Key to Algebra Book 7 Adding and Subtracting Rational Expressions

Key to

&ebra Adding snd Subtracting Rations,I Expressions

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By JulieKingand PeterRasmussen

Name

Class

.at,!

TABLEOF CONTENTS Review .........1 ... AddingFractions witha CommonDenominator............ ...................2 The Oppositeof a Fraction Subtracting Fractions witha Common Denominator.......... ..............7 Addingand Subtracting Fractions withDitferent Denominators..... ................11 Combining Integers and Rational Expressions ..............14 Combining Polynomials andRational Expressions............ ............ 15 MoreAddingandSubtracting Fractions withDifferent Denominators...........................16 LeastCommonDenominators ......... ..............18 UsingCommonDenominators to SolveEquations .......23 Proportions.............. . . . . .2. 6 RatioProbfems. ............. 28 PercentProblems. .........30 TimeProblems .............. 32 Problems aboutRational Numbers ............... 34 Written Work ................. 35 Practice Test........ ......... 36 Youngand Giftedll Manymathematicians triedto solvethegenerallitthdegreeequation from 1545until1820,whentheteenagerNielsHenrikAbelsucceeded where theyhadfailed.He showedthatit is impossible to constructa formulato solveeveryequationof degreefiveor higher. Meanwhilein Franceanotherchildprodigymeta similarfate. gabWAH;l8l1-1832)leda rebellious EvaristeGalois(pronounced life. Hedidn'tattendschooluntilhewas12 yearsold;hismotherprovided him witha classicaleducationuntilthen. Evaristewasonlyan average studentuntil he took a.tnath coursewhoset€xtbookby the famous mathematician Legendiecaughthisfancy.Intwodayshehadfinishedthe bookdesignedfor two yearsof study. 'l knowit." Galoisthenhandedin histextbookwithtre explanation, He was right. WhenGaloissaidhe knewsomething,he meanthe had masteredit. Histeacher,Louis-Paul-Emile Richard,playedan impofiantrolein hislife,particularly sinceGaloisdidn'tgetalongwithmanyotherstudents at the boardingschool. Mr. Richardarrangedfor Galoisto take the examinationfor entranceinto France'smost famousuniversity,the Polytechnic Institute.Unfortunately theyoungGaloisfailedtheoralpaft of theexamwhenhe refusedto answera question,he feltwasimproperly posed. DoyouthinkthatGalois'attitudewasimpertinent?Shouldhe have challengedthe experts? Most mathematicians today agreethat his opinionwas correct,evenif his behaviorwasn'tright. So Galoisretumedto highschoolandworkedind€p€ndently under Mr. Richard.Soon,however,he becameinvolvedin radicalpoliticsand at onepointmadea threaton lhe king'slife. HEwassentto prisonforthe threat,and he remainedthere until ill healthforcedthe authoritiesto translerhimto a sanitorium. Whiletherehe waschallengedto a duel. He accepted. Thatwas verystupid. Galoishadno experience withpistols. The nelt morninghe was killedin the duel. Hewasonly20 yearsold.

Historicalnoteby DavidZitarelli lllustration by Jay Flom

:l,f

Thatis verysad. Fortunatsly Galoisstay€dup theentirenightbeforetheduelwriting his resultsin algebraand a letterto his bestfriend. In it he summarized outlinedth6 directiontrat his researchwouldlead. Wittroutknowingof Abel'swork Galoisprovedexacdythe same result_it is impossibleto constructa formulato solve fifth degree could equations.Buthewentevenfurther.Heknewthatsomeequations be solvedand otherscould not. He introducedgrorps of oblectsto whetheragivenequationof anydegrEecouldbesolvedor not. determine Galoiswas aheadof his time. Hisworkswerenot understoodfor some 25 yearsafter his death. Todaygraduate@ursesin 'abstract "Galois algebra'beginby studyinggrouptheoryand end by str.rdying theory.' On the coverof this bookGaloisprepareslor his duel. Unlikethe "duels'oftheRenaissance, pistols mathematical thisduelwasfoughtwith and broughtthe endto Galois'shortlife.

IMPORTANTNOTICE:This book is sold as a studentworkbookand is notto be used as a duplicating master. No part of this book may be reproducedin any form without the prior written permissionof the publisher. Gopyrightinfringementis a violationof FederalLaw. Copyright@1990by KeyGurriculum Project,Inc.All rightsreserved. @Key to Fractions,Key to Decimals,Key to Percents,Key to Algebra,Key to Geometry,Key to Measurement,and Key to MetricMeasurementareregisteredtrademarksof Key CurriculumPress. Publishedby KeyCurriculumPress,115065thStreet,Emeryville, CA 94608 lsBN 1-559s3-007-3 Printedin the UnitedStatesof Ameri:a 23 22 21 08 07 06 0s

Review In thisbookyou willlearnhowto add andsubtractfractions(bothrationalnumbersand rational multiply expressions). To do thisyouwillhaveto remember howto add,subtract, andfactorpolynomials, howto simplifyrationalexpressions and howto findequivalent expressions in higherterms.Testyourmemoryby doingtheseproblems. Simplify. 3a

3b

3a+3b __

=

a+b

2x* I

x-tl = x'-lb

x"+7x*12

Find an equivalentfraction.

1 = vl 5 l v

x =_ 12 lzxy

3

^4

=1*ffi11

Add.

(x"*4x-6)*(2x,+x-8)= ( x " * 5 ") ( x - l ) = Writethe oppositeof the expression in parentheses.

-(x'"8)=

-(3x'-5x*2)=

Subtract.

3 x- - 5 x =

( 3 x ' * 5 x ) - ( x ' * 4 x= )

-2a' -

( x . * l x * B ) - ( x ' -2 x - 5 ) =

6a' =

Multiply.Makesureyouransweris in simplest form.

(x*7)(x-l)= Bx' *, =

2x'(x- 5)=

5 y ' .( e i )=

Factorcompletely.

1 0 5= l 5 g= 01990 by Ksy CutriculumProiecl.Inc Do not duplicats withoul pormission.

3 x '- 2 7= xL + 3x - BB=

AddingFractionswith a CommonDenominator Whentwo fractionshavethe samedenominator we say thattheyhavea common denominator.Addingfractions witha commondenominator is easy. Denominators tell whatkinds of numbersor expressions are beingadded. Numerators tell howmany, so to addfractionswe just addthe numerators and keepthe denominators the same.

3 T

+

2 +

5=

+

t

+

-+

1 a

a

-=

3 a

Add.

_ 1 -

5

2

r

B

B

5

-7

-4

1 6+ - 2

+

6

6

=

5 q

5_+

7y 1

5

-l

- f

ty

5

+

3=

t

+ -

l

xl

= + _ x+3

n

5

X

3

=

4 x +. 3I v=

3

3

V + + =

=

-t + -t + - 3 =

8

x +3

+

3 1

+

8

a

+ -

2 1

3

r

o

x2

+

3=

1 2 3 + _ = a-l a-l

3

2

+ -

5b

B

3=

5b

7 - 4

xy

+

-

xY 7

2

+ xz-8

= ?('-8

+ r +

5

q2

- t . B x

q

62 + -

1

=

3

+

3

=

Proiecl,hc. @19$ by KeyCurriculum Do nol duplbalewilhoutp€rmission.

Sometimes afteraddlngwe cansimplifythe answer. 5

8 . 1 9 q

I

-2x

5y d

7- x=

+

5y

t

n

r E= 5 r' g 3Y

=

tzY

,

.

L

?

t2-Ll

4 =

f.-4

= x V I

x+z xz- t-l

I

t)z

=

=

l-*4lh-2) I

-

I

t-2

Add. Besureto simplify eachanswer. -3

tz

5 -l 6 + E e

5=

+

tz

+.t,*= - 131v - + _ =

y*4

I++ l

=

+

xt+6t+8

n"+2n

=

at-l

|

5n

+ tlr7

3 r q 4 x +1 2 - 4 x + 1 2

12

+ at-

th

2t

2

3i*It= -5

3 +J =

2x

4

+

5

1z+gx+g

n

nt+2n

=

-q b2+2b-3 Ot990by KeyCufiicrlumProiect,Inc. Do not duplicatewithoutpermissbn.

b z+ 2 b - 3

3

Add. Makesureeachansweris in simplestform.

3

-

?

T

Y

8

2

-

=

Y

31

Y

5v

+ + =

2x

2t

5r

x2+6x

x2+6x

)

5x,

x2-1

-?

V

J

r+rl

x

x+{

-

* xllx -4= -5n

5x2 x2-4?"+3

lfx

xz-4r+3

4x

3x-6

3r -6

-x

3x

Xs+tl

=

x2-q

-ttt

3x2

-

11+5t

a

x-

=

-2* -

d2+4

2 Oz+ 2a + |

4

-6

1z

a

+

tt+4

2 At +2a+ |

=

O19S by Key Cufriqrlum Projecl, Inc 0o not duplicats wilhout p€rmission.

The numerators of theserational expressions arepolynomials. Add. Thensimplify the

answerif youcan. qdd

odd

x1 + 5

2x"-4 "^-3 + x - 3

? ( +l o +

N +z

2x+3

t

x+ |

/ t * + * L - - * Z y= Y*f Y *| 2 -vz 3 vt

T

v 2 + 7= 3 lv

2 x- 3 . x - l ?

x+6

=

x+b

.' - 2E t z - a Ja(x + Zl 3 x( r + 2 ) xz+1x

+

- x z- x ( r t +l ) ( r - 3 ) Ztt+5

(r+l)(r-3) 0199O by Ksy Curridrlum Proiod, Inc. Oo nol dupli:ate without pormission.

7+x-xL

(x+l)(x-3)

5

The Oppositeof a Fraction Subtracting rationalnumbersandexpressions worksthe sameway as subtracting integers. To subtract,we addthe opposite- so we needto knowhowto findthe oppositeof a fraction. The opposite of a fraction has the oppositenumeratorbut the same denominatoras the originalfraction.

-3 - 3 = r f t -6 =6

,,Theopposite ot I is f :' ,,Theopposite of * u *:

? ( x - r + l

x - l :(ar3

= jJ

r--J

,,Theopposire ot **

j'

;s ;f,i{

Writetheoppositeof eachfraction. 5

x-l 3

=

8 -5

-

- - =

8

o +3

=

o-3

=

_-c-r = x-l

r+ | =

3

_g=

x z+2 x - 3

x

_ 2 x +4 = x

-i1 =

_ 2x-4 =

3x'-x++

x

x

4

-?( - - =

lo

x

x-2

xa-l

-

=

5

^2-2x+l

=

Add. -5 + -

8

s=

8

- x x

z 6

?

-

=

2

2-!:l 5

^--2!:'7 5

t.-tfi+3 t+f

+

-Xe+tlr-3 ?(+l

=

0199 by Key Cutriculum Poj6ct, Inc. Do not duplicats rvithoul permission.

SubtractingFractionswith a CommonDenominator fromanother: expression Tosubtract onerational sign. signwithan addition 1. Replacethe subtraction 2. Replacethe second fractionwithits opposite. 3. Go aheadand add. Subtract.

2x -3x + - = 5 5

t 2 5

-{

E

z _tl 15 t5

x+5

x

6

6

=

'2t +'5

*T

n+I

a+r{

a-Z

1 3x ? ( 5=

2x

4r

I _3^= ttr 4r

-L= **y r(*y

x + 3_ I = 1 q

x+4

{ L +

x+5

6x

=

l3x

x+3

@1990by Koy CuriqJlum Prolecl, Inc. 0o not duplbate without p€rmissbn.

=

?(

3r-l

=

r( -f,

2a-6 = a-Z

Zx

- x-2

T

2x+3 x+f

x2+3x-l

?(

t+3

x+3

x" -5x+2

-

?x 7

Subtract.Simplifythe answerif you can.

3x-{

2x-5

3x+3

3t+3

,(+y

=

- . -? ( - y

2x

2x

3x+2 a

3x

4x -12 h

3x

x. + 5x

xz +5r(

lz + 4r(

x+rl

a-5 =

a + 5

qb

ab

-2(

4x

t5

t5

6a 5or+a

=

a-l 5ar+a

t ( z + 3 n - 5 _ xz -2x + lO

lo

8

lo

o l9S by KoyCuriculumPtoied,lnc. wilhoulpomissbn. Oonotduplicato

Hereare somemorerationalexpressions for youto add. Thistimeyouwill haveto multiply the factorsin eachnumerator to findout if thereare any liketermsyou can combine. =

5r-t5 7

3x+18 7

-

-B r + 3

7

{"-2) t-l

5 ( z <2 +) l2x 2(x -3) tOrt

b("" !_) ab

3(r + t{)

x"-|.6 6(a* l) a"-tf

a(c-l) + -

c.-$

=

k+ 7)H-2) + ( x - 5 ) ( x + 2 ) ( x + t i )( x - 3 ) H ++ ) h - 3 ) @1990by Koy Cuniqllum Proiod, Inc. Do not duplicato wilhout pormission.

9

problems Thesesubtraction willbe easiestif you multiply the expressions in the numerators first. Thenchangeeachproblemto an equivalent additionproblem.

3 ( x * l ) _ 2 ( r . - 3 )= 3 x + 3 _ 2 r-5 5 5 5 5

=

. -2^+6

. ? - =

r+9

5

5_I I 4(r - 5) 2x

5 (rr-tf)

2x 2(x + 4)

6fx-l)

,x

^'

tz

?(

( t + t ) ( x- l )

t2

( x + 3 ) ( x- 3 )

r(x + l)

2a+b

ab

Lh-b) _ ab

2 c ( c + 3 )_ c-5 r(t+l

Y. 10

.lc * +) c-5

(x+l\(t - l)

Y, O19S by Key Curridlum Proiect, Inc. Do nol duDlicalewithout o€rmission

Adding and SubilactingFractionswith DifferentDenominators Nowwe willadd andsubtractsomefractionswithdifferentdenominators. To do this,we firsthaveto findequivalent fractions witha commondenominator. ln eachproblembelowyou firstneedto rewriteoneof the fractionsso thatthe two fractions havea commondenominator. Thenyoucango aheadandcombine.

2'2x

21

+ t = tlr7

,

x =5- r 6 5

l

3

4'6

=

2 _ 3= 3 2 1

-+ + :?-

x +5*= t6 8

5 _+

5x

x + 2 3 x x

5 + ? = 2 x x

5x +6

t 5 5

=

l5x

R

=

x2

z^l

'r(

3

9

a\

2 a

|

3 a

?

b

)x

6

-

4

:

=

b

4 +. 3

Y 'r" 'l

@1990by Key Curric1llum Pfojoc{,lnc. Do not duDlicalewithout Dsrmission.

=

11

Add or subtract.

3 + 7 = 2 ' 3 - + / ' = , , 6 , ' . +. 2=kr+x2 =1 , = = 9 ' 4 - , 2h+21 26*21 x+2 2h*2) 2H"2)'2k*21

5 x- 7

3 Ik -71

5 ^ * f k**){rx-t) x - T 3

l

x+5

T

o

h*2)h*5)

6

6k-2)

5

7-Z*=ffi 7(x-2)

2

5

+

x

6x-L,

-L

ffi21=mA+ffit=m

6x-t'l

=

az+Z*

3x+l

Z

X2+6x+9

X+3

r

o

=

+

x-6

x'-36

x

2

x2-9

l -3

l O s'* 6a

a

3

x

2x'-8x

12

5

=

3 a*6

.T

=

Z x-4 o19S by Key Curiculum Pnq.d. lnc. 0o nol duplicale mhoul pt rEaon.

Sometimes denominators thatlookdifferent Be on the arereallyequivalent or opposites. lookoutforthese. lf theyareopposites, it is.easyto makebothdenominators the same.All we haveto do is to multiply onefractionby T. Addor subtract.Lookfor denominators thatareequivalent or opposites.

3

Trt

3++r

2x+5

5+2x

5+2x

e l u i v o le n t

5x 3-x

7x

x-3

7x

7x

^-3

^-3

-5r t . 7 =

x-s

2x x-3

opposites

2 x , .?

3x-l

5x l-3x

(+3 5+r

2x

x-3

-x+3

t2

x +8

t{ -x

x-4

01 990 by Key Curiculum Projscl,Inc. Do not duplicate withoul pormission.

13

CombiningIntegersand RationalExpressions theseintoa single wecanwriteZ * 1 as 2| orwecancombine Inarithmetic

expression. fraction,f . In algebrawe canalsocombinean integerwitha rational Combine.

l' + ?3 =( ! 5x + ?. 9 ,( = a! , "a9 1 a+ 3 l

3 z- r+

-

B-

3

4

=

2+-l.,= W * *, = +? * h ='i:i' |

5 Y +

r1

=

x+2

x

+

=

x- |

2x - l =

x+3

1+ Ll I ".2

2 = x

+ L{ =

x-y

; l

l +x

r

l 1

z

z+ J =

x t x 14

=

+ 'l =

tt +x ! ? +( | z =

* - r . f= Inc. @19S by Ksy CurriculumProiec,t, Do nol duglicatewithoulparmissaon.

GombiningPolynomiafsand RationalExpressions We cancombinepolynomials and rationalexpressions, too. Firstwe haveto writethe polynomial as a rationalexpression withthe samedenominator as the otherexpression. Combine.

-st v *yZ =J ' f y *Z = t * L Y I Y I

=*1!z Y

2 x * x! =

o * t=

b-t = ( x *l ) -

*

=

) ] . rx-5=

(x* 3)* #, - '.lip

* #= ^'*kJ +ft= ^'.#;t

( x *l ) - # =

l L x - T - Z+x - ? = I

ol9s0 by KsyCuiriculum Proi6ct,Inc. Do notduplicatswithouloormission

15

MoreAdding and SubtractingFractionswith DifferentDenominators Foreachproblemon thispageyoufirstneedto rewritealllhefractions so thattheyhavea Youcanalwaysfinda commondenominator by multiplying the commondenominator. natorstogether. denomi

3.?. 1.5 +iF= 3.5 3.5 t5 r + l

7

B

2 _ 7=

=

8

2 x x 5 4 l

Y 7

=

3 t 4

4 _ 3 =

x z y

t+ r

^+ 3 _

q

_

?

'

x+ 2

+

=

(

=

5y 5 12

16

3

a

+ ; = p

r

4

3

5

x

x 5

_

x

=

9._7=

=

4 x ' 3 3x

q

-7 + - 5=

=

5

3x 5

- l -+ 2 = 2 3

3

a

y

f

L-

b

{+

-

y"

5v

+

2

3x

a

3

l

?

+ 7 =

=

^12

=

c

z _J= 7z

+ -+=

I

3

=

2x

019$ by Koy Curiculum Projocl, Inc. 0o not duplbale withoul p€rmissbn.

Finda commondenominator. Thenaddor subtract.

a

ll+ - 3 r ( r + 5 ) ^- 7 ' 2 _ 3 1 1 + l 5 x'= = Fs 5ffilf iiffi) ftffi zJffi)

3

5

3 x -= 2

37iz+l$r+l$

=

6 - a-2

*' 43n

Z

=

9X

x-l

2

.?

x+3

=

x+l

x+ tf _

x

x+3

1+3 , r( T x+5 t+2

k+Z)(x+?l- x6+5)__r.+5r+6 ,

(ffit=

=

ffi+

rl+Sx

fl=6;timi)' 1ffi

_2rr+tor+6

j1= ip5i;;55

x+t{ 3 = + x-4 rr3

r-4

x

x+6

x+7

f

. x-l

=

=

x+Ll

0199Oby KeyCurriclrtum Proisd,Inc. Do notdupli;atewithoutDormi6sbn.

17

Least CommonDenominators Theproduct ofthetwodenominators is notalways theleastor simplest denominator wecoulduse.Thinkaboutaddingfi andf . lf weuse12times8, or 96,as a common denominator, weget 2*Z = -m s ' q "* i4Z : t g= I + o . 8 += t, zl bL l rz-g %*TE Wecouldhaveused24as a common denominator. 5 *J =Y * il = l9 *21 = 3l t2

t?.z

I

3.8

211 24

2q

Theanswers areequivalent, but 3l ir simpler. Addor subtract eachpairof fractions.Tryto findthe leastcommondenominator. i l- - 4 = 6 t 5

?

3 20

-# =

q

b

20

2+

=

| * ! =

l?x 8x 3 . 2 2t

5 ZxL

t x2

5

6, 18

=

r(2

+

3 6^y

=

+

+ _ =

xy

- r lo = @19Qby KsyCurriculum Proioct,Inc. 0o notduplicalswilhoulpormissbn.

Didyoudiscover a wayto findthe leastcommondenominator? Hereis a methodthat alwaysworks.We factorthe denominators. Thenwe multiply by the eachdenominator factorsof the otherdenominator whicharemissing.We multiply by those the numerator factors,too. Findthe leastcommondenominator. Thencombine.

7r.tr=

.2++=?.

5 2.2.3 ooo

8 +' - 2= 35 t+5 ?

2

=

t-15

3 + = 3 +' - L l - b ' 6 - k ; 5 bbc ooo

oo ^/^^1 c b Jleeds )

N e e d so n c .

\*:/

2

4

-

F

lxz

l5r

8

Z

abc

bcd

l

l4x"

3

+

?(

5 a'b

=

=

+ - l

Zlty

2

r

at

13

=

=

3 = ab'

@1990by Key CurriculumProiecl,lnc. Oo not duplicate withoul osrmission.

19

Findthe leastcommondenominator.Thenaddor subtract.

2

x

3.x = q,*6-a-*,. I m)-

2.2

ffil=

zfzx*1 3(zx*a) t

5

x" -3x

4x-l?

m

3x

-+ 3x'+* mt= m)

3 4 + = x- | x"-l

3x-6

x'-4

3

+

x'-25

5 x"-9

4 nz-4

+

5 2x+lO

2 x, -bX+9

nz-5n+6

+

, X z+ 6 X * 5

3x2-l?

20

z

+ _

5

=

+

=

=

1r + {1 + t

z 4x+ 8

=

=

o'l9s by KoyCurriorlum Proiocl,hc, Oo not duplicatewithou(pormbsion.

Add. 2

4 x-6

+

6-x

=

- 7+ a+2

x +3

t0 x'y

+ =

+

xy"

I + x' - 3x

x + + =

v

3 + 2 + -t+ =

a

x"-l

b

+

ob

x'+6x-7

Y + 3 y *| @1990 by KeyCurriculum PO€cl,Inc. DonotduolbalewithoutD€rmission

21

Finda commondenominator andadd. Thensimplify.

+ r- , + 3

t5

x+6 x+5

3 + x-2

+

o-2b = 2sb

x + 2 + -x - 4 = x2-a, x-l

5-x

+ -

5-2x

=

4x+{

5x-25 22

Projocl,Inc. OtgQ by KoyCurriorlum withoutpermbsion. Do notduDlicate

UsingCommonDenominators to SolveEquations Commondenominators are usefulfor solvingequations whichcontainrational expressions. In the equationbelowtherearethreefractions.The leastcommon denominator forthesefractions is 612.lf we multiply bothsidesof the equation by the commondenominator, we willgetan equivalent equation withoutfractions. t t z ' l = !.ooftleleost'.oilondenominotori.ex) + + 2x 1z 6x-

3r

6x"(* + Zx ' /)

.b* . - . - I l 2 x

+

6 * Z l *

=

=

7

6x 7

8x

'bx' x

,& I

3 x + IZ = 7 x

l2 = Hr oo x = 3 0 Solveeachequationby multiplying bothsidesby the leastcommondenominator.

? =L x

x

3

@t990 by Koy Cutriculum Projsc{, Inc. 0o not duplicatewithout Dsrmission

3

.+=Lt

x r ( l + 5 f Z = 5

br{

*f--0

23

Solveeachequation

x+| = x _ | B 2 t +

3

t

=

l

Tx-; 4

7 2

2

v5 t+ (=

3 c<-l

24

=

t

t 2

+ x-l

x+2

?( = !

t 2 z

x.#=ll

1 _ 1 - q 2 2 ^

^ 1 0 5 3'3(x+4) x+Lt

@19S by Key Curricllum Projocl, Inc, Do not duplicato withoul p€rmisgion.

for youto solve. Hereare somemoreequationswith rationalexpressions

V ( f i " +j * ) = ( x * t ) t 2

0 . 8 x= 2 * 0 . 7 x

3*- #*=t

0 . 5+ O . l x= O . B x- 0 . q

o

o o

f O ( 0 . 8 x=) ( 2 + @ . 7 x 1 l O 8x = 2O+7x x=20 0.23x*2.25=O45x-l.05

0.2x-0.08r=1.2

@1900 by KeyCuricllumProjoct,Inc. Donol duplicate withoutp€mission.

0.7x- 0.66x= 2

1 . 2 4 x3=. 2 * 0 . 6 x

25

Proportions equals whichsaysthatonefraction equation A proportionis a verysimplerational another.Eachoftheseis a proportion: 15

x

+ 2 0

2

3

= -

=

5

r+|

1

t

=

+ x+2

by multiplying the if we geta commondenominator Lookat whathappensto a proportion togetherandthenuseit to simplifythe equation. twodenominators 3

1

1 5 z =T

5

4=To q

0

-

_

z ')8 = )s t *

Bs.fr=fr. 8o 2

4

9x= 5'2

2 0 ' 3= l 5 ' +

has"moved" to the othersideof the equation. It looksas if eachdenominator

lvl:

;Y*

+.t\20

? x= 5 ' 2

Z O ' 3= 1 5 ' 4

we endedup. Sincethisis alwaysthe pattern, Thearrowsshowwhereeachdenominator by the opposite eachnumerator usuallyskipthe firststepandgo rightto multiplying We callthiscross multiplying. denominator. to solveeach makesit easyto solveproportions. Usecrossmultiplying Crossmultiplying proportion below. 1 =2 x lto 20

- lO

= ?(

8 t L

29 = 15 B 1

= 16'r 20'',2 2+0= l6x x=15 ? (= 9 r + 8

26

g -= { * l 3

6

@19$ by Key Curiculum Projoc,l,Inc. Do not duDlicatowithoul pormissbn

Solveeachproportion.

!g= 15

x 6

x-6

x

r+8 1 8

= -

g= l+ lt

x+3

1 0 ( r - 6 )= 1 5 ' a lOt-bO = l5r -bO = 5l ?( s 'lZ

3

=

2

7 a 5

t+ 5

- = f -

P = 7

P+r+ 6

v-3 Y+3

x

l

1

x

x-5 x-7

?(+= l z 3 x

-3 n-2

3 n +2

^

=

x+6

r-2 ^+Z

1 = 1t t a

la= l+ tr-rl= O

(r - Z l ( x + Z l = O

x-2=O | ".2=O x'2 | x''2 r+ x+6 -3 x- |

01990 by Key Curiiculum Proioct, Inc, Oo not duplicats without pormission.

d

3

=

5

x-7

2 0= ^ ?

(

5

27

Ratio Problems Wecanuseproportions to solvemanyproblems thatinvolveratios.A ratiois a fraction whichcompares onenumber to another.Theratioof 2 to 3 ir 3 . Wecanwritea proporlion if we knowthattworatiosareequivalent. Writea proportion foreachproblem.Thensolvetheproportion to findtheanswer. lf it takes5 gallonsof gasto drive90 miles, Thecafeteriaused28 bottlesof ketchup howmanygallonswouldit taketo drive in 18days. Howmuchketchupshouldbe 144miles? orderedtor 45 days? qallonr Therolb of {o niht is dwtir lfr ron

Equation:

t

I

xoo

,+'}

l++,5 ' 1 0 ' x 720 = i0x x = I Answer:It tokes I gollonsto drive f$$ miles. A ballplayergets11hitsin 40timesat bat. Howmanyhitswouldyoupredictin 1000 timesat bat?

is Theratioof sandto cementin concrete 5 to 2. Howmanyshovels of sandshould of cement? be mixedwith8 shovels

Equation:

Answer:

Answer:

A typistcantype300wordsin 4 minutes. At thatrate,howlongwouldit taketo typea 13,500worddocument?

lf 12 lb.of coleslaw willfeed100people, howmuchwouldyou needto feed40 people?

Equation:

Equation:

Answer:

Answer:

28

@199 by Key CurriculumProjecl,Inc. Oo nol duolicatewrlhoulosrmission

Solveeachproblemby usinga proportion. The ratioof redto browncandiesin a wellknownbrandis 2 to 3. Chrislikesthe red ones.Jamieprefersbrown.Howmany willChrisbe likelyto get if Jamiegets45?

Thelabelon a 48-lb.bagof lawn fertilizersaysit covers15,000square feet. Howmanypoundswillbe needed to cover50,000squarefeet?

Equation:

Answer:

Answer:

A car'stripmeterregisters 10.2milesfor a 1O-mile measured course.Whenthe meterregisters 153miles,whatdistance hasthe car actuallytraveled?

A cakerecipecallsfor 2 eggsto 1| cups of flour.Howmanyeggsshouldbe usedwith9 cupsof flour?

Equation:

Answer:

Answer:

The ratioof inchesto centimeters in a measurement is about2 to 5. Maria's waistmeasures 70 centimeters. Aboutwhatwouldthatbe in inches?

The ratioof left-handed peopleto all people is about1 to 10. Howmanyleft-handed studentswouldyou expectto findin a highschoolwith450students?

Equation:

Answer: @1990by Key Cuniculum Projoct, Inc. Oo nol duplicatg without p€tmission.

29

PercentProblems Thewordpercentmeans"hundredths" is a ratio or "outof a hundred." A percent witha denominator of 100andso canbewritten or a decimal. aseithera fraction 73/o,fr and.73 allstandfortheratioof 73to 100. Oftentheeasiest problem wayto solvea percent isto writea proportion. Writea proportion foreachproblem. Thensolvetheproportion to gettheanswer. 32 outof 80 peoplein linegotticketsfor theconcert.Whatpercentof the peoplein lineweresuccessful? 32 oulof

Concertticketscost$26each. 22o/"went to charity.Howmuchwentto charity?

80 is

32 = . Equation: fr l . 0 0 . 3 2= gZ0O'

x c

ceuivofcnlfo whot nJmbrrovlof 100?

i1-p" 80.x B0r x = T 0

Answer: 407oof the pcopfcwere successful in gtting f iclets.

Answer:

$15of Dan's$125wageswas withheld fortaxes.Whatpercent of hiswageswaswithheld? Equation:

It 12/" is alwayswithheld fromDan's wages,howmuchwouldbe takenout if he earned$220? Equation:

Answer:

Answer:

A surveyhasshownthat 20h ot teenagers thinktheyareoverweight. In a classof 325,howmanywouldthisbe? Equation:

Thepriceof gasoline was960a gallon beforeit wentup 25/". Howmuchdid it go up? Equation:

Answer:

30

@19S by Ksy CutriculumProiscl, Inc. Do not duplicalo wilhout p€rmission.

Solveeachproblemusinga proportion. A jacketis on salefor $68,whichis 80% of itsoriginalprice.Whatwasthe price beforethe sale?

Theawouldliketo leavea 15%tip. Herbillwas$5.40.Howmuchshould she leave?

Equation:

Equation:

Answer:

Answer:

High Thenumberof students at Plainview wentfrom520to 624in justoneyear. Thisyear'sstudentbodyis whatpercent of lastyear's?

Ted readon a cerealboxthata serving whichis 4/" ot contains 3 gramsof protein, the proteina personshouldeateachday. to this,howmanygramsof According proteinshoulda personeat daily?

Equation:

Answer: Readthe problemaboveagain.Thistime In the sciencelabShadfoundthata candle figureout whatpercentlastyear'sstudent used60 out of 270ml of trappedair before goingout. Whatpercentof the airdidthe bodywasof thisyear's. candleuse? Equation:

Answer: O 1990by Koy Curri@lum Proiecl, Inc. Oo not duplicate wilhoul pormissbn

Answer: 31

TimeProblems I con'l 9o,Jaa. I've 9rl 6 hoursof wort hercto fiaish.

f

'll

You'vcqol lo be kidding;Iock.IfI it shouldfa&elcs; ihan your 6 jvvrs.

hclp.

Riqtri!Wc sharldbc abic+oat feastcul your tinre in holf.Thot't

(,0

Howfongwouldit takeJackandJan workingtogether?We canwritean equationto find the answer.All we haveto do is to thinkabouthowmuchof the workeachonecando in an hour. Total Time

Think:

AmountDonein an Hour

Jack

6 hours

!

Jan

5 hours

1

together

r hours

I

6 5 r

lf we addthe amountseachonedoesin an hour,we shouldgetthe amount theydo togetherin an hour.

Equation:

! + _ l _= l 6 5 x

30^(*

.+)= (*) 3ox

5 x+ 6 x

30

llx

30

X

30 c6 = l l =- / l - l

Answer: Jan was right!Theycan do thejob togetherin lessthanthreehours.

32

Proiocl,Inc. ol9$ by KeyCurricrrlum Do not duplicalewilhoulpetmissbn-

Writean equation for eachproblem.Thensolveit to findthe answer. Jondelivers in 3 hours. Rexeatsa bagof dogfoodin 12days. all hisnewspapers It takeshistrainee4 hoursto coverthe Dukegoesthrougha bagin 6 days. How route. Howlongwillit takethemtogether longwillonebagfeedthetwo of them? if theystartfromoppositeends? Equation:

Tracycan mowthe lawnwitha power mowerin 20 minutes.Haicanmowit with a handmowerin 50 minutes.Howlong willit takethemif theycooperate?

an issue Theoldduplicator couldproduce in 2 hours.Thenewone of the newsletter takes40 minutes.Howlongwillit takeif bothmachines areused?

Equation:

Equation:

Answer: O1990 by Key Curricllum Proioct, Inc. Do not duplicale withoul gormisbn.

33

Problemsabout RationalNumbers Makeup an equationfor eachproblem.Thensolvethe equationto getthe answer. "l'mthinkingof an integer.Whenhalf "l'mthinkingof a number.lf youaddhalf of thisnumberanda thirdof thisnumber, of thisintegeris addedto a thirdof youwillget 35. Whatis my number?" the nextinteger,the resultis 7. Whatis the integer?"

x -, - x = 3 5 2 3

Equation:

( i . $ )= 3 5 . 6 3x+2x 5r Answer:+2

2lo 2lo

X = 2to = * 7 5

"l'mthinkingof a number.lf I addthis I get 2. numberto its reciprocal, Whatis the number?"

Answer: "lf I takea fourthof a numberawayfrom I get10. hallthenumber, Whatisthenumber?"

of ancientGreece.A legendsaystherewasa wasa famousmathematician Diophantus numberpuzzleon histomb- apuzzlethatyou can solvethe way you solvedthe other problemson this page. "ThistombholdsDiophantus.Foronesixthof hislifehe wasa boy,for one twelfth,a youth. Afteroneseventhmorehe marriedandfiveyearslater hada son. Theson livedonlyhalfas longas hisfather,anddiedfouryears live? beforehim.' Howlongdid Diophantus Equation:

Answer:

34

o19S bt l(oyCurtiqrlumProiod,Inc. Do not duplitatowilhdJtpermiE8bn.

Written Work Dotheseproblems onsomecleanpaper.Labeleachpageof yourworkwith yourname,yourclass,thedate,andthebooknumber.Alsonumber eachproblem. Keepthiswrittenworkinsideyourbook,andturnit in wlthyourbookwhenyouarefinished. Pleasedoa neatjob. 1. Findthesum,difference, product andquotient of eachpairof fractions. E

t

f andi

a

2 _ ^ - 4

,

il d and;j= 4

;ano;

4

S

a

b

g and..'-fo r + 9

ffiand,.-z

-

r - f r2

m

, r - 4

and;:fu-

2. Simplify. 2

4

3

5+5-A 5

1 0

i+;+p 1

-

3

2

3*-; 3

a+a+aE i +x+y 1

1

1

1

x + 1

4

5

r - 5

r + 4

3 x - 1

4

+ffi x - 1

ffi-dffi

r 3 r + 2 - 12+2r

t

6i-6i-1t7

x + 9

- -

2

* f i r y1 - f f i

fr

r

.

x2+2r-48 -

+

1 2

12-4x-12

2r+4 x 2 - 1

3. TerryandSandyhadanother argument. Sandysaidthattheanswerto this problem is 11. Hereis Sandy's work: q

l \ I l s ; * t ^ =z ^ l i * n ) I

= 4.2 *4.. I a x l %

=lO*l : l l TerrydisagreedwithSandy'sanswer.Whowas right? Explainwhy. 4. In a surveyof 300students,117 saidtheywouldliketo be liketheirparents. 183saidtheydefinitelydid notwantto be liketheirparents.Writea proportion andthensolveit to findthe answerto eachquestionbelow. a. Whatpercentof the studentswouldliketo be liketheirparents? b. lf theseresultsaretypicalandyou interviewed 500otherstudents,how manywouldyou expectwouldnotwantto be liketheirparents? O19S0by KeyCu|rhulumProl€ct,Inc. Do nol dupli:de s'ithou p€rr*a6bn.

35

PracticeTest youranswerif youcan. Add. Simplify 7 . 8

5'5= 6 x+5

+ _

l z

=

x+5

8 a

5

v

3

2,x

2r

x-2

a+6

x+Z

x+2

-3 + - + =

=

a

P

x . f ,= 2 x+lO

t

=

q L)

1 +

v

3 5x*50

z

z

x'- I

x+3

x

z

x+tf

x-l

=

?(-t

=

=

=

Subtract.Simplifyyouranswerif you can.

l.t 1

2 1

2

- =

5

a-l

2=

2 n n

3 2x+l a , +2 x -4

36

x

x 2?(+l

-

5

?(++ x-5

=

-7

a-l

t

3

+

2x

6x"

x x-3

5

1,

=

=

x x 2- q

=

3 x +| 019S by Key Curticulum Ptoioci. lnc. Do not duplixto wilhout p€rmission.

Solveeachequation.

L 2

3 2x

x-l

x

4

Makeup an equationfor eachproblem.Thensolvethe equationto getthe answer.

lf twoincheson a maprepresents 25 miles, 45ofthe75seniorshavejobs. howmanymileswould7 inchesrepresent?Whatpercent of theseniorsareworking?

Answer: Patcanshelvea cartfullof booksin the libraryin 24 minutesand Kevincan do it in 30 minutes.Howlongwillit takethem if theyworktogether?

"l'mthinkingof a number.lf youaddhalf of thisnumberto a sixthof thisnumber, youwillget 12. Whatis the number?"

Answer: 0199 by KeyCuricrrlumProiect,Inc. Do rd dr,pli:atewilhoutDomissbn.

37

Book L: Operationson Integers Book 2: Vsriables,Termsand Expressfons Book 3: Equations Book 4: Polynomials Book 5; Rqtionsl Numbers Book 6: Multiplying snd Dividing Rational Expressions Book 7: Addingand SubtractingRstionql Expressions Book 8: Graphs Book 9: Systemsof Equations Book 10: SqucreRootsand QuadraticEquations Answersand Notesfor Books l-4 Answersand Notesfor Books5-7 Answersand Notesfor Books8-1O

Key to Key to Key to Key to Key to Key to

Fractions@ Decimals@ Percents@ Geometry@ Measurement@ Metric Measurement@

ilh

CURRICULUMPRESS KEY - \ in Mathematics Education Innovators

'@-

mn ilililru