მათემატიკა 5 კლასი ბექაური (1)

16 gaorkecebuli ucnobi ricxvisa da 1 258 - is jami 2 000 - is tolia. ipo-
ve ucnobi ricxvi.

17 702 - is naturalur ricxvze gayofisas miiRes arasruli ganayofi _ 11
da naSTi 9. ra ricxvze gauyviaT 702?

18 1458-is naturalur ricxvze gayofisas miiRes arasruli ganayofi _ 15
da naSTi 3. ra ricxvze gauyviaT 1 458?

19 daadgine sagnebis masa naxatebis mixedviT: 480g x x

50g 70g x

x xxx x

2000g 150g 2594g

#1 #2 #3

20 A da B punqtebidan, romelTa Soris manZili 260 km - ia, erTmaneTis
Sesaxvedrad ori avtomobili erTdroulad gavida. avtomobilebi erT-
maneTs moZraobis dawyebidan 2 sT - is Semdeg Sexvdnen.A - dan gamosuli
avtomobilis siCqare 10 km/sT - iT meti iyo, vidre B - dan gamosulis
siCqare. gamoTvale B - dan gamosuli avtomobilis siCqare.

21 A punqtidan, erTmaneTis sawinaaRmdego mimarTulebiT motociklisti
da velosipedisti erTdroulad gavidnen. moZraobis dawyebidan 3 sT -
is Semdeg maT Soris manZili 144 km iyo. motociklisti 3-jer meti
siCqariT moZraobda, vidre velosipedisti. gamoTvale velosipedistis
siCqare.

22 ricxvebidan: 37, 40, 22, 30, 33, 36, 80, romelia
a) 2 - is; b) 3 - is; g) 10 - is jeradi?

23 ra cifri unda eweros `* ~- is nacvlad, rom 18 + 34* jami iyos 5 -is
jeradi kenti ricxvi?

24 ra ricxvi unda gavyoT 3 214 - ze, rom miviRoT
a) arasruli ganayofi 3 da naSTi 1 056?
b) arasruli ganayofi 10 da naSTi _ 951.

25 129 Wiqa yuTebSi unda Caalagon, TiToeulSi 10 - 10 Wiqa. ramdeni yuTi
gaivseba?

151

mizani: $ 6.5 wre da misi nawilebi

gavecnoT wres da naxazze mocemulia wrewiri. wrewiri Sekruli wiria.
mis nawilebs. misi yoveli wertili Tanabradaa daSorebuli erTi wer-
tilidan _ wrewiris centridan (naxazze O wertili).

C A
O

B

monakveTs, romelic wrewiris romelime wertils
mis centrTan aerTebs, radiusi ewodeba.

naxazze OA da OB monakveTebi radiusebia. A da B wertilebidan O
centramde manZilebi tolia. amitom, AO = OB.

A da B wertilebiT wrewiri or nawiladaa gayofili. TiToeul maT-
gans wrewiris rkali ewodeba. imisaTvis, rom es rkalebi ganvasxvaoT
erT - erTi maTgani, an orive sami asoTi unda CavweroT.

magaliTad, AB rkali da ACB rkali.

C AA

BB

monakveTs, romelic wrewiris or wertils aerTebs, qorda ewodeba.

A nax.1 - ze mocemuli AB da CD monakveTebi qordebia.
B diametri centrze gamavali qordaa.

C D nax. 2 - ze mocemuli AB qorda diametrs warmoadgens.

nax. 1 AB = AO + OB

A OB radgan AO = OB da TiToeuli maTgani radiusia, Seg-
nax. 2 viZlia davaskvnaT, rom wrewiris diametri radiusze
2 - jer meti sigrZisaa. aRvniSnoT diametri D asoTi,
radiusi R- iT, maSin

D = 2R

figura, romelic wrewiriTaa SemosazRvruli, SenTvis kargad nacnobi
wrea. wrewiri wris sazRvars warmoadgens. Sen garSemo uamrav sagans aqvs
wris forma.

O

wre

daakvirdi naxats. mocemulradiusiani wrewiris
dasaxazad fargliT unda aviRoT radiusis Sesa-
bamisi zoma, Semdeg farglis wveri davamagroT
furcelze, fargali nel-nela SemovatrialoT
ise, rom farglis fanqriani bolo furcels ar
movaSoroT, sanam wrewirs ar miviRebT.

152

daxaze wre da gamoWeri. wre gadakece naxevarwre
ise, rom gadakecvis xazi centrze gadiodes.
gadakecvis xazi diametria. diametri wres
or tol nawilad yofs. TiToeul am nawils
naxevarwre hqvia. (gaWeri wre diametris
gaswvriv da naxevarwreebis toloba Seamowme
maTi erTmaneTze zeddebiT).

makratliT Tu amoWri wris nawils, romelic ori radiusiTaa Semo-

sazRvruli, miiReb wris or nawils. TiToeul am nawils seqtori ewodeba

(nax. 3).

naxazze mocemuli seqtori ase ikiTxeba A

`AOB seqtori~. seqtorebadaa dayofili

magaliTad, stadionis tribunebi, cirkis O

SenobaSi mayurebelTa adgilebi da sxva.

B

nax. 3

K upasuxe kiTxvebs:

1 ras warmoadgens wrewiri? wre?
2 ras warmoadgens qorda? diametri?
3 ra kavSiria radiussa da diametrs Soris?
4 ras warmoadgens seqtori? naxevarwre? rkali?

savarjiSoebi

1 ramdeni radiusis gavleba SeiZleba wrewirSi? diametris? qordis?

2 wrewirze aRebulia M wertili. ramdeni radiusis gavleba SeiZleba
M wertilze? diametris? qordis? Seasrule Sesabamisi naxazebi.

3 daxaze wrewiri, romlis radiusia: a) 2 sm, b) 2 sm 5 mm, g) 4 sm.

4 daasaxele toli monakveTebi naxazis mixedviT.

B

O

AC
ED

5 daxaze saerTo bolos mqone 5 monakveTi, romelTagan TiToeulis sigrZe
2 sm - ia. sad mdebareobs TiToeuli am monakveTis meore bolo?

6 ekuTvnis Tu ara 5 sm radiusis mqone wrewirs wertili, romelic centridan
a) 4sm - iT; b) 5sm - iT; g) 80mm - iT aris daSorebuli?

7 ekuTvnis Tu ara 3sm radiusis mqone wres wertili, romelic centridan
a) 2sm - iT; b) 5sm - iT; g) 30mm - iT aris daSorebuli?

153

8 daasaxele rkalebi. A A

a) b) B
C
B

9 wrewiris radiusis sigrZea a) 3 sm; b) 5 sm 2 mm. ipove diametris sigrZe.

10 wrewiris diametris sigrZea a) 8 sm; b) 7 sm 8 mm. ipove radiusis sigrZe.

11 ras warmoadgens naxazze mocemuli wrewirisaTvis A
O wertili? AB monakveTi? AO monakveTi? daasaxele O

naxazze toli monakveTebi. ra hqvia am monakveTebs? B

12 daxaze saerTo centris mqone ori wrewiri. erTi radiusiT _ 2 sm, xolo
meore radiusiT _ 3 sm. maT Soris moqceuli are gaaferade wiTlad.
ras daarqmev gaferadebul figuras?

13 amoiwere yvela qorda, radiusi da diametri, romelsac naxazze xedav.
AB

O K
D

C
14 moniSne rveulSi erTmaneTisagan 3 sm - iT daSorebuli A da B wertilebi.

daxaze A centris mqone wrewiri, romelic B wertilze gadis. ras udris

am wrewiris diametris sigrZe?

15 gazome 5-TeTriani, 10-TeTriani, 20-TeTriani, 50-TeTriani, 1-la-
riani da 2 - lariani monetebis diametrebi. gamoTvale TiToeuli maT-
ganis radiusi.

16 wrewiris diametris sigrZe 15mm - iT metia radiusis sigrZeze. gamoT-
vale radiusis sigrZe.

17 wrewiris radiusis sigrZe 2sm - iT naklebia diametris sigrZeze. gamoT-
vale diametris sigrZe.

18 wris radiusis sigrZea 11 sm. SeiZleba Tu ara, am wrewiris romelime
qordis sigrZe iyos a) 20sm? b) 23sm? g) 3dm? d) 22sm? e) 5mm?

19 daxaze 2 sm da 5 mm sigrZis radiusis mqone wrewiri. moniSne wrewirze
A wertili da Tu SesaZlebelia, gaavle qordebi: AB = 35 mm, AC = 50 mm
da AD = 55 mm.

20 ramdeni saerTo wertili SeiZleba hqondes or aratol wrewirs? Seas-
rule Sesabamisi naxazi.

21 ramdeni saerTo wertili aqvs 6sm sigrZis radiusis mqone or wrewirs
romelTa centrebs Soris manZilia: a) 12sm? b) 9sm? g) 15sm?

22 Seadgine gantoleba amocanis pirobis mixedviT da amoxseni
a) ra ricxvi unda gavyoT 8 - ze,15000 rom miviRoT?

b) ra ricxvs unda mivumatoT 4825, rom 879654 miviRoT?

g) ra ricxvze unda gavamravloT 250, rom 500000 miviRoT?

d) ra ricxvze unda gavyoT 100000, rom 10 miviRoT?

154

VI Tavis damatebiTi savarjiSoebi

1 qvemoT CamoTvlilTagan romeli ricxvebis gamyofia: 2? 5? 10?
257, 562, 420, 125.

2 dawere iseTi umciresi kenti oTxniSna ricxvi, romelic 5 -ze unaSTod
iyofa.

3 dawere iseTi udidesi SvidniSna ricxvi, romelic 2 - ze unaSTod iyofa.

4 mocemulTagan romelia martivi da romeli Sedgenili ricxvi?
13, 26, 43, 47, 48, 72, 95.

5 dawere 3 ricxvi, romelTagan TiToeuli unaSTod iyofa:
a) 5 - ze; b) 10 - ze; g) 2 - ze da 5 - ze.

6 ipove saerTo gamyofebi:
a) 45 - is da 18 - is; b) 27 - is da 36 - is; g) 50 - is da 75 - is.

7 dawere ori ricxvi, romelTa saerTo gamyofia: a) 3; b) 11; g) 15.

8 arsebobs Tu ara iseTi orniSna martivi ricxvi, romelic bolovdeba:
a) 0 - iT? b) 5 - iT? g) 6 - iT?

9 qvemoT CamoTvlili gamonaTqvamebidan romelia WeSmariti?
a) 31 Sedgenili ricxvia;
b) 27 martivi ricxvia;
g) 35 - isa da 84 - is saerTo gamyofia 7;
d) yvela martivi ricxvi kentia.

10 daxaze rveulSi aseTi naxazi. b)
a)

11 wris diametri 42 sm - is tolia. gamoTvale wris radiusi.

12 daxaze rveulSi 3 wre radiusebiT: 1 sm, 2 sm da 3sm ise, rogorc naxazzea
naCvenebi.

gaaferade: BC
a) wiTlad A, B da C wreebis saerTo nawili;

b) yviTlad A wris is nawili, romelic B da C

wreebis gareTaa; A

g) mwvaned B wris is nawili, romelic A da C

wreebis gareTaa;

d) lurjad C wris is nawili, romelic A da B wreebis gareTaa;

e) yavisfrad B da C wreebis is saerTo nawili, romelic A wris gareTaa;

v) Savad A da C wreebis is saerTo nawili, romelic B wris gareTaa.

romeli wreebis saerTo nawili darCa gauferadebeli?

155

13 daxaze rveulSi olimpiuri simbolo, romelSic 5 toli wre gamoi-
yeneba. wris radiusi 3 sm - is sigrZis aiRe.

14 lelam ricxvis 110 - ze gayofisas miiRo arasruli ganayofi 2 da naSTi
87. ra ricxvi gayo lelam 110 - ze?

15 ramdeni saerTo wertili SeiZleba hqondes or wrewirs, romelTac:
a) toli sigrZis radiusebi aqvT? b) gansxvavebuli sigrZis radiusebi
aqvT? Seasrule Sesabamisi naxazebi.

16 Cavifiqre ricxvi, gavamravle 5 - ze da miRebuli namravli gavadide
123 - iT, miviRe 248. ra ricxvi Cavifiqre?

17 erT qvevrSi 2 - jer meti Rvinoa, vidre meoreSi. oriveSi ki - 3 000 l.
ramdeni litri Rvinoa TiToeul qvevrSi?

18 nikom 2580kg vaSli mokrifa. nawili sakonservo qarxanam Seisyida. dar-
Cenili vaSlis naxevari _ 415 kg nikom bazarze gayida. ramdeni kilog-
rami vaSli Seisyida sakonservo qarxanam?

19 or yuTSi 126 vaSlia. erT maTganSi 24 vaSliT metia, vidre meoreSi.
ramdeni vaSlia TiToeul yuTSi?

20 marTkuTxedis sigrZe 5 - jer metia siganeze, xolo perimetri 60 sm - is
tolia. gamoTvale marTkuTxedis farTobi.

21 marTkuTxedis sigrZe 5sm - iT metia siganeze, xolo perimetri 42sm - is
tolia. gamoTvale marTkuTxedis farTobi.

22 norC naturalistTa wreSi 37 gogona da 28 vaJia. sul mcire, ramdeni
bavSvi unda SeuerTdes maT, rom SesaZlebeli gaxdes maTi 10 gundad ga-
nawileba ise, rom yvela gundSi rogorc gogonebis, ise vaJebis raode-
noba erTnairi iyos? ramdeni bavSvi iqneba TiToeul gundSi? ramdeni
gogona da ramdeni vaJi iqneba TiToeul gundSi?

23 maTematikur wreSi 18 gogona da 21 vaJia. sul mcire, ramdeni bavSvi
unda SeuerTdes maT, rom SesaZlebeli gaxdes maTi 5 gundad ganawileba
ise, rom yvela gundSi rogorc gogonebis, ise vaJebis raodenoba erT-
nairi iyos? ramdeni bavSvi iqneba TiToeul gundSi? ramdeni gogona
iqneba TiToeul gundSi?

24 gamoTvale im orniSna ricxvebis raodenoba, romelic mxolod kenti
cifrebiT iwereba.

K SesaZlebelia, Tu ara

1. ori martivi ricxvis
a) jami; b) sxvaoba; g) namravli martivi ricxvi iyos?

2. erTi martivi ricxvi meore martiv ricxvze unaSTod gaiyos?
3. jami da erT - erTi Sesakrebi unaSTod iyofodes mocemul ricxvze

da meore Sesakrebi ar iyofodes am ricxvze?
4. arc erTi Tanamamravli ar iyofodes 10-ze da namravli iyofodes 10-ze?
5. arc erTi Tanamamravli ar iyofodes 5-ze da namravli iyofodes 5-ze?

156

Tavi 7. wiladi $ 7.1 nawili

mocemuli marTkuTxedebi sxvadasxva raodenobis mizani:
tol nawilebadaa dayofili da maTgan TiTo nawilia
gaferadebuli. viswavloT
a) figuris nawilebis
#1 #2
amocnoba, dasaxeleba,
#3 #4 Cawera da gamoTvla.
b) martivi amocanebis
amoxsna nawilebze.

daakvirdi, marTkuTxedi 1
2
#1 dayofilia 2 tol nawilad. gaferadebulia marTkuTxedis (ikiTxeba:
`erTi meoredi~) nawili, anu naxevari;

#2 dayofilia 3 tol nawilad. gaferadebulia marTkuTxedis 1 (ikiTxeba:
3
`erTi mesamedi~) nawili;

#3 dayofilia 9 tol nawilad. gaferadebulia marTkuTxedis 1 (ikiTxeba:
9
`erTi mecxredi~) nawili;

#4 dayofilia 18 tol nawilad. gaferadebulia marTkuTxedis 1 (ikiTxeba:
18
`erTi meTvrametedi~) nawili.

unmdaSaasdaadvaymoeT, m, aSreTsakbuaTmxiesdadis2,21 , 1 ,1 da a.S. nawilebis misaRebad marTkuTxedi
3, 34 d4a a. S. imden tol nawilad, meramdenedic

gvinda rom miviRoT.
CanawerSi _ 1 ricxvi 18 aRniSnavs, rom figura dayofilia 18 tol nawilad,

xolo ricxvi 118aRniSnavs imas, rom maTgan aRebulia 1 nawili.

amocana 1. Sokoladis fila samma Zmam Tanabrad gainawila. Sokoladis filis

ra nawili ergo TiToeuls?
arapmmaaoosTuxc.xsaminna:aaS.Ta2Tsi.aToTdokaoiemskuei,l16sTnZo16amkwaSnisilawSmiioolkrToioaslvmssamdidigeivrbsiaZRfeToi8oTlk,miims-sTi631ae.linrsaieawTTislioigiknriaeWZr6eirgstoaioa,. lrToonmkalwi?iils asdigurnZde a8
davW-
m - ia.

e. i. mTeli Tokis sigrZe rom vipovoT, 8 m unda gavamravloT 6 - ze. Tokis sig-

rZea 6 × 8 m = 48 m.

pasuxi: mTeli Tokis sigrZea 48 m. 1
9
amocana 3. anas 36 lari hqonda. am Tanxis - iT 2kg alubali SeiZina. ra Rirs
1kg alubali?

amoxsna. amocanis amosaxsnelad jer unda gavigoT ramdeni laris alubali

SeiZina anam. 36 : 9 = 4 (lari).

4 lari gadaixada 2kg alubalSi. maSasadame, 1kg alubali Rirs
4 : 2 = 2 (lari).

pasuxi: 1kg alubali 2 lari Rirs.
aammmaognaxalswniialT. ie4.b2ir-daiassnua71dvirnRiaoswTi4l2ei-rsiTsgia17mnoanwsaiawlTilvil.i?elad 42 unda davyoT 7 tol nawilad da

1 42 : 7 = 6.
7
pasuxi: 42 - is nawili 6 - is tolia.

157

K upasuxe kiTxvebs:

1 rwrraosigsogmvreirCvviaepmnodevebonseTd61ir-inScaiwxirvliisci:axavn)ia31xe6vn?aarw1i?wlrie?? b) 1 nawili? g) 1 nawili?
2 6 36
3
1
4 rogor vipovoT ricxvi, rom1lis 4 nawili 7 - is tolia?
3

savarjiSoebi

1 dedam sazamTro 6 tol nawilad daWra da aleqsandres, maTes, TaTias,
Sios, anas da daTunas gaunawila. sazamTros ra nawili ergo anas?

2 riT hgvanan naxatebi erTmaneTs?

3 AB monakveTis ra nawilia AC monakveTi?
5sm

A 1sm C B

4 AB monakveTis ra nawilia CD monakveTi?

CD

A B

5 erTi kilogramis ra nawilia 250 grami?

6 gamoTvale 18 - is 1 nawili.
3

7 romeli figuris ra nawilia gaferadebuli?

#1 #2 #3 #4

8 #5 #6 #7
9 monakveTis sigrZe 3dm - ia. daxaze monakveTi, romlis sigrZe monakve-
Tis sigrZis 1 iqneba.
10
158 10
daxaze 6 sm sigrZis mqone AB monakveTi. AB monakveTze moniSne C, M da
mmNoownneaakkrvvteeiTTliiseib1yionisasweAi,lBrimo.oamnmaoAkCwveermToeinsTaik61TveonTaeiwuililyioi,nsaxwAoiBlloimsoAnsaNikgvrmeoZTeniasskav12netTniaimweiitlyroies, bAASMiB.

3
daxaze kvadrati, romlis gverdis sigrZe iqneba 6sm. dayavi igi 36 tol
nawilad da sxvadasxva ferad gaaferade misi 1 , 1 , 1 da 1 nawilebi.

36 18 9 12

11 gamoTvale: 1 1 1
5 12 4
a) 1km - is nawili; b) 1sT - is nawili; g) 1 ha - s nawili.

12 darbazis sigrZis 1 nawili aris 4 m. ra sigrZisaa darbazi?
5

13 pirvel dRes moaasfaltes 800 m - is sigrZis gza, rac mTeli gzis 1
nawilia. ra sigrZisaa mTeli gza? 3

14 saxlidan skolaSi mimavalma Teklam 150 m gaiara, rac skolamde
manZilis 1 nawilia. ra manZilia Teklas saxlidan skolamde?
4

15 naxazis mixedviT gamoTvale: a) AM da AB monakveTebis sigrZeebi.
b) AM monakveTi AB monakveTis ra nawils warmoadgens?

14 15 16 17 18 19 20

AM B

16 gigam da Tamazma 6kv. m dabares, rac dasabari nakveTis farTobis 1
nawilia. ra farTobi darCaT dasabari? 10

17 turistebma pirvel dRes gaiares 16km, rac mTeli gzis 1 nawilia.
ramdeni kilometri darCaT turistebs gasavleli? 5

18 rm61aarnfTawakriulTTioxebbdiadiusrkfiajvoiaranmslisuoknbaaivkoivase,?T18is_ sigrZea 12m, xolo sigane _ 8m. nakveTis
kombostos, xolo danarCeni _ lobios.

19 daxaze 4 sm - is sigrZis AB monakveTi da 2 sm 5 mm sigrZis radiusis mqo-
ne wrewiri centriT A wertilSi. monakveTisa da wrewiris gadakve-
Tis wertili aRniSne M- iT. daxaze iseTive radiusis mqone wrewiri
centriT B wertilSi da misi AB monakveTTan kveTis wertili aRniSne
N- iT. gamoTvale: AN, MN da BM monakveTebis sigrZeebi.

20 a) sul mcire, kidev ramdeni kubia saWiro,
raTa mocemuli kubebi erT did kubamde
SevavsoT maTi adgilebis Seucvlelad?

b) aigeba Tu ara mocemuli kubebiT sxva
zomis kubi?

21 mocemuli samkuTxedebidan romelia:
a) marTkuTxa? b) maxvilkuTxa? g) blagvkuTxa?

B
EM

A CD FK N

159

es sainteresoa!

rogor davyoT wrewiri 6 tol nawilad? 3 tol nawilad?

farglis gamoyenebiT SesaZlebelia wrewiris 6 tol nawilad dayofa.
daxaze wrewiri. moniSne masze A wertili. imave radiusiT A wertili-
dan Semoxaze rkali wrewirTan gadakveTamde. gadakveTis wertilebi
aRniSne B da C asoebiT. amis Semdeg imave radiusiT Semoxaze rkali
B da C wertilebidan wrewiris gadakveTamde. gadakveTis wertilebi
aRniSne M da N asoebiT. Semdeg M an N wertilidan imave radiusiT
Semoxaze rkali wrewiris gadakveTamde. gadakveTis wertili aRniSne
K asoTi.

AC AC AC

B NB NB N

M #1 K M #2 K M #3 K

A, B, C, M, N da K wertilebiT wrewiri iyofa 6 tol nawilad, xolo
A, N, M wertilebiT iseve, rogorc B, C, K wertilebiT, wrewiri iyofa
3 tol nawilad. Tu #1 - ze miRebul wertilebs erTmaneTTan mona-
kveTebiT TanmimdevrobiT SevaerTebT, miviRebT wreSi Caxazul eqvs-
kuTxeds (#.3).

AC A AC

BN NB N

MK M MK
#4 #5 #6

Tu 3 tol nawilad dayofil wrewirze mdebare dayofis wertilebs
erTmaneTTan monakveTebiT SevaerTebT, miviRebT wreSi Caxazul sam-
kuTxeds. (#6).
1) ra figura miiReba #1 - ze miRebuli kveTis wertilebis erTis

gamotovebiT SeerTebiT?
2) Camoayalibe, rogor davyoT wrewiri 3 tol nawilad.
3) Seadare erTmaneTs #3 - ze miRebuli eqvskuTxedis gverdebis sig-
rZeebi.
4) Seadare #3 naxazze miRebuli eqvskuTxedis gverdebis sigrZeebi

wrewiris radiusis sigrZes.
5) wrewiris ra nawils warmoadgens #4 - is, MB rkali?
6) wrewiris ra nawils warmoadgens #5 - is AN rkali?

160

$ 7.2 wiladi

mizani:

a) b) g) gavecnoT wilads.

viswavloT misi Cawera,

wakiTxva da modelze

7 11 5 Cveneba.
20 36
9

naxatze mocemuli a) marTkuTxedi 20 tol nawiladaa dayofili. am

toli nawilebidan 7 wiTladaa gaferadebuli. a) marTkuTxedis wiTlad

gaferadebuli nawilis C2a0swaawrermoadadvgieynesnewbiTlawdilisadmnsi:S27v0n.els
, ricxvi
CanawerSi _ 7 da aRniSnavs,
20
rom marTkuTxedi 20 tol nawiladaa dayofili. ricxvi 7 warmoadgens

wiladis mricxvels da aRniSnavs, rom am 20 toli nawilidan gaferade-

bulia 7 nawili.

2w70ilasaediikriToxmebCaa:vw`eSrviodTi, meocedi~. ramden tol nawiladaa mTeli
unda vicodeT,

dayofili da ramdeni aseTi nawilia aRebuli.

7 wiladis mricxveli _ aRniSnavs, ramdeni toli nawilia aRebuli.
20 wiladis xazi.

wiladis mniSvneli _ aRniSnavs, ramden tol nawiladaa mTeli

dayofili.

rTuli araa imis mixvedra, ratom weria b) da g) marTkuTxedebis
11 5
qveS ricxvebi: 36 da 9 , ras aRniSnavs TiToeuli maTgani, an TiToeuli

ricxvi maT CanawerebSi.
380w6ili55kaidiT1k1i0ix40Te5r0bxiaeci:bkx`aivr4:Te68v`1bx9xaieuisbskTaami:miTa`agamsxaseelmxboeuaie:rTTq`mveesosbdaeicmiado~d:;ica~23xd;,uaTer16i,Tmie3ao0T86Ta,sxeads55 i,o~;T11x04m050o,cd4a68m19ecdxarae.dSi. ~.
wiladebs, romelTac toli mniSvnelebi aqvT, tolmniSvneliani wi-

ladebi hqvia. magaliTad, w89i,la49d7 ewbiil: 2a34d,eb24i54s, xv22a44d,as21x94vatmnoilSvmnneilSvinaenli.iani
3 15
wiladebia, xolo 4 , 2 ,

amocana. 1 metris ra nawilia 1 dm? 3 dm?

amoxsna. Tu 1 m sigrZis monakveTs 10 tol nawilad gavyofT, TiToeuli
nnaawwiillsisavsiiRgrebZTe ,imqanTebias1aderm.Teo.is. i1gdrmZe=i110qnemb,ax3odlmo=T13u0
am 10 nawilidan 3
m.

1m =10 dm

1dm = 1 m 3dm = 3 m
10 10

K upasuxe kiTxvebs:

1 rogori ricxvebis Cawera da wakiTxva viswavleT?
2 ras aRniSnavs wiladis mniSvneli? mricxveli?

3 rogor wiladebs hqvia tolmniSvneliani? sxvadasxvamniSvneliani?

161

savarjiSoebi

1 daasaxele mocemuli wiladis mricxveli da mniSvneli. ras aRniSnavs?

a) 5 , b) 3 , g) 1 , d) 18 , e) 2 , v) 15 .

9 10 2 927 7 15

2 waikiTxe wiladebi: a) 2 , b) 15 , g) 18 , d) 19 .

9 27 96 417

3 Cawere wiladis saxiT ricxvi: xuTi meSvidedi, cxra meTeqvsmetedi,
Teqvsmeti meCvidmetedi, ocdaerTi measedi, aTi ormocdamexuTedi.

4 gamosaxe wiladiT mocemuli figuris gaferadebuli nawili.

#1 #2 #3 #4 #5 #6 #7

5 marTkuTxedis ra nawili araa gaferadebeli?

#1 #2 #3 #4

6 torti 12 tol nawiladaa daWrili. tortis ra nawilia 4 naWeri?
7 11 naWeri? 3 naWeri? 7 naWeri? 12 naWeri?

8 medeas dabadebis dReze 11 megobari estumra, 4 vaJi da danarCeni
9 gogonebi. dedam torti Tormetive bavSvs Tanabrad gaunawila. TiTos
10 TiTo naWeri ergo.
11 a) tortis ra nawili ergo medeas?
b) tortis ra nawili ergo vaJebs? gogonebs?
12
13 dawere da waikiTxe 4 wiladi, romelTagan TiToeulis:
14 a) mricxvelia 11; b) mniSvnelia 25.
15
dawere: a) tolmricxveliani 2 wiladi; b) tolmniSvneliani 3 wiladi.
162
moifiqre iseTi wiladi, romlis mricxvelisa da mniSvnelis
a) sxvaobaa 5; b) jamia 8.

moifiqre iseTi wiladi, romlis:
a) mricxveli 2-jer naklebia mniSvnelze;
b) mniSvneli 3-jer metia mricxvelze;
g) mniSvneli mricxvelis tolia.

martis Tvis ra nawilia gasuli, Tu dRes 3 martia? 8 martia? 20 martia?

1 metris ra nawilia 17 sm? 5 dm?

saxlidan skolamde manZilis 3 nawilis gasavlelad sandros 9wT das-
10
Wirda. ra dro dasWirdeba sandros saxlidan skolaSi misasvlelad?

daxaze 20 sm2 farTobis mqone marTkuTxedi. gaaferade marTkuTxedis
2 nawili wiTlad, 3 nawili lurjad.

55

$ 7.3 wiladebis Sedarebis kerZo SemTxvevebi

mizani:

amocana 1. ekam sam dReSi 120-gverdiani wigni waikiTxa. viswavloT
wp_ia21rikvineTalwxiadleRike,asmmewmsaeaitmkieiTrdxaRaoewdsieg_nnoi16bsi31snagnwvaiewlrildi.eibr, imo?emonearlkelddeRbRiee?ss tolmniSvneliani
amoxsna: gamovTvaloT wakiTxuli gverdebis raode- da tolmricxveliani
wiladebis Sedareba.

noba dReebis mixedviT:
mpmeierosavrmeeeldddRRReesseswwaawiiakkiiikTTiTxxaaxawwiiwgginngiinssis121631
nawili, anu 120 : 3 = 40 (gv.);
nawili, anu 120 : 2 = 60 (gv.);
nawili, anu 120 : 6 = 20 (gv.).

20 < 40 < 60.
pasuxi: ekam yvelaze meti raodenobis gverdebi meore dRes waikiTxa,

xolo yvelaze naklebi raodenobis mesame dRes.

Tu 20 < 40 < 60 utolobaSi gverdebis raodenobis nacvlad Sesabamis

nawilebs CavwerT, miviRebT: 1 < 1 < 1 .
6 3 2
rogorc vxedavT, am wiladebs Soris metia is wiladi, romlis mniS-

vnelic naklebia da naklebia is wiladi, romlis mniSvnelic metia.

tolmricxveliani wiladebis Sedarebis es wesi vrceldeba 1 - isagan

gansxvavebuli mricxvelebis mqone wiladebzec.

amocana 2. dedam erTnairi zomis xaWapurebi gamoacxo. erTi xaWapuri

naTiam 4 tol nawilad daWra da sami naWeri SeWama, xolo natom 8 tol

nawilad daWra da sami SeWama. romelma SeWama xaWapuris meti nawili,

naTiam Tu natom?

amoxsna. amocanis kiTxvaze pasuxis gasacemad erTmaneTs unda Sevada-
rnaowiTlwebili, ardaecbig:o43godnaeb83is.
daakvirdi naxats, sadac gamuqebulia wris is
mier SeWmul xaWapuris nawilebs Seesabameba. ro-

gorc vxedavT, naTiam naxevar xaWapurze meti, xolo natom naxevar xaWa-

purze naklebi SeWama, e. i.

1 1 1 1
8 8 4 4
1 3 3
nato 8 8 > 4

1 naTia

4

pasuxi: naTiam natoze meti xaWapuri SeWama.

nax. 1 - ze mocemuli AN monakveTi sxvadasxvanairadaa tol nawilebad
dayofili.

a) A B N
b) A N
D N
g) A C N
N
d) A E nax. 1

e) A K

163

TiToeul SemTxvevaSi aRebulia AN monakveTis garkveuli nawili.

kerZod,

a) AB monakveTi AN monakveTis 3226224421-22--i-isi-isssitstttootoollolliliaiia;aai;.;a;
b) AD monakveTi AN monakveTis
g) AC monakveTi AN monakveTis
d) AE monakveTi AN monakveTis
e) AK monakveTi AN monakveTis
rogorc vxedavT,

AK < AE < AC < AD < AB < AN

Sesabamisad, 2 < 2 < 2 < 2 < 2
24 12 6 4 3
erTnairmricxveliani wiladebi zrdis mixedviTaa dalagebuli.

maT Soris udidesia is, romlis mniSvnelic naklebia da umciresia

is, romlis mniSvnelic metia.

tolmricxvelian wiladebidan is wiladia meti, romlis mniSvne-
lic naklebia da is wiladia naklebi, romlis mniSvnelic metia.

nax. 2 - ze mocemuli AB monakveTi 12 tol nawiladaa dayofili.

AC D KB

nax. 2 7
12
AC monakveTi AB monakveTis 1 nawils Seadgens, AD _ nawils,
11 12
xolo AK _ 12 nawils.

AC < AD < AK < AB

Sesabamisad, 1 < 7 < 11 < 12
12 12 12 12

tolmniSvneliani wiladebidan is wiladia meti, romlis mricx-
velic metia da is wiladia naklebi, romlis mricxvelic naklebia.

nax. 3- ze oTxi toli marTkuTxedi sxvadasxvanairadaa tol nawilebad
dayofili.

1

1 1 1
2 4 2

11 11 1
44 88 4
11
1 1 11 88
8 8 88

nax. 3
2 4 8
naxazidan Cans, rom 1= 2 = 4 = 8 maSasadame,

wiladi 1 - is tolia, Tu misi mricxveli mniSvnelis tolia.

magaliTi 1. romelia meti 457 Tu 324 ?
657 657

amoxsna. radgan am wiladebs toli mniSvnelebi aqvT, xolo 457 - is
mricxveli metia 324 657
657 - is mricxvelze, amitom 457 > 324 .
657 657

164

K upasuxe kiTxvebs:

1 rogor SevadaroT tolmniSvneliani wiladebi?
2 rogor SevadaroT tolmricxveliani wiladebi?
3 rogori wiladia 1 - is toli?
4 SesaZlebelia Tu ara, toli iyos ori wiladi, romelTac toli

mricxvelebi da gansxvavebuli mniSvnelebi aqvT?
5 SesaZlebelia Tu ara, toli iyos ori wiladi, romelTac toli

mniSvnelebi da gansxvavebuli mricxvelebi aqvT?

savarjiSoebi

1 romelia meti:

3 Tu 3 ? 5 Tu 5 ? 11 Tu 16 ?
4 8 3 6 11 16

24 Tu 24 ? 103 Tu 103 ? 9 Tu 9 ?
43 80 104 108 1000 800

32 Tu 32 ? 87 Tu 87 ? 28 Tu 420800?
4 8 89 98 400

2 romelia naklebi:

12 Tu 13 ? 15 Tu 51 ? 100 Tu 1000 ?
4 4 30 30 100 1000

24 Tu 99 ? 103 Tu 53 ? 124 Tu 9900 ?
24 24 56 56 1000 1000

31 Tu 32 ? 59 Tu 58 ? 300 Tu 280 ?
444 444 77 77 500 500

3 ra ricxvi unda eweros `* ~- is nacvlad, rom dawerili toloba an

utoloba swori iyos?

* < 2 ; 3 > * ; 11 > 11 ;
4 4 5 5 * 100

100 = 99 ; 5 = *; 4 < 4;
100 * 24 * 61 *

4 < * ; 1 = 1047; 38 < 38 .
47 47 * * 89

4 anis da Tornikes erTnairi rveulebi aqvT. anim rSevaevusloirs ve1u52lniaswimleti i
7 nawili. romelma
Seavso naweriT, Tornikem _ 12
nawili, anim Tu Tornikem?

5 ngTaiawoTirligaimi?wdaraiakTimTaoTxnaia,ace1Tm4ai4Ta-iagamvmoer_cda95niaaSnnaiwwziilegdnism. evktiiniT?xwuailkioTbxean.mgeitoir, ggiimorwigginmisTu79

6 sagzao brigadas gegmiT liandagi nsaawriklinziegzdoaaggozi. sSe15a90srnuawlila zTeuuanrdaa
daego. brigadam liandagi gzis 21
50
brigadam gegma?

165

7 12 t fqvili or sacxobSi Tanabrad gaanawiles. erTma maTganma daxarja
fqvilis mesamedi, meorem _ meoTxedi nawili. romelma sacxobma da-

xarja meti fqvili? ramdeni toniT meti?

8 daalage zrdis mixedviT wiladebi: 18 , 15 , 2 , 3 , 12 , 16 .
16 16 16 16 16 16

9 daalage klebis mixedviT wiladebi: 52 , 52 , 52 , 52 , 52 , 52 .
1 16 60 87 100 52

10 kinoTeatrma axali skamebi SeiZina. mwvane darbazSi skamebis sk25amnia?wili
dadges, wiTelSi _ 3
5 nawili. romel darbazSi dadges meti

11 ekam, mamiam, TamTam da sandrom TiTo boTli `civi Cai~ SeiZines. ekam
`_c31iv.iviCsaimse~t41i mdaamsiaalm e_vi61? 1
nawili dalia, , TamTam _ 2 , xolo sandrom
`civi Cai~ darCa

12 nikam da laSam 36 soko moagroves. mamam saWmelad vargisi sokoebi ga-

daarCia, danarCeni gadayara. saWmelad uvargisi aRmoCnda yoveli me-9

soko. mogrovili sokos ra nawili aRmoCnda saWmelad vargisi?

13 fexburTis matCis I taims stadionze 1 500 gulSematkivari eswreboda,

meore taims ki, mesamediT meti, vidre pirvels. ramdeni gulSematkivari
eswreboda fexburTis matCis meore taims?

14 klasis 24 moswavlidan gakveTils 1 nawili ar eswreboda. klasis mos-
wavleTa ra nawili eswreboda gakve6Tils? ramdeni moswavle eswreboda

gakveTils?

15 mocemuli naxazis mixedviT gamoTvale AMBC oTxkuTxedis perimetri,
Tu cnobilia, rom AB = 5 sm da wris radiusi r sm - is tolia.

MB
O

A DC

16 daasaxele ori ricxvi, romelTagan TiToeuli jeradia
a) 12 - isa da 14 - is, b) 15 - isa da 60 - is,
g) 40 - isa da 16 - is, d) 7 - isa da 13 - is.

17 me-15 amocanis naxazis mixedviT gamoTvale AMBC oTkuTxedis far-
Tobi, Tu cnobilia, rom r = 3sm.
K SesaZlebelia Tu ara,

aratoli mricxvelebis an aratoli mniSvnelebis mqone wiladebi iyos
toli?

erT finjanSi eteva imis 1 raodenobis wyali, rac CaidanSi.
a) ramdeni finjani wyali 8gaavsebs 4 Caidans?
b) ramden finjans gaavsebs ori Caidani wyali?
b) ramden finjans gaavsebs naxevari Caidani wyali?

166

$ 7. 4 wiladi, rogorc gayofis Sedegi

amocana 1. 2 ha farTobis mqone marTkuTxedis for- mizani:

mis miwis nakveTi samma Zmam Tanabrad gaiyo. ramdeni gavecnoT wiladsa da
ganayofs Soris kavSirs.
heqtari ergo TiToeul Zmas? viswavloT wiladis
Cawera gayofis saxiT
amoxsna. TiToeuli heqtari davyoT 3 tol nawilad. da piriqiT.

TiTo nawilis farTobi iqneba 1 ha. Zmebs or - ori
aseTi nawili, 2 3
anu 3 ha ergoT.

1 ha 1 ha

11 111 1
33 333 3

2 ha 2 ha 2 ha
3 3 3
amocanis amoxsna ase SegviZlia CavweroT: TiToeul Zmas ergo 2:3=32 (ha).
apmmaaSosuacsxaaind:aaTm2ie.T,ao2) e-6uifsli3lZ-mazasSeogk32aonhlayaaodefriig3oar.baivsSv32s ramdeni
. gaunawiles.
Tanabrad

Sokoladi ergo TiToeul bavSvs?

b) 4 fila Sokoladi 5 bavSvs Tanabrad gaunawiles. ramdeni Sokoladi

ergo TiToeul bavSvs?
amoxsna. a) TiToeul bavSvs 6 : 3 = 2 fila Sokoladi ergo.
4mx:o5l=o45d
a) b) TiToeul bavSvs fila Sokoladi ergo. maTSi ricx-
amocana b) amocanisagan imiT gansxvavdeba, rom

vebia sxvadasxva. orive amocana gayofiT amovxseniT.

msgavsi amocanebis amoxsnisas miviRebT, rom

4 : 7 = 4 , 9 : 19 = 9 , 23 : 1 000 = 23 da a. S.
7 19 100

maSasadame, gayofis moqmedeba SegviZlia wiladis saxiT CavweroT. wi-

ladi unda gvesmodes, rogorc mricxvelis mniSvnelze ganayofi, xolo

wiladis xazi _ rogorc gayofis niSani.

amocana 3. elenes sami vaSli hqonda. man vaSlebi Suaze gaWra. wiladis

saxiT CavweroT, ramdeni vaSli aqvs elenes.

amoxsna. radgan elenem samive vaSli Suaze gaWra, mas axla vaSlebis 6
naWeelreinaeqsvsv.aSslulebi26s
vaSli. raodenobisTvis araferi mouklia,
Tavdapirveli
6 = 3. es asec SegviZlia CavweroT: 6 : 2 =3.
e. i. 2

aseve SegviZlia davweroT, rom:

0 = 0, 6 = 3, 25 = 25, 36 = 9, 100 = 50 da a. S.
1 2 1 4 2

ricxvis nulze gayofa ar SeiZleba, maSasadame, wiladis mniSvneli

0 - isagan gansxvavebuli ricxvi unda iyos.
ammaogaxlsniaT. im.oacmoemvuxlsniaTgagnatnotloelbeabSa:eg1xv8iZ=l1i5a. ase CavweroT: x : 18 = 15.

ucnobi gasayofi rom vipovoT, saWiroa ganayofi gavamravloT gamyofze:

x = 18 × 15,
x = 270.
pasuxi: x = 270.

167

K upasuxe kiTxvebs:

1 wiladis gayofis saxiT Cawerisas mniSvneli gasayofia Tu gamyofi?
2 mricxveli?
3 ras udris wiladi, romlis mricxveli nulia?
4 SeiZleba Tu ara wiladis mniSvneli iyos nulis toli?
ras udris wiladi, romlis mricxveli mniSvnelis tolia?
1
2 savarjiSoebi
3
Cawere wiladis saxiT: 3 : 8, 7 :11, 58 : 4, 46 : 1.
4
5 Cawere gayofis saxiT wiladi: 5 , 1 , 12 , 45 , 472 .
3 61 63 9 53
6
7 ras udris wiladi, romlis:
8 a) mniSvneli 1 - is tolia?
b) mniSvneli mricxvelze 10 - jer mcirea?
9 g) mricxveli mniSvnelze 7 - jer metia?

10 dawere sami sxvadasxva wiladi, romelic tolia:
11 a) 2 - is; b) 3 - is; g) 0 - is; d) 100 - is; e) 41 - is.
12
13 amoxseni gantoleba:
168
a) x = 100; b) x = 248 : 4; g) 520 = 26;
25 12 x

d) x = 248; e) 270 = 45; v) 3200 = 1 280 : 2.
100 x x

Cawere naturaluri ricxvis saxiT wiladi:

5 , 40 , 74 , 3200 , 450 .
1 5 74 10 90

amoxseni gantoleba:
x
a) 7 = 324 : 18; b) 250 _ x =125; g) x _ 10 = 6.
10 12 2 3

daalage zrdis mixedviT ricxvebi:

1, 2 , 6 , 3 , 3 .
5354

makam feradi qaRaldebiT samkuTxedi da sami toli
kvadrati gamoWra. samkuTxedis gverdebze miadga
kvadratebi ise, rogorc naxazzea naCvenebi. samkuT-
xedis perimetri 30 sm - ia. gamoTvale, ra farTo-
bis qaRaldi dasWirda makas kvadratebis gamosaWre-
lad?
metris ra nawilia: 2 sm? 3 dm? 8 mm? 11sm? 111mm?

tonis ra nawilia: 1 kg? 1 g? 2 c? 48 kg? 69 g? 485 kg? 1458 g?

saaTis ra nawilia: 4 wT? 60 wm? 15 wT? 48 wT?

ramdeni kvadratuli santimetris tolia: 1 kv. dm ? 1 kv. m ?

$ 7.5 wesieri da arawesieri wiladebi. Sereuli ricxvi

wiladi ricxvebi davyoT sam jgufad: mizani:
I. wiladebi, romelTa mricxveli mniSvnelze
naklebia. aseT wiladebs wesier wiladebs gavecnoT wesier da arawesier
uwodeben. wesieri wiladebia, magaliTad, wiladebs, Sereul ricxvebs.
viswavloT arawesieri wila-
1 , 14 , 89 . dis Cawera Sereuli ricxvis
5 19 104 saxiT. SevZloT wiladis 1-Tan
II. wiladebi, romelTa mricxveli mniSvnelze Sedareba.

metia. aseT wiladebs arawesier wiladebs uwodeben.

III. wiladebi, romelTa mricxveli mniSvnelis tolia. aseTi wiladebic

arawesier wiladebs miekuTvnebian. 5 400 48 , 508 .
5 400 19 104
arawesieri wiladebia, magaliTad, , ,

wesieri wiladi 1 - ze naklebia. magaliTad,

11 < 12 anu 11 < 1; 24 < 57 anu 24 < 1
12 12 12 57 57 57

arawesieri wiladi 1 - ze meti, an 1 - is tolia. magaliTad,
51
48 = 1; 51 > 48 , anu 48 >1
48 48 48
daakvirdi naxats
4 4
4 =1 4 =1 3
4

4 + 4 + 3 = 11 = 2 + 3 = 2 3
4 4 4 4 4 4
3 3
2 + 4 SemoklebiT 2 4 saxiT iwereba da ase ikiTxeba: `ori mTeli sami

meoTxedi~. am saxiT Caweril ricxvebs Sereuli ricxvebi ewodeba. 2 am
rsvynoaixafcxixeTivT_iC,sa2v1m4w1aTer=reilTs2i,4S3menira.vweieiuR.llie.ibia1T,1rxi1-4oc1ilsxv=o4i1s-1z43m:eT4weg=ilaly2ioa(dnfnaaiiwSTiTnlaimwiii3,Rl)x.eoibm.uleToloruienaa14mrS1xTar-siirsv3u,gl_arniowaiygglooanfaradic-si

nawilis mricxveli.

arawesieri wiladis Sereuli ricxvis saxiT Casawerad saWiroa:
a) mricxveli gavyoT mniSvnelze;
b) miRebuli arasruli ganayofi davweroT Sereuli ricxvis mTel

nawilad, xolo naSTi wiladi nawilis mricxvelad;
g) mniSvneli igive gadavitanoT.

mamagoaxlsniaT: im.oCcaevmwuerloiTwiSleradeuilCaivrweircoxvTisgasnaaxyioTfwiislsaadxiiT1:59 .
19
pasuxi: 19 = 3 4 . 5 = 19 :5 =3 (naSTi 4).
5 5

Tu mricxvelis mniSvnelze gayofa unaSTod Sesrulda, es niSnavs,

rom wiladi mTeli ricxvis tolia.

169

yoveli naturaluri ricxvis Cawera SesaZlebelia wiladis saxiT.
=go14go=nam82 9=va14S00li=, x14o600lodsaaam.maS.vaJma
magaliTad, 4 Svidi vaSli
amocana. xuTma

Tanabrad gainawila. vis meti vaSli ergo TiToeul gogonas Tu

TiToeul vaJs?
31laa73mmioons=acxwsa452innlai13.sniTa.ekwei.ii.TiTil..oxiTTve,auiixzTTloeoolpgeeoauuosllguToxivgniTaaosJsogsgeoamu95snelaat=scieve1vamrJaa45Sgdslov_1iae4S52relrTmSigTieoeevm,lraTvdgiieoadlrv,raeixSeTolvTali2Sio13lTdoia-Teskdiu.iTaldmokieevigvudioRvlegaevoSTvlnva,aaiJsSss.-
1
1 4 < 2 3 .
5
sazogadod,

ori Sereuli ricxvidan isaa meti, romlis mTeli nawilic
metia. Tu mTeli nawilebi tolia, maSin metia is Sereuli ricxvi,
romlis wiladi nawilic metia.

magaliTad, 3 4 > 3 1 , radgan 4 > 1 .
5 5 5 5

K upasuxe kiTxvebs:

1 rogor wilads hqvia wesieri? arawesieri?

2 rogori wiladidan gamoiyofa mTeli da wiladi nawilebi?
3 rogori wiladidan gamoiyofa mxolod mTeli nawili?
4 rogor ricxvs hqvia Sereuli ricxvi?
5 rogor CavweroT arawesieri wiladi Sereuli ricxvis saxiT?
6 ra ariTmetikuli moqmedeba igulisxmeba Sereuli ricxvis mTel

da wilad nawilebs Soris?

savarjiSoebi

1 CamoTvlilTagan romeli wiladia wesieri? arawesieri?
2 , 57 , 45 , 580 , 57 , 8 , 10 .
2 5 16 45 90 16 9 47
3
4 dawere 4 iseTi wesieri wiladi, romlis mricxvelia a) 4; b) 7; g) 2.

5 dawere oTxi a) arawesieri wiladi; b) Sereuli ricxvi.
6
waikiTxe ricxvebi da daasaxele maTi mTeli da wiladi nawilebi:
170
3 3 , 4 5 , 12 4 , 100 27 .
5 6 9 1000

Cawere Sereuli ricxvebis saxiT wiladebi: 31 , 96 , 415 , 2007 .
7 16 45 1000

Cawere naturaluri ricxvis saxiT wiladebi: 40 , 80 , 405 , 2000000 .
10 16 15 10000

7 CamoTvlilTagan: 12, 3, 10, 100 TiToeuli ricxvi warmoadgine iseTi
wiladis saxiT, romlis mniSvnelia a) 2; b) 5; g)10.

8 dawere yvela is arawesieri wiladi, romlis mricxvelia 8 da gamosaxe
Sereuli an mTeli ricxvis saxiT.

9 dawere Sereuli ricxvebis saxiT wiladebi:

a) 45 , 64 , 76 , 8 , 91 ; b) 209 , 357 , 234, 87 ; g) 459 , 78459 , 456 .
23 7 6 10 50 100 25 65 10 1000 17

10 amowere mocemulTagan a) wesieri; b) arawesieri wiladebi.
9 , 7 , 3 , 8 , 78 , 37 , 36 , 2 , 6 .
5 10 25 5 30 12 25 59 19

11 amowere mocemulTagan wiladebi, romelic
a) 1 - is tolia; b) 1 - ze metia; g) 1 - ze naklebia.
6 , 49 , 25 , 81 , 1000 , 56 , 6 , 6 .
4 15 25 59 1000 16 103 27

12 Cawere Sereuli ricxvis saxiT wiladi:

a) 45 ; b) 405 ; g) 458 ; d) 36 ; e) 45000 ; v) 87 .
11 20 149 19 16 26

13 aRadgine cxrilSi gamotovebuli monacemebi:

a 2 1 1 5 2
b 7 9
a+b
3 5 12
4 6

2 3 5 4 11 2
4 15 9

14 romelia meti: a) 1 4 Tu 4 1 ? b) 2 6 Tu 5 7 ?
11 11 11 23

15 daalage ricxvebi zrdis mixedviT: 80 , 102 , 45 , 240 .
11 10 5 6

16 daalage ricxvebi klebis mixedviT: 780 , 301 , 180 , 3 , 1200 .
250 340 6 3 300

17 daasaxele ricxvi, romelic a) 1 - ze metia da 2 - ze naklebi; b) 10 -ze
metia da 11 - ze naklebi; g) 0 - ze metia da 1 - ze naklebi.

18 ramden tol nawiladaa dayofili TiToeuli figura?

a) b) g)

171

$ 7.6 wiladi da Sereuli ricxvebis gamosaxva ricxviT sxivze

mizani: yovel ricxvs ricxviT sxivze erTi werti-
li Seesabameba. ricxviT sxivze aigeba ara
viswavloT wiladi da marto naturaluri ricxvebis Sesabamisi
Sereuli ricxvebis Sesabamisi wertilebi, aramed wiladi da Sereuli
wertilebis gamosaxva ricxviT ricxvebis Sesaba-misi wertilebic.
sxivze.

magaliTi 1. avagoT ricxviT sxivze

a) 3 - is; -brs)aa1ddg32aan-1i-0ss<; So83gr)<is21a,65a,a-muiintsdoaSmedsaabrvayimociTsxivi8wTetirotslixlivmeiobsni.aiksvemToanda.kv0e-Tiis,
8

amoxsna. a)

romelic 0

marjvniv gadavTvaloT sami monakveTi da movniSnoT mesame monakveTis
marjvena bolo. es iqneba 3 - is Sesabamisi wertili.
8
3
03 1 2 3 0 < 8 < 1
8
<So1r32isa<a,2u, nadmiatdoamvyorTic3xtvioTli
b) radgan 1 sxivis is monakveTi, romelic
1- sa da 2 -s monakveTad. 1 mTelis marjvniv

gadavTvaloT ori monakveTi da movniSnoT meore monakveTis marjvena
2
bolo, romelic 1 3 - is Sesabamisi wertili iqneba.

0 1 1 2 2 3 1 < 1 2 < 2
3 3

g) radgan 2 s<S2o65ri<saa3,, amitom ricxviTi sxivis is monakveTi, romelic
2- sa da 3 - unda davyoT 6 tol monakveTad. 2 mTelis mar-

jvniv gadavTvaloT xuTi monakveTi da movniSnoT mexuTe monakveTis
5
marjvena bolo, romelic 2 6 - is Sesabamisi wertili iqneba.

0 1 2 2 5 3
6
2 < 2 5 <3
6
iseve, rogorc naturaluri ricxvebis SemTxvevaSi, rac ufro didia

wiladi Tu Sereuli ricxvi, miT ufro marjvnivaa ricxviT sxivze misi

Sesabamisi wertili.

sazogadod,

rac ufro did ricxvs Seesabameba wertili, miT
metadaa igi saTavidan daSorebuli.

magaliTad, 2vi56dr- eis Sesabamisi wertili ricxviT sxivze ufro marjvniv
mdebareobs, 3 2
8 - is, an 1 3 - is Sesabamisi wertilebi.

0 12 3

3 1 2 2 5
8 3 6

ricxviT sxivze cxadad Cans, rom: 3 < 1 2 < 2 5 .
8 3 6

172

K upasuxe kiTxvebs:

1 ramdeni wertili Seesabameba ricxvs ricxviT sxivze?
2 1 - is Sesabamisi wertilis romel mxares moiniSneba

a) wesieri wiladis; b) Sereuli ricxvis Sesabamisi wertili?

savarjiSoebi

1 daasaxele da Cawere ricxvebi, romelTac ricxviT sxivze mocemuli
wertilebi Seesabameba.

AB CD E
3
012 4

2 daxaze ricxviTi sxivi. erTeulad aiRe 5 sm sigrZis monakveTi. moniSne
sxivze 2 , 1 4
5 1 5 , 1 5 ricxvebis Sesabamisi wertilebi.

3 romeli ricxvis Sesabamisi wertilia ricxviTi sxivis sawyis wertil-
2 5
Tan axlos: a) 3 - is Tu 3 9 - is? b) 7 - is Tu 2 6 - is?

4 romeli ricxvis Sesabamisi wertili ufro marjvniv mdebareobs

ricxviT sxivze:

a) 1 - is Tu 1 - is? b) 1 - is Tu 19 - is? g) 1 1 - is Tu 2 -is?
8 9 25 2

d) 36 - is Tu 3 4 - is? e) 0 - is Tu 1 - is?
10 10 550

5 daxaze ricxviTi sxivi. erTeulad aiRe 50 mm sigrZis monakveTi.
moniSne sxivze 6 , 9, 1 1 ricxvebis Sesabamisi wertilebi.
10 5 1 10 , 1 2

6 erTsa da imave ricxviT sxivze moniSne mocemuli ricxvebis Sesabamisi
wertilebi.

1 , 3 , 3 , 1 5 , 1 1 , 5 .
12 6 4 12 4 6

7 naxazze mocemuli monakveTi Seesabameba 1km-s. naxazis mixedviT ujrian
1 5
rveulSi daxaze: 500 m - is, 250 m - is, 3 km - is, 6 km - is Sesabamisi mo-
nakveTebi.

1km

8 erTi saukunis ra nawilia 27 weliwadi?
9 wuTis ra nawilia 19 wami? 30 wami?

173

$ 7.7 tolmniSvneliani wiladebis da Sereuli ricxvebis Sekreba

mizani:

SevZloT naxatze mocemulia kvadrati, romelic 16 mcire

tolmniSvneliani tol kvadratadaa dayofili. mcire kvadratebidan 3 _

wiladebis Sekreba wiTlad, xolo 5 yviTladaa SeRebili. amitom SegviZ-
lxr1o36ailoa+dov1T5e6nyqovnviaabTwTail,l8ra-ido.imsm1e5w6toioTnrallewaiimdalx,riSa.einRvus,eubSSlieelRRSieeeabbRiidlelibdiiialimdckivaiiadrdkievrakakdvtvaraiddsartr13a6ia-s-,
da gamoyeneba
gamoTvlebis dros.

ratebis saerTo
8
tis 16 nawili. miviReT toloba:

3 + 5 = 8
16 16 16

ganxilul SemTxvevaSi SevasruleT tolmniSvneliani wiladebis Sekreba.

tolmniSvneliani wiladebi rom SevkriboT, saWiroa SevkriboT
maTi mricxvelebi, mniSvneli ucvleli davtovoT.

asoebis gamoyenebiT es wesi ase Caiwereba:

a + b = a+b
c c
asmuomzciarnaa.1r. afferarmeTrosbz2e51ahqavs- c d25ahmaz-ezsue m_zimrzae?-

ze xorbali uTesia, xolo 3

daTesili fermers xorbali

amoxsna. imisaTvis, rom gamoviTvaloT, ramden heqtarze aqvs fermers

daTesili xorbali da mzesumzira, am kulturebis mier dakavebuli

farTobebi unda SevkriboT:

2 1 +3 2 = (2 + 1 ) + (3 + 2 ) = (2 + 3) + ( 1 + 2 ) = 5 + 3 = 5 3 (ha).
5 5 5 5 5 5 5 5

pasuxi: fermers xorbali da mzesumzira 4 1 ha - ze aqvs daTesili.
5

Sereuli ricxvebis Sekrebisas mTel nawils mTeli nawili unda
mivumatoT, wilads wiladi da miRebuli Sedegebi SevkriboT.

magaliTi 1. gamovTvaloT jami:

a) 4 + 21 ; b) 2 3 + 5 ; g) 1 13 + 7; d) 8 15 + 7 11 .
25 25 14 14 22 19 19

amoxsna:

a) 4 + 21 = 4 +21 = 25 = 1;
25 25 25 25

b) 2 3 + 5 = 2 + ( 3 + 5 )= 2 + 3+5 = 2 + 8 = 2 8 ;
14 14 14 14 14 14 14

g) 1 13 + 7 = (1 + 13 ) + 7 = ( 1 + 7)+ 13 = 8 + 13 = 8 13 ;
22 22 22 22 22

d) 8 15 + 7 11 = (8 + 7) + ( 15 + 11 ) = 15 + 26 = 15 + 1 7 = 16 7 .
19 19 19 19 19 19 19

174

K upasuxe kiTxvebs:

1 rogor SevkriboT erTnairmniSvneliani wiladebi?
2 rogor SevkriboT Sereuli ricxvebi?
3 rogor SevkriboT Sereuli da wiladi ricxvebi?
4 raSi mdgomareobs Sekrebis a) gadanacvlebadobis Tviseba?

b) jufTebadobis Tviseba?

savarjiSoebi

1 turistebma erT dRes gzis 2 nawili gaiares, meore dRes 1 nawili.
5 5
gzis ra nawili gaiares turistebma orive dRes?

2 ngaikgvoeTpaipsarm aernaTwidlRiesdanbaakrveaTgiisgo17 nawili dabara, meore dRes _ 2 .
papam orive dRes? 7

3 Seasrule wiladebis Sekreba:

7 + 1 ; 4 + 2 ; 9 + 11 ; 2 + 6 ; 3 + 9 ; 13 + 9 ;
16 16 7 7 21 21 13 13 17 17 17 17

7 + 13 ; 28 + 1 ; 15 + 23 ; 41 + 42 ; 11 + 14 ; 47 + 14 .
20 20 30 30 64 64 83 83 33 33 71 71

4 nonam 3 2 m wiTeli da 2 7 m lurji abreSumi SeiZina. ramdeni metri
10 10
abreSumi SeiZina nonam?

5 sgaabdaamix1a3d1a20salbaamrinsavwaiWgrnSeib?i da 2 7 laris rveulebi SeiZina. ra Tanxa
10

6 gamoTvale jami:

4 +116 ; 9 + 41; 4 5 + 9 ; 11 1 + 20 23 ;
21 23 23 42 42

100 + 6 ; 5 + 2 21 ; 1 125 + 3 7 ; 120 5 + 18 9 .
13 37 37 15 42 42

7 lizim 11305000 kg kanfeti SeiZina, misma Zmam ki 650 kg. ramdeni kilog-
1000
rami kanfeti SeiZina da-Zmam erTad?

8 mavTulis gorgals jer 18 3 m mavTuli CamoaWres, Semdeg _ 10 8 m.
10 10
ra sigrZis mavTuli CamoaWres gorgals sul?

9 gamoTvale: 19 9 12 6 15
5 6 23 23 13 13 28 13
7 + 7 ; + ; + ; 2 + 3 28 ;

1 7 + 13 ; 28 + 4 11 ; 23 + 19 ; 24 7 + 8 9 .
15 15 30 30 34 34 11 11

175

10 amoxseni gantoleba:

a) x_ 5 = 1 ; b) x _ 1 2 = 5 ; g) x _ 4 = 7 ; d) x _ 11 1 =5 1 ;
7 7 8 8 10 2 2
11 romelia meti:

a) 1 Tu 2 + 4 + 7 ? b) 12 + 4 + 1 2 Tu 2? g) 3 Tu 21 + 14 + 1 9 ?
14 14 14 20 20 20 30 30 30

12 amoxseni gantoleba: a) x + x = 1; b) y + y = 0.; g) a + a = 3.

13 mgzavrma erT dRes 23 k43ilkmomgeatiarria,gamieaorraemdgzRaevsr3ma41okrmi-vieTdRmeets?i, vidre
pirvel dRes. ramdeni

14 ksrauarmradedneoanSiviawn8ii0g,nw2i1i0agqnnaaiwrai.TlmuailT_ieng51oervnamanawini?ulliernouvsaunli,ednoanvaarniCae,ni41 nawili _ ingli-
_ qarTulenovani.

15 ricxviT sxivze moniSne ricxvebis Sesabamisi wertilebi.

3 , 3 , 1, 6.
48 28

16 Cawere Sereuli ricxvis saxiT: a) 39 ; b) 65 ; g) 415 ; d) 37 .
4 11 400 15

17 ramdeni metria: 1 km? 1 km? 1 km? 1 km?
4 20 10 100

18 ramdeni kilogramia: 1 t? 1 t? 1 t? 1 t?
2 50 500 1000

19 ramdeni wuTia: 1 sT? 1 sT? 1 sT? 1 sT?
2 60 15 30

20 daasaxele naxazis mixedviT:
a) monakveTebi, romelTa erTi bolo emTxveva E wertils;
b) kuTxeebi, romelTa wvero B wertilSia;
g) samkuTxedebi, romelTac K wvero saerTo aqvT.

B TK C

21 A ED g) 1 sT Tu 1 sT?
22 b) 6 sT Tu 6 sT? 12 3
23 romelia meti:
176 a) 1 sT Tu 1 sT? 10 5

54 b) 6 kg Tu 60 kg? g) 5 g Tu 5 g?
1000 1000 10 13
romelia naklebi:
a) 25 kg Tu 25 kg?

500 550

ra ricxvi unda gavyoT 11-ze, rom miviRoT arasruli ganayofi 5 da
naSTi 4?

$ 7.8 Sereuli ricxvis wiladis saxiT warmodgena

magaliTi 1. CavweroT wiladis saxiT Sereuli mizani:

ramiocxxsvnia.4s73aW. iroa vipovoT ricxvi, romlis 7-ze viswavloT Sereuli
ricxvis wiladis saxiT

gayofiT miiReba arasruli ganayofi 4 da naS- warmodgena.
7t3 o=l371m.niSvneliani
Ti 3. es ricxvia 4 × 7 +3 =31, e.i. 4 wiladebis
igive toloba SegviZlia miviRoT

Sekrebis wesis gamoyenebiT. marTlac,

4 3 = 4 + 3 = 28 + 3 = 28 +3 = 31 . 8
7 7 7 7 7 7 11
magaliTi 2. CavweroT arawesieri wiladis saxiT 10 .
amoxsna.
8 11 × 10 + 8 118
10 11 = 11 = 11 .

Sereuli ricxvis wiladis saxiT warmosadgenad saWiroa:
I. mniSvnelis mTelze namravls davumatoT mricxveli da

miRebuli jami davweroT mricxvelSi;
II. mniSvneli igive davtovoT.

K upasuxe kiTxvebs:

1 rogor warmovadginoT Sereuli ricxvi wiladad?
2 rogori wiladi miiReba Sereuli ricxvis wiladad warmodgenis Sedegad?

savarjiSoebi

1 Cawere wiladis saxiT ricxvebi:

1 4 , 5 6 , 10 6 , 9 2 , 4 5 , 24 23 .
5 7 11 9 8 25

2 ramdeni mesamedisagan Sedgeba ricxvi: 12 2 ? 6 1 ? 50 1 ?
3 3 3

3 ramdeni measedisagan Sedgeba ricxvi: 1 21 ? 4 48 ? 6 75 ?
100 100 100

4 axseni Sesrulebuli Canaweris Sinaarsi.

5 4 = 5 + 4 = 45 + 4 = 49 .
9 9 9 9 9

5 Cawere arawesieri wiladis saxiT ricxvebi:

10 9 , 5 6 , 6 6 , 7 7 , 9 4 , 4111050 , 4 23 .
10 7 7 8 25 100

6 ramdeni wuTia a) 1 7 sT? b) 3 11 sT?
60 60

7 ramdeni TeTria a) 1 15 lari? b) 10 70 lari?
100 100

8 ramdeni gramia a) 2 45 kg? b) 11 725 kg?
1000 1000

177

9 Tdgaevmdiosaim3yeS14noakfodileldaaadmgi?samToiiTyeonea.ulSoikoflilada i4stfoillinsawrialmdadendi amyeooTdxaendaimcnaxwvirlisi-

10 manana da dodo erTmaneTis Sesaxvedrad velosipedebiT erTdroulad
gamovidnen. manana 2sT - Si 19km - s, xolo dodo _ 23 km - s gadis.
ra siCqariT uaxlovdebian isini erTmaneTs?

11 sadguridan ori matarebeli erTmaneTis sawinaaRmdego mimarTulebiT
erTdroulad gavida. yovel 4 sT - Si erTi 275 km - s, xolo meore _
241km - s gadioda. ra siCqariT Sordebodnen matareblebi erTmaneTs?

275 km/sT A 241 km/sT
4 sT

12 x - is ra naturaluri mniSvnelobisaTvisaa wiladi wesieri?

a) 4 ; b) 10 7 .
6 _x _x

13 a - s ra naturaluri mniSvnelobisaTvisaa wiladi arawesieri?

a) a 5 ; b) 2 .
+4 a _2

14 romelia meti:

a) 2 Tu 2 ? b) 15 Tu 15 ? g) 65 Tu 65 ?
7 8 11 4 89 84

d) 12 Tu 9 ? e) 25 Tu 26 ? v) 100 Tu 200 ?
19 19 72 72 47 47

15 gamosaxe kilometrebSi:

a) 2km 500 m; b) 6km 400 m; g) 1500 m; d) 2248 m.

16 gamosaxe saaTebSi: g) 95 wT; d) 4sT 15wT.
a) 1sT 20 wT; b) 245 wT;

178

$ 7.9 tolmniSvneliani wiladebis gamokleba

mizani:

naxazze mocemuli kvadrati, mcire zomis 16 viswavloT rogor xdeba
tolmniSvneliani wiladebis
erTnairi kvadratiTaa dafaruli. mocemuli gamokleba. SevZloT wiladis
Tgkvaeafldeirr,aatdxoiesbluo1l86i11a6nakwnvaiawldiilrawitiTigsaeul1f5iae.,r1aa76dmiensbaewmliilix.eidsyvuviilT- 1 - mde damatebis moZebna.

SegviZlia davweroT tolob1a6: 7 + 8 = 15 .
16 16 16
am tolobidan SegviZlia davweroT ori toloba:

15 _ 7 = 8 , 15 _ 8 = 7 .
16 16 16 16 16 16

es tolobebi gviCvenebs tolmniSvneliani wiladebis
gamoklebis wess:

tolmniSvneliani wiladebis gamoklebisas saklebis mricxvels
unda gamovakloT maklebis mricxveli, sxvaoba davweroT
mricxvelad, mniSvneli ucvlelad davtovoT.

am wess asoebiT ase CavwerT:

a _ b = a_ b .
c c c

ganxiluli naxatis mixedviT gaferadebuli da gauferadebeli

nawilebis jami ase Caiwereba:
15 1 16
16 + 16 = 16 .

saidanac SegviZlia davweroT, rom 16 _ 15 = 16 _ 15 = 1.
16 16 16 16
16 15 1 16 SegviZlia 1 miviRebT:
16 _ 16 = 16 sxvaobaSi 16 -iT SevcvaloT,

1 _ 15 = 16 _ 15 = 1 .
16 16 16 16
mnoakxvneeTs insakmvoeTsaixsna13v0i
amocana. nawili. nakveTis ra nawili darCa mosaxnavi?
amoxsna. nawilis gamosaTvlelad unda SevasruloT

gamokleba: 1 _ 3 = 10 _ 3 = 10_ 3 = 7
pasuxi: nakveTis 10 10 10 10 10
7
10 nawili darCa mosaxnavi.

K upasuxe kiTxvebs:

1 rogor SevasruloT tolmniSvneliani wiladebis gamokleba?
2 rogor gamovakloT wiladi 1 mTels?

savarjiSoebi

1 opirrivedlRidsRgeasnmSaevalkeoTbeasS.iggzziissr1a94 nnaawwiilliiSSeeaakkeeTTeess.meaoqerdeandR154esn?awili

2 gogonebi klasSi moswavleTa raodenobis 2 nawils Seadgenen.
5
klasis moswavleTa raodenobis ra nawils Seadgenen vaJebi?

179

3 gdzasisage35bin?awilze daages asfalti. gzis ra nawilze darCa asfalti
4
Seasrule gamokleba:
5
6 41 _ 23 ; 57 _ 49 ; 19 _ 13 ; 78 _ 57 ;
7 50 50 63 63 26 26 87 87
8
9 57 _ 24 ; 14 _ 11 ; 36 _ 3 ; 12 _ 7 .
10 91 91 28 28 49 49 17 17
11
qvevridan amoiRes Rvinis mTeli raodenobis 4 nawili. Rvinis ra
12 nawili darCa qvevrSi? 9
13
Tornikem davalebis 7 nawili Seasrula. davalebis ra nawili darCa
180 Sesasrulebeli? 10

Sesvenebamde 1m54aRnaazwiialmig. aryoiddaiskagratyiodfaimlaRisazmiaarmagmiest1i34karnatwoilfiil, SieSsevsevnee--
bis Semdeg _

nebamde Tu Sesvenebis Semdeg? kartofilis maragis ra nawili darCa

gasayidi?

6000kv. m farTobis mqone nakveTis mesamedi nawili venaxs ukavia, xolo
danarCenis naxevari xexils. ra farTobi ukavia xexils?

romeli sxvaobaa meti:

a) 1 _ 7 Tu 1_ 4 ? b) 1 _ 3 Tu 45 _ 22 ? g) 1_ 64 Tu 5 _ 1 ?
100 100 26 81 81 87 2 2

daalage gamosaxulebebi maTi mniSvnelobis zrdis mixedviT:

1 _ 27 , 1 _ 40 , 1 _ 3 , 1 _ 64 .
50 63 26 87

amoxseni gantoleba:

7 + x = 11 ; 2 + x = 14 ; x + 1 = 6 ;
40 40 25 25 17 17

21 _x = 17 ; 15 _ x = 1 ; 34 _ x= 26 .
38 38 23 23 43 43

sasadiloSi erT dRes kartofilis maragis 5 nawili daxarjes, meore
d114ax-airTjneaskolrebivie, 14 ra na-
dRes vidre pirvel dRes. kartofilis maragis
wili dRes?

gamoTvale gamosaxulebis mniSvneloba.

a) 17 _ 9 + 4 ; b) (1 _ 18 ) _ 20 ; g) 7 + 9 _ 4 ;
56 56 56 63 63 16 16 16

d) 75 + 36 _ 44 ; e) ( 29 + 8 ) _ 23 ; v) 1 _ 7 + 7.
1000 1000 1000 60 60 60 30 30

14 medeam erT dpRiresvewligdniRses2.74wnigawniils irawaniakwiiTlxia,damreoCaremeddReaess 3 nawiliT
meti, vidre 24
wasakiTxi?

15 marim xelTaTmani, qudi da CanTa SeiZina. xelTaTmanSi samive nivTis
1 maqruidmSCia_nT13a6Sin?awili.
Rirebulebis r16a nawili gadaixada, samive nivTis
Rirebulebis nawili gadaixada

16 amoxseni gantoleba:

( )a) 31 +x _ 27 = 11 ; g) (1 _ x) _ 4 = 17 ;
45 45 45 37 37

b) 25 _ ( 12 + x ) = 4 ; d) 70 + x = 56 + 21 .
39 39 39 103 77 77

17 rodesac avzidan benzinis 3 nawili amoiRes, masSi kidev darCa 10 l
4
benzini. ramdeni litri benzini iyo avzSi Tavdapirvelad?

18 rmuamSdebemnaicerwTinaddRResesg,zdiasna41rnCaewniilkii SeakeTes, meore dRes ori imdeni,
mesame dRes. gzis ra nawili Seake-

Tes muSebma mesame dRes?

19 gamoTvale jami:

1 596 + 2 4 ; 3 7 + 5 9 ; 1 + 7 + 4 11 .
56 16 16 30 30

20 gamoTvale im sami ricxvis jami, romelTagan pirvelia 4 1 , xolo
2 5
yoveli momdevno ricxvi winaze 5 - iT metia.

21 erTi saaTis ra nawilia 11 wT? 30wT? 51 wT?

22 1 m - is ra nawilia 43 sm? 50 sm? 25 sm?

23 1 kg - is ra nawilia:
34 g? 500g? 250g? 600g? 6g? 50g? 100mg? 20mg? 4 mg ?

24 erTi kviris ra nawilia 2 dRe?

25 gaizrdeba Tu Semcirdeba ganayofi da ramdenjer, Tu
• gamyofs ucvlelad davtovebT da
a) gasayofs 3 - jer gavzrdiT?
b) gasayofs 5 - jer SevamcirebT?
• gasayofs ucvlelad davtovebT da
a) gamyofs 2 - jer gavzrdiT?
b) gamyofs 3-jer SevamcirebT?
• gasayofsa da gamyofs
a) 3-jer gavzrdiT?
b) 2-jer SevamcirebT?

181

$ 7.10 Sereuli ricxvebis gamokleba

mizani: ayrmiaodmdci.eanndiaRk1ii.slmoagRagnramzaamivialsSoo1b5koa53Slkiga3dS15iokdgkoaSrloCaakdogilashaaqdyoiidndgi?aaigyiadsaa-.

viswavloT Sereuli
ricxvebisa da mTelidan
wiladis gamokleba.

amoxsna. amocanis amosaxsnelad gamovTvaloT

sxvaoba: 1
3 _ 351 3 1 3 5 3_1 2 2
15 5 = (15 + 5 ) _ (3 + 5 ) = (15 _ 3) + ( 5 _ ) = 12 + 5 = 12 + 5 = 12 5 .

pasuxi: maRazias gasayidi darCa 12 2 kg Sokoladi.
5

Sereul ricxvs rom Sereuli ricxvi gamovakloT, saWiroa mTel
nawils gamovakloT mTeli nawili, wilad nawils _ wiladi nawili
da miRebuli Sedegebi SevkriboT.

amocana 2. giorgim 2 sazamTro SeiZina. erTi sazamTro 7 3 kg-s iwonida,
4
orive erTad _ 11kg - s. ramden kilograms iwonida meore sazamTro?

amoxsna. imis gasarkvevad, Tu ramden kilograms iwonida meore sazam-
4j3 a;mis
Tro, unda SevasruloT gamokleba: 11 _ 7 saxiT, xolo 1 mTeli _
11 mTeli warmovadginoT 10 -isa da 1-is
4
4 - is saxiT. miviRebT:

11 _ 7 3 = (10 + 1) _ 7 3 = (10 + 4 ) _ 7 3 = (10 _ 7) + ( 4 _ 3 )=3 + 4 _ 3 = 3 + 1 = 3 1 (kg).
4 4 4 4 4 4 4 4 4

pasuxi: meore sazamTro 3 1 kg -s iwonida.
4

K upasuxe kiTxvebs

1 rogor gamovakloT Sereul ricxvs Sereuli ricxvi?
2 rogor gamovakloT mTel ricxvs wiladi?

savarjiSoebi

1 evrelTSti.ormaamrdaeSnii1k0i1l72okggrbarmiinbjriian,jmieaormeeoSire11t52okmga-riaTSin?aklebi, vidre pir-

2 nnaaTTiiaaz3e8?18050 kg-s iwonis, eka _ 40 kg-s. ramdeni kilogramiT mZimea eka

3 M wertili AB monakveTs ekuTvnis. AB monakveTis sigrZe 12 sm-is, xo-
7
lo MB monakveTis _ 8 12 sm-is tolia. gamoTvale AM monakveTis sigrZe.

4 O wertili AB monakveTs ekuTvnis. AO monakveTis sigrZea 9 13n09a0klm,exboia-.
lo OB monakveTis sigrZe AO monakveTis sigrZeze
gamoTvale AB monakveTis sigrZe. 2 14 m-iT
100

182

5 gamoTvale:

8 9 _ 4 5 ; 19 15 _ 11 9 ; 14 21 _6 17 ;
13 13 17 17 60 60

45 4 _ 24 ; 56 69 _ 49 53 ; 100 _ 99 1 ;
35 100 100 2

18 11 _ 4 5 ; 48 23 _ 45 19 ; 15 _ 15 _ 12 ;
26 26 36 36 27

31 _ 24 5 ; 80 43 _ 74 40 ; 4 125 _ 2.
48 75 75 136

6 amoxseni gantoleba: 1 7 13
3 14 18 18
6 14 _ x = 2 ; 11 + x = 19 ;

x + 4 5 = 12 14 ; 7 29 _ x = 5 11 .
21 21 30 30

7 gamoTvale gamosaxulebis mniSvneloba:

(10 _ 5 3 ) _ 3 4 ; 4 + 6 17 _ 9 11 ;
16 16 18 18

4 5 _ 5 _ 1 15 ; 7 29 + (9 _5 7 ).
22 22 22 34 34

8 rfia.sdraakmldeenbiamldearailTboCammio6af160aselsaarli bRoimri?da, fasdaklebis Semdeg 5 5 la-
10

9 peirrTvemlanzqea.narzaemd3e1n70i5000totnattvviirrTTiiaa, moeroirveezmean14q05a0n00azte?- iT naklebi, vidre

10 4sa15mlb-iidTonnaSkil1e2b15i,lvizderTeipai.rpvierlvSeil. bidonSi 7 2 l zeTia, xolo meoreSi
5 mesame bidonSi?
ramdeni litri zeTia

11 naxazze miTiTebulia detalis zomebi santimetrebSi. gamoTvale x - is
mniSvneloba.
8

x 2

1 4
5

12 smeaom rbeisdaodnSaim1e2sa45melSir_Ze7a45. pirvel da meore bidonSi 7 1 l rZea, xolo
5
l. ramdeni litri rZea TiToeul bidonSi?

13 Cawere arawesieri wiladis saxiT:

7 1 , 12 4 , 1 2 , 3 250 , 7 4 .
5 5 5 1000 5

183

$ 7.11 wiladis sididis cvlis sxvadasxva SemTxveva

mizani: gavarkvioT ra cvlilebas ganicdis wiladi misi

a) gavarkvioT, rogor mricxvelis radenjerme gazrdiT an SemcirebiT.
icvleba wiladis sidide
misi mricxvelis an mniS- rogorc viciT, wiladi SegviZlia ganayofis sa-
vnelis ramdenjerme ga-
didebiT an SemcirebiT. xiT CavweroT. am CanawerSi wiladis mricxveli ga-
b) viswavloT wiladis
naturalur ricxvze gam- sayofs, xolo mniSvneli gamyofs warmoadgens.
ravlebis wesi da gamo-
viyenoT gamoTvlebSi. magaliTad,
4
d9a =4 : 9. amitom, mricxvelis ramdenjerme gazr-
an Semcireba gasayofis imdenjerve gazrdas an

Semcirebas niSnavs. es ki ganayofis, anu wiladis

sididis imdenjerve gazrdas an Semcirebas gamoi-

wvevs.

maSasadame, Tu wiladis mricxvels ramdenjerme gavadidebT,

maSin wiladis sidide imdenjerve gadiddeba.

Tu wiladis mricxvels ramdenjerme SevamcirebT,
maSin wiladis sidide imdenjerve Semcirdeba.

agamSmmaeagogvgaavxardlisniZnxiicldaiTx.loiiv2auT9i01sld.3ra329vm-o0-aajmzsggekae3arvlvng-.aiaajdTTmeriS:rdiavog2l9Ta0ve9gab32daa×0-vsi3jadnde=oiirSTd22.n,e07avTas=m.i31se-2a.7j0Tie.v;ri2s9.0rm×ii3scix=vm9ir2s×i0c33x-.vjeleri unda
gadideba

wiladis naturalur ricxvze gasamravleblad sakmarisia wiladis
mricxveli gavamravloT am ricxvze, miRebuli namravli davweroT
mricxvelad, mniSvneli ucvleli davtovoT.

magaliTi 2. 9 SevamciroT 6 - jer.
20
agmaovaxdsnida.o2T90
magaliTi 3. rom 6 - jer SevamciroT, amisaTvis misi mniSvneli unda
b) 3
6 - jer. 9 = 9 .
20 × 6 120
2 10 .
gamovTvaloT namravli: a) 7 × 11; × 43

amoxsna. a) 2 × 11 = 2 ×11 = 22 = 317 ; b) 3× 10 = 3 ×10 = 30 .
7 7 7 43 43 43

naxazze mocemulia sami toli wre.

4 4 4
16 8 4

mxrolipcioxrvmveeelslaezmbeieztwero_ils44i1a46n, axwnoiallwioil, imaanniugSavmfnTeeelrlaedibiewbrsuxelv. ai1d4,6a,msex4o8var.dedazaev44a_kwv84iilrnadadweieTlbiaism,

wiladebis Sesabamisi nawilebis sididis cvlas naxazze.

184

ram)cdwiirrTiesaw14i6l48naa-dwziiles.iasoqiedrdijadenervaomscrkivjrneierTa ,wmrrcioisrm 4d8weniblaaw.aidlmiozsZee.mbnSnieeSsvaanbmealmidisasasdko,vrn1ij46serodragjsaazebr--
uTebis sxva gza.
1bb46i) Tw-rzwieisl4 -a44djiensraw.siialqdeiidd4ae-nj4ve-arjskevmrneitiTzi,arrdworembiasw.i14l6 andaiwsilmnzieS.vnSeelsaibsa4mi-sjaedr, S44emcmeitriea-

ramdenjerac wiladis mniSvnels SevamcirebT, wiladis sidide
imdenjer gadiddeba da piriqiT, ramdenjerac wiladis mniSvnels
gavadidebT, wiladis sidide imdenjer Semcirdeba.

K upasuxe kiTxvebs:

1 rogor Seicvleba wiladis sidide, Tu mis mricxvels
ramdenjerme gavzrdiT? ramdenjerme SevamcirebT?

2 rogor gavamravloT wiladi naturalur ricxvze?
3 rogor Seicvleba wiladis sidide, Tu mis mniSvnels

ramdenjerme gavzrdiT? ramdenjerme SevamcirebT?
4 rogor gavadidoT wiladi ramdenjerme?
5 rogor SevamciroT wiladi ramdenjerme?

savarjiSoebi

1 gaadide 2 - jer: a) 1 ; b) 2 ; g) 1 .
3 5 2

2 Seamcire 3 - jer: a) 6 ; b)15 ; g) 9 .
11 26 22

3 naxatis mixedviT axseni Canaweris Sinaarsi:

6 × 2 = 6×2 = 12 = 1
24 24 24 2

4 romelia meti da ramdenjer:

a) 3 Tu 12 ? b) 6 Tu 2 ? g) 8 Tu 16 ?
5 5 10 10 8 16

5 AB monakveTis ra nawilia: AD monakveTi? AC monakveTi? KT monakveTi?

AC D M nax. 1 E K TB

6 daasaxele nax. 1 - is mixedviT AB monakveTze
a) 3 - jer; b) 7 - jer; g) 21 - jer mcire monakveTi.

185

7 daasaxele nax. 1 - is mixedviT
a) AC monakveTze 20 - jer; b) AD monakveTze 7 - jer;
g) AM monakveTze 3 - jer; d) DM monakveTze 2 - jer didi monakveTi.

8 naxatis mixedviT upasuxe kiTxvebs:
a) marTkuTxedis ra nawilia gaferadebuli wiTlad?
lurjad?
b) romeli nawilia meti: wiTeli Tu lurji? ramdenjer?

9 Seasrule gamravleba:

a) 3 × 8; b) 6 × 40; g) 5 × 16 ; d) 12 × 10 ;
5 10 80 49

e) 9 × 9 ; v) 6 × 11 ; z) 8 × 4 ; T) 30 × 4 .
20 17 11 9

10 ekam maTematikis davalebis mSoesarnduolmeabadsav13alseTbimsoaSnedsroumal, egbiaasm 2 - jer
meti, vidre _ ekam. ra dro giam?

11 reraTdiroonksaWniirTdeabvza iav11z45issTga-vSsiebiavssembeao, rmeeoornikTani2T-?jer ufro swrafad.

12 daasaxele a) 2 -ze; b) 3 -ze; g) 1 -ze 3 - jer mcire ricxvi.
3 5 2

13 wiladis mricxvelis Seucvlelad gaadide wiladebi 3 - jer:

a) 5 ; b) 2 ; g) 1 ; d) 21 .
6 15 72 81

14 wiladis mniSvnelis Seucvlelad Seamcire wiladebi 2 - jer:

a) 2 ; b) 8 ; g) 40 .
54 24 84

15 daadgine gamoTvlebis gareSe, romelia meti? ramdenjeraa meti?

a) 19 Tu 19 ; b) 12 Tu 12 ; g) 84 Tu 84 ;
5 15 10 40 85 17

16 ipove 2 - ze: a) 3-jer didi ricxvi; b) 3-jer mcire ricxvi.
3

17 ipove 3 - ze: a) 3 - jer didi ricxvi; b) 3 - jer mcire ricxvi.
4

18 orma samodelo firmam erTnairi SekveTa miiRo. pirvelma garkveul
fSeikrvmeaTmisSe4a70srnualwialmie,txiolsaomumSeaoor?erm aimmdaveenjderromSeit_i?2401
droSi Seasrula
nawili. romelma

19 gamoTvale gamosaxulebis mniSvneloba:

a) 11 × 15 + 11 _ 7 ; b) 25 _ 5× 3 ; g) 12 × (2 15 + 1 7 _ 3 15 ).
15 60 60 32 32 16 16 16

20 mmgaaanermZiairlATipanueAnn.qptSueinxdqvatendi,draaxmnodBle opmaurvnaiqnmto31amBkdmepg?uaniqartai,dvaannoemrTmamrainzeTe i2s-jSeersamxevtedi.rraad

186

21 mgaarmoTTkuvaTlxeedmiasrTsikguaTnexe11d12issmp-eirai, mxeotlroi. sigrZe siganeze 6-jer metia.

22 gamoTvale: a) 5 × 22; b) 154× 3 3.
14

23 n naturaluri ricxvia. romelia meti da ramdenjer:

a) 25× n Tu 25 ? b) 12 Tu 12 ? g) 10×n Tu 5×2n ?
42 42 5 5×n 67 67

24 ra ricxvi unda eweros `* ~ - is nacvlad toloba rom swori iyos?

a) 1 = * ; b) 2= * ; g) 5 = * ; d) 4= * ;
5 10 7 25

e) 11 = * ; v) 100 = * ; z) 24 = * ; T) 10 = *.
9 2 9 5

25 daalage zrdis mixedviT wiladebi:

a) 18 , 18 , 18 , 18 , 18 , 18 ;
51 17 31 65 29 26

b) 11 , 1 , 7 , 52 , 3 , 18 .
25 25 25 25 25 25

praqtikuli samuSao

saWiro masala: saxazavi, furceli, 2 feradi fanqari, makrateli.

1) daxaze marTkuTxedi, romlis sigrZe 16 sm - is,xolo sigane 1 sm-is

toli iqneba.

2) marTkuTxedi dayavi 16 tol kvadratad da kvadratebi or ferad

gaaferade ise, rom mezobeli nawilebi erTi feris ar iyos. markuTxe-

dis ra nawils warmoadgens TiToeuli feris nawili?

3) amoWeri 2 kvadrati. marTkuTxedis ra nawili amoWeri?

4) amoWeri 4 kvadrati. marTkuTxedis ra nawili amoWeri?

5) Seadare erTmaneTs amoWrili pirveli da meore nawilebi zeddebiT.
rraommdeelnijaermeata ime1t26iT146u 146?
- is ramdenjer?

mricxveli 2 - is mricxvelze?
16
Camoayalibe daskvna.

6ai)g) iba18m)6moga1T8r)Wd6eu1T)8-r6k11ii82u66sTTm?m-uaxrirres1i4Tda6cmmki?xrdusvierTe1cnl2ax6jxmeivdeddereTilanasujai1461enT186r2a6unkanala-wa14iwien6silbali-kme?liribsseid.bcmairrx?vieoecrmlxTevlime?ailnareianT?mazrdkelaemnzedjbeeiedn:rdjaeeabrniaTaaklnSaeekabldie?abrie?
Camoayalibe daskvna.

187

$ 7.12 wiladis ZiriTadi Tviseba. wiladis Sekveca

mizani: viciT, rom Tu gasayofsa da gamyofs gavamrav-

viswavloT wiladis lebT nulisagan gansxvavebul ricxvze, ganayofi
ZiriTadi Tviseba da ucvleli darCeba. magaliTad, 12 : 3 = 36 : 9 Tu am
SevZloT misi gamoyeneba
wiladis Sesakvecad. tolobaSi ganayofebs wiladebiT SevcvliT,

miviRebT: 12 = 36
3 9

11
22

11 1 1
44 4 4

11 1111 11

88 8888 88

22 22
88 88
naxazze sami toli marTkuTxedia mocemuli. masze kargad Cans, rom
1 2
2 = 4 , (21 - is mricxveli da mniSvneli gamravlebulia 2 - ze, Sedegad
1 - is toli wiladi
miiReba 2 _ 2 );
4
84bmgp21iiaa=rid48=Ri21mqe-84ii,biiT,as(R48(:12te12b-48-o-aiil=ssi48si24mt-rmwoir,iisllci(48tcixavxod-wevliiileslli_imaiw21rddi)diial.ca_maxdnmv84ineiSi)l;Sv_inven24dle)la;iimgngaiyaSomvrnfeailvlliieagbau4y-lozfieai, l4Sei-dazee2g,-aSdzeedm, eiSigeRaddee--
rogorc vxedavT,

wiladis sidide ar Seicvleba, Tu mis mricxvelsa da mniSvnels
gavamravlebT an gavyofT nulisagan gansxvavebul erTsa da imave
ricxvze.

wiladis am Tvisebas wiladis ZiriTadi Tviseba hqvia.

es Tviseba saSualebas gvaZlevs, wiladi SesaZlo umciresi mricxve-

liTa da mniSvneliT CavweroT.

magaliTad: 100 = 100 : 100 = 1 ; b) 240 = 240 : 60 = 4 .
a) 1000 1000 : 100 10 300 300 : 60 5

wiladi misi mricxvelisa da mniSvnelis saerTo gamyofze gayofiT

mis tol wiladad gardavqmeniT. am SemTxvevaSi vambobT, rom:
a) w1i10l000a0dSiesvkSveekcveeTca1S0e0i-Zzlee; ba Tabn)d23a40T00aSneovbkivTecSeeTva6s0r-uzleo. T.

magaliTad: 480 = 48 = 12 = 4 .
360 36 9 3
mivu43i68mR00jeoSTebvuekskviveaec,ceTiTawvjiilderaand1vie0_-mz34oe.i, ZSeebmndoesgm4r-izcexdvealbiosla odsa
_ 3-ze, ris Sedegad
mniSvnelis udidesi

saerTo gamyofi da wiladi masze Seikvecos. am SemTxvevaSi SegveZlo

wiladi Segvekveca 120 -ze (10 × 4 × 3 = 120).

480 = 480 : 120 = 4 .
360 360 : 120 3

188

K upasuxe kiTxvebs:

1 raSi mdgomareobs wiladis ZiriTadi Tviseba?

2 ras niSnavs wiladis Sekveca?

3 rogor wilads ewodeba ukveci wiladi?
4 iqneba Tu ara wiladi ukveci, Tu misi mricxveli da

mniSvneli gansxvavebuli martivi ricxvebia?

savarjiSoebi

1 dawere ramdenime wiladi, romelic tolia

a) 1 - is; b) 3 - is; g) 2 - is; d) 11 - is.
2 4 7 15

2 Secvale mocemuli wiladi iseTi wiladiT, romlis mniSvnelia 48.

1 ; 2 ; 3 ; 5 ; 3 ; 7 .
2 3 4 6 8 12

3 Secvale mocemuli wiladi iseTi wiladiT, romlis mniSvnelia 36.

1 ; 1 ; 3 ; 5 ; 7 ; 11 .
2 3 4 9 18 12

4 eqskavators unda gaeTxara 500 m sigrZis Txrili. pirvel dRes gaTxara
200 m Txrili, meore dRes samuSao mTlianad daasrula. samuSaos ra

nawili Sesrulda pirvel dRes? meore dRes (pasuxebi Cawere ukveci

wiladebis saxiT)?

5 gayofadobis niSnebis gamoyenebiT gansazRvre ra ricxvze Seikveceba
wiladi:

a) 205 ; b) 304 ; g) 2420 ; d) 135 ; e) 2870 ; v) 405 .
2000 698 14700 450 4100 3420

6 Sekvece wiladebi:

a) 20 ; 15 ; 36 ; 25 ; 72 ; 12 ; 22 ; 25 ; 11 .
120 21 40 35 18 102 88 625 121

b) 45 ; 27 ; 36 ; 3 ; 100 ; 80 ; 240 ; 93 ; 105 .
99 90 72 33 1000 640 960 124 140

7 dawere ukveci wiladis saxiT:

a) 20 ; 21 ; 14 ; 42 ; 40 ; 128 ; 315 ; 123 ; 345 .
40 35 35 63 90 160 945 450 210

b) 16 ; 21 ; 27 ; 26 ; 196 ; 360 ; 180 ; 306 .
56 49 72 1300 144 430 108 234

8 `* ~ Secvale ricxvebiT ise, rom swori toloba miiRo:

a) 1 = 2 = 4 = 6 = 8 = 10 ; g) 4 = 16 = * = 80 = * = 100 ;
5 * * * * * 9 * 27 * 63 *

b) 2 = * = * = * = * = * ; d) 5 = 10 = * = 60 = * = 35 .
7 14 28 70 42 77 6 * 30 * 48 *

189

9 mgzavrma gzis 7 nawili gaiara.
20
10 a) gzis ra nawili darCa mgzavrs gasavleli?
11
12 b) romeli manZilia meti _ gavlili Tu gasavleli?
13
14 g) gaiara Tu ara mgzavrma mTeli manZilis naxevari?
15
ukveci wiladis saxiT warmoadgine saaTebSi: v) 90 wT.
16 a) 20 wT; b) 30 wT; g) 14 wT; d) 15 wT; e) 75 wT;
17
18 ukveci wiladis saxiT warmoadgine tonebSi:
19 a) 450 kg; b) 60 kg; g) 920 kg; d) 5 kg; e) 1025kg; v) 3300kg.
20
ukveci wiladis saxiT warmoadgine kilogramebSi: v) 5550 g.
21 a) 30 g; b) 150 g g) 10 g; d) 500 g; e) 1200 g;
22
ukveci wiladis saxiT warmoadgine santimetrebSi: v) 45 mm.
190 a) 5 mm; b) 8 mm; g) 2 mm; d) 4 mm e) 15 mm;

qvemoT CamoTvlilTagan SearCie toli wiladebi:

16 , 125 , 3 , 45 , 70 .
48 225 9 63 245

daiyvane wiladebi tol mniSvnelebze. daalage zrdis mixedviT.

a) 9 ; 2 ; 36 ; 40 ; 50 ; 700 .
10 20 30 40 100 1000

b) 1 ; 12 ; 10 ; 25 ; 16 ; 18 .
12 24 60 60 48 72

marTkuTxedis formis ezos sigrZea 100 m, sigane _ 90 m. gogim am ezoSi
marTkuTxedis formis saTavso aaSena, romlis sigrZe 5 m, xolo sigane
_ 4 m - ia. ezos farTobis ra nawili daikava saTavsom?

ricxviT sxivze aCvene, rom erTi da igive wertili Seesabameba:
1 4 2 6
a) 2 - sa da 8 - s; b) 3 - sa da 9 - s.

cnobilia, rom a -=s?25 da a < 7. ra naturaluri mniSvnelobebis miReba
SeuZlia a -s da
bb

samma zeinkalma 192 detali daamzada. orma maTganma Tanabari raode-
nobis detali daamzada. mesamem ki zustad imdeni, ramdenic danarCenma
orma erTad. ramdeni detali daamzada TiToeulma zeinkalma?

nimuSis mixedviT dawere mocemuli wiladis uaxloesi naturaluri

ricxvebi: 2 < 12 < 3
5

a) ?< 17 < ? b) ?< 4 2 <? g) ? < 1478 < ?
5 5 1000
16 569 4
d) ? < 7 < ? e) ?< 55 < ? v) ?<1 7 <?

amoxseni gantoleba: a) 3x + 1 + x = 45; b) 6x + 6 = 37 × 3 × 8.

ra ricxvi unda eweros x - is nacvlad toloba rom swori iyos?
x x x
a) 5 = 8; b) 11 = 2; g) 10 = 20; d) 85 = 17; e) 121 = 1452 .
x x

$ 7.13 wiladebis Sedareba

naxazze moniSnuli wertilebi Seesabameba ricxvebs: mizani:

1, 5, 1, 2, 3 , 1 5, 2 1 , 2, 3. viswavloT
3 12 2 3 2 6 2 wiladebis Sedareba.
A
11
32

ga31nl0, a1g52eb, u15122l,i23a23OwA1esmioenraikv23weiTli1as65dOe2bdisa 2 1 3
2

Sesabamisi wertilebi ricxviT sxivze

A boloebs Soris, anu 1- is marcxniv.

wesieri wiladi 1 - ze naklebia.

arawesieri wiladebisa da Sereuli ricxvebis Sesabamisi wertilebi

ricxviT sxivze A wertilis marjvnivaa. aqedan vaskvniT, rom yoveli
arawesieri wiladi da Sereuli ricxvi wesier wiladze metia. magaliTad:

4 > 5, 1 < 5 , 3 > 54 , 20 < 1 5 .
3 12 2 3 2 96 21 6

gansxvavebuli mTeli nawilebis mqone Sereuli ricxvebis Sesadare-

blad sakmarisia SevadaroT maTi mTeli nawilebi. metia is Sereuli

ricxvi, romlis mTeli nawilic metia. Tu mTeli nawilebi tolia, ma-

Sin unda SevadaroT maTi wiladi nawilebi. metia is, romlis wiladi

nawilic metia. magaliTad: 1 2 1 2
9 5 5 5 5 5
4 35 > 2 12 , radgan 4 > 2; 3 < 3 , radgan < .

magaliTi 1. romelia meti: a) 3 Tu 4 ? 125 Tu 3
14 7 b) 500 100 ?
viciT, rom tolmniSvneliani wiladebidan metia is wiladi, romlis

mricxvelic metia. amitom mocemuli wiladebi toli mniSvnelebiT unda

CavweroT, anu unda gavaerTmniSvnelianoT da Semdeg miRebuli tolm-

niSvneliani wiladebi SevadaroT.
vnae) l134i dmiav47iywvialnoaTde1b4is- mgdaeer. aTmminsiaSTvvniesl7iaunnedbiasgaTavvaimsrsaavklmoarTis2i-az47e.-wiislmandiiSs-

sidide rom ar Seicvalos, unda gamoviyenoT wiladis ZiriTadi Tviseba
dlabi)am51rm20ii05vcidxyvvaeal1n0o3i0cTw21i0l-130z4a-edm<egdba1iev8s.4a51ma02Srm05emasam=vSaild1az2s0ao5na0riTd.Tea.bml51em02,ai50dv1i34RS15e02e<v50bkTvw74:eicl.47oadT=i1s584-mzn.ei,SvmnievliRieSbeTg:viZ-

3 < 25 maSasadame, 3 < 125 .
100 100 100 500

sxvadasxvamniSvneliani wiladebis Sesadareblad isini tol
mniSvnelamde unda miviyvanoT da miRebuli tolmniSvneliani
wiladebi SevadaroT.

wiladebis Sedarebisas, zogjer tol mricxvelebamde miyvana ufro

mmoagsaaalxmeirSTeximeTb2e.xlvSeeivvaaa.SdiaruomjTowbielsiaadweibli 1a3d7 edbia 6. mricxvelemde miviyvanoT:
t31ol

3 = 6 < 6 .
17 34 31

191

1 K upasuxe kiTxvebs:
2
3 rogor vadarebT ricxvebs ricxviTi sxivis gamoyenebiT?
4 romelia meti _ wesieri wiladi Tu arawesieri wiladi?
5 romelia meti _ naturaluri ricxvi Tu wesieri wiladi?
6 romelia meti _ wesieri wiladi Tu Sereuli ricxvi?
7 rogor SevadaroT tolmniSvneliani wiladebi?
8 rogor SevadaroT sxvadasxvamniSvneliani wiladebi?
rogor SevadaroT tolmricxveliani wiladebi?
1 ratom vamjobineT meore magaliTSi wiladebis tol
mricxvelebamde miyvana?
2
savarjiSoebi
3
romelia meti:
4
a) 3 Tu 4 ? b) 8 Tu 14 ? g) 5 Tu 2 ? d) 11 Tu 13 ?
5 5 15 9 15 6 3 14 15

6 aCvene ricxviT sxivze, rom

7 a) 3 < 3 ; b) 8 > 7 ; g) 1 = 5 ; d) 1 > 3 .
10 5 10 10 2 10 2 10
8
daxaze 16 ujrisagan Sedgenili kvadrati da aCvene, rom
9
10 a) 7 < 1 < 9 ; b) 8 = 4 = 1 ; g) 17 > 1; d) 5 < 3 .
11 16 2 16 16 8 2 16 16 8
12
daalage zrdis mixedviT ricxvebi:

a) 5 , 3 , 1, 5 ; b) 5 , 2 , 11 , 2 1 , 31 .
7 7 14 6 3 12 2 6

daalage klebis mixedviT ricxvebi:

a) 4 , 1 3 , 12 , 1; b) 6 , 12 , 2 1 , 4.
5 5 20 60 30 20 20

mnaektvieTfiasrT52 onbaiwiulkzaveisai,msiondioias daTesili, xolo 3 nawilze soio. romels
Tu siminds? 8

wnaaTikiiaTmxearnTaTdiaRmemsetwiig, npiisrv25e4lndawRielsiTwuaimkeioTrxea, dmReoesr?e dRes _ 7 . rodis
12

sgaanzdisrogmadxaeslaxfaadsiisg1a13daniaxwaidla.iTroemleelqitrgaodeanseraxgaidiis,gxaodlaioxa1d25a nawiliT
sandrom

naklebi, gazis Tu eleqtroenergiis?

ra naturaluri mniSvnelobebis miReba SeuZlia a - s, Tu 5 9 <a<8 5 ?
16 9

dawere 2 -ze didi da 4 -ze mcire wiladi, romlis mniSvnelia 15.
3 5

dawere 3 -ze didi da 5 -ze mcire wiladi, romlis mniSvnelia 24.
4 6

axseni, ratomaa toloba WeSmariti? a) 3 = 60 ; b) 40 = 5 .
4 80 64 8

192

13 gamoiyene qvemoT mocemuli cxrili da gamosaxe:
a) metrebSi: 5 km; 2 km 580 m; 8 km 240 m; 5 km 50 m ; 1 km 9 m.
b) kvadratul metrebSi: 2 km2; 4 km 2; 10 km 2.
g) heqtrebSi: 2 km 2; 4 km 2; 10 km 2.
d) kvadratul santimetrebSi: 5 m 2; 10 m 2; 50 m 2.

sigrZis sazom erTeulebs farTobis sazom erTeulebs
Soris Tanafardoba
Soris Tanafardoba
1 km = 1000m
1 m = 10 dm 1 km2 = 1000 2m 2 = 1 000 000 m 2
1 m = 100 sm 1 m 2 = 10 2dm 2 = 100 dm 2
1 sm = 10 mm 1 m 2 = 100 2sm 2 = 10 000 sm 2
1 sm 2 = 10 2mm 2 = 100 mm 2
1km 2 = 100 ha
1 ha = 100 2m 2 = 10 000 m 2

14 marTkuTxedis formis nakveTis farTobia 24 ha, xolo sigane – 320 m.
gamoTvale nakveTis sigrZe.

VII Tavis damatebiTi savarjiSoebi

1 Rilebis ra nawilia wris SigniT? gareT? wrewirze?

a) b)

2 gaSifre sityva gaferadebuli nawilebis sididis zrdis mixedviT.

be ad

3 gaSifre sityva gaferadebuli nawilebis sididis klebis mixedviT.

l oi d

4 sagnebis ra nawilia Semoxazuli TiToeul naxatze?

193

5 marTkuTxedi dayofilia sxvadasxva feris patara, tol marTkuTxedebad.
marTkuTxedis ra nawilia wiTlad gaferadebuli? yviTlad? lurjad?
marTkuTxedis ra nawilia gauferadebeli?

6 a) figuris ra nawilia wiTlad gaferadebuli?
b) figuris ra nawilia yviTlad gaferadebuli?

a)

b)
7 figuris ra nawilia gaferadebuli?

#1 #4 #5
#3 d) 100 : 15.
8
9 #2
10
dawere Sereuli ricxvis saxiT:
11 a) 17 : 5; b) 29 : 7; g) 60 : 16;
12
13 dawere Sereuli ricxvis saxiT wiladebi: 11 ; 26 ; 52 ; 68 .
194 5 11 28 14

gadaixaze da aRadgine cxrilis gamotovebuli monacemebi.

meoredi mesamedi meeqvsedi

60
48 16
126

ra cifri unda ewros 487 wiladis mniSvnelSi `* ~ -is nacvlad wiladi
4*8
rom: a) wesieri b) arawesieri iyos?

Cawere 1 - is toli wiladi, romlis mniSvnelia: a) 5; b) 100; g) 1200.

3 m sigrZis Toks jer naxevari, Semdeg ki darCenili Tokis mesamedi
CamoaWres. ra sigrZis Toki darCa?

14 daasaxele 13 - ze meti, magram 15 - ze naklebi wiladi.
20 20

15 daxaze marTkuTxedi, romlis gverdebis sigrZeebia 4 sm da 6 sm.
m1ar?TrkaumTdxeendjiesra61a _1
a) gaaferade wiTlad nawili, lurjad 3 nawili.
b) ricxvis 1 metia Tu meti?
36

16 wiladi Sekveces:
a) jer 2-ze, Semdeg 5-ze. ra ricxvze SeukveciaT wiladi?
b) jer 3-ze, Semdeg 2-ze, Semdeg 7-ze. ra ricxvze SeukveciaT wiladi?

17 wiladi 15-ze ikveceba. is 3-ze ukve Sekveces. kidev ra ricxvze Seik-
veceba es wiladi?

18 wiladi 68-ze ikveceba. is 4-ze ukve Sekveces. kidev ra ricxvze Seik-
veceba wiladi?

19 gamoTvale gamosaxulebis mniSvneloba:

a) 3 1 + 5 13 _ 2 6 ; b) 10 + 15 + 7 ;
14 14 14 13 13 13

g) 14 8 + 4 5 _ 3 ; d) 15 + 3 7 + 6 6 .
9 9 9 19 19 19

20 Sekvece wiladi:

16 ; 15 ; 18 ; 20 ; 24 ; 34 ; 36 ; 16 ; 60 ; 100 ; 300 .
20 40 63 15 100 51 54 36 96 125 450

21 gaadide 3 - jer da miRebuli wiladi Cawere ukveci wiladis saxiT:

13 ; 5 ; 11 ; 8 ; 9 ; 70 ; 20 .
21 6 12 30 15 90 27

22 romelia meti: b) 7 Tu 7 ? g) 4 Tu 25 ?
a) 5 Tu 5 ? 12 7 12 6

32 31 e) 4 Tu 5 ? v) 3 1 Tu 23 ?
93 2 6
d) 25 Tu 7 ?
95

es sainteresoa!

ricxvebis saxelwodebebi qarTuli anbanuri numeraciis mixedviT:
10 _ aTi
100 _ asi

1 000 _ aTasi
10 000 _ bevri
100 000 _ bevris - bevri
1 000 000 _ uSqari
10 000 000 _ uSti
100 000 000 _ uStis - uSti

195

kursis gasameorebeli masala

1 dawere oTxi ricxvi Semdegi wesiT:
a) pirveli ricxvia 11, xolo yoveli wina ricxvi 5 -jer pataraa

momdevno ricxvze;

b) pirveli ricxvia 250, xolo yoveli momdevno ricxvi 4-jer metia

winaze;

g) pirveli ori ricxvia 1 da 1, xolo yoveli momdevno ricxvi wina

ori ricxvis jams warmoadgens;

d) pirveli ori ricxvia 1 da 2, xolo yoveli momdevno ricxvi wina

ori ricxvis namravls warmoadgens.

2 samkuTxedis gverdebis sigrZeebia: 5sm, 12sm, 13sm. gamoTvale iseTi
samkuTxedis perimetri, romlis gverdebis sigrZeebi mocemuli samkuTxe-
dis gverdebis sigrZeebze 2 -jer metia.

3 dawere gamosaxuleba jamis an sxvaobis saxiT.
a × (b +5) 4 × (x +8) 6 × (6c _15)
106 × (10c _5) (7 _ b) × 12
16 × (9 _b) (b _27) × 11 (9a +15) × 5
(2a +4) × 6 (26 +2b ) × 4
100 × (b +10000) 5× (y +54)

4 dawere gamosaxuleba namravlis saxiT. 3a + 6a +a
34a _ 17a 15x + 5x +x
3x + 4x 12 +6n 65n _ 15n 25x +25a
2x +x +3x 81m +27n 54a _ 27a 8c _16c
4 + 4a 5x +11x 66c _ 18c
3a _ 6 3a +25a

5 oTxma samSeneblo kompaniam mSeneblobaze 16 788 600 lari daxarja.
erTma kompaniam 4 145 500 lari daxarja, meorem _ 125 500 lariT nak-
lebi, vidre _ pirvelma, xolo mesamem _ 216 800 lariT meti, vidre
meorem. ramdeni lari daxarja meoTxe kompaniam?

6 erT sacavSi Seitanes 22 450 kg kartofili, 10 t kombosto da 1 870 kg
Warxali. meore sacavSi _ 18 450 kg kartofili, 11 500 kg kombosto da

2 t Warxali. romel sacavSi meti bostneuli Seitanes _ pirvelSi Tu

meoreSi?

7 mocemulia sami monakveTi AB, CD da MN. cnobilia, rom AB = CD da
CD = MN. ra SegiZlia Tqva AB da MN monakveTebze?

8 daxaze: a) saerTo wveros mqone maxvili da marTi kuTxeebi;
b) saerTo gverdis mqone maxvili da blagvi kuTxeebi.

9 daxaze saerTo gverdis mqone ori maxvili kuTxe ise, rom maT Seadginon:
a) maxvili kuTxe; b) marTi kuTxe; g) blagvi kuTxe.

10 wignebis maRaziis vitrinis 7 Taroze wignebi Semdegi wesiT awyvia:
pirvel Taroze 5 wigni, meore Taroze _ 8 wigni, mesameze _ 11 wigni,

meoTxeze _ 14 wigni da a. S. ramdeni wigni devs me -7 Taroze?

196

11 Ria onkanidan wuTSi daaxloebiT 10 litri wy-
ali Camoedineba. maris onkanis daketva daavi-
wyda, rac mxolod naxevari saaTis Semdeg gaax-
senda. ramdeni litri wyali daiRvara maris
daudevrobis gamo?

12 norCi ekologis wris wevrebma: elenem, givim, salomem da kaxam daamzades
banerebi warweriT `gaufrTxildi tyes~. elenem yvelaze meti baneri
daamzada, kaxam _ yvelaze naklebi. gogonebma meti baneri daamzades
Tu vaJebma?

13 norCi ekologis wris wevrebma: liam, makam da Tornikem 7 baneri
daamzades warweriT `nu daanagvianeb Sens sofels~ . yvelaze meti bane-
ri liam daamzada, yvelaze naklebi _ makam. vin ramdeni baneri daamzada.

14 norCi reinjerebis razmma 2 dReSi
tyis 1500kv.m farTobi gaasufTava.
ra farTobs gaasufTavebs razmi erTi
kviris ganmavlobaSi, Tu imave tempiT
imSavebs?

15 gelam da nanam SoSiebisTvis 12 galia
daamzades. gelam nanaze 2-jer meti ga-
lia daamzada. ramdeni galia daamzada
nanam?

16 moswavleebma skolis ezos Robis gaswvriv mTel
perimetrze naZvis xeebis nergebi darges. skolis ezos
marTkuTxedis forma aqvs. misi sigrZea 90m, xolo si-
gane _ 78m. mezobel nergebs Soris manZili 3metria.
ramdeni nergi darges moswavleebma?

17 warmoidgine, rom Sen gegmav mogzaurobas avtomobiliT, romelic unda
iqirao. kompania avtomobils aqiravebs aseTi pirobebiT: dRiur ga-

dasaxads, romelic Seadgens 40 lars, emateba 20 TeTri yovel gavlil

kilometrze. ra Tanxis gadaxda mogiwevs, Tu 5 dRis ganmavlobaSi 600

kilometri unda gaiaro?

18 skolidan biblioTekamde misasvlelad arsebobs 3 gza, xolo bib-
lioTekidan levanis saxlamde misasvlelad _ 4 gza. levans surs sko-

lidan saxlSi wasvla ise, rom gzad biblioTekaSi Seiaros. ramdeni

sxvadasxva gziT SeuZlia mas amis gakeTeba? Seasrule Sesabamisi sqema-

turi naxazi.

197

19 erTma Zmam 3t 950kg simindi moiyvana,
meorem _ 800kg -iT meti, vidre pirvel-
ma, xolo mesamem _ 740kg -iT naklebi,
vidre misma orive Zmam erTad. ramdeni
kilogrami simindi moiyvana samive Zmam
erTad?

20 50 litri RviniT erTnairi raodenobis orlitriani da samlitriani
qilebi gaavses. sul ramdeni qila gaivso?

21 Seadgine AB monakveTis sigrZis gamosaTvleli gamosaxuleba da
gamoTvale misi mniSvneloba, Tu

a) AC = 63 sm 5 mm da BC = 42 sm;

b) AC = 5 m 34 sm 5 mm da BC = 3 m 22 sm 8 mm.

AB C

22 ezoSi, romelsac marTkuTxedis forma aqvs, qvis 300 erTnairi,
kvadratuli formis fila daages. ezos sigane 6 m, xolo sigrZe 8 m -ia.
ra zomis filebi daages?

23 erTi monakveTis sigrZea 11 sm 4 mm, meore ki 3 - jer grZelia, vidre
pirveli. ramdeniT grZelia meore monakveTi pirvelze?

24 sinTetikuri qsovili pirvelad
1863 wels Seqmnes.

a) ramdeni weli gavida mas Semdeg

dRemde?

b) romel wels iqneba sinTetikuri

qsovilis Seqmnidan zustad ori

saukune gasuli?

25 natos unda gamoeTvala 26827 _ 1547 sxvaoba. man 8 -is nacvlad kalku-
latorze akrifa cifri 9 da amis gamo sxvaoba SecdomiT gamoTvala.
qvemoT CamoTvlilTagan ra unda moimoqmedos natom Secdomis gamosas-
woreblad?
a) miRebul Sedegs unda gamoaklos 100;
b) miRebul Sedegs unda miumatos 100;
g) miRebul Sedegs unda gamoaklos 10;
d) miRebul Sedegs unda gamoaklos 1000;
e) miRebul Sedegs unda miumatos 1000.

26 levans unda gamoeTvala 14127 _ 856 sxvaoba. man 5 -is nacvlad kalku-
latorze akrifa cifri 9 da amis gamo sxvaoba SecdomiT gamoTvala.

Secdomis gamosasworeblad levanma miRebul Sedegs unda:

a) gamoaklos 40; b) miumatos 40; g) gamoaklos 4;

d) gamoaklos 14; e) miumatos 14.

198

informaciuli da sakomunikacio teqnologiebis dax-
marebiT Sesasrulebeli proeqtebi da davalebebi

1.samkuTxedis praqtikuli gamoyeneba

I etapi: moiZie informacia samkuTxedis sixistis Tvisebis Sesaxeb;
II etapi: daamzade sxvadasxva mravalkuTxedi da aCvene, rom mxolod sam-
kuTxeds aqvs sixistis Tviseba;
III etapi: gaarkvie raSi iyeneben samkuTxedis sixistis Tvisebas;
IV etapi: moiZie fotomasala samkuTxedis samSeneblo konstruqciebSi
gamoyenebis Sesaxeb;
V etapi: gaamzade Segrovili masala saprezentaciod.

2.sigrZis da masis sazomi amerikuli erTeulebi

I etapi: moiZie informacia sigrZisa da masis sazomi amerikuli er-
Teulebis Sesaxeb;
II etapi: Seadgine sigrZis sazomi erTeulebis (inCi, diumi, futi, iar-
di, mili) da masis sazomi erTeulebis (uncia, funti, tona) damokideb-
ulebis cxrilebi;
III etapi: amoxseni amocanebi:
a) 4 uncia namcxvari 2 dolari Rirs. ra Rirs 1 funti namcxvari?
b) lizis saxlidan skolamde manZilia 2 mili. ramdeni iardis tolia
manZili lizis saxlidan skolamde?
V etapi: gaamzade Segrovili masala saprezentaciod

3.mzis sistemis planetebi

I etapi: moiZie informacia mzis sistemis
planetebis Sesaxeb;
II etapi: Seadgine planetebis ricxviTi
maxasiaTeblebis cxrili (mzemde manZili,
masa, diametri, mTvareTa raodenoba);
III etapi: moiZie Sesabamisi fotomasala;
IV etapi:. amoxseni amocanebi:
a) ra dro sWirdeba mzis sxivs dedamiwamde mosasvlelad?
b) ra dro sWirdeba xelovnur Tanamgzavrs, dedamiwas erTxel rom Semoua-
ros?
V etapi: gaaforme proeqti da moemzade prezentaciisaTvis.

4.vardebis xeivnis gaSeneba

bavSvebma skolis ezoSi bilikis gaswvriv
vardebis xeivnis gaSeneba gadawyvites. bi-
likis sigrZe 20 metria. daexmare bavSvebs
Canafiqris sworad dagegmvaSi:
a) ramdeni Ziri nergi unda SeiZinon;
b) ramdenwliani nergebi SeiZinon;
g) ra jiSis vardebi SeiZinon;
d) nergebi sad SeiZinon (SeurCie yvelaze iafi varianti);
e) ra Tanxaa saWiro nergebis SesaZenad.

199

sagnobrivi saZiebeli

a m
adli – 37 mravalkuTxedi – 54
asoiTi gamosaxuleba – 44 mxari – 37
marTi kuTxe – 51
b marTkuTxa paralelepipedi – 130
biji – 37 marTkuTxedi – 100
blagvi kuTxe – 51 martivi ricxvebi – 139
blagvkuTxa samkuTxedi – 54 masa – 35
maxvili kuTxe – 51
g miligrami – 35
gadanacvlebadobis Tviseba – 59 monakveTi – 26
gamokleba – 63 monacemTa analizi – 113
gamravleba – 67 moZraoba sawinaaRmdego
gamyofi – 139 mimarTulebiT – 95
ganrigebadobis Tviseba – 71 moZraoba Semxvedri
gantoleba – 47 mimarTulebiT _ 92
gaSlili kuTxe – 51 mravalkuTxedi – 54
mxari – 37
d
damrgvaleba – 16 f
diagrama – 117 farTobi kvadratis – 107
diametri – 150 farTobi marTkuTxedis – 107
dro, manZili, siCqare – 89 figuris farTobi – 102

k q
kubi – 130 qarTuli anbanuri numeracia _ 10
kuTxe. kuTxeebis saxeebi – 51 qarTuli sazomi erTeulebi _ 37
qorda – 150
n
niSnadoba (ganayofis) _ 80 S
niSnadoba (ricxvis) – 13 Sedareba (naturaluri
numeracia – 6
ricxvebis) _ 13
p Sedareba (wiladebis) _ 189
paleti – 102 Sekveca (wiladebis) _ 186
paraleluri wrfeebi – 23 Sekreba – 61
perimetri – 100 Sereuli ricxvi _ 167
piqtograma – 117
w
r waxnagi – 130
radiusi – 150 wesieri da arawesieri wiladebi _ 167
rkali – 150 wibo – 130
romauli numeracia _ 10 wiladi _ 159
wiladis ZiriTadi Tviseba _ 204
s wrfe – 19
sakoordinato bade – 126
samkuTxedi – 54 x
siCqare – 89 xarisxi – 85
sxivi – 26
j
t jami – 59
texili – 32 jeradi - 139

200