b) 32 g) 64 d) 20.

7 vTqvaT, kalkulatoris gamoyenebiT vpoulobT namravls: 4*2*5.

amave pasuxs miviRebT, Tu vipoviT namravls:

a) 2*6*3 b) 2*8*4 g) 2*5*4 d) 3*2*5.

8 (3 ⋅ 0) ⋅ 9=

a) 9 b) 3 g) 4 d) 0.

saSinao

6 mSoblebma daviTs gauorkeces Tanxa, romelic mas hqonda.

daviTma daxarja 8 lari da darCa 10 lari. ra Tanxa

hqonda daviTs Tavdapirvelad?

7 rusikos yulabaSi 55 lari aqvs. es Tanxa Sedgenilia

20­lariani, 10­lariani da 5­lariani kupiurebiT. maTi

ricxvia 5. ramdeni 5­lariania rusikos yulabaSi?

8 oTxi erTnairi burTis SesaZenad paatam molares 100­lariani

miawoda. molarem mas 8 cali 5­lariani da 8 cali 2­lariani

daubruna. ipoveT burTis fasi.

12

9 (2 ⋅ 4) ⋅ 1=

a) 4 b) 2 g) 8 d) 1.

10 CawereT jamis saxiT: 4 ⋅ 5.

a) 5+5 b) 5+5+5+5 g) 4+4+4 d) 4+4+4+4.

11 8 ⋅ 9 ar aris toli

a) (7 ⋅ 9+9)­is b) (6 ⋅ 9+2 ⋅ 9)­is

g) (5 ⋅ 9+3 ⋅ 9)­is d) (6 ⋅ 9+3 ⋅ 9)­is.

ipoveT namravli:

12 8 ⋅ 1 13 4 ⋅ 4 14 9 ⋅ 3 15 4 ⋅ 8

16 7 ⋅ 7 17 8 ⋅ 0 18 4 ⋅ 5 19 5 ⋅ 5

20 4 ⋅ 6 21 9 ⋅ 8 22 7 ⋅ 0 23 6 ⋅ 6

24 1 ⋅ 9 25 9 ⋅ 0 26 6 ⋅ 2 27 8 ⋅ 8

28 7 ⋅ 9 29 2 ⋅ 7 30 7 ⋅ 6 31 6 ⋅ 9

32 9 ⋅ 4 33 8 ⋅ 5 34 3 ⋅ 8 35 9 ⋅ 7

36 1 ⋅ 4 37 9 ⋅ 5 38 8 ⋅ 6 39 7 ⋅ 5.

ipoveT rac SeiZleba meti wyvili im ricxvebisa, romelTa namravlia:

40 24 41 12 42 25

43 1 44 38 45 81.

amoxseniT amocanebi

46 kalaTburTelTa matCSi TiToeuli gundidan 9 kalaT­

burTelma miiRo monawileoba. sul ramdeni kalaTburTe­

li monawileobda SexvedraSi?

13

47 qarTuli simRerebis konkursSi erT­erTi skolis gundi scenaze

6 mwkrivad ganlagda. TiToeul mwkrivSi 8 moswavle iyo. meo­

re skolis momReralTa gundi ki 5 mwkrivad iyo ganlagebuli,

TiToeul mwkrivSi _ 9 moswavliT. romeli skolis gundSi iyo

meti monawile?

amocanebis amoxsnisas SeadgineT ricxviTi gamosaxuleba, romelic Seicavs gamravlebisa da Sekrebis moqmedebebs da upasuxeT kiTxvebs:

48 mSoblebi vatos 5 dRis ganmavlobaSi sam­sam lars aZlevdnen

skolaSi sauzmisTvis. vato mxolod 1 lars xarjavda. kviris bo­

los mSoblebma mas damatebiT 8 lari aCuqes. sul ramdeni lari

daugrovda vatos?

49 kote kviraSi samjer or­ori saaTi var­

jiSobs fexburTis wreSi. SabaTobiT ki misi

varjiSi 4 saaTs grZeldeba. sul ram deni

saaTi varjiSobs kote kviris ganmavlobaSi?

50 mariamma kuburebi oTx­oTxad daalaga. sul 8 aseTi oTxeuli mi­

iRo da kidev 3 kubura darCa. sul ramdeni kubura hqonia mariams?

51 darbazSi 7 magida dgas. TiToeul magidasTan 4 skamia. aqve, ke­

delTan kidev sami skamia. sul ramdeni skamia darbazSi?

52 ninikos 3 akvariumi aqvs. TiToeul akvariumSi 5 Tevzia. mariams

3 akvariumi aqvs. TiToeul akvariumSi 4 Tevzia. ramdeni Tevzia

orive gogonas akvariumebSi?

53 qeTis saxli 9 sarTuliania. am saxls ori sadarbazo aqvs. I

sadar bazoSi yovel sarTulze 8 binaa, II sadarbazoSi yovel sar­

Tulze 7 binaa. ramdenbiniania qeTis saxli?

14

a) vatom 20 suraTi albomis ramdenime cariel gverdze ase ga­

naTavsa _ TiTo gverdze 4 suraTi. maSasadame, 20 suraTi 4 tol

jgufad gaiyo.

• ra moqmedebiT ipoviT am suraTebiT Sevsebuli gverdebis ode­

nobas?

vmsjelobT _ Tu yovel gverdze 4 suraTia, maSin am suraTebiT

Sevsebuli gverdebis odenobis 4­ze namravli toli unda iyos 20­

is. Tu saZiebeli gverdebis odenobas aRvniSnavT n­iT, maSin n ⋅ 4=20.

aseTi n ricxvi 20­is 4­ze ganayofia _ n=20:4.

cxadia, 5 ⋅ 4=20. amrigad, 20:4=5, n=5.

b) 15 wigns 3 Taroze Tanabrad vanawilebT. ramdeni

wigni iqneba TiToeul Taroze?

• ra moqmedebiT vpoulobT wignebis odenobas TiToeul

Taroze?

wina amocanis amoxsnis analogiurad vmsjelobT:

n ⋅ 3=15

5 ⋅ 3=15 maSasadame, 15:3=5.

• SeadareT a) da b) magaliTebi: erTobliobas (wignebis, suraTe­

bis) jgufebad vyofdiT. ras veZebdiT pirvel SemTxvevaSi? ras ve­

ZebdiT meore SemTxvevaSi?

am suraTs ukavSirdeba gamravlebisa da gayofis

moqmedebebi:

3 ⋅ 6=18 18:6=3

6 ⋅ 3=18 18:3=6 (1)

gayofa SeiZleba sxvadasxvanairad CavweroT:

18 : 3 = 6 18 3

6

gasayofi, gamyofi, ganayofi gayofis komponentebia,

Tanamamravlebi da namravli _ gamravlebis komponentebi.

gasayofi gasayofigamyofi

gamyofi

ganayofi ganayofi

2 gamravleba da gayofa. gayofis Tvisebebi

gakveTilebi 96-99

15

klasSi

• dawereT ricxvebis gayofisa da gamravlebis ma­

galiTebi, romlebic am suraTs ukavSirdeba.

dawerili tolobebiT gamravlebisa da gayofis

komponentebs Soris kavSirebia warmodgenili.

• am tolobebis mixedviT ipoveT ucnobi komponenti, romelic n aso­

Tia aRniSnuli:

1) n ⋅ 6=24 2) n : 6=4

4 ⋅ n=24, 24 : n=6.

g) axla gayofis sxva Tvisebebi warmovadginoT;

magaliTebi: 9 : 1=9, 9 : 9=1, 0 : 5=0.

8 : 0 _ ara aqvs azri.

• SeecadeT am magaliTebiT aRwerili Tvisebebi sityvierad Camoaya­

liboT.

SearCieT swori pasuxi (imuSaveT rveulSi):

1 Tu raime ricxvs gavyofT 1­ze, miviRebT

a) amave ricxvs b) 1­s g) 0­s d) 2­s.

2 Tu nulisgan gansxvavebul ricxvs gavyofT amave

ricxvze, maSin miviRebT

a) nuls b) 1­s g) 2­s d) amave ricxvs.

3 Tu nuls gavyofT nulisgan gansxvavebul ricxvze, miviRebT

a) 0­s b) 1­s g) 2­s d) amave ricxvs.

4 raime ricxvis gayofa nulze

a) mogvcems nuls b) ar SeiZleba

g) mogvcems 1­s d) SeiZleba, Tu es gasayofi nulia.

5 72 : 8=9; am tolobaSi 72 aris

a) gamyofi b) ganayofi g) Sedegi d) gasayofi.

6 63 : 7=9; am tolobaSi 7 aris

a) gamyofi b) ganayofi g) Sedegi d) gasayofi.

16

7 Tu 63 sagans toli odenobis 7 jgufad davyofT, maSin TiToeul

jgufSi sagnebis odenoba iqneba

a) 8 b) 9 g) 7 d) 11.

8 Tu 56 wigns davanawilebT 8 yuTSi ise,

rom yuTebSi toli odenobis wignebi iyos,

maSin TiToeul yuTSi iqneba

a) 8 wigni b) 6 wigni

g) 5 wigni d) 7 wigni.

9 Tu 48 wigns Taroebze ise davawyobT, rom TiTo Taroze 8 wigni

iyos, maSin dagvWirdeba

a) 8 Taro b) 6 Taro g) 7 Taro d) 5 Taro.

SeasruleT gayofa:

10 16 : 4. 11 7 : 7. 12 6 : 3. 13 3 : 1.

14 0 : 3. 15 28 : 4. 16 0 : 5. 17 15 : 3.

18 12 : 4. 19 48 : 6. 20 27 : 9. 21 64 : 8.

22 81 : 9. 23 54 : 6. 24 54 : 9.

dawereT ricxvebis gayofisa da gamravlebis yvela magali-

Ti, romelic SeiZleba mocemul suraTze striqonebisa da

svetebis odenobebs aRwerdes:

25 26 27

28 29

17

gamosaxeT raime sqemiT (suraTiT) ricxvebis gayofa, ipo-

veT ganayofi da miuwereT yvela sxva toloba, romlebiT-

ac gayofis komponentebi erTmaneTs gamravlebiT an gay-

ofiT ukavSirdeba:

30 18 : 9. 31 64 : 8. 32 14 : 7.

33 48 : 8. 34 42 : 6.

ipoveT ucnobi komponenti:

35 5 ⋅ n=45. 36 n : 6=7. 37 28 : n=4. 38 n : 3=8.

amoxseniT amocanebi:

39 ramdeni orTvliani velosipedis dasamzade­

blad aris saWiro 12 Tva li?

40 ramdeni samTvliani velosipedis dasamzade­

blad aris saWiro 27 Tvali?

41 Tedom 24 larad iyida 4 erTnairi bloknoti.

ra Rirs 1 bloknoti?

42 rusudanma Tavisi moswavleebisTvis 12 larad 3 erTnairi wigni

iyida. ramden lars gadaixdida igi imave fasian 5 wignSi?

43 12 enZelas, 18 gvirilis da 30 yoCivardasgan Sekres 3 erTnairi

Taiguli. ramdeni enZela, ramdeni gvirila da ramdeni yoCivardaa

TiToeul TaigulSi?

18

saSinao

SearCieT swori pasuxi (imuSaveT rveulSi):

1 5 : 5=

a) 5 b) 1 g) 0 d) 2.

2 5 : 1=

a) 5 b) 1 g) 2 d) 0.

3 8 : 8=

a) 0 b) 1 g) 2 d) 8.

4 0 : 8=

a) 0 b) 1 g) 8 d) 4.

5 ramdeni xuTeulisgan Sedgeba 30?

a) 6 b) 5 g) 4 d) 7.

6 ramdeni oTxeuli SeiZleba SevadginoT 20­isgan?

a) 4 b) 5 g) 3 d) 20.

ipoveT ganayofi:

7 24 : 6. 8 63 : 9. 9 48 : 6. 10 32 : 8.

11 32 : 4. 12 42 : 7. 13 36 : 6. 14 49 : 7.

15 64 : 8. 16 54 9 17 45 9 18 72 9

19 9 9 20 16 8 21 56 7 22 72 8

23 9 1 24 48 8 25 24 3 26 40 5

27 36 9 28 32 8 29 64 8 30 7 7

31 35 7 32 0 4 33 56 8 34 28 7

35 27 3 36 54 6 37 24 8