მათემატიკა 5 კლასი ბექაური (1)

K upasuxe kiTxvebs:

1 rogor tolobas hqvia gantoleba?
2 ras hqvia gantolebis amonaxsni?

savarjiSoebi

1 amoxseni gantoleba:

a) n + 1 258 = 3005 b) n _ 1300 = 258
x + 4896 = 10258 a _ 6131 = 150
a + 36187 = 110250 836051 _ a = 293
3014 + a = 258963 743001 _ x = 2000
20014 + x = 37000 x _ 2586 = 1089

2 ra ricxvi unda eweros n - s nacvlad, toloba rom swori iyos?

a) 5 × n = 12 000 b) 102 000 : n = 400
3 × n = 66 660 625005 : n = 5
7 × n = 3 500 426000 : n = 42600
100 × n = 900 000 100530 : n = 90
300 × n = 12 300 680000 : n = 4000

3 ucnobi ricxvi asoTi aRniSne, Seadgine Sesabamisi gantoleba da
amoxseni. a) ra ricxvs unda mivumatoT 1 258, rom 6 574 miviRoT?

b) ra ricxvi unda mivumatoT 3 511 - s, rom 4 005 miviRoT?

g) ra ricxvs unda gamovakloT 3 006, rom 25 miviRoT?

d) ra ricxvi unda gamovakloT 567 - s, rom 112 miviRoT?

e) ra ricxvi unda gavamravloT 20 - ze, 8 400 rom miviRoT?

v) ra ricxvi unda gavamravloT 25 - ze, 1000 rom miviRoT?

z) ra ricxvi unda gavyoT 121 - ze, 35 rom miviRoT?

T) ra ricxvze unda gavyoT 5 600 rom 7 miviRoT?

4 Seadgine gantoleba naxatis mixedviT da amoxseni.

200 g 2kg 5kg500 g 5kg

1 kg 5kg 2kg 3kg 2kg 500 g
g)

a) b)

5 amoxseni gantoleba da Seamowme:

3 000 + x = 14 581 + 308 x × 750 = 4 609 + 2 891
4 569 + x = 8 001 _ 873 343 × x = 114 546 _ 45 946
24 086 _ x =3 318 : 158 x : 125 = 1 946 _ 1 688

6 Cawere tolobis saxiT winadadeba da gaarkvie, x - is ra mniSvnelo-
bisaTvis iqneba igi WeSmariti.
a) 444 naklebia 628 - isa da x - is sxvaobaze 41 - iT;
b) 256 - isa da x - is sxvaoba 760 - ze naklebia 644 - iT;
g) x - isa da 518 - is jami 601 - iT metia 467 -ze.

51

7 Cawere tolobis saxiT mocemuli winadadeba: `turistma 100 km - iani
8 gzis x km cxeniT gaiara, xolo danarCeni _ 78 km manqaniT~.

9 jer tolobis marjvena mxares mocemuli moqmedeba Seasrule da
Semdeg amoxseni gantoleba.
10
11 7 × a = 796 _ 12 a + 312 = 43 × 158
12 x : 126 = 1 458 : 9 a _ 4028 = 6 895 : 5
13 2400 : z = 3001 _ 2 989 5 × c = 3578 + 159 × 103

14 jer tolobis marcxena mxares mocemuli moqmedeba Seasrule da
15 Semdeg amoxseni gantoleba.
16
24 036 : 4 = 3 × a 185 040 : 9 _ 301 = 4 419 + a
17 35 942 + 156 558 = 500 × x 324 918 : 6 + 4 009 = a + 8 162
648 005 _ 647 914 = 7 × a 47 896 + 4 587 = x _ 1 547
18
mocemulia 2x _ 3 = 5. gamoTvale 25x _ 18.
52
mocemulia 25x + 370 = 870. gamoTvale 15x _ 17.

erTi fanqari Rirs n TeTri, xolo erTi saSleli m TeTri. Cawere

tolobis saxiT winadadeba: `2 fanqris fasi 3 saSlelis fasze 10

TeTriT metia.~

rgolebSi ricxvebi isea Cawerili, rom marcxena x 8

rgolSi Cawerili ricxvebis jami tolia marjvena 5

rgolSi Cawerili ricxvebis gaorkecebuli jamisa. 4 14 2
risi tolia x ?

nikas da gigis erTad 40 kakali aqvT. Tu nika gigis 12 kakals miscems,
maSin maT erTnairi raodenobis kaklebi eqnebaT. ramdeni kakali aqvs
gigis?

gelam velosipedsa da motociklSi 4200 lari gadaixada. motocikli
velosipedze 2-jer Zviri Rirda. ra Rirda velosipedi?

erT skolaSi 1 033 moswavlea, maTgan 569 gogonaa da 42 warCinebuli
vaJi. ramdeni vaJia skolaSi, romelic araa warCinebuli? qvemoT CamoT-
vlili gamosaxulebebidan romeli Seesabameba amocanis amoxsnas?

a) (1033 _ 569) + 42; b) 1033 + 569 + 42; g) 1033 _ 569;
d) (1033 + 569) + 42; e) (1033 _ 569) _ 42; v) 1033 + 569.

fermerma maRazias 175 kg Warxali, 860 kg kartofili da 120 kg stafi-

lo mihyida. ramdeni kilogrami bostneuli Seisyida maRaziam ferme-

risagan? ras aRniSnavs amocanis pirobebis mixedviT Sedgenili gamo-

saxuleba?

a) 175kg _ 120kg; b) 175kg + 120kg; g) 860kg _ 175kg;

d) 860kg _ 120kg; e) 860kg _ 175kg; v) 860kg + 120kg

gadaixaze da Seavse cxrili:

x 45 42 264 3589
20 + x
x _ 20

$ 2.4 kuTxe. kuTxeebis Sedareba. kuTxeebis saxeebi

movniSnoT O wertili da O wertilidan mizani:
gavavloT OA da OB sxivebi.
gavecnoT kuTxeebis
A saxeebs da viswavloT
maTi Cawera, wakiTxva
da amocnoba.

OB
miviReT am sxivebiT SemosazRvruli geometriuli figura, romelsac

kuTxe ewodeba. OA da OB sxivebi am kuTxis gverdebs warmoadgenen, xo-

lo O wertili wveros. kuTxes ` ~ simboloTi aRniSnaven. naxazze mo-

cemuli kuTxe ase Caiwereba ` O~ (ikiTxeba: `kuTxe O~), an ` AOB~ (ikiT-

xeba: `kuTxe AOB~), an ` BOA~ (ikiTxeba: `kuTxe BOA~). sami asoTi kuT-

xis Cawerisa da wakiTxvisas wveros asos aucileblad SuaSi vwerT da

vkiTxulobT. E
C

O F
A

O CAD D EFB B

nax 4.1

kuTxeebis Sesadareblad isini erTmaneTze ise unda davdoT, rom

wveroebi erTmaneTs SeuTavsdes da TiTo gverdi daemTxves. Tu meore

gverdebic daemTxva, maSin kuTxeebi tolia, Tu ar daemTxva, kuTxeebi

araa toli.

4.1 naxazze mocemuli A da F-is A da F wveroebs erTmaneTs Tu Sevu-

TavsebT, xolo AD gverds ki FB gverds, maSin AC gverdi FE da FB gver-

debs Soris, anu F kuTxis SigniT aRmoCndeba, radgan A < F. es Canaweri

ase ikiTxeba: `kuTxe A naklebia kuTxe F-ze~.

4.2 naxazze AOC da BOC marTi kuTxeebia.

amave naxazze O wertilidan erT wrfeze mdebare ori OA da OB sxivi

gamodis. isini gaSlil AOB kuTxes qmnian.

C

AO B
nax 4.2

gaSlili kuTxe ori marTi kuTxisagan Sedgeba.

marT kuTxeze mcire kuTxes maxvili kuTxe hqvia.
kuTxes, romelic gaSlil kuTxeze mcire da marT kuTxeze

didia, blagvi kuTxe hqvia.

M
C

BD KT
CBD blagvi kuTxea MKT maxvili kuTxea

53

K upasuxe kiTxvebs:

1 rogor davxazoT kuTxe?
2 ras ewodeba kuTxis gverdebi? kuTxis wvero?
3 rogor aRiniSneba kuTxe?
4 rogor xdeba kuTxeebis Sedareba? wakiTxva?
5 rogor kuTxes hqvia gaSlili? blagvi? maxvili?
6 romelia meti, gaSlili kuTxe Tu marTi kuTxe? ramdenjer?
7 romelia meti, maxvili kuTxe Tu blagvi kuTxe?

savarjiSoebi

1 ra saxis kuTxeebia naxazze mocemuli? daasaxele yoveli maTgani.

D O L AC

NE EM K T

O B
nax 4.3

2 daxaze a) marTi kuTxe; b) maxvili kuTxe;

g) blagvi kuTxe; d) gaSlili kuTxe.

3 qaRaldisagan gamoWeri ori b) aratoli kuTxe.
a) toli kuTxe;

4 rveulis furceli gakece Suaze da miRebuli ormagi furceli
kidev gakece Suaze. gaSale da gadakecvis xazebze fanqriT gaavle
wrfeebi. ramdeni kuTxe miiRe? rogori kuTxeebi miiRe?

5 qaRaldisagan gamoWeri blagvi kuTxe, gakece Suaze da gadakecvis
xazze gaWeri. ra saxis kuTxeebi miiRe?

6 gamoiyene mocemuli naxazi da daamzade kuTxis mo- A
deli xisgan, fanerisgan an sxva masalisagan. kuTxis
gverdebi erTmaneTTan ise unda iyos mimagrebuli,

rom SesaZlebe li iyos maTi moZraoba A wertilis
garSemo.

7 rogor kuTxes qmnian saaTis didi da patara isrebi erTmaneTTan
a) 9 : 00 sT - ze? b) 6 : 00 sT - ze? g) 4 : 00 sT - ze? d) 11 : 00 sT - ze?

8 Seadare TvalzomiT nax. 4.3 _ ze mocemuli kuTxeebi da Sedegi
Cawere ,, < ”, ,, > ” da ,, = ” niSnebis gamoyenebiT.

9 daxaze MNT maxvili kuTxe da kuTxis gverdebs Soris gaavle NB
sxivi. ra kuTxeebi miiRe naxazze? romelia maT Soris udidesi?

54

10 daxaze ACB , MNT da EFK kuTxeebi ise, rom ACB kuTxe meti iyos
MNT kuTxeze, xolo MNT kuTxe meti iyos EFK kuTxeze. romelia
meti ACB Tu EFK?

11 daxaze BOC blagvi kuTxe da kuTxis gverdebs Soris gaavle OK
sxivi. ra kuTxeebi miiRe naxazze? romelia maT Soris udidesi?

12 daxaze saerTo wveros mqone 4 marTi kuTxe.

13 daxaze gaSlili kuTxe da gayavi or nawilad.

14 daxaze saerTo wveros mqone blagvi, maxvili da marTi kuTxeebi.

15 daxaze saerTo gverdis mqone blagvi, maxvili da marTi kuTxeebi.

16 daasaxele naxazze mocemuli kuTxeebi da maTi saxeebi.

BC

AD
O
K

17 amocanas dasmul kiTxvaze pasuxis gasacemad aqvs Tu ara yvela
saWiro monacemi? Tu ara aqvs, daamate da amoxseni.
`SoTikom 3 skidan amoRebuli TafliT,
36 qila unda gaavsos. ramdeni Tafli
eteva TiToeul qilaSi? ~

18 deda Svilze n wliT ufrosia. ramdeni wlisaa Svili?
gadaixaze da Seavse cxrili amocanis pirobis gaTvaliswinebiT.

dedis asaki 36 28 50

n 30 21 24 30 25

Svilis asaki 21 10

K SSeseasZalZlebeebleliaiaT,uTauraa,ra

kvadrati daiyos or oTxkuTxedad ise, rom TiToeul oTxkuTxeds
hqondes a) erTi blagvi kuTxe? b) ori blagvi kuTxe?

55

$ 2.5 samkuTxedi

mizani: naxazze mocemulia sami mdgenisagan Sedgenili

1) gavecnoT samkuTxeds Caketili texiliT SemosazRvruli geometriuli
da mis saxeobebs
kuTxeebis mixedviT. figura _ samkuTxedi. B

2) samkuTxedis gverdebis AC
Tvisebebs. A, B da C wertilebi samkuTxedis wveroebia,

xolo maTi SemaerTebeli AB, AC da CB monakveTebi _ gverdebi. radgan

viciT, rom texilis Semadgeneli monakveTebis sigrZeTa jami metia misi

boloebis SemaerTebeli monakveTis sigrZeze, amitom vwerT:

AB+ BC > AC, aseve AB+ AC > BC da AC+ BC > AB.

samkuTxedis nebismieri ori gverdis sigrZis jami
mesame gverdis sigrZeze metia.

naxazze mocemuli samkuTxedis kuTxeebia:

ABC, ACB da BAC anu B, C da A.
samkuTxedi rom davasaxeloT, unda CamovTvaloT misi wveroebi nebis-
mieri mimdevrobiT. naxazze mocemul samkuTxeds ase davasaxelebT:

BAC, an ABC an ACB ...
unda ganvasxvaoT marTkuTxa, blagvkuTxa da maxvilkuTxa
samkuTxedebi.

ABC marTkuTxaa, radgan misi erT-erTi kuTxe ( C) marTia.

B

AC

MNK blagvkuTxaa, radgan misi erT-erTi kuTxe ( N) blagvia.

K

NM

EFT maxvilkuTxaa, radgan misi samive kuTxe maxvilia.
F

ET
scade da daxaze iseTi samkuTxedi, romelsac
a) ori kuTxe marTi aqvs;
b) ori kuTxe blagvi aqvs;
g) erTi kuTxe marTi aqvs da meore _ blagvi.
darwmundebi, rom arcerTi aseTi samkuTxedi ar arsebobs.

K upasuxe kiTxvebs:

1 rogor samkuTxeds hqvia marTkuTxa? maxvilkuTxa? blagvkuTxa?
2 ra geometriuli figuraa samkuTxedis wvero? gverdi?
3 ra geometriuli figuraa samkuTxedis sazRvari?
4 samkuTxedis erTi kuTxe blagvia. rogori iqneba danarCeni kuTxeebi?
5 samkuTxedis erTi kuTxe marTia. rogori iqneba danarCeni kuTxeebi?

56

savarjiSoebi

1 daxaze TvalzomiT: a) maxvilkuTxa; b) blagvkuTxa; g) marTkuTxa
samkuTxedi. Seni varaudi Seamowme marTkuTxa saxazavis saSualebiT.

2 daxaze samkuTxedi da moniSne wertili, romelic
a) ekuTvnis samkuTxeds; b) ar ekuTvnis samkuTxeds.

3 daasaxele naxazze mocemuli samkuTxedi, misi gverdebi da kuTxeebi.
ra saxis samkuTxedia kuTxeebis mixedviT? daasabuTe Seni varaudi.
G

TR

4 daxaze marTkuTxa samkuTxedi, romlis marTi kuTxis gverdebis
sigrZeebia 3 sm da 4 sm. gazome mesame gverdis sigrZe.

5 samkuTxedis erTi gverdis sigrZea 12 sm, meoris _17 sm, xolo mesamis
_ 14 sm. gamoTvale samkuTxedis perimetri.

6 78 sm perimetris mqone samkuTxedis formis detalis erTi gverdis
sigrZe 23 sm, meoris _ 37 sm. ra sigrZisaa detalis mesame gverdi?

7 samkuTxedis erTi gverdis sigrZea 47 sm 8 mm, danarCen or gverds
toli sigrZe aqvs. samkuTxedis perimetri 128 sm - is tolia. gamoT-
vale samkuTxedis danarCeni gverdebis sigrZeebi.

8 samkuTxedis erTi gverdis sigrZea 31 sm, xolo perimetri _ 88 sm.
gamoTvale danarCeni ori gverdis sigrZe, Tu cnobilia, rom erTi
maTgani meoreze 2 - jer grZelia.

9 samkuTxedis ori gverdis sigrZea 8 sm da 2 sm. qvemoT CamoTvlil-
Tagan romeli SeiZleba iyos samkuTxedis mesame gverdis sigrZe?
a) 3 sm; b) 6 sm; g) 9 sm; d) 14 sm.

10 manZili giorgis saxlidan skolamde 400 m - ia, xolo skolidan skveram-
de _ 500 m. giorgis saxli, skola da skveri isea ganlagebuli, rogorc
naxatzea naCvenebi. qvemoT CamoTvlilTagan romeli SeiZleba iyos
manZili giorgis saxlidan skveramde?

500 m 400 m

a) 100 m; ?m d) 850 m.
b) 900 m; g) 950 m;

11 samkuTxedis ori gverdis sigrZea 3sm da 4sm. qvemoT CamoTvlilTagan
romlis toli SeiZleba iyos samkuTxedis perimetri?

a) 12 sm; b) 6 sm; g) 8 sm; d) 14 sm.

57

12 manZili Teas saxlidan daTos saxlamde 300 m - ia, xolo daTos
saxlidan anas saxlamde _ 400 m.
a) ra udidesi manZili SeiZleba iyos Teas saxlidan anas saxlamde?
b) ra umciresi manZili SeiZleba iyos Teas saxlidan anas saxlamde?
g) SeiZleba Tu ara Teas saxlidan anas saxlamde manZili 500 m - is
toli iyos?

13 ramdeni marTkuTxa samkuTxedia naxazze?

14 ramdeni maxvilkuTxa samkuTxedia naxazze?

15 daxaze kvadrati da dayavi sam samkuTxedad, romelTagan ori iqneba
marTkuTxa, xolo mesame a) maxvilkuTxa; b) blagvkuTxa.

16 samkuTxedis erTi gverdis sigrZea 10 sm, meoris _ 7 sm. CamoTvlili
sidideebidan
2 sm, 3 sm, 4 sm, 16 sm, 17 sm, 25 sm
SearCie mesame gverdis sigrZe ise, rom miiRo:
a) udidesi perimetris mqone samkuTxedi;
b) umciresi perimetris mqone samkuTxedi.

17 Cawere winadadeba gantolebis saxiT da amoxseni:
a) x ricxvi 147 - ze 568 - iT metia;
b) 40 gaorkecebul x ricxvze 10 - iT metia;
g) 580 metia 4 - ze x - jer;
d) erTi ricxvi meore ricxvze 3 - jer naklebia, xolo orives jami
52 - ia.

a) gansazRvreT TvalzomiT manZili romelime or punqts Soris.
b) igive manZili gazomeT nabijebiT da Sedegi gamosaxeT metrebSi

da santimetrebSi.
g) igive manZili gazomeT sigrZis sazomi romelime xelsawyoTi.
d) gamoiangariSeT:

1) gansxvaveba Tvalzomisa da nabijebiT gazomvis Sedegebs Soris;
2) gansxvaveba Tvalzomisa da sigrZis sazomi xelsawyoTi gazomvis

Sedegebs Soris.
3) gansxvaveba nabijebiT gazomvisa da sigrZis sazomi xelsawyoTi

gazomvis Sedegebs Soris.
v) SeadgineT jgufis wevrebis sia Sedegebis mixedviT. siis TavSi

dawereT misi gvari, visac aRmoaCnda manZilis TvalzomiT Sefasebis
ukeTesi unari.

58

II Tavis damatebiTi savarjiSoebi

1 erTi avtomati erT wuTSi 450 kanfets fuTavs, meore _ 540 - s. ramden
kanfets SefuTavs orive avtomati naxevar saaTSi? Seadgine gamosa-
xuleba mocemuli amocanis pirobis mixedviT.

2 gamoTvale gamosaxulebis mniSvneloba:
a) 720 000 : 8 _ 705 × 12 + 453;
b) 423 160 : 40 _ 51 000 : 8;
g) 372 950 : 2 _ 9 048 + 325 × 106;
d) 70 109 _ (453 _ 387) × 136.

3 Tinikom TojinaSi 14 lari gadaixada, xolo wignSi _ n lari. ramdeni
lari gadaixada Tinikom navaWrSi? Seadgine gamosaxuleba mocemuli
amocanis pirobis mixedviT da gamoTvale misi mniSvneloba, roca
n = 10; 12.

4 erTi paketi rZe 900 g - s iwonis. ramden grams iwonis n cali ase-
Ti paketi rZe? Seadgine gamosaxuleba mocemuli amocanis pirobis
mixedviT da gamoTvale misi mniSvneloba, roca n = 5; 20.

5 Cawere gamosaxulebis saxiT da gamoTvale misi mniSvneloba:
a) 64 542 - is 3 - ze ganayofisa da 54 - is 126 - ze namravlis jami;
b) 25 892 - is ganayofi 4 002 - isa da 3 998 - is sxvaobaze.

6 daasaxele naxazis mixedviT yvela
a) marTi kuTxe; b) maxvili kuTxe; g) blagvi kuTxe; d) gaSlili kuTxe.
B
A
D
O
E

7 gamoTvale 21 630 : x _ 1085 gamosaxulebis mniSvneloba, roca x = 7; 5.

8 gamoTvale 2 569 × x _ 1085 gamosaxulebis mniSvneloba, roca x = 100;
206.

9 moZebne x -is mniSvneloba, romlisTvisac 15 + x = 67.

10 x -is ra mniSvnelobisaTvis sruldeba toloba:
a) 789 _ x = 128? b) 1024 + x = 2007? g) 105x = 1050? d) x : 18 = 1035?

11 rogor kuTxes Seadgenen erTmaneTTan saaTis isrebi:
a) 3 sT - ze? b) 7 sT - ze? g) 10 sT - ze?

12 daxaze ori kuTxe ise, rom maTi saerTo nawili iyos:
a) samkuTxedi; b) oTxkuTxedi.

13 daxaze ori toli AB da CD monakveTi ise, rom maT:
a) hqondes mxolod erTi saerTo wertili;
b) ar hqondes saerTo wertili.

14 daxaze ori wrfe ise, rom maT:
a) hqondes mxolod erTi saerTo wertili;
b) ar hqondes saerTo wertili.

59

15 daasaxele naxazis mixedviT: B C
16 a) marTkuTxa samkuTxedi; A D
b) maxvilkuTxa samkuTxedi;
17 g) blagvkuTxa samkuTxedi E

18 qvemoT CamoTvlili pirobebidan SearCie erTi, romlis mixedviT Sed-
genil amocanaSi davsvamT kiTxvas:
19
20 `ramdeni bavSvia gaerTianebuli cekvis wreSi?~
21 a) cekvis wreSi 25 moswavlea gaerTianebuli. maT Soris 13 gogonaa
60 b) cekvis wreSi gaerTianebulni arian gogonebic da vaJebic. maT

Soris gogonebis raodenoba erTiT metia vaJebis raodenobaze.
g) cekvavs 10 gogona da 10 vaJi.
d) cekvavs 10 gogona da gogonebze 4-iT naklebi vaJi.
e) cekvis wreSi gaerTianebulia 14 gogona da gogonebze 2-iT meti

vaJi.

mocemulia teqsti:
`avtosadgomze 84 avtomanqana idga. ramdeni avtomanqana darCa avto-
sadgomze?~
qvemoT mocemuli monacemebidan SearCie is, romlis damatebiT Sedgeba
amocana.
a) avtosadgomidan imaze 6-iT meti avtomanqana gavida, vidre iq Tav-

dapirvelad idga;
b) avtosadgomze saRamos 12 avtomanqana Semovida;
g) avtosadgomidan jer 35 avtomanqana gavida, xolo Semdeg _ 26.

Secvale amocanis teqsti ise, rom igi amoixsnas gamosaxulebiT:
`9_ 6~

a) or skamze 6 gogona ijda. erT maTganze ki _ 9. ramdeni gogona ijda
meore skamze?

b) baRSi mocvis 9 buCqi da mocvis buCqebze 6-iT meti mocxaria. mocxaris
ramdeni buCqia baRSi?

g) erT gaCerebaze avtobuss 9 mgzavri eloda, meoreze 6-iT naklebi,
vidre _ pirvelze. ramdeni mgzavrieloda avtobuss orive gaCere
baze?

avtosadgomze 12 satvirTo da 20 msubuqi manqana idga. avtosadgo-
midan gavida 17 manqana.
a) darCa Tu ara avtosadgomze erTi msubuqi manqana mainc?
b) darCa Tu ara avtosadgomze erTi satvirTo manqana mainc?

yuTSi sul 20 burTia _ Savi da TeTri. cnobilia, rom yuTidan Tu 10
burTs amoviRebT, maT Soris erTi mainc aucileblad iqneba TeTri,
xolo Tu 12-s amoviRebT, maT Soris erTi mainc aucileblad iqneba
Savi. ramdeni TeTri da ramdeni Savi burTia yuTSi?

gigam CxirebiT samkuTxedi unda Seadginos. mas
5 Cxiri aqvs zomiT: 3 sm, 4 sm, 8 sm, 20 sm da 27 sm.
romeli CxirebiT Seadgens giga samkuTxeds?

Tavi 3. ariTmetikuli moqmedebebi naturalur ricxvebze da maTi Tvisebebi

$ 3.1 mravalniSna ricxvebis Sekreba. Sekrebis Tvisebebi

moqmedebas, romliTac jamis gamoT- mizani:
vla xdeba, Sekreba ewodeba.
a) CamovayaliboT da asoebis
10 + 7 =17 gamoyenebiT CavweroT Sekrebis
gadanacvlebadobis da jufTeba-

Sesakrebebi jami dobis Tvisebebi.

naturaluri ricxvebis Sekreba Tan- b) ganvixiloT Sekrebis kerZo
rigobrivad xdeba: erTeulebs erTeule- SemTxvevebi.
bi emateba, aTeulebs _ aTeulebi, aseu-
lebs _ aseulebi da a. S. g) viswavloT Sekrebis Tvisebebis
magaliTad, gamoyeneba gamoTvlebSi.

d) viswavloT, rogor xdeba
milionze didi naturaluri

+ 67 1 14 8 5 32 ricxvebis Sekreba.

47 59 0 97 7 3

1 147 0 58 30 5

Tu ori Sesakrebidan erT - erTi nulis tolia,
maSin jami udris meore Sesakrebs.

Tu erT - erT Sesakrebs a asoTi aRvniSnavT, maSin es wesi ase
a +0 =a
Caiwereba: 0 +a =a



magaliTad,

4 523 + 0 = 4 523, 2 006 + 0 = 2 006, 0 + 65 003 = 65 003, 0 + 70 089 = 70089.

viciT, rom Sekrebis dros gamoTvlebis Sedegi SesakrebTa Tanmim-

devrobaze araa damokidebuli. Sekrebis am Tvisebas gadanacvlebado-

bis Tviseba hqvia.

SesakrebTa gadanacvlebiT jami ar icvleba.

magaliTad,

7 + 3 = 10 7+3=3+7 3 + 7 = 10

asoebis gamoyenebiT es kanoni nebismieri a da b ricxvebisaTvis ase

Caiwereba: a+b =b+a

sami Sesakrebis SemTxvevaSi SegviZlia jufTebadobis TvisebiT
visargebloT, romelic SemdegSi mdgomareobs:

ori ricxvis jams mesame ricxvi rom mivumatoT, SegviZlia
pirvel ricxvs mivumatoT danarCeni ori ricxvis jami.

asoebis gamoyenebiT es Tviseba nebismieri a, b da c ricxvebisaTvis

ase Caiwereba:

(a + b) + c = a + (b + c)

61

(6 + 12) + 8 = 26 6 + (12+ 8 ) = 26 12 + (6 + 8 ) = 26

mocemul naxatze dakvirvebiT SegviZlia davweroT:

(6 + 12) + 8 = 6 + (12 + 8) = 12 + (6 + 8).
Sekrebis ganxiluli Tvisebebidan SegviZlia gamoviyvanoT Sedegi:

ori ricxvis jams rom mesame ricxvi mivumatoT, sakmarisia, erT rome-

lime ricxvs mivumatoT danarCeni ori ricxvis jami. asoebis gamo-
yenebiT es Sedegi nebismieri a , b da c ricxvebisaTvis ase Caiwereba:

a + b + c = (a + b) + c = a + (b + c) = b + (a + c)
sabolood SegviZlia CamovayaliboT daskvna:

ramdenime ricxvis Sekreba SeiZleba nebismieri TanmimdevrobiT
Sesruldes.

am Tvisebis gamoyenebiT gamoTvlebis Catareba zogjer martivdeba.
magaliTad,

1063 + 645 + 1237 = (1063 + 1237) + 645 = 2300 + 645 = 2945

K upasuxe kiTxvebs:

1 raSi mdgomareobs SesakrebTa gadanacvlebadobis Tviseba?
2 ras udris ori ricxvis jami, rodesac erTi maTgani 0 - is tolia?
3 ra mniSvnelobis miReba SeuZlia a - s tolobaSi: a + 0 = a ?
4 ra mniSvnelobis miReba SeuZlia a, b da c - s (a + b) + c = a + (b + c)

tolobaSi?
5 ra hqvia 259 + 26 = 285 CanawerSi 259 - s? 26 - s? 285- s?

savarjiSoebi

1 gamoTvale jami: 33 657 148 + 49 182 537
345 800 + 1 258 259 20 215 863 + 456 798 312
298 600 + 995 050 + 1 350 + 700 000
24 589 + 96 458 120

589 614 + 1 025 809

2 Seasrule Sekreba:

28 147 + (964 + 1 589) 75 296 103 + 22 478 009

(4782_3529) +1287563 (55 789 + 458) + (9 632 + 61)

(420 651 + 5 184) + 75 082 (1 258 + 654 789) + (20 005 + 65 478)

3 gamoTvale ioli gziT:

(47 + 18) + 12 1 521 + (491 + 479)

(53 + 85) + 15 (7 117 + 2 397) + 2 883

(134 + 291) + 66 15 879 + (2 452 + 121 + 4 000)

4 gamoTvale: 288 + 9940 +4060 +1212
87 + 32 + 13 1054 +2805 +7195 +646
333 + 251 + 67 4 257 + 8 335 + 1 665 + 343
453 + 46 + 287

62

5 gamoTvale:
a + 1489 657 gamosaxulebis mniSvneloba, roca a = 258 979; 100 058.

6 erT qalaqSi sami bankia. erT bankSi mosaxleobis anabari 12 milion 785
aTas 800 laria, meoreSi _ 11 milion 624 aTas 600 lari, xolo mesameSi
_ 18 milion 900 aTas 500 lari. ramden lars Seadgens mosaxl eobis
anabari samive bankSi?

7 gamoTvale:
a) udidesi SvidniSna da umciresi SvidniSna ricxvebis jami.
b) umciresi rvaniSna da udidesi rvaniSna ricxvebis jami.

8 ori wlis win qalaqSi 5756 800 adamiani cxovrobda. axla 250 700-iT
meti cxovrobs. ramdeni adamiani cxovrobs axla qalaqSi?

9 amoxseni gantoleba: x _ 58 910 075 = 4 638 959
x _ 2 547 = 1 927 531 a _ 13 058 609 = 10 227 447
a _ 35 759 = 148 006 c _ 6 209 474 = 77 845 102
c _ 58 309 = 148 006 x _ 58 309 = 148 006
x _ 3 999 = 2 000 000

10 Secvale mocemuli gamosaxuleba iseTi gamosaxulebiT, romelic
Sekrebis mxolod erT moqmedebas Seicavs.

452 002 + (869 005 + a ) (2 719 546 + 869 005) + a
(x + 400 500) + 845 697 23 578 + x + 1 205

11 Rvinis qarxanam gasca 35840 l `saferavi~, 86875 l `manavis mwvane~
da 105450 l `rqawiTeli~. ramdeni litri Rvino gasca qarxanam sul?

12 erT qalaqSi 750 850 mcxovrebia, meoreSi 250 600-iT naklebi, vidre
_ pirvelSi, xolo mesameSi 198 750-iT naklebi, vidre _ meoreSi. ramdeni

mcxovrebia samive qalaqSi erTad?

13 gamoTvale gamosaxulebis mniSvneloba iolad moqmedebaTa Tanmimdev-
robis SerCevis gziT:

1242 + 1867 + 1458 + 1333 1008 + 994 + 200 + 6 + 92
872 + 25269 + 50128 + 731 1159308 + 2411365 + 392 + 635
1200705 + 457 + 300295 225200 + 427427 + 5373
6554327 + 345550 + 123 11611 + 7544 + 12456
(1700 + 12800) + 41200 (5470 + 690) + 1530
121000 + (19000 + 150000) 861000 + (5000 + 1134000)
1500000 + (258000 + 242000) 4328000 + (1251000 + 372000)
8887 +11419 + 725 + 411+145 11+12+13+14+15+16 +17 +18 +19

14 amoxseni gantoleba Sekrebis Tvisebebis gamoyenebiT:

(a + 43) + 37 = 128 3678 + (4002 + x) = 12369

(x + 4089) + 20001 = 50093 238960 + (7896 + a ) = 359657

(783 + x) + 20007 = 54790 (6719 + x) + 2281 = 71000

63

15 cxrilSi naCvenebia 3 dRis ganmavlobaSi skolis bufetSi zogierTi
sakvebi produqtis gayidviT Semosuli Tanxa.

kvebis orSabaTi samSabaTi oTxSabaTi sul
produqti

katleti 24 lari 25 lari 60 TeTri 20 lari 40 TeTri

orcxobila 80 TeTri 1 lari 60 TeTri 1 lari 60 TeTri

funTuSa 15 lari 25 TeTri 12 lari 50 TeTri 13 lari 25 TeTri

cxrilis mixedviT upasuxe kiTxvebs:
* ra Tanxis katleti gaiyida bufetSi am sami dRis ganmavlobaSi?
* ra Tanxis orcxobila gaiyida bufetSi am sami dRis ganmavlobaSi?
* ra Tanxis funTuSa gaiyida bufetSi am sami dRis ganmavlobaSi?
* ra Tanxa miiRo bufetma am sami dRis ganmavlobaSi dasaxelebuli

produqtebis gayidviT?
* romel sakveb produqtSi aiRo bufetma danarCenebTan SedarebiT

meti Tanxa?

16 saskolo inventaris SesaZenad ganaTlebis saministrom wels 2 587 800
lari gamoyo. momdevno saswavlo wlisTvis gaTvaliswinebulia am
Tanxis 550 600 lariT gazrda.
a) ramdeni lari iqneba momaval wels gamoyofili saskolo inventaris
SesaZenad?
b) ramdeni lari iqneba gankuTvnili saskolo inventaris SesaZenad
orive wlis ganmavlobaSi?

17 miwis erT nakveTze 3 500 Ziri vazi darges, meoreze 655 ZiriT meti,
vidre _ pirvelze, xolo mesameze _ 4 260 Ziri. ramdeni Ziri vazi
darges samive nakveTze?

18 1724 589 marcxena svetSi mocemuli ricxvebidan
1705 788 SearCie erTi Sesakrebi, marjvena sveti-
1748 579 dan meore da Seadgine, rac SeiZleba 20148457
1245 789 27142357
a) didi jami. 20448357
b) mcire jami. 24145357

1425 789 20148357

19 ra cifrebi unda eweros `* ~ - ebis nacvlad, toloba rom swori iyos?

*84* 1*5 5*17
+ 2**3 + **17 + *4*8

6529 581* 681*
*0846

20 saxlidan skolaSi misvlas Zma 30 wT-s andomebs, da _ 40 wT-s. da
saxlidan Zmaze 10 wT-iT adre gavida. qvemoT CamoTvlilTagan rome-
lia SeuZlebeli mocemuli pirobebis mixedviT?
a) skolaSi ar daugviania arc das, arc Zmas.
b) skolaSi Zmam daagviana, dam _ ara.
g) skolaSi orivem daagviana.

64

$ 3.2 mravalniSna ricxvebis gamokleba

mizani:

amocana 1. ninos xelfasi 800 lars Sead- a) viswavloT, rogor xdeba
gens. man 86 lari gadasaxadebSi gadaixada. milionze didi naturaluri
ramdeni lari darCa ninos xelfasidan? ricxvebis gamokleba.
amoxsna. imis gamosaTvlelad, Tu ramde-
ni lari darCa ninos, unda SevasruloT b) ganvixiloT gamoklebis
gamokleba: 800 lari _ 86 lari. anu unda kerZo SemTxvevebi.
gamoviTvaloT 800 _ 86 gamosaxulebis
mniSvneloba: g) viswavloT jamidan ricxvis
gamokleba da ricxvidan
800 _ 86 =714 jamisa da sxvaobis gamokleba.

d) SevZloT gamoklebis

saklebi maklebi sxvaoba Tvisebebis gamoyeneba
pasuxi: ninos xelfasidan 714 lari darCa. gamoTvlebSi.

amocanis amosaxsnelad SevadgineT gamo-

saxuleba gamoklebis moqmedebis gamoyenebiT, sadac 800 - s akldeba 86.

ras niSnavs 800 - s gamovakloT 86? 800 - s gamovakloT 86 niSnavs,

vipovoT iseTi ricxvi, romelic 86 - Tan jamSi gvaZlevs 800 - s. aseTi

ricxvi aris 714. moqmedebas, romliTac sxvaobis gamoTvla xdeba,

gamokleba ewodeba. Tu saklebs a asoTi aRvniSnavT, xolo maklebs _

b asoTi, maSin maTi sxvaoba iqneba:

a _ b.

Tu saklebi da maklebi tolia, maSin maTi sxvaoba 0 - is tolia.
a _a =0

ricxvi ar Seicvleba, Tu am ricxvs nuls gamovaklebT.
a _0 =a

davubrundeT ganxilul amocana 1 - is amosaxsnelad gamoyenebul
tolobas:

800 _ 86 = 714
Tu sxvaobas (Tanxas, romelic ninos darCa _ 714) mivumatebT
maklebs (Tanxas, romelic ninom gadaixada _ 86), miviRebT saklebs
(ninos xelfass _ 800), anu

714 + 86 = 800
asea yovelTvis da ara mxolod am amocanaSi.

nebismieri a , b da c ricxvebisaTvis, Tu a _ b = c, maSin a = b + c

ucnobi saklebi rom vipovoT, saWiroa sxvaobas mivumatoT maklebi.

magaliTad, tolobidan
a _ 2 458 963 = 3 147 852
a = 3 147 852 + 2 458 963
a = 5 606 815

aseve SegviZlia CamovayaliboT ucnobi maklebis povnis wesi:

ucnobi maklebi rom vipovoT, saWiroa saklebs gamovakloT sxvaoba.

asoebis gamoyenebiT es ase SegviZlia CavweroT:

nebismieri a , b da c ricxvebisaTvis, Tu a _ b = c, maSin b = a _ c.

65

magaliTad, 1000258 _ b = 847

b =1000258 _ 847

b =999411

mravalniSna ricxvebis gamokleba ise, rogorc Sekreba, Tanrigob-

rivad xdeba. magaliTad,

791005 32
_759097 73

31907 59

ramdenad sworad SevasruleT gamokleba, amis Semowmeba SegviZlia
ornairad:
1. SekrebiT

7 59 097 7 3
+ 3 1 907 59

7 9 1 00 5 32

2. gamoklebiT

7 9 1 00 5 32
_ 3 1 907 59

7 59 097 7 3

rogorc wina paragrafidan viciT, Tu gamosaxuleba mxolod

Sekrebis moqmedebebs Seicavs, moqmedebaTa Tanmimdevroba nebismi-

erad SegviZlia SevcvaloT. moqmedebaTa Tanmimdevrobis Secvla

gamoklebis Semcvel gamosaxulebaSic SeiZleba, magram am SemTxveva-

Si garkveuli wesebis dacvaa saWiro. magaliTad,

(7_3)_2 =7_(3_2) toloba arasworia, radgan 2= 6,

xolo (7_3)_2 =7_ (3+2) toloba sworia, radgan 2 =2.

qvemoT mocemuli tolobebi nebismieri a, b da c ricxvebisTvisaa

marTebuli (Seamowme konkretuli ricxvebis CasmiT):

(a +b)_c =(a _c )+b =a +(b _c );

a _(b +c )=(a _b )_c =(a _c )_b ;

a _(b _c )=(a _b )+c =(a +c )_b .

es tolobebi SegviZlia gamoTvlebis gasamartiveblad gamoviyenoT.

magaliTi 1. gamovTvaloT: a) (107895+32198)_7895;

amoxsna. b) 52931_(52931_73508).

a) (107895+32198) _7895=(107895_7895) +32198 =100000 +32198 =132198;

b) 52931_(52931_73508)=(52931_52931) +73508 =0 +73508 =73508.

magaliTi 2. vipovoT a _(b +7), Tu a _b =13.

amoxsna. a _(b +7) =(a _b ) _7=13_7 =6.

K upasuxe kiTxvebs:

1 romelia sxvaoba, maklebi da saklebi tolobaSi: a _ b = c?

2 gaizrdeba Tu Semcirdeba sxvaoba, Tu saklebs gavzrdiT? SevamcirebT?

3 gaizrdeba Tu Semcirdeba sxvaoba, Tu maklebs gavzrdiT? SevamcirebT?
4 ras udris maklebi, Tu saklebi sxvaobis tolia?

66

savarjiSoebi

1 daasaxele
• saklebi tolobaSi: a) 900 _ 6 = 894; b) 15 _ 12 = 3.
• maklebi tolobaSi: a) 200 _ 9 = 191; b) 7 _ 2 = 5.

2 ipove sxvaoba: a) x _ x; b) 2a _ a ; g) 10c _ 8c.

3 Seasrule gamokleba da Sedegi Seamowme SekrebiT.
1 578 963 _ 1 478 952, 4 000 000 _ 2 589 574, 5 200 654 _ 4 658 231.

4 Seasrule gamokleba da Sedegi Seamowme gamoklebiT.
23 579 841 _ 1 5785 698, 8 458 700 _ 725 896, 40 005 607 _ 23 040 771.

5 ramdeniTaa 2 000 000 meti
a) 2 458 - ze? b) 100 349 - ze? g) 1 999 887 - ze?
d) udides xuTniSna ricxvze? e) udides luw eqvsniSna ricxvze?

6 ramdeniT naklebia 1 248 357 a) 3 000 000 - ze? b) 2 671 585 - ze?

7 ipove x - is mniSvneloba cxrilis gamoyenebiT:

saklebi maklebi sxvaoba
14 256 357 7 444 193 x

1 840 597 x 425 654
x
980 000 8 000 000

8 ipove ucnobi komponentebi cxrilis gamoyenebiT:

I Sesakrebi 607 a 3331 12627 z c
II Sesakrebi x 2025
3000 y b 115600 823465
jami 1204
7623 35328 300726 999999

9 Seamcire:
a) 475 789 - isa da 598 148 - is jami 15 247 - iT;
b) umciresi rvaniSna ricxvi erTi milioniT;
g) udidesi SvidniSna ricxvi erTi milioniT.

10 gamoiyene toloba: 2 376 807 _ 1 304 009 = 1 072 798 da gamoTvale:
a) 2 376 807 _ 1 072 798; b) 1 304 009 + 1 072 798.

11 amoxseni gantoleba: 4 789 + x = 10 091 665 _ 4 758
a + 40 542 = 3 299 487

526 813 = 1 554 008 _ b y _ 3 657 121 = 2 565 457 + 4 349 356

d + 14 705 542 = 72 222 159 c + 4 012 542 = 12 587 657 _ 3 255 451

x_ 247 806 = 937 421 3 009 402 _ a = 432 745 + 781 606

12 ra ricxvs unda mivumatoT 7, rom:
a) umciresi xuTniSna ricxvi miviRoT?
b) udidesi xuTniSna luwi ricxvi miviRoT?

67

13 gamoTvale gamosaxulebis mniSvneloba moqmedebaTa jer mocemuli,
xolo Semdeg xelsayreli TanmimdevrobiT:

a) (49 +37) _ 49; b) 127_ (27 + 40); g) 347_ (247_ 91).

14 gamoTvale moqmedebaTa xelsayreli Tanmimdevrobis SerCeviT:

a) 13749+28017_ 8017; b) 39837+163_27500;

g) 25014 _(22014 +150); d) 17145_ (145 _57).

15 kalaTaSi n cali msxali da a cali vaSlia, maTgan b wiTeli vaSlia.

kalaTidan amoiRes yvela wiTeli vaSli. dawere gamosaxuleba, ro-
melic giCvenebs kalaTaSi darCenili xilis raodenobas.

16 bankis molarem erT dRes gasca 754 896 lari, meore dRes _ 800 690
lari, xolo mesame dRes _ 47 580 lariT naklebi, vidre _ meore
dRes. ramdeni lari gasca molarem samive dRes?

17 erT beRelSi 1 356 000 kg xorbalia, meoreSi _ 647 000kg-iT naklebi,
vidre _ pirvelSi, xolo mesameSi _ 860 000 kg-iT meti, vidre _ meo-
reSi. ramdeni kilogrami xorbalia samive beRelSi?

18 sam qalaqSi 6580500 mcxovrebia. erT qalaqSi 2125700 mcxovrebia, meo-
reSi _ 802500-iT meti, vidre _ pirvelSi. ramdeni mcxovrebia mesame qalaqSi?

19 aRadgine gamotovebuli cifrebi.

a) _8* * 6 * 5 b)_ * 4 * 8 * g) _3* 3 5 0* d) +9 * 9 4 *
4 * 7 * 5 * 9 * 6 * 9 *5 3 * * 8

28457 38574 18*12 *00275

20 fulis erT SekvraSi 784 lariT meti iyo, vidre _ meoreSi. pirveli
Sekvridan ramdenime xuTlariani, xolo meoredan imdenive orlari-
ani amoiRes. amis Semdeg pirvel SekvraSi 421 lariT meti darCa, vid-
re _ meoreSi. ramdeni lari amoiRes TiToeuli Sekvridan?

21 cnobilia, rom a +m = 30. ras udris
a) a + m + 4? g) a + (m _ 10)?
b) (a _ 6) + m? d) 45 _ (a + m)?

22 cnobilia, rom (m _ n) + c = 60. gamoTvale:
a) (m _ n) + (c + 10); b) m _ (n + 4) + c; g) (m _ 5) _ n + c.

23 samkuTxedis perimetri 66 sm - ia, xolo ori gverdis sigrZe _ 25 sm
7mm da 18 sm 8mm. gamoTvale samkuTxedis mesame gverdis sigrZe.

K SesaZlebelia, Tu ara
s.xvaoba iyos maklebis toli?

dadebiTi pasuxis SemTxvevaSi moitane saTanado magaliTi.

68

$ 3.3 mravalniSna ricxvebis gamravleba.
gamravlebis gadanacvlebadobisa da jufTebadobis Tvisebebi
mizani:

daakvirdi naxats, masze mocemulia a) CamovayaliboT da asoebis
ormoebi, romelTagan TiToeulSi ciram gamoyenebiT CavweroT gamrav -
badrijnis TiTo CiTili unda dargos. lebis gadanacvlebadobisa
ramdeni CiTili unda dargos ciram? da jufTebadobis Tvisebebi.

b) ganvixiloT gamravlebis

kerZo SemTxvevebi.

g) viswavloT gamravlebis

Tvisebebis gamoyeneba

gamoTvlebSi.

d) viswavloT, rogor xdeba

milionze didi naturaluri

ricxvebis gamravleba.

Sen ukve ici, rom tol SesakrebTa jami SegviZlia namravliT

SevcvaloT. ciras mier darguli badrijnis CiTilebis raodenoba

ornairad SeiZleba gamoviTvaloT: 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 + 6 = 6 × 9

an 9 + 9 + 9 + 9 + 9 + 9 = 9 × 6 gamosaxulebebiT. cxadia, TiToeuli am

gamosaxulebis mniSvneloba CiTilebis erTsa da imave raodenobas

Seesabameba. 9+9+9+9+9+9=9× 6

sadac Sesakrebi `9~ eqvsjer meordeba, 9 × 6 =54.

Tanamamravlebi namravli

6+6+6+6+6+6+6+6+6=6× 9
sadac Sesakrebi `6~ cxrajer meordeba, 6× 9= 54.

9×6=6×9

gamravlebis am Tvisebas gadanacvlebadobis Tviseba hqvia.

TanamamravlTa gadanacvlebiT namravli ar icvleba.

nebismieri a da b ricxvebisaTvis gadanacvlebadobis Tviseba ase

Caiwereba: a× b = b×a

Tu erT-erTi Tanamamravli 0 -is tolia, namravlic nulis tolia.

a × 0 = 0, 0 × a = 0 .

ricxvi rom erTze gavamravloT, miviRebT imave ricxvs.

a ×1=a, 1×a =a

Tu badrijnis CiTilebis nacvlad cira ormoebSi kartofils CaTesavs

ise, rom TiTo ormoSi 4 bolqvi Caflas, maSin kartofilis bolqvebis

saWiro raodenoba iqneba:

4× 9+4× 9+4× 9+4× 9+4× 9+4× 9 =(4× 9)× 6 = 216
an 4× 6+4× 6+4× 6+4× 6+4× 6+4× 6+4× 6+4× 6+4× 6 = (4× 6)× 9 = 216.

rogorc vxedavT, (4 × 9) × 6 = (4 × 6) × 9,

aseve (4 × 9) × 6 = 4 × (6 × 9).

maSasadame,

namravli rom ricxvze gavamravloT, sakmarisia erT - erTi Tana-
mamravli gavamravloT ricxvze da miRebuli namravli gavamravloT
meore Tanamamravlze.

69

am wess gamravlebis jufTebadobis Tviseba hqvia da asoiTi gamosa-
xulebis saxiT ase Caiwereba:

a × b × c = a × (b × c) an a × b × c = (a × b) × c an a × b × c = (a × c) × b

gamravlebis jufTebadobis Tviseba vrceldeba ara marto 3 Tana-
mamravlze, aramed TanamamravlTa nebismier raodenobaze. jufTeba-
dobis Tvisebis gamoyeneba gamoTvlebs gviadvilebs.

magaliTad, ganvixiloT 369 × 5 × 2 namravlis gamosaTvleli ori gza.
I. jer 369 × 5 = 1845 namravli gamoviTvaloT, xolo Semdeg 1845× 2=3690
II. jer 5 × 2 = 10 namravli gamoviTvaloT, xolo Semdeg 369 × 10.

amoxsnis romeli gza ufro iolia _ I Tu II?
Sen ici, 1000-is farglebSi rogor gavamravloT nulebiT dabo-

loebuli (e. w. mrgvali) ricxvebi. igive wesi vrceldeba aTasze did
ricxvebzec. magaliTad,

6000 × 300 = (6 × 1000) × 6000 da 300 warmovadgineT namravlis
× (3 × 100) = saxiT cifrebis poziciuri mniSvnelo-
= 6 × 1000 × 3 × 100 = bis mixedviT.
gamravlebis gadanacvlebadobis Tvise-
= 6 × 3 × 1000 × 100 = bis mixedviT.
gamravlebis jufTebadobis Tvisebis
= (6 × 3 × (1000 × 100) = mixedviT.

= 18 × 100000 = 1 800 000

maSasadame, nulebiT daboloebuli ricxvebis gasamravleblad nu-
lebs yuradReba ar unda mivaqcioT, unda gavamravloT maT win
mdgomi cifrebiT Sedgenili ricxvebi da miRebul namravls marjvniv
mivuweroT imdeni nuli, ramdeni nulicaa orive TanamamravlSi erTad.

magaliTad, 4000000 × 120 = (4× 12)× 1000000 = 480000000;
2500000 × 40 = (25× 4)× 1000000 = 100000000.

gamravlebis Tvisebebs viyenebT asoiTi gamosaxulebis gamartivebis
drosac.

magaliTi 1.

2 × n × 3 = (2 × 3) × n = 6 × n = 6n * asoiTi Tanamamravlebi erTad davajgu-
feT da ricxviTi Tanamamravlebi _
aseT SemTxvevaSi amboben erTad.

rom 2× n× 3 gamosaxuleba * simartivisaTvis 6× n CanawerSi gamov-
tovebT gamravlebis `× ” niSans da
gavamartiveT. 6× n - is nacvlad davwerT 6n.

2×n×3 =6n

magaliTi 2.

3 × x × 5 × y = (3 × 5) × (x × y) = 15 × x × y = * asoiTi Tanamamravlebi erTad
= 15 = 15xy
davajgufeT da ricxviTi
aseT SemTxvevaSi amboben, rom
Tanamamravlebi – erTad.
3 × x× 5 × y
* simartivisaTvis 15×x×y
gamosaxuleba gavamartiveT.
CanawerSi gamovtovebT
3 × x× 5 × y = 15xy
gamravlebis `× ” niSans

da davwerT 15xy.

70

ricxvebis gamravleba milionis farglebs gareT iseve sruldeba,
rogorc milionis farglebSi.
magaliTad,

4580 6 245 8 06 2 4 5 8060
× 12 × 1 20 × 1 02

9161 2 + 49161 2 + 4 9 1 612
+4 5 806 24 5806 245 8 06

54967 2 29 496720 2 5 0 7 221 20

K upasuxe kiTxvebs:

1 ra hqvia ricxvebs, romelTac vamravlebT?

2 ra hqvia gamravlebis Sedegad miRebul ricxvs?

3 ras udris ricxvis 1 - ze namravli?

4 ras udris ricxvis 0 - ze namravli?

5 rogor gavamravloT nulebiT daboloebuli ricxvebi?

6 Camoayalibe gamravlebis a) gadanacvlebadobis Tviseba;
b) jufTebadobis Tviseba.

7 sityvierad Camoayalibe a × b = a + a + . . . + a tolo-

bis Sinaarsi. b Sesakrebi

savarjiSoebi

1 Secvale ricxvebis Sekreba gamravlebiT:
a) 25 + 25 + 25 + 25 + 25 + 25 + 25
b) 1 000 000 + 1 000 000 + 1 000 000 + 1 000 000 + 1 000 000 + 1 000 000

2 gamoTvale: 908407 × 4 458000 × 17
46800 × 70 70070 × 2080
125 × 8 400708 × 9 235700 × 4500
425 × 32
8649 ×120

3 gamoTvale ioli gziT:

20 × (843 × 50) 8 ×16 ×125 40 ×512 ×25
80 ×(13 ×125)
5 × (329 × 50) 4 ×(111 ×25) 2500 ×(47 × 4)

89 ×4 × 25 2 ×307 ×50

4 gaamartive gamosaxuleba:
5 ×a ×3 × 2 5y × 3x 11 + 3 ×a × 5 17 _ 4 ×a ×2 × b
10 ×a × 4 × 5
3x × 2 5a × 2c × 3 5 ×12 × a × 3 × y 7m ×2 _ 4
2n × 3m × 4
3m × 2n × 3 3n ×5 + 8

5 gaadide 267
a) 3 - jer; b) 40 - jer; g) 500 - jer; d) 8 000 - jer.

6 gamoTvale gamosaxulebis mniSvneloba:
6004 × 308 4007 × 409 90005 ×1005 852 × 20005

4032 × 261 450 × 4 000 2008 × 3040 40300 × 7080

7 ra ricxvi unda eweros a - s nacvlad, toloba rom swori iyos?
a : 458 = 589 a : 205 = 1 800 a : 2 005 = 14 a : 349 = 42 579

71

8 gamoTvale 24 × x + 48 910 gamosaxulebis mniSvneloba, Tu x = 25; 107.
9
10 gamoTvale 32 × x _ 451 × 35 gamosaxulebis mniSvneloba, Tu x = 1200.
11
gamoTvale 320 000 _ 451 × x gamosaxulebis mniSvneloba, Tu x = 80;
12
13 Tanamamravlebi daamrgvale umaRles Tanrigamde da ise gamoTvale
namravli. Seamowme gansxvaveba TanamamravlTa damrgvalebiT miRebul
14 Sedegsa da zust namravls Soris.

15 798 × 8 132 99 421 ×187
16
17 Cawere mokled: 9 066 × 821 420 201 × 975
18
19 a) 5 × a; b) 104 × c; g) a × x; d) a × c × x; e) 5 × a × 5.

20 skolis samsarTuliani Senobis yovel sarTulze a saklaso oTaxia
21 da TiToeul saklaso oTaxSi _ n merxi. sul ramdeni merxia skolaSi?
22
Seadgine saWiro gamosaxuleba da gamoTvale misi mniSvneloba, roca
23 a = 14, n = 12.
24
Cawere gamosaxulebis saxiT:
72
a) a Seamcire 15 - iT; b) c gaadide 7- iT; g) y - s miumate a da c

ricxvebis jami; d) x - s gamoakeli a da c ricxvebis namravli;

puris sacxobidan maRaziebSi 800 - gramiani 500 puri waiRes. ramdeni
kilogrami puri waiRes sacxobidan maRaziebSi?

erTi puris gamocxobas 600 g fqvili sWirdeba. ramdeni kilogrami
fqvilia saWiro 700 puris gamosacxobad?

a ricxvi gaadides 4 - jer da miRebul Sedegs miumates 14. ra ricxvi

miiRes? dawere saZiebeli ricxvis Sesabamisi gamosaxuleba da gamoT-
vale misi mniSvneloba, roca a = 205; 1200; 200087.

Secvale ricxvebis Sekreba gamravlebiT:
a ) 12 +106 +1 2 +1 0 6 +1 2 +1 0 6 +1 2 +1 0 6 +1 2 +1 0 6 +1 2 +1 0 6 +1 2 +1 0 6 ;

b) a +a +a +a +a +a; g) 3 × a + 3 × a + 3 × a + 3 × a.

aRadgine gamotovebuli cifrebi: a) * 8 * b) * 8 *
× *8 × *8
4*** 4**0
*** ***
**5*0 ** 4* 0

Seadgine 1 an 2 - moqmedebiani amocana, romelSic mocemuli ricxvi
ramdenjerme unda gadiddes.

vaJam gamravlebis cxrili zepirad ar icis, magram kargad asrulebs

Sekrebasa da gamoklebas. rogor gamoTvalos vaJam 4 × 9 namravli?

maRaziam gayida 324 m CiTi, romlis erTi metri 6 lari Rirda da 156 m
abreSumi. erTi metri abreSumi erT metr CiTze 3 - jer Zviri Rirda.
ramdeni laris CiTi da abreSumi gayida maRaziam?

marTkuTxedis sigane 16 m 48 sm - ia, xolo sigrZe 2 - jer metia siga-
neze. gamoTvale marTkuTxedis perimetri.

daxaze marTkuTxedi, romlis sigane 2 sm 8 mm - ia, xolo sigrZe 3 - jer
metia siganeze. gamoTvale marTkuTxedis perimetri.

$ 3.4 ganrigebadobis Tviseba

mizani:

amocana 1. baRSi xeebi 8 rigadaa darguli. viswavloT gamravlebis
TiToeul rigSi 5 vaSli da 4 msxalia. sul ganrigebadobis Tviseba
ramdeni xea baRSi darguli? jamisa da sxvaobis
amocanis amoxsna SesaZlebelia ori xerxiT. mimarT da gamoviyenoT
I. cal - calke gamoviTvaloT Tu ramdeni gamosaxulebis gamarti-
vebisa da misi mniSvne-
vaSlis da ramdeni msxlis xea darguli lobis gamosaTvlelad.
baRSi da miRebuli Sedegebi SevkriboT:

5 × 8 = 40 (vaSlis)

4 × 8 = 32 (msxlis)

40 + 32 = 72 (vaSlisa da msxlis erTad).

II. jer gamoviTvaloT ramdeni xea TiToeul rigSi, Semdeg ki

gavarkvioT, sul ramdeni xea darguli baRSi:

5 + 4 = 9 (rigSi)

9 × 8 = 72 (baRSi).

orive xerxiT miviReT, rom baRSi 72 xea, anu

maSasadame, (5 + 4) × 8 = 5 × 8 + 4 × 8.

jami ricxvze rom gavamravloT, sakmarisia, TiToeuli Sesakrebi
gavamravloT am ricxvze da miRebuli Sedegebi SevkriboT.

am Tvisebas gamravlebis ganrigebadobis Tviseba hqvia. igi nebis-

mieri a, b da c ricxvebisaTvis ase Caiwereba:

(a + b) × c = a × c + b × c
amocana 2. TaTiam da nikam 10 -10 rveuli SeiZines. TaTiam TiTo rveul-

Si 60 TeTri gadaixada, nikam _ 45 TeTri. nikaze ramdeniT meti Tanxa

gadaixada TaTiam?

amocanis amoxsna SesaZlebelia ori xerxiT.

I. cal-calke gamoviTvaloT Tu ra Tanxa gadaixades navaWrSi TaTiam

da nikam da TaTias mier gadaxdil Tanxas gamovakloT nikas mier

gadaxdili Tanxa: 60 × 10 = 600 (TeTri TaTiam)

45 × 10 = 450 (TeTri nikam)

600 _ 450 = 150 (TeTri = 1 lari 50 TeTri).

II. jer gamoviTvaloT ramdeni TeTriT meti Rirs 1 rveuli, romelic
TaTiam SeiZina, Semdeg ki, gavarkvioT nikaze ramdeniT meti Tanxa

gadaixada TaTiam:
60 _ 45 = 15 (TeTriT meti 1 rveuli).

15 × 10 = 150 (TeTri = 1 lari 50 TeTri).

orive xerxiT miviReT, rom TaTiam nikaze 1 lariTa da 50 TeTriT
meti gadaixada.

(60 _ 45) × 10 = 60 × 10 _ 45 × 10

maSasadame,

sxvaoba ricxvze rom gavamravloT, sakmarisia, am ricxvisa da
saklebis namravls gamovakloT am ricxvisa da maklebis namravli.

es Tviseba nebismieri a, b da c ricxvebisaTvis ase Caiwereba:
(a _ b) × c = a × c _ b × c.

ganrigebadobis Tvisebis gamoyeneba amartivebs gamoTvlebs.

73

magaliTad,

12×10082 =12×(10000+80+2)=12×10000+12×80+12 ×2 =120000+960+24 =120984.

ganxilul magaliTebs Tu daakvirdebi, dainaxav, rom ganrigeba-
dobis Tvisebis gamoyenebis Sedegad miRebuli gamosaxulebebi ufrCx-
ileboa. anu frCxilebian gamosaxulebebSi frCxilebi gavxseniT.

maSasadame, gamonaTqvami `gavxsnaT gamosaxulebaSi frCxilebi~ _
mocemuli gamosaxulebis tol, ufrCxilebo gamosaxulebad gardaqm-
nas niSnavs.

magaliTi 1. gavxsnaT frCxilebi gamosaxulebaSi: a) (5 + b) × 3;
b) 5× (2x + 4).

amoxsna. gamravlebis ganrigebadobis kanonis gamoyenebiT vwerT:

a) (5 + b) × 3 = 3 × 5 + 3 × b =15 + 3b
b) 5 × (2x + 4) = 5× 2x + 5× 4 =10x + 20
magaliTi 2. mocemuli namravli CavweroT jamis saxiT: 3 × (15 + a).

amoxsna. gamravlebis ganrigebadobis kanonis gamoyenebiT vwerT:

3 × (15 +a) = 3×15 + 3 × a = 45 + 3a.

K upasuxe kiTxvebs:

1 Camoayalibe gamravlebis ganrigebadobis kanoni

a) jamis mimarT; b) sxvaobis mimarT.

2 raSi viyenebT gamravlebis kanonebs?

3 ras niSnavs gamonaTqvami `frCxilebis gaxsna~?

4 rogor gavamravloT mravalniSna ricxvi erTniSna ricxvze
gamravlebis ganrigebadobis kanonis gamoyenebiT?

5 qvemoT CamoTvlilTagan romeli tolobaa mcdari?
a) a(b +c) =ab +ac b) a(b _c) =ac _ab

g) a(b +c) =ac +ab d) a(b _c) =ab _ac

6 qvemoT CamoTvlilTagan romeli tolobaa marTebuli?
a) 3(5 +x) =15x + 3 b) 2(7 _ 4a) = 10a
g) 8(x _4) =8x _ 4 d) 5(6 + a) = 5a + 30

savarjiSoebi

1 Cawere namravlis saxiT:
2
3 a) 8+8+8+8+8+8; b) 3+3+3+3+3+3+3+3+3+3+3+3+3+3+3;
74 e) a+a+a+a+a.
g) 12+12+12; d) 3+3+3+ . . . +3;

n

romeli tolobaa swori da romeli araswori? gaaswore Secdoma.

6 × (4 +a) =24 +6b 25 × (n +4) =25n +4n
23 × (a _40) =23a _23
42 × (105 +258) =42 ×105 +42× 258
15 ×(57 +x) =15 × 57 +x 5 × (12 _x) =5 ×12 +5 ×x =60 +5x

axseni rogoraa gamravleba Sesrulebuli:

a) 325× 5=(300+20+5)× 5=300× 5+20× 5+5× 5=1500+100+25=1625;
b) 994× 12=(1000_6)×12=1000×12_6×12 =12000 _72 =11928.

4 cotne da luka saxlidan erTi mimarTulebiT erTdroulad gavidnen.
cotne yovel wuTSi 85 m-s, xolo luka _ 95m-s gadioda. ra manZili

iqneba maT Soris naxevari saaTis Semdeg?

5 sandro skolidan da saba saxlidan erTmaneTis Sesaxvedrad erTdrou-
lad gamovidnen. sandro 1 wT-Si 80 m-s, xolo saba – 90 m-s gadioda.
isini erTmaneTs 25 wT-Si Sexvdnen. ra manZilia skolasa da sabas saxls
Soris?

6 gamoTvale gamosaxulebis mniSvneloba:
5× (23+40) (28+24) × 5 120 × (80 +9)

(18 +8) × 25 25 × (44 +72) 5 × (22000 +120)

7 gamoTvale ganrigebadobis Tvisebis gamoyenebiT:
79 × 6 254 × 4 125 × 8

58 × 9 2025 × 2 1025 × 4

4040 × 25 252 × 4 5020 × 5

8 gaamartive gamosaxuleba: (4a +c ) × 3
15 × 4a 3x × 7 (12 +4a) × 2
5 × (x +12) 2a × 5c × 2 5m × 6n

2 × (13 _n) 2x × 3y × 7

9 gaxseni frCxilebi gamosaxulebaSi:
5 × (x + 4) (8 + y) × 5 a × (8 +9)
(1+m) × 3 n × (22+12)
4 × (a +5) 2 × (4 +b) (3 + m) × 4
5 × (a +x)

10 marim 4 rveuli da 4 saweri kalami SeiZina. TiTo rveuli 75 TeTri
Rirda, TiTo kalami _ 2 lari 50 TeTri. ra Tanxa gadaixada marim
navaWrSi?

11 ra unda eweros `*~-is nacvlad, toloba rom swori iyos?

a) (* _ *) × 11 =99 _77a ; b) (6 + *) ×5 = * + 5a ;
g) (* + *) + 30 =450a +30; d) (a _*) ×12 = *a _84.

12 gamravlebis romeli Tvisebaa gamoyenebuli gamoTvlebSi?
a) 10 × 235 =235 × 10 =2350;
b) 612× 8 =(600 +12) ×8 =600 × 8 +12 × 8 =4800 +96 =4896;
g) 9 ×899 =9 × (900 _1) =8100 _9 =8091;
d) 25 × 312 × 4 =312 × (25× 4) =312 × 100 =31200.

13 TaTiam 5 funTuSa da amdenive ,,ekleri” iyida. erTi funTuSa 45
TeTri, xolo ,,ekleri” – n TeTri Rirda. dawere TaTias mier
navaWrSi gadaxdili Tanxis gamosaTvleli gamosaxuleba da
gamoTvale misi mniSvneloba, roca n = 60.

14 gamoTvale ioli gziT: 4 × 360 × 5
50 × 325 × 2
25 × 135 × 4 250 × 12 × 4
2 × 320 × 5
125 × 8 × 7

75

15 ipove da gaaswore Secdoma:

6 × (72 + 38) = 6 × 72 + 6 × 38 = 600
125 × (120 + 120) = 125 ×120 + 120 = 15120
20 × (1200 + 800) = 1200 × 20 _ 20 × 800 = 40000
30 × (81 _ 79) = 81 × 30 _ 109 = 2321

16 romeli niSani unda eweros ujris nacvlad: `< ~, `> ~ Tu `= ~?
Seadare gamosaxulebaTa mniSvnelobis gamoTvlis gareSe.

12 × (45 + 75) 12× 45 +12 × 75 25×(862 _ 842) 862 × 25 _25 × 842
145 × (78 _ 50) 144× 78 _ 145 × 50 17×14 +16×17 17 ×14 +15×17
11× 18 +11× 23 12 × 18 +11× 23 245× (35 _18) 245 × 35 _245 × 18

17 qarsacav zolSi xeebi sam rigadaa darguli. TiToeul rigSi 154 Wer-
mis da 46 alublis xea. sul ramdeni xea darguli qarsacav zolSi?

18 gaamartive gamosaxuleba da gamoTvale misi mniSvneloba:
a) 15 × a × 2, Tu a = 20;

b) (5 +a) × 3, Tu a = 10;

19 sawyobSi 120 t xorbali da 100 t mzesumzira inaxeboda. yoveldRiu-
rad sawyobidan 5 tona xorbali da 3 t 500 kg mzesumzira gahqondaT.
ramdeni tona marcvleuli iqneba darCenili sawyobSi 12 dRis Semdeg?

20 Tu erTi fanqari n TeTri Rirs, xolo erTi saSleli _ m TeTri,
maSin ra eRireba 10 fanqari da 10 saSleli erTad?

21 a - s ra mniSvnelobisTvisaa mocemuli toloba swori?

a) 7 × a =7; b) 12 ×a = 0; g) 0 × a = 6;
d) a ×12 =12; e) a ×1= 0; v) a × a =a;

22 moifiqre amocana, romlis amosaxsnelad gamoiyeneb gamosaxulebas:

4 ×(10 + 8) an 4 ×10 + 4 ×8

23 16 dRis ganmavlobaSi mSeneblobaze 2 brigada mu-Saobda _ erTi 18 -ka-
ciani, meore ki 9 -kaciani SemadgenlobiT. TiToeul muSas dReSi 35
lars uxdidnen. ramdeni lari aiRo yvela muSam erTad?

24 Tinikos da anas toli raodenobis Tanxa hqondaT. Tinikom Tanxis na-
xevriT 40 - TeTriani rveulebi SeiZina, xolo danarCeni TanxiT erTi
albomi. anam mTeli Tanxa 80 - TeTrian rveulebSi gadaixada. romelma
meti nivTi SeiZina, Tinikom Tu anam?

25 fizkulturis gakveTilze, rodesac maris klaselebi simaRlis mixed-
viT erT mwkrivad moewyvnen, mwkrivSi marize dabali aRmoCnda 12 mos -
wavle. kalaTburTis wreSi Caewera mwkrivSi mdgari 9 yvelaze maRali
moswavle, romelTagan 4 marize dabali iyo. ramdeni moswavlea maris
klasSi?

76

$ 3.5 gamosaxulebis gardaqmna.
saerTo mamravlis frCxilebs gareT gatana

(a +b ) × c =a × c + b × c mizani:

toloba CavweroT aseTi saxiT: viswavloT gamosaxulebis
gamartiveba saerTo mam-
a × c + b × c = (a + b ) × c. ravlis frCxilebs gareT
gatanis gziT.
visargebloT miRebuli tolobiT da
gardavqmnaT gamosaxuleba:

3 × 14 + 3 × 16.
3×14+3×16 =3× (14 +16) =3 × 30 = 90

aseT SemTxvevaSi amboben, rom saerTo mamravli gavitaneT frCxilebs
gareT. saerTo mamravlis frCxilebs gareT gatanas gamoTvlebis gasa -
martiveblad viyenebT.

magaliTi 1. gamoviTvaloT 17×19 _34× 9 gamosaxulebis mniSvneloba.

amoxsna.

17 × 19 _ 34 × 9 =17 × 19 _17 × 2 × 9 =17 × (19 _2 × 9) =17 × (19 _18) =17 × 1 =17

magaliTi 2. saerTo mamravli gavitanoT frCxilebs gareT:
a) 36 + 4a ; b) 7a + 5a ; g) 2m + 9mn; d) 6n + 15m.

amoxsna. a) 36 + 4a gamosaxulebaSi saerToa ricxviTi mamravli 4.
gavitanoT 4 frCxilebs gareT:

36 + 4a = 4 × (9 +a );

b) 7a + 5a gamosaxulebaSi saerToa asoiTi mamravli a .
gavitanoT a frCxilebs gareT:

7a + 5a = a × (7 + 5) = 12a;

g) 2m + 9mn gamosaxulebaSi frCxilebs gareT unda gavitanoT saerTo

asoiTi mamravli m, miviRebT: 2m + 9mn = m × (2 +9n)

d) 6n + 15m gamosaxulebaSi saerTo ricxviTi mamravlia 3,

vinaidan 6 = 3 × 2 da 15 =3 × 5. gavitanoT 3 frCxilebs gareT,
miviRebT: 6n + 15m = 3 × (2m +5).

magaliTi 3. gavamartivoT gamosaxuleba: 12 x + 45 + 8x.
amoxsna.

12 x + 45 + 8 x = gamoviyenoT Sekrebis jufTebadobis Tviseba;
= (12x + 8 x ) + 45 = saerTo mamravli x gavitanoT frCxilebs gareT;

=(12+8)×x +45= SevasruloT Sekreba frCxilebSi;
= 20 × x + 45 =
20 × x CavweroT 20x saxiT.
= 20 x + 45

miviReT: 12x + 45 + 8 x = 20x + 45

K upasuxe kiTxvebs:

1 rogor gesmis gamonaTqvami: ,,gamosaxulebis gamartiveba~?
2 SeiZleba Tu ara saerTo mamravlis frCxilebs gareT gatana

sxvaobaSi?

77

savarjiSoebi

1 daasaxele gamosaxulebaSi saerTo mamravli:
2
a) 5x + 5; b) 20x + 10; g) 20x + 4; d) 20x + 45.
3
4 gaitane frCxilebs gareT saerTo mamravli da gamoTvale gamosaxulebis
5 mniSvneloba:

6 1685 × 20 _ 20 × 995 100 × 67 _ 67 × 99 81 × 16 + 19 ×16
7 5106 × 43 _ 5097 × 43 85 × 11 _ 11 × 77 47 × 34 + 13 × 34
8 17 × 1294 + 1506 × 17 25 × 76 _ 25 × 36 71 × 17 + 71 × 43
9
10 ostati 1 sT-Si 12 detals Carxavs, Segirdi ki _ 8-s. ramden detals
gamoCarxaven ostati da Segirdi erTad muSaobiT 7 sT-Si?
11
12 biZinam 65 -TeTriani 24 rveuli SeiZina, ris Semdegac 1 lari 30 TeTri
13 darCa. ra Tanxa hqonda biZinas Tavdapirvelad?
78
gamoTvale saerTo mamravlis frCxilebs gareT gataniT:

15 × 29 +15 16 × 34 + 8 × 232 5 × 89 + 5 × 41
16 × 20 _ 20 81 × 16 + 27 × 27 85 × 11 _ 17 × 54
9 × 42 _ 7 × 53 25 × 39 _ 50 × 15 18 × 1294 + 36 × 853

saeqskursio xarjebisaTvis ocdaerTma moswavlem 15-15 lari Sekriba.
aRmoCnda, rom 42 lari akldebodaT. sul ramdeni lari unda Seegrove-
binaT moswavleebs?

gogis 12 cali xuTlariani, 11 cali aTlariani da 4 cali oclariani
aqvs. sul ra Tanxa aqvs gogis?

mananas 9 cali aTTeTriani, 7 cali ocTeTriani da 3 cali ormocdaaT-
TeTriani aqvs. ra Tanxa aqvs mananas?

saerTo mamravli gaitane frCxilebs gareT:

25m + 25n 3a + 9x a + 4a
18n + 18m 5x + 2x
4y + 12x

gaamartive gamosaxuleba:

85c _ 4c 25b + 18b 15x + x _ 7x
46a + 8a 20y _ 14y 14m + 5m + 9m
64n _ 25n 64d _ 48d 200c _ 140c +16c

saerTo mamravli gaitane frCxilebs gareT:

12m +18n 14a +21d 16n +12m 18y + 9x

jer gaamartive gamosaxuleba, Semdeg gamoTvale misi mniSvneloba,
roca x = 45; 28; 103:

6x + 4x 46x + 84 x 14x + 5x + x 425x + 575x

gamoTvale ioli gziT:

35 × 63 _ 63 × 25 26 × 50 _ 6 × 50 5 × 34 _ 5 ×14 +90
40 × 54 _ 54 × 10 12 × 18 + 88 × 18 105 × 77 _ 105 × 27

14 nikos xelfasi m lars Seadgens. nikos meuRles ki, nikoze 80 lariT
naklebi xelfasi aqvs. ramden lars Seadgens nikosa da misi meuRlis
xelfasebi erTad?

15 samkuTxedis erTi gverdis sigrZea a sm, meorisa 1 sm-iT meti, vidre _
pirvelis da mesamisa 1 sm-iT meti, vidre meoris. dawere samkuTxedis

perimetris gamosaTvleli gamosaxuleba.

16 nino bebias n mamali da 6 -jer meti dedali qaTami hyavs. sul ramdeni
qaTami hyavs nino bebias?

17 gamoTvale: 3a + 12 + a + 27 _ 3a, Tu a) a = 6; b) a = 12.
18 giorgis 11 Savi da n - jer meti TeTri kurdReli hyavs. sul ramdeni

kurdReli hyavs giorgis?

19 erTmaneTis momdevno sami naturaluri ricxvidan umciresia n. dawere
am sami ricxvis jamis gamosaTvleli gamosaxuleba da gamoTvale misi
mniSvneloba, roca n = 5.

20 erTmaneTis momdevno sami naturaluri ricxvidan udidesia n. dawere
am sami ricxvis jamis gamosaTvleli gamosaxuleba da gamoTvale misi
mniSvneloba, roca n = 40.

21 samkuTxedis erTi gverdi meore gverdze 3 sm - iT, xolo mesame gverdze
4 sm - iT naklebia. amasTan, umciresi gverdi m- is tolia. dawere sam-
kuTxedis perimetris gamosaTvleli gamosaxuleba da gamoTvale misi
mniSvneloba, roca m = 12.

22 gaitane frCxi lebs gareT udidesi saerTo mamravli:
45a + 63; 1 6x + 48; 56 a _ 49x ; 80x _ 60y.

23 gamoTvale gamosaxulebis mniSvneloba:

a) 7a + 7x, Tu a +x = 20 g) 16a + 32x , Tu a + 2x =11
b) 2a _ 12x, Tu a _6x = 65 d) 10m _ 15n, Tu 2m _ 3n = 40

24 erT dRes maRaziaSi toli raodenobis 8 -lariani da 14 - lariani
CanTebi gaiyida, sul 132 laris.
a) aRniSne x - iT gayiduli erTi saxis CanTebis raodenoba. dawere
gayiduli CanTebiT Semosuli Tanxis Sesabamisi asoiTi gamosaxuleba
da gaamartive;
b) gamoiyene gamosaxuleba da gaarkvie, TiToeuli saxis ramdeni
CanTa gaiyida.
g) ra Tanxis 14 - lariani CanTebi gaiyida?

25 xuTlariani banknotebis sami Sekvraa. erT maTganSi 820 laria, meoreSi _
685 lari, xolo mesameSi _ 500 lari. ramdeni banknotia samive SekvraSi
erTad? amoxseni gamosaxulebis SedgeniT.

26 aTlariani banknotebis sami dastaa. erT maTganSi 1000 laria, meoreSi _
980 lari, xolo mesameSi _ 640 lari. ramdeni banknotia samive dastaSi

erTad? amoxseni gamosaxulebis SedgeniT.

79

$3.6 gayofa. gayofis kerZo SemTxvevebi

mizani:

a) ganvixiloT gayofis kerZo ricxvebis gayofa gamravlebis moqmedebis

SemTxvevebi da CavweroT gamoyenebiT ganimarteba. magaliTad, 24 gavyoT

asoebis gamoyenebiT. 6-ze _ niSnavs vipovoT iseTi ricxvi, romelic

b) gaviRrmaoT codna da 6-ze gamravlebisas mogvcems 24-s. aseTia ricx-

gaviwafoT gayofis ucnobi vi 4. amitom, 24 : 6 = 4

komponentebis povnaSi.

Tu gasayofs a asoTi aRvniSnavT, gamyofs -
b asoTi, xolo ganayofs c asoTi, maSin
a: b = c

gasayofi gamyofi ganayofi

ucnobi gasayofisa da ucnobi gamyofis gamosaTvleli wesebi asoe-

biT ase Caiwereba: a = b × c; b = a : c.

a : 0 gamosaxulebas azri ara aqvs. magaliTad, 16 : 0 niSnavs vipovoT

iseTi ricxvi, romelic 0 - ze gamravlebisas 16 - s mogvcems. aseTi ricx-

vi ar arsebobs, radgan nebismieri ricxvis 0 - ze namravli 0 -is tolia.

maSasadame,

ricxvis 0 -ze gayofa ar SeiZleba.

ricxvis 1 -ze da 0 -ze gamravlebis Tvisebebidan:
a × 1 = a da a × 0 = 0

gamomdinare vaskvniT:
I. ricxvis erTze gayofisas imave ricxvs miviRebT.

a :1 =a.
II. 0 -gan gansxvavebuli ricxvis Tavis Tavze gayofisas miviRebT 1 -s.

a :a =1
III. 0 - is gayofa 0 -gan gansxvavebul ricxvze gvaZlevs 0 -s.

0:a=0

0 -ebiT daboloebuli ricxvis (mrgvali ricxvis) 0 -ebiT daboloe-

bul ricxvze gayofisas:

a) gasayofs da gamyofs CamovaciloT imdeni 0, ramdeni 0 -iTac

bolovdeba gamyofi;

b) gavyoT miRebuli ricxvebi.

magaliTad, 72000000 : 9000 = 72000 : 9 = 8000.
8000 : 800 = 80 : 8 = 10,

K upasuxe kiTxvebs:

1 daasaxele 84 : 4 = 21 tolobaSi gasayofi, gamyofi da ganayofi.
2 ra Sedegs iZleva ricxvis Tavis Tavze gayofa?
3 ra ricxvze gayofaa SeuZlebeli?
4 rogor vipovoT ucnobi gasayofi? gamyofi?
5 ratom moviTxovT meore da mesame TvisebaSi, rom gamyofi nulisagan

gansxvavebulia?

80

savarjiSoebi

1 Seadgine ori toloba 3 200 : 400 = 8 tolobis gamoyenebiT.

2 Seasrule gayofa: 14550 : 2 24540 : 3 0 : 458
12450 :1 21510 : 9 148890 : 10 215000 : 8
156420 : 5

3 amoxseni gantoleba:

1248 : x = 3 a : 124 = 4001 a : 204 = 6000 12400 : x =124
20505 : a = 5 x : 35 = 3004 x : 320 = 250 240800 : a = 400

4 gamoTvale: (1500 × 31) : 5 (6000 _ 4200) : 60
(800 + 160) : 4 (15 × 40) : 5

5 gamoTvale:
(5600 + 8 125) : 25 (372 + 144) : 12 (8100 + 5350) : 50
(11000 + 84700) : 55 (4500 + 7128) : 36 (11000 + 8024) : 8
(4700 _ 1292) : 4 (5304 _ 1299) : 15 (180000 _ 16024) : 8

6 Seasrule gayofa:

2000 : 10 42000000 : 7 6000000 : 50000 400000 : 4000
6800 : 400 7500000 : 60 5600000 : 200000 800000 : 20000

7 gamoTvale sxvadasxva xerxiT:

(9 × 6) : 3 30 : (2 × 15) (5920 × 600) : 40 7920 : (2 × 11)
(9075 × 621) : 3 37800 : (60 × 9) (5225 × 8125) : 25 36000 : (2 × 60)

8 eqvsi ,,ekleri” imdeni Rirs, ramdenic oTxi `Su~. ra Rirs erTi `Su~, Tu
erTi `ekleri~ 90 TeTri Rirs?

9 kinoseanss erTi klasidan 5 -iT meti gogona daeswro, vidre _ vaJi.
gogonebma bileTebSi 120 lari gadaixades, vaJebma – 80 lari.

ra Rirda kinos 1 bileTi?

10 lelam da naTiam erTnairi rveulebi SeiZines. amasTan, lelam 4 rveu-
liT meti SeiZina, vidre _ naTiam. rveulebSi lelam 11 lari 20 TeTri
gadaixada, naTiam _ 8 lari.
a) ra Rirda erTi rveuli? b) ramdeni rveuli SeiZina orivem erTad?

11 ori muSa 8 sT -Si 120 detals amzadebs. erTi maTgani 1 sT -Si 7 detals
amzadebs. ramden detals amzadebs meore muSa 1 sT -Si? qvemoT CamoTv-

lilTagan romeli gamosaxuleba ar Seesabameba amocanis amoxsnas?

a) (120 _ 8 × 7) : 8; b) 120 : 8 _ 7; g) (120 _ 8 × 7) : 7.

12 gamoTvale x - is mniSvneloba:

x : 50 = 25000 6488 : x = 8 3645 : x = 3645 x : 11 =3300
x : 70 = 3550 1242 : x = 9 4764 : x = 1 98124 : x = 6

13 2 kg msxali imdeni Rirs, ramdenic _ 3 kg vaSli. gegam 3 kg msxalSi 9

lari gadaixada. ra Rirs 1 kg vaSli?
14 maswavlebelma 68 calxaziani da 51 ujriani rveuli klasis moswavleebs

Tanabrad gaunawila. ramden moswavles daurigda rveulebi? ramdeni

rveuli Sexvda TiToeul moswavles?

15 ipove naSTi: 54 : 4 = 13 (naSTi ?) 364 : 100 = 3 (naSTi ?)
191 : 8 = 23 (naSTi ?) 49 : 9 = 5 (naSTi ?)

81

$3.7 mravalniSna ricxvebis gayofa

mizani: amocana. qarxnis 384 muSas xelfasis saxiT erT
TveSi 941184 lari ergo. ramdeni lari ergo
viswavloT naturaluri TiToeul muSas saSualod?
ricxvis orniSna, samniSna amoxsna: amocanis amosaxsnelad unda Sevasru-
da mravalniSna ricxvze loT gayofa: 941184 : 384.
gayofa.

ganvixiloT, rogor xdeba mravalniSna ricxvis samniSna ricxvze gayofa.

_ 941184 : 384 = 2451 1) 941 -Si 384 Tavsdeba 2 -jer. 2 -s vwerT ganayofis
768 pirvel cifrad.

_ 1731 2) 384 × 2 = 768. 768 -s vwerT 941 -is qveS Tanrigobrivad.
1536 3) 941 - s gamovakloT 768. miRebuli naSTia 173. 173 -s

_11992508 marjvnidan mivuweroT 941-is momdevno cifri _ 1 .
miviRebT 1731 -s.
_ 384 4) 1731-Si 384 moTavsdeba 4-jer. 4 -s vwerT ganayofis
384 meore cifrad.
5) 384×4=1536. 1536-s Tanrigobrivad vwerT 173-is qveS.
0 6) 1731-s gamovakloT 1536. miRebuli naSTia 195. Camo-
vitanoT momdevno cifri _ 8 da 195 -s marjvnidan
Semowmeba: 2451 mivuweroT. miviRebT 1958 -s.
× 384 7) 1958-Si 384 moTavsdeba 5-jer. 5-s vwerT ganayofis
mesame cifrad.
9804 8) 384 × 5 = 1920. 1958 _ 1920 = 38
9) 38-s marjvnidan mivuweroT 8-ianis momdevno cifri
19608 - 4, miviRebT 384 -s.
10) 384 -Si 384 moTavsdeba 1-jer. 1-s vwerT ganayofis
7353 meoTxe cifrad.
maSasadame, 941184 : 384 = 2 451.
941184

pasuxi: 2451 lari.

magaliTi. SevasruloT gayofa 6432 : 67

amoxsna: 6432 : 67 = 96 1) 64-Si 67 ar Tavsdeba;
603 2) 643 -Si 67 Tavsdeba 9 -jer. 9 -s vwerT

402 ganayofis pirvel cifrad.
402 3) 67 × 9 = 603. 603 -s vwerT 643 -is qveS

0 Tan rigobrivad.
Semowmeba: 67 4) 643 _ 603 = 40. naSTia 40. 40 -s marjvnidan

× 96 mivuwerT 643 -is momdevno cifrs _ 2 -s.
402 miviRebT 402 -s.
603 5) 402 -Si 67 moTavsdeba 6 -jer. 6 -s vwerT
6432 ganayofis meore cifrad.
6) 67 × 6 = 402. 402 -s Tanrigobrivad vwerT
402 -is qveS.
7) 402 _ 402 = 0. miRebuli naSTi 0 -ia, rac
imas niSnavs, rom gayofa damTavrebulia.
maSasadame, 6 432 : 67 = 96.

82

rodesac viciT gasayofi da gamyofi, maSin winaswar SegviZlia ganvsaz-
RvroT ganayofSi cifrTa raodenoba (ganayofis niSnadoba). imis gas-
arkvevad, Tu ramdenniSna ricxvi unda iyos ganayofi, Semdegnairad unda
moviqceT:
1. gasayofSi marcxnidan marjvniv mZimiT movniSnoT imdeni cifri, ramdeni

cifricaa gamyofSi.
2. davakvirdeT, romelia didi, gasayofSi moniSnuli ricxvi Tu gamyofi?

a) Tu gamyofi didia, maSin 2 352 : 6 = 392
ganayofi imdeni cifrisagan 18
Sedgeba, ramdeni cifricaa
moniSnulis (mZimis) marjvniv 55
2 352 : 6 = ... ganayofi samniSna 54
ricxvia. 12
12
0

b) Tu gamyofi pataraa, maSin 2 352 : 12 = 196
ganayofis cifrTa raodenoba _1 2
moniSnulis marjvniv mdgom _110158
cifrTa raodenobaze 1-iT 72
metia. 2 352 : 12 = ... ganayofi _ 72
samniSna ricxvia.
0

K upasuxe kiTxvebs:

1 ras niSnavs erTi ricxvis meoreze gayofa?
2 rogor davadginoT ganayofis niSnadoba winaswar?
3 rogor SevamowmoT gayofis Sedegad miRebuli Sedegis siswore?
4 ra SegiZlia Tqva: a) 0-is ricxvze gayofaze?

b) ricxvis 0-ze gayofaze?

g) ricxvis 1-ze gayofaze?

d) ricxvis Tavis Tavze gayofaze?

savarjiSoebi

1 ramdenniSna ganayofi miiReba?
1024: 6; 3366 : 6; 2025 : 5; 5025 : 5; 32125 : 5;
5025 : 25; 250125 : 25; 144144 : 12; 144144 : 144; 1000000 : 125.

2 Seasrule gayofa: 12 096 : 8; 360 : 9;
9 984 : 6; 23 324 : 4.

3 1080 fanqari yuTebSi gaanawiles. TiToeul yuTSi 108 fanqari Cades.
ramdeni yuTi dasWirdaT sul?

4 laSam, nikam da demetrem 37 lari da 20 TeTri Tanabrad gainawiles.
ra Tanxa ergo TiToeuls?

5 ra droSi gaivlis avtomanqana 288km -s, Tu yovel saaTSi:
a) 72 km-s gaivlis? b) 48 km-s gaivlis?

83

6 Seasrule gayofa: 23324 : 119
9984 : 6 12096 : 8 360 : 9
2400 : 40 420 : 35 2300 : 25 7881 : 37
9855 : 365 19710 : 365 95936 : 16 95892 : 262

7 156 sm sigrZis mavTulisagan kubis karkasi daamzades. ra sigrZis
wibo aqvs kubs?

8 gamoTvale gamosaxulebis mniSvneloba:
9000 : 12 23142 : 19 46284 : 38 88160 : 145
12628 : 41 9963 : 9 19926 : 18 19926 : 27
28536 : 41 328032 : 402 247818 : 103 262626 : 13

9 Seadare gayofis moqmedebis Sesrulebis gareSe da Seni varaudi
Seamowme gayofiT:
a) 58448 : 26 da 120144 : 24; b) 285120 : 160 da 14256 : 16.

10 amoxseni gantoleba: 2024506 : a = 253
11948 : n = 29 m : 501 = 2070
352 × a = 12672 x : 3500 = 14017500
595560 : x = 709

11 Tedo qalaqidan sofelSi 2 sT -Si Cavida avtomobiliT, romelic
yovel saaTSi 70 km - s gadioda. ra droSi Cava Tedo soflidan

qalaqSi motocikliT, Tu yovel saaTSi 35 km -s gaivlis?

12 ostati erT saaTSi akeTebs 12 qanCs, Segirdi ki 8 -s. ramden saaTSi
gaakeTebs orive erTad 100 qanCs?

13 sakvebad dReSi Zroxas 18 kg, xolo cxens 12 kg Tiva esaWiroeba.
ramden dRes eyofa 240 kg Tiva or Zroxas da or cxens?

14 ipove iseTi x ricxvi, romlis 57 -ze gayofiT miiReba isev x ricxvi.

15 ra cifri unda CavweroT `* ~- is nacvlad 9 * 8 < 998 utolobaSi,
rom mcdari utoloba miviRoT?

16 ra cifri unda CavweroT `*~- is nacvlad 22 * 8 + 3 393 = 5 621 tolobaSi,
WeSmariti toloba rom miviRoT?

17 gamoTvale x - is mniSvneloba cxrilis monacemebis mixedviT.

I Sesakrebi II Sesakrebi jami
x
954 248 999855
x 458987 750048
803
458792 x

dawereT nebismieri samniSna ricxvi da marjvnidan miuwereT igive
ricxvi. miRebuli eqvsniSna ricxvi gayaviT 1 001 -ze. igive gaimeoreT
kidev or sxva samniSna ricxvze. ra daskvna SegiZliaT CamoayaliboT
aseTi saxis eqvsniSna ricxvebze?

84

§3.8 amocanebi oTxive ariTmetikul moqmedebaze

1 gamoTvale gamosaxulebis mniSvneloba mizani:
martivi gziT nimuSis mixedviT:
ganvimtkicoT codna
a) 215 + 10 596 + 285 + 404 = (215 + 285) + ariTmetikuli moqmede-
+(10596 + 404) = 500 +11000 =11500 bebisa da maTi Tvisebebis
Sesaxeb. gaviwafoT maT
b) 75 × 32 × 375 = (25 × 3) × (4 × 8) × (125 × 3) = efeqturad gamoyenebaSi.
= (25 × 4) × (125 × 8) × (3 × 3) = 100 × 1000 × 9 = 900000

g) 378 × 239 + 378 × 462 _ 378 = 378 × (239 + 462 _ 1) = 378 × 700 = 264600

268 + 183 + 302 + 217 + 130 4 × 8 × 3 × 25 × 125
365 + 73 + 25 + 235 + 2 25 × 13 × 8 × 4 × 250
25 + 285 + 430 + 215 + 270 + 75 27 × 9 + 27 × 2 _ 27

2 gamoTvale gamosaxulebis mniSvneloba ariTmetikuli moqmedebebis

Tvisebebis gamoyenebiT:

76 × 19 + 19 × 24 26 × 65 + 74 × 65

313 × 308 _ 313 × 298 113 × 84 + 187 × 84

140 × 915 + 140 × 85 300 × 444 + 556 × 300

537 × 489 _ 537 × 128 _ 537 × 272 24 × 47 + 46 × 47 + 30 × 47

29 × 110 _ 19 × 110 + 790 × 110 357 × 187 + 143 × 187 _ 400 × 187

507 × 227 + 456 × 227 + 37 × 227 1 041 × 24 + 1 041 × 29 + 147 × 1041

3 maRaziam 12 kg-iani 20 yuTi wiwibura gayida. 1kg wiwibura 1 lari da 75
TeTri Rirs. ra Tanxis wiwibura gayida maRaziam?

4 sworadaa miTiTebuli moqmedebaTa Sesrulebis rigi?
1 _2 1 _4 2402: 12 +3 19;
a) 127 15 20; b) 5 30 2: 1 3+ _1 2: 10 3 4_
+ ×
× ×
g) 1200 60 5 10; d) 700 300 5 27

5 gamoTvale ricxviTi Tanamamravlebis namravli da Sedegi asoiTi
Tanamamravlebis win dawere.

2 ×b×5 3×y× 8 6×4 ×b ×5
a × 4 × 50 6 × x × 20 × n 125 × 8 × m

6 Tornikem saxlidan mdinaremde manZilis gazomvis mizniT saxlidan gas-
vlisas jibeSi `nabijmzomi~ Caido, romelic fexiT mosiarulis mier
gadadgmuli nabijebis raodenobas aRricxavs. Tavdapirvelad `nabi-
jmzomi~ 0 -s aCvenebda. ra Cveneba iqneba `nabijmzomze~ 1 sT 40 wT - is
Semdeg, Tu Tornike yovel wuTSi 125 nabijs dgamda?

7 velosipeds borbalze mricxveli miamagres, romelic gavlil manZils
aRricxavs kilometrebSi. velosipedis gasvlis win mricxveli 0 -s
aCvenebda. ra Cveneba iqneba mricxvelze moZraobis dawyebidan 1 sT
20 wT -is Semdeg, Tu velosipedisti yovel wuTSi 250 m-s gadis?

8 gamoTvale gamosaxulebis mniSvneloba:
4 692 : (489 _ 466) × 30 (1 593 : 27 + 214) × 27
5 356 : (416 _ 390) + 458 6 732 : 12 × 43 _ 589
(1 536 _ 9 045 : 45) × 101
10 000 _ (546 + 54) × 14
74 089 _ (235 × 51 _ 68 890 : 83) 4 × 324 + 72 : 9 _ 6
22 792 : 22 + 81 720 : 90 + 1002 4 × (324 + 72) : 9 _ 6
79360 : 32 + 7000 : 28 + 5555 : 101 4 × (324 + 72 : 9) _ 6

85

9 manqanis Zaraze 25 yuTi moaTavses. TiToeul yuTSi TafliT savse 12
10 qilaa, TiToeul qilaSi _ 800 g Tafli. ramdeni kilogrami TafliT
11 datvirTes manqana?
12
13 natom klasisTvis Savi fanqrebis 6, xolo feradis 12 kolofi SeiZina.
14 Savi fanqrebi kolofebSi 12 - 12, xolo feradebi _ 24 - 24 cali ido.
sul ramdeni fanqari SeiZina natom?
15
matarebelma 11 sT -Si 924 km gaiara, TviTmfrinavma ki 5 sT -Si _ 4 200
16 km. ramdenjer aRemateba TviTmfrinavis mier 1 sT -Si gavlili manZili
matareblis mier 1 sT - Si gavlil manZils?
17
18 erTi eleqtrosadenis sigrZe meoreze 4 -jer naklebi, xolo mesameze _
19 12 sm -iT metia. ra sigrZisaa samive sadeni erTad, Tu pirvelis sigrZe
20 42 m -ia?
21
22 erTma mgzavrma manqaniT 135 km gaiara, fexiT 9 -jer naklebi, vidre
86 manqaniT da cxeniT 3 - jer meti, vidre fexiT. sul ramdeni kilometri
gaiara mgzavrma?

vagonebidan 5 manqana brinji da 7 manqana Saqari CamotvirTes. TiToeul
manqanaze brinji 65 tomara iyo, xolo Saqari _ 60 tomara.
a) ramdeni tomara brinji da Saqari CamotvirTes sul?
b) ramdeni kilogrami brinji da ramdeni kilogrami Saqari CamotvirTes

sul, Tu TiToeuli tomara brinji 40 kg -s, xolo Saqari 50 kg -s
iwonida?

gamoTvale x - is mniSvneloba: (8x _ 88) : 2 = 525 × 4
(x _ 18) × 36 = 25380 101 × x = 10605 : 105
20 × x × 8 = 200480 (a : 11) × 69 = 2346 : 34
21 × 12 × a = 2520

dawere x - is yvela naturaluri mniSvneloba, romlisTvisac

mocemuli utoloba WeSmaritia:
x <5 0<x <4 25 < x < 27
x <2 1<x <3 102 < x < 108
x < 11 15 < x < 19 30007 < x < 30010

ra ricxvze unda gavyoT 21 840, rom miviRoT 12 - is 70 - ze namravli?

ra ricxvi unda gavamravloT 309 -ze, rom miviRoT 59328 -is 48 -ze ga-
nayofi?

ori ricxvis namravli 4 -ianiT bolovdeba. pirveli Tanamamravli
orniSna ricxvia, romlis cifrTa jami 12 - is tolia. meore Tanamamrav-
lis bolo cifria 3. ipove pirveli Tanamamravli.

jemi qiliT 4 lari 50 TeTri Rirs. ra Rirs jemi qilis gareSe, Tu igi
qilaze 14 - jer Zviria?

gamosaxulebaSi 200 × 10 _100 : 4+16 Casvi frCxilebi ise, rom
a) Seicvalos moqmedebaTa rigi; b) ar Seicvalos moqmedebaTa rigi.

samkuTxedis Sedgenis ramdeni SesaZleblobaa monakveTebiT, romelTa
sigrZeebia: 2 sm, 3 sm, 10 sm, 11 sm?

$3.9 ricxvis xarisxi

mizani:

viciT, rom tol SesakrebTa jami SegviZlia 1. gavecnoT ricxvis xarisxs.
namravlis saxiT mokled CavweroT. 2. viswavloT 10-is xarisxebis
magaliTad,
gamoyeneba didi ricxvebis
3+3 +3 +3 +3 +3=3×6 Casawerad.
aseve SegviZlia mokled CavweroT toli Tana-

mamravlebis namravli.

magaliTad, 3 × 3 × 3 × 3 × 3 × 3 = 36.
36 gamosaxulebas xarisxi hqvia da ase ikiTxeba: `sami meeqvse xarisxSi”.

36 gamosaxulebaSi:

a) 3 aris Tanamamravli, romelic 6 -jer meordeba da mas xarisxis fuZe

hqvia.

b) 6 - s xarisxis maCvenebeli hqvia da gviCvenebs, Tu ramdenjeraa ricx-

vi 3 Tanamamravlad aRebuli. magaliTad, 24 gamosaxuleba gviCvenebs,

rom namravlSi Tanamamravli _ 2 oTxjer meordeba. e. i.

2 4 = 2 × 2 × 2 × 2.

ricxvis meore xarisxs sxvanairad ricxvis kvadrats uwodeben.
magaliTad, 72 gamosaxuleba warmoadgens 7 - is kvadrats.

ricxvis mesame xarisxs sxvanairad ricxvis kubs uwodeben.

magaliTad, 53 gamosaxuleba warmoadgens 5 - is kubs.
sazogadod, a ricxvis n xarisxSi axarisxeba niSnavs a ricxvis

Tavis Tavze n - jer gamravlebas.
an = a × a × ... × a

n-jer

rodesac frCxilebSi Cawerili gamosaxulebis axarisxeba xdeba, maSin

jer frCxilebSi moqceul moqmedebebs vasrulebT, Semdeg axarisxebas.

magaliTad, (10 + 3) 2= 132 = 13 × 13 = 169

rodesac ufrCxilebo gamosaxuleba xarisxs Seicavs, maSin gamosa-

xulebis mniSvnelobis gamosaTvlelad jer xarisxi unda gamoviTvaloT,

xolo Semdeg vagrZelebT gamoTvlas moqmedebaTa Sesrulebis rigis

mixedviT.

magaliTad,

a) 10 + 32 = 10 + 3 × 3 = 10 + 9 = 19;
b) 8 × 53 = 8 × (5 × 5 × 5) = 1000;
g) 400 : 22 = 400 : (2 × 2) = 400 : 4 = 100.

K upasuxe kiTxvebs:

1 rogor ikiTxeba Canaweri: 87? 42? 23?
2 ras gviCvenebs cifri 5 CanawerSi: 52? 25?
3 romeli moqmedeba Sesruldeba pirvelad a) 56 _ 52 b) (56 _ 5)2

gamosaxulebis mniSvnelobis gamoTvlisas?

savarjiSoebi

1 Cawere xarisxis saxiT:
a) 4 × 4 × 4 × 4 × 4 × 4 × 4 × 4; b) 11 × 11 × 11 × 11; g) 24 × 24 × 24 × 24 × 24.

2 Cawere namravlis saxiT: 26, 34, 73, 17, 102, 95.

87

3 waikiTxe gamosaxuleba: 22, 23, 52, 54, 55.
4
5 ramdenjeraa Tanamamravlad aRebuli ricxvi 6 gamosaxulebaSi:
6 d) 6n; e) 6m?
a) 62; b) 66; g) 611;
7
8 daasaxele xarisxis fuZe da xarisxis maCvenebeli:
9 g) a3.
10 a) 72; b) 53;

11 aRadgine cxrilSi monacemebi: xarisxi 32 74
12
13 xarisxis fuZe 3 4 10

14 xarisxis maCvenebeli 43 3

15 Cawere kvadratis an kubis saxiT namravli:
16
4 × 4 × 4; 9 × 9; 11 × 11; 100 × 100; 6 × 6 × 6; 1 × 1 × 1.
17
Cawere mokled gamosaxuleba: g) 10 + 10 + 10 + 10 + 10 + 10;
88 a) 8 + 8 + 8 + 8 + 8; d) 10 × 10 × 10 × 10 × 10 × 10.
b) 8 × 8 × 8 × 8 × 8;

gamoTvale: 23, 33, 72, 110, 102, 94.

gadaxaze rveulSi da Seavse ricxvTa kvadratebisa da kubebis cxrili
nimuSis mixedviT.

x 1 2 3 4 5 6789
x 2 81
x 3 64

Seadare gamosaxulebaTa mniSvnelobebi: g) 5 × 3 da 53;
a) 2 + 2 + 2 da 2 × 2 × 2; b) 4 × 2 da 42;

Cawere 10 - is xarisxis saxiT: 10; 100; 1 000; 10 000; 100 000; 1 000 000.

warmoadgine mocemuli ricxvi namravlis saxiT 10 - is xarisxebis gamo-
yenebiT nimuSis mixedviT: 840 000 = 84 × 104

2000, 3000000, 150000000, 1002000.

warmoadgine ricxvi jamis saxiT 10 - is xarisxebis gamoyenebiT
nimuSis mixedviT.

78 493 = 70 000 + 8 000 + 400 + 90 + 3 = 7 × 104+ 8 × 103+ 4 × 102+ 9 × 10 + 3
357489, 24582, 3208, 4002324

ra ricxvia Cawerili namravlis saxiT?
27 × 104, 5 × 102, 34 × 103, 7 × 109 , 111 × 105, 608 × 108.

ra ricxvia Cawerili jamis saxiT?

3 × 104+ 4 × 102+ 9 × 10 + 7, 1 × 105+ 5 × 104+ 8 × 103+ 9 × 102+ 6,
2 × 104+1 × 103+ 2 × 102+ 4 × 10 + 5, 5 × 105+ 8 × 104+ 2 × 103+ 1 × 102+ 7 × 10.

gamoTvale: 6 : 14 (5 _ 3)3 (2 × 3)2
8 + 32 10 _ 23 (17 _ 16)3
23 + 4 (2 + 3)3 7 × 42 42 _ 8
23 + 33
(1 + 4)2

18 romelia meti: 72 Tu 27? 32 Tu 3 × 2?
52 Tu 5 × 2? 23 Tu 2 × 3? 42 Tu 4 + 4?
23 Tu 2 × 2 × 2?

19 kinoTeatris darbazSi x rigia da TiToeul rigSi _ a skami. ramdeni
skamia kinoTeatris darbazSi? Seadgine gamosaxuleba da gamoTvale
misi mniSvneloba, roca x = 20; a = 30.

20 gamoTvale gamosaxulebis mniSvneloba:

6 : 14 (6 : 1)4 (10 _ 2)3 800 + (12 × 5)2
10 _ 23 (2 + 4)2 (19 _ 16)3 (800 + 12) × 52
42 _ 8 (5 _ 3)3 (2 + 3 + 5)4 20 000 + 502 _ 402

21 gamoTvale a2, Tu a = 15; 22; 100.

22 gamoTvale x3, Tu x = 4; 6; 10.

23 badeze patara kvadratebis raodenoba Cawere xarisxis saxiT:

a) b) g)

24 ipove x- is ricxviTi mniSvneloba:

23 _ x = 1 800 _ a = 112
34 _ x = 50 33 + x = 120
625 _ x = 53 8x = 72 + 7

25 ra ricxvis kvadratia: 4? 25? 16? 49? 900? 2 500?

26 ipove ricxvi, romlis kubia a) 1; b) 1 000; g) 27; d) 8; e) 64.

27 gamoTvale gamosaxulebis mniSvneloba:

240 _ 23 × 10 857 _ 8 × 32 6 058 + 70 × 43
23 × 104 _ 40 × 33 (2 + 3) × (2 + 3)3 (2 × 3 × 5)2 × (2 × 3 × 5)
(63 _ 22 × 4) + 32 (63 _ 22 ) × 4+ 32 63 _ 22 × (4+ 32 )

28 aRadgine tolobaSi gamotovebuli cifrebi:

(1*)2 = 1*4 (2*)2 = 4*1
(1*)2 = **5 (7*)2 = 6**1
(1*)2 = **0 (*3)2 = 1**

29 M da N wertilebi Sesabamisad AB da BC monakveTebis Suawertilebia.
MN = 16 sm. gamoTvale AC monakveTis sigrZe.

AM B NC

89

III Tavis damatebiTi savarjiSoebi

1 Seasrule moqmedebebi: 4 589 × 231 2 450 × 2 100
124 672 : 256 2 058 × 308 230 156 : 326
2 76 389 + 2 589 642 150 500 : 125 3 240 324 : 324
7 000 089 _ 49 985 100 466 : 263 14 515 023 _ 256 479
3 256 571 + 5 400 890
4
moqmedebis Sesrulebis gareSe, mxolod pasuxis niSnadobis mixedviT
5 daadgine, romeli moqmedebaa arasworad Sesrulebuli:

6 51 054 : 127 = 42 46 048 : 426 = 18
7 124 000 : 4 000 = 310 324 : 18 = 18
8
9 romelia araswori toloba? g) (ab)c =(ac) ×(bc);
10 a) (a +b)c =ac +bc; d) a _(b +c) =a _ b _c.
11 b) (a _ b)c =ac _bc;
12
13 amoxseni gantoleba: 3x + 9 = 5 700
14 5x = 2 005 7 580 : x = 1 895
15 2 304 : x = 48 12 597 : x = 4 199
16 2 499 = 3x + 1 263
90
romelia meti: 56 000 : 1120 Tu 170 000 : 34 000?
576 24 : 98 Tu 7 350 : 1225?
40 000 : 25 Tu 200 000 : 125?

gamoTvale gamosaxulebis mniSvneloba:

143 × 24 +24 ×143 16 × 34 + 32 ×33 98 × 15 +49 × 70

gamoTvale:

(1 + 2)2, (12 _ 10)3, (11 _ 7)2, (1 + 1)3, (4 + 5)2, (2 + 8)3.

romelia meti: 23 Tu 32? 103 Tu 210?
400 Tu 602? 300 Tu 182?

TiToeul saaxalwlo nobaTSi a vaSli da 5 mandarinia. ramdeni vaSli
da mandarini iqneba 21 aseT nobaTSi?

xelTaTmani n lari Rirs, CanTa qudze 3 -jer, xolo qudi xelTaTmanze
2 - jer Zviria. ra Rirs CanTa?

rogor Seicvleba jami, Tu erT Sesakrebs 12 -iT gavadidebT, xolo
meores 9 -iT SevamcirebT?

rogor Seicvleba jami, Tu erT Sesakrebs 20 -iT gavadidebT, xolo
meores 25 -iT SevamcirebT?

gasayofi gaadides 2 - jer, gamyofi Seamcires 3 - jer. rogor Seicvala
ganayofi?

Tu erTi fanqari 20 TeTri Rirs, xolo erTi saSleli 15 TeTri, ma-
Sin ra eRireba n fanqari da n saSleli erTad?

Tu n kenti ricxvia, maSin romeli ricxvi iqneba luwi?
a) n +2; b) 2n + 1; g) n _2; d) 3n +1.

qilaSi moaTavses baqteria, romelic yovel erT wamSi 2 baqteriad
iyofa. ramdeni baqteria iqneba qilaSi 10wm-is Semdeg?

Tavi 4. amocanebi moZraobaze. farTobi. $ 4.1 siCqare

mizani:

amocana 1. nika qalaqidan sofelSi avtobusiT gavecnoT siCqaris cnebas,
2 saaTSi Cavida. qalaqidan soflamde 140 km - ia. siCqaris damokidebulebas
avtobusi yovel saaTSi erTnair manZils gadioda. drosTan da manZilTan.
ramden kilometrs gadioda avtobusi erT saaTSi? viswavloT siCqaris
amoxsna. Tu avtobusma 2 sT-Si 140 km gaiara, gamoTvla.
maSin 1 sT-Si 2 -jer nakleb manZils gaivlida.

140 : 2 = 70
pasuxi: avtobusi erT saaTSi 70 km -s gadioda.

winadadeba: ,,avtobusi erT saaTSi 70 km -s gadioda~ sxvanairad ase
SegviZlia CamovayaliboT: `avtobusis moZraobis siCqare iyo 70 kilometri
saaTSi~. mokled es ase iwereba: 70 km/sT.

siCqares v asoTi aRniSnaven. Cvens SemTxvevaSi v = 70km/sT.
siCqaris sazomi erTeulebia: km/sT, m/wT, m/wm da a. S.
amocana 2. giorgi yovel saaTSi 60 km -s gadioda da daniSnulebis ad-
gilze 6 saaTSi mivida. ramdeni kilometri gauvlia giorgis?
amoxsna. giorgi Tu erT saaTSi 60 km -s gadioda, 6 saaTSi 6 -jer met
manZils gaivlida. e.i. giorgi 6 saaTSi 60 × 6 km -s, anu 360 km -s gaivlida.
pasuxi: giorgis 360 km gauvlia.
amocanaSi mocemuli iyo giorgis moZraobis siCqare _ 60km/sT da moZ-
raobaze daxarjuli dro _ 6 sT. maSasadame, sxeulis mier gavlili man-
Zili droisa da siCqaris namravliT gamoiTvleba.
daimaxsovre toloba:

= ×gavlili daxarjuli
dro
manZili
moZraobis
siCqare

Tu gavlil manZils s -iT, xolo daxarjul dros t - Ti aRvniSnavT, ma-
Sin es toloba ase Caiwereba:

s = v .t
amocana 3. ereklem 60 km/sT siCqariT 180 km Seusveneblad gaiara. ra dro
dasWirda erekles am manZilis gasavlelad?
amoxsna. s = v . t tolobidan t =s : v. maSasadame, erekles mier daxar-
juli drois gamosaTvlelad manZili unda gavyoT siCqareze.

t = 180 : 60 = 3(sT)
pasuxi: erekles 3 sT dasWirda.

K upasuxe kiTxvebs:

1 rogor gamoiTvleba moZraobis siCqare, rodesac viciT moZravi
sxeulis mier gavlili manZili da daxarjuli dro?

2 rogor gamoiTvleba moZravi sxeulis mier gavlili manZili,
rodesac viciT moZraobis siCqare da daxarjuli dro?

3 rogor gamoiTvleba moZraobaze daxarjuli dro, rodesac viciT
moZravi sxeulis siCqare da mis mier gavlili manZili?

91

savarjiSoebi

1 laSam avtomanqaniT 240km 3 sT -Si gaiara ise, rom yovel saaTSi erTi
da imave manZils gadioda. ra siCqariT moZraobda laSa?

2 velosipedistma 3 wuTSi 750 metri gaiara. ra siCqariT moZraobda ve-
losipedisti?

3 TaTiam 90 m 15 wamSi gairbina. ra siCqariT garboda TaTia?

4 gadaixaze da Seavse cxrili:

avtomanqanis mier avtomanqanis mier avtomanqanis moZraobis

gavlili manZili daxarjuli dro siCqare

300 km 5 sT

100 m 4 wm
2000 m 2 wT

Camoayalibe wesi, riTac gamoTvale avtomanqanis moZraobis siCqare.

5 xSirad mogismenia radioTi da televiziiT informacia qaris moZraobis
siCqaris Sesaxeb. magaliTad,
a) ,,mosalodnelia susti qari _ 5 m/wm siCqariT~;
b) ,,mosalodnelia Zlieri qari _ 13 m/wm siCqariT~;
g) ,,mosalodnelia grigali _ 35 m/wm siCqariT~.
ris mixedviT afaseben qars sityvebiT: ,,susti~, ,,Zlieri~ da sxva?

6 daasaxele siCqaris sazomi erTeulebi.

7 kus moZraobis siCqare daaxloebiT 5 m/wT -ia. ra manZils gaivlis ku 15
wuTSi? naxevar saaTSi? 1 saaTSi?

8 ra SemTxvevaSi ityvian `kus nabijiT midiso~?
9 gadaixaze da Seavse cxrili:

avtomanqanis mier avtomanqanis mier avtomanqanis moZraobis
gavlili manZili daxarjuli dro siCqare

5 sT 62 km/sT
4 wT 1500 m/wT
1 wm
12 m/wm

Camoayalibe wesi, riTac gamoTvale avtomanqanis mier gavlili manZili.

10 velosipedistma 12 km/sT siCqariT Seusveneblad gaiara 60 km. ra dro
dasWirda velosipedists am manZilis gasavlelad?

11 gadaixaze da Seavse cxrili:

manZili siCqare dro

150 km 50 km/sT

450 km 90 km/sT

300 m 15 m/wT

Camoayalibe wesi, riTac gamoTvale moZraobaze daxarjuli dro.

92

12 motociklisti 3 saaTis ganmavlobaSi 65 km/sT siCqariT moZraobda. ra
manZili gaiara motociklistma sul?

13 goderZim 60 metri 6 m/wm siCqariT gairbina. ra dro dasWirda goderZis
saamisod?

14 avtomobilma 3 sT -Si 180 km gaiara. ra manZils gaivlida avtomobili
imave siCqariT 45 wT -Si?

15 or qalaqs Soris manZili avtobusma 2 sT -Si gaiara. ra droSi gaiv-
lida avtobusi imave manZils Tu
a) 2 -jer meti siCqariT imoZravebda?
b) 3 -jer naklebi siCqariT imoZravebda?

16 gadaixaze da Seavse cxrili:

fasi 5 lari 9 lari 1 lari da 80
TeTri

raodenoba 7 11

Rirebuleba 36 lari 12 lari da 32 lari da
10 TeTri 40 TeTri

17 ricxvis 25 -ze zepirad gamravlebisaTvis sakmarisia am ricxvs mar-
jvnidan ori 0 miuwero da miRebuli ricxvi 4 -ze gayo. axseni wesis

marTebuloba da am wesiT gamoTvale:

24 × 25 840 × 25 8004 × 25
56 × 25 144 × 25 168400 × 25
120 × 25 8004 × 25 1280000 × 25

18 daasaxele naxazze mocemuli figura, misi gverdebi da kuTxeebi.

CD

BE

AF

TK

19 miliardebis klass mosdevs trilionebis klasi. waikiTxe ricxvebi:
2 738 492 790 122; 324 000 062 902 233.

20 winaswari gaTvlebiT mecnierebi varaudoben, rom msoflios mosaxleo-
ba 2050 wlisTvis 9 miliards miaRwevs. dawere es ricxvi cifrebis
gamoyenebiT.

21 speqtakli 9-is 15 wT-ze daiwyo da 10-is nax-
evarze damTavrda. ramden xans gagrZelda
speqtakli?

93

$4.2 moZraoba Semxvedri mimarTulebiT

mizani: amocana 1. A da B qalaqebidan erTmaneTis Sesax-
vedrad ori avtomobili gamovida. isini erTma-
amovxsnaT moZraobis amo- neTs moZraobis dawyebidan 3sT-is Semdeg Sexvdnen.
canebi, romlebSic ori A-dan gamosuli avtomobili 75 km/sT siCqariT
sxeuli moZraobs Semxvedri moZraobda, xolo B-dan gamosuli _ 80 km/sT siC-
mimarTulebiT.

qariT. ra manZilia A da B qalaqebs Soris?

amoxsna. amocanis amosaxsnelad gamoviyenoT sqematuri naxazi.

sT -Si 75km sT -Si 80km

AB
amocana amovxsnaT ori xerxiT:

I xerxi. A qalaqidan gamosuli avtomobili Sexvedris adgilamde gaiv-
lida 75 × 3 =225 (km). B qalaqidan gamosuli avtomobili Sexvedris adgi-
lamde gaivlida 80 × 3 =240 (km). maT mier gavlil manZilebs Tu SevkrebT,
miviRebT A qalaqidan B qalaqamde manZils: 225 +240 = 465 (km)

II xerxi. erTmaneTisadmi Semxvedri mimarTulebiT moZravi sxeule-
bi erTmaneTs uaxlovdebian siCqariT, romelic maTi siCqareebis jamis
tolia. radgan A qalaqidan gamosuli avtomobilis siCqarea 75 km/sT,
xolo B- dan gamosulis _ 80km/sT, amitom isini yovel saaTSi erTmaneTs
(75 + 80) km/sT siCqariT uaxlovdebian. qalaqebs Soris manZilia:

(75 + 80) × 3 = 155 × 3 = 465 (km).
pasuxi: A da B qalaqebs Soris manZili 465km - ia.
amocana 2. baTumidan Tbilisisaken dilis 8 sT - ze 55km/sT siCqariT gavida
satvirTo matarebeli, xolo imave dros Tbilisidan baTumisaken _ 70km/sT
siCqariT samgzavro matarebeli. baTumsa da Tbiliss Soris manZili
375km - ia. romel saaTze Sexvdnen matareblebi erTmaneTs?
amoxsna. matareblebi erTmaneTs (55 + 70)km/sT = 125km/sT siCqariT uax-
lovdebian. isini Sexvedramde daxarjavdnen (375 : 125)sT = 3sT. matareb-
lebi dilis 8 sT - ze gamovidnen da 3sT - is Semdeg Sexvdnen. Sexvedrisas
iqneboda 8sT + 3sT = 11sT.
pasuxi: matareblebi erTmaneTs 11 sT - ze Sexvdnen.
amocana 3. M qalaqidan N qalaqamde 855 km - ia. am qalaqebidan Semxvedri
mimarTulebiT erTdroulad ori matarebeli gamovida da erTmaneTs 9
saaTis Semdeg Sexvda. M qalaqidan gamosuli matareblis siCqare 48 km/sT
iyo. ra siCqariT moZraobda N qalaqidan gamosuli matarebeli?
amoxsna. amocana amovxsnaT sqematuri naxazis gamoyenebiT.

48km /sT 9 sT 9 sT xkm/sT
M N

855 km

unda gavigoT, ramden kilometrs gadioda erT saaTSi N qalaqidan gamo-
suli matarebeli. amisaTvis unda vicodeT dro da manZili, romelic man
Sexvedramde gaiara. dro cnobilia: 9 sT. manZili ucnobia, magram misi
gamoTvla SeiZleba. igi MN manZilis is nawilia, romelic M qalaqidan
gamosul matarebels darCa gasavleli Sexvedris adgilidan N qalaqamde.
mis gamosaTvlelad mTel manZils _ 855km-s unda gamovakloT M qalaqi-

94

dan gamosuli matareblis mier gavlili manZili. analizidan gamomdi-
nare, SegviZlia amocana amovxsnaT Semdegi gegmis mixedviT:
1) ra manZili gaiara Sexvedramde M qalaqidan gamosulma matareblma?

48 × 9 = 432 (km).
2) ra manZili gaiara Sexvedramde N qalaqidan gamosulma matarebelma?

855 _ 432 = 423(km)
3) ra siCqariT moZraobda N qalaqidan gamosuli matarebeli?

423 : 9 = 47(km/sT)
ganxiluli amocana sxva xerxiTac amovxsnaT:
1) ramden kilometrs gadioda orive matarebeli erTad erT saaTSi?

855 : 9 = 95 (km/sT)
2) ra siCqariT moZraobda N qalaqidan gamosuli matarebeli?

95 _ 48 = 47(km/sT)
pasuxi: N qalaqidan gamosuli matarebeli 47km/sT siCqariT moZraobda.

K upasuxe kiTxvebs:

1 ra saxis amocanebis amoxsna viswavleT am paragrafSi?
2 rogor gamovTvaloT ra siCqariT uaxlovdebian erTmaneTs

Semxvedri mimarTulebiT moZravi sxeulebi?

savarjiSoebi

1 gamoiyene sqemebi da gamoTvale ra siCqariT uaxlovdebian erTmaneTs
moZravi sxeulebi.

6 km/sT 4km/sT
a)

60km/sT 54km/sT
b)

2 punqtebidan, romelTa Soris manZili 27km-ia, ori fexiT mosiarule
erTmaneTis Sesaxvedrad erTdroulad gamovida. erTi maTgani 4km/sT,
xolo meore _ 5km/sT siCqariT moZraobda. ramdeni sT-is Semdeg Sexvd-
nen isini erTmaneTs?

3 A da B punqtebidan ori motociklisti erTmaneTis Sesaxvedrad erT-
droulad gamovida da erTmaneTs 4sT-is Semdeg Sexvda. erTi maTgani
26 km/sT, xolo meore – 29 km/sT siCqariT moZraobda. ra manZilia A
da B punqtebs Soris?

4 ori qalaqidan, romelTa Soris manZili 216km-ia, erTmaneTisSesaxvedrad
erTdroulad ori avtomanqana gamovida. avtomanqanebi erTnairi
siCqariT moZraobdnen. qalaqebidan ra manZilze Sexvdnen isini erTmaneTs?

5 T da M qalaqebidan, romelTa Soris manZili 318 km-ia, Semxvedri
mimarTulebiT erTdroulad ori matarebeli gamovida. T qalaqidan
gamosuli matarebeli saaTSi 52 km-s gadioda, xolo M-idan gamosuli
_ 54km -s. ramdeni saaTis Semdeg Sexvdebian matareblebi erTmaneTs?

95

a) Seasrule amocanis Sesabamisi sqematuri naxazi: gaavle wrfe da masze M
da T asoebiT moniSne punqtebi, saidanac matareblebma daiwyes moZrao-
ba. isrebiT aCvene TiToeuli qalaqidan gamosuli matareblis moZrao-
bis mimarTuleba. kargad gaiazre, romel qalaqTan ufro axlos iqneba
Sexvedris adgili: M- Tan, Tu T- sTan da aRniSne igi C asoTi.
b) ra manZils gaivlis Sexvedramde M-idan gamosuli matarebeli?
g) ra sigrZisaa gzis CT monakveTi?

6 erTmaneTTan Sesaxvedrad avtomobiliT mzia Tbilisidan, xolo san-
7 dro _ quTaisidan erTdroulad gamovidnen da 2 saaTSi Sexvdnen. san-
8 dro saaTSi 65km -s gadioda, xolo mzia 55km -s. ra manZilia Tbilisidan
9 quTaisamde?
10
gamoiyene sqemebi da gamoTvale Semxvedri mimarTulebiT erTdroulad
11 gamosuli satvirTo matareblebis mier Sexvedramde daxarjuli dro.
12
13 a) 56 km/sT 44 km/sT b) 43 km/sT 42 km/sT
14
96 400 km 255 km

Seadgine amocana mocemuli sqemis mixedviT da amoxseni.

A 60km/sT 90km/sTB

3sT 1sT 30wT

?km

A da B qalaqebidan Semxvedri mimarTulebiT ori matarebeli erT-
droulad gavida. isini erTmaneTs 3 sT - Si Sexvdnen. A- dan gamosulma
Sexvedramde 180 km gaiara, xolo B- dan gamosulma 120km. ra droSi Cava
A- dan gamosuli matarebeli B qalaqSi?

A qalaqidan B qalaqisaken, romelTa Soris manZili 315km-ia, dilis 8
sT-ze 65 km/sT siCqariT gaemgzavra avtobusi. dilis 9 sT-ze B qalaqi-
dan A- sken imave gziT 60 km/sT siCqariT gaemgzavra meore avtobusi.
a) romel saaTze Sexvdebian avtobusebi erTmaneTs?
b) B- dan meramdene kilometrze Sexvdebian avtobusebi erTmaneTs?
g) ramdeni kilometri eqneba gavlili Sexvedris momentSi A qalaqidan

gamosul avtobuss.

A qalaqidan B qalaqisaken, romelTa Soris manZili 400 km-ia, dilis 8
sT-ze 70 km/sT siCqariT gaemgzavra avtobusi. dilis 10 sT-ze B qalaqi-
dan A-ken imave gziT gaemgzavra meore avtobusi, romelic 60 km/sT
siCqariT moZraobda. romel saaTze Sexvdebian avtobusebi erTmaneTs?

Seavse saWiro cifrebiT: 14**08 _ 64853
+ 35** 1*1**
1524 19
*9*55

patara andro da misi da _ sofo erTmaneTisaken erTdroulad gaemarT-
nen da 5 wuTSi Sexvdnen erTmaneTs. Sexvedramde androm 100m gaiara,
xolo sofom _ 120m. ra siCqariT moZraobda TiToeuli maTgani?

turisti 5km/sT siCqariT Tu ivlis, turbazaSi 6sT-Si dabrundeba. ra
siCqariT unda iaros turistma, turbazaSi 5sT-Si rom dabrundes?

$ 4.3 moZraoba sawinaaRmdego mimarTulebiT

amocana 1. sadguridan erTmaneTis sawinaaRm- mizani:
dego mimarTulebiT erTdroulad ori mata-
rebeli gavida. erTi maTgani 50 km/sT siCqariT amovxsnaT moZraobis amo-
moZraobs, meore 40 km/sT siCqariT. ra manZili canebi, romlebSic ori sxe-
iqneba matareblebs Soris 4 sT-is Semdeg? uli moZraobs erTmaneTis
amoxsna. amocanis amosaxsnelad gamoviyenoT sawinaaRmdego mimarTulebiT.
sqematuri naxazi.

50 km/sT 40 km/sT
A

amocana amovxsnaT ori xerxiT. 4 sT

I xerxi. amoxsna CavataroT Semdegi gegmiT:

a) ramden kilometrs gaivlis pirveli matarebeli 4 saaTSi?

50 × 4 = 200 (km)

b) ramden kilometrs gaivlis meore matarebeli 4 saaTSi?

40 × 4 = 160 (km)

g) ra manZili iqneba matareblebs Soris moZraobis

dawyebidan 4 saaTis Semdeg?

200 + 160 = 360 (km).

II xerxi. amoxsna CavataroT Semdegi gegmiT:

a) ra siCqariT Sordebodnen matareblebi erTmaneTs?

50 + 40 = 90 (km/sT)

b) ra manZili iqneba matareblebs Soris 4 sT -is Semdeg?

90 × 4 = 360 (km).

pasuxi: moZraobis dawyebidan 4 sT -is Semdeg matareblebs Soris

manZili iqneba 360 km.

K upasuxe kiTxvebs:

1 ra saxis amocanebis amoxsna viswavleT?
2 rogor gamoviTvaloT, ra siCqariT Sordebian erTmaneTs

sawinaaRmdego mimarTulebiT gasuli sxeulebi.

savarjiSoebi

1 sqemis mixedviT gaarkvie ra manZili iqneba A sadguridan erTdroulad
gasul matareblebs Soris sqemaze miTiTebuli drois Semdeg.
56km/sT A 44km/sT

a)
3 sT

61km/sT A 68km/sT
b)

5 sT

97

2 Tea da daTo erTdroulad erTmaneTis sawinaaRmdego mimarTulebiT
gaemarTnen. Tea yovel saaTSi 5km -s, daTo _ 6km -s gadioda.
a) ra manZiliT iqnebian erTmaneTs daSorebulni Tea da daTo moZraobis
dawyebidan 1 sT -is Semdeg?
b) ra manZiliT iqnebian Tea da daTo erTmaneTs daSorebulni moZraobis
dawyebidan 2 sT -is Semdeg?
amocanis amosaxsnelad gamoiyene sqematuri naxazi.

Tea 5km/sT 6km/sT daTo

1sT

3 sqemis mixedviT gaarkvie ra manZili iqneba A sadguridan erTdroulad
gasul matareblebs Soris moZraobis dawyebidan 2 sT-is Semdeg?

46 km/sT A 54 km/sT b) 43 km/sT A 47 km/sT
a) ? km

? km

4 or avtobuss Soris manZilia 200km. ra manZili iqneba maT Soris 1sT-is
moZraobis Semdeg, Tu cnobilia, rom isini moZraoben 65km/sT da 80km/sT
siCqariT erTmaneTis a) Semxvedri mimarTulebiT? b) sawinaaRmdego mimar-
TulebiT?

5 erTi punqtidan erTmaneTis sawinaaRmdego mimarTulebiT erTdroulad
gasul or velosipedists Soris manZili 2 sT-Si 46 km iyo. ra manZili
iqneba maT Soris 3sT-Si, Tu isini moZraobas imave siCqariT gaagrZele-
ben?

6 erTi punqtidan erTmaneTis sawinaaRmdego mimarTulebiT erTdroulad
gasul or motocikls Soris manZili 1sT-Si 44 km iyo. ra manZili iqneba
maT Soris 4 sT- Si, Tu isini moZraobas imave siCqariT gaagrZeleben?

7 erTi punqtidan sawinaaRmdego mimarTulebiT erTdroulad gasul mgzav-
rebs Soris manZili 1 sT-Si 10km-ia. ra manZili iqneba maT Soris kidev
naxevari saaTis Semdeg, Tu isini moZraobas imave siCqariT gaagrZeleben?

8 sqemaze naCvenebia erTmaneTis sawinaaRmdego mimarTulebiT erTi adgi-

lidan erTdroulad gamosul or avtomanqanas Soris manZili da dro,

ra droSic am manZiliT daSordnen isini erTmaneTs. sqemaze mocemulia,

agreTve, erT-erTi avtomanqanis siCqare. gamoTvale meore avtomanqa-

nis siCqare. 5 sT
4 sT

65 km/sT ? km/sT 80 km/sT ? km/sT

a) b)

548 km 775 km

98

9 A da B qalaqebidan, romelTa Soris manZili 520 km-ia, erTdroulad
ori matarebeli gavida. erTi 78 km/sT, xolo meore 53 km/sT siCqariT
moZraobda. ra manZili iqneba matareblebs Soris 2 sT-is Semdeg, Tu
isini imave siCqariT imoZraveben a) Semxvedri mimarTulebiT?
b) sawinaaRmdego mimarTulebiT?

10 avtosadguridan avtobusi 65 km/sT siCqariT gavida. 2 sT Semdeg sawi-
naaRmdego mimarTulebiT imave sadguridan mikroavtobusi gavida.
mikroavtobusis gasvlidan 2 sT Semdeg avtobussa da mikroavtobuss
Soris manZili 410 km iyo. ra siCqariT moZraobda mikroavtobusi?

11 matarebeli sadguridan 60 km/sT siCqariT gavida. 3 sT-is Semdeg imave
sadguridan sawinaaRmdego mimarTulebiT gavida meore matarebeli
70 km/sT siCqariT.
a) ra manZili iyo matareblebs Soris meore matareblis gamosvlis
momentSi?
b) ra manZili iqneba matareblebs Soris meore matareblis gasvlidan
2 saaTis Semdeg, Tu isini imave siCqariT imoZraveben?

12 avtomobilma erT saaTSi gaiara gasavleli manZilis mesamedi, momdevno
saaTSi imdenive, ramdenic wina erT saaTSi, mesame saaTSi _ darCenili
65 km. ra manZili gaiara avtomobilma sul?

13 naxazze mocemuli ABCD da ABD texilebidan romels aqvs meti sigrZe?
BC

AD

14 ori velosipedisti erTmaneTis Sesaxvedrad
erTdroulad gaemarTa. maT Soris manZili
75km iyo. erTi 12km/sT, xolo meore 13km/
sT siCqariT moZraobda. ramden saaTSi Sex-
vdebian isini erTmaneTs?

15 nikas sami ormocdaaTlariani, xolo laSas ormocdaaTi xuTlariani
banknoti aqvs. ramdeni banknoti unda misces laSam nikas imisaTvis,
rom maT Tanabari Tanxa hqondeT?

16 Seavse Canaweri saWiro cifrebiT:

+ 31*04 _1825*3*
1*42* 1*44

*01*2

17 Tbilisidan da Telavidan erTdroulad erTmaneTis Semxvedri mimar-
TulebiT ori mikroavtobusi gamovida. Telavidan gamosuli mikroav-
tobusis siCqarea 50km/sT, xolo Tbilisidan gamosulis siCqare 70km/sT.
gamosvlidan ra droSi Sexvdebian mikroavtobusebi erTmaneTs, Tu
manZili Tbilissa da Telavs Soris 180 km-ia?

99

$ 4.4 wyalze moZraobis amocanebi

mizani:

viswavloT rogoraa amocana 1. TaTias burTi mdinareSi Cauvarda. mdinaris
damokidebuli dinebis dinebis siCqarea 2 km/sT. ra manZilze waiRebs wyali
siCqareze wyalze moZ- burTs naxevar saaTSi?
ravi sxeulis siCqare. amoxsna. amocanis amoxsnisas unda gaviTvaliswinoT,
rom burTs wyalze moZraobis sakuTari siCqare ar

gaaCnia, amitom wyalSi Cavardnis adgils igi mdina-

ris dinebis siCqariT Sordeba. mdinaris dinebis siCqarea 2km/sT, ese

igi burTi erT saaTSi 2km-is tol manZilze gadaadgildeba. naxevar
saaTSi ki, Cavardnis adgils burTi 2 : 2 = 1(km ) - iT daSordeba.

pasuxi: naxevar saaTSi wyali burTs 1 km manZilze waiRebs.

amocana 2. kateri A navsadguridan B navsadguramde mdinaris dinebis

mimarTulebiT 3sT-is ganmavlobaSi micuravda. mdinaris dinebis siCqare

2 km/sT - ia, xolo kateris sakuTari siCqare 14 km/sT.

a) ra manZilia navsadgurebs Soris?

b) ra droSi dabrundeba kateri B navsadguridan A navsadgurSi?

amoxsna. a) dinebis mimarTulebiT kateris siCqare iqneba: 14 + 2 = 16 (km/sT).

sami saaTis ganmavlobaSi kateri gaivlida 16 × 3 = 48 (km).

pasuxi: navsadgurebs Soris 48 km - ia.

b) radgan kateri A navsadguridan B navsadguramde mdinaris dinebis

mimarTulebiT moZraobda, amitom B navsadguridan A navsadgurSi

dabrunebisas katers dinebis sawinaaRmdego mimarTulebiT mouwevs

moZraoba. kateris siCqare dinebis sawinaaRmdego mimarTulebiT iqneba:
14 _ 2 = 12(km/sT). am siCqariT 48km-is gavlas (48 : 12)sT = 4sT moandomebs.

pasuxi: kateri B navsadguridan A navsadgurSi dabrunebas 4 sT - s

moandomebs.

K upasuxe kiTxvebs:

1 rogor gamoviTvaloT mdinareze moZravi sxeulis siCqare, roca igi
a) dinebis mimarTulebiT moZraobs?
b) dinebis sawinaaRmdego mimarTulebiT moZraobs?

2 ra siCqariT imoZravebs tivi mdinareze?

savarjiSoebi

1 tivi yovel saaTSi 4000m -s curavs. ra siCqariT moZraobs tivi? pasuxi
gamosaxe km/sT - Si.

2 menave A punqtidan B punqtSi Cavida da ukan, A punqtSi dabrunda.
rogorc dinebis mimarTulebiT, ise ukan dabrunebisas, menaves navis
sakuTari siCqare ar Seucvlia. romeli mimarTulebiT moZraobaze
daxarjavda menave met dros dinebis mimarTulebiT Tu dinebis sawi-
naaRmdego mimarTulebiT? ratom?

3 mdinaris dinebis siCqare 2km/sT-ia. ra siCqariT imoZravebs mdinareSi
Cagdebuli nafoti?

100