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sarCevi<br />
Tavi1<br />
racionaluri ricxvebi<br />
1.1 simravle 6<br />
1.2 simravleTa gaerTianeba da TanakveTa 13<br />
1.3 dadebiTi da uaryofiTi ricxvebi 19<br />
1.4 ricxviTi wrfe 23<br />
1.5 mopirdapire ricxvebi. ricxvebis Sedareba 26<br />
1.6 ricxvis moduli 31<br />
1.7 ricxvebis Sekreba 34<br />
1.8 ricxvebis gamokleba 40<br />
1.9 ricxvebis gamravleba da gayofa 45<br />
1.10 ricxviTi gamosaxulebis mniSvnelobis gamoTvla 51<br />
1.11 kalkulatori 55<br />
I Tavis mimoxilva 59<br />
testi 64<br />
Tavi 2<br />
mravalkuTxedebi<br />
2.1 geometriuli figurebi 67<br />
2.2 sxivi, monakveTi, texili 72<br />
2.3 mosazRvre da vertikaluri kuTxeebi 75<br />
2.4 mravalkuTxedi 83<br />
2.5 samkuTxedi 85<br />
2.6 agebis amocanebi 91<br />
2.7 paralelur wrfeTa Tvisebebi 98<br />
2.8 samkuTxedis kuTxeTa Tvisebebi 101<br />
2.9 tolferda samkuTxedi 105<br />
2.10 samkuTxedebis toloba 110<br />
2.11 samkuTxedebis ageba 116<br />
2.12 paralelogrami 121<br />
2.13 wrewiris qorda da mxebi 127<br />
2.14 praqtikuli amocanebi 131<br />
II Tavis mimoxilva 135<br />
testi 143<br />
Tavi 3.<br />
monacemebi<br />
3.1 monacemTa saxeebi da maTi mopovebis saSualebebi 150<br />
3.2 ricxvis procentuli Cawera 155<br />
3.3 rogor vipovoT ricxvi misi procentis mixedviT? 160<br />
3.4 ori sididis fardobis procentuli Cawera 165<br />
3.5 monacemTa ricxviTi maxasiaTeblebi 172<br />
III Tavis mimoxilva 176<br />
4 Tavi 1 • racionaluri ricxvebi
testi 180<br />
Tavi 4<br />
erTwevri da mravalwevri<br />
4.1 cvladiani gamosaxuleba 183<br />
4.2 toli gamosaxulebebi 187<br />
4.3 naturalurmaCvenebliani xarisxi 192<br />
4.4 xarisxis Tvisebebi 197<br />
4.5 erTwevri da mravalwevri 201<br />
4.6 moqmedebebi mravalwevrebze 208<br />
4.7 mravawevris mravalwevrze gamravleba 213<br />
4.8 mravalwevris mamravlebad daSla 218<br />
4.9 gantoleba 224<br />
4.10 wrfivi gantoleba 228<br />
4.11 amocanis amoxsnis algebruli meTodi 232<br />
4.12 mimdevrobebi 238<br />
4.13 ariTmetikuli progresia 243<br />
IV Tavis mimoxilva 248<br />
testi 254<br />
Tavi 5<br />
koordinatTa sistema<br />
5.1 wertilis koordinatebi 257<br />
5.2 wertilis ageba misi koordinatebiT 261<br />
5.3 grafiki 264<br />
5.4 pirdapirproporciulobis grafiki 270<br />
5.5 figuraTa paraleluri gadatana 275<br />
V Tavis mimoxilva 279<br />
testi 283<br />
pasuxebi 286<br />
5
Tavi 1 racionaluri ricxvebi<br />
1.1 simravle<br />
mizani:<br />
6 Tavi 1 • racionaluri ricxvebi<br />
gavecnoT simravlisa da qvesimravlis cnebebs.<br />
xSir SemTxvevaSi Cveni msjelobis sagani aris ara calkeuli obieqti,<br />
aramed maTi garkveuli jgufi, krebuli, erToblioba. magaliTad,<br />
SesaZloa vmsjelobdeT:<br />
� ara calkeul ferwerul tiloebze, aramed maT koleqciaze;<br />
� ara calkeul fexburTelebze, aramed safexburTo gundze;<br />
� ara sibrtyis calkeul wertilebze, aramed maT iseT<br />
erTobliobaze, rogoricaa monakveTi.<br />
sxvadasxva saxis erTobliobis, jgufisa Tu krebulis aRsaniSnavad<br />
maTematikaSi gamoiyeneba erTiani termini - simravle. swori warmodgena<br />
simravleze, rogorc maTematikur cnebaze, SeiZleba konkretuli<br />
magaliTebisa da simravleTa zogadi Tvisebebis gacnobiT CamoviyaliboT.<br />
magaliTad, ganxilvis sagani SeiZleba iyos:<br />
� erovnuli biblioTekis wignTsacavSi daculi wignebis simravle;<br />
� saklaso oTaxis merxebis simravle;<br />
� naturaluri ricxvebis simravle;<br />
� mocemuli samkuTxedis wertilTa simravle;<br />
� oTxkuTxedebis simravle da a.S.<br />
yoveli simravle elementebisgan Sedgeba. naturaluri ricxvebis<br />
simravlis elementebia ricxvebi 1, 2, 3 da a.S. ricxvi 9 aris naturaluri<br />
ricxvebis simravlis elementi, xolo ricxvi 2,5 ar aris naturaluri<br />
ricxvebis simravlis elementi.<br />
simravlis elementebis miTiTeba moxerxebulia maTi sruli<br />
CamonaTvalis moyvaniT. simravle, romelic Sedgeba a, b, c da d<br />
elementisagan, ase CavweroT: {a, b, c, d}. kerZod, erTniSna naturaluri<br />
ricxvebis simravle ase Caiwereba: {1, 2, 3, 4, 5, 6, 7, 8, 9}.<br />
elementebis CamonaTvalis saxiT SesaZlebelia mocemul iqnas mxolod<br />
is simravleebi, romelTa elementebis raodenoba sasrulia, anu sasruli<br />
simravleebi.<br />
usasrulo simravlis mosacemad uTiTeben iseT Tvisebebs, romelic<br />
gaaCnia am simravlis yovel elements da ar gaaCnia arcerT sxvas.<br />
magaliTad, 5-is jeradi kenti naturaluri ricxvebis simravle aris<br />
usasrulo simravle, romlis elementebia ricxvebi 5, 15, 25, 35, 45 ... .<br />
simravle gansazRvrulia, Tu garkveviTaa cnobili, romeli elementi<br />
ekuTvnis am simravles da romeli - ara.<br />
simravleebs, rogorc wesi, didi laTinuri asoebiT aRvniSnavT: A, B, C, ... ,<br />
simravleTa elementebs ki patara laTinuri asoebiT: a, b, c, ... . is faqti,<br />
rom x aris A simravlis elementi, Semdegnairad Caiwereba: x∈A. ikiTxeba:
“x ekuTvnis A-s”. y∉B Canaweri ikiTxeba: “y ar ekuTvnis B-s” da niSnavs,<br />
rom y ar aris B simravlis elementi.<br />
ganvixiloT 3-is jeradi luwi naturaluri ricxvebis simravle. es<br />
simravle A-Ti aRvniSnoT. A simravle usasruloa. misi elementebia<br />
ricxvebi: 6, 12, 18, 24 ... . B-Ti aRvniSnoT 6-is jeradi naturaluri<br />
ricxvebis simravle. B simravlis elementebia: 6, 12, 18, 24 ... . rogorc<br />
vxedavT, A da B simravle erTi da imave elementebisgan Sedgeba _ es<br />
simravleebi tolia, A = B.<br />
simravleebs, romlebic erTi da imave elementebisgan Sedgeba,<br />
toli simravleebi ewodeba.<br />
vTqvaT, A romelime skolis meSvideklaselTa simravlea, xolo B amave<br />
skolis moswavleTa simravle. yoveli meSvideklaseli, cxadia, skolis<br />
moswavlec aris. amitom, A simravle B simravlis nawili, anu B_s<br />
qvesimravlea.<br />
A simravles ewodeba B simravlis qvesimravle, Tu A simravlis<br />
yoveli elementi amave dros B simravlis elementicaa.<br />
is faqti, rom A simravle B simravlis qvesimravlea, ase Caiwereba: A⊂B.<br />
simravleebs Soris mimarTebis TvalsaCinod warmosadgenad venis<br />
diagramas iyeneben.<br />
magaliTad, A ⊂ B mimarTebis SemTxvevaSi venis<br />
diagrama marjvniv moTavsebul naxazzea<br />
warmodgenili. am naxazze A simravlis elementebs<br />
pirobiTad mcire wris wertilebi, xolo B simravlis<br />
elementebs – didi (momcveli) wris wertilebi<br />
Seesabameba.<br />
simravles, romelic arcerT elements ar Seicavs carieli simravle<br />
ewodeba. carieli simravlis magaliTebia: 2-is jeradi kenti ricxvebis<br />
simravle, mTvareze mcxovreb adamianTa simravle da a.S. cariel simravles<br />
∅ simboloTi aRniSnaven. carieli simravle nebismieri simravlis<br />
qvesimravles warmoadgens.<br />
sasruli A simravlisTvis n(A)-Ti A simravlis elementebis raodenobas<br />
aRniSnaven. magaliTad, Tu A = {a, b, g}, maSin n(A) = 3, xolo Tu A = {a, b,<br />
g, 5, 7,}, maSin n(A) = 5. cxadia, rom n(∅) = 0.<br />
upasuxe kiTxvebs<br />
1. aris Tu ara ricxvi 42 Svidis jeradi orniSna naturaluri ricxvebis<br />
simravlis elementi? miekuTvneba Tu ara am simravles ricxvi 7? ricxvi 54?<br />
2. rogor ikiTxeba da ras niSnavs Canaweri x∈E ? Canaweri c∉F ?<br />
3. rogor simravleebs ewodeba sasruli simravleebi?<br />
4. rogor simravleebs ewodeba toli simravleebi?<br />
5. rogor ganimarteba simravlis qvesimravle ?<br />
6. rogor ikiTxeba da ras niSnavs Canaweri D⊂E ?<br />
B<br />
A<br />
Tavi 1 • racionaluri ricxvebi 7
savarjiSoebi<br />
1 a) daasaxele saqarTvelos qalaqebis simravlis ramdenime<br />
elementi;<br />
b) daasaxele zRvispira qalaqebis simravlis ramdenime elementi;<br />
g) daasaxele qalaqi, romelic ekuTvnis saqarTvelos qalaqebis<br />
simravles, magram ar ekuTvnis zRvispira<br />
qalaqebis simravles;<br />
d) daasaxele elementi, romelic ekuTvnis<br />
zRvispira qalaqebis simravles da<br />
ar ekuTvnis saqarTvelos qalaqebis<br />
simravles.<br />
2 ganvixiloT S = {a, b, g, d, e, 2, 5, 3, 1, 0}<br />
simravle.<br />
a) ramdeni elementia S simravleSi?<br />
b) Cawere S simravleSi Semavali laTinuri asoebis simravle.<br />
ramdeni elementia am simravleSi?<br />
g) S simravlis elementebidan ramdenia qarTuli anbanis aso?<br />
d) S simravlis elementebidan ramdeni ar aris naturaluri<br />
ricxvi?<br />
e) S simravlis elementebis saerTo raodenobis ra nawilia<br />
laTinuri asoebi? naturaluri ricxvebi?<br />
3 daasaxele ocze naklebi martivi ricxvebis simravlis yvela<br />
elementi. ramdeni elementia am simravleSi? Cawere es simravle<br />
elementebis CamonaTvalis saxiT.<br />
4 daasaxele evropis saxelmwifoTa simravlis ramdenime elementi.<br />
ekuTvnis Tu ara am simravles germania? brazilia? espaneTi?<br />
CineTi?<br />
5<br />
vTqvaT, F aris ukveci wesieri wiladebis simravle.<br />
sasrulia Tu ara es simravle?<br />
Semdegi Canawerebidan romelia marTebuli?<br />
a) 1 – 2 ∈ F ; b) 2 – 5 ∉F ; g) 5 – 3 ∈F ; d) 3 – 24 ∈F .<br />
6 vTqvaT, A qarTuli xmovani asoebis simravlea. Semdegi<br />
Canawerebidan romlebia marTebuli?<br />
7<br />
8 Tavi 1 • racionaluri ricxvebi<br />
a) b ∈ A; b) u ∈A; g) d ∉ A. d) u ∈A;<br />
qvemoT CamoTvlil simravleTagan romelia sasruli da romeli -<br />
usasrulo?<br />
a) orniSna naturaluri ricxvebis simravle;<br />
b) kenti naturaluri ricxvebis simravle;<br />
g) 100-is gamyofebis simravle;
d) 3-is jeradi naturaluri ricxvebis simravle;<br />
e) monakveTis wertilTa simravle;<br />
v) samkuTxedis wveroebis simravle.<br />
8 qvemoCamoTvlil simravleTagan romelia sasruli da romeli -<br />
usasrulo?<br />
a) mTeli ricxvebis simravle;<br />
b) luwi naturaluri ricxvebis simravle;<br />
g) 1000-is gamyofebis simravle;<br />
d) 10-is jeradi naturaluri ricxvebis simravle;<br />
e) mocemuli sxivis wertilTa simravle;<br />
v) mocemuli oTxkuTxedis gverdebis simravle.<br />
9 daasaxele 5-is jeradi naturaluri ricxvebis simravlis ramdenime<br />
elementi. aris Tu ara sasruli es simravle?<br />
10 gansazRvre sityvierad usasrulo simravle, romlis<br />
elementebia:<br />
a) 1, 3, 5, 7, ... ;<br />
b) 1, 2, 3, 4, ... ;<br />
g) 1, 4, 9, 16, ... ;<br />
d) 10, 100, 1000, 10000, ...<br />
11 gansazRvre sityvierad simravle, romlis elementebia:<br />
a) 2, 3, 5, 7, 11, 13 ... ;<br />
b) 9, 18, 27, 36, ... ;<br />
g) a, b, g, d, ... ;<br />
d) a, b, c, d, ...<br />
12 mocemulia oTxi simravle:<br />
A={b, r, a, z, i}, B={b, z, a, r, i}<br />
C={a, z, r, i} D={z, a, r, i}<br />
Semdegi Canawerebidan romeli Canawerebia marTebuli?<br />
a) A=B; b) A= C; g) B⊂ C; d) D⊂ A; e) b ∈A; v) b∉ B.<br />
13 ganixile A ={a, b, c, d, e} simravle da Cawere:<br />
a) A simravlis xmovnebis qvesimravle;<br />
b) A simravlis Tanxmovnebis qvesimravle;<br />
g) A simravlis ramdenime erTelementiani qvesimravle;<br />
d) A simravlis ramdenime oTxelementiani qvesimravle.<br />
14 Cawere A ={20, 3, 7 – , 0, 4, 5, 9<br />
8 1 – } simravlis<br />
3<br />
a) mTeli ricxvebis qvesimravle;<br />
b) naturaluri ricxvebis qvesimravle;<br />
g) kenti ricxvebis qvesimravle;<br />
d) 3-is jeradi ricxvebis qvesimravle;<br />
Tavi 1 • racionaluri ricxvebi 9
e) ramdenime orelementiani qvesimravle;<br />
v) ramdenime samelementiani qvesimravle.<br />
15 mocemuli diagramis mixedviT gaarkvie, romeli Canaweria<br />
marTebuli:<br />
16 vTqvaT, A skolis moswavleTa simravlea, B skolis warCinebul<br />
moswavleTa simravle. mocemuli diagramebidan romeli gamosaxavs<br />
A da B simravleebs Soris mimarTebas?<br />
a) b) g) d)<br />
A<br />
17 mocemuli diagramebidan romeli gamosaxavs A da B simravleebs<br />
Soris mimarTebas, Tu:<br />
a) A naturalur ricxvTa simravle, xolo B luw naturalur<br />
ricxvTa simravlea?<br />
b) A luw ricxvTa simravle, xolo B kent ricxvTa simravlea?<br />
g) A Svidis jeradi ricxvebis simravle, xolo B rvis jeradi<br />
ricxvebis simravlea<br />
a) b) g) d)<br />
18 ras udris n(A), Tu A<br />
a) qarTuli anbanis Tanxmovani asoebis simravlea;<br />
b) 9-is jeradi orniSna ricxvebis simravlea;<br />
g) 100-is gamyofebis simravlea;<br />
d) 30-is martivi gamyofebis simravlea.<br />
19 vTqvaT M M marTkuTxedebis simravlea,K K kvadratebis simravle,<br />
xolo O O oTxkuTxedebis simravle. Semdegi diagramebidan romeli<br />
gamosaxavs M M, K K da O O simravleebs Soris mimarTebas?<br />
a) b) g)<br />
M<br />
10 Tavi 1 • racionaluri ricxvebi<br />
O<br />
C<br />
A<br />
B<br />
K<br />
B<br />
A<br />
M<br />
a) A⊂B; b) A⊂C;<br />
g) C⊂A; d) B⊂A.<br />
B<br />
A<br />
B<br />
A<br />
B B A<br />
A<br />
B<br />
K<br />
O<br />
B<br />
A<br />
O<br />
A<br />
K<br />
M<br />
B
20 vTqvaT, A 3-is jeradi naturaluri ricxvebis simravlea, B 6-is<br />
jeradi naturaluri ricxvebis simravle, xolo C 15-is jeradi<br />
naturaluri ricxvebis simravle. Semdegi naxazebidan romeli<br />
gamosaxavs A, B da C simravleebs Soris mimarTebas?<br />
a) b) g)<br />
21 ra simravleebi SeiZleba iyos gamosaxuli mocemul naxazze?<br />
a) A - adamianebi, B - skolis moswavleebi,<br />
C – studentebi;<br />
b) A- cxovelebi, B- lomebi, C - vefxvebi;<br />
g) A - ricxvebi, B - martivi ricxvebi,<br />
C – luwi ricxvebi.<br />
B<br />
A<br />
d) A - ricxvebi, B - mTeli ricxvebi,<br />
C - naturaluri ricxvebi.<br />
C<br />
22 vTqvaT M M 2-is jeradi, F 5-is jeradi, xolo K K 10-is jeradi<br />
ricxvebis simravlea. gamosaxe am simravleebs Soris mimarTeba<br />
diagramis saSualebiT.<br />
23<br />
Cawere Sereuli ricxvis saxiT:<br />
a) 7 – 2 b) 1<br />
–9<br />
e) 17 – v)<br />
16<br />
13 –<br />
9<br />
24<br />
Sekvece wiladi:<br />
a) 7 – 14 b) 8 – 16<br />
e) 18 – 54<br />
C<br />
A<br />
B<br />
v) 27 – 63<br />
C<br />
g) 23 – 13<br />
z) 23 – 11<br />
g) 25 – 35<br />
z) T)<br />
d) 69 – 25<br />
T) 68 – 24<br />
d) 69 – 46<br />
25<br />
ricxvebis: 1, 2, 3, gamoyenebiT Seadgine yvela SesaZlo<br />
a) wesieri wiladi; b) arawesieri wiladi.<br />
26<br />
rogor Seicvleba wiladis mniSvneloba, Tu mis mricxvels:<br />
a) gavzrdiT samjer? b) SevamcirebT orjer?<br />
27<br />
rogor Seicvleba wiladis mniSvneloba, Tu mis mniSvnels:<br />
a) gavzrdiT samjer? b) SevamcirebT orjer?<br />
A<br />
28 mocemuli wiladis mricxveli gaamravles 15-ze. ra ricxvze unda<br />
gaamravlon wiladis mniSvneli, rom miiRon mocemul wiladze<br />
B<br />
A<br />
B<br />
C<br />
Tavi 1 • racionaluri ricxvebi 11
a) 5-jer didi ricxvi?<br />
b) 3-jer patara ricxvi?<br />
29 Seadare ricxvebi:<br />
a) 1 – 2 da 3 – 4 b) 3 – 10 da 2 – 15 g) 9 – 16 da 9 – 11 d)1 1 – 2 da 1 3 – 4<br />
e) 11 – da<br />
21 3 – v)<br />
14 2 – da<br />
5 13 – z) da T) 7 da<br />
40 35 –<br />
6<br />
30<br />
eka, Tamuna da naTia biblioTekars wignebis aRdgenaSi miexmarnen.<br />
ekam wignebis 1 – aRadgina, Tamunam _<br />
80 5 – , xolo naTiam _<br />
16 3 – .<br />
8<br />
romelma gogonam aRadgina meti wigni?<br />
31<br />
gigim 20 amocanidan sworad 15 amocana amoxsna, nikam ki 10<br />
amocanidan mxolod 3 amocana amoxsna SecdomiT. romeli<br />
moswavlis Sedegia ukeTesi?<br />
32<br />
cxrilSi mocemulia giorgis saskolo weris Sedegebi sam saganSi:<br />
1 gamoTvaleT<br />
2 gamoTvaleT<br />
3 wrewirze<br />
sagani<br />
12 Tavi 1 • racionaluri ricxvebi<br />
qulaTa<br />
maqsimumi<br />
dagrovili<br />
qulebi<br />
qarTuli ena 100 83<br />
qimia 50 42<br />
maTematika 25 20<br />
gaarkvie, romeli sagani icis giorgim ukeTesad.<br />
jgufuri samuSao<br />
n(A), Tu A<br />
a) 12-is gamyofebis simravlea;<br />
b) 10-is jeradi orniSna naturaluri ricxvebis simravlea;<br />
g) 35-is martiv gamyofTa simravlea.<br />
n(A) - n(B), sadac<br />
a) A samniSna naturaluri ricxvebis, xolo B orniSna<br />
naturaluri ricxvebis simravlea;<br />
b) A im orniSna ricxvebis simravlea, romelTa CanawerSi<br />
mxolod kenti cifrebia gamoyenebuli, xolo B im orniSna<br />
ricxvebis simravle, romelTa CanawerSi mxolod luwi cifrebia<br />
gamoyenebuli.<br />
moniSnulia 5 wiTeli da 5 lurji wertili. A im<br />
monakveTebis simravlea, romelTa boloebidan erTi wiTeli,<br />
meore ki lurji wertilia, xolo B im monakveTebis simravle,<br />
romelTa boloebi lurji wertilebia. romelia meti n(A) Tu<br />
n(B)? ramdeniT?
3.2 ricxvis procentuli Cawera<br />
mizani:<br />
aTwiladis 100-ze<br />
gamravlebisas mZime<br />
ori cifriT marjvniv<br />
gadagvaqvs.<br />
viswavloT:<br />
1. ricxvis procentuli Cawera;<br />
2. procentis Cawera wiladis saxiT;<br />
3. ricxvis procentis gamoTvla.<br />
monacemebis gadmosacemad xSirad ricxvis procentul<br />
Canawers iyeneben.<br />
ricxvis procentuli Canaweri gviCvenebs ricxvSi<br />
measedebis raodenobas, magaliTad: 1 = 50<br />
2 100 =50%;<br />
0,75=75%.<br />
procentis saxiT Casawerad ricxvs vamravlebT 100-ze da vuwerT<br />
procentis niSans: `%„.<br />
aTwiladis 100-ze<br />
gayofisas mZime ori<br />
cifriT marcxniv<br />
gadagvaqvs.<br />
magaliTi 1. CavweroT procentis saxiT ricxvi: a) 3<br />
1 ;<br />
4<br />
b) 0,432.<br />
3 3<br />
amoxsna: a) 1 = (1 × 100)% = 175% ;<br />
4 4<br />
b) 0,432 = (0,432× 100)% = 43,2% .<br />
aseve unda moviqceT nawilis procentuli CawerisaTvis. magaliTad,<br />
ricxvis 3<br />
5 nawili igivea, rac am ricxvis 3 × 100 procenti, anu 60%; 0,36<br />
5<br />
nawili igivea, rac 0,36× 100 procenti, anu 36%. piriqiT,<br />
ricxvis 1% am<br />
ricxvis measedi<br />
nawilia.<br />
procentis saxiT Cawerili ricxvis wiladis saxiT Casawerad<br />
sakmarisia, procentis ricxvi gavyoT 100-ze.<br />
magaliTi 2. CavweroT wiladis saxiT: a) 48%; b)<br />
325,8%.<br />
12<br />
amoxsna: a) 48% = 48:100 = ;<br />
25<br />
b) 325,8%=325,8:100 = 3,258.<br />
amocana 1. levanis klasSi 30 moswavlea, maTgan 60% gogonaa. ramdeni<br />
gogonaa levanis klasSi?<br />
ricxvis 100% TviT<br />
am ricxvis tolia.<br />
60 3<br />
amoxsna: CavweroT 60% wiladis saxiT: 60% = = .<br />
100 5<br />
e. i. leva nis TanaklaselTa 3<br />
nawili, anu<br />
3<br />
30× = 18<br />
5<br />
moswavle gogonaa.<br />
5<br />
pasuxi: levanis klasSi 18 gogonaa.<br />
a ricxvis p procenti a ricxvis p - 100 nawilis, anu<br />
magaliTi 3. vipovoT 120-is 2,5%.<br />
120× 2,5<br />
amoxsna: = 3 .<br />
100<br />
a x p<br />
- 100 -is tolia.<br />
Tavi 3 • monacemebi 155
upasuxe kiTxvebs<br />
156 Tavi 3 • monacemebi<br />
1. rogor CavweroT ricxvi procentis saxiT?<br />
2. rogor CavweroT procentulad Cawerili ricxvi wiladad?<br />
3. ra kavSiria ricxvis nawilsa da procents Soris?<br />
4. rogor gamovTvaloT mocemuli ricxvis procenti?<br />
5. Camoayalibe aTwiladis procentuli Caweris wesi.<br />
6. Camoayalibe procentis aTwiladad Caweris wesi.<br />
savarjiSoebi<br />
1 Cawere mocemuli ricxvi procentis saxiT:<br />
a) 1; b) 4 1<br />
7<br />
13 16<br />
; g) ; d) ; e) ; v) ;<br />
5 10 10 20 25<br />
z) 19<br />
; T)<br />
1<br />
1 ; i)<br />
50 2 3<br />
2 ; k) 0,5; l) 0,26; m) 0,256;<br />
4<br />
n) 1,01; o) 1,15; p) 2,2356.<br />
2 procentulad mocemuli ricxvi Cawere wiladis saxiT:<br />
a) 3%; b) 17%; g) 10%; d) 1%; e) 25%; v) 5%; z) 50%; T) 85%;<br />
i) 115%; k) 27,5%; l) 32,215%; m) 1<br />
3 % ; n)<br />
3<br />
1<br />
33 % ; o) 0,01%.<br />
3<br />
3 ricxvis ramdeni procentia am ricxvis<br />
a) 1<br />
1<br />
11<br />
nawili? b) nawili? g) nawili? d) 0,2 nawili?<br />
2 4 20<br />
e) 0,75 nawili? v) 0,99 nawili? z) 0,125 nawili? T) 1<br />
3 nawili?<br />
4 ricxvis ra nawilia am ricxvis<br />
a) 25%? b) 50%? g) 10%? d) 100%? e) 67%? v) 2,5%? z) 0,1%?<br />
5 gamoTvale:<br />
a) 20-is 25%; b) 45-is 10%; g) 200-is 5%; d) 7,2-is 50%;<br />
e) 3,5-is 1%; v) 1,7-is 80%; z) 1<br />
1<br />
-is 3%; T) 3 -is 33%.<br />
3 33<br />
6 naxazis mixedviT daadgine, wris ramdeni procentia<br />
gaferadebuli.<br />
a) b) g) d)
7 romelia meti:<br />
a) 12-is 25% Tu 20-is 10% ?<br />
b) 10-is 5% Tu 15-is 3%?<br />
g) 3,6-is 50% Tu 50-is 3,6%?<br />
d) 1<br />
1<br />
-is 100% Tu 100-is<br />
10 10 %?<br />
8 Seavse cxrili<br />
procenti aTwiladi wiladi<br />
10%<br />
0,25<br />
3/8<br />
15,6%<br />
0,268<br />
2/25<br />
9 Saqris Warxlisagan miRebuli Saqris wona Warxlis wonis 13%-ia.<br />
ramdeni kg Saqari miiReba 200kg Saqris Warxlisagan?<br />
10 Saqris lerwmisagan miRebuli Saqris wona lerwmis wonis 15%-ia.<br />
ramdeni kg Saqari miiReba 1tona Saqris lerwmisagan?<br />
11 saqarTvelos farTobi 69700 kvadratuli kilometria. misi<br />
teritoriis 54% mTebs ukavia. ramdeni kvadratuli kilometri<br />
ukavia mTebs?<br />
12 klasis 25 moswavlidan gakveTils eswreboda<br />
80%. ramdeni moswavle ar eswreboda<br />
gakveTils?<br />
13 ramdeni TeTria 2 laris 20%?<br />
14 ramdeni kilometri darCeba gasavleli<br />
mgzavrs gzis 40%-is gavlis Semdeg, Tu<br />
mTeli gzis sigrZe 5 kilometria?<br />
15 klasis 30 moswavlidan 60% gogonaa. gogonaTa 50% friado sania.<br />
ramdeni gogonaa friadosani?<br />
16 cxrilSi mocemulia, saqarTvelos teritoriis ramdeni procenti<br />
ukavia mis xuT regions:<br />
qarTli 19,3%<br />
kaxeTi 15,6%<br />
afxazeTi 12,7%<br />
imereTi 8,7%<br />
samegrelo 6,5%<br />
Tavi 3 • monacemebi 157
a) gadaixaze cxrili, monacemebis mixedviT gamoTvale regionebis<br />
farTobebi da miRebuli Sedegebi Seitane cxrilis mesame svetSi<br />
(gamoTvlebis gasamartiveblad gamoiyene kalkulatori).<br />
b) saqarTvelos teritoriis ramden procents Seadgens danarCeni<br />
regionebi?<br />
g) miRebuli Sedegebis mixedviT Seadgine svetovani diagrama.<br />
17 yuTSi 20 burTia. burTebis 20% TeTria, 25% _ Savi, danarCeni ki<br />
_ wiTeli. burTebis ramdeni procentia: a) wiTeli? b) TeTri an<br />
Savi? g) ramdeni burTia TeTri an wiTeli?<br />
18 maRaziaSi erTi Tvis ganmavlobaSi<br />
12 t bostneuli<br />
gaiyida. svetovan<br />
diagramaze naCvenebia<br />
gayiduli zogierTi<br />
saxeobis bostneulis wili<br />
procentebSi.<br />
a) ramdeni procentia im<br />
bostneulis wili, romelic<br />
araa diagramaze<br />
naCvenebi?<br />
b) TiToeuli saxeobis ramdeni tona bostneuli gaiyida?<br />
19 ramdeni laria 15 laris 2<br />
nawilis 15%?<br />
3<br />
20 daasabuTe, rom a ricxvis b procenti da b ricxvis a procenti<br />
erTmaneTis tolia.<br />
21 gias 10 lari hqonda. am Tanxis 30%-iT man ori fila Sokoladi<br />
iyida, darCenili Tanxis 20%-iT ki _ 14 cali saReWi rezina.<br />
158 Tavi 3 • monacemebi<br />
a) ra Rirda erTi fila Sokoladi?<br />
b) ra Rirda erTi saReWi rezina?<br />
g) ra Tanxa darCa gias?<br />
gayiduli bostneuli procentebSi<br />
35<br />
30<br />
25<br />
20<br />
15<br />
10<br />
5<br />
0<br />
kartofili<br />
kombosto<br />
stafilo<br />
22 gzis sworxazovani monakveTis gaswvriv dargulia oTxi xe. manZili<br />
mezobel xeebs Soris 10, 45 da 60 metria. sul cota, ramdeni<br />
xis dargva mogviwevs damatebiT, nebismier or mezobel xes Soris<br />
manZili toli rom aRmoCndes?<br />
23 gvanca, lana da salome erTi saxlis pirvel, meore da mesame<br />
sarTulze cxovroben (ara aucileblad am Tan mimdevrobiT). maTgan<br />
erTi meeqvse, meore meSvide da mesame ki – merveklaselia.<br />
• meeqvseklaseli gogona cxovrobs pirvel sarTulze.<br />
xaxvi
• meSvideklaseli gogona cxovrobs gvancas binis zemoT.<br />
• merveklaseli gogona cxovrobs lanas binis qvemoT.<br />
romeli gogona romelklaselia da meramdene sarTulze<br />
cxovrobs?<br />
24 nikam daxarjuli bunebrivi gazis safasuri 125 lari gadaixada.<br />
gadaxdilma Tanxam misi xelfasis 0,12 nawili Seadgina. ra xelfasi<br />
aqvs nikas?<br />
25 salomem 135 lari eleqtroenergiis, 38lari dasufTavebis da 12<br />
lari sasmeli wylis moxmarebis safasuri gadaixada. gadaxdilma<br />
Tanxam misi xelfasis 0,2 nawili Seadgina. ra xelfasi aqvs<br />
salomes?<br />
26 ipove proporciis ucnobi wevri:<br />
2 4<br />
2 4<br />
x 2 7 14 5 y<br />
a) = ; b) = ; g)1, = ; d) 3 = 5 ; e)<br />
9 3 t 9 4 1, 2 1<br />
1<br />
z<br />
2<br />
27 ricxvi 512 dayavi<br />
a) 5-is, 11-is da 16-is proporciul nawilebad<br />
b) 4 : 5 : 7 SefardebiT.<br />
28 gadaixaze da Seavse cxrili<br />
a) Tu cnobilia, rom x da y sidideebs Soris<br />
pirdapirproporciuli damo kidebulebaa:<br />
4 × 1, 5 1, 4 : 0, 7<br />
=<br />
.<br />
x 0,75 + 1,25<br />
x 1 2 0,5 5 30 3 0,2<br />
y 3 15 36 0,6 2 0,3<br />
b) Tu cnobilia, rom x da y sidideebs Soris ukuproporciuli<br />
damo kidebulebaa:<br />
x 2 1 0,5 5 11 3 0,2<br />
y 10 15 100 0,5 2 0,9<br />
Tavi 3 • monacemebi 159
3.3 rogor vipovoT ricxvi misi procentis mixedviT<br />
mizani:<br />
160 Tavi 3 • monacemebi<br />
viswavloT:<br />
1. ricxvis gamoTvla misi procentiT;<br />
2. mocemuli procentiT gazrdiT an SemcirebiT<br />
miRebuli ricxvis gamoTvla;<br />
3. praqtikuli amocanebis amoxsna fasis momatebadaklebasa<br />
da xarjTaRricxvaze.<br />
ganvixiloT amocana, romlis amosaxsnelad dagvWirdeba ricxvis povna<br />
misi mocemuli procentiT.<br />
amocana 1. 20% saSemosavlos daqviTvis Semdeg maswavleblis xelfasma<br />
360 lari Seadgina. ramdenia maswavleblis xelfasi daqviTis gareSe?<br />
amoxsna: maswavleblis xelfasi daqviTvis gareSe CavTvaloT 100%-ad.<br />
maSin 20%-is daqviTvis Semdeg darCeboda xelfasis 80%, rac amocanis<br />
pirobis Tanaxmad 360 laris tolia.<br />
amocanis amosaxsnelad saWiroa vipovoT ricxvi, romlis 80% 360-is<br />
tolia.<br />
vTqvaT, es ricxvia x, maSin misi 80% iqneba 0,8x. amocanis pirobis<br />
Tanaxmad vwerT:<br />
0,8x = 360<br />
Tu am tolobis orive mxares gavyofT 0,8-ze, miviRebT:<br />
x = 360 : 0,8 = 450.<br />
pasuxi: maswavleblis xelfasi daqviTvis gareSe 450 laria.<br />
imisaTvis, rom vipovoT x ricxvi, romlis p procenti a ricxvis<br />
tolia, saWiroa:<br />
p<br />
I. p% CavweroT wiladis saxiT (miviRebT -s).<br />
100<br />
p<br />
II. a ricxvi gavyoT wiladis saxiT Caweril p% -ze (anu -ze).<br />
100<br />
magaliTi 1. vipovoT x ricxvi, romlis<br />
a) 25% udris 20-s; b) 3,125% udris 1,5-s.<br />
I. 25%=0,25 I. 3,125%=0,03125<br />
I I. x = 20 : 0,25 = 80 I I. x = 1,5:0,03125 =48<br />
praqtikaSi xSirad gvxvdeba amocana, romelSic<br />
mocemuli ricxvi (sidide) izrdeba an mcirdeba mocemuli<br />
procentiT. magaliTad:<br />
amocana 2. benzinis fasma 10%-iT moimata. ra Rirs litri<br />
benzini amJamad, Tu momatebamde misi fasi 2,1 lari iyo?<br />
amoxsna: 2,1 laris 10%-iT gazrdaSi igulisxmeba 2,1 laris-<br />
Tvis misive 10%-is damateba, anu 2,1l + 2,1l × 0,1=2,31l.<br />
ricxvis 10%iT<br />
gazrda niSnavs<br />
am ricxvis 1,1-ze<br />
gamravlebas.
imave amocanis amoxsna SegviZlia Semdegi msjelobiT:<br />
Tu momatebamde benzinis fass, anu 2,1 lars, CavTvliT 100%-ad, maSin<br />
10%-is momatebis Semdeg fasi gaxdeba 110%. 2,1-is 110% ki udris:<br />
2,1× 1,10 = 2,31 .<br />
pasuxi: amJamad benzini 2lari da 31 TeTri Rirs.<br />
a ricxvis p% -iT gazrdiT miRebuli ricxvis gamosaTvlelad<br />
saWiroa:<br />
I. gamovTvaloT a ricxvis p % , anu .<br />
I I. a ricxvs davumatoT misi p % , anu , miviRebT<br />
magaliTi 2. ra ricxvi miiReba, Tu12-s 5% -iT gavzrdiT?<br />
12× 5<br />
I. 12-is 5% udris =0,6.<br />
100<br />
I I. 12 + 0,6 = 12,6.<br />
am magaliTSi imave Sedegis misaRebad sakmarisia gamoviTvaloT 12-is<br />
(100 + 5)%, anu 12 gavamravloT 1,05-ze:<br />
12× 1,05 = 12,6.<br />
amocana 3. maRaziam saSobaod 60%-iani fasdakleba gamoacxada. ra fasSi<br />
gaiyideba fexsacmeli, romelic fasdaklebamde 120 lari Rirda?<br />
amoxsna: 60% -iani fasdakleba niSnavs Zveli fasis<br />
Semcirebas am fasis 60%-iT. gamovTvaloT 120-is<br />
60%: 120× 0,60 = 72 . e. i. fasi 72 lariT Semcirebula.<br />
amitom axali fasi iqneba<br />
120- 72 = 48 (lari).<br />
SevniSnoT, rom es amocana SeiZleba Semdegi<br />
msjelobiTac amovxsnaT: sawyisi fasi 120 lari<br />
CavTvaloT 100%-ad. maSin 60%-iT fasdaklebis Semdeg<br />
miviRebT 40%-s. 120-is 40% ki udris: 120× 0,40 = 48 .<br />
ricxvis 60%-iT<br />
Semcireba niSnavs pasuxi: fexsacmeli 48 larad gaiyideba.<br />
am ricxvis 0,4-ze<br />
a<br />
gamravlebas.<br />
ricxvis p% -iT SemcirebiT miRebuli ricxvis gamosaTvlelad<br />
saWiroa:<br />
a× p<br />
I. gamovTvaloT a ricxvis p % , anu .<br />
100<br />
a× p<br />
I I. a ricxvs gamovakloT misi p % , anu a − .<br />
100<br />
magaliTi 3. ra ricxvi miiReba, Tu 80-s SevamcirebT 25%-iT?<br />
I. 80-is 25% udris 80× 0,25=20.<br />
I I. 80 - 20 =60.<br />
am magaliTSi imave Sedegis misaRebad sakmarisia gamoviTvaloT 80-is<br />
(100 – 25)%, anu 80 gavamravloT 0,75-ze: 80× 0,75 = 60.<br />
Tavi 3 • monacemebi 161
upasuxe kiTxvebs<br />
1. rogor gamoviTvaloT ricxvi misi procentis mixedviT?<br />
2. ras niSnavs ricxvis gazrda mocemuli procentiT?<br />
3. ras niSnavs ricxvis Semcireba mocemuli procentiT?<br />
4. ra ricxvi miiReba a ricxvis p% -iT gazrdiT?<br />
5. ra ricxvi miiReba a ricxvis p% -iT SemcirebiT?<br />
6. rogor gamoiTvli umoklesi gziT 6-is 40%-iT gazrdiT miRebul ricxvs?<br />
7. rogor gamoiTvli umoklesi gziT 6-is 40%-iT SemcirebiT miRebul<br />
ricxvs?<br />
8. miiReba Tu ara igive a ricxvi, Tu a -s gavzrdiT 10%-iT, Semdeg miRebul<br />
ricxvs SevamcirebT 10%-iT?<br />
9. ramdeni procentiT gaizrdeba sabolood ricxvi, romel sac gavzrdiT<br />
jer 10%-iT, xolo Semdeg kidev 20%-iT?<br />
savarjiSoebi<br />
1 ipove ricxvi, romlis<br />
a) 10% udris 5-s; b) 5% udris 10-s; g) 25% udris 25-s;<br />
d)<br />
162 Tavi 3 • monacemebi<br />
1<br />
33 %<br />
3 udris 100-s; e) 0,1% udris 1-s; v) 120% udris 12-s.<br />
2 ra ricxvi miiReba<br />
a) 10-is 10%-iT gazrdiT? b) 36-is 50%-iT<br />
gazrdiT?<br />
g) 30-is 200%-iT gazrdiT? d) 0,48-is 1%-iT<br />
gazrdiT?<br />
e) 10-is 10%-iT SemcirebiT? v) 36-is 50%-iT<br />
SemcirebiT?<br />
z) 30-is 20%-iT SemcirebiT? T) 200-is 13%-iT<br />
SemcirebiT?<br />
3 vanos klaselTa 25% qeraTmiania. ramdeni moswavlea vanos klasSi,<br />
Tu maT Soris 8 moswavlea qeraTmiani?<br />
4 ninos klaselTa 20% friadosania. ramdeni moswavlea ninos<br />
klasSi, Tu maT Soris 6 moswavlea friadosani?<br />
ricxvis p%-iT<br />
gazrda igivea, rac<br />
am ricxvis 100 + p<br />
-ze<br />
100<br />
gamravleba, xolo<br />
ricxvis p%-iT<br />
Semcireba igivea, rac<br />
am ricxvis 100 − p<br />
-ze<br />
100<br />
gamravleba.<br />
5 medeam kviras Tavisufali drois 40% kompiuterul TamaSebs<br />
dauTmo. ramdeni saaTi hqonda Tavisufali im dRes medeas, Tu is<br />
2 saaTis ganmalobaSi kompiuteriT TamaSobda?<br />
6 Tanaklaselebma eqskursiisaTvis Segrovili Tanxis 30% samgzavro<br />
xarjebisaTvis gadades, darCenili 140 lariT ki sursaTi SeiZines.<br />
ra Tanxa Seagroves Tanaklaselebma eqskur siisaTvis?
7 moswavlis CanTa 40%-iani fasdaklebis<br />
Semdeg 18 larad gaiyida. ra Rirda<br />
CanTa fasdaklebamde?<br />
8 50%-iani saSobao fasdaklebiT<br />
fexsacmlis fasma 45 lari Seadgina. ra<br />
Rirda fexsacmeli fasdaklebamde?<br />
9 policielis xelfasma 40% -iT gazrdis<br />
Semdeg 1260 lari Seadgina. ra xelfasi<br />
hqonda policiels gazrdamde?<br />
10 bankSi Setanil Tanxas yovelwliurad<br />
9% emateba. ra Tanxa daugrovdeba<br />
laSas erT welwadSi, Tu man bankSi 500 lari Seitana?<br />
11 wignis maRazia gamomcemlobidan miRebul wignebs gamomcemlobis<br />
fasze 25%-is damatebiT yidis. ra mogeba darCeba maRazias 1000<br />
wignis gayidvis Semdeg, Tu gamom cemlobis mier dadgenili erTi<br />
wignis fasi 8 laria?<br />
12 yava moxalvisas wonis 12%-s kargavs. ramdeni grami moxaluli yava<br />
miiReba 200 grami nedli yavisagan?<br />
13 xorblisagan misi masis 75% fqvili miiReba. ramdeni kg xorbalia<br />
saWiro 3kg fqvilis misaRebad?<br />
14 puris comi gamocxobisas wonis 20%-s<br />
iklebs.<br />
a) ramdeni 400 -gramiani puri<br />
gamocxveba 10 kg comisagan?<br />
b) ramdeni kg comia saWiro 5 kg puris<br />
gamosacxobad?<br />
15 qalaqSi amJamad 10 000 mosaxlea. mosaxleobis ricxvi<br />
yovelwliurad 10%-iT izrdeba. ramdeni mosaxle iqneba am qalaqSi<br />
a) erTi wlis Semdeg?<br />
b) ori wlis Semdeg?<br />
16 biznesmenma Tavisi fulis 40% Rvinis warmoebaSi daabanda,<br />
saidanac 60% mogeba miiRo, danarCeni Tanxa sakonservo<br />
warmoebaSi Cado, saidanac 50% mogeba miiRo. ramdeni laris<br />
mflobeli gaxda biznesmeni, Tu misi sawyisi Tanxa 20 000 lars<br />
Seadgenda?<br />
17 ramdeni procentiT gaizrdeba marTkuTxedis formis miwis nakveTis<br />
farTobi, Tu mis sigrZes 10, xolo siganes 20 procentiT gavzrdiT?<br />
Tavi 3 • monacemebi 163
18 Tambaqos nawarmi jer 20, Semdeg 40 procentiT gaZvirda. sul<br />
ramdeni procentiT gaZvirda Tambaqos nawarmi?<br />
19 wriul diagramaze mocemulia<br />
saqarTvelos mosaxleobis ricxovneba<br />
procentebSi religiuri aRmsareblobis<br />
mixedviT 2002 wlis aRweris<br />
monacemebiT.<br />
a) diagramis saSualebiT gamo Tvale<br />
sxvadasxva aRmsareb lobis mqone<br />
mosaxleobis rao denoba (saqarTvelos<br />
mTeli mosaxleoba 2002 wlis aRweriT<br />
4 356 000 kacs Seadgenda).<br />
164 Tavi 3 • monacemebi<br />
b) mocemuli diagramis mixedviT Seadgine<br />
cxrili, romelic gvi Cvenebs mosaxleobis<br />
rao deno bas aRmsareblobis mixedviT<br />
rogorc procentebSi, ise ricxobrivad.<br />
20 gamoTvale:<br />
a) 1sT-is ra nawilia 3wT; b) 1m-is ra nawilia 20 sm;<br />
g) 1kg-is ra nawilia 300g; d) 1t-is ra nawilia 250kg;<br />
e) 1m2 –is ra nawilia 25dm2 ; v) 1ha-is ra nawilia 4000m2 ;<br />
z) 1dm3-is ra nawilia 10sm3 ; T) 1m3 i) 1sT-is ra nawilia 1wm;<br />
–is ra nawilia 100l;<br />
aba, scade!<br />
daexmare ganawilebaSi<br />
1. maranSi 7 qvevri savsea RviniT, 7 ki – naxevradsavse, xolo<br />
danarCeni carielia. Rvino da qvevrebi samma Zmam Tanabrad<br />
unda gainawilos ise, rom TiToeuls toli raodenobis qvevrebi<br />
da Rvino unda ergos. ganawilebisas qvevridan qvevrSi Rvinis<br />
gadasxma ar SeiZleba. rogor u nda gainawilon?<br />
2. rvalitriani qila rZiTaa savse. rZe orma qalma Tanabrad unda<br />
gainawilos. erTs mxolod 5-litriani, xolo meores 3-litriani<br />
carieli bidoni aqvs. rogor gainawilon?<br />
4%<br />
1%<br />
10% 1%<br />
84%<br />
marTlmadidebluri<br />
kaTolikuri<br />
grigorianuli<br />
mahmadianuri<br />
sxva
3.4 ori sididis fardobis procentuli warmodgena<br />
mizani:<br />
viswavloT:<br />
1. ori sididis fardobis procentuli Cawera;<br />
2. monacemTa gamosaxva fardobis procentiT.<br />
praqtikuli amocanebis amoxsnisas sidideebis Sefardebas xSirad<br />
procentebiT gamosaxaven.<br />
amocana 1. rZe maRalcximianad iTvleba, Tu masSi cximis Semcve loba<br />
aranakleb 4,5%-ia. qarxanaSi Semosuli rZis cximianobis dasadge nad<br />
Seamowmes 1,5kg rZis sinji. aRmoCnda, rom igi 63g cxims Seicavda.<br />
SeiZleba Tu ara, es rZe maRalcximianad CaiTvalos?<br />
amoxsna. gamovsaxoT rZisa da masSi Semavali cximis wona erTi da imave<br />
erTeulebSi:<br />
1,5kg=1500g.<br />
gamoTvlebi<br />
kalkulatoriT<br />
vipovoT, rZis wonis ra nawils warmoadgens cximis<br />
wona da gamovsaxoT igi procentebSi:<br />
63 ÷ 1500 x 100 = 4,2<br />
.<br />
daskvna: am rZeSi cximis Semcveloba 4,2%-ia. maSasadame, rZe maRalcximiani<br />
ar aris.<br />
amocana 2. erTma kalaTburTelma matCis ganmavlobaSi<br />
20 tyorcna ganaxor ciela, romelTagan 15 tyorcna<br />
Sedegiani iyo, xolo meorem 40-jer tyorcna, aqedan 28<br />
Caagdo kalaTSi. romeli kalaTburTeli ufro zustad<br />
tyorcnis?<br />
amoxsna. imis gasarkvevad, Tu ra sizustiT tyorcnidnen<br />
kalaT burTe lebi, daviTvaloT maT mier natyorcni<br />
burTebis ramdeni procenti moxvda kalaTSi.<br />
gamoTvlebi CavataroT or etapad:<br />
I etapi: gavigoT, TiToeuli kalaTburTelis mier natyorcni<br />
burT ebis raodenobis ra nawils Seadgens Cagdebuli<br />
burTebis raodenoba:<br />
pirveli kalaTburTeli: 15 : 20 = 0, 75<br />
meore kalaTburTeli: 28 : 40 = 0, 7<br />
II etapi: miRebuli ricxvebi CavweroT procentis saxiT:<br />
pirveli kalaTburTeli: 0, 75 = ( 0, 75 × 100)% = 75%<br />
meore kalaTburTeli: 0, 7 = (0, 7 × 100)% = 70%<br />
daskvna: pirveli kalaTburTelis tyorcnis sizustea 75%, meoresi ki<br />
_ 70%. maSasadame, pirveli kalaTburTeli ufro zustad tyorcnis<br />
burTs.<br />
Tavi 3 • monacemebi 165
imisaTvis, rom gavigoT, a sidide b sididis ramdeni procentia,<br />
saWiroa:<br />
I. Tu es sidideebi zomis gansxvavebul erTeulebSia mocemuli,<br />
gadaviyvanoT isini erTsa da imave erTeulebSi.<br />
II. gavyoT a sidide b -ze da ganayofi warmovadginoT procentis saxiT,<br />
a<br />
e. p%<br />
i. = ×gavamravloT 100 100-ze:<br />
b<br />
a<br />
p%<br />
= × 100<br />
b<br />
ama Tu im narevSi (xsnarSi, naerTSi) raime nivTierebis procentis saxiT<br />
Caweril wils koncentracia ewodeba. magaliTad, ganxilul amocana1-Si<br />
gamoTvlili cximis koncentracia rZeSi 4,2%-is toli iyo.<br />
amocana 3. 90%-iani koncentraciis spirtis 3 litr xsnars 2 litri<br />
sufTa wyali daamates. ra koncentraciis xsnari miiRes?<br />
amoxsna. 3 litri xsnari 90%, anu 3⋅ 0,9 =2,7litr spirts Seicavda.<br />
wylis damatebis Semdeg xsnaris moculoba gaxda 5 litri, spirtis<br />
raodenoba ki ar Secvlila. amitom miRebuli xsnaris koncentracia<br />
iqneba<br />
2,7<br />
× 100 = 54% .<br />
5<br />
pasuxi: miRebuli xsnaris koncentraciaa 54%.<br />
gaxsovdes!<br />
procentebiT SeiZleba gamoisaxos mxolod erTi da imave erTeulebiT<br />
gazomili sidideebis Sefardeba.<br />
upasuxe kiTxvebs<br />
1. ra SemTxvevaSi SeiZleba warmovadginoT ori sididis Sefardeba<br />
procentis saxiT?<br />
2. rogor gamovsaxoT erTi sidide meore sididis procentebiT?<br />
3. Seicvleba Tu ara pirveli amocanis pasuxi, Tu cximisa da rZis wonas<br />
gamovsaxavT wonis sxva erTeulebiT (magaliTad, kilogramebiT)?<br />
4. ras ewodeba narevis koncentracia?<br />
5. Semcirdeba Tu gaizrdeba spirtis xsnaris koncentracia da ramdenjer,<br />
Tu xsnars imave moculobis wyals SevurevT?<br />
166 Tavi 3 • monacemebi
savarjiSoebi<br />
1 gamoTvale, a ricxvi b ricxvis ramdeni procentia, Tu:<br />
a) a = 9, b = 36 b) a = 15, b = 75<br />
g) a = 2, 4, b = 4,8 d) a = 10, b = 5<br />
2 gamoTvale, a sidide b sididis ramdeni procentia, Tu:<br />
a) a = 12 wT, b = 60wT b) a = 1, 5 kg, b = 30kg<br />
5<br />
g) a = dm, b = 2m d) a = 0, 3 lari, b = 40TeTri<br />
3 qvemoT CamoTvlilTagan romeli sidideebis fardoba<br />
SeiZleba gamoisaxos procentebiT? SesaZleblobis<br />
SemTxvevaSi gamoTvale:<br />
a) 3,5 kg da 4 lari; b) 16 wuTi da 1,2 saaTi;<br />
g) 2m 2 da 3,2m; d) 7sm da 2dm.<br />
4 cxrilSi moyvanilia <strong>VII</strong> klasis 25 moswavlis gamokiTxvis<br />
Sedegebi, Tu vin sportis romel saxeobas misdevs:<br />
sportis moswavleTa moswavleTa<br />
saxeoba raodenoba saerTo<br />
raodenobis<br />
procenti<br />
fexburTi 8<br />
kalaTburTi 3<br />
Widaoba 2<br />
Wadraki 4<br />
curva 1<br />
arc erTi 7<br />
a) gadaixaze cxrili da Seavse misi bolo sveti;<br />
b) monacemebis mixedviT aage svetovani diagrama;<br />
g) Seadgine analogiuri cxrili Seni klasis mixedviT.<br />
5 gamoTvale a da b ricxvebis procentuli fardoba, Tu:<br />
a) a ricxvi b ricxvis 2/5 nawilia;<br />
b) a ricxvi b ricxvis 3/10 nawilia;<br />
g) a ricxvi b ricxvis 13/14 nawilia;<br />
d) a ricxvi b ricxvis 0,9 nawilia;<br />
e) a ricxvi b ricxvis meoTxedia;<br />
v) a ricxvi orjer metia b ricxvze.<br />
Tavi 3 • monacemebi 167
6 klasSi 12 biWi da 15 gogonaa. klasis moswavleTa ramdeni<br />
procentia gogona?<br />
7 benzingasamarT sadgurs 3 tona benzinis maragi hqonda.<br />
sadgurma 0,54 tona benzini gayida. gamoTvale, mTeli<br />
maragis ramdeni procenti darCa gasayidi.<br />
8 cxrilSi mocemulia dedamiwis kontinentTa farTobi<br />
kvadratul kilometrebSi.<br />
kontinentis<br />
kontinentis<br />
farTobi<br />
kontinenti xmeleTis saerTo<br />
farTobi farTobis<br />
procenti<br />
evrazia 55119000<br />
CrdiloeT amerika 242300000<br />
samxreT amerika 17814000<br />
afrika 30218000<br />
avstralia 8503000<br />
antarqtida<br />
xmeleTis saerTo<br />
farTobi<br />
14245000<br />
gamoTvale xmeleTis saerTo farTobi, gadaixaze da Seavse<br />
cxrili.<br />
9 gamoTvale, ( a+ b)-s<br />
ramdeni procentia ( a− b),<br />
Tu a = 1, 9 ,<br />
b = 0,6 .<br />
10 Senadnobi 60g vercxlis da 40 g oqrosa da 100g spilenZisagan<br />
Sedgeba. gamoTvale am SenadnobSi Semavali TiToeuli<br />
metalis koncentracia.<br />
11 ra koncentraciis xsnars miviRebT, Tu 100g marils 300g<br />
wyalSi gavxsniT?<br />
12 200g zRvis wyals, romelic 3% marils Seicavda, 400g<br />
sufTa wyali Seuries. ra koncentraciis xsnari miiRes?<br />
13 ramdeni grami wyali unda davumatoT 30g marils, rom<br />
miviRoT 25%-iani koncentraciis marilwyali?<br />
168 Tavi 3 • monacemebi
14 figuris farTobis ramden procents Seadgens<br />
gaferadebuli nawilis farTobi?<br />
a<br />
d<br />
b g<br />
e v<br />
15 ramdeni procentiT gaZvirda eleqtroenergia (anu<br />
sawyisi fasis ramdeni procentia namati), Tu adre<br />
1 kilo vati 12 TeTri Rirda, xolo amJamad erTi<br />
kilovatis fasia 18 TeTri?<br />
16 ramdeni procentia fasdakleba maRaziaSi, Tu labada,<br />
romelic 200 lari Rirda, 120 larad gayides?<br />
17 cxrilSi mocemulia saqarTvelos mosaxleobis<br />
raodenoba 1913, 1970 da 2002 wlebis aRweriT.<br />
wlebi mosaxleobis raodenoba cvlilebebi<br />
procentebSi<br />
1913 2 601 000<br />
1970 4 686 000 moimata<br />
2002 4 356 000 daiklo<br />
gamoTvale procentebSi mosaxleobis raodenobis cvlileba<br />
momdevno aRwerisas wina aRwerasTan SedarebiT.<br />
18 aleqsande erTi wlis gamavlobaSi 4 santimetriT gaizarda da<br />
misma simaRlem 164 sm-s miaRwia. ramdeni procentiT moimata<br />
simaRleSi aleqsandrem erTi wlis gamavlobaSi?<br />
19 wriul diagramaze mocemulia fermeris mier miRebuli sxvadasxva<br />
jiSis yurZnis mosavlis raodenobebi tonebSi.<br />
gamoTvale:<br />
daimaxsovre<br />
1<br />
1%<br />
100 =<br />
1<br />
10%<br />
10 =<br />
1<br />
20%<br />
5 =<br />
1<br />
25%<br />
4 =<br />
1 1<br />
= 33 %<br />
3 3<br />
1<br />
50%<br />
2 =<br />
1 =<br />
100%<br />
a) mTliani mosavlis ramden procents Seadgens saferavis jiSis<br />
yurZnis mosavali;<br />
b) ra gardusuli zomis seqtori Seesabameba rqawiTelis yurZnis<br />
Tavi 3 • monacemebi 169
aodenobas.<br />
170 Tavi 3 • monacemebi<br />
wriul diagramaze aRniSnuli monacemebis mixedviT aage<br />
Sesabamisi svetovani diagrama.<br />
20 svetovan diagramaze mocemulia firmis TanamSromeli qalebisa da<br />
mamakacebis raodenobebi.<br />
20<br />
15<br />
10<br />
5<br />
0<br />
qalebi<br />
mamakacebi<br />
diagramis mixedviT upasuxe kiTxvebs:<br />
a) ramdeni TanamSromelia sul firmaSi?<br />
b) TanamSromelTa ramdeni procentia qali?<br />
g) ramdengradusiani seqtori Seesabameba Sen mier agebul wriul<br />
diagramaze mamakacebis raodenobas?<br />
aage mocemuli diagramis Sesabamisi wriuli diagrama da miuTiTe<br />
masze firmis TanamSromeli qalebisa da mamakacebis raodenobebi<br />
procentebSi.<br />
21 gaavle wrfe da masze aRniSne A da B wertilebi, romelTa Soris<br />
manZili 8 sm-ia. amave wrfeze moniSne iseTi C wertili, rom BC<br />
mona kve Tis sigrZe AB monakveTis sigrZis 40% iyos. ramdeni aseTi<br />
wertili arse bobs? ras udris AC monakveTis sigrZe?<br />
22 ramdeni procentiT gaizrdeba ricxvi, Tu mas samjer gavzrdiT?<br />
23 ramdeni procentiT Semcirdeba ricxvi, Tu mas aTjer<br />
SevamcirebT?<br />
24 ramdeni g wyali unda davasxaT 30% koncentraciis mqone 200g<br />
marilwyals, rom miviRoT 20% koncentraciis xsnari?<br />
25 Seuries 500grami 80% koncentraciis da 300grami 70%<br />
koncentraciis spirits xsnarebi. ra koncentraciis xsnari miiRes?<br />
26 mas Semdeg, rac oqrosa da vercxlis Senadnobs 50g oqro<br />
daumates, SenadnobSi oqros koncentracia 30%-dan 80 %-mde<br />
gaizarda. ramdeni grami iyo Senadnobi oqros damatebamde?<br />
27 ipoveT udidesi samniSna ricxvi, romelic iyofa 3-zec da 5-zec.<br />
28 risi tolia iseTi wiladebis mricxvelebis jami, romelTa<br />
mniSvnelia 12, xolo maTi mniSvnelobebi 1/3-sa da 5/6-s Sorisaa?
29 gamoTvale 2-is, 2-is da 3-is saSualo ariTmetikuli.<br />
30 ipove x, Tu x-sa da 4,2-is saSualo ariTmetikuli 5-is tolia.<br />
31 1, x, 5, y, z …ricxvTa mimdevrobis meore wevridan dawyebuli<br />
yoveli wevri misi mezoblebis saSualo ariTmetikulis tolia.<br />
ipove x, y da z.<br />
wyvilebSi samuSao<br />
TiToeulma Tqvenganma damoukideblad SearCieT ori aranulovani<br />
ricxvi. gamoTvaleT, meore ricxvis ramdeni procentia pirveli<br />
ricxvi. Semdeg gamo TvaleT, pirveli ricxvis ramdeni procentia meore<br />
ricxvi. miRebuli ori ric xvi gadaamravleT, pasuxi daamrgvaleT<br />
mTel ricxvamde da CaiwereT. Sea dare Sen mier miRebuli Sedegi Seni<br />
megobrebis mier miRebul Sedegebs. ra kanonzomierebas amCnevT? axseniT<br />
es kanonzomiereba.<br />
gamoTvlebisas SegiZliaT isargebloT kalkulatoriT.<br />
qarTuli xuroTmoZRvrebisa da<br />
arqiteqturis nimuSebi<br />
gelaTis samonastro<br />
kompleqsi<br />
quTaisis maxloblad, wyalwiTelas<br />
xeobaSi aago mefe<br />
daviT aRmaSenebelma gelaTis<br />
monasteri da akademia, romelsac<br />
Tanamedroveni “axal ierusalimsa<br />
da meore aTens”<br />
uwodebdnen.<br />
daarsda 1106 wels.<br />
yaurmis xidi<br />
Zv. qarTuli xuroTmoZ-<br />
Rvrebis erT-erTi SesaniSnavi<br />
Zeglia qvis xidi mdinare<br />
faravanze. igi XII - XIV saukuneebiT<br />
TariRdeba.<br />
a) romel geometriul figurebs xedav nagebobebSi?<br />
b) ramdeni saukunisa gelaTis kompleqsi? yaurmis xidi?<br />
Tavi 3 • monacemebi 171
3.5 monacemTa ricxviTi maxasiaTeblebi<br />
mizani:<br />
172 Tavi 3 • monacemebi<br />
davadginoT sxvadasxva saxiT warmodgenili<br />
monacemebis ricxviTi maxasiaTeblebi: sixSire,<br />
diapazoni, saSualo, moda.<br />
dedis SekiTxvaze, Tu ra niSnebze swavlobs, nikam upasuxa: — cxrianze,<br />
radgan Cems mier miRebul niSnebSi 9 qula meti raodenobiTaa.<br />
dedam dRiuridan amowera sxvadasxva saganSi semestris ganmavlobaSi<br />
nikas mier miRebuli niSnebi:<br />
7, 8, 9, 6, 9, 7, 9, 6, 7, 6, 9, 6, 8, 9, 7, 9, 6, 9, 10, 8.<br />
Semdeg daalaga miRebuli niSnebi zrdis mixedviT da Seadgina sixSireTa<br />
cxrili:<br />
niSani 6 7 8 9 10<br />
sixSire 5 4 3 7 1<br />
saSualo qulis gamosaTvlelad dedam qulebis maT sixSirebze<br />
namravlebi Sekriba da miRebuli Sedegi sixSireTa jamze gayo:<br />
6 ⋅ 5 + 7 ⋅ 4 + 8 ⋅ 3 + 9 ⋅ 7 + 10 ⋅ 1<br />
saSualo = = 7,75<br />
5+ 4+ 3+ 7+ 1<br />
miuxedavad imisa, rom nikas SefasebebSi 9 qulis sixSire yvela sxva<br />
qulis sixSires sWarbobs, semestris manZilze nikas mier miRebuli<br />
qulaTa saSualo 8 qulaze naklebi aRmoCnda. udidesi sixSiris monacems<br />
moda ewodeba. ganxululi monacemebis modaa 9.<br />
nikas mier miRebuli qulebidan udidesi qulaa 10, umciresi ki 6. maTi<br />
sxavoba 4 gviCvenebs, Tu ramdenadaa gabneuli monacemebi. sazogadod,<br />
udidesi da umciresi monacemis sxvaobas ricxviTi monacemebis gabnevis<br />
diapazoni ewodeba.<br />
T(CO )<br />
amocana1. Tiniko sami dRis<br />
ganmavlobaSi 9 00 saaTidan<br />
saRamos 21 00 saaTamde<br />
yovel 3 saaTSi erTxel<br />
iwerda haeris temperturas<br />
da miRebuli monacemebis<br />
safuZvelze Seadgina xazovani<br />
diagrama. diagramis mixedviT<br />
gamovTvaloT monacemTa<br />
ricxviTi maxasiaTeblebi.<br />
amoxsna.<br />
8<br />
7<br />
6<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
9 saaTi 12 saaTi 15 saaTi 18saaTi 21saaTi<br />
I dRes umciresi temperatura iyo +20 dilis 900 saaTze, udidesi +70 _<br />
1500 saaTze. am dRes gabnevis diapazoni 50-is tolia. monacemTa modaa 50 ,<br />
xolo saSualo temperatura dRis ganmavlobaSi<br />
(20 +50 +70 +50 +30 ): 5 = 4,40 .<br />
I dRe<br />
I I dRe<br />
I I I dRe
I I dRes umciresi temperatura iyo +1,5 0 dilis 9 00 saaTze, udidesi +6 0<br />
_15 00 saaTze. am dRes gabnevis diapazoni 4,5 0 -is tolia. monacemTa modaa<br />
4 0 , xolo saSualo temperatura dRis ganmavlobaSi (1,5 0 +4 0 +6 0 +4 0 +1 0<br />
):5=3,3 0 .<br />
mesame dRis ricxviTi maxasiaTeblebi gamoTvale damoukideblad.<br />
daviTvaloT gabnevis diapazoni samive dRis ganmavlobaSi: am<br />
SemTxvevaSi umciresia III dRis monacemi+ 0,5 0 , xolo udidesi _ I dRis<br />
monacemi 7 0 . amitom gabnevis diapazoni iqneba 6,5 0 .<br />
rogorc vxedavT, gabnevis diapazoni samive dRis ganmavlobaSi<br />
aRemateba calkeuli dReebis gabnevis diapazonebs. SevniSnoT, rom<br />
sazogadod, ricxviTi simravlis gabnevis diapazoni yovelTvis<br />
aranaklebia, vidre misi nebismieri qvesimravlis gabnevis diapazoni<br />
(axseni, ratom?).<br />
upasuxe kiTxvebs<br />
1. ra monacems hqvia moda?<br />
2. rogor daiTvli monacemTa saSualos?<br />
3. rogor daiTvli monacemTa gabnevis diapazons?<br />
4. eTanxmebi Tu ara nikas mosazrebas, rom is cxrianze swavlobs?<br />
5. SeiZleba Tu ara ricxviT monacemTa gabnevis diapazoni naklebi iyos<br />
vidre am monacemTa nawilis gabnevis diapazoni?<br />
savarjiSoebi<br />
1 paragrafSi mocemuli diagramis mixedviT ipove samive dRis<br />
monacemTa moda.<br />
2 paragrafSi mocemuli diagramis mixedviT gaarkvie romeli<br />
winadadebaa marTebuli:<br />
a) samive dRis SemTxvevaSi udidesi temperatura 15 saaTze iyo;<br />
b) samive dRis SemTxvevaSi umciresi temperatura 9 saaTze iyo;<br />
g) 15-dan 21 saaTamde SualedSi samive dRes temperatura<br />
klebulobda;<br />
d) 9-dan 15 saaTamde SualedSi samive dRes temperatura<br />
matulobda.<br />
3 mocemuli sixSireTa cxrilis mixedviT ipove monacemTa saSualo,<br />
moda da diapazoni;<br />
monacemebi -1,4 +2,3 -4 +1,4 +6 -4,4 +3,3 4<br />
sixSire 1 3 2 5 2 3 4 1<br />
4 ras udris x, Tu x, 5, 3, 8, 4, 5, 3, 10 ricxviTi monacemebis<br />
modaa 3.<br />
5 ras udris a, Tu zrdis mixedviT dalagebuli a, 5, 6,5, 8,4, 11<br />
ricxviTi monacemebis gabnevis diapazonia 7,4 .<br />
Tavi 3 • monacemebi 173
6 ras udris b, Tu klebis mixedviT dalagebuli b, 5,1, 4,7, 3,4,<br />
-1 ricxviTi monacemebis gabnevis diapazonia 8.<br />
7 ras udris x, Tu 2,5, -3, 7, x, -4,2, -5, 3, 1,7 ricxviTi<br />
monacemebis saSualoa 1,25?<br />
8 dasavleT saqarTvelos ianvris Tvis haeris temperaturis gabnevis<br />
diapazoni 8 gradusi, xolo aRmosavleT saqarTveloSi imave<br />
Tvis gabnevis diapazoni 12 gradusi iyo. SeiZleba Tu ara mTel<br />
saqarTveloSi igive maCvenebeli iyos a) 7 gradusi? b) 11 gradusi?<br />
g) 13 gradusi?<br />
9 cxrilSi mocemulia eqvsi Tanaklaselis<br />
Sedegi 100 m-ze sirbilSi.<br />
a) daalage es monacemebi klebis mixedviT;<br />
b) gamoTvale monacemTa saSualo da<br />
gabnevis diapazoni;<br />
174 Tavi 3 • monacemebi<br />
g) romel moswavles aqvs saukeTeso<br />
Sedegi?<br />
d) romeli moswavlis Sedegia<br />
saSualosTan yvelaze axlos?<br />
10 xazovan diagramaze mocemulia wisqvil-kombinatis mier sami<br />
wlis ganmavlobaSi gadamuSavebuli xorblis raodenoba (tonebSi)<br />
kvartlebis mixedviT.<br />
diagramis mixedviT<br />
daadgine TiToeuli<br />
wlis da mTlianad sami<br />
wlis ganmavlobaSi<br />
monacemTa ricxviTi<br />
maxasiaTeblebi:<br />
a) udidesi da umciresi<br />
monacemebi;<br />
b) monacemTa saSualo;<br />
g) moda;<br />
d) gabnevis diapazoni.<br />
60<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
I kvartali<br />
I I kvartali<br />
I I I kvartali<br />
11 sami sazamTros saSualo masaa 7kg, meoTxes masa 9kg. ra saSualo<br />
masa aqvs oTxive sazamTros?<br />
12 mocemulia erTi klasis moswavleebis mier maTematikis<br />
sakontroloSi miRebuli, zrdis mixedviT dalagebuli, qulaTa<br />
sixSiruli cxrili.<br />
qula a 4 5 6 7 8 b<br />
sixSire 1 2 3 5 8 5 1<br />
salome 16,2 wm<br />
giga 14,7 wm<br />
naTia 15,9 wm<br />
luka 15,3 wm<br />
kaxa 17,4 wm<br />
lela 19,2 wm<br />
I V kvartali<br />
2009w.<br />
2010w.<br />
2011w.
gamoTvale a da b qulebis mniSvnelobebi, Tu cnobilia, rom<br />
qulaTa saSualoa 6,4, xolo gabnevis diapazoni — 7.<br />
13 16 sm 3 moculobis brinjaos zodi 162 grams iwonis. ramdens iwonis<br />
brinjaos birTvi, romlis moculoba 24 sm 3 -ia?<br />
14 or qalaqs Soris manZili 55 km-ia, xolo am qalaqebs Soris<br />
manZili rukaze 2,2 sm-is tolia. rogoria am rukis masStabi?<br />
15 ABC samkuTxedSi ∠ A = 68 0 , ∠ B = 42 0 . qvemoT CamoTvlili<br />
utolobebidan romelia WeSmariti?<br />
a) AC < AB < BC; b) AB < AC < BC; g) BC < AB < AC; d) AC < BC < AB.<br />
16 gamoTvale:<br />
a)<br />
; b)<br />
16 baRSi msxlis, alublisa da atmis xeebia. xeebis saerTo<br />
raodenobis nawili msxlisa da alublis xeebia, xolo atmis<br />
xeebis raodenoba xeebis raodenobis -s Seadgens. gamoTvale<br />
ramdenia baRSi atmis da ramdeni msxlis xe, Tu alublis xe sul<br />
oria.<br />
SeiZleba Tu ara?<br />
mocemulia zrdis mixedviT dalagebuli monacemebi:<br />
x; 3; 5; 6; 8; y.<br />
SeiZleba Tu ara, rom am monacemTa gabnevis diapazoni iyos<br />
a) 4? b) 7?<br />
.<br />
Tavi 3 • monacemebi 175
I I I Tavis mimoxilva<br />
ra viswavleT am TavSi?<br />
1. raodenobrivi da Tvisobrivi monacemebis mopovebis saSualebebi;<br />
2. ricxvis Cawera procentis saxiT da procentis saxiT Cawerili ricxvis<br />
Cawera wiladis saxiT;<br />
3. ricxvis procentis povna da ricxvis povna misi procentiT an<br />
nawiliT ;<br />
4. fardobis procentuli Cawera da gamoyeneba;<br />
5. sididis procentuli zrda-kleba da martivi xarjTaRricxva;<br />
6. monacemTa mowesrigeba da warmodgena cxrilisa da diagramis saxiT;<br />
7. monacemTa ricxviTi maxasiaTeblebis (saSualo, moda, gabnevis<br />
diapazoni) gamoTvla.<br />
upasuxe kiTxvebs<br />
176 Tavi 3 • monacemebi<br />
• ra saxis monacemebs icnob?<br />
• monacemTa mopovebis ra saSualebebi arsebobs?<br />
• rogor Cawer ricxvs procentis saxiT?<br />
• rogor Cawer procentis saxiT Caweril ricxvs wiladis saxiT?<br />
• ra kavSiria ricxvis procentsa da nawils Soris?<br />
• rogor ipovi a ricxvis p procents?<br />
• rogor ipovi ricxvs, romlis p procenti a ricxvis tolia?<br />
• rogor ipovi a ricxvze p procentiT met (nakleb) ricxvs?<br />
• rogor ipovi a ricxvi b ricxvis ramdeni procentia?<br />
• ra aris monacemTa moda?<br />
• ra aris monacemTa sixSiruli cxrili?<br />
• rogor daiTvli monacemis sixSires?<br />
• rogor daiTvli ricxviT monacemTa saSualos?<br />
• ra aris ricxviT monacemTa gabnevis diapazoni?<br />
• ra saxis diagramebi ici?<br />
• rogor aageb xazovan (svetovan) diagramas?
SeurCie marcxena svetSi mocemul winadadebas sityva marjvena<br />
svetidan<br />
1. ricxviT moncemTa udidesi da umciresi monacemis<br />
sxvaobas gabnevis ------- ewodeba<br />
2. udidesi sixSiris monacems ----- ewodeba<br />
3. ricxvis p/100 nawils am ricxvis p ------- ewodeba<br />
4. monacemis gameorebaTa raodenobas am monacemis<br />
------- ewodeba<br />
5. ricxviTi monacemebis jamis monacemTa raodenobaze<br />
ganayofs monacemTa ------ ewodeba<br />
6. ricxvis erTi procenti misi -------- nawilia<br />
savarjiSoebi<br />
moda<br />
sixSire<br />
diapazoni<br />
saSualo<br />
procenti<br />
measedi<br />
1 gamoTvale zepirad:<br />
a) 5-is ra nawilia 2,5; b) 27-is ra nawilia 3;<br />
g) 55-is 2<br />
nawili; d) 20-is 0,7 nawili ;<br />
11<br />
e) ricxvi, romlis 3<br />
4<br />
udris 6-s ; v) ricxvi, romlis udris 8-s;<br />
7 9<br />
z) 50-is 10% ; T) ricxvi, romlis 50% udris 60-s;<br />
i) 1kg-is ramdeni procentia 300g; k) 1sm-is ramdeni procentia 6 mm;<br />
l) 1sT-s ramdeni procentia 12 wT; m) 1t-s ramdeni procentia 200kg.<br />
2 gamoTvale:<br />
a) 50 ramdeni procentiT metia 40-ze;<br />
b) 40 ramdeni procentiT naklebia 50-ze.<br />
3 gazarde 20%-iT<br />
a) 3 lari; b) 5 tona; g) 1,2 metri; d) 25 kg; e) 3,6<br />
4 Seamcire 40%-iT<br />
a) 5 lari; b) 3 tona; g) 1,5 metri; d) 2,5 kg; e) 36sm.<br />
5 TaTias klasSi 25 moswavlea. maTi 60% gogonaa. gogonebis 1/5<br />
nawili friadosania. a) ramdeni friadosania TaTias klasSi? b) ra<br />
sididis (gradusebSi) seqtori Seesabameba vaJebis raodenobas im<br />
wriul diagrmaze, romelzec TaTias Tanaklaseli gogonebisa da<br />
vaJebis raodenobebi iqneba naCvenebi? aage es diagrama.<br />
6 ornitologi (frinvelTa yofis<br />
Semswavleli mecnieri) akvirdeba,<br />
dReSi ramdenjer tovebs SaSvi budes<br />
bartyebisTvis saWmlis misatanad. aRricxva<br />
rom gaudvildes, dakvirvebaTa dRiurSi<br />
budidan SaSvis yovel gamofrenas igi xazis<br />
CamosmiT aRniSnavs (ix. qvemoT).<br />
Tavi 3 • monacemebi 177
178 Tavi 3 • monacemebi<br />
00 00<br />
06 −12<br />
00 00<br />
12 −18<br />
am monacemebis mixedviT upasuxe kiTxvebs:<br />
a) monacemTa mopovebis romel xerxs mimarTavs ornitologi?<br />
b) ramdenjer datova bude Savma dRis pirvel naxevarSi?<br />
g) dRis romel naxevarSia SaSvi ufro aqtiuri?<br />
d) saaTSi saSualod ramdenjer tovebs SaSvi budes dRis<br />
ganmavlobaSi?<br />
7 ramdeni procentiT Camofasda qurqi, romlis sawyisi fasi 500<br />
lari iyo, xolo Camofasebis Semdeg 350 larad gaiyida?<br />
8 ramdeni procentiT gaZvirda fqvili, Tu adre 1kilogrami 80<br />
TeTri Rirda, axla ki misi Rirebuleba 1lari da 20 TeTria?<br />
9 tolferda samkuTxedis fuZis sigrZe perimetris 30%-ia.<br />
perimetris ramdeni procentia ferdis sigrZe?<br />
10 qalaqis mcxovrebTa erTi nawili laparakobs mxolod qarTulad,<br />
nawili mxolod rusulad, danarCeni qarTuladac da rusuladac.<br />
qarTulad laparakobs am qalaqis mcxovrebTa 80%, rusulad_<br />
60%. qalaqis mcxovrebTa ramdeni procenti laparakobs orive<br />
enaze?<br />
11 cxrilSi mocemulia kamaTelze mosuli qulebis sixSireTa cxrili<br />
qula 1 2 3 4 5 6<br />
sixSire 4 3 5 6 4 3<br />
cxrilis mixedviT upasuxe kiTxvebs:<br />
a)sul ramdenjer gaagores kamaTeli? b) ras udris monacemTa<br />
diapazoni?<br />
g) ras udris monacemTa moda?<br />
d) ras udris monacemTa saSualo?<br />
12 wriul diagramaze mocemulia erTi klasis 25 moswavlis moswrebis<br />
Sedegebi pirveli trimestris mixedviT procentebSi. diagramis<br />
mixedviT daadgine:<br />
a) ramdeni warCinebulia klasSi?
) ramdeni moswavlis Sedegia<br />
damakmayofilebeli an kargi?<br />
g) ramden gradusiani<br />
seqtori Seesabameba<br />
aradamakmayofilebeli<br />
Sedegis mqone moswavleebs?<br />
d) rogori Sefasebis moswavleebi<br />
Warboben klasSi?<br />
aradamakmayofilebeli<br />
12%<br />
damakmayofilebeli<br />
36%<br />
13 SekiTxvaze, ramdeni jariskacia mis<br />
razmSi, leitenantma upasuxa, rom misi<br />
razmis 2<br />
1<br />
nawili sadarajozea,<br />
7 7 nawili<br />
hospitalSi, xolo danarCeni 28 jariskaci<br />
mwyobrSia. ramdeni jariskacia razmSi?<br />
14 SekiTxvaze, ramdeni moswavlea mis<br />
klasSi, levanma upasuxa, rom misi klasis<br />
moswavleTa 20% friadosania, 40%<br />
oTxosani, 25% samosani, xolo 3 moswavle<br />
orosania. ramdeni moswavlea levanis<br />
klasSi?<br />
Sedegebi<br />
15 samxedroTa or A da B banaks Soris manZili 20<br />
kilometria. A dajgufebis meTaurma jariskacs<br />
gadasca amanaTi, ubrZana, mietana igi B dajgufebis<br />
meTaurisaTvis da maSinve ukan gamobrunebuliyo.<br />
jariskacma 2 saaTSi gaiara banakebs Soris manZilis<br />
60%. A banakidan ra manZilze iqneba jariskaci<br />
gamgzavrebidan 4 saaTis Semdeg, Tu is imave<br />
siCqariT imoZravebs?<br />
warCinebuli<br />
20%<br />
kargi<br />
32%<br />
16 dedam Tamriko wamlis sayidlad gagzavna. manZili saxlidan<br />
afTiaqamde 1,5 km-ia. am manZilis 55% Tamrikom 10 wuTSi gaiara.<br />
afTiaqidan ra manZilze iqneba Tamriko gagzavnidan 20 wuTis<br />
Semdeg, Tu imave siCqariT ivlis?<br />
17 xorbali iZleva 80% fqvils, xolo SoTis puri mis gamosacxobad<br />
saWiro fqvilze 20%-iT mZimea. ramdeni grami xorbalia saWiro<br />
erTi 480-gramiani SoTis puris gamosacxobad?<br />
18 mas Semdeg, rac fexburTelTa gundi erTma Tavdamsxmelma datova,<br />
gundSi TavdamsxmelTa raodenoba 25%-dan 20%-mde Semcirda.<br />
ramdeni fexburTeli iyo gundSi Tavdamsxmelis wasvlamde?<br />
19 mas Semdeg, rac xsnaridan 2kg wyali aaorTqles, xsnaris<br />
koncentracia 80%-dan 90%-mde gaizarda. ramdeni kilogrami iyo<br />
xsnari wylis aorTqlebamde?<br />
Tavi 3 • monacemebi 179
20 qalaqi sami A, B da C<br />
raionisagan Sedgeba.<br />
wriul diagramaze<br />
naCvenebia raionebis<br />
mosaxleobis wili<br />
procentebSi, xolo<br />
4000<br />
svetovan diagramaze<br />
1000<br />
mosaxleobis raodenoba<br />
A raionSi. gamoTvale<br />
mosaxleobis raodenoba B da C raionebSi.<br />
180 Tavi 3 • monacemebi<br />
aba scade!<br />
A raioni<br />
C raioni<br />
45%<br />
B raioni<br />
35%<br />
ipove naturaluri ricxvi, romlis Tavisive 32%-ze gayofis Sedegad<br />
arasrul ganayofad amave ricxvis 12% miiReba.<br />
testi<br />
cxrilSi Setanilia im qveynebis monacemebi, romlebic 2004 wels<br />
Sediodnen pirvel aTeulSi erTi maCvenebliT mainc: farTobiT,<br />
mosaxleobiT, erToblivi Sida produqtiT.<br />
qveyana dedaqalaqi farTobi mosaxleoba<br />
erToblivi<br />
Sida produqti<br />
(aTasi km2 ) (mln) (mlrd $)<br />
avstralia kanbera 7 6 8 7 2 0 , 1 692,4<br />
argentina buenos-airesi 2 7 6 7 3 9 , 5 172,1<br />
aSS vaSingtoni 9 6 2 9 2 9 5 , 7 12438,9<br />
bangladeSi daka 1 4 4 1 4 4 , 3 61,9<br />
brazilia brazilia 8 5 1 3 1 8 6 , 1 732,1<br />
germania berlini 3 5 7 8 2 , 4 2906,7<br />
didi<br />
britaneTi<br />
londoni 2 4 4 6 0 , 4 2295,1<br />
espaneTi madridi 5 0 4 4 0 , 3 1120,3<br />
iaponia tokio 3 7 7 1 2 7 , 4 4799,1<br />
indoeTi deli 3 2 8 8 1 0 8 0 , 3 749,4<br />
indonezia jakarta 1 9 1 9 2 4 1 , 9 284,1<br />
italia romi 3 0 1 5 8 ,1 1836,4<br />
kanada otava 9 9 8 4 3 2 , 8 1098,4<br />
nigeria abuja 9 2 3 1 2 8 , 8 91,6<br />
pakistani islamabadi 8 0 3 1 6 2 , 4 90,3<br />
ruseTi moskovi 1 7 0 7 5 1 4 3 , 4 755,4<br />
safrangeTi parizi 5 4 7 6 0 , 7 2216,3<br />
sudani xartumi 2 5 0 6 4 0 , 2 25,4<br />
yazaxeTi astana 2 7 1 7 1 5 ,2 51,9<br />
CineTi pekini 9 5 9 7 1 3 0 6 , 3 1843,1
cxrilis mixedviT upasuxe 1 _ 5 kiTxvebs:<br />
1. msoflios romel qveyanaSia mosaxleoba yvelaze meti?<br />
a) ruseTSi; b) CineTSi; g) iaponiaSi; d) indoeTSi.<br />
2. romeli qveyanaa teritoriiT yvelaze didi?<br />
a) ruseTi; b) CineTi; g) iaponia; d) aSS;<br />
3. romeli qveynis erToblivi Sida produqtia yvelaze meti?<br />
a) aSS b) CineTi g) iaponia d) germania<br />
4. ramdeniT naklebia mosaxleoba safrangeTSi, vidre nigeriaSi?<br />
a) 58 mln; b) 59,7 mln; g) 68,1 mln; d) 45 mln.<br />
5. ramdeni dolariT metia erToblivi Sida produqtis warmoeba<br />
germaniaSi, vidre ruseTSi?<br />
a) 1258,3 mlrd. $; b) 2151,3 mlrd. $; g) 1580,4 mlrd. $; d) 980,8<br />
mlrd. $.<br />
6. Tu x, 1, 7, 5, 4, 5, 3, 7 ricxviTi monacemebis modaa 7, maSin<br />
ras udris x =,<br />
a) 1; b) 7; g) 5; d)3.<br />
7. Tu zrdis mixedviT dalagebuli a, 3, 5, 8, 10 ricxviTi<br />
monacemebis gabnevis diapazonia 8 , maSin a =<br />
a) 2; b) 5; g)3; d) 8.<br />
8. ras udris x, Tu -3, 5, x, -5, 3, 7 ricxviTi monacemebis<br />
saSualoa 1?<br />
a)- 5; b) 5; g) -1; d) 1.<br />
9. <strong>VII</strong> klasis moswavle gogonebis simaRleTa gabnevis diapazoni 13smia,<br />
vaJebisa ki 10 sm. ramdeni sm SeiZleba iyos simaRleTa gabnevis<br />
diapazoni am klasSi?<br />
a) 14 sm; b) 11 sm; g) 10 sm; d) 9 sm.<br />
10. 15 laris 40% tolia:<br />
a) 1l 40 T; b) 60 l; g) 200 T; d) 6 l.<br />
11. erTi saaTis 20% aris:<br />
a) 1<br />
20<br />
sT; b) 1<br />
5<br />
wT; g) 12 wT; d) 20 wT.<br />
12. ricxvi, romlis 5% 6-is tolia, aris:<br />
a) 120; b) 56; g) 360; d) 30.<br />
13. 20 aris 80 -is<br />
a) 4%; b) 25%; g) 30%; d) 5%.<br />
14. ra ricxvis 10% -ia 24-is 3/8 nawili?<br />
a) 160-is; b) 120-is; g) 72-is; d) 90-is.<br />
15. ramdeni procentiT gaizrdeba ricxvi, Tu mas gavadidebT 5-jer?<br />
a) 10%-iT; b) 900%-iT; g) 400%-iT; d) 90%-iT.<br />
16. ramdeni procentiT Semcirdeba ricxvi, Tu mas SevamcirebT 5-jer?<br />
a) 5%-iT; b) 80%-iT; g) 500%-iT; d) 50%-iT.<br />
Tavi 3 • monacemebi 181
17. 10% -iT gaZvirebis Semdeg litri benzinis fasma 2,2 lari<br />
Seadgina. ra Rirda litri benzini gaZvirebamde?<br />
a) 2 lari; b)1l 99 TeTri; g) 1l 10T; d) 1l90 TeTri .<br />
18. puris fasi 40 TeTridan 50 TeTramde gaizarda. ramdeni<br />
procentiT gaZvirda puri?<br />
a) 35%-iT; b) 10%-iT; g) 25%-iT; d) 50%-iT;<br />
19. ramdeni kilometria realuri manZili qalaqebs Soris, Tu am<br />
qalaqebs Soris manZili 1: 500 000 masStabis rukaze 7 sm-ia?<br />
a) 12 km; b) 35 km; g) 350 km; d) 60km.<br />
20. 200 mililitri xsnari 40 miligram wamals Seicavs. ramden wamals<br />
Seicavs 300 mililitri igive xsnari?<br />
a) 60 mg; b) 500 mg; g) 400 mg; d) 80 mg.<br />
21. rogori iyo qalaqSi mosaxleobis raodenoba erTi wlis win,<br />
Tu axla qalaqSi 220 000 mcxovrebia, xolo wliuri namati 10%-s<br />
Seadgens?<br />
a) 198000; b) 210000; g) 120000; d) 200000.<br />
22. marTkuTxedis sigane perimetris 20%-ia. perimetris ramdeni<br />
procentia am marTkuTxedis sigrZe?<br />
a) 15%; b) 25%; g) 30%; d) 40%.<br />
diagramze mocemulia erT-erT maRaziaSi gayiduli kompiuterebis<br />
raodenoba kvartalebis mixedviT. diagramis mixedviT upasuxe # 23,<br />
# 24, 25 kiTxvebs:<br />
182 Tavi 3 • monacemebi<br />
60<br />
50<br />
40<br />
30<br />
20<br />
10<br />
I kvartali<br />
II kvartali<br />
III kvartali<br />
IV kvartali<br />
23. ramdeni kompiuteri gaiyida am maRaziaSi wlis ganmavlobaSi?<br />
a) 160; b) 165; g) 170; d) 180.<br />
24. ramdeni procentiT naklebi kompiuteri gaiyida pirvel kvartalSi,<br />
vidre meoreSi?<br />
a) 20%-iT; b) 25%-iT; g) 10%-iT; d) 40%-iT.<br />
25. wlis ganmavlobaSi gayiduli kompiuterebis raodenobis ra nawili<br />
gaiyida III kvartalSi?<br />
a) 1 ; b) 2 ; g)<br />
4<br />
5<br />
1 ; d) 1 .<br />
5<br />
6
4.11 amocanis amoxsnis algebruli meTodi<br />
mizani:<br />
232 Tavi 4 • erTwevri da mravalwevri<br />
viswavloT:<br />
1. algebruli meTodiT amocanis amoxsnis safexurebi;<br />
2. cxrilis gamoyeneba.<br />
amocanis amoxsnis algebruli meTodi gulisxmobs amocanis pirobebis<br />
Sesabamisi gantolebis Sedgenas da am gantolebis amoxsnis safuZvelze<br />
amocanis gadawyvetas. es meTodi Sedgeba Semdegi safexurebisagan:<br />
1. sityvieri modelis Sedgena;<br />
2. ucnobis Semotana da monacemebTan dakavSireba;<br />
3. algebruli modelis (gantolebis) Sedgena;<br />
4. gantolebis amoxsna;<br />
5. amocanis SekiTxvaze pasuxis gacema.<br />
amovxsnaT ramdenime amocana algebruli meTodis gamoyenebiT.<br />
amocana1. antilopas wamSi 20 metris garbena<br />
SeuZlia. jiqi masze swrafi cxovelia. mas wamSi 25<br />
metris garbena ZaluZs, magram am siCqariT jiqs<br />
mxolod 20 wamis ganmavlobaSi SeuZlia sirbili. ra<br />
manZilze ar unda miuSvas antilopam jiqi, rom Tavi<br />
usafrTxod igrZnos?<br />
amoxsna:<br />
1. sityvieri modeli<br />
antilopas<br />
ganarbeni<br />
usafrTxo<br />
manZili<br />
+ =<br />
2. ucnobis Semotana da monacemebTan dakavSireba:<br />
vTqvaT, usafrTxo manZili jiqsa da antilopas Soris<br />
x metria. 20 wamSi antilopa gairbens 20× 20 = 400<br />
metrs, xolo jiqi _ 25× 20 = 500 metrs.<br />
3. gantolebis Sedgena:<br />
400 + x = 500<br />
4. gantolebis amoxsna:<br />
x = 100 .<br />
5. pasuxi:<br />
jiqis<br />
ganarbeni<br />
manZili, romel ze axlos antilopam jiqi ar unda miuSvas, aris 100 m.
amocana 2. moswavleTa sazafxulo banakma SeiZina 1490 laris Rirebulebis<br />
oTxadgiliani da eqvsadgiliani 16 karavi. erTi oTxadgiliani karvis fasia<br />
80 lari, xolo eqvsadgilianisa 110 lari. ramdeni oTxadgiliani da ramdeni<br />
eqvsadgiliani karavi SeiZina banakma?<br />
amoxsna:<br />
1. sityvieri modeli:<br />
oTxadgilian<br />
karvebSi<br />
gadaxdili<br />
Tanxa<br />
eqvsadgilian<br />
karvebSi<br />
gadaxdili<br />
Tanxa<br />
+ =<br />
2. ucnobis Semotana da monacemebTan dakavSireba:<br />
mTliani<br />
Tanxa<br />
vTqvaT, oTxadgiliani karavi x cali iyides. maSin eqvs adgi lians<br />
(16- x ) cals iyididnen. oTxadgilian karvebSi gadaixdid nen 80x lars,<br />
eqvsadgilianebSi _ 110(16- x ) lars. yvela karavSi erTad gadaixdidnen<br />
(80 x + 110(16- x )) lars.<br />
1. gantolebis Sedgena:<br />
80 x + 110 (16- x ) = 1490<br />
2. gantolebis amoxsna; 80x+ 1760 − 110x= 1490<br />
− 30x =−270<br />
x = 9<br />
5. pasuxi: banakma SeiZina 9 oTxadgi li ani da 7 eqvsadgiliani karavi.<br />
ganxiluli amocanis amoxsnisas ganto lebis Sedgena gagiadvildeba, Tu<br />
Seadgen cxrils.<br />
oTxadgiliani<br />
karvebi<br />
eqvsadgiliani<br />
karvebi<br />
karvebis erTi karvis yvela karvis<br />
raodenoba Rirebuleba Rirebuleba<br />
x 80 80x<br />
16–x 110 110(16–x)<br />
amocana 3. erT benzingasamarT sadgurs 2 300 litri benzini hqonda,<br />
meores _ 2800 litri. dRis ganmavlobaSi meore sadguridan gaiyida<br />
xuTjer meti benzini, vidre pirvelidan, ris Semdeg orive benzingasamarT<br />
sadgurSi Tanabari raodenobis benzini aRmoCnda. ramdeni litri benzini<br />
gaiyida TiToeuli sadguridan?<br />
amoxsna:<br />
1. sityvieri modeli<br />
pirvel sadgurSi<br />
darCenili benzini<br />
=<br />
meore sadgurSi<br />
darCenili benzini<br />
Tavi 4 • erTwevri da mravalwevri 233
2. ucnobis Semotana da monacemebTan dakavSireba:<br />
vTqvaT, pirveli sadguridan gaiyida x litri benzini, maSin meore<br />
sadguridan gaiyideboda 5x litri. pirvelSi darCenili benzinis<br />
raodenobaa (2 3000- x ) litri, xolo meoreSi (2 800-5x ) litri.<br />
3. gantolebis Sedgena:<br />
4. gantolebis amoxsna:<br />
234 Tavi 4 • erTwevri da mravalwevri<br />
2300 − x= 2800 − 5x.<br />
4x = 500 , x = 125.<br />
5. pasuxi: pirveli saduridan gaiyida 125 l benzini, meoridan — 125×<br />
5 l=625 l.<br />
gantolebis Sedgena am amocanaSic cxrilis gamoyenebiT SeiZleba:<br />
benzingasamarTi<br />
I sadguri<br />
benzingasamarTi<br />
I I sadguri<br />
iyo gaiyida darCa<br />
2 300 l xl (2 300–x)l<br />
2 800 l 5xl (2 800–5x)l<br />
algebruli meTodi warmatebiT SegviZlia gamoviyenoT geometriuli<br />
amocanebis amosaxsnelad.<br />
amocana 4. AC fuZis mqone tolferda ABC samkuTxedSi gavlebulia<br />
AD biseqtrisa. C wverosTan mdebare gare kuTxe 1,5-jer metia ADC<br />
kuTxeze. vipovoT ABC samkuT xedis kuTxeebi.<br />
B<br />
moc: ΔABC, AB = BC ,<br />
AD biseqtrisaa,<br />
∠ DCK = 1, 5∠ADC<br />
amoxsna:<br />
u.g. ∠A, ∠B, ∠C<br />
I. sityvieri modeli:<br />
mocemulobis Tanaxmad ∠ DCK = 1, 5∠ADC<br />
II. ucnobis Semotana da monacemebTan dakav Sireba:<br />
A<br />
C K<br />
0<br />
vTqvaT ∠ ACD = x . maSin,<br />
0<br />
∠ BAC = x ,<br />
0<br />
x<br />
∠ DAC = ,<br />
2<br />
0<br />
0 0 x<br />
∠ ADC = 180 − ( x + ),<br />
2<br />
0 0<br />
∠ DCK = 180 −x.<br />
III. gantolebis Sedgena:<br />
0<br />
0 0 0 0 x<br />
180 − x = 1,5(180 − ( x + )).<br />
2<br />
IV. gantolebis amoxsna:<br />
0 0 0 9 0<br />
180 − x = 270 − x ,<br />
4<br />
9 0 0 0 0<br />
x − x = 270 −180<br />
4<br />
5 0 0 0 0<br />
x = 90 , x = 72 .<br />
0 0 4<br />
0<br />
V. pasuxi: ∠ A=∠ C = 72 , ∠ B= 180 −( ∠ A+∠ C)<br />
= 36 .<br />
D
upasuxe kiTxvebs<br />
1. ra safexurebs gulisxmobs amocanis amoxsnis algebruli meTodi?<br />
2. ra kavSiria amocanis amoxsnis pirvel da mesame safexurebs Soris?<br />
3. ris safuZvelze SevadgineT amocana 1-is amoxsnis sityvieri modeli?<br />
4. amocana 1-is mixedviT SeZlebs Tu ara jiqi antilopas daWeras, Tu<br />
manZili maT Soris iqneba 95 metri? 120 metri?<br />
savarjiSoebi<br />
1 erTi ricxvi meoreze 3-jer mcirea. ipove es ricxvebi, Tu maTi<br />
sxvaobaa 16<br />
2 erTi ricxvi meoreze 4-jer didia. ipove es ricxvebi, Tu maTi<br />
jamia 85.<br />
3 erTi mosazRvre kuTxe meoreze 360-iT didia. ipove am kuTxeebis<br />
gradusuli zomebi.<br />
4 erTi mosazRvre kuTxe meoreze 4-jer didia. ipove am kuTxeebis<br />
gradusuli zomebi.<br />
5 erT bidonSi orjer meti rZea, vidre meoreSi. ramdeni litri<br />
rZea TiToeul bidonSi, Tu oriveSi 30 litri rZea?<br />
6 erTi kilogrami pirveli xarisxis brinji 0,2 lariT iafia<br />
umaRlesi xarisxis brinjze. ra Rirs TiToeuli xarisxis erTi<br />
kilogrami brinji, Tu Tinikom TiTo kilogram orive xarisxis<br />
brinjSi erTad 3,6 lari gadaixada?<br />
7 saaxalwlod erT maRaziaSi ori yuTiT meti maSxala gaiyida,<br />
vidre meoreSi. ramdeni yuTi maSxala gaiyida TiToeul maRaziaSi,<br />
Tu oriveSi erTad 22 yuTi gaiyida?<br />
8 or yuTSi 72 kilogrami vaSlia. erT maTganSi 3-jer meti vaSlia,<br />
vidre meoreSi. ramdeni kilogrami vaSlia TiToeul yuTSi?<br />
9 dilidan Sesvenebamde maRaziam 64 kilogrami Saqris fxvnili<br />
gayida. Sesvenebis Semdeg kidev Semoitanes 100 kilogrami da<br />
gayides 38 kilogrami. saRamos maRaziaSi darCenili iyo 136<br />
kilogrami Saqris fxvnili. ramdeni kilogrami Saqris fxvnili iyo<br />
maRaziaSi diliT?<br />
10 imis Semdeg, rac avtobusSi avida 10 mgzavri da Camovida 6,<br />
avtobusSi 28 mgzavri aRmoCnda. ramdeni mgzavri iyo avtobusSi<br />
Tavdapirvelad?<br />
Tavi 4 • erTwevri da mravalwevri 235
11 marTkuTxedis perimetri 48 santimetria. misi erTi gverdis<br />
sigrZe 5-jer metia meoreze. gamoTvale marTkuTxedis gverdebis<br />
sigrZeebi.<br />
12 marTkuTxedis formis samosaxlo nakveTis sigrZe 20 metriT metia<br />
mis siganeze. nakveTze Semovlebuli Robis sigrZe 160 metria.<br />
gamoTvale nakveTis farTobi.<br />
13 navma dinebis mimarTulebiT sami, xolo sawinaaRmdego<br />
mimarTulebiT ori saaTi imoZrava. gamoTvale mdinaris dinebis<br />
siCqare, Tu navis sakuTari siCqarea 8,5 km/sT da navma orive<br />
mimarTulebiT 44,5 kilometri gaiara.<br />
14 navma dinebis mimarTulebiT ori, xolo sawinaaRmdego<br />
mimarTulebiT erTi saaTis ganmavlobaSi 43 kilometri gaiara.<br />
gamoTvale navis sakuTari siCqare, Tu dinebis siCqarea 2,5 km/sT.<br />
15 vaSlis Ciris miRebisas vaSli Tavisi wonis 85%-s kargavs.<br />
a) ramdeni kilogrami nedli vaSlia saWiro 20 kilogrami Ciris<br />
misaRebad?<br />
b) ramdeni kilogrami Ciri miiReba 200 kilogrami nedli<br />
vaSlisagan?<br />
16 yava moxalvisas Tavisi wonis 12%-s kargavs.<br />
a) ramdeni kilogrami nedli yavaa saWiro 5 kilogrami moxaluli<br />
yavis misaRebad?<br />
b) ramdeni kilogrami moxaluli yava miiReba 20 kilogrami nedli<br />
yavisagan?<br />
17 nato 15 wliT ufrosia nanaze. aTi wlis Semdeg nato 2-jer<br />
ufrosi iqneba nanaze. ramdeni wlisaa TiToeuli gogona?<br />
18 mama 50 wlisaa, Svili 28-is. ramdeni wlis win iyo Svili mamaze<br />
2-jer umcrosi?<br />
19 samkervalo firmas SekveTa 10 dReSi unda Seesrulebina. firma<br />
yoveldRiurad imaze 27 nawarmiT mets amzadebda, vidre<br />
SemkveTTan iyo SeTanxmebuli da SekveTa 7 dReSi Seasrula.<br />
amasTanave sxva SemkveTsac gaumzada 54 aseTi nawarmi. dReSi<br />
ramden nawarms amzadebda firma.<br />
20 giam erT dRes romanis 3<br />
5<br />
nawili waikiTxa, meore dRes _ darCenilis<br />
naxevari da ukanaskneli 20 gverdi. ramden gverdiani romani<br />
waikiTxa giam?<br />
21 ori qalaqidan, romelTa Soris manZili 330 kilometria,<br />
erTmaneTis Sesaxvedrad erTdroulad ori avtomobili gamovida.<br />
erTi avtomobilis siCqare 10km/sT-iT metia meore avtomobilis<br />
236 Tavi 4 • erTwevri da mravalwevri
siCqareze. 2 saaTis Semdg avtomobilebs Soris manZili 30<br />
kilometrs udrida. ra siCqariT moZraobdnen avtomobilebi?<br />
22 336 mm sisqis Zeli 30 mm sisqis ficrebad unda daixerxos. ramdeni<br />
ficari miiReba daxerxvis Semdeg, Tu TiToeuli gaxerxvisas<br />
naxerxis saxiT 4 mm sisqis masala ixarjeba?<br />
23 fexburTis gundis 16 moTamaSis saSualo asaki 23 weli iyo. mas<br />
Semdeg, rac gundidan veterani fexburTeli wavida, saSualo asaki<br />
22 weli gaxda. ramdeni wlis iyo veterani fexburTeli?<br />
24 daamtkice, rom gemis sakuTari siCqare dinebis mimarTulebiT<br />
siCqarisa da sawinaaRmdego mimarTulebiT siCqaris saSualo<br />
ariTmetikulis tolia.<br />
25 gemi A da B navsadgurebs Soris manZils dinebis mimarTulebiT 8<br />
saaTs, xolo sawinaaRmdego mimarTulebiT 10 saaTs andomebs. ra<br />
dros moandomebs gemi amave manZilis gavlas mdgar wyalSi?<br />
26 marTkuTxedi, romlis sigrZe 80sm, xolo sogane 60sm-ia, 48 tol<br />
kvadratad aris dayofili. ipove TiToeuli kvadratis gverdis<br />
sigrZe.<br />
27 ori kubidan erTis moculoba 27-jer metia meore kubis<br />
moculobaze. ramdenjer metia pirveli kubis waxnagis farTobi<br />
meore kubis waxnagis farTobze?<br />
aba scade!<br />
daamtkice, rom toli perimetrebis mqone marTkuTxedebs<br />
Soris udidesi farTobi kvadrats aqvs.<br />
Tavi 4 • erTwevri da mravalwevri 237