Matematika Za Prvi Razred Gimnazije 4n6b6b

I

1IJl1

ITPOITOPU:HOHAJIHOCT epe H nponopn;Hje (82) •.\ flojaM pa3Mepe H rrporropu;Hje; ocHOBHa ocoforna rrporropqaja; rrpoA)')KeHa rrporropu;Hja (82)

E OcttoBtte oco6Htte rrpo}J;}')Kema rrporropu;Hja (85) eKTHa H o6pll)'Ta nponopn;HOHaJIHOCT (86) •.\ ,[(HpeKTHO H o6pttyTO rrporropu;HOHaJIHe BeJIWilflie; rrpHMepH H r paqrnK (86)

pHMeHe nponopn;Hja (90)

------

A flpocT asMepttH pa'l}'H; cno:>KeH a3MepttH pa'I)'H; rrpou;eHTir J.f rrpOMHJIH; paqytt MeIIIalba; KaMaTHH paqytt (rrpocTa H CJIO:>KeHa K~faTHa opMyna) (90)

· ,*·~·'-. J~ ....

..

BO~

_ ~

.t.

Pa3Mepe :u nponop~:uje UeHe pa3mf'Illnrx m1K0Balha Bop;e je,n;Hor 11cTor rrpo113Bol)aqa p;aTe cy y cne.n;ehoj Ta6em1. 1,5£

24wrn

35wru

~

UeHa

I

o,5e

KoHwnrna

t

5£

I

I

86p;11H

I

3a Koje ce IIaKOBaH:.e rpe5a orrpep;eJIMTlf aKo HaM je IIOTpecStto 10£ Bop;e? HajrroBOJbirnjy KYIIOBlfHY o.n;pel)yje ogHoc KOJIWIJflfe 11 u;ette: 05 15 • ==0,021, ' ==0,043, ~==0,058. 24 35 86 HapaBHo, Behli O)l;HOC (KoJIMq11He Bop;e 3a 1 .n;11ttap) je rroBOJbH11j11 3a Kyrru;a.

+

Pa.3Mepa )];Ba 5poja pa3JIMqJ1Ta op; ttyJie jecTe KOJilft{Hl1K Ta p;Ba 5poja.

a

AKo je Ha :;t:O H b #0, ottp;a pa3Mepy cSpojeBa a J1I b 03HaqaBaMO ca a: b J1IJI11 f;·

IlpJ111ep 2. Pa3MepoM ce 113pIDKaBa op;Hoc Mel)y BaJIYTaMa pa3JIHtIMTHX 3eMaJba. TaKo, KpajeM HOBeM5pa 2010. rop;MHe, O)l;HOC USD: EUR($/€) je 8110 1,35. To 3Haqv. CMO 3a 1 eBpo p;ofo1jaJrn 1 .n;onap J1 35 u;ettrn, aJIM 11 .n;a cMo 3a 100 eBpa, .n;oc511jaJIJ11 i:: .n;onapa. YormITe, C)'Me HoBu;a op; D$ H E€ Bpe.n;e rro.n;je.n;ttaKo aKo cy 8pojeB11 D 11 E y pa3Mep11 1,35, Tj. aKo je D: E = 1,35. IlpIDteH>yjyhli jeAHOCTaBHe aJITe5apcKe 3aKOHJ1TOCTl1, JiaKO rrpeTBapaMo HOBau; 113 jeAHe BanyTe y .n;pyry. KoJIMKo .n;onapa Bpe)l;lf 10€?

KoJIHKO eBpa Bpep;H 200$?

Kon11Ko eBpa Bpep;H 1$?

D: 10=1,35 D= 10 · 1,35

200: E= 1,35 E=200: 1,35 E== 148,15

1: E = 1,35

D=l3,5

E=-11,35

E==0,74

+

3a;i;aTa

y HCTOM rrepHop;y Kao y rrpeTXO)l;HOM rrpHMepy, O)l;HOCl1 eBpa ca OCTaJil1M BOp;ehMM CBeTCKHM BaJIYTaMa (8p11TaHCKa yttTa, jarraHCKl1 jeH, IIIBaju;apCKl1 pairnK 11 aycTpan11jCKl1 .n;onap) 811Jil1 cy: GBP : EUR (£/€)

0,85

JPY: EUR(¥/€)

111,26

CHF:EUR

1,33

EUR:AUD

1,37

_ KoJIMKo eBpa Bpep;H Kao 100¥?

3 KoJIMKO 5pHTaIICK11X yttTH Bpe.n;11 Kao 100¥?

I 11

I1

1 ''

2) Kon11KH je o.n;ttoc GBP : JPY? 4) Kon11Ko je o.n;ttoc CHF : AU -

II AKo cy jeAJlaI
loj

:JpeMa ,n;eqmttwu;wjw, rrocrnjw 8ecKOHaq}fo MHoro

cyMe OA

:rapoBa 8pojeBa qwja je pa3Mepa wcra, Tj. q}ljw je

cyMe op; XS H

.·01rnqHHK HCTH.

D X -=E y

I

I

DS H E€ H

Y€, OH.Ila je

a H b je,n;ttaxa pa3MepH 8pojeBa c H d, Tj. axo je a c a: b=c: d, OAJIOCHO,-=-,

AKo je pa3Mepa ,n;Ba 8poja

b d

;ieqnmmmia

K(l)l(eMO p;a cy OBH rrapoBH nponop~OHaJDIJI. ,lJ,aTy je,n;Ha.KOCT Ha3JiBa.'dO nponop~joM. ~-

IlpHMep3.

a b

a H b o8pa3yje rrporropu;Hjy ca rrapoM c H d, H aKo cy II03HaTa

a

_-\Ko nap 8pojeBa

b.

rpw 8poja, qeTBpTH ce je,n;ttocTaBHO op;pel)yje Ha ornoBy IT03Harnx aJITe8apcKHX 3aKOHHTOCTH.

a

7 1 1 - : x=-:6 7 3 7 3 -:x=-

6

aqw,n;a

0,12 : 0,4 =X: 0,01 0,3 =X : 0,01 x=0,3 · 0,01 =0,003

7

7 3 49 x=-:-=6 7 18

135 Ey

H

b

•

Hapep;Ha TeopeMa Kao w 3a,n;aTaK HaBep;eH rrocrre lhe p;ajy ocofome nponopqirja Koj e cy :seoMa Kopwctte. H3

Y? D?

AKo cy a,

b, c H d pearrttw 8pojeBw, CBH pa37IH'IHTH op; ttyrre, ra.ua je a : b =c : d aKKo a ·'d =b · c.

OCHOBHa ocofou1a i::ponopuma

a c b d ·ep;HaKOCTH ITOMHO)IG{MO ca bd, ,n;o8HjaMO ~. (bd) =~. (bd), b d

,'101.:a3. Jep;HaKOCT a : b = c : d 3arrHcaneMo y oblIHKJ - =-. AKo o8e Cipatte oBe

op;aKrre crrep;H (HaKoH CKpallHBalha) ,n;a je TaKol)e, aKo o8e cTpatte je,n;ttaKOCTH

a

c

b

d

ad= be.

ad= be rro,n;errwMo ca bd, p;o8wjaMo ad bd

=~. db

a : b =c : d

O,IJ;HOCHO, - =-. •

~ ) H)lp3fillhH

CIIOJballllh.11

3ap;aTaK

'IJJ3HOBH

a, b, c H d 8pojeBH pa3JIJ{qHTH o,z:i ttyrre. J;oKa)J(H p;a je je,n;HaKOCT a : b = c : d eKBHBaJieHTHa CBaKOj

.

-

--. .

'lil3HOBJI

HeKa cy

O.z:l crre,z:ienHX je,n;HaKOCTH:

ad

=

be

) b:a=d:c; 2) a : c = b : d; 3) d: b =c: a;

(YttyTpamlhH 'UiaHOBH nporropu;:uje Mory 3aMeHHTH MeCTa.) (CrrOJbamlbH 'UiaHOBJ1 npoIIopqHje Mory 3aMeHJ1TH Mecra.)

_) ak : bk= c: d, 3a cBaKH 8poj k pa3JIWil1T o,z:i HYJie; 5) ak : b =ck : d, 3a cBaKH 8poj k pa37IH'IHT o,z:i ttyrre.

83

Je,UHaxocni rpH u BJrme pa3Mepa Ha3JIBa.MO npop;yxceeoM upouopqHjoM. AKo cy pa3Mepe a : b, c : d u e :f .!'-.1el)yco8Ho je,IV1aKe, oHp;a jep;HaKocrn a : b = c : d = e :f, a c e . . ogHocHo, - =- =-, npe,n;craBJbaJY npo,n;pi<etty nponopqHJY TPH p;aTe pa3Mepe. b d f 3a npoA)')Ketty nponoprvrjy a : b = c : d = e :f KopHcTH ce H rnep;etrn 3a1mc a : c : e = b : d :f, npH qei\ty p;Be ra~e y norne,D;JDeM 3a1mcy He npep;cTaBlhajy 3HaK 3a p;elhelbe sen ce KOpHcTe caMo p;a DH pa3p;BojJ1IJie ,,npse" (,,ropibe") op; ,,,n;pymx" (,,p;olhHX") 'illaHoBa pa3Mepa Koje 08pa3yjy p;aTy rrpop;pt<eHy rrporropq11jy.

fl>op tpau.e upo.:i;y; eue uponopunje o~ ofoPIHJtX

nponopw1ja

IlpHMep 4. 08pa3yjMo npop;pt<etty nporropq11jy op;: a: b =2; 3, b: c =4: 5 H c: d =6: 7. IIpeqiBIDfje, OA-peAJ!Mo '!eM)' je je,D;HaKo a : b : c : d. TA • ~ • • Q b b C n3 npBe ,qBe nporropqHJe p;ouHJaMO Je,D;HaKOCTH - =- H - =-. 2 3 4 5 .Ua bHCMO OBe jep;HaKOCTJII ,,IIOBe3a1m'', IIOTpe8Ho je H3jep;Ha'IHTJII op;roBapajyne CTpaEjeJJ,HaKOCTit: y Kojmrn ce rrojaslhyje HCTa rrpoMeHJhHBa. .N1Ho)KelheM m.teHHJiaqa o8e cTpaHe rrpBe jep;HaKOCTJ1 ca 4 H MHO)KelbeM HMeHHJiaqa . . a b b c o8e CTpatte A-pyre Je,D;HaKOCTJf ca 3, p;obHJaMO - =- H - =-. 8 12 12 15 L{alrne, a: b: c=8: 12: 15. Cap;a je rroTpe8Ho OBY npop;y)Ketty rrporropqHjy ycarrracJr."C." - . c d cac: d = 6 : / ,TJ.Ca-=-.

6

7

}fa _a_ =-b- =-c- H _c_ =-d- p;o8HjaMo p;a je a: b: c: d = 16: 24: 30: 35. 8 . 2 12 . 2 15 . 2 6 . 5 7 . 5

3a;\aTaK OA-pe,D;Jf a : b : c : d aKO je a : b = 7 : 2, b : c = 3 : 4 H d : a= 4 : 3. IlpHMep 5. 3c5np yr;ioBa y "IpO)TIIY je;:max je

180°.

OA-peAJ!MO yrrroBe a, p11 y HeKor Tpoyrrra aKO je a : p: y = 2 : 3 : 4. Ha oCHosy p;aTe npop;pt<eHe rrporropqHje 3aKJhyqyjeMo p;a je

5:=~=1=k,

2 3 4 3a HeKo k, rra je a= 2k, p= 3k, y = 4k. KaKo je a+ p+y = 180°, p;o8Hja.Mo p;a je 2k +3k +4k = 180°, Tj. 9k = 180°. )J;aKJie, k =20°, rra je a =2k =40°, p=3k =60° Hy =4k =80°.

Ilp11 pap;y ca rrpop;y:>KeH11M rrporropQ11jaMa Kop11ctto je yaecrn 03HaKy 3a bpoj KOMe cy jep;HaKe CBe pa3Mepe Koje obpa3yjy TY rrpop;y)l(etty rrporropQ11jy.

+

3 aT OA-pe,D;Jf 8pojeBe x, y Hz aKo je x: y: z=3: 2: 5112x+5y+3z=124.

8

E 3ar1aTaK AKocy

=e :f,

AKo je a : b = k: l u. c: d = m : n, ,n;a Jil1 je OHAa a : b : c: d = k : l: m : n? 08pa3JIO)l(Jl o,n;roBop.

iepe.

n bpoie:sa KOpUCHTIUIO CilOBa

YormITe, aKo je n HeKH npu.po,n;att 8poj, a a 1 H b1, a2 H b2, ... ,an u bn napOBH 8pojeBa
)la bJICMO 03IDl'DL!ll

ca 1nueK01Ma: a 1, a:, ... , !l •

HeKa cy a1, a2, ... ,an H b1, b2' ... , bn peaJIH11 8pojeBH, CBH pa3IDf'UfTH o,n; ttyne, TaKBH ,n;a je a 1 : a2 : ••• : an= b1 : b2 : ••• : bn. Ta,n;a je

klal +k2a2 + ... +knan - a l - a2 - - a,, klbl+k2b2+ ... +knbn -b.-b2 - ... -~ 3a 8u.no Koje 8pojeBe k1, k2' ... , kn TaKBe )J.a cy o8a 'IJiaHa rrpBe pa3Mepe

TeOpeMa

pa3JIH'U1Ta OJJ. Hyne.

CTpaHe rnau;a

i{oKa3. HeKa je k 8poj TaKaB ,n;a je ~ =~ = ... =~ = k.

bl

b2

bn

3811p o8mn

k1a 1 +k2a 2 + ... .:...k~a

Ta,n;a je a 1 =kb 1, a2=kb2' ... ,an =kbn, rra je

k 1a 1 +k2a2+ ... +knan k 1kb 1+~kb2 + ... +kJbn k1b1 +k2b2+ ... +knbn k1b1 +k2b2 + ... +knbn

k(k 1b1+~b2 + ... +knbn) =k, klbl +k2b2 + ... +knbn

Ha311Ba ce JIHHeapaa

KOMOKHaqHja 8pojesa al, a2' .•. , an.

aCHTH qu.Me je TBpljelbe ,n;oKa3aHo.

35 . •

•

0Ba TeopeMa u.Ma 8pojHe norne,!V'1Il;e. Ha rrpHMep, aKo je a: b =c: d, OHAa je • (a +c): (b +d) =a: b, no,n; ycnoBOM ,n;a je b +d ;tO; • (a-c): (b-d) =a: b, no,n; yrnoBoM ,n;a je b-d ;tO; • (2a -3c) : (2b- 3d) =a : b, rro,n; ycnoBoM ,n;a je 2b- 3d = rel="nofollow">=O; HTA. 3ap;aTaK HeKa cy al, a2, ... , an H bl' b2' ... , bn peaJIHJi 8pojeBH, CBJi pa3IDf'Ilrrll 04 H)'Jle, TaKBU ,n;aje a 1 : a 2 : •• • : an =b 1 : b2 : ••. : bn. )J;OKaJKH p;a je

+ ~a2 + ... + knan - l1a1 + ~a2 + ... +/nan klbl +k2b2+ ... +knbn llbl +[2b2-t- ... +[nbn

k 1a 1

3a 8u.no Koje 8pojeBe k 1, k2' ... ,kn' le 12' ... , l" TaKBe )J.a cy 8pojeBH Koju o8pa3)iY nponopu;u.jy pa3nu.qu.n1 OJJ. Hyne. Ilponopu;u.je cy 3ay3HMane u;eHTpanHo MeCTO y cTaporpqKoj MaTeMarnnu. J13Meljy ocTanor, ,n;peBHH MaTeMaTWiapu. (IlnaToH, Ey,n;oKc, Ilu.rnropa HT.I{.) npO)>qaBaJJH cy u. nponopu;Hje o8nu.Ka

c-a b ---=-.

~

c-b a 1. O,n;pe,n;u. HeKOJIHKO TpojKH 8pojeBa (a, b, c) Koje 3a,n;oBOJbaBajy ,n;ary rrponopuHjy. 2. 3aHHMJbHBO je }:\a cBaKa TPH y3acronHa 'lJiaHa a, b, c HH3a l, 1, 2, 3, :5, 8, 13, 21, 34 ...

HID t'ip01esa 'llrjll

je CBaKH 'LJ:a.H, no
:s:u~ma

ce <1>1u5oH
1, 1,2,3,5,&,

13, 21, 34, 55, 89, ...

3a,n;oB0JbaBajy ,n;aTy nporr'?pu;Hjy. 3aIIITo?

85

)J;HpeKTHa H obpnyTa nponop:qHOHaJIHOCT ilnpeicrna H ocSpHyTa rrpouopqH.omurnocr rrpep,cTaBJbajy ttajjep,HOCTaBHHje Bpcre 3aBUCHOCTif Mel)y rrpm1eIUhJIBID! Benwnma.Ma. y OBOM op,eJbKY IIOCMaTpaneMO UCKJb}'DIBO Be,lJ!'Ilme 'llrje cy BpeAHOCTif YBeK uo3uiUu8He. iliq>e~'THO

npo11op1I110H3.iIHe B e.;JJ[11JCHe

X ~

x1

x3 ~

}'

Y:

Y;

Yi

x

x.

x

x·a

x,

~=~=~= k

·a

x:a f

:a

,i!Be 3aBJ1CHe Bemf'illHe cy p;upeKTHO nponop~OHa.JIHe aKO je KOJIWIHHK O~OBapajyhnx Bpe;:t,Hocrn CTaJiaH, rj. KOHCTaHTaH. 0Ba KOHCTaHTa ce Ha3HBa Koe<}nn~aj eHT ~eKTHe npon opqitoHaJIHOCTII. HJrje TeIIIKo 3aKJbJ'DITlf p,a YKOJIJ1KO cy ,D,Be upoMeHJbJIBe BeJIJ1'1HHe p,ttpeKTHO rrponOpf\1'iOH
IlpBMepl. Cnep,eb.e 3aBHCHe BeJIM1Ilfile cy ,D,HpeKTHO rrporropn;HoHaJIHe: • C}'Ya HOBn;a H KOJIJf'IKHa pocSe (HCTe BpCTe) - KOHCTaHTHa pa3Mepa OBe ]!,Be BeJIWIJ1He je n;eHa; • rrpel)eHH rryT Ji rrpoTeKJIO BpeMe rrpH paBHOMepHOM KpeTCllhy- KOHCTaHTHa pa3Mepa oBe ,!(Be BeJIJf'Ilffie Ha3HBa ce 8p3HHa; • Maca H 3aupeMKHa TeJia Ca'IJ11beHJ1X 0)::1 HCTOf MaTepHjaJia - KOHCTaHTHa pa3Mepa OBe ,D,Be BeJIH'Dffie je rycTIIBa MaTepHjana. Pa3MOrpHMo p,eTaJbirnje uocJie]J,lbH rrpHMep.

2 ... ..

.:·: :

1 ······1(Vv ~1) :

0 m

m

4 m

_··· = 2 =3 =0,8 (kg/dm3)

v; v2 v3

AKo 3a HeKOJIJ1Ko KOD;KH ttanpaslbeHHX op, J1CTOr MaTepHjana y IIpBOM KBap,paHTY Koopwmarnor c:HcTeMa (jep cy Mace H 3arrpeM:HHe II03JfTHBHe) HaD;pTaMO Ta'!Ke 'IHje cy rrpBe KOOPAHRaTe 3arrpeMHRe, a APyre KOOpAJ1HaTe op,roBapajyhe Mace, yo
7

m (ykg)

6 ..... .

v

(yck:

e

3aµ,aTaK :-- KOOp,D;HHaTHOM CHCTeMy rrpHKa.>KJ1 rpacl>HK 3aBHCHOCTH rrpet)eeor rryra o,n; npoTeKJior ~ eMeHa rrpH paBHOMepHOM Kpernlhy 8p3HHOM o,n; 2m/s. J

.:/iflCiflBo. Oce HMeHoBaTH ca s (y MeTpKMa) 11 t (y ceK)'HA~ra) .

3ap;aTaK 8poj

1X;rnq11He X 11 Y cy AHpeKTHO rrporropqHOHaJIHe. HeKoJIHKo op;roBapajyhH:x Bpe,!J;Ilocnt 3a X H

Y ,n;aTe cy y ttape}:{Hoj Ta8e1m.

x

1,8

0,3

0,5

y

2,16

0,36

0,6

0,25 .

0,3

ffqHHe HaqpTaj rpacl>wK 3aBHCHOCTH BeJIWJHHa X H Y. _ A.Ko je X = 2,4, o}:{pe,n;w op;roBapajyhy BpeAHOCT BeJIH'IHHe Y.

SMepa

AKO

a OBe

je y = 2,4, OJ:{pe,n;H o,n;roBapajyhy Bpe,L{HOCT BeJIH'IHHe x.

--·onwKo ,n;Be ,n;wpeKTHO rrporropqwottantte BeJIWJHHe Mepm.ro pa3mf'IlrrnM jegm1~aMa

lfaBe~eHe

je:mmme »epe

epe, ott,n;a ce Koecl>wqHjeHT rrporropqwoeanttocTH KMeiryje op;roBapajyhoM H3BegeeoM ~:urnwqoM Mepe Koja rrpe,n;cTaBJba KOJIWJHHK jeAHHHIJ;a Mepe KOjI'll'da

cy Mepeee

c:nna3He Ben11q11He.

'i • m 2) x

-V - 3arrpeMHHa y m 3

m-

(KybHHM MeTpHMa)

V - 3arrpeMHHa y cm 3

k=Y/X

Maca y

--

·

l

p = m/V - rycnma y kg/m

3

(

rrpeJ)eHH rryT y m

t- BpeMe y

(MeTpHM~

s-

rrpeJ)eHH rryT y

p =m/V - rycnma y g/cm

s (ce.KYH,n;a.Ma)

~3Jmaym/s

v =s/t-

t- BpeMe y h (caTHMa)

(KHJIOMeTpHMa)

~

K - KOJIHqHHa po8e y m 2)

(nHTPHMa)

e

(AHnapmrn)

__,_

c =PIK - QeHa y AJIB./kg

km

:OOOm

h

3600s m

1 - = --

zQ,28S

l m = O,OOlkm

s

__ l_h 3600

p - Bpe,L{HOCT pobe y $ i

t>

paBHOMepHa -----1

(KwnorpaMHMa)

lOClCJC'-0...-m-

bp31ma y km/h

p - Bpe}:{HOCT po8e y AJIB.

K - Kon11q11Ha po8e y kg

m

3

v=s/t- paBem1epea

--!--

km

10000-

=0,0: -gcmJ

)

)

s-

ko

__£_ 3

-

c=PIK- qeea y $/i

(Aonap11Ma)

km =36, h

IlpHMep2. lip3HHa ayToMobHna je

90km/h. OgpeJ:{HMo lberoBy bp3HHY y MerpJrna y MHHYTY

m/min) .

[y dm 3)

•

kg

s:: ) m - Maca y g rpaMHM:a KyuHHM qeHTHMeTpHMa_,____ _ _ _ _ _ _ __

3

~

y

(KHROrpaMHMa)

-.-

-

(y dm

I'

-

90 km =90 · lOOOm h60min

=1500~

+

min

87

;::{Be 3aBHCHe Bt!J"IWrnHe cy oOpJIYrO nponop.qHOHaJIHe aKO je rrpOH3BO,ll; O,ll;fOBapajytmx Bpe;:ui:OCTil CTallaH, Tj. KOHCTaHTaH. 0Ba KOHCTaHTa ce Ha3JIBa Koeqnru;ujeHT oOpll)'Te

OOpHJio npouopUIIOHa;me Be.Il{q]fHe

X

x1

x.

x3

Y

Y1

Y:

Y1

nponopqHOHaJIHOCTK. ,ilaKne, a.KO

cy ;:i;Be npoMeHJbHBe BeJilllUUie obpHyro nporropQHOHaJIHe, rrpH rroBehalby

jeµ;He o;:i; IDJtx o;wel)eH Bpoj nyra, ,n;pyra BeJIH'IIDia ce TOJIHKO rryTa cMalbyje.

\0

II

1tl

"
x·_ = x-a)-(y:a) = x: a}· (y ·a)

1,5

IlpHMep3. BpeMe rroTpe5Ho ,n;a ce o6aBH HeKH rrocao H

Bpoj pa,n;HHKa KojH cy aHra)l(OBaHH Ha

TOM IIOCJIY npeACTaBJbajy Hajje,ll;HOCTaBHHjH rrpHMep ,n;Be obpHyTO rrporropQHOHaJIHe BeIDl'IJllie (HapaBHO, IIOA npeTIIOCTaBKOM ,n;a CBH pa,n;HHQH pa,n;e HCTOM 5p3HHOM).

2 pa,ll;HHKa obaBe HeKH rrocao 3a 4 caTa, oH,n;a he 1 pa,n;HHK Taj HCTH rrocao obaBHTH 3a 8 caTH, ,n;oK he 4 paAffHKa rrocao 3aBpurnTH 3a 2 caTa. AKo

2

:Opoj pa,n;HHKa n BpeMe 3aBHCHOCT H3Mel)y n H

3a Koje BpeMe he

-+---1-

4h

t

8h

L !_ __ '.

s

~

lh

t H3pa)l(aBaMO ¢oPMYJIOM: n · t=8.

3 pa,a;ID1Ka obaBHTH rrocao?

AKo y npeTXo,n;Hoj ¢opMYJIH

n 3aMeHHMO ca 3, ,n;o5HjaMo 3 · t=B, 2

o,n;aKJie CJie,n;H ,n;a je

t =2,666 ... =2 3' o,n;HocHo, ,n;a he rrocao bHTH o6aBJbeH 3a 2 caTa H 40

MHH)'Ta.

KoJIKKo pa,ll;HHKa rpe6a attra)l(OBaTH ,n;a CH nocao fo.10 o5aBJbeH 3a 3 cam? Ca,n;a t Tpe5a 3cn.1eHMTI1 ca 3, IIITO Hae ,n;oBo,n;H ,n;o je,n;HaqHHe

n · 3=8, 'Ilfje je pernelbe

n =2,666 ... =2,(6).

0Bora rryTa MopaMo rrruIGhl

~Ia'UITH ,n;oBHjeHH

pe3}'JI'TaT 5y;u;ytrn ,n;a pernelbe rrpo5neMa Tpe5a ,n;a 5y,n;e rrpHpOTiaE - :: Ha rrocTaBJbeHo IrnTalbe rnacH: ,n;a bH nocao fo.10 obaBJbeH 3a 3 aJITa)l(OBarn HajMalbe 3 pa,n;HHKa H rrocao he bHTH obaBJbeE

88

j. 0,voBop

cna. -rr:pe5Ho je

iI?C

e3a.Hor po Ka. ·

+

-II

ajyn 11x 1ffyTe

i

flalby

3arrpeM11He Tena q11je cy Mace lkg o8pHYTo cy rrporropu;110HaJIHe rycnrnM MaTepHjana OA Koj11x cy HarrpaBJbeHa, jep je rrpo113BOA 3arrpeMHHe Tena H rycTMHe MaTep11jana KOHCTaHTaH H H3HOCH lkg. (foMHJia BYHe Mace lkg 3ay311Ma MHoro BHIIle rrpocTopa OA KOMaAa rnmI
llpHMep4.

JeAHOCTaBHO je
vi =0,5

Vym 3 p ykg/m

3

P, =2

-

~=l

"-3 =0,4

P2 = 1

p3=2,5

lBJ1TJ1

l

;t

p(kg/m')

6

t

y = -k , x E -_..._ \ _{O}, oe x

je k Heim HXCHpaH peanaH Opoj pa3;nf'DIT OJ:i H)'lle, IIpe!lCTaBlba

yoITIIITeH>e o6pH)• e rrponopm10H

KaKo 113rneAa rpa 11K 3aBHCHOCTJ1 HacnynyjeMo u;pTalbeM HeKOJIMKo TaqaKa y KoopAHHaTHOM c11cTeMy, q11je cy KoopAHHaTe rrapoBH o,woBapajytrn:x BpeAHOCTH cn11Ka AOJie neBo). HapaBHO, IIITO BHIIle TaqaKa HaqpTaMO, AOfo1jaMo bOJbH YBHA KaKo Kp11Ba 3aBHCHOCTH H3rJieAa (cJIMKa AOJie AeCHo). He

3aBHCHOCT o6muca

fpanK OBe 3aBHCHOCTH

je XHIIep&ma xojy 'IHHe ;me rpatte.

- r -- -

p(kg/m3)

--...t..__

5-fi

I \ I

x

x

4-t-1 3-t!

3-t

2-t!--H V,, p,)

2H

1-tt+--rlv2' 0 .

1-'2~..

I

I

I

1

2

3

I

(Vl' P1)

1~P,) •_ I _ (V,, p 3)

- '

4

(v·P=l]

I

5

6'V(m3)

0

•-

~:r

1

2

3

,·

tY-

4

5

6 ) V(m3)

Kp11Ba 3aBHCHOCTH ABe o8pHyTo rrporropu;110HaJIHe Benwnrae Koje Y3HMajy ca.Mo II03J1TJ1BHe BpeAHOCTJ1 rrpeACTaBJba je.n;HY rpaHY nmep8one. • J1 40

1eHJ1

3a.i;aTaK

JJ;aTa je Kou;Ka Mace lkg. AKO je 3arrpeMJlfHa OBe KOIJ;Ke jeAHaKa l,6m3 , OApeAJ1 rycTHHY MaTep11jana OA Kojer je Kou;Ka HarrpaBJbeHa. AKo je 3arrpeM11Ha oBe Kou;Ke Malba o,n; l,6m3 , KaKBa je rycTHHa MaTep11jana o,n; Koj er je Kou;Ka HarrpaBJbeHa? ) AKo je 3arrpeMHHa oBe Kou;Ke Bena o,n; l,6m3 , KaKBa je rycTHHa MaTepnj ana o~ Koj er j e Kou;Ka HarrpaBJbeHa? 1) AKo je KOD;Ka HarrpaBJbeHa o,n; .n;pBeTa <JJ.1ja je rycTMHa H3Mel)y 500kglm3 11 750kg/m3 , rrpou;eHH 3arrpeMHHY OBe KOIJ;Ke .

• 89

IIpHMeHe nponopn;Hja Ilpocr a3XepHI1 pa'lpl

3a;ia.J:Of y Kojmia ce rrojaBJbyjy p;Be (AJipeKTHo mm: o5pH)'TO) rrporropu;110HaJIHe

BeJIWUme yrnaBHOM ce pernaBajy rrpMMeHOM rrporropu;Hja. Crreu;11jantto, aKo je ,n;aT '

je,n;a11 nap op;rosapajyhKx spe,n;HoCTH (x 1,y1) HeKe p;Be rrporropu;Hottantte sen11q11He H HeKa iJPyra spemiocT caMO je,n;He o,n; rrocMaTpami:x BeJIMq.MHa, Ha rrpHMep

x 2, ott,n;a ce

jemiocraBHO O.D;pel)yje H op;rosapajyha Bpe,n;HOCT y 2 .D;pyre sen11q11He. ,1J;aTH rro,n;au;H ce rrpmca3)jy y Ta8em1, rrpH tieMY ce ,n;HpeKTHa rrporropu;HoHaJIHOCT 03Haqasa rrapoM CIJ>em~a JtcTor cMepa, p;oK ce o5pttyTa rrporropu;HoHaJIHocT 03Haqasa rrapoM

cyrrpoTHo ycMepeHHX crpeJIH.u;a. IlomTyjyhH rrocTaBJbeHe crpen11u;e, je,n;ttocTaBHO ce
X1=2:'.!. x2 Y2

rrporropu;11ottaJIHOCT ,n;11peKTHa

~: =~:)

xi

I

Y1

Xi '+'

o6pttyTa

I

xi

I Y11

Xi '+'

Yi '+'

X1 =11_ x2 Y1 (x1 · Y1 =x2 · Y2)

Y2

IlpHMep 1. AKo ce cymelbeM

70kg CBe)l(er rpo~a ,11;0811ja 14kg CJBOr, KOJIHKO he ce KHnorpaMa

CJBOr rpo~a ,n;o5HTH cymeIDeM CBe)l(e

cyso

70kg

14kg

SOkg '+'

x

I .....

11/llllllllll:

SOkg cse)l(er? 70

14

so

x

-=-;

x=lOkg. •

IlpKMep2. A.J. Hry 3a

12 ,n;aHa, 3a KOJIHKO

811 ,n;a.Ha rrpo'IHTao KJDHry ,n;a je tIHTao TPH caTa ,n;HeBtto?

,n;aHH

earn

2 3

!

12 ...

x

I

2

x

3

12

- = - ; x=8,n;aHa. •

3cmaTaK 3a

15 earn HerrpeKH,n;HOr pa,n;a qeTHpH CllaBHHe

(H3 KOjHX so,n;a HCTHqe HCTOM 5p3HHOM)

Harrytte 5a3eH. 3a KOJIHKO caTH he 5HTH HaIIYibeH 5a3eH aKO ra rryHH rnecT CJiaBHHa (H3 CBaKe HCTOM 5p3HHOM HCTH'Ie sop;a Kao y rrpeTXO,ll;HOM cnyqajy)?

3a;:i;a-raK 3a

12 rranaq.MHKH nOTpe5Ho je 200 rpaMa 5paIIIHa. KonHKO 5panrna je noTpe5Ho 3a 30

rrana'IlflII(H?

90 I'

!!55!!!111!"'

....

,11.11n11111111111, ____________________

~ ~

IlpHMep3. J1 ce

)M

ce

Crroxea a3llepHH paqyH

Tue rycKe 3a ,n;Ba ,n;aHa CHecy ,n;Ba jajern. KoJIMKO jaja he rnecT rycaKa ,n;a CHecy 3a rnecT J'aHa? :JocTaBJbeHM 3a,n;arnK je BepoBaTHO Hajrro3Han1j11 3a,n;arnK oBe BpcTe. qecTo ce CBpcrnBa y ,,TpHK 3a,n;aTKe': 8y,n;yh11 ,n;a BeJIHKH 8poj Jbyp;H ,n;aje rrorpernaH o,n;roBop. .TpMK" Koj11 ce rrpeB11,n;11 rrp11 pernaBalby jecTe y TOMe ,n;a ce 8poj jaja He rroBenaBa 3 ;iyrn, Ben 3 · 3 rryrn, o,n;HOCHO, 9 rryTa. HaHMe, TPM rryrn ce noBehao 8poj rycaKa, an11 ce ":pM rryrn rroBehao 11 8poj ,n;aHa. rycKe

·3(:

p;aHH

!·3(:!

jaja

~ ! )·9

J;aKJie, rnecT rycaKa he 3a rnecT ,n;aHa ,n;a cHecy 18 jaja. JJ;a je 3a11crn TaKO, MO)l(eMo ce 7BepMTJ1 11 pa3MHIIIJbajyh11 Ha cne,n;etm HaqMH: 2 rycKe 3a 2 ,n;aHa CHecy 2 jajern, 1 rycKa 3a 2 ,n;aHa cHece 1 jaje, 6 rycaKa 3a 2 ,n;aHa CHecy 6 jaja, 6 rycaKa 3a 6 ,n;aHa CHecy 18 jaja; ;m11

2 rycKe 3a 2 ,n;aHa cHecy 2 jajeTa., 2 rycKe 3a 1 ,n;aH cHecy 1 jaje, 6 rycaKa 3a 1 ,n;aH CHecy 3 jajern, 6 rycaKa 3a 6 ,n;aHa CHecy 18 jaja.

+ ryCKe

3a.a;aTaK

IKO

AaHJ1

,.- 2 ; - /' 2 • :> 4 .... 3..10

J;Be rycKe 3a ,n;Ba ,n;aHa CHecy ,n;Ba jajeTa. _) KoJIMKO jaja he qeT11p11 rycKe ,n;a cHecy 3a ,n;eceT ,n;aHa? 2) Kon11Ko jaja he ocaM rycaKa ,n;a CHecy 3a rreT ,n;aHa? ~ruyiflciUBo je ,n;aTO Ha MaprHHH.

ryCKe

·4

;lamf

/' 2 I ( 2 -2,5 .:t.. g t 5

jaj a

~ !)·10 j aj a

2 x

)·10

BapHjaHTe aa,n;aTKa KOjH cMo pernaBaJIH y rrpeTXo,n;HoM rrp11Mepy 1:1ecTo ce Mory ecTH y eHl.trMaTCKHM 1:1acorr11cHMa am1 H Ha TeCTOBHMa 3a aanocJielbe. opMynaI\Hje cy npH TOMe 1:1ecTO maJbHBe.

!OM)

L KoKa H no aa AaH H no cHece jaje H no. KoJIJ1Ko jaja he Aa CHecy mecT KOKa aa wecT AaHa? (0,llroBop: 24 jajern.)

(113

10

1

~

a6pHKa H no aa AaH H no npoHaBeAe TOHY H no 'IOKOJia,lle. KoJIHKo TOHa 'IOKOJia,n;e he ,Ila np0H3Be,lle ,lleBeT a6pHKa aa ,lleBeT AaHa? (0AfOBOp: 54 TOHe 'IOKOJiap;e.)

91 I

IIpHMep4. Tpn :Mannme 3a 10 ;iaHa, pa.udrn 6 cant ,a;HeBHO, 3anaK)'jy 9 000 rraKeTa tteKor rrpmt3Bo;:ra 3a KOID1KO ,a;aHa be 4 Manrn:He 3anaKOBaTJ1 11 200 rraKeTa pap;ehn 8 caTH ,llleBBO? PaaMUIIIJb
3 MamttHe 3a 1 Mannrna 3a 1 MaIIrnHa 3a 1 Manrntta 3a

10 ;:raBa, pa,ll;e!m 6 cam ;uteBHO, 3anaK)'jy 9 000 rraKeTa, Iv ,uatta, pap;errn: 6 caTu .ITHeBHO, 3anaK)'je 3 000 rraKeTa, 1 µ;aH, pap;erm 6 cant .UHeBHO, 3arraK)'je 300 naKeTa, 1 .uatt, pa.uefll1 1 caT }J;HeBHO, 3anaK)'je 50 naKeTa,

1 Mannrna 3a 1 .uatt, pap;ehn 8 caTH p;tteBHO, 3arraKyje 400 naKeTa, 4 Maunrne 3a 1 .uatt, pa.ueh11 8 caTH p;HeBHO, 3anaKyjy 1 600 naKeTa. KaKo je 11200: 1600=7, 3aKJbyqyjeMo p;a:

4 . iarrnrne 3a -; .uaHa, pap;e11.JI 8 can1 p;HeBHO, 3anaKyjy 11 200 naKeTa. IIonaBJbeHJ11 3ap;aTaK MO)l(eMo pemaBaTH H Ha octtoBy cnep;ehe rneMe .

.--- - -

IlOBehaBaH>eM 8poja p;aHa 3a

He1rn nocao, CMa:Ebyje ce 8poj

HeKJI rrocao, rroBehaBa ce H

pawrnx caTH no .n;atty.

IloBeilaBa:EbeM 8poja .n;atta 3a Heim nocao, cMalDyje ce 8poj Mannma Koje rpe8a .n;a

------

IIoBehaBa:EbeM 8poja .n;aHa 3a

pa~~

.--J

8poj 3aIIaKOBaHHX rraKeTa.

----MarnHHe

!I

p;aHH

l~ t

caTH

:I

IIaKeTH

9 ooo

11 200

I i

Cmfq}lo Kao y rrpMMepy 3, .n;o8njaMo p;a je

X=l0· ~ · ~ · ll200 _7 . • 4

8

9 000

)J;pyrM Ha'Ilfll pemaBa:Eba 3ap;aTKa H3 rrpeno.n;ttor rrpHMepa je yo8nqajeH H qnHe ra p;B OCHOBHa KOpaKa: • npBo Tpe8a YIBPAHTM .n;a JIM je Benwnrna qnja je jep;Ha Bpep;HOCT Herro3HaTa .n;upeKTHO HJIJ1 o8pliJTO nporrOpl\MOHaJIHa ca CBaKOM p;pyroM BeJIHqHHOM rrojep;HHa'Illo, ~ a 3aTI1M opMHpaTII op;roBapajyhy rrporrop~Mjy H HailJi Herro3HaTH qJiaH.

3a,::i;aTa TpM MaIIIHBe 3a 10 p;atta, pa.n;efll1 6 caTH ,a;HeBHO, 3arraKyjy 9 000 rraKeTa HeKor npOH3BO}J;a

I KoJIHKo naKeTa he fo.1n1 3aIIaKOBaHO aKo 3 MamHHe pap;e 7 p;aHa no 4 caTa p;HeBHo " KoJIHKo caTH p;:HeBHO Ipe8a p;a pa.n;e 2 MarnHHe p;a 8n 3a 10 p;aHa 3arraKOBaJie 10 OOC naKeTa?

92

Ilpo~eHTH 11 npoMrum cy 3arrn:rn 5pojeBa

l1

sa

LOj11 cy rroce5tto rroro,n;H11 y pa3HJ1M ::rpaKTJ1qHJ1M c11Tyan;11jaMa. Tipon;ettTHMa :e 5poje CTOTJ1 ,n;eJIOBJ1 HeKe BeJIJ1qJ1He, ,ll;OK :e rrpoM11JI11Ma 5poje XJ1Jba)l;J1TJ1 ,n;eJIOBH.

p% =_.P___, p%o =-P-. 100 1000 cyrnT11H11, paqytt ca rrpon;eHTHMa HJIJ1 rrpoM11JI11Ma rrpe,n;cTaBJba yo811qajett pa'I)'H ca pa3JIOMD;J1Ma 3arr11catt11M y rrocebHOM nbJIHKy.

11poqeHTll

1 1%=100 2%=-2-=_l_ 100 50 ,__ 20 1 20%=-=5 100

' 2%o- 2 - 1 1 - 1000 - 500

50%=~=_!_ 2

500o/oo = 5 00 =_!_ 1000 2

100%= lOO =l 100

1000%o = lOOO = 1 1000

>---

- - -

100 -

1%o=-l1000

JI

IIpOMlLTII

•

-

20%o=~=-11---

-

1000

50

3a

= o;:t

(Kp03) CTO

naT. pro mille = o::r (Kp03) XHJblO,Y

IlpHMep S. :1p11MeT11Mo ,n;a cy opMYnan;11je ttape,n;Ha p;Ba 3a,n;aTKa 11cTe.

Axo je pa3Mepa ue;ume Uetta je,n;ttor rrpo113Bo,n;a je CHH)l(eHa 3a 20% 11 ca,n;a H3HOCJ1 2 450 ,n;11ttapa. Kon11Ka je fa.ma ~eHa po5e rrpe rrojeT11H:.eH:.a?

Uetta jep;Hor rrpo113Bo,u.a je CHJ1)1(eHa 3a IIeTMH)' If cap;a H3HOGf 2 450 ,IJ.HHapa. KoJIJfKa je farna Qetta po5e rrpe rrojeTHH>eiba?

{

~

11aT. per centum

AKo ca

G n ;:te.'Ia P jegHaxa p%. muaje G:P=lOO:p mm G: lOO=P:p.

x 03Haq11Mo Qetty rrpe rrojeT11H:.e1na, ,n;o811jaMo ,n;a je X-

1

20%x = 2 450, O,ll;HOCHO, X - - X = 2 450,

5 J1JIJ1 eKBJ1BaJieHTHO

4 80%x = 2 450, Op;HOCHO, - X = 2 450. 5 }fa IIOCJiep;H:.e je,n;ttaKOCTJ1 J1MaMO ,n;a je x = 3 062,5 ,ll;HHapa. •

AKo je pa3Mepa ue.nrne G H ;ie.na P je;ma.Ka ~ oH.;Ia je G: P=lOOO :p.

fipHMep6. Hajrro3HaT11ja yrroTpe5a rrpoMHJia je IIpHJIHKOM 113pa)l(aBCUba Kon:wnme am
.. .

93

Ilpi111q>7. 3aCT)'IIJheHOCT AEO KpBIDiX rpyna y fo1jH je cne,n;eha:

•A- 41,8%, • 0-34,9%, • B - 16,2%, • AE- 7,1%. ,IJ;a 8HcMo 3aCTyIIJbeHocr KpBIDiX rpyna y 8HjH CJIHKOB11r11je rrp11Ka3aJI11, Kop11cr11Mo TaK03BaHe Kp)"A
41,8%. 360°=150,48° 34,9% . 360° = 125,64° 16,2% . 360° = 58,32° 7,1%. 360°=25,56° . •

3a,lara · ITopep; rro,n;ene ADO KpBHHX rpyna, Y3HMa ce y ofoHp H TaK03BaH11 pe3yc-aKTop (Rh-aKTop). CBH Jhy,ll;H ce ,n;e11e Ha Rh+ (PX-II03J1THBHe) J1 Rh-(PX-HeraTHBHe), rra CBaKa ABO KpBHa rpyna HMa p;Be nop;rpyne: A-, A+, 0-, 0+, B-, B+, AB-, AB+. KpBHa rpyna 0-je yttMBep3aJIHH ,n;aeaJiaQ KpBH Hy 811j11 HMa rrp118JIJ1)1(HO 5,5% lhyp;11 oBy KpBHY rpyny. Y3HMajyhH y ofo.Hp rrop;aTKe H3 npeTXop;Hor rrpHMepa, rrpoQeHH KOJIHKO npoQeHara lh)'Wi y 8HjH, KOjH HMajy 0 KpBtty rpyrry, HMa HeraT.HBaH Rh-cl>aKTop, a KOJIHKO II03HTHBaH.

Ilp11Mep 8. IJ;eHa jewrnr npoH3Bo,n;a je y p;Be npop;aBHHQe 8.HJia 11cTa, 520 p;HHapa. 0Hp;a je y jep;Hoj npop;aBHHQM QeHa Hajnpe cIDDKeHa 3a 15%, a 3aTHM 3a 10%, a y p;pyroj rrpop;aBHHQH je CaMO jep;am:ryr CH.H)l(eHa 3a 25%. y KOjoj npo,n;aBHHQH je cap;a OBaj rrpOH3BOp; j eTIIBaj H? Ilpsa rrpo,n;aBHHQa

Hoea ueaa rrocne

Cttlf)l(elbe 15%

IIOBefta.Ba.J:ba, 0.llHOCHO,

ueae x 3a p% je4HaKa je x±p% · x=(l00±p)% · x CHJDKa.Ba.J:ba

CHM)l(elbe 10%

15%. 520 =78 520-78=442 10%. 442 =44,2 442 -44,2 = 397 ,8

Cttlf)l(elhe 25%

25% . 520 = 130 520 -130 = 390

,[{aKJie, IIpOH3BO,ll; je jeTMHHjH y npop;aBHHQH y Kojoj je QeHa caMO jep;attrryr CHM)l(eHa 3a 25%.

3aJJ;arar AKo je HeKH npoH3BO,n; nocKyneo 25%, KOJIHKO rrpoQeHaTa Hosy Qetty Tpe8a CHH3HTM p;a 8H ce p;o8.HJia rrpsofo1THa Qetta?

94

-

-~~-~-~

I

. IIpHMep9.

Pa'IJ'B Memau.a

Y KOM op;Hocy Tpd5a noMernani pacrnope anKoxona op; 60% H 85% p;a 8H ce p;o8Ho pacrnop op; 70%? AKo je xkg KOnHqHHa pacrnopa anKoxona op; 60%, oHp;a y lboj HMa 60% · x = 0,6x anKoxona. :niMo rpyr pyra

AKo je ykg KOnJllqHHa pacrnopa anKoxona op; 85%, OHp;a y lbOj HMa 85% · y = 0,85y anKoxona. Kap;a OBe KOnHqHHe IIOMernaMO H p;o8HjeMO (x +y )kg HOB Or paCTBOpa, y lbeMJ HMa 0,6x + 0,85y anKoxona. ,[(a 8H oBa cMerna npep;cTaBJbana pacrnop am<0xona op; 70%, IIOTpe8Ho je p;a B
0,6x + 0,85y = 0,7(x + y). yfa OBe jep;HaKOCTH p;aJbe p;o8HjaMO p;a je (0,7 -0,6)x=(0,85-0,7)y, Tj. O,lx=0,15y, op;aKne cnep;H p;a je

YK)'IIHa xo:nrnrna: 750m

Anxoxona: 40% · 750mf =300mf

x :y=0,15: 0,1=3:2. 3a,IJ;itTaK Koje KOnHqHHe pacrnopa anKoxona op; 60% H pacrnopa anKoxona op; 85% TPe8a IIOMelllaTH p;a 8H Ce p;o5HnO

50i paCTBOpa anKOXOna Op; 70%?

(Rh-

:BaKa

a:a

IIpHMep 10.

PaqyH no;:{ene

TpH nopop;HQe cy 3aKynHne Ha Hep;eJby p;aHa Kyhy Ha Mopy 3a 75 000 AKHapa.

OBY

TpornKoBe cy nop;enHnH a3MepHo 8pojy qnaHoBa. AKo jep;Ha nopop;~a JiMa TP"

MKO

qnaHa, p;pyra qeTHpH a Tpeha neT, op;peAJfMO CYMJ Kojy 3a 3aKyn TPe8a p;a H3p;BojH

p,a

CBaKa nopop;HQa. AKo ca

x, y, z 03HaqHMo CJMe Koje Tpe8a pep;oM p;a H3p;Boje TpoqnaHa, qernopoqnaHa H

neTOqnaHa nopop;HQa, oHp;a p;o8HjaMo p;a je

+ y + Z = 75 000 H X : y : Z = 3 : 4 : 5. ¥13 p;pyre jep;HaKocrn: p;o8HjaMo p;a je x = 3k, y = 4k, z = 5k, 3a HeKo k, X

HOj

~H

a 3arn:M, J3HMajytrn:

y ofoHp Ji npBy jep;HaKOCT, HMaMO p;a je

3k+4k+5k=l2k=75 000.

k = 75 000 : 12 = 6 250, rnTo 3Haq11 p;a TpoqnaHa nopop;HQa Tpe8a p;a nnaTH x = 3 · 6 250 = 18 750 p;HHapa, qernopoqnaHa y = 4 · 6 250 = 25 000 p;HHapa, a neTOqnaHa

,[(aKne,

z = 5 . 6 250 = 31 250 AH Hapa. IIpHMep 11. TpH nopop;HQe Tpe8a p;a nop;ene TpomKoBe KOIIalba 8yttapa Koju cy H3HOCJurn 140 000 p;HHapa. ,[(orOBop je p;a nop;eny H3Bprne o8pttyTO a3MepHo )'AaJbeHocrn 8yttapa op; CBOjHX Kyha, npH qeMy je jep;Ha Kyha op; 8yttapa )'AaJbeHa 300 MeTapa, p;pyra 600 MeTapa H Tpeha 1200 MeTapa.

Ke Ha


AKo ca x, y,

z 03HaqHMO op;roBapajyhe CJMe, OHp;a p;ofo.1jaMo p;a je + y + Z = 140 000

1

1

1

y: Z =--: - - : - -. 300 600 1200 Op;aBp;e, Kao y npeTXop;HoM npH.Mepy, pa~a..\io p;a je: x = 80 000 p;HHapa, y =40 000 p;HHapa H z = 20 000 p;Jrnapa. • X

H X:

95

Kalilanm pa11JH

IIo3ajMJbHBalbe u ynaralbe HOBn;a npep;cTaBJbajy p;eo CBaKOp;HeBHOr nornoBalba. IIpH

TOMe, je~a CTparui p;aje HoBan; p;pyroj, oqel<}'jyffii p;a joj ce HOBan; 3a HeKo BpeMe Bparn )'BehaH 3a o,u;pel)eH)' CJMY. Ha npm.1ep, ~'Balbe HOBn;a y 5~ je yn:aralbe 3a oco8y qJ1jl1 je HOBan;, p;oK je TO 3a

baHKJ II03ajMJbJIBalbe HOBn;a: yn:araq p;aje HOBan; battn;M, a baHKa Il03ajMJbeHJ1 HOBan; Kopucrn 3a COIICTBeHO IIOCJIOBalbe.

c ,u;pyre CTpaHe, Il03ajMJbJ1Balbe HOBu;a (y3J1Malbe

Kpewrra) op; baHKe 3a baHKY npep;cTaBJba yn:arafbe. Y CBaKOM CJI)"Iajy, OHaj KO yn
3a Bpe

II03ajMJbeHOf HOBIJ;a.

}'Be

Yno)KeHa cyMa HoBn;a Ha3MBa ce maBmm;a, p;oK ce oqeKHBaHa p;obHT HaKOH o,n;pel)eHor BpeMeHa Ha3J1Ba Ka.MaTa J1JIJ1 HHTepec. KonHKa he KaMaTa (p;obHT) bJ1TJ1 3aBJ1CJ1 op; .n;oroBopeHe Ka.MaTHe cT011e Kaja ce H3pa)KaBa y npon;eHTJ1Ma J1 npep;cTaBJba p;obHT Kojy

fot p;oaeno HHBeCTHpalbe CYMe op; 100 ,ll;J1Hapa (eBpa HJIM HeKe ,n;pyre BanyTe Kaja ce KopJtcTH) 3a rOJJ.lflfY p;atta J1 BpeMeHcKor 11epuop;a H3pa)KeHor y rop;HHaMa.

ynor

10 0 ,lJ.J1H G ,!J.J1H

l

J.1Majyffii y BMlzy 3Haqefue KaMaTHe CTone p%, je,n;HocTaBHO paqyHaMo p;obHT rnaBIDtD;y

K 3a

G J1 rrepHop; op; t rop;HHa:

K t p 1

G

-=-·--,

100

Tj.

K = G · _E_ · t = G · p% · t. 100

OB Ila a.KO

IIocne.z~1na ¢oM)'Jla ce Ha3J1Ba IIpOCTa Ka.MaTHa c}>opMYJia. -

Y npaKrn ce npocTa KaMaTHa ¢opMyna yrnaBHOM KopHCTJ1 3a nepHop;e KojH HHCY p;y)i(J1 o.u. ro'AJIBY p;atta. TaKo, aKO je HOBan; yn:o)KeH Ham Mecen;H, npH qeMy je 1 ::;; m::;; 12, KaMaTa H3HOCH:

m

K=G·p%·-, 12 a aKO je yno2KeH Ha d .n;atta, 1 ::;; d ::;; 360, oHp;a je KaMaTa: d K=G·p%·-, 360 rrop; npenmcraBKOM (38or jeJJ.HOCTaBHHjer paqytta) .n;a cBaKM Mecen; HMa 30 p;aHa, a rowma 360 p;atta.

IlpH111ep 12. HeKo je Y3eo Kpep;HT op; 150 000 AJtHapa Ha fOAHH)' p;aHa, ca KaMaTHOM cTonoM op; 5,1 %. KOJIID
1. ,UaKJie,

K=G·p· t=l50000·0,051·1=7650.

+

a.KO

PPl1 BpaTJ1

fipHMep 13. - ~o je opoqJ1o 320 000 p;l1Hapa Ha TPH Mecen;a, ca fOAfillllbOM KaMaTHoM crorroM o.:i; KonHKO he HoBn;a rro,n;HhH no l1CTeK)' .n;oroBopeHor rrepHo,n;a?

pa [Ban; ~aH>e

G=320 000,p=7%=0,07, t=0,25; K = 320 000 · 0,07 · 0,25 = 5 600. ,::u<.n:e, ynaraq he rro,n;HhH 320 000 + 5 600 = 325 600 .n;1rnapa.

+

ra)l(e eH>a

~~eHor

o.n; iHT Kojy ce

BpeMeHcKe rrepHo,n;e .n;y)l(e o.n; ro.n;irny ,n;aHa, HaKoH acTeKa CBaKe rop;Jrne rnaBinn~a ce ehaBa 3a KaMaTy, n:a ce ,n;aJbe paqytta ca HOBOM rnaBHHn;OM. Ha Taj Ha'IJrn .n;ofotja.Mo 7aK03BaHy CJIO)l(eHy KaManry OPMYJIY·

~

_:...xo je HeKo opo'IHO CYMY G ca KaMaTHOM cTorroM p% l1 rrop;IDKe

HOB~ TeK no HCTeKY

n

-ug1rna, OH,n;a je JKyrrHa CYMa HaKoH:

• 1 ro,n;HHe je,n;HaKa G(l +p%), • 2 ro,n;HHe je,n;HaKa G(l + p%)(l + p%), • 3 ro,n;HHe je,n;HaKa G(l +p%)(l +p%)(l +p%),

• n ro,n;HHaje,n;HaKa G(l +p%)". HHje TernKo 3aKJbyqHTl1 .n;a, YKOJIHKO 5J1 HCTa oco5a rrocne CBaKe rowrne rrowf3a.n:a KaMaTy, rrocne n ro,n;HHa 5J1 YKYIIHO rrop;l1rna:

G(l +np%).

llp.RMep 14. HeKo je opo'IHO 100 000 ,n;HHapa Ha 3 ro,n;HHe ca KaMaTHOM CTOIIOM 4,2%. KomfKo HOBn;a he IIO,ll;l1hl1 HaKOH J1CTeKa OBOf rrepHo,n;a:

1) aKO IIOCJie CBaKe J1CTeKJie f0,ll;l1He IIO,n;IDKe KaMary; 2) aKo rro,n;H)l(e HOBan; HaKoH TPH ro.n;1rne?

r .n;y)l(J1

2,

y rrpBOM cnyqajy K0pl1CTJ1MO rrpocTy KaMaTtty ¢opMYJIY, a y .n;pyroM CJIO)l(etty KaMaTH)· ¢opMyny.

1) 100 000. (1+3. 0,042) = 112 600. 2) 100 000. (1+0,042) 3 =113 136,6088 . •

,a

;

5,1 %.

eHCKJ1

97

3aµ;a~H

Pa3Mepe n

A

nponop~Hje

ieJUIBJrqa Mepe 3a JJ;)')IGIBY. 1 inch {in) [mPi]

-----

1 foot
12in 3ft

1 yard 1yd) LJapA] -·-1 mile mi) [MIDba]

....,.__

- -

---+--

-

..

25,4mm 30,48cm 0,9144m

··-+

-~

l

l,609km

5 280ft

1) ,futjaromu1a TB eKpaHa ,izyra'!Ka je 37 HH'Ia. KoJIHKa je H>etta µ;pmtta y QeHTID.teTpHMa? :: PacTOjalbe H3Me~y Baumttrrotta H lhyjopKa, Ba3,qyrnttoM JIHHHjOM l13HOCH 204 MHJbe. KonHKO je TO pacTOjaH>e y KHJIOMeTpJ.·iMa? 3) KonHKO npHOJIIDKHo HH'Ia HMa jeAaH QeHTliMeTap?

- · opmq>aj nponOpl.{Mjy, aKO je Moryne, O;:rpegJ[ opoj x, aKO je: 1 2 3 11 x: -=- :-; L.J 0,3: x=0,6: 0,2; 3 5 4

0)1;

OpojeBa 15, 18, 35 H 42. 1

2

3

x : 1 - = 1- : 1 -; 4) 0,01 : 0, 1 = x : 0,01. 3 5 4

OAJle;ut npoA)')KeHY nponopQHjy, aKo je x : y = 3 : 5, y : z = 2 : 3 H z : u = 4 : 3. - ,UoKaJKM CJJ.eAehe cneQHjanHe cnyqajeBe TeopeMe ca CTpatte 85. HeKa cy a, b, c H d pea.mm: opojeBH, CBH pa3JIH'IHTH 0)1; ttyne, TaKBH µ;a je a : b = c : d. Ottµ;a je: (a+ c) : (b + d) =a : b, noµ; ycnoBoM µ;a je b + d ;t. O; :. 'a -c): (b-d) =a: b, noµ; ycnoBoM µ;a je b-d ;t. O; : (2a - 3c) : (2b- 3d) =a : b, noA yrnoBOM µ;a je 2b- 3d ;t. O; "*I (ax-'-cy): (bx+dy) =a: b, 3a CBe OpojeBe X HJ TaKBe µ;a je bx-dy :;t 0.

3.".iHp }'T.ilOBa

KOHBeKCHOf 'ieTBOJ><>Jma je!Illax je 360°.

6. HeKa cy a, b, C, d, a, p, y H l) peamrn OpojeBH, CBH pa3JIH'IJ.fTII O:I H}n e. AKO je a: b =c: d, AOKIDKH Aa je (aa +pb): (ya +8b) =(ac+pd): ( c7. O)l;pe,IU1 )'TJIOBe a, p, y H 8 KOHBeKcHor '!eTBopoyrna, aKo ·e a: _ :

O)l;pe)Ui opojeBe X, y Hz, aKO je x: y: z=3: 5: 7 H x+2y - 3==5 . Ogpew1 opojeBe x, y H z, aKO je x : y = 5 : 6, y : z = 2 : 3 " x -

E

I I HeKa cy a., al' ... , a,, H b., bl' ... , bn peaJIHH opo"eB

]:{a je a 1 : a 2 :

••• :

a,, =b 1 : b2 :

••• :

k1b1

b". ,UoKCUKH )!;a ·e (a- -_-_ - - -

+k2b2 + ... +knbn r b. - 1_ - -

3a OHJJ.o Koje 8pojeBe k1, k2, ••• ,kn' 11, 12' ... , 1 nponopQHjy pa3JIH'IHTH oµ; ttyne.

---

=0, 2b +d :;t: 0 H

IO. HeKa cy a, b, c H d 8pojeBH pa3JIWIHTH oµ; Hyne T (a - 3c) : (b-3d) = (2a + c) : (2b + d). ,UoKCUKH µ;a ie a:

k 1a 1 +kia2 + ... +knan

98

8=1: 2: 3: 4.

-

12. 0ApeA11 HeKoJI11Ko TpojK11 8pojeBa (a, b, c) Koje 3aAOBOJbaBajy nponopQ11jy

b-a c-b

c a

----

'2

'

,IbtpeKTHa H obpHyTa nporropn;HoHaJIHOCT

hr

Ta je noBOJbH11je Kyn11T11, 8oQy ca 0,5e coKa no QeH11 OA 64 Al.fHapa 11n11 8oQy ca l,25e 11CTOr COKa no QeH11 OA 134 A11Hapa?

A

( · c\ HaQpTaj rpacl>11K 3aB11rnocT11 113Mel)y npel)eHor nyTa (y MeTp11Ma) 11 npoTeKJior

J

BpeMeHa (y ceKYHAaMa) np11 paBHOMepHOM KpeTaH>Y 8p311HOM OA 0,5m/s.

1') Ep311Ha ryceH11Qe A je lOdm/h, 8p311Ha ryceH11Qe B je lOcm/min, a 8p311Ha ryceHHQe C je lOmm/s. Koja ryceH11Qa je Haj8p)l(a, a Koja je Hajcnop11ja? ) HaQpTaj rpacl>11K 3aB11cHocT11 113Mel)y npoTeKJior BpeMeHa (y caT11Ma) 11 npocewe 8p311He KpeTaH>a (y Kl1JIOMeTp11Ma Ha caT) np11n11KOM npenacKa nyTa AyratIKor 2km.

llpHMeHe nponopn;Hja a nanatI11HKe je noTpe8Ho: 200g 8panIHa, 200me MJieKa, 200me BOAe, 2 jajeTa H 30g yJba. ,D;a 011CMO HanpaBl1Jll1 CMecy 3a naJia'Il1HKe OA 5 jaja, KOJil1KO je noTpeOHO ocTamu cacTojaKa? f

(J 3a TpaHcnopT rpal)eBl1HCKOr MaTep11jana noTpeOHO je 15 KaMHOHa HOC"130CTH OA

\_j

3 TOHe. KOJil1KO KaMl10Ha HOCl1BOCTl1 OA 5 TOHa je noTpeOHO 3a TpaHcnopT 11CTe KOJI11tI11He MaTep11jana?

~OJil1KO c11jaJI11Qa OA lOOW Aaje 11CTO OCBeTJbelbe Kao 24 rnjaJUfQa OA 75W?

G ~A

5kg 8paurna 11cneqe ce 10 XJieOoBa OA no 750g. Kon11Ko XJieOoBa OA no lkg ce MO)l(e A0011Tl1 OA 6kg 8pau1Ha?

@BT

11 Kpojaqa 3a 5 AaHa, paAetrn no 4 caTa AHeBHO, canrnjy 10 KanyTa. 3a KOJil1KO AaHa lie 6 Kpojaqa canrnT11 25 KanyTa aKo paAe 5 caT11 AHeBHo? ' Kon11Ko KanyTa lie canrnT11 4 Kpojaqa 3a 3 AaHa aKo paAe 8 caT11 AHeBHO? 3) Kon11Ko caT11 AHeBHO Tpe8a Aa paAe ABa Kpojaqa Aa 811 3a 6 AaHa camHmt 10 KanyTa? •) Kon11Ko Kpojaqa lie 3a 9 AaHa, paAeli11 no 5 caT11 AHeBHO, canrnT11 30 Kanyra?

Can11H11TeT eA03eMHOr Mopa je 38%0. Kon11Ko 11Ma con11 y 2t BOAe 113 oBor Mopa?

2 \./JeAaH npo113BOA je ABa nyTa nojecl>T11H110 3a 11cT11 npou;eHaT. IlpBH nyT je QeHa CH11)1(eHa 3a 250 A11Hapa, a Apyr11 nyT 3a 200 A11Hapa. 3a KOJil1KO npoQeHaTa je QeHa CHl1)1(eHa cBaK11nyT11 KOJIHKa je 811na QeHa npo113BOAa npe npBor CHJ1)1(eH>a?

A

~flpeMa ITOgau;HMa H3 2010. rogHHe, 3a ITpHCTyrr HHTepHeTy HajBHUJe ce KOpHCTH (_,,hirefox ca ygenoM og qaK 42,2%. Ha gpyroM MecTy je Internet Explorer ca ygenoM Ha Tp:>KHUJTY og 25,8%, goK je Ha TpeheM MecTy Google Chrome KojH gp:>KH 25% J,p:>KHUJTa.

_) y KOM rrpou;eHTY cy 3aCT)'IIJbeHH OCTaJIH ITpeTpa:>KHBaqw?

VOgpe.ITTf rrpwbJIH:>KHe BpegHOCTH u;eHTpanHHX yrnoBa Kpy:>KHHX 11ceqaKa KPY:>KHor AJtj arp(l}.ia 3acTyrrJbeHOCTI1 HHTepHeT ITpeTpa:>KHBaqa Ha Tp:>KHUJTy.

25 KapaTIL'da ce H3pa:>KaBa qnrnoha nerypa 3JiaTa Koje ITopeg 3JiaTa yrnaBHOM cagp:>Ke 5aKap j ep OH He KBapH 5ojy 3JiaTa, a ITOBehaBa MY qBpcTOhy. Epoj KapaTa X ce ozq>el;iyj e
X = 24 ~z, rge je M , Maca q11cTor 3JiaTa y neryp11, a M" je u

yKy1raa . 1aca n erype. qHCTO 3JiaTo (ITpou;eHaT 3JiaTa je 100%) HMa 24 KapaTa. Kom1Ko ITpou;eHaTa 3JiaTa cagp:>KH 18-KapaTHa nerypa 3JiaTa?

0 y

KO. [ o,nHocy Tpe5a ITOMelIIaTH qHCTO 3JiaTO ca 18-KapaTHHM ga fo.1 ce gobHJIO

22-KapaTHo 3JiaTo?

(5 Ha cKJiaµHrrrry H

a Senor 5paUJHa ITO u;eHH og 50 gHHapa ITO KHJiorpaMy H u;pHor

5paUJHa IIO u;emt OA 35 AHHapa ITO KHJIOrpaMy.

y KOM OgHocy Tpe5a ITOMeUJan1

oBe gBe BpcTe 5pamHa ga fo.1 ce go5Hna MeUJaBHHa ITO u;eHH og 40 AHHapa no KJ.rnorpa.'dY?

12

·

OJIJ.1KO rpaMa pacTBopa CYMnOpHe KHCeJIHHe og 96% Tpe5a gogaTH y 15,5f Boge ga

"---" 5u ce go5110 pacrnop og 3%?

.,9. Ce.ri;a.'d J11.1Tapa cnattor pacTBopa og 4% noMeUJaHo je ca gpyr11M cnaHHM pacTBopoM J[

~o5uj eHO j e 11 JIHTapa HOBOr pacTBopa. AKO je 5% COJIH y HOBOM pacTBopy,

o;rpe.ri;H j a'!JfHY gpyror pacTBopa. qeTiipH pagHHKa cy 3apagIDia 98 400 AHHapa. IlpBH pagHHK je pag110 10 gaHa no 6 caTH )UieBHO, gpyrH pagHHK je pagHO 8 gaHa no 8 caTH gHeBHO, TpehH pagHHK je pa.ITT!O 12 ;laHa IIO 6 cant p;HeBHO H qeTBPTH je pagHO 10 gaHa no 5 caTH gHeBHO. KoJIJtKo CBaKJt paWJHI<: Tpe5a ga gofo.1je?

3J

'n:o:>KeHo j e 100 000 ]J;JtHapa y 5aHKY Koja IIJiaha KaMaTy 4,2% rogHUJlhe.

6>. KonKKa he K~ laTa fon11 HaKOH 8 Meceu;11? - KOJIHKO HOBu;a ce MO:>Ke ITO,ITT!IlH HaKOH 3 rogHHe aKO ce KaMaTa paqytta CJIO)KeHO [ KaMaTHO. [ opM)'JIOM?