eroberogeday

lwm pl:ún nig Esn Bisidæ briBaØabR&tEpñkKNitviTüa

គណះកមមករនពន្ឋ នង ិ ិ េរៀបេរៀង េ

េ

ក លម ឹ ផលគន ុ

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គណះកមមករ្រតួតពនតយបេចច កេទស ិ ិ េ េ េ

ក លម ឹ ឆុន

េ

ក នន់ សុខ

េ

ក្រសី ទុយ រ ី

េ

ក អុង ឹ សំ

ក ទតយ ិ េម៉ង

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គណះកមមករ្រតួតពនយអកខ ិ េ

ក លម ឹ មគក ិ សរិ

ក អុង ឹ សំ

កញញ លី គុ

វរុិ ទ្ឋ

រចនទំពរ័ នង ិ ្រកប

ករយកុំ ពយទ័ ិ ូ រ

េ

ង

ង

េ

្ណ ក -i-

ក ្រពំ ម៉

Garmáfa esovePAkMENlMhat; KNitviTüafñak;k;TITI7 kmµviFIfµIenH at;KNi rYmman 10 emeron KW cMnYnKt; Kt;- tYEck nig BhuKuNrYm-

RbPaKrgVal; RbPaK-cMnYnTsPaKTsPaK-PaKryPaKry-rgVas; s;rgV l;-kenSamBICKNitsmIkardWeRkTImYymanmYyGBaØat nig vismPaB . kñúgemeronnImYy² eyIgxJMú)andkRsg; ykmkeFIVdMeNaHRsayKMrUya:a:g andkRsg;lMlhM at; at;ykmkeFI ek, cgcaM . ek,aHk, aHk,ayEdlGac[ ayEdlGac[GñksikSagayyl; Sagayyl; nig qab; qab;cgcaM edayGectnaR))akd esovePAenHminl¥hYseKhYsÉgenaHeT kMhusqÁgedayGectnaR Caman TaMgbec©keTs nig GkçraviruTæ GaRs½yehtuenH eyigxMJúrgg;;caMcaMCanic©nUv mtiriHKn; EbsßabnaBIsMNak; eBlevla . Kn;Ebsß ak;GñGkñ sikSaRKb; SaRKb;eBlevla CaTIbBa©Ba©b;eyI eyIgxJMúsUmeKarBCUnBrGñksikSaTaMgGs; Gs;mansu mansuxPaBl¥ nig TTYl)aneCaKC½yCanic©kñúgkarsikSa nig muxrbrepSg² rbrepSg² .

)at )at; at;dMdbM g 26 vicäikar 2010 Gñkeroberog lwm plÁún Tel: 017 768 246

- ii -

matikarerOg emeronTI1 emeronTI2 emeronTI3 emeronTI emeronTI4 emeronTI5 emeronTI6 emeronTI7 emeronTI8 emeronTI9 emeronTI10 10

cMNgeCIgemeron

TMB½r

cMnYnKt; Kt; tYEck nig BhuKuNrYm tYEck nig BhuKuNrYm RbPaK cMnYnTsPaK PaKry rgVas; s;rgV rgVal; l; kenSamBICKNit smIkardWeRkTImYymanmYyGBaØat vismPaB

01 08 16 27 38 43 46 49 56 73

- iii -

emeronTI1

cMnYnKt;

lMhat; !>k-rkcMnYnKt; n Edl 43 < n < 67 . x-cMnYnKt;ess a Edl 55 ≤ a < 65 . @> KNnaRbmaNviFIénkenSamxageRkam ³ k> 40 × 5 + 50 × 6 − 7 × 60 x> (57 + 43 − 7 × 5) × 20 + 4 × 90 K> 56 ÷ 8 + (47 − 17) ÷ 5 − 13 X> [4 × 15 + 72 ÷ 8 − (47 − 23) ÷ 6] × 2 g> 75 − 38 ÷ 2 + 75 ÷ 5 × 7 + 81 ÷ 3 ÷ 9 × 7 − 15 + 6 × 7 #> enAGnuviTüal½ymYymansisSRbus ^&( nak; nig sisSRsI %&*nak; kñúgmYyqñaMsisSEdlGanesovePAenAkñúgbNÑal½y)an $ k,al b¤eRcInCag $k,almancMnYn *@$ nak; . etIsisSEdl)anGanesoVePAenAkñúgbNÑal½yticCag $ k,alman cMnYnb:unµannak; ? $> b‘uNÑa)anTijEpøemon % cegáam EdlkñúgmYycegáam man !0 Epø . Kat;)anjaMGs; * Epø ehIyEpøemonEdlenAsl;)anEbgEck[b¥Ún² Kat; ^ nak; . etIb¥ÚnKat;mñak;²TTYl)anemonb:unµanEpø ? -1-

emeronTI1

cMnYnKt;

%> enAkñúgfñak;eronmYymansisSRbus !* nak; nig sisSRsIxøH . ebIsisSmñak;²TTYl)anesoveTAlMhat; @k,al ehIyesovePAEdl )anTijTaMgGs;mancMnYn &^ k,al . rkcMnYnsisSRsI ? ^>bgÁt;cMnYnxageRkamykRtwmxÞg;db; xÞg;ry nig xÞg;Ban; ³ k> !$((0% x> #*$000 K> @((0#$^ X> ^&%$%%0 &>eFVIkar):an;sµantémøén ³ k> 398 + 527 x> 3648 × 999 K> 4201 ÷ 58 *> KNnatémøén k> 2 × 5 x> 5 + 3 K> 8 − 3 X> 3x ebI x = 5 (>sRmYl k> 9 × 16 x> 4 − 4 K> 100 × 196 X> 5 + 364 − 8 2

3

2

2

3

2

2

2

3

3

3

3

-2-

emeronTI1

cMnYnKt;

dMeNaHRsay !>k-rkcMnYnKt; n Edl 43 < n < 67 cMnYnKt; n EdlenAcenøaH 43 nig 67 KW ³ 44 , 45 , 46 , 47 , 48 ,49 50, 51 , 52 , 53, 54, 55 , 56 , 57,58,59,60,61,62,63,64,65,66 . x-cMnYnKt;ess a Edl 55 ≤ a < 65 cMnYnKt;ess a EdlenAcenøaH 55 nig 65 KW 55 , 57 , 59 , 61,63 . @> KNnaRbmaNviFIénkenSamxageRkam ³ k> 40 × 5 + 50 × 6 − 7 × 60 = 10 × (4 × 5 + 5 × 6 − 7 × 6) = 10 × ( 20 + 30 − 42) = 10 × 8 = 80

x> (57 + 43 − 7 × 5) × 20 + 4 × 90 = (100 − 35) × 20 + 360 = 65 × 20 + 360 = 1300 + 360 = 1660

K> 56 ÷ 8 + (47 − 17) ÷ 5 − 13 = 7 + 30 ÷ 5 − 13 = 7 + 6 − 13 = 13 − 13 = 0

-3-

emeronTI1

cMnYnKt;

X> [4 × 15 + 72 ÷ 8 − (47 − 23) ÷ 6] × 2 = [60 + 9 − 24 ÷ 6] × 2 = (69 − 4) × 2 = 65 × 2 = 130

g> 75 − 38 ÷ 2 + 75 ÷ 5 × 7 + 81 ÷ 3 ÷ 9 × 7 − 15 + 6 × 7 = 75 − 19 + 15 × 7 + 27 ÷ 9 × 7 − 15 + 42 = 75 − 19 + 105 + 21 − 15 + 42 = 243 − 34 = 209

#>cMnYnsisSEdl)anGanesoVePAenAkñúgbNÑal½yticCag $ k,alman cMnYn ³ 679 + 578 − 824 = 433 nak;. $> b‘uNÑa)anTijEpøemon % cegáam EdlkñúgmYycegáam man !0 Epø . Kat;)anjaMGs; * Epø ehIyEpøemonEdlenAsl;)anEbgEck[b¥Ún² Kat; ^ nak; . etIb¥ÚnKat;mñak;²TTYl)anemonb:unµanEpø ? -cMnYnEpøemonsrub ³ 5 × 10 = 50 Epø -cMnYnEpøemonenAsl; ³ 50 − 8 = 42 Epø -cMnYnEpøemonEdlb¥Ún²TTYl)an ³ 42 ÷ 6 = 7 Epø .

-4-

emeronTI1

cMnYnKt;

%> enAkñúgfñak;eronmYymansisSRbus !* nak; nig sisSRsIxøH . ebIsisSmñak;²TTYl)anesovePAlMhat; @k,al ehIyesovePAEdl )anTijTaMgGs;mancMnYn &^ k,al . rkcMnYnsisSRsI ? -cMnYnsisSRsI ³ 76 ÷ 2 − 18 = 38 − 18 = 20 nak; ^>bgÁt;cMnYnxageRkamykRtwmxÞg;db; xÞg;ry nig xÞg;Ban; ³ k> !$((0% -ykRtwmxÞg;db;KW: ³ 149905 ≈ 149910 -ykRtwmxÞg;ryKW ³ 149905 ≈ 149900 -ykRtwmxÞg;Ban;KW ³ 149905 ≈ 150000 x> #*$000 -ykRtwmxÞg;db;KW: ³ 384000 ≈ 384000 -ykRtwmxÞg;ryKW ³ 384000 ≈ 384000 -ykRtwmxÞg;Ban;KW ³ 384000 ≈ 384000 K> @((0#$^ -ykRtwmxÞg;db;KW: ³ 2990346 ≈ 2990350 -ykRtwmxÞg;ryKW ³ 2990346 ≈ 2990400 -5-

emeronTI1

cMnYnKt;

-ykRtwmxÞg;Ban;KW ³ 2990346 ≈ 2990000 X> ^&%$%%0 -ykRtwmxÞg;db;KW: ³ 6754550 ≈ 6754550 -ykRtwmxÞg;ryKW ³ 6754550 ≈ 6754600 -ykRtwmxÞg;Ban;KW ³ 6754550 ≈ 6755000 &>eFVIkar):an;sµantémøén ³ k> 398 + 527 ≈ 400 + 500 = 900 x> 3648 × 999 ≈ 4000 × 1000 = 5000 K> 4201 ÷ 58 ≈ 4200 ÷ 60 = 70 *> KNnatémøén k> 2 × 5 = 4 × 25 × 5 = 100 × 5 = 500 x> 5 + 3 = 125 + 9 = 134 K> 8 − 3 = 64 − 9 = 55 X> 3x ebI x = 5 eK)an 3x = 3 × 5 = 3 × 25 = 75 (>sRmYl k> 9 × 16 = 3 × 4 = 12 2

3

3

2

2

2

2

2

2

-6-

emeronTI1

cMnYnKt;

x> 4 − 4 = 16 − 2 = 14 K> 100 × 196 = 10 × 14 = 140 X> 5 + 364 − 8 = 125 +274 − 2 = 127 27 2

3

3

3

3

-7-

emeronTI2

tYEck nig BhuKuN

lMhat; !>cUrbBa¢ak;fa ³ k> plbUk 78 + 120 Eckdac;nwg 2 x> pldk 4140 − 720 Eckdac;nwg 3 K> plbUk 165 + 270 Eckdac;nwg 5 X> pldk 5130 − 342 Eckdac;nwg 9 @> eKmanmYycMnYnmanelxR)aMxÞg; 734
0 bMeBjelxkñúgRbGb;edIm,I[cMnYnenaHEckdac; ³ k> nwg 2 pg nig 5 pg x> nwg 3 pg nig 9 pg . #> eKmancMnYn 660 , 540 , 645 nig 610 . etIcMnYnNaxøHCaBhuKuNén 30 ? $> eKmancMnYn 11 , 15 , 17 , 18 , 19 , 20 , 21 , 22 , 23 nig 29 k>cUrrktYEckéncMnYnnImYy² x>etIcMnYnNaxøHCacMnYnbfm %> bMEbkcMnYn 63 , 100 , 129 , 225 , 567 , 1980 , 2097 , 50336 nig 127008 CaplKuNktþabfm . -8-

emeronTI2

tYEck nig BhuKuN

^>rktYEckrYmFMbMput PGCD éncMnYnxageRkam ³ x> 10 nig 108 k> 9 nig 15 K> 128 nig 324 X> 192 , 160 nig 96 g> 48 , 72 nig 132 c> 36 , 168 , 144 nig 252 &>rkBhuKuNrYmtUcbMput (PPCM b¤ LCM) éncMnYnxageRkam ³ k> 81 nig 225 x> 120 nig 35 K> 34 , 420 nig 245 X> 70 , 21 nig 28 g> 512 , 18 nig 20 c> 88 , 220 nig 528 *> rkcMnYn x nig y edaydwgfa PGCD(x, y ) = 12 nig x + y = 72 Edl 0 < x < y . (>rkcMnYn x EdltUcCageKbMputedaydwgfa PPCM(6 , x) = 24

-9-

emeronTI2

tYEck nig BhuKuN

dMeNaHRsay !>cUrbBa¢ak;fa ³ k> plbUk 78 + 120 Eckdac;nwg 2 eKman 78 + 120 = 198 Eckdac;nwg 2 eRBaHcMnYnenHmanelx cugeRkayCacMnYnKU . x> pldk 4140 − 720 Eckdac;nwg 3 eKman 4140 − 720 = 3420 Eckdac;nwg 3 eRBaHplbUk elxlMdab; 3 + 4 + 2 + 0 = 9 Eckdac;nwg 3 . K> plbUk 165 + 270 Eckdac;nwg 5 eKman 165 + 270 = 435 Eckdac;nwg 5 eRBaHelxcugeRkaybBa©b; edayelx 5 . X> pldk 5130 − 342 Eckdac;nwg 9 eKman 5130 − 342 = 4788 Eckdac;nwg 9 eRBaHplbUkelx lMdab; 4 + 7 + 8 + 8 = 27 Eckdac;nwg 9 .

- 10 -

emeronTI2

tYEck nig BhuKuN

@> eKmanmYycMnYnmanelxR)aMxÞg; 734
0 bMeBjelxkñúgRbGb;edIm,I[cMnYnenaHEckdac; ³ k> nwg 2 pg nig 5 pg edaycMnYnenHmanelxcugeRkayCaelx 0 enaHvaEckdac;nwg 2 ehIynwg 5 Canic© . dUcenHelxEdlRtUvbMeBjkñúgRbGb;enaHKW ³ 0,1,2,3,4,5,6,7,8,9 . x> nwg 3 pg nig 9 pg . edayplbUkelxlMdab; 7 + 3 + 4 + 0 = 14 edIm,I[cMnYn 734
0 Eckdac;nwg 3 pg nig 9 pgluHRtaEt vaEckdac;nwg 9 . dUcenHelxEdlRtUvbMeBjKW 4 . #> eKmancMnYn 660 , 540 , 645 nig 610 . etIcMnYnNaxøHCaBhuKuNén 30 ? cMnYnEdlCaBhuKuNén 30 KW 660 , 540 . $> eKmancMnYn 11 , 15 , 17 , 18 , 19 , 20 , 21 , 22 , 23 nig 29 k>cUrrktYEckéncMnYnnImYy² -cMnYn 11 mantYEck 1 nig 11 -cMnYn 15 mantYEck 1 , 3 , 5 nig 15 - 11 -

emeronTI2

tYEck nig BhuKuN

-cMnYn 17 mantYEck 1 nig 17 -cMnYn 18 mantYEck 1 , 3 , 6 , 9 nig 18 -cMnYn 19 mantYEck 1 nig 19 -cMnYn 20 mantYEck 1 , 2 , 4 , 5 , 10 nig 20 -cMnYn 21 mantYEck 1 , 3 , 7 nig 21 -cMnYn 22 mantYEck 1 , 2 , 11 nig 22 -cMnYn 29 mantYEck 1 nig 29 x>etIcMnYnNaxøHCacMnYnbfm %> bMEbkcMnYn 63 , 100 , 129 , 225 , 567 , 1980 , 2097 , 50336

CaplKuNktþabfm . 63 = 1 × 32 × 7 100 = 1 × 22 × 52 129 = 1 × 129 225 = 1 × 32 × 52 567 = 1 × 34 × 7 1980 = 1 × 22 × 32 × 5 × 11 2097 = 1 × 32 × 239

- 12 -

nig 127008

emeronTI2

tYEck nig BhuKuN

50336 = 1 × 25 × 11 × 143 127008 = 1 × 25 × 34 × 7 2

^>rktYEckrYmFMbMput PGCD éncMnYnxageRkam ³ k> 9 nig 15 PGCD(9 , 15 ) = 3

x> 10 nig 108 PGCD(10 , 108) = 2

K> 128 nig 324 PGCD(128 , 324 ) = 4

X> 192 , 160 nig 96 PGCD(192 , 160 , 96 ) = 32

g> 48 , 72 nig 132 PGCD(48 , 72 , 132 ) = 12

c> 36 , 168 , 144 nig 252 PGCD( 36 , 168 , 144 , 252 ) = 12

- 13 -

emeronTI2

tYEck nig BhuKuN

&>rkBhuKuNrYmtUcbMput (PPCM b¤ LCM) éncMnYnxageRkam ³ k> 81 nig 225 PPCM (81 , 225) = 2025

x> 120 nig 35 PPCM (120 , 35) = 840

K> 34 , 420 nig 245 PPCM ( 34 , 420 , 245) = 2940

X> 70 , 21 nig 28 PPCM (70 , 21 , 28 ) = 420

g> 512 , 18 nig 20 PPCM (512 , 18 , 20 ) = 22940

c> 88 , 220 nig 528 PPCM (88 , 220 , 528) = 2640

*> rkcMnYn x nig y edaydwgfa PGCD(x, y ) = 12 nig x + y = 72 Edl 0 < x < y . eday PGCD(x, y ) = 12 enaH x = 12p , y = 12q Edl PGCD(p, q) = 1 . - 14 -

emeronTI2

tYEck nig BhuKuN

eKman x + y = 72 eK)an 12p + 12q = 72 b¤ p + q = 6 eday 0 < x < y enaH 0 < p < q eKTaj p = 1 , q = 5 dUcenH x = 12 , y = 60 . (>rkcMnYn x EdltUcCageKbMputedaydwgfa PPCM(6 , x) = 24 eday 24 = 6 × 4 . dUcenH x = 24 CatémøtUcCageKEdl PPCM(6 , x) = 24 .

- 15 -

emeronTI3

cMnYnKt;rWLaTIhV

lMhat; !>sresrcMnYnKt;rWuLaTIb − 7, 5 , − 10 , 0 , − 22 , 3 , 8 , − 2 , 9 , 1 , 23 34 , − 13 . k>tamlMdab;eLIg x>tamlMdab;cuH @> edAcMnuc A(−4) , B(−6) , C(3) nig D(5) elIbnÞat;cMnYn rYcKNna RbEvg AB , AC nig BC . #> bMeBjsmPaBxageRkam ³ k> | −4 |= ... x> | +2 |= ... K> a = −7 nig b CacMnYnpÞúyén a enaH | b |= ... $> KNnaplbUkxageRkamedayeRbIbnÞat;cMnYn ³ k> 4 + 6 x> − 5 + (−2) K> + 7 + (−3) X> 0 + 5 g> (−4) + 3 %> bMeBjsmPaBxageRkam ³ k> 9 + ... = 0 x> 5 + ... = 2 K> − 8 + ... = −12 X> 15 + ... = −18 g> − 6 + 5 + ... = −3 c> − 7 + (−3) + ... = −12 - 16 -

emeronTI3

cMnYnKt;rWLaTIhV

^>KNnapldk ENnaM ³ 10 − 6 = 10 + (−6) b¤ 15 − (−9) = 15 + (9) k> 7 − 4 x> 17 − 22 K> − 10 − 3 X> 20 − (−5) c> − 13 − 5 g> − 12 − (−10) q> 23 − (−3) &>KNnakenSamxageRkam ³ k> − 22 + 10 − (−7) x> 4 − 12 + (−6) K> 7 − (−6) + 4 X> − 8 − (−2) + 1 g> − 4 − 6 + (−5) c> 5 − 11 − (−8) *> KNnarYceRbobeFob k> 10 − 3 nig 3 − 10 x> (−9) − (−5) nig (−5) − (−9) etIviFIdkmanlkçN³RtLb;b¤eT ? K> [4 − (−7)] − 2 nig 4 − [(−7) − 2] X> (12 − 7) − 3 nig 12 − (7 − 3) etIviFIdkmanlkçN³pþMúb¤eT ? - 17 -

emeronTI3

cMnYnKt;rWLaTIhV

(>KNnaplKuNxageRkam ³ k> (−6)(−4) K> (−−9)(7) g> (−500)(−230) q> − 31×× (59) !0> bMeBjcenøaHkñúgsmPaBxageRkam ³ k> − 5 = (−1)(...) K> − 24 = (−1)(...) g> (...)(13) = −78 q> (−7)(...) = 0 !!> KNnaplEckxageRkam ³ k> (−36) ÷ 9 K> (−45) ÷ 9 g> −1144 q> (−7 9− 20)

- 18 -

x> (−12)(−3) X> (10)(−9) c> (−−15)(400) x> (−1)(...) = −2 X> (0)(...) = −4 c> (−3)(...) = −12 C> (...)(−5) = 30 x> 50 ÷ (−5) X> −−405 c> [−1 +3(−8)] C> [−5−+10(−5)]

emeronTI3

cMnYnKt;rWLaTIhV

!@> KNna x> 20 − 12 + 8 k> 6 + 9 + (−2) K> 16 − 9 − 5 X> 3 × 12 × 4 g> 7 × (−5) × 2 c> 18 ÷ 3 × 2 C> 50 ÷ [5 × (−2)] q> 11 × [5 × (−2)] Q> 280 ÷ 20 ÷ 7 !#> KNnakenSamelxxageRkam ³ k> 28 ÷ [7 × (−3 + 5)] x> [40 + 3(1 − 2)] × 6 K> [(21 + 25) ÷ 7] × (21 + 28 ÷ 7) X> [5700 − 43(88 + 12)] ÷ 2 ÷ (8 − 2) g> 300 ÷ {[150 + (40 ÷ 8)] × [−7 − (9)]} !$> bUNamanXøI % RKab; bgKat;[XøIEfm # RKab;eTot . bUNa)an elgXøICamYysMcaj;Gs;XøI !0RKab; . etIbUNaRtUvCMBak;XøIsMcMnYnb:unµanRKab; ?

- 19 -

emeronTI3

cMnYnKt;rWLaTIhV

dMeNaHRsay !>sresrcMnYnKt;rWuLaTIb − 7, 5 , − 10 , 0 , − 22 , 3 , 8 , − 2 , 9 , 1 , 23 34 , − 13 . k>tamlMdab;eLIg − 22,−13,−10,−7,−2,0,1,3,5,8,9,23,34 . x>tamlMdab;cuH 34,23,9,8,5,3,1,0,−2,−7,−1,−13,−22 . @> edAcMnuc A(−4) , B(−6) , C(3) nig D(5) elIbnÞat;cMnYn rYcKNna RbEvg AB , AC nig BC . eK)an AB = −4 − (−6) = −4 + 6 = 2 AC = 3 − ( −4) = 3 + 4 = 7

BC = 3 − ( −6) = 3 + 6 = 9

- 20 -

emeronTI3

cMnYnKt;rWLaTIhV

#> bMeBjsmPaBxageRkam ³ k> | −4 |= 4 x> | +2 |= 2 K> a = −7 nig b CacMnYnpÞúyén a enaH | b |=| 7 |= 7 . %> bMeBjsmPaBxageRkam ³ k> 9 + (−9) = 0 x> 5 + (−3) = 2 K> − 8 + (−4) = −12 X> 15 + (−33) = −18 g> − 6 + 5 + (−2) = −3 c> − 7 + (−3) + (−2) = −12 ^>KNnapldk ENnaM ³ 10 − 6 = 10 + (−6) b¤ 15 − (−9) = 15 + (9) k> 7 − 4 = 3 x> 17 − 22 = −5 K> − 10 − 3 = −13 X> 20 − (−5) = 25 - 21 -

emeronTI3

cMnYnKt;rWLaTIhV

g> − 12 − (−10) = −2 c> − 13 − 5 = −18 q> 23 − (−3) = 26 &>KNnakenSamxageRkam ³ k> − 22 + 10 − (−7) = −22 + 10 + 7 = −22 + 17 = −5

x> 4 − 12 + (−6)

4 − 12 − 6 = 4 − 18 = −14

K> 7 − (−6) + 4

= 7+ 6+ 4 = 7 + 10 = 17

X> − 8 − (−2) + 1

= −8 + 2 + 1 = −8 + 3 = −5

g> − 4 − 6 + (−5)

= −4 − 6 − 5 = −15

c> 5 − 11 − (−8)

= 5 − 11 + 8 = 13 − 11 = 2

- 22 -

emeronTI3

cMnYnKt;rWLaTIhV

*> KNnarYceRbobeFob k> 10 − 3 = 7 nig 3 − 10 = −7 dUcenH 10 − 3 = −(3 − 10) x> (−9) − (−5) = −9 + 5 = −4 nig (−5) − (−9) = −5 + 9 = 4 dUcenH (−9) − (−5) = −[(−5) − (−9)] viFIdkKµanlkçN³RtLb;eT. K> [4 − (−7)] − 2 = 4 + 7 − 2 = 11 − 2 = 9 nig 4 − [(−7) − 2] = 4 − (−7 − 2) = 4 + 7 + 2 = 13 dUcenH [4 − (−7)] − 2 ≠ 4 − [(−7) − 2] X> (12 − 7) − 3 = 12 − 7 − 3 = 12 − 10 = 2 nig 12 − (7 − 3) = 12 − 4 = 8 dUcenH (12 − 7) − 3 ≠ 12 − (7 − 3) viFIdkKµanlkçN³pþMúeT . (>KNnaplKuNxageRkam ³ k> (−6)(−4) = +24 x> (−12)(−3) = +36 - 23 -

emeronTI3

cMnYnKt;rWLaTIhV

K> (−9)(7) = −63 X> (10)(−9) = −90 g> (−500)(−230) = +115000 c> (−15)(400) = −6000 q> − 31 × (59) = −1829 !0> bMeBjcenøaHkñúgsmPaBxageRkam ³ k> − 5 = (−1)(5) x> (−1)(2) = −2 K> − 24 = (−1)(24) X> (0)(...) = −4 ¬ minGacman ¦ g> (−6)(13) = −78 c> (−3)(4) = −12 q> (−7)(0) = 0 C> (−6)(−5) = 30 !!> KNnaplEckxageRkam ³ k> (−36) ÷ 9 x> 50 ÷ (−5) - 24 -

emeronTI3

cMnYnKt;rWLaTIhV

K> (−45) ÷ 9 = −5 X> −−405 = +8 g> −1144 = −4 c> [−1 +3(−8)] = −3 q> (−7 9− 20) = −3 C> [−5−+10(−5)] = 1 !@> KNna k> 6 + 9 + (−2) = 15 − 2 = 13 x> 20 − 12 + 8 = 28 − 12 = 16 K> 16 − 9 − 5 = 16 − 14 = 2 X> 3 × 12 × 4 = 3 × 48 = 144 g> 7 × (−5) × 2 = 7 × (−10) = −70 c> 18 ÷ 3 × 2 = 6 × 2 = 12 q> 11 × [5 × (−2)] = 11 × (−10) = −110 C> 50 ÷ [5 × (−2)] = 50 ÷ (−10) = −5 Q> 280 ÷ 20 ÷ 7 = 14 ÷ 7 = 2 - 25 -

emeronTI3

cMnYnKt;rWLaTIhV

!#> KNnakenSamelxxageRkam ³ k> 28 ÷ [7 × (−3 + 5)] = 28 ÷ 14 = 2

x> [40 + 3(1 − 2)] × 6 = (40 − 3) × 6 = 240 − 18 = 222

K> [(21 + 25) ÷ 7] × (21 + 28 ÷ 7) (46 ÷ 7 ) × ( 21 + 7 ) = 184

X> [5700 − 43(88 + 12)] ÷ 2 ÷ (8 − 2)

= (5700 − 4300) ÷ 2 ÷ 6 = 1400 ÷ 2 ÷ 6 = 700 ÷ 6 = 350 ÷ 3

g> 300 ÷ {[150 + (40 ÷ 8)] × [−7 − (9)]} = 300 ÷ [155 × ( −16)]

= 300 ÷ ( −2480) = −

15 124

!$> bUNamanXøI % RKab; bgKat;[XøIEfm # RKab;eTot . bUNa)an elgXøICamYysMcaj;Gs;XøI !0RKab; . etIbUNaRtUvCMBak;XøIsMcMnYnb:unµanRKab; ? cMnYnXøIEdlbUNaCMBak;sM KW 2 . eRBaH 5 + 3 − 10 = −2 - 26 -

emeronTI4

RbPaK

lMhat; !>bMeBjcMnYnkñúgRbGb;xageRkam ³ k> 53 = (...) 20 − 24 21 K> 43 = (...) = = − 8 (...) (...) @> sRmYlRbPaKxageRkam ³ k> 69 K> −19684 2232 g> 4464 #> KNna x nig y xageRkam ³ k> 7x = 216

− 28 x> (...) = 8 32

x> 18 33 625 X> −−1000 x> −y5 = 20 28

K> x. 73 = 23 X> x ÷ 118 = 113 $> eRbobeFob nig erobryRbPaKxageRkamtamlMdab;cuH k> 117 , 56 , 49 , 23 x> 12 , 23 , 43 , 78 , 127

- 27 -

emeronTI4

RbPaK

%> bMeBjsBaØa ( < , > ) kñúgRbGb;xageRkam ³ k> −98
−97 x> −31
−32 X> −113
110 K> 73
−76 ^>KNnarYcsRmYl 7 7 4 k> 14 + x> 6. − 3. 20 20 8 8 7 −8 + − 25 25 8 11 7 + + 15 15 15 4  4 + −  5  18 

K> g> q> &>KNnarYcsRmYllTæpl k> 20 × 45 13  4 K> − 59  13 −−  28 28  9 

X> c> C>

q>

3

x> 117 × −413 × 117 X> (−8. 13 ) × (2. 52 )



c> 12. 14 ÷  − 143 

g> 289 ÷ 76 3  − 2 ÷−  4  3 

2

 1  2 −  +   4  3 3 4 5. − 3. 10 − 18 19 1 61 1 2. − + − 2 18 72 36



2

- 28 -



emeronTI4

RbPaK

*> KNnarYcsRmYl k>  15 − 12  ÷  − 23 × 18  K> X> g>

11  22 7  x> 12 ×− +   33 11 

    3  1 1  1 1 ×  − −  ÷  − 2 . + 1.  5  4 6  3 4 1 1 3  1 3 2 ( −1. × 2. × 1 ) ÷ 1 ( −2 ) × 1  6 2 5  4 10 3  1 7   7 2 1 +  ÷  +   5 22   15 5 

c>

3 2 5 5 + 2 ×1 4 3 16 2 4 + 5 15

q>

1 7 1 1    − 1 − 1−    − 18 2 6 3 2 7  ÷ ×  × × 1 4 1 10   2 1 +   1 +       2 3 7



(>BinitülMnaMKMrU

1 1 1 1 1 1 1 1 1 = − , = − , = − 1× 2 1 2 2 × 3 2 3 3 × 4 3 4

tamlMnaMKMrUenHcUrKNna ³

1 1 1 1 1 + + + + ..... + 1× 2 2 × 3 3 × 4 4 × 5 1995 × 1996

- 29 -



emeronTI4

RbPaK

!0> BinitülMnaMKMrU ³ 1 1 1 2 1 1 1 4 = , = , = , = 1 1 2 1+ 3 3 1+ 3+ 5 4 1+ 3+ 5 1 1 1 1 , , 5 6 7 8

sresrRbPaK

nig tamlMnaMKMrUxagelI ?

- 30 -

emeronTI4

RbPaK

dMeNaHRsay !>bMeBjcMnYnkñúgRbGb;xageRkam ³ k> 53 = 12 20 x> −87 = −3228 21 K> 43 = −− 68 = −− 24 = 32 28 @> sRmYlRbPaKxageRkam ³ k> 69 = 23 6 x> 18 = 33 11 K> −19684 = −73 625 5 X> −−1000 = 8 2232 1 g> 4464 = 2 #> KNna x nig y xageRkam ³ k> 7x = 216 eK)an x = 721× 6 = 2 . - 31 -

emeronTI4

RbPaK

x> −y5 = 20 28 eK)an y = − 520× 28 = −7 . K> x. 73 = 23 eK)an x = 23 ×× 73 = 149 X> x ÷ 118 = 113 eK)an x = 113 × 118 = 83 . $> eRbobeFob nig erobryRbPaKxageRkamtamlMdab;cuH k> 117 , 56 , 49 , 23 eK)an 49 < 117 < 23 < 56 nig erobryRbPaKxageRkamtamlMdab;cuH 56 , 23 , 117 , 49 . x> 12 , 23 , 43 , 78 , 127 eK)an 12 < 127 < 23 < 43 < 78 nig erobryRbPaKxageRkamtamlMdab;cuH 78 , 43 , 23 , 127 , 12 . - 32 -

emeronTI4

RbPaK

%> bMeBjsBaØa ( < , > ) kñúgRbGb;xageRkam ³ k> −98 < −97 x> −31 > −32 X> −113 < 110 K> 73 > −76 ^>KNnarYcsRmYl 7 14 + 7 21 k> 14 + = = 20 20 20 20 x> 6 78 − 3 48 = (48 + 7) −8 (24 + 4) = 55 −8 28 = 278 K> − 725 + −258 = − 725− 8 = −2515 = − 53 X> g> c> q> C>

1 8 − 27 + 128 101  1  2 −  +  = − + = = 4 3 16 27 432 432     8 11 7 8 + 11 + 7 26 + + = = 15 15 15 15 15 3 4 3 4 53 58 53 × 9 + 58 × 5 5 −3 =5 +3 = + = 10 − 18 10 18 10 18 90 767 = 90 4  4  4 2 36 − 10 26 + −  = − = = 5  18  5 9 45 45 19 1 61 1 23 1 61 1 2 − + − = − + − 2 18 72 36 2 18 72 36 828 − 4 + 61 − 2 883 = = 72 72 2

3

- 33 -

emeronTI4

RbPaK

&>KNnarYcsRmYllTæpl k> 20 × 45 = 805 = 16 ) 3 =− x> 117 × −413 × 117 = 7 ×11(−×341) ××(11 7 41 13  4 ( −5) × (13 − 13) × (4) K> − 59  13 −  = =0 28 28 9 9 × 28 × 9   X> (−8 13 ) × (2 52 ) = − 325 × 125 = −20 g> 289 ÷ 76 = 289 × 76 = 83 c> 12 14 ÷  − 143  = 494 × (− 143 ) = − 218   2 3  − 2 3 9 27 ÷−  = × = 4  3  4 4 16

q> *> KNnarYcsRmYl k>  15 − 12  ÷  − 23 × 18  

x>

   18  3 =  −  × (− 12 ) = 5  10  11  22 7  ×− +  12  33 11  11 − 22 + 21 1 = × =− 12 11 12

- 34 -

emeronTI4

RbPaK

K> 53 ×  − 14 − 16  ÷  − 2 13 + 1 14  

   3  − 5  − 7 5 = × +  ÷ 5  12   3 4  3  1  12 = − × =  4  − 13 13 1 1 3  1 3 2 ( −1 × 2 × 1 ) ÷ 1 ( −2 ) × 1  10 3 6 2 5  4  − 7 5 8   5  − 23  5  = × ×  ÷  × ×   6 2 5   4  10  3   14   − 115  14 × 24 112 = − ÷ = =  3   24  3 × 115 115  1 7   7 2 1 +  ÷  +   5 22   15 5   6 7   7 + 6 =  + ÷  5 22 15     132 + 35 15 = × 5 × 22 13 501 = 286

X>

g>

c>

3 2 5 23 8 21 23 7 37 5 + 2 ×1 + × + 4 3 16 = 4 3 16 = 4 2 = 4 = 111 2 4 6+4 10 2 8 + 5 15 15 15 3

- 35 -

emeronTI4

q>

RbPaK

1 7 1 1    − 1 1 − −   − 18   2 6 3 2 7  ÷ ×  × × 1 4 1 10 2  1 +   1 +     2 3 7 1 5  6   − 1 5 − 9 8 1 9 6  ÷ 7 =  × × =2× × × = −  3 4 8 3 8 5 2 5   6   2 3  7

(>BinitülMnaMKMrU

1 1 1 1 1 1 1 1 1 = − , = − , = − 1× 2 1 2 2 × 3 2 3 3 × 4 3 4

tamlMnaMKMrUenHcUrKNna ³

1 1 1 1 1 + + + + ..... + 1× 2 2 × 3 3 × 4 4 × 5 1995 × 1996 1 1 1 1 1 1 1 1 1 1 = ( − ) + ( − ) + ( − ) + ( − ) + ... + ( − ) 1 2 2 3 3 4 4 5 1995 1996 1 1995 = 1− = 1996 1996

!0> BinitülMnaMKMrU ³

1 1 1 2 1 1 1 4 = , = , = , = 1 1 2 1+ 3 3 1+ 3+ 5 4 1+ 3+ 5+ 7 1 1 1 1 , , 5 6 7 8

sresrRbPaK

nig tamlMnaMKMrUxagelI

- 36 -

emeronTI4

RbPaK

eK)an 15 = 1 + 3 + 55 + 7 + 9 1 6 = 6 1 + 3 + 5 + 7 + 9 + 11 1 7 = 7 1 + 3 + 5 + 7 + 9 + 11 + 13 1 8 = 8 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15

- 37 -

emeronTI5

cMnYnTsPaK

lMhat; !>sresrRbPaKnImYy²xageRkamCacMnYnTsPaK ³ 13 k> 107 x> 100 59 2451 K> 1000 X> − 10000 g> − 13 52 @> sresrcMnYnTsPaKnImYy²xageRkamCaRbPaKcRmuH CaTRmg;Edl bRgYm)an ³ x> − 0.065 k> 0.8 K> 8.375 X> − 0.248 g> 8.7 #> KNnatémøxageRkam ³ k> 1.48 + 16.943 x> 24.765 + 185 + 3.89 K> 0.096 + 17.67 + 103.598 X> 31.64 − 17.186 g> 14.3 − 8.56 c> 5.28 − (− 0.7) − (− 9.16)

- 38 -

emeronTI5

cMnYnTsPaK

$> rktémøxageRkam ³ x> 2.4 × 8 × 0.059 k> 4.27 × 13 K> 14.2 × (−0.8) × (−1.34) X> − 52 × [5.9 − (−3)] g> 9.2 × [(−4.1) − 0.7] − 2.62 c> − 0.32 − [− 199 + (−2001)]× (−0.005) q> (4 + 0.117) ×  − 14  (0.06 × 3.2)   %> KNnatémøxageRkam ³ k> 0.8 ÷ 4 x> 3.41 ÷ 6 K> (−1.27) ÷ (−0.5) X> 14.6 ÷ 138 g> 0.8 ÷ [−1.54 − (−1.38)] × 0.002 ^>BIraCFanIPñMeBjeTAextþeBaFisat;mancm¶ay 189.65km . extþkMBugqñaMgsßitenAcenøaHPñMeBj nig extþeBaFisat; . ebIkMBugqñaMgmancm¶ay 91.45km BIPñMeBj . etIBIextþkMBg;qñaMgeTAextþeBaFisat;mancm¶ayb:unµan km ?

- 39 -

emeronTI5

cMnYnTsPaK

dMeNaHRsay !>sresrRbPaKnImYy²xageRkamCacMnYnTsPaK ³ k> 107 = 0.7 13 x> 100 = 0.13 59 K> 1000 = 0.059 2451 = −0.2451 X> − 10000 1 g> − 13 = − = −0.25 52 4 @> sresrcMnYnTsPaKnImYy²xageRkamCaRbPaKcRmuH CaTRmg;Edl bRgYm)an ³ k> 0.8 = 108 = 45 65 13 x> − 0.065 = − 1000 =− 200 8375 67 K> 8.375 = 1000 = 8 248 31 X> − 0.248 = − 1000 =− 125 87 g> 8.7 = 10 - 40 -

emeronTI5

cMnYnTsPaK

#> KNnatémøxageRkam ³ k> 1.48 + 16.943 = 18.353 x> 24.765 + 185 + 3.89 = 213.655 K> 0.096 + 17.67 + 103.598 = 121.364 X> 31.64 − 17.186 = 14.454 g> 14.3 − 8.56 = 5.74 c> 5.28 − (− 0.7) − (− 9.16) = 5.28 + 0.7 + 9.16 = 15.14

$> rktémøxageRkam ³ k> 4.27 × 13 = 55.51 x> 2.4 × 8 × 0.059 = 1.1328 K> 14.2 × (−0.8) × (−1.34) = 15.2224 X> − 52 × [5.9 − (−3)] = −3.56 g> 9.2 × [(−4.1) − 0.7] − 2.62 = −46.78 c> − 0.32 − [− 199 + (−2001)]× (−0.005) = −11.32 q> (4 + 0.117) ×  − 14  (0.06 × 3.2) = −0.197616 



- 41 -

emeronTI5

cMnYnTsPaK

%> KNnatémøxageRkam ³ k> 0.8 ÷ 4 = 0.2 x> 3.41 ÷ 6 = 0.5683 K> (−1.27) ÷ (−0.5) = 2.54 X> 14.6 ÷ 138 = 14.68× 13 = 23.725 g> 0.8 ÷ [−1.54 − (−1.38)] × 0.002 = 0.8 ÷ [( −0.16)(0.002)] = 0.8 ÷ ( −0.00032) = −2500

^>BIraCFanIPñMeBjeTAextþeBaFisat;mancm¶ay 189.65km . extþkMBugqñaMgsßitenAcenøaHPñMeBj nig extþeBaFisat; . ebIkMBg;qñaMgmancm¶ay 91.45km BIPñMeBj . etIBIextþkMBg;qñaMgeTAextþeBaFisat;mancm¶ayb:unµan km ? eK)an ³ 189.65km − 91.45km = 98.20km dUcenHBIextþkMBg;qñaMgeTAextþeBaFisat;mancm¶ay 98.20km .

- 42 -

emeronTI6

PaKry

lMhat; !>sresrPaKrynImYy²CaRbPaK nig CacMnYnTsPaK ³ k> 48% x> 28% K> 37.5% X> 66% c> 110% g> 99% @> bþÚrRbPaK nig cMnYnTsPaKnImYy²CaPaKry ³ k> 76 x> 17 K> 0.78 20 X> 0.095 g> 1.35 c> 1 256 #> Rkumh‘unmYy)ankat;bnßybuKÁlikcMnYn 24 nak;ecjBIbuKÁlik 400 nak; . etIbuKÁlikEdl)ankat;bnßymanb:unµanPaKry ? $> enAmNÐle)aHeqñatmYykEnøgmanGñkcuHeQµaHe)aHeqñat 8500 nak; ehIyenAéf¶e)aHeqñatman 5% énGñkcuHeQµaHmin)anmke)aHeqñat . KNnacMnYnmnusSEdl)anmke)aHeqñat ? %> BUsM)anepJIR)ak; 20 000 000 erolenAFnaKarmYyedayTTYl)an GRtakarR)ak; 7% . etIryHeBlb:unµaneTIbKat;TTYl)anR)ak; TaMgedImTaMgkar)an 21 925 000 erol ? - 43 -

emeronTI6

PaKry

dMeNaHRsay !>sresrPaKrynImYy²CaRbPaK nig CacMnYnTsPaK ³ 48 12 k> 48% = 100 = = 0.48 25 28 7 x> 28% = 100 = = 0.28 25 375 3 K> 37.5% = 1000 = = 0.375 8 66 33 = = 0.66 X> 66% = 100 50 99 g> 99% = 100 = 0.99 11 c> 110% = 110 = = 1.10 100 10 @> bþÚrRbPaK nig cMnYnTsPaKnImYy²CaPaKry ³ k> 76 = 85.71% x> 17 = 85% 20 K> 0.78 = 78% X> 0.095 = 0.95% g> 1.35 = 1.35% 31 c> 1 256 = 25 = 124% - 44 -

emeronTI6

PaKry

#> Rkumh‘unmYy)ankat;bnßybuKÁlikcMnYn 24 nak;ecjBIbuKÁlik 400 nak; . etIbuKÁlikEdl)ankat;bnßymanb:unµanPaKry ? 24 buKÁlikEdl)ankat;bnßyman ³ 400 = 0.06 = 6%

$> enAmNÐle)aHeqñatmYykEnøgmanGñkcuHeQµaHe)aHeqñat 8500 nak; ehIyenAéf¶e)aHeqñatman 5% énGñkcuHeQµaHmin)anmke)aHeqñat . KNnacMnYnmnusSEdl)anmke)aHeqñat ?

KNnacMnYnmnusSEdl)anmke)aHeqñat ³

8500 − 8500 × 5% = 8500 × 95% = 8075

nak; .

%> BUsM)anepJIR)ak; 20 000 000 erolenAFnaKarmYyedayTTYl)an GRtakarR)ak; 7% . etIryHeBlb:unµaneTIbKat;TTYl)anR)ak; TaMgedImTaMgkar)an 21 925 000 erol ? 21925000 ryHeBlEdlKat;epJI ³ 20 000 = 15.66 . 000 × 7%

- 45 -

emeronTI7

rgVas;rgVal;

lMhat; !>BUpleFVIm¢úledrmYyedImmanRbEvg 35cm . k> ebIKat;manEdkRbEvg 4 620mm etIKat;eFIVm¢úl)anb:unµanedIm ? x> ebI 20% énEdkRtUvxateBleFVI etIKat;eFIVm¢úl)anb:unµanedIm ? @> lkçikavas;kMNat;)anmYypab;edayeRbIEm:RteQImanRbEvg 98cm rYcnagvas;pab;dEdledayeRbIEm:RtsMBt;EdlmanRbEvg 99cm kñúgkarvas;elIkeRkayenHnageXIjfapab;kMNat;xøICagmun 1cm . etIkMNat;BitR)akdmanRbEvgb:unµanEm:Rt ? #> mnusSeBjv½ymñak;manQam 5.5 lIt . ebIeKdwgfaQam 1mm maneKalikaRkhm % lan . rkcMnYneKalikarRkhmTaMgGs; ? $> rfynþmYysIusaMgGs; 9 lIt kñúg 100km . ebIFugsaMgrfynþmanragRbelBIEb:tEkgEdlmanvimaRt ³ 85cm , 45cm , 12cm edayvas;BIxageRkA . ebIcMNuH10% ticCagmaDEdl)anKNnatamrUbmnþ . k> etIFugsaMgrfynþenaHmancMNuHBitb:unµanlIt ? x>etIrfynþenaHGaceFVIdMeNI)anb:unµanKILÚEm:Rt ? 3

- 46 -

emeronTI7

rgVas;rgVal;

dMeNaHRsay !>BUpleFVIm¢úledrmYyedImmanRbEvg 35cm . k> ebIKat;manEdkRbEvg 4 620mm etIKat;eFIVm¢úl)anb:unµanedIm ? x> ebI 20% énEdkRtUvxateBleFVI etIKat;eFIVm¢úl)anb:unµanedIm ? cemøIy k> cMnYnm¢úlEdlKat;eFIV)an eday 4620mm = 462cm nigm¢úledrmYyedImmanRbEvg 35cm dUcenH 462 ≈ 13 edIm 35 x> ebI 20% énEdkRtUvxateBleFVI enaHKat;eFIVm¢úl)anKW ³ 462 − 462 × 20% 462 × 80% = ≈ 10 edIm 35 35 #> mnusSeBjv½ymñak;manQam 5.5 lIt . ebIeKdwgfaQam 1mm maneKalikaRkhm % lan . rkcMnYneKalikarRkhmTaMgGs; ? edayQam 5.5 lIt = 5.5dm = 5500 000mm dUcenHcMnYneKalikarQamTaMgGs;man ³ 5500 000 × 5 lan = 27 500 000 lan .

3

3

- 47 -

3

emeronTI7

rgVas;rgVal;

$> rfynþmYysIusaMgGs; 9 lIt kñúg 100km . ebIFugsaMgrfynþmanragRbelBIEb:tEkgEdlmanvimaRt ³ 85cm , 45cm , 12cm edayvas;BIxageRkA . ebIcMNuH10% ticCagmaDEdl)anKNnatamrUbmnþ . k> etIFugsaMgrfynþenaHmancMNuHBitb:unµanlIt ? x>etIrfynþenaHGaceFVIdMeNI)anb:unµanKILÚEm:Rt ? cemøIy k> cMNuHBitKitCalIténFugsaMgrfynþ ³ -maDrbs;FugsaMg ³ 85 × 45 × 12 = 45900cm = 45.9 lIt ebIcMNuH10% ticCagmaDEdl)anKNnatamrUbmnþdUcenHcMNuHBit KitCalIténFugsaMgKW 45.9 × 90% = 41.31 lIt. x>cm¶ayEdlrfynþenaHGaceFVIdMeNIr)an ³ 41.31 × 100km = 459km . 9 3

- 48 -

emeronTI8

kenSamBICKNit

lMhat; !>sresrCakenSamBICKNittaml,HxageRkam ³ k> bYndk x x> y bUk 10 K> kaerénplbUk x nig y X> 2b bUknwgplKuN m nig n g>plKuNrvag x bUk y nig x dk y . @> k> plbUkénBIrcMnYnesµInwg 80 . ebIcMnYnTImYyesµInwg x . cUrsresrkenSamBICKNitsRmab;cMnYnTIBIr ? x> pldkénBIrcMnYnesµInwg 10 ebIcMnYnEdltUcCageKtageday y sresrCakenSamBICKNitsRmab;cMnYnmYyeTot ? #> KNnatémøkenSamBICKNit x(3y + 2) cMeBaH ³ x> x = −3 nig y = −1 k> x = 7 nig y = 2 K> x = 5 nig y = −4 $> rfynþmYymanel,ÓnmFüm 60km / h . sresrrUbmnþcm¶aycr d kñúgryHeBl t rYcKNnacm¶aycr kalNaeKebIkbrkñúgryHeBl t = 2h , t = 2h20mn , t = 30mn nig t = 1h5mn . - 49 -

emeronTI8

kenSamBICKNit

%> ctuekaNEkgmYymanbeNþayesµInwg 15m nig TTwg x m . k> sresrrUbmnþbrimaRt P énctuekaNEkg ? x> KNnabrimaRtctuekaNEkgcMeBaH x = 5m , x = 8m , x = 10m ^> KNna x> 2x − 5x k> 3x + 7x K> 3x × 6x X> 4x × (−8x) &> BnøatkenSam ³ k> (x + 1)(x − 1) x> (x + 3)(x − 2) K> (2x − 3)(5 − 3x) X> (x + 3)(x + 3) g> (2x − 3)(y + 2) c> 4(x − 3) + (x − 5)(x + 1) q> − 2(2 − x) − (x + 2)(3x + 1) *> dak;CaplKuNktþa ³ k> ax + bx x> ax − ab K> ab − 3a X> 5x + 10x g> 3a + 3b + 3c c> 8x + 12y − 4z q> 3(x − 4) + x(x − 4) C> (x + 5)(x + 2) − (3x + 1)(x − 5) Q> xy + 3x − 4y − 12 j> 3ab + 7b − 3ac − 7c 2

- 50 -

emeronTI8

kenSamBICKNit

dMeNaHRsay !>sresrCakenSamBICKNittaml,HxageRkam ³ k> bYndk x sresr 4 − x x> y bUk 10 sresr y + 10 K> kaerénplbUk x nig y sresr (x + y ) X> 2b bUknwgplKuN m nig n sresr 2b + mn g>plKuNrvag x bUk y nig x dk y sresr (x + y )(x − y ) @> k> plbUkénBIrcMnYnesµInwg 80 . ebIcMnYnTImYyesµInwg x . cUrsresrkenSamBICKNitsRmab;cMnYnTIBIr ? x> pldkénBIrcMnYnesµInwg 10 ebIcMnYnEdltUcCageKtageday y sresrCakenSamBICKNitsRmab;cMnYnmYyeTot ? cemøIy k> sresrkenSamBICKNitsRmab;cMnYnTIBIr tag y CacMnYnTIBIr eK)an x + y = 80 dUcenH y = 80 − x . x> sresrCakenSamBICKNitsRmab;cMnYnmYyeTotKW 10 + y . 2

- 51 -

emeronTI8

kenSamBICKNit

#> KNnatémøkenSamBICKNit x(3y + 2) cMeBaH ³ k> x = 7 nig y = 2 eK)an x(3y + 2) = 7(6 + 2) = 7 × 8 = 56 x> x = −3 nig y = −1 eK)an x(3y + 2) = (−3)(−3 + 2) = 3 K> x = 5 nig y = −4 eK)an x(3y + 2) = (5)(−12 + 2) = −50 $> rfynþmYymanel,ÓnmFüm 60km / h . sresrrUbmnþcm¶aycr d kñúgryHeBl t rYcKNnacm¶aycr kalNaeKebIkbrkñúgryHeBl t = 2h , t = 2h20mn , t = 30mn nig t = 1h5mn . cemøIy sresrrUbmnþcm¶aycr d kñúgryHeBl t KW d = 60t -ebI t = 2h enaH d = 60 × 2 = 120km . 20 7 7 -ebI t = 2h20mn = 2 + 60 = h enaH d = 60 × = 140km 3 3 -ebI t = 30mn = 12 h enaH d = 60 × 12 = 30km . - 52 -

emeronTI8

kenSamBICKNit

%> ctuekaNEkgmYymanbeNþayesµInwg 15m nig TTwg x m . k> sresrrUbmnþbrimaRt P énctuekaNEkg ? x> KNnabrimaRtctuekaNEkgcMeBaH x = 5m , x = 8m , x = 10m cemøIy k> sresrrUbmnþbrimaRt P énctuekaNEkg eK)an P = 2(15 + x) = 30 + 2x x> KNnabrimaRtctuekaNEkgcMeBaH x = 5m , x = 8m , x = 10m -ebI x = 5m enaH P = 30 + 2 × 5 = 40m -ebI x = 8m enaH P = 30 + 2 × 8 = 46m -ebI x = 10m enaH P = 30 + 2 × 10 = 50m ^> KNna k> 3x + 7x = 10x x> 2x − 5x = −3x K> 3x × 6x = 18x X> 4x × (−8x) = −32x &> BnøatkenSam ³ k> (x + 1)(x − 1) = x − x + x − 1 = x − 1 2

2

2

2

- 53 -

emeronTI8

kenSamBICKNit

x> (x + 3)(x − 2) = x − 2x + 3x − 6 = x + x − 6 K> (2x − 3)(5 − 3x) = 10x − 6x − 15 + 9x = −6x + 19x − 15 X> (x + 3)(x + 3) = x + 3x + 3x + 9 = x + 6x + 9 g> (2x − 3)(y + 2) = 2xy + 4x − 3y − 6 c> 4(x − 3) + (x − 5)(x + 1) 2

2

2

2

2

= 4x − 12 + x 2 + x − 5x − 5

q>

2

= x 2 − 17 − 2( 2 − x ) − ( x + 2)( 3x + 1) = −4 + 2x − 3x 2 − x − 6x − 2 = −3x 2 − 5x − 6

*> dak;CaplKuNktþa ³ k> ax + bx = x(a + b) x> ax − ab = a(x − b) K> ab − 3a = a(b − 3) X> 5x + 10x = 5x(x + 2) g> 3a + 3b + 3c = 3(a + b + c) c> 8x + 12y − 4z = 4(2x + 3y − z ) q> 3(x − 4) + x(x − 4) = (x − 4)(3 + x) 2

- 54 -

emeronTI8

kenSamBICKNit

C> (x + 5)(x + 2) − (3x + 1)(x + 5) = ( x + 5)( x + 2 − 3x − 1) = ( x + 5)( −2x + 1)

Q> xy + 3x − 4y − 12

= x( y + 3) − 4( y + 3) = ( y + 3)( x − 4)

j> 3ab + 7b − 3ac − 7c

= b( 3a + 7 ) − c( 3a + 7 ) = ( 3a + 7 )(b − c )

- 55 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

lMhat; !>eKmansmIkar 2x + 3 = 1 , 5 − (x − 2) = 8 , x + 4x − 1 = 0 nig x(x − 5) − 3 = 1 . etIsmIkarNaxøHCasmIkardWeRkTImYymanmYyGBaØat ? @> edaHRsaysmIkarxageRkamrYcepÞógpÞat;cemøIy ³ k> 15 + x = 21 x> 40 + n = 70 K> m + 1.2 = 1.5 X> y − 14 = 19 g> 81 − y = 56 c> 16x = 48 q> 102 = 17x C> 7x = 18 Q> 0.t11 = 5 #>edaHRsaysmIkarxageRkam k> 5(n + 2) − 15 = 4 − (2n − 5) x> 2x + 5 = 6x − 3 K> 5y − 9y + 2 = 10 − 13 + y X> 2(4x − 1) + x = 16 − 2x − 40 g> 0.05x + 0.4 = 0.15 c> 3(2y − 1) − 5 = −4y + 22 q> 2 − 5(3 − 2y ) = −7 + 4(2 + y ) 2

- 56 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

$> edaHRsaysmIkarxageRkamrYcepÞógpÞat;cemøIy ³ x> 5 + 4(t − 2) = 2(t + 7) + 1 k> 5x + 10(x − 2) = 40 K> x −3 2 + 1 = 7x X> 3 − 2x3− 3 = 5 −2 x g> x +4 3 − x −2 4 = 83 c> 2x3− 1 + 4 = 2x6x− 1 C> x +2 1 − x −5 3 = x +4 2 q> x3 + 24x − 35x = 1 %> ctuekaNEkgmYymanbeNþayesµInwg 24m ehIyépÞRkLaesµInwg épÞRkLakaerEdlmanRCugesµInwg 12m . KNnaTTwgctuekaNEkgenaH ? ^> m:asIunftmYymantémø 72 duløa bnÞab;BIbBa©úHtémø 20% . etItémøedImrbs;m:asIunftenaHesµInwgb:unµan ? &> FugBIrmansaMg 150 lIt . ebIeKyksaMg 13 lItBIFugTImYy nig yk 35 lItecjBIFugTIBIrenaHFugTaMgBIrenAsl;saMgesµIKña . etIFugnImYy²mansaMgb:unµanlIt ? *> KNnabrimaRtRtIekaNmYy ebIRtIekaNenaHmanrgVas;RCugTImYy esµInwg 16m rgVas;RCugTIBIresµInwg 72 énbrimaRt nigRCugTIbIesµInwg 1 énbrimaRt . 3 - 57 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

(>LanmYyecjdMeNIrBIPñMeBjeTAkMBg;camEdlmancm¶ayesµInwg 124km edayel,Ón 65km / h ehIyLanmYyeTotecjdMeNIr Rcas;KñaBIkMBg;cammkPñMeBjedayel,Ón 45km / h . eKdwgfaLanTaMgBIrecjdMeNIrenAeBlEtmYy . k> rkcm¶ayBIPñMeBeTAkEnøgCYbKña ? x> etIb:unµanem:ageRkaymkeTIbLanTaMgBIrCYbKña ? !0> plbUkBIrcMnYnesµInwg 24 ehIyBIrdgéncMnYnTI! bUknwgcMnYnTI@ esµInwg 26 . KNnacMnYnTaMgBIrenaH ? !!> vIr³ekµgCag»Bukrbs;Kat; 24 qñaM . eKdwgfaryHeBl@qñaMeTotpl bUkGayuGñkTaMgBIresµInwg 40 qñaM . rkGayu»Buk nig Gayurbs;vIr³ >

- 58 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

dMeNaHRsay !>eKmansmIkar 2x + 3 = 1 , 5 − (x − 2) = 8 , x + 4x − 1 = 0 nig x(x − 5) − 3 = 1 . etIsmIkarNaxøHCasmIkardWeRkTImYymanmYyGBaØat ? smIkarEdlCasmIkardWeRkTImYymanmYyGBaØatKW³ 2x + 3 = 1 nig 5 − ( x − 2) = 8 . @> edaHRsaysmIkarxageRkamrYcepÞógpÞat;cemøIy ³ k> 15 + x = 21 eK)an x = 21 − 15 2

x=6

epÞógpÞat; 15 + 6 = 21 21 = 21

Bit

dUcenH x = 6 . x> 40 + n = 70 eK)an n = 70 − 40 n = 30

epÞógpÞat; 40 + 30 = 70 - 59 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat 70 = 70

Bit

dUcenH n = 30 . K> m + 1.2 = 1.5 eK)an m = 1.5 − 1.2 m = 0. 3

epÞógpÞat; 0.3 + 1.2 = 1.5 1.5 = 1.5 Bit dUcenH m = 0.3 . X> y − 14 = 19 eK)an y = 19 + 14 y = 33

epÞógpÞat; 33 − 14 = 19 19 = 19 Bit dUcenH y = 33 . g> 81 − y = 56 eK)an − y = 56 − 81 y = 25 - 60 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

epÞógpÞat; 81 − 25 = 56 56 = 56 Bit dUcenH y = 25 . c> 16x = 48 48 eK)an x = 16 x=3

epÞógpÞat; 16(3) = 48 48 = 48

dUcenH x = 3 . q> 102 = 17x eK)an x = 102 17

Bit

x=6

epÞógpÞat; 102 = 17(6) 102 = 102

Bit

dUcenH x = 6 .

- 61 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

C> 7x = 18 eK)an x = 18 × 7 x = 126 126 = 18 7

epÞógpÞat;

18 = 18

dUcenH x = 126 . Q> 0.t11 = 5 eK)an t = 5 × 0.11 t = 0.55 0.55 =5 0.11

Bit

epÞógpÞat;

5=5

Bit

dUcenH t = 0.55 . #>edaHRsaysmIkarxageRkam k> 5(n + 2) − 15 = 4 − (2n − 5) 5n + 10 − 15 = 4 − 2n + 5 5n − 5 = −2n + 9 5n + 2n = 9 + 5 - 62 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

7n = 14 14 7 n=2 n=

x> 2x + 5 = 6x − 3

2 x − 6 x = −3 − 5 − 4 x = −8 −8 −4 x=2 x=

K> 5y − 9y + 2 = 10 − 13 + y

5y − 9y − y = 10 − 13 − 2 − 5y = −5 −5 y= −5 y =1

X> 2(4x − 1) + x = 16 − 2x − 40 8x − 2 + x = 16 − 2x − 40

8x + x + 2x = 16 − 40 + 2 11x = −22 22 11 x = −2

x=−

- 63 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

g> 0.05x + 0.4 = 0.15

0.05x = 0.15 − 0.4 0.05x = −0.25 0.25 x=− 0.05 x=5

c> 3(2y − 1) − 5 = −4y + 22

6y − 3 − 5 = −4y + 22 6y − 8 = −4y + 22 6y + 4y = 22 + 8 10y = 30 30 y= 10 y=3

q> 2 − 5(3 − 2y ) = −7 + 4(2 + y ) 2 − 15 + 10y = −7 + 8 + 4y − 13 + 10y = 1 + 4y 10y − 4y = 1 + 13 6y = 14 14 8 y= = 6 3

- 64 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

$> edaHRsaysmIkarxageRkamrYcepÞógpÞat;cemøIy ³ k> 5x + 10(x − 2) = 40 5x + 10x − 20 = 40

15x = 60 60 x= =4 15

x> 5 + 4(t − 2) = 2(t + 7) + 1 5 + 4t − 8 = 2t + 14 + 1 4t − 3 = 2t + 15 4t − 2t = 15 + 3

K>

2t = 18 18 t= =9 2 x−2 x +1= 3 7 x− 2+ 3 x = 3 7 x+1 x = 3 7 7 x + 7 = 3x 4 x = −7 7 x=− 4

- 65 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

X> 3 − 2x3− 3 = 5 −2 x

9 − 2x + 3 5 − x = 3 2 − 2x + 12 5 − x = 3 2 − 4x + 24 = 15 − 3x − 4x + 3x = 15 − 24

g>

− x = −9 x=9 x+3 x−4 3 − = 4 2 8 2( x + 3) − 4( x − 4) 3 = 8 8 2x + 6 − 4x + 16 3 = 8 8 − 2x + 22 3 = 8 8 − 2x = 3 − 22

− 2x = −19 19 x= 2

- 66 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

c> 2x3− 1 + 4 = 2x6x− 1 lkçxNÐ 2x − 1 ≠ 0 b¤ x ≠ 12 eK)an 3 + 24x(2−x1− 1) = 2x6x− 1 3 + 8x − 4 = 6x

8x − 6x = 4 − 3 2x = 1 1 x= 2 x 2x 3 x + − =1 3 4 5 20x + 30x − 36x =1 60 14x =1 60 60 x= 14 30 x= 7 x+1 x−3 x+ 2 − = 2 5 4 10( x + 1) − 4( x − 3) = 5( x + 2) 10x + 10 − 4x + 12 = 5x + 10

minyk

q>

C>

6x + 22 = 5x + 10 x = −12

- 67 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

%> ctuekaNEkgmYymanbeNþayesµInwg 24m ehIyépÞRkLaesµInwg épÞRkLakaerEdlmanRCugesµInwg 12m . KNnaTTwgctuekaNEkgenaH ? cemøIy KNnaTTwgctuekaNEkg tag x CaTTwgrbs;ctuekaNEkgenH tambRmab;eK)ansmIkar ³ 24x = 12 × 12 24x = 144 144 24 x=6 x=

^> m:asIunftmYymantémø 72 duløa bnÞab;BIbBa©úHtémø 20% . etItémøedImrbs;m:asIunftenaHesµInwgb:unµan ? cemøIy tag x CatémøedImrbs;m:asIunft 20 eK)ansmIkar x − 100 x = 72 eKTaj)an x = 72 ×80100 = 90 duløa . - 68 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

&> FugBIrmansaMg 150 lIt . ebIeKyksaMg 13 lItBIFugTImYy nig yk 35 lItecjBIFugTIBIrenaHFugTaMgBIrenAsl;saMgesµIKña . etIFugnImYy²mansaMgb:unµanlIt ? cemøIy tag x nig y CacMNuHsaMgkñúgFugTImYy nig FugTIBIrerogKña tambRmab;eK)ansmIkar ³ x + y = 150 (1) ehIy x − 13 = y − 35 naM[ y = x − 13 + 35 = x + 22 (2) yksmIkar (2) CMnYskñúg (1) eK)an x + x + 22 = 150 2x = 150 − 22

2x = 128 128 = 64 x= 2 y = 64 + 22 = 86

ehIy dUcenHFugTImYymansaMg 64 lIt nigFugTIBIrmansaMg 86 lIt .

- 69 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

*> KNnabrimaRtRtIekaNmYy ebIRtIekaNenaHmanrgVas;RCugTImYy esµInwg 16m rgVas;RCugTIBIresµInwg 72 énbrimaRt nigRCugTIbIesµInwg 1 énbrimaRt . 3 cemøIy KNnabrimaRténRtIekaN ³ tag p CabrimaRténRtIekaN eK)an p = 16 + 72 p + 13 p 21p = 336 + 6p + 7p 21p = 336 + 13p 8p = 336 p=

336 = 42cm 8

dUcenHbrimaRtRtIekaNKW 42cm .

- 70 -

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

(>LanmYyecjdMeNIrBIPñMeBjeTAkMBg;camEdlmancm¶ayesµInwg 124km edayel,Ón 65km / h ehIyLanmYyeTotecjdMeNIr Rcas;KñaBIkMBg;cammkPñMeBjedayel,Ón 45km / h . eKdwgfaLanTaMgBIrecjdMeNIrenAeBlEtmYy . k> rkcm¶ayBIPñMeBjeTAkEnøgCYbKña ? x> etIb:unµanem:ageRkaymkeTIbLanTaMgBIrCYbKña ? cemøIy k> rkcm¶ayBIPñMeBjeTAkEnøgCYbKña tag x Cacm¶ayBIPñMeBjeTAkEnøgrfynþTaMgBIrCYbKña eK)an 124 − x Cacm¶ayBIkMBg;cameTAkEnøgCYb . eK)ansmIkar 65x = 12445− x 9x = 13(124 − x ) 9x = 1612 − 13x

x>cMeBaH

22x = 1612 1612 806 x= = km 22 11 806 x 806 806 x= t= = = h 11 65 65 × 11 715

eK)an

- 71 -

.

emeronTI9

smIkardWeRkTImYymanmYyGBaØat

!0> plbUkBIrcMnYnesµInwg 24 ehIyBIrdgéncMnYnTI! bUknwgcMnYnTI@ esµInwg 26 . KNnacMnYnTaMgBIrenaH ? cemøIy KNnacMnYnTaMgBIr tag x CacMnYnTImYy nig 24 − x CacMnYnTIBIr eK)ansmIkar 2x + 24 − x = 26 naM[ x = 2 . dUcenHcMnYnTImYyesµI 2 nigcMnYnTIBIr 24 − 2 = 22 . !!> vIr³ekµgCag»Bukrbs;Kat; 24 qñaM . eKdwgfaryHeBl@qñaMeTotpl bUkGayuGñkTaMgBIresµInwg 40 qñaM . rkGayu»Buk nig Gayurbs;vIr³ ? cemøIy tag x CaGayurbs;vIrH nig x + 24 CaGayurbs;»Buk eK)ansmIkar (x + 2) + (x + 24 + 2) = 40 2x + 28 = 40 b¤ x = 6 dUcenHvIrHmanGayu 6 qñaM nig»BukmanGayu 6 + 24 = 30 qñaM . - 72 -

emeronTI10

vismPaB

lMhat; !>eRCIserIsvismPaBkñúgsMeNrxageRkam ³ − 2 < −1 ; 8 > 5 ; a = b ; c < d ; a + c < a + d , a + c = a + d

@> bMeBjsBaØa > b¤ < , = enAkñúgRbGb;xageRkam ³ k> 10 < 20 enaH 10 − 4
20 − 4 x> a < b enaH a − c
b − c K> a > b enaH a − c
b − c X> 1 < 12 enaH 1 ÷ 4
12 ÷ 4 g> − 20 < −2 enaH − 20 ÷ (−4)
− 2 ÷ (−4) c> a < b enaH a + c
b + c q> a > b enaH a + c
b + c cMeBaH c > 0 C> a > b enaH a + c
b + c cMeBaH c < 0 #>bgðajfaebI a > b nig c > d naM[ a + c > b + d rYc[]TahrN_CaelxbBa¢ak; . $> bgðajfaebI a > b nig c < d naM[ a − c > b − d rYc[]TahrN_CaelxbBa¢ak; . - 73 -

emeronTI10

vismPaB

%>bgðajfaebI a > b nig b > c naM[ a > c rYc[]TahrN_CaelxbBa¢ak; . ^> eRboeFobcMnYn ³ k> 53 nig 73 x> 23 nig 17 27 7 55 17 K> 14 ni g X> ni g 11 5 23 11 &> a = 10 , b = −1 . epÞógpÞat;fa | a + b |≤| a | + | b | ?

- 74 -

emeronTI10

vismPaB

dMeNaHRsay !>eRCIserIsvismPaBkñúgsMeNrxageRkam ³ − 2 < −1 ; 8 > 5 ; a = b ; c < d ; a + c < a + d , a + c = a + d

cemøIy sMeNrEdlCavismPaBman ³ − 2 < −1 , 8 > 5 , a < d ,

nig a + c < a + d . enAkñúgRbGb;xageRkam ³

@> bMeBjsBaØa > b¤ < , = k> 10 < 20 enaH 10 − 4 < 20 − 4 x> a < b enaH a − c < b − c K> a > b enaH a − c > b − c X> 1 < 12 enaH 1 ÷ 4 < 12 ÷ 4 g> − 20 < −2 enaH − 20 ÷ (−4) > − 2 ÷ (−4) c> a < b enaH a + c < b + c q> a > b enaH a + c > b + c cMeBaH c > 0 C> a > b enaH a + c > b + c cMeBaH c < 0

- 75 -

emeronTI10

vismPaB

#>bgðajfaebI a > b nig c > d naM[ a + c > b + d rYc[]TahrN_CaelxbBa¢ak; . cemøIy eKman a > b naM[ a − b > 0 ehIy c > d naM[ c − d > 0 eK)an (a − b) + (c − d) > 0 dUcenH a + c > b + d . ]TahrN_Caelx ³ yk a = 10 , b = 7 , c = 9 , d = 5 eK)an 10 > 7 nig 9 > 5 dUcenH 10 + 9 > 7 + 5 19 > 12 Bit . $> bgðajfaebI a > b nig c < d naM[ a − c > b − d rYc[]TahrN_CaelxbBa¢ak; . cemøIy eKman a > b naM[ a − b > 0 ehIy c < d naM[ d − c > 0 - 76 -

emeronTI10

vismPaB

eK)an (a − b) + (d − c) > 0 dUcenH a − b > b − d . ]TahrN_Caelx ³ yk a = 23 , b = 17 , c = 9 , d = 15 eK)an 23 > 17 nig 9 < 15 dUcenH 23 − 9 > 17 − 15 14 > 2 Bit . %>bgðajfaebI a > b nig b > c naM[ a > c rYc[]TahrN_CaelxbBa¢ak; . cemøIy eKman a > b naM[ a − b > 0 ehIy b > c naM[ b − c > 0 eK)an (a − b) + (b − c) > 0 a−c> 0

dUcenH a > c . ]TahrN_CaelxbBa¢ak; ³ yk a = 27 , b = 20 , c = 15 eKman 27 > 20 nig 20 > 15 dUcenH 27 > 15 Bit . - 77 -

emeronTI10

vismPaB

^> eRboeFobcMnYn ³ k> 53 nig 73 eday 5 < 7 dUcenH 53 < 73 . x> 23 nig 17 27 2 17 eKman 23 = 18 eday 18 > 17 dUcenH > . 27 3 27 7 K> 14 ni g 11 5 70 7 77 eKman 14 = ni g = 11 55 5 55 7 < . eday 70 < 77 dUcenH 14 11 5 17 X> 55 ni g 23 11 55 605 17 221 eKman 13 = ni g = 143 11 143 55 17 dUcenH 13 > . 11 &> a = 10 , b = −1 . epÞógpÞat;fa | a + b |≤| a | + | b | eday | a + b |=| 10 − 1 |= 9 , | a | + | b |=| 10 | + | −1 |= 11 dUcenH | a + b |≤| a | + | b | . - 78 -