SSRP Self Learning Guide Maths Class 9 - In Hindi

10 September 2015

lkekU; ifjp; vk S j mi;k sx d s rjhd s

xf.kr ds vn~Hkqr lalkj esa vkidk Lokxr gS!

xf.kr dh nqfu;k cgqr gh [kwclwjr gS& ;g fMtkbu ,oa iSVuZ ls Hkjh iM+h gSA xf.kr ,d izdkj dh

Hkk’kk gSA vad vkSj izrhd blds o.kZ ;k v{kj gSA

rqe rks tkurs gh gks& rqe bldh Hkk’kk dks le> ldrs gks vkSj bls fcuk fdlh dh lgk;rk ls

vuqHko dj ldrs gks vkSj rc rqe bldh lqUnjrk dh iz”kalk djksxsA ;gkW vkids fe= gS tks blesa

vkidh enn djsaxsa& jkspd iz”uksa dk gy] muesa tqM+ko] vo/kkj.kk cukuk] bR;kfnA

tc vki iz;kl djsxsa vkidks etk vk;sxk vkSj vki xyfr;k¡ djsaxsaA blls ?kcjk;s ughA D;k vki

tkurs gS fd ;s xyfr;k¡ vkidks csgrj lh[kus esa enn djsxhA

bl ;k=k esa vki dbZ vo/kkj.kkvksa dk irk yxkus tk jgs gSA ,d Mk;jh bl fdrkc ls tqM+h gq;h

gSA vki Mk;jh esa viuh] Hkkoukvksa] lansg ds loky bR;kfn fy[k ldrs gSaA vki bls vius nksLrksa

,oa f”k{kd ds lkFk lk>k dj ldrs gSA blls vkidks lh[kus es csgrj enn feysxhA

;g yfuZax xkbZM vkidks xf.kr dh cqfu;knh phtksa dks le>us esa vkidh enn djsxhA blls vki

vius vki xf.kr ¼;k dksbZ Hkh fo’k;½ lh[kus dk rjhdk le> tk;sxsaA

;gka nh xbZ lkexzh dks iwjk djus esa vkidks 45 ls 50 fnu yxsaxs ;fn vki gj fnu de ls 30&40

feuV dk;Z djsaxs] oSls vki tYnh Hkh iwjk dj ldrs gSaA vkSj ;fn blls T;knk le; Hkh yx jgk gS

rks dksbZ ckr ugha gS] ij bls le> ds lkFk gh iwjk djuk Bhd gksxkA

yfuZax xkbZM esa eksVs rkSj ij nks fgLLks gSa& Lkh[kus ds Lrjokj fcUnq vkSj muij vk/kkfjr Lrjokj

Lok/;k; vH;kl o vkdyu lq>koA

lh[ku s d s fcUn q

;gka N% Lrj ds yfuZax vkCtsfDVo fn, x, gSA gj ,d Lrj iwjk djus ds ckn vki ;gka fn, x,

y{;ksa ls feyku dj ldrs gSa fd ml Lrj esa fdruk lh[k ik, gSaA ;fn yxrk gS fd fdlh fcUnq

ds ckjs esa vkSj vH;kl djuk gS] rks LkkIrkfgd d{kk esa f”k{kd vkSj vU; lkfFk;ksa ls ckr djsaA

Lrj Lk h[ ku s d s fcUn q (yfu Z ax vkCt s fDVO k) 1 1- pkjksa cqfu;knh lafØ;kvksa dh xgjh le> vkSj muds varj lEc)rk dks tkuuk

2- pkjksa cqfu;knh lafØ;kvksa dks djus ds fy, fofHkUu rjhdksa dh le>

3- vius vki ls fu;ekas ,oa vo/kkj.kkvksa dks cukuk

4- la[;k iSVuZ dk vuqlj.k ,oa lkSan;Z dh ljkguk djuk

5- oMZ izkCye dks le>us vkSj mUgas gy djus ds fy, fofHkUu rjhdksa dk irk

yxkukA

SSRP LG Math-IX 2014-15

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2" !

!

2 1- fHkUu] n”keyo ,oa izfr”kr ds vk/kkjHkwr fopkjksa dks izkIr djus esa

2- LFkkuh; eku ,oa iw.kkZdksa dh vo/kkj.kkvksa dks iqu% rktk cukus esa

3- lehdj.k ,oa lehdj.k ds larqyu ds vo/kkj.kk dks Li’V djuk

4- chtxf.kr ds vk/kkjHkwr vo/kkj.kkvksa dks le>us esa

5- xf.krh; izrhdksa ij lk/kkj.k izkstsDV djus esa

6- oxhZdj.k] iSVuZ irk djus esa vkSj vo/kkj.kkvksa dks fudkyus esa

3 1- fHkUu la[;kvksa ds chp lafØ;kvksa dh le>

2- fHkUu vkSj n”keyo la[;kvksa ds chp lafØ;kvksa esa vo/kkj.kkvksa dks dSls fl)

fd;k tkrk gS& blds ckjs esa le>A

3- fofHkUu vo/kkj.kkvksa ds chp ds vUrj&lEcU/k le>uk] tSls& xq.kk] foHkkT;rk

ds fu;e] y- l- ,oa e- l-A

4- iw.kkZadksa ds lkFk lafØ;k,a& lk/kkj.k igsfy;ka cukuk ,oa mudks gy djuk

5- f=Hkqt ds ckjs esa dqN vk/kkjHkwr fopkjksa dh le>A

4 1- Lka[;kvksa ds [ksy

2- Ikzfr”kr

3- okf.kT; xf.kr

4- Lkk/kkj.k C;kt

5 1- {ks=Qy] /kkfjrk vkSj vk;ru

2- vuqikr vkSj lekuqikrh Hkkx

3- lehdj.k

6 1- xzkQ

2- cgqinksa dk tksM vkSj ?kVkuk

3- cgqinksa dk xq.kk


!

3" !

!

Lok/;k; vH;kl vk S j xfrfof/ k;k a

gj ,d esa Lrj esa dh tkus okyh xfrfof/k;ksa ,ao vH;kl ds ckjs eas funsZ”k fn;s x, gSaA bls

tYnckth esa u djsa] cfYd ,d Lrj dk vH;kl iwjk djus esa 10 ls 12 fnu dk le; ysa] bl rjg

ls ;g vkB ls nl lIrkg dk dk;Z gks ldrk gSA

;gka ls vkids Lok/;k; vH;kl “kq# gksrs gSaA ;gka fn, x, vH;klksa dks ,d vyx uksVcqd esa djsaA

vkSj vius vuqHkoksa dks jkstuk viuh Mk;jh esa fy[krs jgsaA

;gka igys pkj Lrj ds fy, yfuZax xkbZM nh xbZ gSA bu ij dke djrs gq, vkxs ds lh[kus ds

y{;ksa ds ckjs esa T;knk Li’Vrk gksus ij f”k{kd ds lkFk feydj vkxs ds nks Lrj ¼Lrj 5 o 6½ dh

yfuaZax xkbZM fodflr gksxhA

Lok/;k; vH;kl vk S j xfrfof/ k;k a Lrj 1

1-1- la[;k ig syh % irk djks eSa dkSu g¡w\ v- eSa ,d rhu vadksa dh la[;k g¡wA

c - eSa ,d le la[;k g¡wA

l- rqe eq>s fcuk “ks’k ds 25 vkSj 50 ls foHkkftr dj ldrs gksA

n- eSa rhu vadh; lcls NksVh la[;k gWwA crkvks eS dkSu g¡w\

,slh gh dqN la[;k igsyh cukvksA D;k vkids fe= igsyh dk tokc ns ldrs gSa] muls iwNasA ,d

la[;k ds ckjs eas ,d ls vf/kd okD; fy[kus dh dksf”k”k djksA igsyh ds var esa dqN lqjkx nsuk er

HkwyukA

vc bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks& 1- vkidks ;g djrs gq, dSlk yxk\

2- vkius vius fdu&fdu nksLrksa ds lkFk lk>k fd;kA mudks dSlk yxk \

3- D;k vius nksLr dh igsyh dk tokc ns ik, \ nksLrksa ds }kjk iwNh xbZ fdl igsyh dk

tokc nsus esa lcls T;knk eqf”dy gqbZ] D;ksa\

1 -2 - 'k Cnk s a dk oxh Zdj.k % D;k vki bckjrh ¼ftlesa o.kZu gks½ iz”uksa dks dfBu eglwl djrs

gks\ ;g xfrfof/k vkidh enn djsxhA bu “kCnksa dks i<+sa vkSj mUgsa uhps nh xbZ lkfj.kh esa

oxhZ—r djrs gqq, fy[ksa&

nsuk] ysuk] tkuk] vkuk] Hkkx] tksM+] “ks’k] T;knk] de] j[kuk] lewg] dqy] Hkjuk] mM+syuk] [kkyh

djuk] ckdh] Hkjk gqvk] cdk;k] vUrjA

tk sM + ? kVko x q. k k H k kx


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4" !

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D;k rqEgkjs fe= us Hkh mlh rjg oxhZ—r fd;k ftl rjg ls rqeus fd;k Fkk\ muds lkFk ppkZ

djksA



3- D;k vius nksLrksa us Hkh ;gh mRrj fy[kk \ irk djksA

1-3- tk sM + d s ckj s e s a % ;g vklku gS ;k eqf”dy\

300 $ 400] 450 $ 200] 325 $ 375] 260 $ 338

bu lokyksa ds ckjs esa D;k jk; gS\ D;k rqe fcuk isij vkSj isu dk mi;ksx fd;s tksM+ ldrs gks\

367 $ 438] 453 $ 249] 305 $ 157 $ 284

v- Åij iw.kZ djus ds ckn rqe bu lokyksa dks djus dk iz;kl djks&

26 $ 38] 36 $ 28] 24 $ 40 c- la[;kvksa ds cVokjs ls rqe vkjke ls budks tksM+ ldrs gks tSlk rqe pkgrs gksA ;gk¡ ,d

mnkgj.k gS& ;fn rqe 467 $ 378 dks tksM+uk pkgrs gks rks rqe la[;kvksa dks bl izdkj ck¡V

ldrs gks&

7867300400281735045087370460

+++

+++

+++

gk¡] bu rhuksa lokyksa dk ,d gh tokc gSA bu rhuksa esa ls dkSu lk ,d tksM+us esa vklku gS\ rhu

vadh; la[;kvksa blh izdkj fcuk dkih&isu ds tksM+us dk iz;kl djsaA



3- D;k rqe rhu vadh; la[;kvksa dks fcuk isij vkSj isu ds tksM+ ldrs gks\

4- D;k rqe dqN loky cuk ldrs gks ftudk ,d gh mRrj gks\


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5" !

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1-4- vf/ kdre vFk Z

ge 445 $ 440 ¾ 885 ls D;k le> ldrs gS\

⇒ 445 vkSj 440 feydj 885 cukrs gSaA

⇒ 440 vkSj 445 feydj Hkh 885 cukrs gSaA

⇒ ;fn ge 885 ls 445 dks fudkyrs gSa rks gesa 440 feysxkA

⇒ ;fn ge 440 dks 885 ls nwj ys tkrs gS rks gesa 445 feysxkA

mijksDr okD;ksa dks xf.krh; okD;ksa esa bl izdkj fy[krs gSa &

440445885445440885885445440885440445

=−

=−

=+

=+

;g mnkgj.k dsoy tksM ij vk/kkfjr gSA ,sls gh ?kVko] xq.kuQy vkSj Hkkx ds dqN mngkj.k cuk;s

,oa mues ,d ls vf/kd vFkZ irk yxkus dh dksf”k”k djsaA

vc bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&

1- vkidks ;g djrs gq, dSlk yxk\


3- tksM+ vkSj ?kVko dSls tqMs+ gq, gS\

4- xq.kk vkSj Hkkx dSls tqM+s gq, gS\ vius mRrj dks nksLr ds lkFk ppkZ djksA mnkgj.k ds lkFk

O;k[;k nksA

5- D;k vki xq.kk vkSj Hkkx es laca/k irk dj ldrs gks\

6- D;k vki vkSj cuk ldrs gS\

1 -5 - pk; oky s dk cgh[k krk

foØe dh gekjs Cykd vkWfQl ds ikl ,d pk; dh nqdku gSA Cykd ds deZpkjh ogk¡ ls pk; ihrs

gaS ysfdu iSlk os eghus ds vUr esa pqdkrs gaS] tc os osru ikrs gaSA blfy, foØe ,d cgh[kkrk

j[krk gSA

Okg la[;kvksa dks bl izdkj fy[krk gS& 15 $ 20 $ 35 $ 12 $ 74----mls 30 ls 40 rd la[;k

blh rjg fy[kuk gksrk gSa vkSj mls og mlh rsth ls tksM+ Hkh ysrk gSA

la[;kvksa dks {kSfrt esa fy[ksa vkSj tksM+us dk iz;kl djsaA ;g ,d vadh;] nks vadh;] rhu vkSj pkj

vadh; la[;kvksa ds lkFk djsaA


!

6" !

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3 -1 - ? kVkuk

;g xfrfof/k vkidks vklkuh ls ?kVkuk lh[kus esa enn djsxkA fuEufyf[kr lokyksa ds fy, mRrj

irk djsaA

........78156..........278356

=−

=−

.........10104........70164

=−

=−

..........16204

.........86274...........486674

=−

=−

=−

1- vkius blls D;k lh[kk\ bls fy[kus dk iz;kl djsa vkSj vius fe= ds lkFk ppkZ djsaA

;gk¡ ?kVkus dk ,d vkSj rjhdk gSA

...........278453 =−

igyh (cMh la[;k) l[;k dks rksM+sa (300+ 153)

278 dks 300 ls ?kVk;s (300-278=22)

vc 22 esa 153 dks tksM+s (153+22=175)

2- dqN iz”u fy[ksa vkSj nksukas rjhds ls mRrj izkIr djsaA dkSu lk rjhdk rqEgs T;knk ilan

vk;k\ ;k iz”u dh izd`fr ds vuqlkj lgh rjhdk pquuk T;knk lgh gS\

3- dqN iz”u fy[ksa vkSj fcuk isij vkSj isu ds mlds mRrj dks izkIr djus dk iz;kl djsaA

3 -2 - nk s l a fØ;k, a %&

efYydk ds ikl jk/kk dks nsus ds fy, 57 #i,] eksgu ds fy, 128 #i, ] m’kk ds fy, 375 #i,

vkSj vkfej ds fy, 250 #i;s gSA efYydk ds ikl dqy 1000 #i;s gSA bu lHkh dks #i;s nsus ds

ckn efYydk ds ikl dqy fdrus #i;s cpsxsa\ D;k rqe bls dsoy nks lafØ;k dk iz;ksx djrs gq,

gy dj ldrs gksA



3- D;k rqe bls nks lafØ;kvksa ds lkFk gy dj ik,\

4- D;k vki lokyksa dks i<+dj le> ikus esa l{ke gSa\

5- D;k vki ,sls loky dks vklku ikrs gS\


!

7" !

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4 -1 - [k qn d s fu;e cukuk

;g xfrfof/k vkidks dqN xf.krh; vo/kkj.kkvksa dks Lo;a irk yxkus esa enn djsxhA

1511513131551

=×

=×

=×

**;fn ge fdlh la[;k dk xq.kk 1 esa djrs gS rks ogh la[;k izkIr gksrh gSA** Åij fn, x, xq.kk ds

rF;ksa dks ns[k dj ge cgqr vPNh rjg ls ;g fu;e le> ldrs gaSA

fuEufyf[kr xq.kk ds rF;ksa dks ns[ksa vkSj fu;e cukus dk iz;Ru djsaA

**xq.kt ×!xq.kd ¾ ifj.kke** xq.kk dk ,d lkekU; fu;e gSA bl vk/kkj ij vki uhps fn, x,

xf.krh; dFkuksa ds fy, fu;e cukb,A bls vius uksVcqd esa fy[kuk er HkwyukA

1) 3003100 =× 120012100 =× 5005100 =×

2) 12342438

=×

=×

305660512

=×

=×

324864416

=×

=×

3) 30653056

=×

=×

8010880810

=×

=×

9015690615

=×

=×

4) )52413()2847()2446( =×==×+=×

)126718()70710()5678( =×==×+=×



3- D;k rqe fu;e cuk ik;s\

4- D;k rqe iz”u dks i<+dj le> ik;s\

5- D;k rqeus dksbZ u;k fu;e Lo;a ls cuk;k\

6- D;k rqeus vius fopkjksa dks fy[kk\


!

8" !

!

4 -2 - U kb Z x q. ku rk fydk

D;k rqe fdlh Hkh pkgs gq, la[;k dh xq.ku rkfydk cukuk pkgrs gks\ ;g xfrfof/k vkidh enn

djsxkA mnkgj.k ns[ksa&

162189126187541831081861441887218436182901851801810

=×

=×

=×

=×

=×

=×

=×

=×

=×

,d la[;k pqusA dksbZ nks vadh; la[;k vPNk jgsxkA xq.ku rkfydk cuk;saA isij] isu ;k dSydqysVj

dk iz;ksx u djsaA ;fn vkids ikl ?kM+h gks rks ns[ksa & ;g djus esa fdruk le; yxk\ vf/kd

la[;kvksa dk mi;ksx djds vH;kl djksA



2- D;k rqe u;k xq.ku rkfydk cuk ikus esa l{ke gks ldsa\

3- D;k rqeus le; dh x.kuk dh\

4- D;k rqeus dbZ la[;kvksa ds lkFk vH;kl fd;k] fdlds lkFk T;knk eqf”dy gqbZ\

4 -3 - ,d l s vu sd

xq.kuQy ds ,d rF; ds vk/kkj ij ge ,sls dbZ cuk ldrs gSaA ;gk¡ ,d mnkgj.k gS&

45153 =× bl rF; ls dj ge dbZ vkSj lokyksa dks Hkh gy dj ldrs gSaA tSls &

;fn rhl ckj iUnzg dks ge x.kuk djrs gSa rks 4501530 =×

2251515 =× (450 dk vk/kk 225)

;fn ge bu nksuksa rF;ksa dks tksM+s rks ge vPNh rjg ls 45 ckj 25 dh x.kuk dj ldrs gSaA

6751545 =×


!

9" !

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xq.kk ds rF;ksa ds lkFk xq.kuQy Hkjsa &

........1560........159

............1518

=×

=×

=×

,d xq.kk dk rF; dk pquko djsaA blds vk/kkj ij ,d ls vf/kd xq.kk ds rF;ksa dks cukus dk

iz;kl djksA le; dks fQDl djks rqe fdrus cuk ldrs gks\ vius fe= ds ckjs esa D;k [;ky gS\



2- D;k rqeus la[;kvksa dks nqxquk ;k vk/kk djrs le; vkjkenk;d eglwl fd;k\

3- ,d feuV esa rqe fdrus xq.kkRed rF;ksa dks cuk ik;s\

5 -1 - U k kprh l a[;k, a

D;k vkius eap ij urZfd;ksa ds lewg dks ukprs gq, ns[kk gS\ os iafDr;kW cuk ldrh gaSA dqN

urZfd;kW iafDr;ksa esa lkeus ls vkrh gSa vkSj dqN ihNs pyh tk;asxhA urZfd;ksa dh rjg ;gk¡ la[;k,¡

ukp dj jgh gSA fuEufyf[kr rF;ksa ds xq.kuQy dk irk yxk,W vkSj vadksa ds LFkku dks ns[ksaA

....................876923

....................676923....................1176923....................576923....................776923....................276923

=×

=×

=×

=×

=×

=×


2- D;k vki bl rjg ds fdlh xq.kuQy ls lacaf/kr jkspd iSVuZ dks tkurs gSa\

6 -1 - ,d l a fØ;k & nk s vFk Z

315÷ ,d loky gS] blls D;k irk pyrk gS\

⇒ 15 esa dqy fdrus 3 gSa\

⇒ ;fn ge 15 dks 3 cjkcj Hkkx esa ckWVs rks ,d Hkkx D;k gksxk\

⇒ D;k nksuksa ckrsa lgh gaS\

vc rqe 5315 =÷ “kkfCnd okD; ¼bckjrh loky½ esa dSls cnyksxs\


!

10" !

!


2- D;k vki xf.krh; okD;ksa dks “kkfCnd okD;ksa esa cny ikus esa l{ke gks lds\

6 -2 - fu;e cukuk

D;k rqEgsa ;kn gS fd dSls xq.kuQy ds dqN rF;ksa ij vk/kkfjr fu;e cukrs gS\ mlh izdkj ;gk¡ dqN

Hkkx ds rF;ksa ds dqN lsV gSaA dqN fu;e cukus dk iz;kl djksA **HkkT; » Hkktd ¾ HkkxQy**

vkidks fu;e cukus esa enn djsxkA

1) 12112 =÷ 18118 =÷ 33133 =÷

2) 5210 =÷ 10220 =÷ 20240 =÷

3) 20360 =÷ 10660 =÷ 51260 =÷

4) 51680 =÷ 5840 =÷ 5420 =÷

fHkUu ls Hkkx fdl izdkj lEcaf/kr gS\ Hkkx ds rF; ds pkSFks lsV dks ns[ksaA ;g le fHkUu ls fdl

izdkj lacaf/kr gS\



2- D;k rqe fu;e cukus esa l{ke gks lds\ gkW ;k uk

3- D;k rqe Kkr dj ldrs gks fd Hkkx dh fØ;k fdl izdkj xq.kk ls lacaf/kr gS\

4- D;k rqe la[;kvksa ds Hkkx ls lacaf/kr dqN vkSj fu;e dks irk dj ikus esa l{ke gks ldsa\

6 -3 - H k kx ;k x q. k k dh l q Unjrk\

379333376222373111

=÷

=÷

=÷

333937222637111337

=×

=×

=×

v- bl lwph dks c<+k;asA D;k ;g pkj vadh; la[;kvksa esa Hkh ykxw gS\


!

11" !

!

c- uhps nh xbZ la[;kvksa dk HkkxQy Kkr djksA blls D;k vk”p;Zpfdr gks jgs gks\ vius fe= ds

lkFk ppkZ djksA

..................18222222222...................9111111111

=÷

=÷



2- D;k rqeus fyLV dks c<+k;k\

3- D;k vkius vius mRrj dh tk¡p Hkktd vkSj HkkxQy dks xq.kk djds fd;k\

4- D;k rqeus bls vU; ds lkFk lk>k fd;k\

6 -4 - NqVh g qb Z l a[;k dk s irk djuk

^^HkkT; » Hkktd ¾ HkkxQy vkSj “ks’kQy** ds vk/kkj ij NwVh gqbZ la[;kvksa dk irk djksA

1- HkkT; gS 45 vkSj HkkxQy 9 gS] rks Hkktd gksxk --------------------------

2- Hkktd gS 8 vkSj HkkxQy gS 7 rks HkkT; gksx ---------------------------------

3- 47 dks 9 ls iwjk&iwjk foHkkftr ugha fd;k tk ldrk] “ks’kQy gksxk -------------------

vc vki **HkkT;] Hkktd] HkkxQy vkSj “ks’kQy** ds chp fdrus lEca/kksa dk irk yxk ldrs gaS\

vius fe= ds lkFk fudys fu’d’kksZa ds ckjs esa ppkZ djsaA


2- fdrus lEca/kksa dks vkius Kkr fd;k\

3- D;k rqeus vkSj mnkgj.kksa ij dke fd;k\


2-1 - ,d e s a nk s

1.1, 1.2, 1.3...

;|fi nks la[;k,¡ gaS] os vdsys xfrfof/k dh vksj ladsr djrh gSaA 1-1 eryc igys mnns”; esa igyh

xfrfof/kA 1-2 dk eryc gS igys mn~ns”; ds fy, nwljh xfrfof/kA blh izdkj fHkUuksa esa Hkh nks

la[;k,W gSA ysfdu os ,d la[;k dks iznf”kZr djrh gSaA ns[ksa ge fHkUu ls D;k le> ldrs gaSA


!

12" !

!

53

dk D;k eryc gksxk\

⇒ nksuksa 3 vkSj 5 ,d gh la[;k dk ladsr djrs gSaA

⇒ ,d la[;k ;k oLrq dks ik¡p cjkcj Hkkx esa ck¡Vk x;k gS vkSj muesa ls rhu Hkkx fy;k x;k

gSA

⇒ ;fn ge ik¡p cjkcj Hkkxksa esa ls rhu Hkkx ij fopkj djrs gaS rks tks cprk gS oks nks Hkkx gSA

bls ge ,sls mYys[k dj ldrs gSa 52

⇒ ;fn ge 52vkSj

53 dks tksM+rs gS rks ge iwjh oLrq ;k la[;k ik tk;saxsA

⇒ ;fn ge ik¡p Hkkx esa ls rhu Hkkxksa ij fopkj dj jgs gS rks bldk eku ,d ;kfu iw.kZ ls

de gaSA

⇒ fHkUu esa tks la[;k js[kk ls mij gksrh gS mls va”k dgrs gS vkSj tks la[;k js[kk ds uhps gksrh

gS mls gj dgk tkrk gSA

⇒ D;k fopkj gS ge fHkUu 74 ls D;k eryc fudky ldrs gS\ viuh uksV cqd esa fy[kksA vius

fopkj dks vius nksLrksa ds lkFk lk>k vkSj ppkZ djksA



2-2 - f H k Uuk s a dk le wg

31

74

311

55

743

89

36

34

75

1213

1211

710

1257

1224

714

1212

77

28

• fdrus fHkUuksa dk eku ,d ls vf/kd gS ¼iw.kZ½ • fdrus dk eku ,d ls de gS\ • fdrus fHkUUk ,d nwljs ds cjkcj gSa\


!

13" !

!

viuh uksVcqd esa rhuksa lokyksa ds vk/kkj ij rhu dkye cukvksa vkSj fn, x, fHkUuksa dks rhu lewg esa

ckWVksaA vius mRrj dks vius fe= ds mRrj ls tkWp djks ;fn vkidh #fp gks rks dqN vkSj fHkUuksa dks

fy[kksa vkSj mUkds lkFk ;gh dke nksLrksa ls feydj djus dk iz;kl djksA



2-3 - lgh dk s fVd djuk

uhps dqN xf.krh; dFku fn, tk jgs gSaA lgh ds lkeus ^lgh* vkSj xyr ds lkeus ^xyr* “kCn

fy[kks&

• fHkUu 52 esa ;fn gj leku jgs vkSj va”k dks Øe”k% 3] 4] 5 ----- c<+k;k tk; rks fHkUu dk

eku ?kVsxkA

• fHkUu 52

esa ;fn va”k leku jgs vkSj gj dks Øe”k% 6] 7] 8 ----- c<+k;k tk; rks fHkUu dk eku leku

jgsxkA

• ;fn va”k vkSj gj nksuks leku gks rks fHkUu dk eku ,d ds cjkcj gksxkA

• fdlh Hkh la[;k dks fHkUu ds :i esa iznf”kZr dj ldrs gaSA • la[;k 5 dks fHkUu ds :i esa fy[ksa rks mldk gj 1 gksxkA

• 55 ,d fefJr fHkUu gSA

• vius mRrj dh vius fe= ds mRrj ls tk¡p djksA

• 74 !dks

52 ds LFkku ij j[kdj mijksDr iz”uksa ds mRrj nsus ds iz;kl djksA



2 -4 - vk Sj H k h rjhd s

fHkUu esa ,d dk eryc gS **,d iw.kZ oLrq ;k la[;k** nks eryc ,d gh eku ds nks iw.kZ oLrq ;k

la[;kA fuEu fHkUuksa dks ns[kksA mlls rqe D;k fopkj fudky ldrs gks\


!

14" !

!

.....48,

36,

242

.....44,

33,

221

=

=!

1- ,slh rhu vkSj fHkUu fy[kks tks 3 ds cjkcj gks&

2- Åij dh vo/kkj.kk ds vk/kkj ij fuEu iz”ukas ds tokc nsus dk iz;kl djksA

a) 522− ! b)!

5214− !! c)!

523+ ! ! d)

542

534 +

!bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&



2 -5 - Lkc fey s rk s ,d

31!dk D;k eryc gS\ gesa 15 dks rhu cjkcj Hkkx esa ckWVuk gS vkSj mlesa ls ,d Hkkx dks vyx

djuk gS] blfy, mRrj 5 gSA blh izdkj fuEufyf[kr lokyksa ds tokc Kkr djksA mRrj dk

fujh{k.k djksA fHkUuksa dks {kSfrt fy[ksA muds va”k vkSj gj dk fujh{k.k djsAA rqe vo/kkj.kk fudky

ldrs gksA

.............602010

.............60105

.............6063

.............6042

..............6021

=

=

=

=

=

of

of

of

of

of

gkW] ;fn ge leku la[;k ls va”k vkSj gj esa xq.kk djsa rks ge leku fHkUu ;k mlh fHkUu ds leku

eku dh nwljh fHkUu ik;ssaxsA

1- D;k va”k vkSj gj dks leku vad ls Hkkx nsus ij leku fHkUu izkIr gks ldrh gS\ 2- D;k ge leku fHkUu ikrs gS ;kfn ge leku la[;k dks fHkUu ds va”k vkSj gj ls tksMrs ;k

?kVkrs gaS\


!

15" !

!

3- D;k ge 53 ds leku 5 vkSj fHkUu fy[k ldrs gS\

bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&



2 -6 - dk Su Bhd \

Åij fn, x, fp= dks ysdj eqerkt vkSj r#.k us vius&vius rjhds ls fHkUu ds :i dks crk;kA

e qerkt% fHkUu tks nksuksa :iksa dks iznf”kZr djrh gS% 56 ;k

106

r:.k % ugh] rqe xyr gksA os nl Hkkxksaa esa foHkkftr gSaA mlesa ls 6 Hkkxksa dks jax fn;k x;k gS]

blfy, jaxs Hkkx ds fHkUu tks :i iznf”Zkr djrh gS og gS 106

vPNk vki gh crkb, &

1- dkSu lgh gS\ r:.k ;k eqerkt\ 2- vkids mRrj dk D;k dkj.k gS\ 3- D;k vkidk nksLr Hkh leku n`f’Vdks.k j[krk gS\

4- vki 311 dks fp= ls dSls iznf”kZr djsxsa\




2 -7 - tkn qb Z ox Z

QkmaMs”ku dSEi esa tknqbZ oxZ ds ckjs esa ckr gqbZ Fkh] D;k vkidks ;kn gS\ ;gkW oSLkk gh ,d nwljk

tknqbZ oxZ gSA ysfdu iz”u FkksM+k dfBu vkSj pqukSrhiw.kZ gSA D;k vius igys gh iz;kl esa bls lgh

iwjk dj ldrs gks\ fn, x, lokyksa ds vk/kkj ij lgh [kkus dks irk djks\


!

16" !

!

1 2 3

4 5 6

7 8 9

1- ;fn rqe bl la[;k esa 43tksM+rs gks rks rqEgsa 4 feysxkA

2- rhu vk/ks fgLls fey dj bls cukrs gSSA

3- ;fn rqe 3 esa ls 41?kVkrs gks rks rqe bl la[;k dks ikvksxsA

4- bldk leku fHkUu 510 gSA

5- ;fn rqe bl fefJr fHkUu dks vleku fHkUu esa cnyrs gks rks rqEgsa 25izkIr gksxkA

6- 12 ds vk/ks dk vk/kk\

7- bl la[;k dk nqxquk 214 gS

8- )412()

216( ×+×

9- bl la[;k vkSj 2 dk varj 41 gSA

10- la[;kvksa dks {kSfrt [kMh+ vkSj frjNk tksM+ksaA D;k lHkh mRrj leku gS\ D;k rqe ,d u;k

tknqbZ oxZ cuk ldrs gks\ vius nksLrks ls ppkZ djksA ;fn vkidks enn dh vko”;drk gS

rks —i;k vius v/;kid ls iwNasA bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&



2 -8 - U k kprh l a[;k, a & fQj l s !

D;k vkidks ukprh la[;k,a ;kn gS& Lrj ,d okyh xfrfof/kA ;gka tokc fn, x, gSa] mudkss /;ku

ls ns[kks fQj uhps fn, x, iz”uksa dk mRrj nksA

153846 538461 384615


!

17" !

!

846153 461538 615384

1- lHkh Ng la[;kvksa dks ,d ckj esa fcuk xyrh ds i<us dh dksf”k”k djksA vc bls fcuk ns[ks

cksydj nksgjkvksA fdrus iz;klksa esa ;g dj ik,\ 2- fdl la[;k esa 1 dk LFkkuh; eku gtkj gS\ 3- vadksa ds LFkku dk eku dc c<+rk gS\ 4- vkSj ,sls gh 10 loky cukvksa ftlds mRrj bu la[;kvksa esa fNis gSa\




2 -9 - vk/ k kj g S 10

uhps fn, x, la[kkvksa ds iSVuZ dks /;ku ls nsf[k,&

1

10

100

1000

10000

100000

gka] tc la[;k esa dksbZ vad vius LFkku ls ck;ha vksj tkrk gS rks mldk eku c< tkrk gS &

• vius LFkku ls ,d vad ck;ha vksj tkus ij mldk eku fdrus xquk c<rk gS\

• nk;ha vksj ,d LFkku c<us ij bldk eku fdrus xquk de gksrk gS\

• la[;kvksa dk LFkkuh; eku 10 ls fdl izdkj lacaf/kr gS\

bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks& 1- vkidks ;g djrs gq, dSlk yxk\


2 -10 - bdkb Z ] l a[;k vk S j v ad

la[;k 25067 esa fdruh bdkbZ;ka gS\

⇒ ;fn rqEgkjk mRrj 7 gS rks ;g xyr gSA lgh mRrj gS iPphl gtkj ljlB bdkbZ;kaA


!

18" !

!

⇒ ij rqEgkj tokc 7 Hkh fdlh loky dk mRRkj gS\ mRRkj 7 ikus ds fy, D;k gksxk\

⇒ ;fn rqe iwNrs gks** la[;k 25067 ds bdkbZ ds LFkku dk vad D;k gS\ rqe tokc ds :i esa

7 ikvksxsA

;gka dqN vkSj loky fn, x, gSa] mudks le>ks vkSj mRrj nksA

1- la[;k 25067 esas fdrus lSdM+s lfEefyr gSa\ 2- la[;k 25067 esas fdrus gtkj gSa\ 3- la[;k esa ngkbZ dh la[;k Kkr djusa dk lcls ljy rjhdk D;k gS\ 4- fdrus lSdMk feydj djksM+ cukrs gSa\ 5- ;fn fdlh la[;k essa 634 lSdM+k gS rks mlesa fdrus gtkj gksxsa\ 6- dqN vkSj iz”u cukus dk iz;kl djks vkSj mldk mRrj vius fe=ksa ls iwNks




2 -11 - nk su k a a s vk s j foLrkj

10000 1000

)1010000( ÷ 100

)101000( ÷ 10

)10100( ÷ 1

)1010( ÷ 101

)101( ÷ 1001

)1001( ÷

Åij nh xbZ lkfj.kh esa ,d iSVuZ gSA la[;k 1 okys dkye ds nksuksa vksj ,d [kkl Øe esa la[;kvksa

dk fOkLrkj gSA bl vk/kkj ij uhps fn, x, lokyksa dk mRrj nksA

1- rqe bls nksukas rjQ fdruk foLrkj dj ldrs gks\ 2- D;k ;g vkidks fiNys xfrfof/k esa iwNs x;s iz”uksa ds tokc ikus esa enn djrk gS\




2 -12 - Tkc LF k ku cnyrk g S !

ge tkurs gS fd la[;k esa vad dk LFkku cnyus ls bldk eku cny tkrk gSA ,d vad fofHkUu

inksa esa vyx&vyx vFkZ iznf”kZr djrk gSA 5 dh la[;k fn[kkbZ ns jgk gS mlds fofHkUu inksa dks

ns[kksA blds ckjs esa lkspks vkSj iz”u ds mRrj irk djus dk iz;kl djksA

5 51

65

65× 57× 52 ο5 -5 .5 +5


!

19" !

!

1- izR;sd la[;k esa ik¡p ds ckjs esa D;k irk pyrk gS\

2- muesa ls dkSu 105

ds cjkcj gS\

3- rqe 1005

dks n”keyo :i esa dSls fy[kksxs\

4- D;k rqe dksbZ vU; LFkku lksp ldrs gks ftlesa ik¡p dks LFkku fn;k tk lds\ bl dke ds vk/kkj ij uhps fn, x, lokyksa dk mRrj fy[kks&



2 -13 - fcUn q ;k v ad

ckrphr dks i<+ks vkSj le>ks vkSj uhps fn, x, lokyksa ds mRrj fy[kksA

fouk sn % ;fn rqe fdlh n”keyo la[;k dks 100 ls xq.kk djuk pkgrs gks rks fcUnq dks nks LFkku

nkfguh rjg ys tkvkasA 2.5743100432.57 =× lk su h % rqe xyr gksA ;g fcUnq ugha tks ?kwerk gSA fcUnq mlh txg jgrk gSA ;g rks la[;k dk

LFkkuh; eku gS tks cnyrk gSA ;fn rqe 100 ls xq.kk djrs gks rks ikp¡ ngkbZ feydj ik¡p gtkj gks

tkrs gSA lkr bdkbZ lkr lkS cu tkrk gSA

fouk sn % nwljs la[;kvksa esa D;k cnyko gksrk gS\

lk su h % eq>s ,d feuV lkspus nks\\

1- fdldk rØ lgh gS\ 2- D;k fcUnq cnyrk gS\ 3- lksuh us dSls O;k[;k fn;k fd la[;k dk eku cnyrk gS tks bdkbZ ;k nlosa LFkku ij gksrh

gSa\ 4- ;fn ge 57-432 dks 100 ls Hkkx nsa rks D;k mRrj izkIr gksxk\ 5- rqe bls dSls O;k[kk djksxs\




2 -14 - thjk s g S ghj k s ! bl la[;k 00500-01020 ds vk/kkj ij uhps fn, x, lokyksa dk mRrj D;k gksxk\

1- dkSu lk “kwU; la[;k dk eku c<+krk gS\ 2- vU; “kwU; D;ksa ugh\


!

20" !

!

3- D;k rqe oks rhu LFkku crk ldrs gks tgka “kwU; ghjks gksrk gS\ 2 -15 - i z fr”kr & 100 gj okyh f H k U U k l a[;k

^^Ikzfr”kr 100 gj ds lkFk ,d fHkUu gSA** D;k ;g dFku lgh gS\ bls dqN mnkgj.kksa ls irk djksA

rc fuEufyf[kr iz;kl djks

1- dqN izfr”kr la[;k,¡ yks vkSj mUgsa fHkUu esa cnyksA

2- blh rjg dqN fHkUu la[;kvksa dks izfr”kr eas cnyksA

Ikzfr”kr dks fHkUu esa cnyuk vkSj fHkUu dks izfr”kr eas cnyus ds fy, ge ,d vo/kkj.kk dks ykxw

djrs gS tks ge bl Lovf/kxe lkexzh esa fn;s xfrfof/k esa lh[k pqds gSaA



2 -16 - pkj vFk Z

uhps nh xbZ rkfydk esa [kkyh LFkku HkjksA vo/kkj.kkvksa ds chp lacU/k irk djks vkSj viuh dkih esa

fy[kksA

Fraction

21

41

...........

...........

411

..................

Percentage

%5010050

502501

==×

×

............................... ............................. 20% ......................... ...........................

Decimal

5.50.10050

==

................... .3 ..................... ................... ...................

Ratio 1: 2 ............. ............ ........... ......... .................





!

21" !

!

2 -17 - D;k l ar qyu cjkcj g S !

lkSjHk cktkj ls dqN lfCt;k¡ [kjhnus x;kA nqdkunkj us ,d fdyks I;kt rkSykA D;k nqdkunkj us

Hkkj dk lgh x.kuk fd;k\ D;k larqyu lgh gS\ lkSjHk dks “kd FkkA bls dSls tk¡ps\ mlus ,d

feuV lkspk vkSj ,d fopkj mlds eu esa vk;kA mlus nqdkunkj ls I;kt vkSj ck¡V ds rjktw dks

cnyus ds fy, dgkA

vc uhps fn, x, lokyksa dk mRrj nksA

1- lkSjHk us ,slk D;ksa dgk\

2- vki larqyu ds ckjs esa D;k dg ldrs gS\

3- rjktw dh lqbZ vkSj xf.kr dk ladsr cjkcj gSA D;k rqe lger gks\ dSls\

4- D;k rqEgsa fx¶V ckDl dk mnkgj.k ;kn gS ftlds ckjs esa dSEi esa ppkZ gqbZ Fkh\ og blls

dSls tqM+rk gS\

5- D;k rqe bl vo/kkj.kk dks ,d mnkgj.k ftlesa (3+ 2 = 5) la[;k dk iz;ksx gks O;k[;k

dj ldrs gks



2 -18 - eku yk s !

eku yks Ο=Δ rks uhps fn, x, dFkuksa esa ls dkSu lk dFku lgh gS\

a) )32(4 ++Ο=+Δ

b) 21

2×Ο=

Δ

c) 2%50 ×Ο=×Δ


!

22" !

!

d) )21

21(

4××Ο=

Δ

e) 15)10(5 −+Ο=−Δ

f) 105

21

+Ο=+Δ

g) Ο×=×Δ 55 h) Ο−=−Δ 55

1- mRrj ds ckjs esa nksLrksa ds lkFk ppkZ djksA

2- ;fn rqe viuk mRrj tk¡puk pkgrs gks rks rqe D;k djksxs\



2 -19 - chtxf.krh; dFku

Ckhtxf.kr ,oa xf.kr dh Hkk’kk ds ckjs esa ;kn djks ftlds ckjs esa dSEi esa ppkZ dh xbZ FkhA ;gk¡

dqN dFku gSaA bUgsa chtxf.krh; okD;ksa esa cnyus dk iz;kl djksA

• v** dks pkj ckj tksM+uk

• ^v* dks pkj ls xq.kk djks

• pkj ckj **v**

• ,d la[;k dk rhu xquk ,oa nwljs la[;k dk pkj xquk 24 gksrk gSA

• fdlh la[;k dk ,d frgkbZ 5 gS

• fdlh la[;k ds pkj xquk esa 5 tksM+us ij 30 feyrk gS

• la[;k dk vk/kk vkSj mldk ,d frgkbZ feydj 5 gksrk gSA

• ,sls ikap okD; vkSj cukdj mudks chtxfr.kh; okD; eas cnyksA




2 -20 - xf.kr dFkuk s a dh bckjr

fn, x, mnkgj.k dh rjg chtxf.krh; dFku dks bckjrh ¼”kkfCnd½ dFku esa cnyksA

aa =×1 : fdlh la[;k esa 1 ls xq.kk djus ij xq.kuQy og la[;k gh gksrh gSA


!

23" !

!

1) aa =÷1 2) aaaaa 4=+++ 3) acabcba +=+ )(

vc dqN lw= dks ;kn djks vkSj bldks “kkfCnd dFku esa cnyksA rc dFku dks vius nksLrksa ls

i<+okvksA fQj bu xf.krh; lw=ksa dks dkih eas fy[kksA




2 -21 - eku D;k \

uhps fn, x, chtxf.krh; dFkuksa dk Hkk’kkbZ dFkuksa esa cnydj fy[kksA fQj gj ,d ^a’!esa dk eku

Kkr djksA

1) 295 =+a

2) 152=

a

3) 362 =a

4) 752=+

aa

5) 433)5( =+×a nksLrksa ls ppkZ djks&

1- vki vius mRrj dh tk¡p dSls djsxsa\

2- 1 vkSj 0 dh le> dSls vkidks bu leL;kvksa dks lqy>kus es enn djrk gS\




2 -22 - eku dh vnyk&cnyh

;fn a = 4 vkSj b = 6 rks uhps fn, x, chtxf.krh; dFkuksa dk eku irk dhft,A vkidh enn ds

fy, ,d mnkgj.k fn;k x;k gSA

1064 =+=+ ba 1) ...........2 =+ ba

2) ........)21()

41( =×+× ba

3) .........=ab 4) ......2)43( =−+ ba


!

24" !

!

vc bUgsa Hkh iwjk djks&

1- dqN vU; chtxf.krh; dFku cukvks vkSj vius nksLrksa ls cnydj gy djksA

2- pjksa ds fy, vU; eku nksA

3- pj dk eku fHkUu la[;k ekudj Hkh ,sls dFku cukvks vkSj gy djksA



2 -23 - X k f. kr dh H k k k " k k (i z k st sDV dk; Z)

xf.kr la[;kvksa ,o izrhdks dh Hkk’kk gS A Hkk’kk esa tSls&TkSls vkidk “kCn HkaMkj c<+sxk vkidh

okrkZyki “kSyh lq/kjsxhA blh izdkj tSls&tSls ge xf.krh; ladsrksa ds izfr tkx:d gksrs gaS] ge

xf.kr dh vo/kkj.kkvksas dks Li’V ,oa vklkuh ls le> ldrs gSaA

uhps dqN dk;Z crk, x, gSa muds vuqlkj ;g izkstsDV iwjk gks ldrk gSA ;g dk;Z djus esa ,d

lIrkg yxsaxs] ;fn 30 ls 40 feuV jkst yxkvksA

1- xf.krh; ladsrksa dks ;kn djsa vkSj fy[ksa] tSls& etc...,, 〈÷× !

2- vius nksLrksa dks viuh fyLV fn[kkvks vkSj ns[kks rqe D;k Hkwy x, Fks\ !3- vc d{kk 7] 8 vkSj 9 dh ikB~;iqLrd ns[kks] D;k vkSj Hkh dqN ladsr gSa] mUgsa Hkh viuh

fyLV esa tksMksA !4- vc ;g Hkh fy[kks fd fdl ladsr dk mi;ksx fdlfy, gksrk gS vkSj mldk eryc D;k gS\!5- vc viuh QkbZuy fyLV dks Vhpj ls lk>k djksA !



2 -24 - rc D;k gk sr k\

cgqr xehZ gSA rkieku 43 fMxzh rd igq¡p x;kA vki dejs esa cSBs ilhuk cgk jgs gSaA vki bl

fLFkfr ds ckjs esa D;k lkspsaxs\ fuf”pr :i ls BaMh gokvksa ds ckjs esaA tc ge fdlh pht dks [kksrs

gSa rc gesa mlds ewY; dk irk pyrk gSA

blh rjg dh fLFkfr dh dYiuk djks vkSj vius fopkjksa ds ckjs esa vius fe= ds lkFk ppkZ djksA

1- ;fn gekjh la[;k iz.kkyh esa “kwU; ugha gksrk gesa D;k eqf”dy vkrh\ lkspks vkSj fyLV

cukvksA


!

25" !

!

2- ;fn udkjkRed la[;k,a u gksrha] rks ge xf.kr eas fdu fopkjksa dk O;Dr ugha dj ikrs\




2 -25 - ? k kV s e s a ? k kVk

D;k rqEgsa Lrj 1 dh xfrfof/k 1.2 ;kn gSA ml xfrfof/k esa geus dqN “kCnksa dks pkj vk/kkjHkwr

lafØ;kvksa ds vk/kkj ij lewg esa ck¡Vk FkkA vc dqN fLFkfr;ksa ij fopkj djksA D;k rqe bUgs

ldkjkRed ;k udkjkRed Js.kh esa ckWV ldrs gks\

1- iznhi ds ikl 10000 :i;s gSA ysfdu mlus bls vius fe= eksbu ls m/kkj fy;k

(+10000/ - 10000) 2- mlus NksVk lk O;kikj fd;k vkSj 2000 #i;s dek;sA (-2000/+2000) 3- mlus yksu ds fgLls ds Hkqxrku ds fy, 2000 :i;s iznhi dks fn;sA (-2000/+2000) 4- vxys o’kZ mlus vius O;kikj dks c<+kus dh lksphA mlus cSad ls 20000 dk yksu fy;kA 5- ml lky iznhi us 12000 dk ykHk dek;kA (-12000/+12000) 6- mlus cSad dks nl gtkj vkSj eksbu dks nks gtkj :i;s fn;sA

vc bu lokyksa ds mRrj crkvks& v- iznhi us fdruk #i;k m/kkj fy;k\

c- mlus fdruk ykHk dek;k\

l- orZeku esa mls yksu pqdkus ds fy, vkSj fdruk dekuk gksxk\


3-1 -1 - , slk D;k s gk sr k g S\

xf.kr dh d{kk esa lewg esa dk;Z gks jgk FkkA vafdr vkSj jsgkuk ,d lewg esa FksA mUgsa fHkUu la[;k esa

fHkUu dk xq.kk djuk FkkA

v a fdr% fHkUu ls fHkUu dk xq.kk djus ds fy, va”k vkSj gj nksuksa esa xq.kk djuk pkfg,A

j sg kuk % D;k ;g lgh gS\ rqe dSls tkurs gks\ D;k rqe bls fl) dj ldrs gks\

v a fdr% vPNk! rqeus <sjkas iz”u iwN fy;kA eq>s iz;kl djus nksA

vafdr us fy[kuk vkjEHk fd;kA

a

221

21

21

21

=+++

2214 =× (xq.kk] tksM dk gh laf{kIr :i gSA) !


!

26" !

!

2.............21

28

==× (4 ds LFkku ij mlds leku fHkUu la[;k j[kus ij½

j s s g kuk % eq>s [kkyh LFkku Hkjus nksA

mlus ,d {k.k lkspk vkSj viuh uksVcqd esa fy[kk 248

2218

21

28

==×

×=×

j sg kuk % D;k ;g lgh gS\

vafdr% fcydqy Bhd! D;k rqe xq.kd ds fy, mlds leku mlds leku dksbZ vkSj fHkUu j[kdj ;g

xq.k dj ldrh gksA jsgkuk] D;k rqe 21ds LFkku ij

63dk j[kd ;g gy dj ldrh gks\

jsgkuk% eq>s iz;kl djus nksA vkSj mlus gy djuk “kq: fd;kA

21224

6238

63

28

2..........63

28

==××

=×

==×

v a fdr% cgqr vPNk! Rkqe le> xbZA

j sg kuk % ysfdu vafdr] tc ge xq.kk djrs gS] lkekU;r% xq.kuQy] xq.kd vkSj xq.; ls cM+k gksxkA

D;k ;gka Hkh ,slk gh gS\ v a fdr% **izk;% gksrk gS ij gj le; ughaA bl xq.kuQy dks ns[kks] rqe Bhd ls le> tkvksxh eSa

D;k dguk pkgrk gw¡A

mlus jsgkuk dh uksV cqd esa xq.kuQy ds dqN rF; fy[ks&

1414

2214

4148241234

=×

=×

=×

=×

=×

jsgkuk gka] eSa le> xbZA ;fn xq.kd ,d ls de gS] rks xq.kuQy xq.; ls de gksxkA D;k eSa lgh gw¡\

vafdr% gka rqe Bhd dg jgh gksA

vc x q. k k dju s dh dk s f”k”k djk s& 1- ,d lefHkUu ,oa fo’ke fHkUu dk

2- ,d le fHkUu vkSj ,d fefJr fHkUu dk

3- nks vleku fHkUu

4- nks feJr fHkUu

vc bu lHkh esa xq.kd ds LFkku ij mlds leku dksbZ nwljh fHkUu la[;k j[kdj ;gh dk;Z djks]

vkSj vius mRrj dh tkap Hkh djksA


!

27" !

!

3 -1 -2 - D;k r qe fl) dj ldr s gk s !

;fn rqe fHkUu la[;k dks fHkUu ls Hkkx nsuk pkgrs gks rks bls nwljs fHkUu ds O;qRØe ¼myVs½ fHkUu ls

xq.kk djuk gksrk gSA rqe bl fu;e dks tkurs gksA ysfdu D;k bls dHkh VsLV djus dk iz;kl

fd;k\ rqe bls dSls fl) dj ldrs gks\ ;gkW ,d mnkgj.k gSA bls /;ku ls ns[kks vkSj le>ks!

⇒ igys bls ge xf.kr dh Hkk’kk esa j[krs gaSA

cd

ba

dc

ba

×=÷

⇒ vc vafdr ds crk, x, rjhds ls dks viukrs gSa &

⇒ ,d xf.krh; dFku dk p;u djks] tSls fd 6213 =÷ . ¼;gka rhu esa N% vk/ks fgLls gSa ;k N%

vk/ks fgLls feydj rhu cukrs gaS½

⇒ vc 3 ds LFkku ij mlds leku nwljh fHkUu la[;k j[kks 621

412

=÷

⇒ vc 21 ds O;qRØe ls xq.kk djks

424

12

412

=×

⇒ leku fHkUu ikus ds fy, va”k vkSj gj dks 4 ls Hkkx nks 644424=

÷÷

⇒ vc 21 ds LFkku ij blds leku nwljh dksbZ fHkUu j[kks 6...........

105

412

==÷

⇒ vc 21 ds O;qRØe fHkUu ls xq.kk djks

20120

510

412

=×

⇒ ;g gS rqEgkjs loky dk gy 616

202020120

==÷

÷

;gk¡ dqN fHkUu la[;k,a nh xbZ gSaA bu fHkUu la[;kvksa dks fdlh nwljh fHkUu la[;k ls Hkkx nksA Hkkx

ds fy, vyx&vyx rjg ds fHkUuksa dks pquksA mlds ckn ,d fHkUu pquks] bls leku fHkUu esa cnyks

vkSj fQj ls Hkkx djds ns[kksA vius mRrj dh tkap t:j djrs jgksA

416

75

1012

89

1512

653

vc fdlh fefJr fHkUu dks vleku fHkUu esa cnyks vkSj fQj Hkkx nksA vius mRrj dh tkWp Hkh

djksA


!

28" !

!

3 -1 -3 - nk s f H k Uuk s a d s chp dh fH k Uu

D;k rqe 31ls cM+h vkSj

21ls NksVh fHkUu fy[k ldrs gks\

^^leku fHkUu** vo/kkj.kk dh le> ls rqEgas blesa enn feysxhA ftlds ckjs esa rqe igys gh le>

pqds gksA D;k rqEgsa ;kn gS] dgka vkSj dc\

3 -2 -1 - n”keyo d s lkF k l a fØ;k, a ,d fHkUu la[;k dks izfr”kr] n”keyo vkSj vuqikr esa cnyk tk ldrk gSA ¼Lrj 2 esa ns[ksasa½ blfy, n”keyo la[;kvksa ds xq.kk ds fy, mUgsa fHkUu esa cnydj xq.kk fd;k tk ldrk gSA ns[ksa

mnkgj.k& 1.10.10010

102

1052.5. ===×=×

v- uhps fn, x, lokyksa dks ,sls gh xq.kk djds mRrj ns[kksA fQj mUgsa fcuk fHkUu esa cnys xq.kk

djus dk iz;kl djksA

002.005.02.005.002.05.02.05.2.05.

×

×

×

×

×

c- uhps nh xbZ n”keyo la[;kvksa ds Hkkx ds fy, leku rjhds dks viukvksA

03.015.03.15.315.3.5.135.15315

÷

÷

÷

÷

÷

=÷

igys ,d ;k nks n”keyo la[;kvksa dks fHkUu esa cnyks rc Hkkx nksA mRrj ikus ds ckn HkkxQy dh

tkap djus dk iz;kl djksA

l- D;k rqeus bu lokyksa ds fy, dqN j.kuhfr fudkyh\ mUgssa viuh dkih ij fy[kuk ,oa vius

fe= ds lkFk ppkZ djuk u HkwyukA


!

29" !

!

3 -2 -2 - n”keyo l[;kvk s a dk tk sM + ,o a ? kVkuk

n”keyo la[;kvksa dk tksM+&?kVkuk Hkh lkekU; la[;kvksa ds ftruk gh ljy gSA uhps fn, x, pj.kksa

dks viukrs gq, ,slk djks] rqe [kqn gh blds rjhds le> tkvksxs&

⇒ dqN n”keyo la[;kvksa dks fy[kks ...)007.0,3.3,45.0,02.1,5(.

⇒ muesa ls dksbZ nks la[;k,a pquks 02.1,5. ⇒ mUgsa vadksa ds LFkkuh; eku ds vuqlkj fy[kksA ;fn ,slk vklkuh ls ugha dj ik jgs gks rks

uhps fn, x, ckDl tSls ,d LFkkuh; eku ckDl cuk yksA vkSj mlesa lgh dkye esa

la[;kvkas dks fy[k yksA tSlk fd 0-5 vkSj 1-02 fy[kk x;k gSA vc rqe fdlh Hkh la[;k dk

LFkkuh; eku vklkuh ls tku ikvksxsA

⇒ rc la[;kvksa dks tksM+ks A dbZ ckj ,slk vH;kl gksus ds ckn rqe fcuk LFkkuh; ckDl ds Hkh

;g dk;Z dj ikvksxsA

v)- D;k ?kVkuk Hkh ,sls gh gks ldrk gS\ vkt+ek dj ns[kks] cl ,d dke djuk gksxk cMh la[;k

¼ftlesa ls ?kVk;k tkuk gS½ dks ckDl esa Åij fy[kuk gksxkA vkSj ?kVus okyh la[;k dks mlds uhps

okyh iafDr esaA

c)- ,d vk Sj vH;kl% dqN vkSj n”keyo la[;kvksa dks fy[k ysaA vc vuqeku djsa fd buesa ls

dkSu lh nks ;k rhu ,slh la[;k,a gksaxha ftudk tksM 50 vkSj 60 ds chp gks ldrk gSA vc tksM+dj

nsf[k,] vki mRrj ds fdrus djhc gSa\ ,slk dbZ ckj djds nsf[k, vki ik;saxs fd vkidk vuqeku

vlyh mRrj ds djhc gksrk tk jgk gSA

3 -3 -1 - x q. kt vk S j vioR; Z

vkidks igkM+k ;kn gS] uhps nsf[k, ;gka 2] 3] 4] vkSj ikap ds igkMs+ gh gSa !

2, 4, 6, 8, 10, 12, 14... 3, 6, 9, 12, 15, 18, 21... 4, 8, 121, 16, 20, 24, 28... 5, 10, 15, 20, 25, 30, 35...

bdkbZ 101

1001

5

1 0 2


!

30" !

!

gj ,d iafDr esa fy[kh la[;k,a ml iafDr dh igyh la[;k ds xq.kt gSaA ;s lHkh la[;k,a 2]3]4]5] dh

xq.kt gSaA budks D;k vkSj dqN Hkh dgrs gSa\ bUgsa vkSj Bhd rjhds ls dSls dg ldrs gSa\

uhps fn, x, xf.krh; dFkuksa dks ns[kksA le>us esa eqf”dy gks jgh gks rks vius vklikl fdlh ls

ppkZ djks] ;k fiNyh d{kk dh ikB~;iqLrdksa dks Hkh ns[k ldrs gksA ge bl ckjs esa vkSj fdrus vkSj

fdl izdkj ds dFku cuk ldrs gS\

⇒ ;fn abba =× rks “ab”, “a” vkSj “b” dk xq.kuQy gksxkA

⇒ “ab” dks “a” vkSj “b” ls fcuk “ks’kQy ds Hkkx ns ldrs gSA

⇒ “a” vkSj “b” dks “ab” dk vioR;Z dgk tkrk gSA

⇒ fdlh Hkh xq.kuQy esa nks ;k mlls vf/kd xq.kt ¼vioR;Z½ gksrs gSaA

⇒ fos”ks’k iz—fr dh la[;kvksa dk xq.kuQy Hkh fo”ks’k gksrk gSA ¼le la[;kvksa dk xq.kuQy \½

⇒ 2 ds lHkh xq.kt le gksaxsA

vc uhp s fn, x, i z”uk s a d s mRrj nk sA 1- D;k 4 ds lHkh vioR;Z le la[;k,a gS\

2- 1 dks dkeu QSDVj (lekioR;Z) D;kas dgrs gaS\

3- D;k 4 ds lHkh vioR;Z 2 ds Hkh vioR;Z gS\

4- D;k 10 ds lHkh vioR;Z 2 vkSj 5 ds Hkh vioR;Z gS\

5- 1] 2 ]3] 4] 5] vkSj 6 esa ls dkSu 12 dk viorZd ugh gSA

6- D;k 12] 12 dk vioR;Z ;k viorZad gS ;k nksuksa gS\

3 -3 -2 - H k kT; l a[;k ,o a vHk kT; x q. ku[k.M

tSlk fd rqe tkurs gks vHkkT; la[;kvksa ds nks xq.ku[k.M gksrs gaS% 1 vkSj Lo;a og la[;kA

mnkgj.k ds fy, 17 dks ge fcuk “ks’kQy ds dsoy 1 vkSj 17 ls foHkkftr dj ldrs gSA blhfy,

17 dks vHkkT; la[;k dgrs gSA

v- vc bu iz”uksa ds mRrj nsus dk iz;kl djksA

1- lcls NksVh vHkkT; la[;k dkSu gS\

2- D;k le la[;k,W vHkkT; la[;k gksrh gS\

3- 1 vkSj 100 ds chp esa fdruh vHkkT; la[;k,W gS\ 25 ;k 26\

4- D;k 1 vHkkT; la[;k gS\ vius mRrj dk dkj.k crkvksA


!

31" !

!

c- vc uhps fn, x, xf.krh; dFkuksa ij /;ku nks vkSj uhps fn, x, lokyksa ds mRrj nksA

7139132222296

33222725522100352260

32212

×=

×××××=

××××=

×××=

×××=

××=

1- vioR;kZs ds D;k fo”ks’k xq.k gksrs gSa\

2- bl izfØ;k dks D;k dgrs gSa\ D;k bldk dksbZ [kkl uke Hkh gS\

3- gesa ;g lh[kus dh D;ksa vko”;drk gS\

3 -3 -3 - ok fr Zd ;k bckjrh i z”u

uhps fn, x, bckjrh lokyksa dks i<ks] le>ks vkSj gy djksA bu okfrZd iz”uksa ds mRrj irk djksA

D;k rqEgs y/kqRke lekioR;Z dh vo/kkj.kk irk gS] rc rks ;s loky rqEgkjs fy, fcydqy gh vklku

gSa!

v- jatw ikWp fnu esa ,d ckj iqLrdky; tkrh gSA latw pkj fnuksa esa ,d ckj iqLrdky; tkrh

gSA vkt os iqLrdky; esa feysA os vxyh ckj dc feysaxs\

c- nknh ekW Vksdjh esa vke ys tk jgh FkhA ,d lkbfdy lokj us mUgsa VDdj ekj nhA Vksdjh

fxj iM+hA nksuks us vke dks bdV~Bk fd;kA vkneh us pqipki vkeksa dh fxurh dh vkSj

iwNk& rqEgkjh Vksdjh esa fdrus vke Fks\ nknh us dgk& eSa ugha tkurh ysfdu ;fn eS bls 6

ds lewg esa j[kwa rks blesa 5 cp tk;saxsA vkSj ;fn eS bls 5 ds lewg esa j[krh gwa rks pkj cp

tkrs gSaA ;fn bls eS pkj ds lewg esa j[krh gwW rks 3 cpsaxsA vkneh us lkspk vkSj cksyk&

geus lkjs vke bdV~Bs dj fy, gSA vc rks crkvks] Vksdjh esa fdrus vke Fks\ rqe rks crk

gh nksxs] rks tYnh crkvks& nknh dh Vksdjh eas fdrus vke Fks\

l- vkeksa dh D;k la[;k gksxh ;fn nknh ekW dgrh& ;fn eS bls 6 ds lewg esa j[kwW rks 4 cp

tkrs gaSA 5&5 ds lewg cukÅa rks 3 cp tkrs gS vkSj ;fn 4 dss lewg esa j[kwa rks 2 cprs gSa

vkSj 3 ds lewg esa j[kus ij 1 vke cp tkrk gSA

n- Vksdjh esa fdrus vke gksxsa ;fn nknh ekW dgrh& ;fn eSa lHkh vkeksa dks 6] 5] 4 ] 3 vkSj 2

ds lewg esa j[kwa rks 1 vke cprk gSA


!

32" !

!

3 -3 -4 - y?k qre lekior Zd (y-l-) vk S j egRre lekior Zd (e-l-)

60 vkSj 40 dk y- l- = 12032522 =×××× ftldk eryc gS fd 120 og NksVh ls NksVh la[;k gS tks 60 vkSj 40 ls iw.kZr;k foHkkftr gSA

60 vkSj 40 dk e- l = 20522 =×× ftldk eryc gS fd 20 og cM+h ls cM+h la[;k gS ftlls 60 vkSj 40 nksuks dks fcuk “ks’kQy ds

foHkkftr dj ldrs gksA

;gkW ,d vkSj rjhdk gS

rqEgkjs fy, dkSu lk rjhdk vklku gS\

v- ;gkW dqN la[;k,W nh x;h gSA muds y- l- vkSj e- l- nksuks rjhdksa ls Kkr djksA 48, 50, 75, 120, 96, 80, 150, 160...

c- nks ;k rhu la[;kvkas dks pquksA blds y- l- vkSj e- l- dks dk vuqeku djks] gy djks fQj vius

vuqeku ls feyku djksA


!

33" !

!

3 -3 -5 - D;k tku ik, \

v)- nks Øekxr la[;kvksa dks pquks] tSls 6] 7 ;k 20] 21- vc mudk y- l- vkSj e- l- Kkr djksA

⇒ rqeus blls D;k le>k\

c)- nks vHkkT; la[;k,a pquks] tSls 13] 17 ;k 19] 29 bR;kfnA vc mudk y- l- vkSj e- l- Kkr

djksA ⇒ rqeus blls D;k le>k\

l)- nks la[;k,¡ yksA mudk y- l- vkSj e- l- Kkr djksA vc mu nksuksa la[;kvksa dk xq.kuQy

Kkr djksA muds y- l- vkSj e- l- dk xq.kuQy Kkr djksA

⇒ nksuks xq.kuQy dks ns[kksA D;k ikrs gks\

l)- nks la[;kvksa dk e- l- gS 3

y- l- gS 120 ,d la[;k gS 24

nwljh la[;k D;k gksxh\

;fn rqe bl leL;k dks gy u dj ikvksa rks nks ;k rhu la[;kvksa dk tksM+k pquksA mudk y- l-

vkSj e- l- irk djksA vc y-l- vkSj e- l- dk xq.kuQy Kkr djksA la[;kvksa ds chp ds laca/kksa dks

Kkr djksA vc irk djks& D;k ;s lEca/k fdUgha Hkh nks ;k rhu la[;kvksa ds lsV ij ykxw gksrs gSaA ;k

lHkh l[;kvksa ds lkFk\ vc rks vkidks lqfuf”pr gks tkuk pkfg, fd vkidk gy lgh gSA

3 -3 -6 - foH k kT;rk d s fu;e

D;k vki 1001 ds xq.ku[k.M dks Kkr dj ldrs gSa\ ¼;k ,slh la[;k ftlls ge mls fcuk “ks’kQy

ds iwjk dk iwjk foHkkftr dj ik,aA½ ;g xfrfof/k ls vki la[;kvksa ds foHkkftr gksus ds fu;e dks tYnh gh le> tk;saxsA blls tYnh

ls irk djuk vklku gks tk,xk fd dksbZ la[;k iwjh iwjh foHkkT; gS ;k ugha\

10] 20] 30] 40] 50 --------;s 10 ds xq.kt gSA ;g iSVuZ Li’V gSA bdkbZ ds LFkku ij “kwU; la[;k gSA

blfy, ;fn fdlh la[;k ds bdkbZ ds LFkku ij “kwU; gksxk rks og la[;k 10 ls foHkkT; gks ldrh

gSA

vc fuEufyf[kr vioR;ksZa ij /;ku nks vkSj la[;kvksa dh foHkkT;rk ds fu;e cukvksA

5, 10, 15, 20, 25, 30...

2, 4, 6, 8, 10, 12, 14...


!

34" !

!

3, 6, 9, 12, 15, 18, 21...

9, 18, 27, 36, 45, 54...

rqe dqN vkSj fu;e Hkh cuk ldrs gksA ;gka ,d ladsr gS&

3 vkSj 9 ds vioR;ksaZ ds vadksa dks tksM+ksA ¼3 ds vioR;Z& 1$2] 1$5] 2$1 vkSj 9 ds& 1$8] 2+7½ D;k dksbZ iSVuZ feyrk gS\ vc fy[kksa ,slh la[;kvksa dh foHkkT;rk ds D;k fu;e gksaxs\

0] 121] 132] 143] 154----------- ds fu;e cukus ds fy, 11 ds vioR;ksZa ds Øfed vadksa dk tksM+ Kkr

djksA foHkkT;rk fu;eksa dks fy[kuk u HkwyukA

vc] fuEufyf[kr iz”uksa ds mRrj nks&

1- D;k 3 dk vioR;Z 1456 gS\ 2- bls 3 dk vioR;Z cukus ds fy, rqe blesa fdruk vkSj tksM+ksxs\ 3- bls 3 dk vioR;Z cukus ds fy, rqe blesa ls fdruk ?kVkvksxsa\ 4- pkj ;k ik¡p rhu vadh; la[;kvksa dks fy[kksA bls 4 ls xq.kk djksA blds bdkbZ vkSj ngkbZ

ds LFkku ds vadksa dk fujh{k.k djksA rqeus D;k ik;k\ 3 -4 -1 - i w. k k Z ad k s a d s lkF k l a fØ;k, a

(v)- bu dFkuksa dks iwjk djks&

1- nks /kukRed la[;kvksa dk ;ksx-----------------------------------------------------------------------------------------------------------

2- nks xq.kkRed la[;kvksa dk ;ksx----------------------------------------------------------------------------------------------------------

3- ,d xq.kkRed vkSj ,d /kukRed la[;k dk ;ksx-------------------------------------------------------------------------

4- nks /kukRed la[;kvksa dk vUrj------------------------------------------------------------------------------------------------------------

(c)- Åij dh rjg xq.kk vkSj Hkkx dh fØ;kvksa ds fy, fu;e fy[kksA

(l)- fdlh ckr dks fl) djuk ,d vPNk xf.krh; dkS”ky gSA uhps nh xbZ xfrfof/k vkidks ;g

le>us esa enn djsxh& nks la[;kvksa dk xq.kuQy ,d /kukRed la[;k gksrh gSA

Ikgys dh rhu vo/kkj.kkvksa dks ;kn j[ksaA

aa ×=1 ;g 1 dh fo”ks’krk gS

acabcba +=+ )( forj.k dk fu;e

1, 2, 3, 4, 5.... = +1, +2, +3, +4, +5.... lHkh izk—frd la[;k,a ;k fxufr;ka /kukRed gksrs gSA

uhps lR;kiu fn;k x;k gS A izR;sd pj.k esa mi;ksx ykbZ x;h vo/kkj.kkvksa dks igpkuus dk iz;kl

djksA


!

35" !

!

1)11(0)1()1(0)1(.............

0)11(.............0)11()11(

)0(1)11(1011

0)1(1)1(1()0(1)11(1

0)11(

+=−×−

=−++

=−+

=+×−+

=+×−+−×−

−=+−−

=+−

=+×+−×

=+−

=+−

( )11()11( −×=+×− )

(n)- vc rqe fl) djus dh dksf”k”k djks 1)1()1( +=−÷−

3 -5 -1 - fdru s f=dk s . k

f=dks.k dh dqy la[;k Kkr djus dh dksf”k'k djksA vius mRrj dks nksLrksa ds lkFk lk>k djksA D;k

muds Hkh mRrj ogh gSa\


!

36" !

!

3 -5 -2 - cuk, a fp= ig syh

1- ,d f=Hkqt cuk;asA vc fdlh dks.k ls mlds lkeus okyh js[kk dks feykrs gq, ,d js[kk

[khasfp, ¼fp= 1½A vc f=Hkqtksa dh la[;k dh x.kuk djksA D;k ;g 3 gS\

2- vc nwljs fp= dks ns[kks blesa ,slh gh js[kk nks dks.kksa ls [khaph xbZ gS A f=Hkqtksa dh la[;k

dh x.kuk djksA D;k rqeus blesa Nqis gq, iSVuZ dks tkuk\

3- vc bUgha dks.kksa ls ,d&,d js[kk vkSj [khpksa ¼fp= 3½A irk djksa fd ;g iSVuZ lgh gS ;k

ugh\

(fp= 1) (fp= 2) (fp= 3)

4- fdlh f=Hkqt ds vk/kkj dks.k ls rhu js[kk, [khpksaA fp= dks vius fe= dks fn[kkvksa vkSj

dgks fd os dqy f=Hkqtksa dh la[;k irk djsaA

3 -5 -3 - f=H k qt d s ckj s e s a

;gk¡ f=Hkqt ls lEcaf/kr dqN dFku fn;s x;s gSaA muds lkeus lR; ;k vlR; fy[kksA fQj vius

fe=ksa ds lkFk mRrj dks lk>k djksA

1- f=Hkqt ds lHkh dks.k vf/kd dks.k gks ldrs gSaA

2- f=Hkqt ds rhu esa ls nks dks.k ledks.k gks ldrs gSaA

3- f=Hkqt dh yEckbZ vkSj mlds dks.kksa dh eki chp lEca/k gksrk gSA

4- rqe 3 lseh] 4 lseh vkSj 8 lseh Hkqtkvksa okys f=Hkqt cuk ldrs gksA

5- nks ledks.k f=Hkqt feydj ,d oxZ cukrs gSaA

6- ,sls gh dqN vkSj okD; cukvksA

.


!

37" !

!


4-1 -1 - la[;kvk s a e s a vk—fr;k a

1- tc ge fdlh oLrq ds ckjs esa lksprs gSa& dqN vk—fr;ka gekjs eu esa vkrh gSaA tSls tc ge

f[kM+dh ds ckjs esa lksprs gS rks vk;r dh vk—fr eu esa mHkjrh gSA blh rjg pwfM+;ksa ls

o`Rr] xsan ls xksyk bR;kfnA ysfdu tc ge fdlh la[;k ds ckjs esa lksprs gSa rks D;k dksbZ

vk—fr gekjs eu esa mHkjrh gS\ bu fp=ksa dks /;ku ls ns[ksa&

2- D;k ge 3] 6] 10--- dks f=Hkqt la[;k,aW dg ldrs gSa\ D;k vki vxyk f=Hkqt cuk ldrs gS\

D;k 22 ,d f=Hkqt laa[;k gS\

3- ;gka fNis ,d vkSj iSVuZ dk irk yxk ldrs gSa%

(1 + 2 = 3; 3 + 3 = 6; 6 + 4 = 10.....)

4- vc la[;kvksa ds vU; iSVuZ dks ns[ksaA D;k vki muds fy, mfpr vk—fr lksp ldrs gS\

4, 9, 16, 25...

5- ;fn 3] 6] 10------ dks f=Hkqt la[;k dgk tkrk gS rks 4] 9] 16------- dks ge D;k dgsxsa\

6- D;k vk;r ,d la[;k dks iznf”kZr dj ldrk gS\

7- D;k vki vk;r vkSj la[;kvksa ds chp dksbZ iSVuZ irk dj ldrs gS\

4 -1 -2 - xf.kr& Nk sV k s e s a cMh ckr

ge tkurs gS fd xf.kr vU; rjg dh **Hkk’kk** gSA ;g ges cM+s fopkjksa dks cgqr NksVs esa crkuk

fl[kkrk gSA **ikWp dk lkr ckj tksM+** dks xf.kr esa cgqr gh vklkuh ls vkSj NksVs :i eas fy[k

ldrs gSa% 7 ×!5

,d vkSj mnkgj.k ysaA **;fn 50 dks 5 leku fgLlksa esa ckWVrs gS rks ,d Hkkx 10 gksxk** dks xf.kr

dh Hkk’kk esa fy[k ldrs gSa%

105051

=×


!

38" !

!

mijksDr dh rjg ge “a x a x a x a x a x a x a x a?” dks dSls fy[k ldrs gaS\ vkSj bls ge

“a+a+a+a+a+a+a+a?” ls dSls vyx dj ldrs gaS\

bldk mRrj bl ckr ij fuHkZj gS fd 7 dks dgk¡ j[kk tkrk gSA rjhdk gS igys iz”u dks fy[kus

dk rjhdk gksxk “ 7a ” vk rks nwljs dk D;k eryc gksxk\

1- D;k vki fuEufyf[kr dk foLrkj dj ldrs gS\

...............................)5(..................................4....................................5....................................

4

5

5

=

=

=

=

bbbb

4 -1 -3 - chth; O; atdk s dk s tk sM +u k + vk S j ? kVkuk

;gk¡ dqN chtxf.krh; dFku gSaA mUgsa /;ku ls ns[ksaA uhps fn, pj.kokj funsZ”kksa ds vuqlkj vH;kl

djsa&

.........)()()()(

........)()()()(........23)()(.........)()()()(........23)()(

23

23

23

=+=×+××

=−=×−××

=−=+−++

=×=××××

=+=++++

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

⇒ mRrj dk vuqeku yxkb;sA

⇒ pj “a” dk eku 2 j[ksaA

⇒ vius mRrj dk eku Kkr dhft,

⇒ vkids fdrus mRrj lgh gS\

⇒ D;k vki xyr mRrj okys lokyksa ds fy, nksckjk iz;kl djuk pkgrs gaS\

⇒ vki blls D;k lh[ks\

⇒ vius lkfFk;ksa ds lkFk vius lh[ks x, fu’d’kksZa ds ckjs esa ppkZ djsaA

gks ldrk gS vkids vfUre nks iz”uksa ds mRrj gksaA vxyh xfrfof/k esa blds ckjs esa vkSj csgrj

le>us esa enn feysxhA


!

39" !

!

4 -1 -4 - leku ,o a vleku in

abmnaa

bmnaba

32312

345

÷+×

−+

−

⇒ Åij ds dFku esa “ a5 ” , ,d in gSA

⇒ igys dFku esa nks in gSA nwljs esa rhu in rFkk rhljs esa pkj in gSA

vc fuEufyf[kr iz”uksa ds mRrj irk djus dk iz;kl djsa&

1 - 3 lkbfdy $ 2 “kVZ 2 - 6 cksry $ 4 feBkbZ 3 - 10 eksckby Qksu $ 5 eksckby Qksu

4 - ........34 =+ ba

5 - ......45 =+ aa

6 - .......52 33 =+ aa 7 - igys nks iz”u esa ge phtksa dks D;ksa ugh tksM+ ldrsa\ 8 - vfUre nks iz”uksa ds ckjs esa D;k fopkj gS\

9 - D;k 23,aa leku in gS ;k vleku in\

10 - ;g dSls irk djksxs\

D;k vki bl vo/kkj.kk ij vius fe= dh le> dks tkWpuk pkgrs gS\ mlds le{k rØ nsa fd 23,aa leku in gSa vkSj bl lanHkZ esa mlds rØ ij /;ku nsaA

4 -1 -5 - cgl

jksfgr us efyd ds lkFk ,d pky pyus dk fu.kZ; fy;kA ;|fi efyd vleatl esa Fkk fQj Hkh

og jkth gks x;kA nksuksa ds chp bl izdkj ckrphr gqbZ&

jk s fgr % eSa nks xf.kr ds dFku fy[kus tk jgk gw¡A D;k rqe crk ldrs gks fd muesa ls dkSsu lgh gS

vkSj dkSu xyr\


!

40" !

!

efyd% gk¡] crk ldrk gw¡A

jk s fgr% ;s nks dFku gSa 2

2

aaaaaa

=×

=+

efyd% igyk xyr gSA

jk s fgr% vPNk vc pj dk eku 2 j[kksa vkSj tkap djksA

E k fyd% vksg! nksuksa dk mRrj rks ,d gh gS vkSj lgh gSA

jk s fgr % rks D;k rqe lger gks fd igyk dFku lgh gS\

efyd% ;|fi mRrj lgh gS ysfdu igyk dFku xyr gS D;ksafd ----------

efyd us D;k dgk gksxk] \ blls rqEgkjs eu esa D;k ckrsa Li’V gqbZa] viuh uksVcqd esa fy[kksA

4 -1 -6 - x q. ku rjhd s

lkekU;r% xf.krK fdlh vutku dkjd dks iznf”kZr djus ds fy, “x” dk iz;ksx djrs gaSA ysfdu

;g ,d leL;k mRiUu djrk gSA

bls ns[kks] D;k ;g lUnsg ugha iSnk djrk\

xxxxx ××××

rc xf.krKksa us xq.kk dks iznf”kZr djus ds fy, ,d nwljk rjhdk lkspk tks nks pjksa ds chp fcUnq

j[kukA

“ cba .. ” dk eryc gS “ cba ×× ”

bl rjhds us Hkh Hkze iSnk fd;k [kkldj rc tc ge n”keyo dk iz;ksx djrs gSaA ge ¼5-4 xq.kk 3½ dks dSls fy[k ldrs gS\

“5.4.3?”

rc mUgksaus ,d vU; rjhds dk irk yxk;kA mUgksaus fu.kZ; fy;k& tc xq.kk ds izrhd ds LFkku ij

dks’Bd dk iz;ksx djsaA

5 (6) (4) = 5 x 6 x 4.

bl rjhds us Hkh leL;k [kM+h dj nhA tc Nk=ksa us bl rjg ds dFku 2 + (6+7) – 5 dks ns[kk]

mUgksaus FkksM+k eqf”dy eglwl fd;kA

vUr esa os ,d vU; jkspd rjhds ij vk;sA xq.kk ds fy, fdlh Hkh izrhd dk iz;ksx u djsaA


!

41" !

!

“abc” eryc “ cba ×× ”

blfy, xf.kr esa vktdy viuh bPNkuqlkj bu rhuksa rjhdksa dk iz;ksx djrs gSaA vc fuEufyf[kr

dFku dks foLrkj djus dk iz;kl djrs gaSA

a(b) = ............

a.b = .............

ab = .............

4.5 = ..............

vc bu lokykasa ij lkspks vkSj mRrj nks&

1- ;g djrs gq, vkius dSlk eglwl fd;k\

2- D;k vkius xq.kk djus ds fy, igys bu rhuksa fof/k;ksa dk iz;ksx fd;k Fkk\

3- D;k vki rhuksa rjhdks dk iz;ksx dj ldsA

4 -2 -1 - f H k Uu l s i z fr”kr

v- bl xfrfof/k dks djus ls igys iwoZ dh yfuZax xkbZM dk lanHkZ ysaA mu vo/kkj.kkvksa dks ;kn

djasA

1- ge fHkUu la[;kvksa ds ckjs esa dkSu&dkSu lh vo/kkj.kk,a le> ldrs gSa\

2- leku fHkUu dh vo/kkj.kk

3- fHkUu dks n”keyo] vuqikr esa cnyuk

4- izfr”kr dh ifjHkk’kk ¼,slk fHkUu ftldk gj 100 gks] izfr”kr dgykrk gSA½

c- ;g xfrfof/k vkidks fHkUu la[;kvksa dks izfr”kr esa cnyuk lh[kus esa enn djsxhA ;g

n”keyo fHkUu dks izfr”kr esa cnyus ds rjhdks dks irk djus esa Hkh enn djsxhA

ge tkurs gaS %2010020

= vkSj %3510035

= ...

bl vk/kkj ij ge 21dks izfr”kr esa dSls cnysxsa\

,d rjhdk gS **leku fHkUu** dh vo/kkj.kk dk iz;ksxA va”k vkSj gj esa 50 ls xq.kk djus ij gesa

10050

izkIr gksxkA rc ge vklkuh ls dg ldrs gS fd %5021=

ysfdu 31

dks izfr”kr esa dSls cnysaxs\


!

42" !

!

fuEufyf[kr xf.krh; lehdj.kksa dks /;ku ls ns[ksasA

%10010050

502501

21

==×

×= )

2100210050( =÷= %5050

2100

21001

===×

%2010020

205201

51

==×

×= )

5100510020( =÷= %2020

5100

51001

===×

%2510025

254251

41

==×

×= )

4100410025( =÷= %2525

4100

41001

===×

Åij fn;s x;s lehdj.kks ls ges ,d vo/kkj.kk dk irk pyrk gS& **fHkUu la[;kvksa dks izfr”kr esa

cnyus ds fy, muesa 100 ls xq.kk djuk gksxkA**

l- fuEu vc uhps fn, x, pj.kokj funsZ”kksZa dk ikyu djrs gq, iz”uksa ds mRrj nks&

1- izR;sd Js.kh dh nks fHkUu la[;k,a fy[kks& le fHkUu] fo’ke fHkUu vkSj feJ fHkUu

2- mUgsa izfr”kr esa cnyks

3- rc 5 n”keyo la[;k,a fy[kks

4- mUgsa izfr”kr esa cnyksA

5- mRrj dks ns[kksA D;k vo/kkj.kk cu jgh gS] viuh dkih esa fy[kksA

6- bl vo/kkj.kk ij vius fe= ds lkFk ppkZ djksA

4 -2 -2 - la[;k d s fuf”pr i z fr”kr

;g xfrfof/k vkidks la[;kvksa ds dqN fuf”pr izfr”kr ,oa mlds T;knk vFkZ ikus esa enn djsxkA

;kn jgs fd izfr”kr eryc ,d ,slh fHkUu la[;k gS ftldk gj 100 gSA blfy, fHkUu la[;kvksa ds

lHkh xq.k ,oa fØ;kfof/k;ka izfr”kr ij Hkh ykxw gksrh gSA

ge tkurs gS fd 60 dk vk/kk 30 gksrk gS vkSj vk/kk dk eryc gS 50 %.

blfy, 60 dk 50 % gksrk gS 30

xf.kr esa ge bls O;Dr djrs gSa 302160

1005060%5060 =×=×=×

154160

1002560%2560 =×=×=×

blfy, 60 dk 25 % gksxk 15


!

43" !

!

mijksDr ls ge D;k le> ikrs gSa!

60%10045%75

%75%25%100451560

%2541

6015

15%2560

=

=

=−

=−

==

=×

vc fuEufyf[kr iz”uksa dks gy djksA

................).........430%25.).........3

12.........60)2..........%1050)1

=×

=×

=×

=×

bu dFkukas ds ckjs esa vkSj irk djus dh dksf”k”k djks vkSj mRrj dks vius nksLrksa ls lk>k djksA

4 -2 -3 - i z fr”kr vk/ k k fjr l a fØ;k, a

;g xfrfof/k fdlh la[;k dk fuf”pr izfr”kr Kkr djus ds rjhds dks pquus esa vkidh enn djsxhA

fuEu dFkuksa dks ns[ksa vkSj fn, x, lokyksa dk mRrj nsa&

6%10606530

%10%5%50

=×

=÷

=÷

18%30601836

%303%10

=×

=×

=×

36%60606%106030%5060

%60%10%50

=×

=×

=×

=+

24630%40606%106030%5060

%40%10%50

=−=×

=×

=×

=−

1- 240 dk 55% Kkr djus dk lcls vklku rjhdk D;k gS\

2- 240 dk 80% rqe dSls Kkr djksxs\

3- fdrus rjhdksa ls rqe 240 dk 95% Kkr dj ldrs gks\

4- ,sls gh T;knk ls T;knk iz”u cukvks vkSj vius nksLrksa ds lkFk gy djrs gq, bu rjhdksa ij

ppkZ djksA

4 -3 -1 - ok f. kT; xf. kr

;g xfrfof/k vkids fopkj tks vkius ykHk vkSj gkfu ds ckjs esa lh[kk gS dks rktk djus esa vkidks

enn djsxhA vkSj lw=ksa ds vk/kkj ij vkidksa vius lw= cukus esa Hkh enn djsxhA

fuEu inksa dh ifjHkk’kk fy[kus djus dk iz;kl djks] tSls& Ø; ewY;] foØ; ewY;] ykHk] gkfu

bR;kfnA vius lkFkh ls bu ifjHkk’kkvksa dh tkWp djuk u HkwyukA


!

44" !

!

vius ifjHkk’kk ds vk/kkj ij vf/kdre iz”u cukus dk iz;kl djksA ;gkW dqN mnkgj.k gSa&

⇒ ;fn Ø; ewY;] foØ; ewY; ls vf/kd gS rks ge ik;saxs fd------

⇒ ;fn] foØ; ewY;] Ø; ewY; ls vf/kd gS rks ge ik;saxs------

⇒ ;fn gkfu dks fo+Ø; ewY; esa tksM+rs gSa rks ge ik;saxs-------

cuk; s x; s mRrj d s vk/ k kj ij l w= cuku s dk i z;kl djk sA

(CP – cost price; SP – selling price; P – profit; L – loss)

o foØ; ewY; & Ø; ewY; ¾ ykHk SP – CP = P

o Ø; ewY; & foØ; ewY; ¾ gkfu CP – SP = L

o Ø; ewY; $ ykHk ¾ -------- ----------------------------

4 -3 -2 - dk Su lgh \

Qk:[k vkSj lq”khy ds chp ds okrkZyki dks i<+ksA cgl dks iwjk djus esa lq”khy dh enn djksA

Qk:[k % 100 izfr”kr eryc 100100

, D;k eS lgh gwa\

l q”k hy% gkW] rqe fcydqy lgh gksA

Qk:[k % ;fn fHkUu dk va”k vkSj gj leku gksrk gS rks eku 1 gksrk gSA

l q”k hy% gkW] blesa dksbZ lansg ughA

Qk:[k % blfy, lkS izfr”kr eryc gS 1- ;fn rqe xf.kr esa lkS izfr”kr vad ikrs gks rks bldk

eryc gS rqeus dsoy 1 vad izkIr fd;kA D;k rqe lger gks\

l q”k hy% D;k\ ugha] rqe xyr gks D;ksafd--------------------------------------------------------------------------------------------------------

;fn fdlh fu’d’kZ ij ugha igWqp ik jgs rks dksbZ ckr ughaA vxyh xfrfof/k vkidh enn djsxhA


!

45" !

!

4 -3 -3 - i z fr”kr e s a ykH k vk S j gk W fu

rkfydk ns[kksA fjDr LFkkuksa dh iwfrZ djus dk iz;kl djksA ;gkW dqN ladsr fn;s x;s gS

SPCP

SPPCPCP

=×

=+

=+

=×

%115%115%15100

%)15(%100%100

Cost price Profit Loss Selling pirce

400 15% .....................

550 10% .....................

.................... 15% 690

500 .................... 600

................... 15% 340

vius mRRkj dks nksLrksa ds lkFk lk>k djuk er HkwyukA

4 -5 -1 - Lk k / k kj. k C;kt

D;k rqeus **C;kt** “kCn lquk gS\ rqe blls D;k le>rs gks\ D;k rqEgkjs fe= Hkh ogh le>rs gSa]

tSlk rqe\ muds lkFk ppkZ djksA

**C;kt ,d fVdV ds leku gS tks ge ;k=k ds le; cl ;k Vsªu esa [kjhnrs gSA** bl dFku vkSj

C;kt ds ckjs esa rqEgkjs fopkjksa esa fdruh lekurk,W rqe <Ww< ldrs gks\

cl gekjh ugha gksrhA ge dqN le; bldk iz;ksx djrs gSA ge blds fy, dqN “kqYd vnk djrs

gSA blh izdkj ge fdlh ds iSls dk dqN le; ds fy, mi;ksx djrs gSa rks mlds fy, C;kt nsrs

gSaA

;fn rqe yEch nwjh dh ;k=k djrs gSa rks fVdV esa vf/kd iSls yxsaxsA blh izdkj ;fn rqe fdlh ds

iSLkksa dk yEcs le; rd iz;ksx djrs gks rks blds fy, T;knk C;kt nsuk gksxkA

fdl Js.kh vkSj fdl izdkj ds cl esa ;k=k dj jgs gks] blds vuqlkj Hkh “kqYd cny tk;sxkA blh

izdkj C;kt esa Hkh O;fDr ;k cSad ftlus rqEgsa iSls fn;s gS muds vuqlkj cnyko gksxkA

blfy, C;kt dh x.kuk rhu phtksa ij fuHkZj djrk gS&


!

46" !

!

⇒ ftl iSls dk mi;ksx gksrk gS ¼ewy/ku½

⇒ ftrus le; ds fy, iSls dk iz;ksx fd;k tkrk gS ¼fnu@eghuk@o’kZ dh la[;k½

⇒ vkSj C;kt dh njA

C;kn dh nj izk;% izfr”kr esa O;Dr fd;k tkrk gSA ,d mnkgj.k ysrs gSaA C;kt dh nj 5% gSA blds rhu igyw gSa&

1- ge 100 #i;s dk mi;ksx dj jgs gS

2- ge bls ,d o’kZ ds fy, mi;ksx dj jgs gSA

3- tc ge 100 :i;s dks ,d o’kZ ckn okil djsxssa rks ge 5 #i;s vkSj nsus gksaxsA ¼;fn ge lkS

#i;s dks cSd esa tek djrs gS tks 5% dk C;kt nsrk gS rks ge ,d o’kZ ckn 5 #i;s vf/kd

ik;sxsa½

vc fuEufyf[kr i z”uk s a d s mRrj nk s&

1- **C;kt dh nj 8 izfr”kr gS** dFku ls D;k rkRi;Z gS\

2- ;fn ge 1000 #i, cSad esa tek djrs gSa tks 8 izfr”kr dk C;kt nsrk gS A rks ge ,d o’kZ

ds i”pkr~~ fdrus #i, ik;saxs\

3- ge 4000 #i, dk yksu cSad ls ysrs gSa] tks 9 izfr”kr dk C;kt ysrk gSA ,d o’kZ ds ckn

gesa fdruk C;kt okil djuk gksxk\

4- ge 2000 #i, dk yksu cSad ls ysrs gS] tks 9 izfr”kr dk C;kt ysrk gSA nks o’kZ ds ckn gesa

fdruk C;kt okil djuk gksxk\

5- ;fn ge 2000 #i;s dk yksu cSad ls 18 izfr”kr C;kt dh nj ls ysrs gSa rks] ,d o’kZ ds

ckn gesa fdruk C;kt okil djuk gksxk\

6- vki vfUre rhu iz”uksa ls D;k le>s\

7- ;g rhu la[;kvksa ds xq.kuQy ls dSls lacaf/kr gS\

8- D;k rqe C;kt dh x.kuk djus ds fy, lw= cuk ldrs gksa\

4 -5 -2 - vKkr dk irk yxkuk

ge tkurs gS fd C;kt dh x.kuk djus ds fy, ewy/ku] o’kZ vkSj C;kt dh nj dk xq.kk djuk gksrk

gSA ge bls ,sls j[k ldrs gSa&

“ PNRI = ”

;g xfrfof/k ges lw= esa NwVs gq, dks irk djus esa enn djsxhA

blds fy, D;k izfØ;k viuk,axs\ blds fy, ,d dFku ysrs gS


!

47" !

!

P = 1000 rupees; N = two years; R = 8%

ge tkurs gSa fd mRrj 160 #i;s gaSA

bl dFku dks mRrj lfgr ,sls fy[krs gSa% 821001000160 ××=

eku ysrs gSa fd le; ugha fn;k x;k gS rc lw= dks fuEu izdkj ls fy[krs gq, le; dk irk dj

ldrs gSa%

2212

80180

801160

80160810160

810160

=

=

×=

××=×

×=

××=

××=

NNN

N

NN

N

vc fuEufyf[kr dks gy djks&

82100

160

21001000160

××=

××=

P

R

D;k rqe vo/kkj.kk dks le> ik, ftlds dkj.k ge NwVs gq, ¼vKkr½ dk irk dj ik,A

,sls vkSj Hkh iz”u cukb;s] mRrj Kkr dhft, ,d lwpuk dks fNikb;s vkSj vius fe= ls mls irk

djus dks dgsaA

tc vki iz”u cuk jgsa gksa rks bls vkSj dfBu djus ds fy, fuEu rjhdksa dk iz;ksx djsaaA

v- o’kZ dh la[;k dks eghus esa cnfy,&

1 eghuk =121

o’kZ

2 eghuk =61

o’kZ


!

48" !

!

73 fnu =51

o’kZ

146 fnu = 52o’kZ bR;kfn

c- C;kt dh nj dks fefJr fHkUu esa cnydj loky cuk,a& 218 %

Lrj 5 vk S j Lrj 6

f”k{kd }kjk lq>k, x, funsZ”kksZa ds vuqlkj Lrj 5 vkSj 6 ds vH;kl djus gSaA

fiNys pkj Lrj ij dke djrs gq, rqeus vusd izdkj dh xf.krh; vo/kkj.kkvksa ds ckjs esa lkspk]

le>k gksxkA D;k vHkh Hkh ,slk yxkrk gS fd dqN phtksa ds ckjs esa vkSj tkuus&le>us dh t:jr

gSA

,sls Yk{;ksa] ftuds ckjs esa rqEgsa nksckjk vH;kl dh t:jr gskxh mUgsa ;gka fy[k yksA bl xkbZM dh

“kq#vkr esa Lrj 5 vkSj 6 ds fy, igys ls gh dqN y{; r; fd, x, gSaA bu nksuksa dks feykdj

Lrj 5 vkSj 6 ds fy, yfuZax xkbZM cusxhA

5 4- {ks=Qy] /kkfjrk vkSj vk;ru

5- vuqikr vkSj lekuqikrh Hkkx

6- lehdj.k

6 4- xzkQ

5- cgqinksa dk tksM+ vkSj ?kVkuk

6- cgqinksa dk xq.kk

vc vius f”k{kd ls feyksA mUgsa vius fy, Lrj 5 o 6 ds fy, fudkys x, lh[kus ds fcUnqvksa dks

fn[kkvksA os rqEgsa bu nksuksa gh Lrjksa ds fy, vH;kl r; djus esa enn nsaxsA

Education

SSRP Self Learning Guide Maths Class 9 - In Hindi