13_direct and indirect proportions

8/14/2019 13_Direct and Indirect Proportions

1/16

AE]CDQJLA ELXC]TC W]IWI]QEILT =7:

:4: Elq}iasdqeil

Oinjl w}cwj}ct qcj hi} neotc`h jla net tetqc} Nc stct 477 o@ ih

rjqc}, = twiilt ih tsbj}, : twiil ih qcj `cjxct jla 37 o@ ih oe`k

Nir osdn zsjlqequ ih cjdn eqco re`` nc lcca, eh nc njt qi ojkc qcj

hi} hexc wc}tilt9

Eh qri tqsaclqt qjkc =7 oelsqct qi j}}jlbc dnje}t hi} jl jttcomù,

qncl nir osdn qeoc risà hexc tqsaclqt qjkc qi ai qnc tjoc fim9

Rc dioc jd}itt ojlu tsdn teqsjqeilt el is} aju/qi/aju ehc, rnc}c rc

lcca qi tcc xj}ejqeil el ilc zsjlqequ m}elbelb el xj}ejqeil el qnc iqnc}

zsjlqequ

Hi} c~jow`c6.e% Eh qnc lsomc} ih j}qed`ct ws}dnjtca eld}cjtct, qnc qiqj` ditq j`ti eld}cjtct

.ee% Oi}c qnc oilcu acwiteqca el j mjlk, oi}c et qnc elqc}ctq cj}lca

.eee% Jt qnc twcca ih j xcned`c eld}cjtct, qnc qeoc qjkcl qi dixc} qnc tjoc aetqjldc

acd}cjtct

.ex% Hi} j bexcl fim, oi}c qnc lsomc} ih ri}kc}t, ctt re`` mc qnc qeoc qjkcl qi diow cqc

qnc ri}k

Imtc}xc qnjq dnjlbc el ilc zsjlqequ cjat qi dnjlbc el qnc iqnc} zsjlqequ

R}eqc hexc oi}c tsdn teqsjqeilt rnc}c dnjlbc el ilc zsjlqequ cjat qi dnjlbc eljliqnc} zsjlqequ

Nir ai rc hela isq qnc zsjlqequ ih cjdn eqco lccaca mu Oinjl9 I}, qnc qeoc hexc

tqsaclqt qjkc qi diow`cqc qnc fim9

Qi jltrc} tsdn zsctqeilt, rc lir tqsau tioc dildcwqt ih xj}ejqeil

:4= Ae}cdq W}iwi}qeil

Eh qnc ditq ih : kb ih tsbj} et ]t :>, qncl rnjq risà mc qnc ditq ih 4 kb tsbj}9 Eq et ]t 31

Ae}cdq jla Elxc}tcW}iwi}qeilt

DNJWQC]

:4


2/16

=7= OJQNCOJQEDT

Teoe`j}ù, rc djl hela qnc ditq ih 3 kb i} > kb ih tsbj} Tqsau qnc hi`ìrelb qjm`c

Imtc}xc qnjq jt rcebnq ih tsbj} eld}cjtct, ditq j`ti eld}cjtct el tsdn j ojllc} qnjq

qnce} }jqei }cojelt diltqjlq

Qjkc ilc oi}c c~jow`c Tswwitc j dj} stct 1 eq}ct ih wcq}i` qi q}jxc` j aetqjldc ih

87 ko Nir hj} re`` eq q}jxc` stelb := eq}ct9 Qnc jltrc} et :>7 ko Nir aea rc dj`ds`jqc

eq9 Teldc wcq}i` diltsoca el qnc tcdila eltqjldc et := eq}ct, ec, qn}cc qeoct ih 1 eq}ct, qnc

aetqjldc q}jxc``ca re`` j`ti mc qn}cc qeoct ih 87 ko El iqnc} ri}at, rncl qnc wcq}i`

diltsowqeil mcdioct qn}cc/hià, qnc aetqjldc q}jxc``ca et j`ti qn}cc hià qnc w}cxeist

ilc @cq qnc diltsowqeil ih wcq}i` mc~ èq}ct jla qnc di}}ctwilaelb aetqjldc q}jxc``ca mcu ko Lir, diow`cqc qnc hi`ìrelb qjm`c6

Wcq}i` el èq}ct .~% 1 > := :3 =7 =3

Aetqjldc el ko .u% 87 :>7

Rc hela qnjq jt qnc xj`sc ih~ eld}cjtct, xj`sc ihu j`ti eld}cjtct el tsdn j rju qnjq qnc

}jqei~

uaict liq dnjlbc; eq }cojelt diltqjlq .tju k% El qnet djtc, eq et

:

:3.dncdk eq*%

Rc tju qnjq~ jlau j}c el ae}cdq w}iwi}qeil, eh ?~

ku

i}~ ?ku,

El qnet c~jow`c, 1 :=87 :>7

? , rnc}c 1 jla := j}c qnc zsjlqeqect ih wcq}i` diltsoca el

èq}ct .~% jla 87 jla :>7 j}c qnc aetqjldct .u% el ko Ti rncl~ jlau j}c el ae}cdq

w}iwi}qeil, rc djl r}eqc : =

: =

~ ~

u u? [u:,u= j}c xj`sct ihu di}}ctwilaelb qi qnc xj`sct~:,

~=ih~ }ctwcdqexcù_

Qnc diltsowqeil ih wcq}i` jla qnc aetqjldc q}jxc``ca mu j dj} et j djtc ih ae}cdq

w}iwi}qeil Teoe`j}ù, qnc qiqj` joislq twclq jla qnc lsomc} ih j}qed`ct ws}dnjtca et j`ti

jl c~jow`c ih ae}cdq w}iwi}qeil


3/16

AE]CDQJLA ELXC]TC W]IWI]QEILT =74

AI QNET

Qnelk ih j hcr oi}c c~jow`ct hi} ae}cdq w}iwi}qeil Dncdk rncqnc} Oinjl [el qnc eleqej` c~jow`c_ re``

qjkc 237 o@ ih rjqc}, 3 twiilt ih tsbj},:

==

twiilt ih qcj cjxct jla :=3 o@ ih oe`k qi w}cwj}c qcj hi}

hexc wc}tilt* @cq st q}u qi slac}tqjla hs}qnc} qnc dildcwq ih ae}cdq w}iwi}qeil qn}isbn qnc hi`ìrelb jdqexeqect

.e% Qjkc j dìdk jla he~ eqt oelsqc njla jq :=

]cdi}a qnc jlb`c qs}lca qn}isbn mu qnc oelsqc njla h}io eqt i}ebelj` witeqeil

jla qnc qeoc qnjq njt wjttca, el qnc hi`ìrelb qjm`c6

Qeoc Wjttca .Q% .Q:% .Q

=% .Q

4% .Q

1%

.el oelsqct% :3 47 13 87

Jlb`c qs}lca .J% .J:% .J=% .J4% .J1%.el acb}cc% 57

Q

J

Rnjq ai uis imtc}xc jmisq Q jla J9 Ai qncu eld}cjtc qibcqnc}9

EtQ

Jtjoc cxc}u qeoc9

Et qnc jlb`c qs}lca qn}isbn mu qnc oelsqc njla ae}cdqù w}iwi}qeilj`

qi qnc qeoc qnjq njt wjttca9 Uct*

H}io qnc jmixc qjm`c, uis djl j`ti tcc

Q:6 Q

=? J

:6 J

=, mcdjstc

Q:6 Q

=? :3 6 47 ? :6=

J:6 J

=? 57 6 :>7 ? :6=

Dncdk eh Q=6 Q

4? J

=6 J

4jla Q

46 Q

1? J

46 J

1

Uis djl }cwcjq qnet jdqexequ mu dniitelb uis} irl qeoc elqc}xj`

.ee% Jtk uis} h}ecla qi he ` qnc hi`ìrelb qjm`c jla hela qnc }jqei ih net jbc qi qnc

di}}ctwilaelb jbc ih net oiqnc}

Jbc W}ctclq Jbc

hexc ucj}t jbi jbc jhqc} hexc ucj}t

H}eclat jbc .H%

Oiqnc}t jbc .O%

H

O

Rnjq ai uis imtc}xc9

Ai H jla O eld}cjtc .i} acd}cjtc% qibcqnc}9 EtH

Otjoc cxc}u qeoc9 Li*

Uis djl }cwcjq qnet jdqexequ reqn iqnc} h}eclat jla r}eqc airl uis} imtc}xjqeilt


4/16

=71 OJQNCOJQEDT

Q]U QNCTC

Qnst, xj}ejm`ct eld}cjtelb .i} acd}cjtelb% qibcqnc} lcca liq j`rjut mc el ae}cdq

w}iwi}qeil Hi} c~jow`c6

.e% wnutedj` dnjlbct el nsojl mcelbt idds} reqn qeoc msq liq lcdcttj}eù el j w}cacqc}/

oelca }jqei.ee% dnjlbct el rcebnq jla ncebnq joilb elaexeasj t j}c liq el jlu klirl w}iwi}qeil jla

.eee% qnc}c et li ae}cdq }c`jqeiltnew i} }jqei mcqrccl qnc ncebnq ih j q}cc jla qnc lsomc}

ih cjxct b}irelb il eqt m}jldnct Qnelk ih tioc oi}c teoe`j} c~jow`ct

: Imtc}xc qnc hi` irelb qjm`ct jla hela eh~ jlau j}c ae}cdqù w}iwi}qeilj`

.e% ~ =7 :2 :1 :: > 3 =

u 17 41 => == :8 :7 1

.ee% ~ 8 :7 :1 :> == =8 47

u 1 > := :8 =7 =1 =>

.eee% ~ 3 > := :3 :> =7

u :3 =1 48 87 2= :77

= W}eldewj` ? ]t :777, ]jqc ? ># wc} jllso He` el qnc hi` irelb qjm`c jla hela

rnedn quwc ih elqc}ctq .teow`c i} diowisla% dnjlbct el ae}cdq w}iwi}qeil reqn

qeoc wc}eia

Qeoc wc}eia : ucj} = ucj}t 4 ucj}t

Teow`c Elqc}ctq .el ]t%

Diowisla Elqc}ctq .el ]t%W : W:77

q}

+

W

:77

} q

Eh rc he~ qeoc wc}eia jla qnc }jqc ih elqc}ctq, teow`c elqc}ctq dnjlbct w}iwi}qeilj`ù

reqn w}eldewj` Risà qnc}c mc j teoe`j} }c`jqeiltnew hi} diowisla elqc}ctq9 Rnu9

@cq st dilteac} tioc ti`xca c~jow`ct rnc}c rc risà stc qnc dildcwq ihae}cdq w}iwi}qeil

C~jow`c :6 Qnc ditq ih 3 ocq}ct ih j wj}qeds`j} zsjèqu ih dìqn et ]t =:7 Qjms`jqc qnc

ditq ih =, 1, :7 jla :4 ocq}ct ih dìqn ih qnc tjoc quwc

Ti`sqeil6 Tswwitc qnc clbqn ih dìqn et~ ocq}ct jla eqt ditq, el ]t, etu

~ = 1 3 :7 :4

u u=

u4

=:7 u1

u3

QNELK, AETDSTT JLA R]EQC


5/16


Jt qnc clbqn ih dìqn eld}cjtct, ditq ih qnc dìqn j`ti eld}cjtct el qnc tjoc }jqei Eq et

j djtc ih ae}cdq w}iwi}qeil

Rc ojkc stc ih qnc }c`jqeil ih quwc: =

: =

~ ~

u u?

.e% Nc}c~:

? 3, u:

? =:7 jla~=

? =

Qnc}chi}c, : =

: =

~ ~

u u? bexct

=

3 =

=:7 u? i} 3u

=? = =:7 i} =

= =:7

3u

? ? >1

.ee% Eh ~4

? 1, qncl4

3 1

=:7 u? i} 3u

4? 1 =:7 i} 4

1 =:7

3u

? ? :8>

[Djl rc stc4=

= 4

~~

u u? nc}c9 Q}u*_

.eee% Eh~1

? :7, qncl1

3 :7=:7 u

? i} 1:7 =:7

3u

? ? 1=7

.ex% Eh ~3

? :4, qncl3

3 :4

=:7 u? i} 3

:4 =:7

3u

? ? 318

= 1 :7 3Liqc qnjq nc}c rc djl j`ti stc i} i} el qnc w`jdc ih

>1 :8> 1=7 =:7

C~jow`c =6 Jl c`cdq}ed wi`c, :1 ocq}ct nebn, djtqt j tnjair ih :7 ocq}ct Hela qnc

ncebnq ih j q}cc qnjq djtqt j tnjair ih :3 ocq}ct slac} teoe`j} dilaeqeilt

Ti`sqeil6 @cq qnc ncebnq ih qnc q}cc mc~ ocq}ct Rc hi}o j qjm`c jt tnirl mcìr6

ncebnq ih qnc imfcdq .el ocq}ct% :1 ~

`clbqn ih qnc tnjair .el ocq}ct% :7 :3

Liqc qnjq oi}c qnc ncebnq ih jl imfcdq, qnc oi}c risà mc qnc clbqn ih eqt tnjair

Ncldc, qnet et j djtc ih ae}cdq w}iwi}qeil Qnjq et,:

:

~

u?

=

=

~

u

Rc njxc:1

:7?

:3

~.Rnu9%

i}:1

:3:7 ?~

i}:1 4

=

?~

Ti =: ?~

Qnst, ncebnq ih qnc q}cc et =: ocq}ct

J`qc}ljqcù, rc djl r}eqc : =

: =

~ ~

u u? jt : :

= =

~ u

~ u?


6/16


7/16


AI QNET

Teldc qnc twcca et slehi}o, qnc}chi}c, qnc aetqjldc dixc}ca risà mc ae}cdqù

w}iwi}qeilj` qi qeoc

.e% Rc njxc

23

87 =7

~?

i}23

=787

?~

i} ~ ? =3

Ti, qnc q}jel re`` dixc} j aetqjldc ih =3 ko el =7 oelsqct

.ee% J`ti,23 =37

87 u?

i}=37 87

23u

? ? =77 oelsqct i} 4 nis}t =7 oelsqct

Qnc}chi}c, 4 nis}t =7 oelsqct re`` mc }czse}ca qi dixc} j aetqjldc ih =37 keìocq}ct

J`qc}ljqexcù, rncl~ et klirl, qncl ilc djl acqc}oelcu h}io qnc }c`jqeil=37

=7

~

u?

Uis klir qnjq j ojw et j oelejqs}c }cw}ctclqjqeil ih j xc}u j}bc }cbeil J tdj`c et

stsj`ù bexcl jq qnc miqqio ih qnc ojw Qnc tdj`c tnirt j }c`jqeiltnew mcqrccl

jdqsj` clbqn jla qnc clbqn }cw}ctclqca il qnc ojw Qnc tdj`c ih qnc ojw et qnst qnc

}jqei ih qnc aetqjldc mcqrccl qri wielqt il qnc ojw qi qnc jdqsj` aetqjldc mcqrccl

qri wielqt il qnc j}bc }cbeil

Hi} c~jow`c, eh : do il qnc ojw }cw}ctclqt > ko ih jdqsj` aetqjldc [ec, qnc tdj`c et

: do 6 > ko i} : 6 >77,777_ qncl = do il qnc tjoc ojw re`` }cw}ctclq :8 ko

Ncldc, rc djl tju qnjq tdj`c ih j ojw et mjtca il qnc dildcwq ih ae}cdq w}iwi}qeil

C~jow`c 36 Qnc tdj`c ih j ojw et bexcl jt :647777777 Qri deqect j}c 1 do jwj}q il

qnc ojw Hela qnc jdqsj` aetqjldc mcqrccl qnco

Ti`sqeil6 @cq qnc ojw aetqjldc mc~ do jla jdqsj` aetqjldc mcu do, qncl

:647777777 ?~ 6u

i} 2:

4 :7?

~

u

Teldc ~ ? 1 ti,2

:

4 :7?

1

ui} u ? 1 4 :72 ? := :72 do ? :=77 ko

Qnst, qri deqect, rnedn j}c 1 do jwj}q il qnc ojw, j}c jdqsj`ù :=77 ko jrju h}io

cjdn iqnc}

Qjkc j ojw ih uis} Tqjqc Liqc qnc tdj`c stca qnc}c Stelb j }s`c}, ocjts}c qnc ojw

aetqjldc mcqrccl jlu qri deqect Dj`ds`jqc qnc jdqsj` aetqjldc mcqrccl qnco


8/16

=7> OJQNCOJQEDT

C^C]DETC :4:

:, Hi`ìrelb j}c qnc dj} wj}kelb dnj}bct lcj} j }je`rju tqjqeil swqi

1 nis}t ]t 87> nis}t ]t :77

:= nis}t ]t :17

=1 nis}t ]t :>7

Dncdk eh qnc wj}kelb dnj}bct j}c el ae}cdq w}iwi}qeil qi qnc wj}kelb qeoc

=, J oe~qs}c ih wjelq et w}cwj}ca mu oe~elb : wj}q ih }ca weboclqt reqn > wj}qt ih mjtc

El qnc hi`ìrelb qjm`c, hela qnc wj}qt ih mjtc qnjq lcca qi mc jaaca

Wj}qt ih }ca weboclq : 1 2 := =7

Wj}qt ih mjtc >

4, El Zsctqeil = jmixc, eh : wj}q ih j }ca weboclq }czse}ct 23 o@ ih mjtc, nir osdn}ca weboclq tnisà rc oe~ reqn :>77 o@ ih mjtc9

1, J ojdnelc el j tihq a}elk hjdqi}u he``t >17 miqq`ct el te~ nis}t Nir ojlu miqq`ct re``

eq he el hexc nis}t9

3, J wniqib}jwn ih j mjdqc}ej cl`j}bca 37,777 qeoct

jqqjelt j clbqn ih 3 do jt tnirl el qnc aejb}jo

Rnjq et qnc jdqsj` `clbqn ih qnc mjdqc}ej9 Eh qnc

wniqib}jwn et cl`j}bca =7,777 qeoct ilù, rnjq

risà mc eqt cl`j}bca clbqn9

8, El j oiac` ih j tnew, qnc ojtq et 5 do nebn, rne`c

qnc ojtq ih qnc jdqsj tnew et := o nebn Eh qnc clbqn

ih qnc tnew et => o, nir ilb et qnc oiac` tnew9

2, Tswwitc = kb ih tsbj} dilqjelt 5 :78 d}utqj`t

Nir ojlu tsbj} d}utqj`t j}c qnc}c el .e% 3 kb ih tsbj}9 .ee% := kb ih tsbj}9

>, ]jtnoe njt j }ija ojw reqn j tdj`c ih : do }cw}ctclqelb :> ko Tnc a}exct il j

}ija hi} 2= ko Rnjq risà mc nc} aetqjldc dixc}ca el qnc ojw9

5, J 3 o 87 do nebn xc}qedj` wi`c djtqt j tnjair 4 o =7 do ilb Hela jq qnc tjoc qeoc

.e% qnc clbqn ih qnc tnjair djtq mu jliqnc} wi`c :7 o 37 do nebn .ee% qnc ncebnq ih j

wi`c rnedn djtqt j tnjair 3o ilb

:7, J ijaca q}sdk q}jxc`t :1 ko el =3 oelsqct Eh qnc twcca }cojelt qnc tjoc, nir hj}

djl eq q}jxc` el 3 nis}t9

AI QNET : Il j tzsj}ca wjwc}, a}jr hexc tzsj}ct ih aehhc}clq teactR}eqc qnc hi`ìrelb elhi}ojqeil el j qjms`j} hi}o

Tzsj}c/: Tzsj}c/= Tzsj}c/4 Tzsj}c/1 Tzsj}c/3

@clbqn ih j teac .@%

Wc}eocqc} .W%

@

W


9/16


J}cj .J%

@

J

Hela rncqnc} qnc clbqn ih j teac et el ae}cdq w}iwi}qeil qi6

.j% qnc wc}eocqc} ih qnc tzsj}c

.m% qnc j}cj ih qnc tzsj}c

= Qnc hi`ìrelb elb}caeclqt j}c }czse}ca qi ojkc nj rj hi} 3

wc}tilt6

Tsfe)]jrj ? =37 b, Tsbj} ? 477 b,

Bncc ? =77 b, Rjqc} ? 377 o@

Stelb qnc dildcwq ih w}iwi}qeil, ctqeojqc qnc

dnjlbct el qnc zsjlqequ ih elb}caeclqt, qi

w}cwj}c nj`rj hi} uis} d`jtt

4 Dniitc j tdj`c jla ojkc j ojw ih uis}

d`jtt}iio, tnirelb relairt, aii}t,

m`jdkmij}a cqd .Jl c~jow`c et bexcl nc}c%

QNELK, AETDSTT JLA R]EQC

Qjkc j hcr w}im`cot aetdsttca ti hj} slac} ae}cdq xj}ejqeil Ai uis qnelk qnjq

qncu djl mc ti`xca mu sleqj}u ocqnia9

:44 Elxc}tc W}iwi}qeil

Qri zsjlqeqect oju dnjlbc el tsdn j ojllc} qnjq eh ilc zsjlqequ eld}cjtct, qnc iqnc}

zsjlqequ acd}cjtct jla xedc xc}tj Hi} c~jow`c, jt qnc lsomc} ih ri}kc}t eld}cjtct, qeoc

qjkcl qi heletn qnc fim acd}cjtct Teoe`j}ù, eh rc eld}cjtc qnc twcca, qnc qeoc qjkcl qi

dixc} j bexcl aetqjldc acd}cjtct

Qi slac}tqjla qnet, cq st iik elqi qnc hi`ìrelb teqsjqeil

Vjnccaj djl bi qi nc} tdnii` el his} aehhc}clq rjut Tnc djl rj`k, }sl, dud`c i} bi mu

dj} Tqsau qnc hi`ìrelb qjm`c


10/16

=:7 OJQNCOJQEDT

Imtc}xc qnjq jt qnc twcca eld}cjtct, qeoc qjkcl qi dixc} qnc tjoc aetqjldc acd}cjtct

Jt Vjnccaj aism`ct nc} twcca mu }sllelb, qeoc

}casdct qi nj`h Jt tnc eld}cjtct nc} twcca qi qn}cc

qeoct mu dudèlb, qeoc acd}cjtct qi ilc qne}aTeoe`j}ù, jt tnc eld}cjtct nc} twcca qi :3 qeoct,

qeoc acd}cjtct qi ilc hehqcclqn .I}, el iqnc} ri}at

qnc }jqei mu rnedn qeoc acd}cjtct et elxc}tc ih qnc

}jqei mu rnedn qnc di}}ctwilaelb twcca eld}cjtct%

Djl rc tju qnjq twcca jla qeoc dnjlbc elxc}tcù

el w}iwi}qeil9

@cq st dilteac} jliqnc} c~jow`c J tdnii` rjlqt qi twcla ]t 8777 il ojqncojqedt

qc~qmiikt Nir ojlu miikt disà mc misbnq jq ]t 17 cjdn9 D`cj}ù :37 miikt djl mc

misbnq Eh qnc w}edc ih j qc~qmiik et oi}c qnjl ]t 17, qncl qnc lsomc} ih miikt rnedn

disà mc ws}dnjtca reqn qnc tjoc joislq ih oilcu risà mc `ctt qnjl :37 Imtc}xc qnc

hi` irelb qjm`c

W}edc ih cjdn miik .el ]t% 17 37 87 23 >7 :77

Lsomc} ih miikt qnjq :37 :=7 :77 >7 23 87

djl mc misbnq

Rnjq ai uis imtc}xc9 Uis re`` jww}cdejqc qnjq jt qnc w}edc ih qnc miikt eld}cjtct,

qnc lsomc} ih miikt qnjq djl mc misbnq, kccwelb qnc hsla diltqjlq, re`` acd}cjtc

]jqei mu rnedn qnc w}edc ih miikt eld}cjtct rncl bielb h}io 17 qi 37 et 1 6 3, jla qnc

}jqei mu rnedn qnc di}}ctwilaelb lsomc} ih miikt acd}cjtct h}io :37 qi :=7 et 3 6 1

Qnet ocjlt qnjq qnc qri }jqeit j}c elxc}tct ih cjdn iqnc}Liqedc qnjq qnc w}iasdq ih qnc di}}ctwilaelb xj`sct ih qnc qri zsjlqeqect et diltqjlq;

qnjq et, 17 :37 ? 37 :=7 ? 8777

Eh rc }cw}ctclq qnc w}edc ih ilc miik jt~ jla qnc lsomc} ih miikt misbnq jtu, qncl

jt~ eld}cjtctu acd}cjtct jla xedc/xc}tj Eq et eowi}qjlq qi liqc qnjq qnc w}iasdq~u

}cojelt diltqjlq Rc tju qnjq~ xj}ect elxc}tcù reqnu jlau xj}ect elxc}tcù reqn~ Qnst

qri zsjlqeqect ~ jla u j}c tjea qi xj}u el elxc}tc w}iwi}qeil, eh qnc}c c~etqt j }c`jqeil

ih qnc quwc~u ? kmcqrccl qnco, k mcelb j diltqjlq Ehu:, u

=j}c qnc xj`sct ih u

di}}ctwilaelb qi qnc xj`sct~:,~

=ih~}ctwcdqexcù qncl~

:u

:?~

=u

=.? k%, i} : =

= :

~ u

~ u?

Rc tju qnjq~ jlau j}c el elxc}tc w}iwi}qeil

Ncldc, el qnet c~jow`c, ditq ih j miik jla lsomc} ih miikt ws}dnjtca el j he~ca

joislq j}c elxc}tcù w}iwi}qeilj` Teoe`j}ù, twcca ih j xcned`c jla qnc qeoc qjkcl qi

dixc} j he~ca aetqjldc dnjlbct el elxc}tc w}iwi}qeil

Qnelk ih oi}c tsdn c~jow`ct ih wje}t ih zsjlqeqect qnjq xj}u el elxc}tc w}iwi}qeil Uis

oju lir njxc j iik jq qnc hs}leqs}c j}}jlbelb w}im`co, tqjqca el qnc elq}iasdqi}u wj}q

ih qnet dnjwqc}

Nc}c et jl jdqexequ hi} mcqqc} slac}tqjlaelb ih qnc elxc}tc w}iwi}qeil

Os`qewèdjqexc elxc}tc ih j lsomc}

et eqt }cdew}idj` Qnst, :=

et qnc

elxc}tc ih = jla xedc xc}tj .Liqc

qnjq: :

= = := =

? ? %


11/16

AE]CDQJLA ELXC]TC W]IWI]QEILT =::

Q]U QNCTC

AI QNET

Qjkc j tzsj}ca wjwc} jla j}}jlbc 1> dislqc}t il eq el aehhc}clq lsomc} ih }irt jt

tnirl mcìr

Lsomc} ih .]:% .]

=% .]

4% .]

1% .]

3%

]irt .]% = 4 1 8 >

Lsomc} ih .D:% .D

=% .D

4% .D

1% .D

3%

Di`solt .D% := >

Rnjq ai uis imtc}xc9 Jt ] eld}cjtct, D acd}cjtct

.e% Et ]:

6 ]=? D

=6 D

:9 .ee% Et ]

46 ]

1? D

16 D

49

.eee% J}c ] jla D elxc}tcù w}iwi}qeilj qi cjdn iqnc}9

Q}u qnet jdqexequ reqn 48 dislqc}t

Imtc}xc qnc hi`ìrelb qjm`ct jla hela rnedn wje} ih xj}ejm`ct .nc}c~ jlau% j}c el

elxc}tc w}iwi}qeil

.e% ~ 37 17 47 =7 .ee% ~ :77 =77 477 177

u 3 8 2 > u 87 47 =7 :3

.eee% ~ 57 87 13 47 =7 3

u :7 :3 =7 =3 47 43

@cq st dilteac} tioc c~jow`ct rnc}c rc stc qnc dildcwq ih elxc}tc w}iwi}qeil

Rncl qri zsjlqeqect~ jlau j}c el ae}cdq w}iwi}qeil .i} xj}u ae}cdqù% qncu j}c j`ti r}eqqcl jt~u

Rncl qri zsjlqeqect~jlau j}c el elxc}tc w}iwi}qeil .i} xj}u elxc}tcù% qncu j}c j`ti r}eqqcl jt~:

u


12/16

=:= OJQNCOJQEDT

C~jow`c 26 8 wewct j}c }czse}ca qi he`` j qjlk el : nis} =7 oelsqct Nir ilb re`` eq

qjkc eh ilù 3 wewct ih qnc tjoc quwc j}c stca9

Ti`sqeil6

@cq qnc acte}ca qeoc qi he`` qnc qjlk mc~oelsqct Qnst, rc njxc

qnc hi`ìrelb qjm`c

Lsomc} ih wewct 8 3

Qeoc .el oelsqct% >7 ~

@cttc} qnc lsomc} ih wewct, oi}c re`` mc qnc qeoc }czse}ca mu

eq qi he`` qnc qjlk Ti, qnet et j djtc ih elxc}tc w}iwi}qeil

Ncldc, >7 8 ?~ 3 [~:u

:?~

=u

=_

i}>7 8

3

~

?

i} ~ ? 58

Qnst, qeoc qjkcl qi he`` qnc qjlk mu 3 wewct et 58 oelsqct i} : nis} 48 oelsqct

C~jow`c >6 Qnc}c j}c :77 tqsaclqt el j nitqc` Hiia w}ixeteil hi} qnco et hi} =7

ajut Nir ilb re`` qnctc w}ixeteilt jtq, eh =3 oi}c tqsaclqt fiel qnc b}isw9

Ti`sqeil6 Tswwitc qnc w}ixeteilt jtq hi}u ajut rncl qnc lsomc} ih tqsaclqt et :=3

Rc njxc qnc hi`ìrelb qjm`c

Lsomc} ih tqsaclqt :77 :=3

Lsomc} ih ajut =7 u

Liqc qnjq oi}c qnc lsomc} ih tqsaclqt, qnc tiilc} risà

qnc w}ixeteilt c~njstq Qnc}chi}c, qnet et j djtc ih elxc}tc

w}iwi}qeil

Ti, :77 =7 ? :=3 u

i}:77 =7

:=3

?u i} :8 ?u

Qnst, qnc w}ixeteilt re`` jtq hi} :8 ajut, eh =3 oi}c tqsaclqt fiel qnc nitqc`

J`qc}ljqcù, rc djl r}eqc~:u

:?~

=u

=jt : =

= :

~ u

~ u?

Qnjq et, ~:

6~=

?u=

6u:

i} :77 6 :=3 ?u 6 =7

i} u ?:77 =7

:8:=3

?

C~jow`c 56Eh :3 ri}kc}t djl mseà j rj`` el 1> nis}t, nir ojlu ri}kc}t re`` mc

}czse}ca qi ai qnc tjoc ri}k el 47 nis}t9

Ti`sqeil6

@cq qnc lsomc} ih ri}kc}t cowìuca qi mseà qnc rj`` el 47 nis}t mcu


13/16


14/16

=:1 OJQNCOJQEDT

.e% J}c qnc lsomc} ih twikct jla qnc jlb`ct hi}oca mcqrccl qnc wje}t ih

diltcdsqexc twikct el elxc}tc w}iwi}qeil9

.ee% Dj`ds`jqc qnc jlb`c mcqrccl j wje} ih diltcdsqexc twikct il j rncc` reqn :3

twikct

.eee% Nir ojlu twikct risà mc lccaca, eh qnc jlb`c mcqrccl j wje} ih diltcdsqexc

twikct et 179

1, Eh j mi~ ih trccqt et aexeaca joilb =1 dneà}cl, qncu re`` bcq 3 trccqt cjdn Nir

ojlu risà cjdn bcq, eh qnc lsomc} ih qnc dneà}cl et }casdca mu 19

3, J hj}oc} njt clisbn hiia qi hcca =7 jleoj`t el net djqq`c hi} 8 ajut Nir ilb

risà qnc hiia jtq eh qnc}c rc}c :7 oi}c jleoj`t el net djqq`c9

8, J dilq}jdqi} ctqeojqct qnjq 4 wc}tilt disà }cre}c Fjtoelac}t nistc el 1 ajut Eh,

nc stct 1 wc}tilt eltqcja ih qn}cc, nir ilb tnisà qncu qjkc qi diow`cqc qnc fim9

2, J mjqdn ih miqq`ct rc}c wjdkca el =3 mi~ct reqn := miqq`ct el cjdn mi~ Eh qnc tjoc

mjqdn et wjdkca stelb =7 miqq`ct el cjdn mi~, nir ojlu mi~ct risà mc he``ca9

>, J hjdqi}u }czse}ct 1= ojdnelct qi w}iasdc j bexcl lsomc} ih j}qed`ct el 84 ajut

Nir ojlu ojdnelct risà mc }czse}ca qi w}iasdc qnc tjoc lsomc} ih j}qed`ct el

31 ajut9

5, J dj} qjkct = nis}t qi }cjdn j actqeljqeil mu q}jxc`èlb jq qnc twcca ih 87 ko)n Nir

ìlb re`` eq qjkc rncl qnc dj} q}jxc`t jq qnc twcca ih >7 ko)n9:7, Qri wc}tilt disà heq lcr relairt el j nistc el 4 ajut

.e% Ilc ih qnc wc}tilt hc`` e`` mchi}c qnc ri}k tqj}qca Nir ilb risà qnc fim

qjkc lir9

.ee% Nir ojlu wc}tilt risà mc lccaca qi heq qnc relairt el ilc aju9

::, J tdnii` njt > wc}eiat j aju cjdn ih 13 oelsqct as}jqeil Nir ilb risà cjdn

wc}eia mc, eh qnc tdnii` njt 5 wc}eiat j aju, jttsoelb qnc lsomc} ih tdnii` nis}t qi

mc qnc tjoc9


15/16

AE]CDQJLA ELXC]TC W]IWI]QEILT =:3

AI QNET

: Qjkc j tnccq ih wjwc} Hià eq jt tnirl el qnc hebs}c Dislq qnc lsomc} ih wj}qt jlaqnc j}cj ih j wj}q el cjdn djtc

Qjms`jqc uis} imtc}xjqeilt jla aetdstt reqn uis} h}eclat Et eq j djtc ih elxc}tc w}iwi}qeil9 Rnu9

Lsomc} ih wj}qt : = 1 > :8

J}cj ih cjdn wj}q j}cj ih qnc wjwc}

:

= qnc j}cj ih qnc wjwc}

= Qjkc j hcr dilqjelc}t ih aehhc}clq tevct reqn de}ds`j} mjtct He` qnc tjoc joislq ih

rjqc} el cjdn dilqjelc} Liqc qnc aejocqc} ih cjdn dilqjelc} jla qnc }ctwcdqexc

ncebnq jq rnedn qnc rjqc} cxc` tqjlat Qjms`jqc uis} imtc}xjqeilt Et eq j djtc ih

elxc}tc w}iwi}qeil9

Aejocqc} ih dilqjelc} .el do%

Ncebnq ih rjqc} `cxc` .el do%

RNJQ NJXC RC AETDSTTCA9

:, Qri zsjlqeqect~ jlau j}c tjea qi mc el ae}cdq w}iwi}qeil eh qncu eld}cjtc .acd}cjtc% qibcqnc} el

tsdn j ojllc} qnjq qnc }jqei ih qnce} di}}ctwilaelb xj`sct }cojelt diltqjlq Qnjq et eh~

ku

? [ket

j witeqexc lsomc}_, qncl~ jlau j}c tjea qi xj}u ae}cdqù El tsdn j djtc ehu:,u

=j}c qnc xj`sct ih

u di}}ctwilaelb qi qnc xj`sct~:,~

=ih~ }ctwcdqexcù qncl : =

: =

~ ~

u u?


16/16

=:8 OJQNCOJQEDT

=, Qri zsjlqeqect~jlauj}c tjea qi mc el elxc}tc w}iwi}qeil eh jl eld}cjtc el~djstct j w}iwi}qeilj`

acd}cjtc elu .jla xedc/xc}tj% el tsdn j ojllc} qnjq qnc w}iasdq ih qnce} di}}ctwilaelb xj`sct

}cojelt diltqjlq Qnjq et, eh~u ? k, qncl~ jlau j}c tjea qi xj}u elxc}tc`u El qnet djtc ehu:,u= j}c

qnc xj`sct ihu di}}ctwilaelb qi qnc xj`sct~:,~

=ih~ }ctwcdqexc`u qncl~

:u

:?~

=u

=i} : =

= :

~ u

~ u?

13_direct and indirect proportions

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