MATH10 Lesson 2

8/17/2019 MATH10 Lesson 2

1/19

MATH10

ALGEBRA

RATIO, VARIATION AND

PROPORTION

Week 3 Day 1 Ratio, Variation and Proportion (Ale!ra and Triono"etry, #o$n %nd


2/19

TODAY’SOBJECTIVE

• )*e ratio and proportion in *ol+in pro!le"* in+ol+inte",

• -denti.y te di/erent type* o. +ariation,

• )nder*tand te di/erene !eteen diret +ariation andin+er*e +ariation,

• )nder*tand te di/erene !eteen o"!ined +ariation

and 2oint +ariation, and• De+elop "ate"atial "odel* $*in diret +ariation,

in+er*e +ariation, o"!ined +ariation and 2oint +ariation

At te end o. te le**on te *t$dent* aree4peted to5

Week 3 Day


3/19

Defnition

RATIO

A ratio i* an indiated 6$otient o. to 6$antitie*

E+ery ratio i* a .ration and all ratio* an !ede*ri!ed !y "ean* o. a .ration Te ratio o. x andy i* ritten a* x : y, it an al*o !e repre*ented a*

T$*,

Defnition

RATIO

A ratio i* an indiated 6$otient o. to 6$antitie*

E+ery ratio i* a .ration and all ratio* an !ede*ri!ed !y "ean* o. a .ration Te ratio o. x andy i* ritten a* x : y, it an al*o !e repre*ented a*

T$*,y

x

y

4y54 =

Week 3 Day


4/19

1 E4pre** te .olloin ratio* a* *i"pli7ed .ration*5

a 8 5 %0

! 94%4(5&4&4( %%

−++−

%. Write te .olloin o"pari*on* a* ratio* red$edto loe*t ter"* )*e o""on $nit* ene+erpo**i!le

a & *t$dent* to 9 *t$dent*

! & day* to 3 eek*

8 .eet to % yard*

d A!o$t 10 o$t o. &0 *t$dent* took Mat Pl$*

E:AMPLE

Week 3 Day


5/19

Defnition

PROPORTION

A proportion i* a *tate"ent indiatin te e6$ality o.to ratio*

T$*, , , areproportion*

-n te proportion x : y = m : n, x and n are alled teextremes, y and m are alled te means. x and m arete alled te antecedents, y and n are alled teconsequents

-n te e+ent tat te "ean* are e6$al, tey are alled

te mean proportional.

Defnition

PROPORTION

A proportion i* a *tate"ent indiatin te e6$ality o.to ratio*

T$*, , , areproportion*

-n te proportion x : y = m : n, x and n are alled te

extremes, y and m are alled te means. x and m arete alled te antecedents, y and n are alled teconsequents

-n te e+ent tat te "ean* are e6$al, tey are alled

te mean proportional.

n

m

y

x= n:m

y

x= n:my:x =

Week 3 Day


6/19

1 ;ind te "ean proportional o.

% Deter"ine te +al$e o. 4 in te .olloinproportion5

a % 5 8 < 4 5 %0

!

.25::225 x x =

4

1

x20

x=

−

E:AMPLE

Week 3 Day


7/19

Defnition

VARIATION

A +ariation i* te na"e i+en to te *t$dy o. te e/et*o. ane* a"on related 6$antitie*

Variation de*ri!e* te relation*ip !eteen +aria!le*

Defnition

VARIATION

A +ariation i* te na"e i+en to te *t$dy o. te e/et*o. ane* a"on related 6$antitie*

Variation de*ri!e* te relation*ip !eteen +aria!le*

Week 3 Day


8/19

Direct Variation

Wen one 6$antity i* a on*tant "$ltiple o. anoter6$antity, e *ay tat te 6$antitie* are directly proportional to one anoter

Let 4 and y repre*ent to 6$antitie* Te .olloin aree6$i+alent *tate"ent*5

• y < k4, ere k i* a non=ero on*tant

• y varies irect!" it 4

• y i* irect!" proportiona! to 4

Te on*tant k i* alled te constant o# variation or teconstant o# proportiona!it".

Direct Variation

Wen one 6$antity i* a on*tant "$ltiple o. anoter6$antity, e *ay tat te 6$antitie* are directly proportional to one anoter


• y < k4, ere k i* a non=ero on*tant

• y varies irect!" it 4

• y i* irect!" proportiona! to 4


De7nition pae 30&

Week 3 Day


9/19

Write an e6$ation tat de*ri!e* ea +ariation

17.d i* diretly proportional to t d=r en t=1.

19.V i* diretly proportional to !ot l and w V=6h enw=3 6nd h=4.

%& W i* diretly proportional to !ot R and te *6$are o. W=4 en R=1!! and =!."#.

(Exercises page 309)

E:AMPLE

Week 3 Day


10/19

1 -n te )nited >tate*, te o*t* o. eletriity i* diretlyproportional to te n$"!er o. kiloatt o$r* (kW

$*ed -. a o$*eold in Tenne**ee on a+erae $*ed30?9 kW per "ont and ad an a+erae "ontlyeletri !ill o. @1???, 7nd a "ate"atial "odeltat i+e* te o*t o. eletriity in Tenne**ee in ter"*

o. te n$"!er o. kW $*ed (E4a"ple 1 pae 30& % Hooke* La *tate* tat te .ore needed to keep a*prin *treted x un$ts !eyond it* nat$ral lent i*diretly proportional x. Here te on*tant o.proportionality i* alled a *prin on*tant

a Write Hooke* La a* an e6$ation! -. a *prin a* a nat$ral lent o. 10 " and a .ore

o. &0 C i* re6$ired to "aintain te *prin *tretedto a lent o. 18 ", 7nd te *prin on*tant

Wat .ore i* needed to keep te *prin *treted to

a lent o. 1&" ( E4eri*e %3 pae 1?1 .ro" Ale!ra nd

E:AMPLE

Week 3 Day


11/19


12/19

1 A !roter and *i*ter a+e eit (po$nd* tat +arie*a* te $!e o. te $!e o. eit (.eet and tey *are

te *a"e proportionality on*tant Te *i*ter i* F .eettall and ei* 10 po$nd* Her !roter i* F& tallHo "$ doe* e ei

(#o$r T$rn pae 30F

E:AMPLE

Week 3 Day


13/19

Inverse Variation


• , ere % i* a non=ero on*tant

• y varies inverse!" it x • y i* inverse!" proportiona! to x.


Inverse Variation


• , ere % i* a non=ero on*tant

•

y varies inverse!" it x • y i* inverse!" proportiona! to x.


De7nition pae 30F

4

ky=

Week 3 Day


14/19

1 Te n$"!er o. potential !$yer* o. a o$*e derea*e*a* te prie o. te o$*e inrea*e* (*ee te rap on

te !elo -. te n$"!er o. potential !$yer* o. ao$*e in a parti$lar ity i* in+er*ely proportional tote prie o. te o$*e, 7nd a "ate"atial e6$ationtat de*ri!e* te de"and .or te o$*e* a* it relate*

to te prie Ho "any potential !$yer* ill tere !e.or a @% "illion o$*e (E4a"ple 3 pae 30F

E:AMPLE

%00 &00 F00 900

%00

F00

&00

900

1000 •

•

•

•

(100,1000

(%00,800

(&00,%80

(F00,1F

Prie o. te o$*e (in to$*and* o. dollar*

D e " a n d ( n $ " ! e r o . p o t e n t i a l ! $ y e r *

Week 3 Day


15/19

Inverse Variation $it% Po$ersInverse Variation $it% Po$ers

De7nition pae 30

43o.po1erntteto

alproportionin+er*elyi*yor4,o.po1er nt te1itin+er*ely

+ariey tat*ay1eten,4

kye6$ationte!yrelatedareyand4-.

n=

Week 3 Day


16/19

Joint Variation an Co&'ine Variation

• Wen one 6$antity i* proportional to te prod$t o. toor "ore oter 6$antitie*, te +ariation i* alled (ointvariation.

E4a"ple5 >i"ple intere*t i i* de7ned a* • Wen diret +ariation and in+er*e +ariation o$r at te

*a"e ti"e, te +ariation i* alled co&'ine variation

E4a"ple5 o"!ined a* la in e"i*try,

Joint Variation an Co&'ine Variation

• Wen one 6$antity i* proportional to te prod$t o. toor "ore oter 6$antitie*, te +ariation i* alled (ointvariation.

E4a"ple5 >i"ple intere*t i i* de7ned a* • Wen diret +ariation and in+er*e +ariation o$r at te

*a"e ti"e, te +ariation i* alled co&'ine variation

E4a"ple5 o"!ined a* la in e"i*try,

De7nition pae 30

V

T

kP=

tPr-=

Week 3 Day

E:AMPL k


17/19

1 Te a* in te ead*pae o. a *oda !ottle a* a

+ol$"e o. ?0 "l, pre**$re o. % at" (at"o*pere*,and a te"perat$re o. %?9I (*tandard roo"te"perat$re o. J; -. te *oda !ottle i* *tored in are.rierator, te te"perat$re drop* to appro4i"ately%?I (&%J; Wat i* te pre**$re o. te a* in te

ead*pae one te !ottle i* illed (E4a"ple & pae 309

E:AMPLE

Week 3 Day

W k 3 D


18/19

S)**ARY

Diret, in+er*e, 2oint and o"!ined +ariation an

!e $*ed to "odel te relation*ip !eteen to6$antitie* ;or to 6$antitie* x and y e *ay tat5

•

•

Koint +ariation o$r* en one 6$antity i* diretlyproportional to to or "ore 6$antitie*

o"!ined +ariation o$r* en one 6$antity i*diretly proportional to one or "ore 6$antitie* andin+er*ely proportional to one or "ore oter

6$antitie*

k4yi.4toalproportiondiretlyi*y =

4

k

yi.4toalproportionin+er*elyi*y =

Week 3 Day

W k 3 D


19/19

Week 3 Day

LA>>WRI

HMEWRI

* %%, 3%, 33,3F, 3, 3?,&0,&%,&3,& pae 30?'313

* pae %0, %,&F,83 pae30?'310

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MATH10 Lesson 2