rubrikas “maTematika gakveTilebis Semdeg”
waruZRveba zurab aRdgomelaSvili
maTematika gakveTilebis Semdeg
( nawili II )
Tema 3. amocanebi siTxis gadasxmaze
amocana 1.
(puasonis amocana). erT kacs aqvs 12 pinti Rvino (1 pinti
= 0,568 l) da surs misi naxevari aCuqos axlobels, magram mas aqvs ori
WurWeli: erTi 8 pintis, xolo meore – 5 pinti tevadobis. rogor unda
gadmoasxas 8 pinti tevadobis WurWelSi 6 pinti Rvino?
vidre amovxsnidiT puasonis amocanas, urigo ar iqneboda amogvexsna
ramodenime martivi amocana.
amocana 2. gvaqvs onkaniani 100 litriani cisterna savse wyliT da ori
WurWeli, romelTagan erTis tevadobaa 3 litri, xolo meorisa ki – 5 litri.
am WurWlebis meSveobiT unda miviRoT 1 litri wyali?
amoxsna. es amocana SeiZleba amoixsnas zepirad. SevavsoT 3-litriani
WurWeli, gadavasxaT es wyali xuTlitrian WurWelSi. Semdeg kidev SevavsoT
samlitriani WurWeli da gadavasxaT iqidan wyali xuTlitrian WurWelSi
manamdis ar gaivseba xuTlitriani. amis Semdeg samlitrian WurWelSi
darCeba 1 litri wyali. es yoveli sqematurad ase SegviZlia gamovsaxoT
(SeniSvna: misaRebi Sedegi axlac da Semdegac Casmuli iqneba marTkuTxedSi).
100
100 – 3=97
97
97 – 3=94
3-litriani
WurWeli
0
3
3 – 3= 0
3
94
3 – 2 = 1
cisterna
5-litriani
WurWeli
0
0
0 + 3=3
3
SeniSvna
(miviReT)
3 + 2=5
amocana 3. wina amocanis pirobiT miviRoT am mocemuli WurWlebidan
romelimeSi: 2 l; 3 l; 4 l; 5 l; 6 l; 7 l; 8 l.
amoxsna. mogvyavs erTi SesaZlo amoxsna. yvelaze advilia miviRoT 3; 5
an 8 l. amisaTvis sakmarisia SevavsoT pirvel or SemTxvevaSi TiTo WurWeli
mocemulebidan, xolo mesamis misaRebad ki – orive. rom miviRoT 2 litri,
amisaTvis saWiroa SevavsoT xuTlitriani WurWeli da amiT gadavavsoT
samlitriani WurWeli. xuTlitrian WerWelSi dagvrCeba 2 litri wyali. Tu
axla davcliT samlitrian WurWels, Semdeg xuTlitrianidan masSi
gadmovasxavT 2 litrs, Semdeg gavavsebT xuTlitrians, maSin miviRebT 7
litr wyals, amasTan samlitrianSi kidev aris adgili 1 litrisaTvis.
gadavasxaT masSi xuTlitriani WurWlidan gavsebamde. amis Semdeg
xuTlitrianSi darCeba 4 litri wyali. 6 litris miReba gacilebiT advilia.
avavsoT patara WurWeli da gadavasxaT cariel xuTlitrianSi. Semdeg isev
avavsoT patara WurWeli. orive WurWelSi miviReT 6 litri wyali. es
yovelive sqematurad ase SegviZlia warmovadginoT (SeniSvna: miRebuli,
misaRebi Sedegi, axlac da Semdegac Casmuli iqneba marTkuTxedSi).
5-litriani
WurWeli
0
SeniSvna
(miviReT)
100
3-litriani
WurWeli
0
100 – 5=95
0
0 + 5 = 5
5
95
0 + 3= 3
5 _ 3 = 2
3 , 2
95 + 3=98
98
3 – 3=0
0 + 2 = 2
2
2 – 2 =0
98 – 5=93
2
0 + 5 = 5
2 + 5 = 7
93
93 + 4 = 97
97
2 + 1=3
3
3 – 3= 0
5 _ 1 = 4
4 – 4=0
0 + 3=3
4
97 – 3=94
0 + 3 =3
3
3 + 3 = 6
94 – 2=92
3
cisterna
3 + 2=5
3 + 5 = 8
arsebobs kidev ufro mokle sqematuri gamosaxvac. Sev-TanxmdeT, rom
siTxis raodenoba WurWlebSi CavweroT Semdegi saxiT (V; x ; y), sadac V aris
mTlianad mocemuli siTxe litrebSi, x – siTxis raodenoba litrebSi,
romelic aris mcire WurWelSi, xolo y ki – did WurWelSi (zogierT
SemTxvevaSi V-s gamosaxva ar gvWirdeba).
axla CavweroT amocanaSi 3 -Si, rogor icvleba WurWlebSi wylis
raodenoba. Tu visargeblebT zemoT moyvanili sqemiT:
(cisterna; 3-litriani; 5-litriani)  (100; 0; 0)  (95; 0; 5 )  (95; 3 ; 2 ) 
(98; 0; 2)  (93; 2; 5 )  (93; 3; 4 )  (97; 3; 0)  (97; 0; 3)  (94; 3; 3 )  (92; 3; 5 )
amocana 4. gvaqvs 100 litriani cisterna savse wyliT da ori carieli,
WurWeli, romelTa tevadobaa Sesabamisad 9 litri da 12 litri. mxolod
maTi gamoyenebiT unda miviRoT 4 litri wyali.
amoxsna. SevniSnoT rom 9-sa da 12-is udidesi saerTo gamyofia 3. am
WurWlebiT nebismieri operaciiT miRebuli siTxis raodenoba warmoadgens
martivi operaciebis romeliRac mimdevrobas, romelic aRwerilia wina
msjelobaSi. aq unda SevniSnoT, rom siTxis moculoba yoveli aseTi
operaciis Semdeg TiToeul WurWelSi siTxis raodenoba litrebSi unda
iyofodes 3-ze. vinaidan Tavdapirvelad orive WurWeli carieli iyo da maTi
tevadobebis raodenoba litrebSi iyofa 3-ze, amitom pirveli operaciis da
rogorc advili
misaxvedria yoveli aseTi operaciis Semdeg TiToeul
maTganSi wylis raodenoba litrebSi unda gaiyos 3-ze, magram 4 ar iyofa
unaSTod 3-ze, amitom 4 litris miReba mxolod am WurWlebis meSveobiT
SeuZlebelia.
analogiuri msjelobiT SegviZlia davrwmundeT, rom ramdenime
WurWliT miRebuli siTxis raodenoba (igulisxmeba, rom am WurWlebs aqvT
mTeli raodenobis tevadobebi raime erTeulSi) aucileblad unda iyofodes
am WurWlebis tevadobis erTeulSi gamomsaxveli mTeli ricxvebis udides
saerTo gamyofze. winaaRmdeg SemTxvevaSi amocanas amonaxsni ar gaaCnia.
saSinao davalebis amocanebi
amocana 5. amoxseniT amocana 3, Tu onkanidan SegviZlia avavsoT
mxolod didi WurWeli da cisternaSi Caclis ufleba gvaqvs mxolod patara
WurWlidan. SeadgineT Sesabamisi sqemebi.
amocana 6. rogor miviRoT 100 litriani tevadobis onkaniani
cisternidan ori WurWlis meSveobiT, romelTa tevadobaa Sesabamisad 7
litri da 12 litri, 1 litri wyali.
amocana 7. kasrSi gvaqvs aranakleb 10 litri benzini. rogor
gadmovasxaT am kasridan 6 litri benzini mxolod ori WurWlis meSveobiT,
romelTa tevadobaa Sesabamisad 5 litri da 9 litri?
amocana 8. 8-litriani saTli savsea rZiT. am saTlidan unda
gadmovasxaT 4 litri ori WurWlis meSveobiT, romelTa tevadobaa
Sesabamisad 3 litri da 5 litri. rogor gavakeToT es?
amocana 9. kasrSi gvaqvs 12 litri navTi. rogor gadmovasxaT am
kasridan 6 litri navTi ori WurWlis meSveobiT, romelTa tevadobaa 5
litri da 8 litri?
amocana 10. kasrSi gvaqvs aranakleb 13 litri benzini. rogor
gadmovasxaT am kasridan 8 litri, ori WurWlis meSveobiT, romelTa
tevadobaa Sesabamisad 5 litri da 9 litri?
amocana 11. gvaqvs kasrSi 16 litri wyali. rogor gadmovasxaT misgan
8 litri ori WurWlis meSveobiT, romelTagan erTis tevadobaa 6 litri,
xolo meorisa ki 11 litri?
amocana 12. amoxseniT
raodenobis gadasxmebiT.
puasonis
amocana,
rac
SeiZleba
naklebi
amocana 13. gvaqvs sami WurWeli, romelTagan pirvelis tevadobaa 6
litri, meorisa _ 3 litri, mesamisa ki – 7 litri. pirvelSi asxia 4 litri
Rvino, mesameSi – 6 litri, xolo meore ki – carielia. mxolod am sami
WurWlis meSveobiT pirvel da mesame WurWelSi miviRoT 5-5 litri Rvino.
saswavlo aRmzrdelobiTi mizani. am ciklis amocanebis amoxsna
moiTxovs mixvedrilobas, aviTarebs kombinatorul azrovnebas.
meToduri
miTiTebebi.
Tavdapirvelad CamoayalibeT
puasonis
amocana, magram ar amouxsnaT. amocanebis amoxsna daiwyeT amocana 2idan. aucileblad gaamaxvileT yuradReba zogad meTodze.
Tema 4. maTematikuri rebusebi
amocana 1. ЧАЙ : АЙ = 5
amoxsna: amocanis pirobidan gamomdinare ЧАЙ = АЙ x 5, anu Ч x 100 + АЙ = АЙ
x 5, saidanac Ч x 100 = АЙ x 4 =Ч x 25 = АЙ vinaidan ricxvi АЙ orniSnaa, amitom Чm SeiZleba miiRos Semdegi mniSvnelobebi 1. 2 an 3. Ч-s yovel mniSvnelobas
Seesabameba gansazRvruli amonaxsni. Tu Ч =1, maSin АЙ = 25, e.i. А = 2 da
Й
= 5; Tu Ч =2, maSin АЙ = 50; Tu Ч =3, maSin АЙ = 75; amgvarad, Canaweris gaSifvra
SeiZleba sami xerxiT: ЧАЙ=125; ЧАЙ=250 an ЧАЙ=375.
amocana 2. ЛИК x ЛИК=БУБ ЛИК.
amoxsna: tolobis orive mxares gamovakloT ЛИК miviRebT ЛИК x (ЛИК-1)=
БУБ x 1000. ricxvebi ЛИК da ЛИК-1 warmoadgenen or momdevno naturalur
ricxvebs, amitom isini urTierTmartivia. vinaidan maTi namravli iyofa 1000ze (1000=555222), amitom erT-erTi maTgani iyofa 125-ze, magram ar iyofa
2-ze, xolo meore iyofa 222=8-ze da ar iyofa 5-ze. kent samniSna ricxvebs
Soris 125-ze iyofa mxolod 125, 375, 625, da 875. ori momdevno am ricxvebs
Soris (1-iT meti, an 1-iT naklebi) 8-ze iyofa, mxolod 376 da 624 Semowmeba
gviCvenebs, rom pirveli am ricxvebidan akmayofilebs amocanis pirobas,
xolo meore ki – ara. maSasadame miviReT rebusis gaSifvra:
376  376=141376.
amocana 3. МЯУ x МЯУ=МЯУЯК+УЯЯ.
amoxsna: marjvniv gvaqvs jami xuTniSna da samniSna ricxvisa, amitom
МЯУ x МЯУ ar aRemateba 99999+999, rac naklebia 101000-ze. vinaidan
400x400=160000>101000, amitom М SeiZleba iyos: 1, 2 an 3. aqedan gamomdinare,
marjvena jami ar aRemateba 39999+999, rac naklebia 41000-ze. vinaidan
300x300=90000>41000, amitom М3, e.i. М=1 an М=2 maSasadame, marjvena jami ar
aRemateba 29999+999 da naklebia 31000. vinaidan 200x200=40000, rac metia 31000ze, amitom М2 amgvarad М=1.
miviReT (100+10Я+У)  (100+10Я+У) = (10000+1000Я+100У+10Я+К) + (100У+10Я+Я).
frCxilebis gaxsniT da msgavsi wevrebis SeerTebiT calkeul mxareebSi
miviRebT 10000+100Я2+У2+2000Я+20ЯУ+200У=10000+1000Я+200У+20Я+К+Я. aqedan
100Я2+У2+1000Я+20ЯУ=20Я+К+Я (*) vinaidan М=1, amitom Я1. Tu Я aranaklebia
2-ze, maSin tolobis marcxena mxare ar aris naklebi 1004+10002=2400-ze, im
SemTxvevaSi, roca marjvena mxare nebismieri У, Я, К-saTvis ar aRemateba
209+9+9=198-s. dagvrCenia erTaderT SemTxvevaRa, roca Я =0.
(*) tolobaSi Я=0 Casmisas miviReT, rom У2=К, vinaidan 42=16>9 , amitom
У-ma SeiZleba miiRos mxolod ori mniSvneloba. У=2 an У=3, maSin К=4 an К=9
Sesabamisad. amgvarad arsebobs rebusis ori gaSifvra: МЯУ=102 an МЯУ=103.
amocana 4. БАРБОС+БОБИК=СОБАКИ
amoxsna: gaSifvra SeiZleba vawarmooT Semdegnairad. Sekreba aseul
aTaseulebis svetSi С=Б+1. Sekreba aTeul aTaseulebis svetSi gviCvenebs,
rom А+Б=10+"0" an A+B+1=10+`0~ ("0" aris aso "o" da ara nuli). erTeulis
svetSi miviRebT С+К=И an С+К=10+И. ganvixiloT pirveli SemTxveva.
aTeulebis svetSi "0"+И="0"+С+К>К"0"+ И=10+ К"0"+С+К=10+ К = С +"0"=10,
Б+1+"0"=10"0"=9-Б. aseulebis svetidan Cans, rom 2Б+1=10+А. am gantolebebis
kombinirebiT TiToeul gantolebasTan, romelic miviReT aTeul aTaseulebis svetis ganxilvisas ("0"=9-Б-is gaTvaliswinebiT), miviRebT 4 sistemas
orucnobiani ori gantolebisas, romlebis amoxsnac gvaZlevs А=5, Б=7 da
aqedan gamomdinare С=8, "0"=2 gaSifruli cifrebis Casmisas
+
7
8
5 P 7 2 8
7 2 7 И K
2 7 5 К И
aTaseulebis svetidan miviRebT: Р=4. vinaidan 8+К=И, amitom К=1, И=9. e.i.
miviReT:
+
7
8
5
7
2
4
2
7
7
7
5
2
9
1
8
1
9
axla ganvixiloT meore SemTxveva: С+К=10+И, maSin "0"+И="0"+С+К–10.
magram "0"+И+1=К an "0"+И+1=10+К vinaidan "0"+ С =19, toloba SeuZlebelia,
amitom "0"+ И+1= К da "0"+С=9, "0"=8-Б ukanasknel Tanafardobas aTeul
aTaseulebis svetSi SekrebasTan ganxilviT miviRebT, rom А + 2Б=17 an А+2Б=18.
vinaidan aTeulebis da aseulebis svetidan Sekrebisas araferi ar gadadis
Semdeg TanrigSi, amitom 2Б=А an 2Б=10+А. Tu aviRebT TiTo gantolebas
TiToeuli wyvilisas, miviRebT 4 orucnobian wrfiv gantolebaTa sistemas.
maTgan mxolod erTs gaaCnia naturalur ricxvebSi amonaxsni: Б= 7, А=4. aqedan
gamomdinare, С = Б+1=8, "0"=8-Б=1. gaSifruli cifrebis CasmiT
+
7
8
4 P 7 1 8
7 1 7 И K
1 7 4 K И
aTaseulebis svetidan miviRebT: Р=5. Semdgom 8+К=10+И (erTeulebis svetidan),
К=2+ И, amasTan К, И, asoebiT daSifruli Semdegi cifrebidan: 0; 2; 3; 6; 9
romeliRacaa. am SemTxvevaSi gvaqvs erTaderTi SesaZlebloba К=2 da И=0.
miviReT meore (bolo) gaSifvra:
+
7
8
4
7
1
5
1
7
7
7
4
1
0
2
8
2
0
saswavlo aRmzrdelobiTi mizani
am amocanebSi umetes gaaCnia ramodenime swori gaSifvra. es keTdeboda
specialurad, rom moswavleebi gadaCveuliyvnen mxolod gadasinjvis xerxis
gamoyenebas.
TiToeuli am tipis amocanebisTvis saWiroa: a) ukidures SemTxvevaSi
vipovoT romelime amonaxsni mainc, amasTan, rac SeiZleba bevri amonaxsni; b)
sasurvelia vipovoT yvela amonaxsni da davamtkicoT, rom sxva amonaxsnebi
ar gaaCnia.
b) punqtis sworad damtkicebisaTvis ZiriTadad aucilebelia ganvixiloT
yvela SemTxveva datotvil logikur sqemaSi. maTematikuri rebusebi moxerxebuli obieqtia sakmaod rTul (datotvil, Sromatevad) logikur
msjelobaSi, romelSic aucilebelia gavarCioT yvela SesaZlo SemTxveva.
rogorc praqtika gviCvenebs, VI-VII-VIII klaselebs ZiriTadSi xelewifebaT
srulad Caataron da Caweron msgavsi msjelobebi, mxolod ramodenime
swored warmoebuli gaSifvra mogvcems sasurvel Sedegs. is gadaaCvevs
moswavleebs brmad SerCevas cifrebisas da uCvenebs maT mkacri logikuri
msjelobis aucileblobas.
meToduri SeniSvnebi
CanawerSi gamravlebis niSani xSirad gamotovebulia, amitom moswavleebi
xSirad ureven samniSna ricxv ABC-s AxBxC namravlSi. rom ar davuSvaT
areuloba, xSirad ricxvs aRniSnaven zemodan xaziT. am SemTxvevaSi samniSna
ricxvi Caiwereba ase: ABC .
unda mieqces yuradReba am tipis amocanebis amoxsnisas, rom Catardes
sruli mkacri logikuri garCeva yvela SesaZlo SemTxvevisa.
saSinao davalebis amocanebi
amocana 5. СУКxСУК=БАРСУК
amocana 6. КАКАКxКО=КОКОКО
amocana 7.
+
amocana 8.
А Н Д Р Е Й
Ж А Н Н А
Д Р У Ж Б А
+
К О Р О В А
Т Р А В А
М О Л О К О
amocana 9. БЕРУ4 = БУЕР
aq  gamravlebis niSania
amocana 10. ТРУД + ВОЛЯ = УДАЧА damatebiTi pirobaa: ricxvi ТР da ВО
unaSTod iyofa 13-ze.
amocana 11. КО+ЛЯ=ОЛ – Я. a) gaSifreT damatebiTi pirobiT К+О+Л+Я=21.
2) gaSifreT damatebiTi pirobis gareSe (10 pasuxze meti).
amocana 12.
amocana 13.
Tema 5. amocanebi “waSlil cifrebze”
moiTxoveba aRvadginoT Canaweri ariTmetikuli moqmedebebisa, romelSic
cifrebis nawili Secvlilia varskvlavebiT. aRdgena SeiZleba vawarmooT
ramdenime xerxiT. ganvixiloT ramdenime amocana.
amocana 14.
+
3
*
8
*
2
0
8
*
4
6
7
*
amoxsna: vinaidan 6+7=13, amitom jami tolia 8 043. Semdeg cifrebs
aRvadgenT mimdevrobiT – Tavdapirvelad aTeulebis svetSi, Semdeg –
aseulebisaSi, Semdeg – aTaseulebisaSi. erTeulebis svetidan aTeulebis
svetSi gadaitaneba 1-iani, amitom 1+8+*=14. aTeulebis svetSi * unda
SevcvaloT cifriT 5. aseulebis svetSi 1+*+2=10, amitom *-is nacvlad unda
CavsvaT cifri 7. da bolos, aTaseulebis svetSi 1+3=*=8, saidanac *=4.
pasuxi: 3 786+4 257=8 043.
amocana 15.
-
7 *
* 6
*
3
* 0
*
8 *
* *
0
*
*
*
4
amoxsna: Tu CamovitanT "0"-s, miviRebT meore gamoklebaSi Semdeg Canawers:
-
* 8 0
3 * *
0
aqedan gamomdinare akldeba ricxvi 380. is tolia gaoTxkecebuli
gamyofisa. e.i. gamyofi tolia 380 : 4 = 95. aseulebis cifri pirvel sxvaobaSi
an 6-ia an 7. davakvirdeT 95-is ra ricxvebze gamravlebiT miiReba samniSna
ricxvi aseulebSi cifriT 6 an 7. aseTi ori ricxvia: 7 da 8. 957=665, 958=760.
aqedan gamomdinare saZiebeli ricxvi SeiZleba iyos 7 980 an 7 030. ricxvebs
957 da 958 aTeulebis TanrigSi aqvT cifri 6, romelic emTxveva
Tavdapirvelad mocemul CanawerSi. advili Sesamowmebelia, rom Canaweridan
7 980 : 95 da 7 030 : 95 SegviZlia miviRoT sawyisi Canaweri “waSlili
cifrebiT”. e.i. Canaweris aRdgena SesaZlebelia ornairad.
amocana 16.
x
+
*
*
* *
* 3 7
* * *
*
*
2
*
*
*
*
*
0
*
*
* 7
* *
* 6
3
*
amoxsna: gamravlebis tabulidan Cans, 7-is erTniSna ricxvze namravli
bolovdeba cifriT 6, mxolod - 78, xolo cifriT 3 _ 79. aqedan gamomdinare
mamravlis bolo ori cifria 98. ra cifria samravlis aTeulebis TanrigSi?
aRvniSnoT is aso X-iT. CvenTvis cnobilia, rom (10X+7) 9= =10(9X+6)+3
bolovdeba 03-iT. e.i. 9X+6 bolovdeba 0-iT xolo 9X – cifriT 4. gamravlebis
tabulidan Cans, rom arsebobs X-sTvis erTaderTi varianti X=6. samravlis
aseulebis cifri aRvniSnoT Y-iT. Cven vxedavT, rom 9(100Y+67) bolovdeba
203-iT, e.i 9Y+6 bolovdeba cifriT 2, anu 9Y bolovdeba cifriT 6. e.i. Y=4. ra
cifria mamravlis aseulebSi? aRvniSnoT is aso A-Ti. maSin pirobidan
gamomdinare aseulebis cifri 467A tolia 7-is. vinaidan 4672=934, 4673=1401,
4674=1868, 4675=2 335, 4676=2 802, 4677=3 269, 4678=3 736, 4679=4 203, amitom
A=8. samravlis aTaseulebis cifri aRvniSnoT Z-iT. unda vipovoT iseTi Z,
romlisTvisac ricxvSi 8(1000Z+467) aTaseulebis cifri iqneba Z. vinaidan
4678=3 736, amitom 8Z+3 unda bolovdebodes 3-iT, anu 8Z unda bolovdebodes
0-iT. e.i. Z=5.
pasuxi: 5467898=4909366.
amocanebi saSinao davalebisaTvis
17.
18.
2 3

*
+ * * *
* * *
* * 5 6
5
*
*
19.
2
*
*
7 *
7 *

+
*
ზურაბ აღდგომელაშვილი
მასწავლებელთა
დახელოვნებისა
*
2
8

+
8
*
*
*
*
*
*
6
*
*
*
*
*
*
*
6
-- 1997-2007 წლებში ქ.თბილისის
და
მეთოდური
უზრუნველყოფის
ინსტიტუტის მათემატიკის სწავლებისა და მეთოდიკის ლაბორატორიის
გამგე;
2007-2013 წლებში
საქართველოს პედაგოგთა კვალიფიკაციის
ამაღლებისა და გადამზადების ინსტიტუტის მათემატიკის მიმართულების
ხელმძღვანელი, ასოცირებული პროფესორი.