Math Review Packet you answer all the questions? Does you answer make sense? Did you work the...

WordProblems

Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafterreviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.

SimplifyingWordProblems

Mostofusareintimidatedbywordproblems.Butifwetakeawaytheextrawords,alotoftheconfusiongoeswiththem.Usethesefourstepsasanoutlinetoguideyouthroughtheproblem:

Step1‐UnderstandtheProblem.Readthroughtheproblemtogetageneralideaofthesituation.Whatishappeninginthestory?Whatquestion s doyouneedtoanswer?Whatinformationcanbedisregarded?

Step2‐DeviseaPlan.Readthroughtheproblemagain,lookingfordetails.Decidewhatoperationstouse addition,multiplication,etc. .Setuprelationships,translatefromEnglishtomath,makelists,drawandlabelpictures.Useanystrategyyouknowtoorganizetheinformationintoaworkableformat.

Step3‐WorkthePlan.Nowthatyouhaveaplan,workit.Dothemath,crunchthenumbers.Someproblemsrequireseveralsteps,sobesuretoworkthemall.

Step4‐LookBack.Nowthatyouaredone,areyoureallydone?Didyouanswerallthequestions?Doesyouanswermakesense?Didyouworktheproblemcorrectly?

Someextraadvice

Manywordproblemshavemorethanonepartthatneedstobeworkedout.Breakdowntheproblemtoitsindividualpartsandworkoneatatime.Ifsomethingseemstobemissinginordertoworkasub‐problem,setitasideandworkanotherpart.Youmayfindthemissinginformationasyouworkothersub‐problems.

Problemswithlistsofthesametypeofthingtendtobeadditionproblems.“Johnworked5hoursonMonday,12hoursonTuesday,6hoursofWednesdayand7hoursand30minutesonThursday.Howmanyhourshasheworkedthisweek?”Besurethattheunitsmatchbeforeadding.

o Units:hours,hours,hours,hoursandminutes.

Problemswithmixedunitstendtobemultiplicationordivisionproblems.“Anairplanetravelsat350milesperhour.Howfardoestheairplanetravelin6hours?”

o Units:milesperhour,hours,miles.

Vocabulary

TherearemanykeywordsinEnglishthatcanhelpusdecidewhatmathematicaloperationorrelationshiptouse.Herearesomeexamples.

Words,Phrases Symboloroperation

plus,sumof,addedto,joinedwith,increasedby,more,morethan,and,total

minus,differencebetween,subtractedfrom,decreasedby,reducedby,less,lessthan,exceeds,takeaway,remove,purchase

times,timesmorethan,multipliedby,product,equalamountsof,of,at,each,every,total

∗

dividedby,divides,quotient,ratio,compared to,separatedintoequalparts,per,outof,goesinto,over

equals,isequalto,is,isthesameas,was,were,willbe,resultsin,makes,gives,leaves

twotime,double,twice 2 ∗

half,halfasmuch,halves,onehalfof 12∗

What number,part,percent,amount,price,etc. ,anumber,thenumber,how many,much,often,far,few,etc.

Thisisasymbol placeholder forthenumberyouaretryingtofindasyouconstructyourformula.

, , ? , _______

Example

BestBakeryishavingasaleonpumpkinpies.Eachpieiscutinto8equal

slices.Theyhave2 piesleft.Jimwantstopurchase1 piesfordinner.

Howmanypieswillbeleft?At$0.75perslice,howmuchwillthepieJimisbuyingcost?

Step1‐UnderstandtheProblem.Youcandothisbycrossingoutunnecessaryinformation,organizingtheinformationintocharts,andwritingoutrelationships.

Bestbakeryishavingasaleonpumpkinpies.Eachpieiscutinto8equal

slices.Theyhave2 piesleft.Jimwantstopurchase1 piesfordinner.

Howmanypieswillbeleft?At$0.75perslice,howmuchwillthepieJimisbuyingcost?

Step2‐DeviseaPlan.Translatekeywordsandphrasesintotheirpropermathematicalsymbols,breakdifficultproblemsintosubproblems,andwritedownanypertinentformulasorcautions.

Question1‐Howmanypieswillbeleft?

Theproblemisaskingustocomparethenumberatthebeginningtothenumberaftersomepiesaretakenaway byJim .Thisisasubtractionproblem.

Numberofpiesinthestore

takeawayNumberofpiesbought

resultsinNumberofpiesleft.

214 1

38

? ?

Step3‐WorkthePlan.

214

138

78

Step4‐LookBack.Justbecauseyouhaveananswer,doesn’tmeanthatitistherightanswer.Sometimesstoryproblemstakeseveralstepstocomplete,sobesurethatyouhaveansweredallthequestionsthatwereasked.Workbackwardsoruseestimationtocheckyouranswer,makesurethatyouranswerisreasonable.Mostimportantly,answerthequestionthatwasasked.

78 ofapieisleft.

Question2‐Howmuchwillthepiecost?

Theproblemisaskingforatotalcost,butweknowthecostforeachslice.Thisisamultiplicationproblem.

Costofaslice foreachnumberofslices

resultsin TotalCost

$. 75 ∗ ? ? ? ?

Thisproblemhasasubproblem,becausewedon’tactuallyknowhowmany

slicesofpieJimbought.Jimbought1 ofapie,andeachpiewascutinto8slices.

Thisisalsoamultiplicationproblem.

Numberofslices

foreach pie resultsin totalslices

8 ∗ 138

? ?

Step3‐WorkthePlan.

8 ∗ 138

11

Jimbought11slices,sousethisanswerintheoriginalquestion.

Costofaslice foreachnumberofslices

resultsin TotalCost

$. 75 ∗ 11 ? ?

$. 75 ∗ 11 $8.25Step4‐LookBack.Didyouanswertheoriginalquestion?Didyouincludethe

appropriateunits?Doesyouranswerseemlogical?

Howmuchwillthepiecost?Thepiewillcost$8.25.

Thepiecosts$.75foronesliceofpie.Wecanestimatethateachslicecostsalittlelessthan$1.Usingthisestimate,weknowthatifhebought11slicesfor$1each,hewouldpay$11.Becauseouractualpriceforeachsliceislessthanourestimate,ourtotalislessthantheestimatedtotal.$8.25islessthan$11,sotheanswermakessense.

Practice

1. AplanetakesofffromapointonthefloorofDeathValleywhichis200feetbelowsealevel.Theplanemustclearan800ftmountainby300ft.Howmanyfeetmusttheplaneclimbaftertakeofftosafelyclearthemountain?

a. 900ftb. 1300ftc. 1100ftd. 500ft

2. Johnpaid$45.00forashirt.Thisrepresentsadiscountof25%.Whatwasthe

originalpriceoftheshirt?a. $60.00b. $180.00c. $11.25d. $33.75

3. If37%ofanumberis36,whatisthenumber?

a. 64 b. 32 c. 15 d. 97

4. Whatpercentofanhouris36minutes?

a. 60%b. 40%c. 24%d. 4%

5. 5/8expressedasapercentis?

a. 56%b. 62.5%c. 87.5d. 114.3%

6. Aplanetravelsatanaveragespeedof600milesperhourfor4hours.Howfardid

theplanetravel,inmiles?a. 150milesb. 240milesc. 2400milesd. 1500miles

7. Mariaismakingascaledrawingofherbasement,whichis20ftlong.Ifherscaleis

½in 1ft,howmanyincheslongshouldshemakeherdrawing?a. 20inchesb. 6inchesc. 10inchesd. 15inches

8. Theweightsof4childrenare80,85,90,and75lbs.Whatistheaverageweight,in

pounds,ofthefourchildren?a. 90lbsb. 75lbsc. 80.4lbsd. 82.5lbs

9. Allofthefollowingarewaystowrite20percentofnEXCEPT

a. . 2

b.

c.

d. 20

10. Whichofthefollowingisclosestto√10.5?a. 3b. 4c. 5d. 8

11. Threepeoplewhoworkpart‐timearetoworktogetheronaproject,buttheirtotal

timeontheprojectistobeequivalenttoonlyonepersonworkingfulltime.Ifoneofthepeoplebudgeted½ofhistimefortheprojectandthesecondpersonbudgeted1/3ofhertime,whatfractionofhistimeshouldthethirdpersonputintotheproject?

a. 1/3b. ¼c. 1/6d. 1/8

Key1. B2. A

3. D4. A

5. B6. C

7. C8. D

9. D10. A

11. C

Integers


Integersarepositiveandnegativewholenumbers.ApplicationswithintegersinvolveOrderofOperations.Operationsarewaystocombinetwonumbers.Theoperationsweusemostoftenareaddition,subtraction,multiplicationanddivision.

OrderofOperations

1. Dooperationsinsidegroupingsymbolsbeforetheoperationsoutsideofthegroupingsymbols.

a. Workgroupingsymbolsfromtheinsidetotheoutside.3 12 5

3 7 21

b. Groupingsymbolsinclude parentheses , brackets , braces ,aswellas

operationssuchas ,|absolutevalues|,and√roots.

5 7 3

5 √45 27

c. Onceanoperationsuchasarootoranabsolutevaluehasbeenperformed,it

mayhelptoreplaceitwithparenthesestoavoidmistakes.3| 6|3 6 18

2. Evaluateexponentsandrootsbeforeotheroperations.Followtheappropriaterules

ofexponents.12 3 12 9

3

3. Performmultiplicationanddivisionbeforeadditionandsubtraction.Worklefttoright,andworkmultiplicationanddivisiontogether.DONOTperformallthemultiplicationandthenallthedivision.

4 20 2 ∗ 54 10 ∗ 54 5054

4. Additionandsubtractionisthelaststepintheorderofoperations.Rememberthat

numberscanbeaddedinanyorder.a. Itisoftenhelpfultochangetheordertomakesumsthatareeasytowork

with,suchasmultiplesof10.53 14 753 7 460 1474

b. Tochangetheorderforsubtraction,keepthenegativeorthesubtraction

signwiththenumberthatfollowsit.10 15 1210 12 1522 15

7

5. Rememberspecialproperties,suchastheDistributiveProperty.Theseallowyoutoworkinaslightlydifferentorderwithoutchangingtheanswer.

4 25 3 124 25 4 3 12100 12 12

100

RulesofIntegers

1. Whenaddingtwointegers,thelargersignwins:a. Ifthesignsarethesame,addandkeepthesign.

3 5 83 5 8

b. Ifthesignsaredifferent,subtractandkeepthesignofthelargernumber.7 4 37 4 3

2. Whensubtractingtwointegers,changesubtractiontoadditionoftheopposite.13 4 13 4

173. Whenmultiplyingintegers,twonegativesmakeapositive:

a. Whenmultiplyingtwopositives,theanswerispositive.3 8 24

b. Whenmultiplyingtwonegatives,theanswerispositive.3 8 24

c. Whenmultiplyingapositiveandanegative,theanswerisnegative.3 8 243 8 24

4. Formorethantwointegersbeingmultipliedtogether,countthesigns:

a. Anevennumberofnegativesignswillmultiplytoapositivenumber.2 5 4 1 40

b. Anoddnumberofnegativesignswillmultiplytoanegativenumber.2 5 4 40

5. Todividetwointegers,therulesarethesameasmultiplication.6. Anabsolutevalueisthedistanceanumberisfromzero.Distanceisnevernegative,

soabsolutevalueswillneverbenegative.a. |5| 5b. | 3| 3c. |0| 0

Practice1. 27 3 ∗ 2

2. 17 15 13

3. 3 10 2 7

4. ‐4 5 3 12

5. 5 4 ∗ 3

6. 5 4 ∗ 3

7. 2|5 9|

8. √9 16

9. 3 4 5 6 ‐1

10. 10 15 6 7 8

11. 4 2

12. 15 5 ∗ 3

Key1. 182. 153. 314. ‐28

5. 176. 277. 88. 5

9. 110. 011. 812. 9

AddingandSubtractingDecimals

Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafterreviewingthecontentsyoustilldon’tpass, youshouldenrollintheappropriatemathclass.

AddingandSubtractingDecimals

1. Writetheproblemincolumnform.2. Lineupthedecimals.3. Addzerostotherighttofillinmissingplacevalues.4. Addorsubtractjustlikeawholenumber.Carrythedecimalstraightdownintothe

answer.

:24.62 3.4

:24.62 3.40

21.22

Practice

1. 24.62 3.42. 61.4 0.34 1.23. . 4 .023 .064. 1.6 0.7

5. 2.42.3

6. 5.1. 321.456

7. 2.12.32

8.

. 41. 023.68

Key1. 21.222. 62.94

3. .4834. 2.3

5. 2.126. 6.876

7. 1.88. 1.113

Line up the decimal

Fill in zeros to the right

Bring the decimal straight down

MultiplyingandDividingDecimals


MultiplyingDecimals

Multiplyingdecimalsisverysimilartomultiplyingwholenumbers,exceptthatyouneedtodeterminewheretoputthedecimalintheanswer.Followthesestepstomultiplydecimalnumbers:

1. Multiplythenumberasthoughyouwereworkingwithwholenumbers.2. Countthetotalnumberofdigitsafter totherightof thedecimal.3. Beginningatthefarright,countthesamenumberofdigitstotheleft,sothatthe

answerhasthesamenumberofdigitsafterthedecimalasthetotalofthenumberofdigitsintheproblem.

Example

:1.03 ∗ 2.5

: 103 ∗ 25 2575

1.03 ∗ 2.5 2.575

Practice1. 1.03 ∗ .25

2. 2.4 ∗ 1.2

3. . 06 ∗ .63

4. 0.4 ∗ 1.5

5. 3 ∗ 1.42

6. 12.3 ∗ .005

7. 300 ∗ .04

8. 0.0007 ∗ .002

2 digits after

the decimal

1 digit after the

decimal

3 digits after the

decimal

Key1. 0.25752. 2.883. .03784. 0.6

5. 4.266. 0.06157. 128. 0.0000014

DividingDecimals

Dividingdecimalssimilartodividingwholenumbers.Againtheconcerniswheretoplacethedecimal.Followthesestepstodividedecimals.

1. Writetheprobleminlongdivisionformat.Besuretoidentifytheplacementofthedecimalinboththedivisor outside andthedividend inside .

2. Inthedivisor,movethedecimaltotherightuntilallnon‐zerodigitsareinfrontofthedecimal.Movethedecimalinthedividendthesamenumberofplacestotheright.Thedecimalsshouldbemovedthesamedistanceandthesamedirection.

3. Addzerostotheend afterthedecimal ifneeded.4. Divide.Bringthedecimalinthedividendstraightupintothequotient answer .5. Rememberthatdivisiondoesnotalwayscomeouteven.Asaruleofthumb,you

shouldroundyouranswertofourdecimalplaces.Youmayneedtoaddzerostotheendofthedividendsothatyouhaveenoughdigits.

Example

38.8 1.2isthesameas1.238.8

32.333333…

12.388.0000

38.8 1.2 32.3333

Add zeros if

needed

Move the decimal so that the

divisor is a whole number

Move this decimal the same distance and

direction, and then carry it straight up.

Round your answer

to 4 decimal places.

Practice1. 37.5 4

2. 2.123 0.2

3. 471.012 2.3

4. 1.159 4.8

5. 0.055 11

6. 725 0.005

7. 1.25 50

8. 453 0.7

Key1. 9.3752. 10.6153. 204.78784. 0.2415

5. 0.0056. 1450007. 0.0258. 647.1429

AddingandSubtractingFractions


TheLowestCommonDenominator LCD

Whenaddingandsubtractingfractions,thedenominatorsofthefractionsmustbeequal,orlike.Followthesestepstofindacommondenominator:

1. Identifythelargestdenominator.TheLCDcannotbesmallerthanthisnumber,andmustbeamultipleofthisnumber.

2. TherearepossibilitieswhenlookingforanLCD:a. Thesmallernumberdividesevenlyintothelargernumber.Inthiscase,the

largernumberistheLCD.

Example: and .

8 4 2.8dividesevenlyby4

8istheLCD

b. Thenumbersmaynotdivideevely,buttheyhaveacommonfactor.Dividethesmallernumberbythecommonfactorandmultiplytheresulttothelargernumber.ThisistheLCD.

Example: and

9 6 ? , butboth6and9divideevelyby3

6 3 2, 2 ∗ 9 18.

18istheLCD

c. Thenumbersdonotdivideevenlyandhavenofactorincommon otherthan1 .TofindtheLCD,multiplythetwonumbers.

Example: and

4 3 ? andtheyhavenofactorsincommon

4 ∗ 3 12

12istheLCD

AddingandSubtractingFractions

1. FirstidentifytheLCD.Multiplyeachfractionby1 1 2/2or3/3or4/4etc. tobuildacommondenominator.

2. Add orsubtract thenumerators tops andkeepthedenominator bottom .3. Simplify.Thefractionmustbereducedtolowestterms.

Example

13

25

TheLCDof3and5is15.Multiplyeachfractionbytheappropriateversionof1.

13∗55

25∗33

515

615

Nowthatwehavecommondenominators,addthetopsandkeepthebottom.

5 615

1115

Reduceifpossible.Inthiscase,thefractiondoesnotreduce.

AddingandSubtractingMixedNumbers

1. Writetheadditionorsubtractionproblemincolumnform.Makesurethefractionpartshaveacommondenominator.

2. Addandsubtractthefractionpartsfirst.a. Foraddition,ifthefractionsadduptoanumber1orlarger,youwillneedto

changetheanswertoamixednumberandcarrythewholenumberparttothewholenumbercolumn.

b. Forsubtraction,ifthefirstfractionissmallerthanthesecondfraction,youwillneedtoborrow1fromthewholenumbercolumn.Rememberthat1 1/1or2/2or3/3,etc.

3. Addorsubtractthewholenumbercolumn.4. Besurethatyouranswerisinlowestterms.Allfractionanswersmustbereduced

tolowestterms.

Example

313

112

Writetheproblemincolumnformat.Findacommondenominatorforthefractionparts.

313∗22→ 3

26

112∗33→ 1

36

Because2/6issmallerthan3/6wewillneedtoregrouporborrowfromthewholenumber.

2126→ 2

86

136

Performthesubtraction.

156

Reducethefractionpart,ifpossible.Inthiscase,thefractionisalreadyinlowestterms.

Practice

1.

2.

3.

4.

5. 1 2

6. 5 3

Key

1. 1

2.

3.

4.

5. 4

6. 1

MultiplyingandDividingFractions


Whenmultiplyinganddividingfractions,itisnotnecessarytogetacommondenominator.However,multiplyingworksmuchbetterifyouworkwitheitherproperorimproperfractions.Donotattempttomultiplymixednumberswithoutfirstchangingthemtoimproperfractions!

MultiplyingFractions

1‐ Changeanymixednumberstoimproperfractions2‐ Multiplystraightacross,numeratortonumeratoranddenominatortodenominator.3‐ Reducethefraction.

Example

45∗ 1

23

45∗53

4 ∗ 55 ∗ 3

4 ∗ 5 55 5 ∗ 3

43

Firstchangeanymixednumberstoimproperfractions

Multiplystraightacross.Becausewewillbereducingthefraction,don’tactuallycarryoutthemultiplication.Instead,lookforcommonfactors.

Dividethenumerator top anddenominator bottom bythesamenumber.

Youcanalsoreducebeforeyoumultiplyandmanypeoplepreferthismethod.Becautioustoonlyreducethiswayformultiplication.Thisdoesnotworkwithanyotheroperation.

157∗ 1

59

127∗149

12 37 7

∗14 79 3

41∗23

83

Changemixednumberstoimproperfractions

Reducebeforemultiplying.Besuretodivideanumeratorandadenominatorbythesamenumber.

Multiplystraightacross.Checktoseeifyouranswerreduces‐ theremayhavebeenacommonfactorthatwasmissed.

DividingFractions

1‐ Changeanymixednumberstoimproperfractions.2‐ Changedivisiontomultiplicationbymultiplyingbythereciprocal3‐ Followtherulesformultiplyingfractions.

Example

56

223

56

83

56∗38

56 3

∗3 38

52∗18

516

First, changemixednumberstoimproperfractions.

Second,changedivisionbyafractiontomultiplicationbythereciprocal.

DONOTREDUCEFRACTIONSTHATAREBEINGDIVIDED.

Nowthatwehavetwofractionsbeingmultiplied,wecanreduceandmultiply.

Practice

1. 512

∗34

2.134∗ 2

3.213∗ 3

23

4. 13∗ 1

45

5.214∗ 3

23

6. 12

2

7.214

23

8.325

69. 3

41516

10. 34

916

11.158

815

12.213

137

Key

1. 516

2.312

3.859

4. 35

5.814

6. 14

7.338

8. 1730

9. 45

10.113

11. 65328

12.

11930

Conversions


ConvertingFractionstoDecimals

Everyfractionisreallyadivisionproblem.Tochangeafractiontoadecimal,dothedivision.

45isthesameas4 5

0.85 4.0 4

50.8

ConvertingDecimalstoFractions

Therearetwotypesofdecimalsthatcanbeconvertedintofractions,thedecimalsthatterminate stop ,andthedecimalsthatrepeat.

Terminatingdecimals‐tochangeaterminatingdecimaltoafractionsayitsname.Besuretoreducethefractiontolowestterms.

. 25 25100

14

Noticethatthedecimalformofthenumberhastwodigitsafterthedecimal,andthedenominatoroftheoriginalfractionhastwozerosaftertheone.

Twodecimaldigits twozeros.

Repeatingdecimals‐tochangearepeatingdecimaltoafraction,firstdeterminewhatisrepeating.Thisisthenumeratorofthefraction.

. 27272727…27?

Becausetherearetwodigitsthatrepeat,thedenominatorwillbe99.Besuretoreduceyourfractiontolowestterms.

Twodigitsrepeating twonines.

. 27272727…2799

311

Thisruleappliesregardlessofhowmanydigitsarerepeating.

. 2222222…29

Onedigitrepeating onenineinthedenominator.

. 358358358…358999

Threedigitsrepeating threeninesinthedenominator.

ConvertingPercentagestoDecimals

Apercentisaspecialfractioninwhichthedenominatorisalways100.Percentliterallymeansdivideby100.Tochangeapercenttodecimal,justdivideby100.Theeasiestwaytodothisistomovethedecimaltwoplacestotheleftanddropthepercentsign.

325% 325. 100 3.25

ConvertingaDecimaltoaPercent

Tochangeadecimalintoapercent,reversetheprocessofchangingapercentintoadecimal.Thismeansthatinsteadofdividingby100andremovingthepercentsign,youwillneedtomultiplyby100andaddthepercentsign.Todothis,movethedecimaltwoplacestotherightandputapercentsignontheendofthenumber.

0.357 .357 ∗ 100% 35.7%

Rememberthatwearedividing by100,sothedecimalshouldappearsmaller thanthepercent.

Herewearemultiplyingby100,sothepercentshouldappearlargerthanthedecimal.

ConvertingaPercenttoaFraction

Rememberthatpercentmeansdivideby100.Whenworkingwithfractions,wechangedivisionintomultiplicationbythereciprocal,somultiplyby1/100.

35% 35 100351∗

1100

35100

720

Simplifyyouranswer.Ifthereisadecimalinthenumerator,youwillneedtomovebothdecimalsthesamedistanceandthesamedirectionuntiltherearenodigitsafterthedecimal. sameruleasdecimaldivision .

2.4%2.4100

241000

3125

ConvertingaFractiontoaPercent

Ifyoucanmakethedenominatorequalto100,thenumeratoristhepercent

310

30100

30%

Ifyoucannot,firstchangethefractionintoadecimal,thenconvertthedecimalintoapercent.

17

.1429 14.29%

Practice

Completethechartbyconvertingeachnumbertotheothertwoforms.

Fraction Decimal Percent

Ex134

1.75 175%

1. 17

2. 0.125

3. 12%

4. 58

5. 0.33333…

6. 3.5%

7.525

8. 2.6

9. 485%

Key

Fraction Decimal Percent

Ex134

1.75 175%

1.

. 142857… 14.29%

2. 18

. 12.5%

3. 325

0.12 %

4.

0.625 62.5%

5. 13

. … 33.33%

6. 7200

. 035 . %

7. 5.4 540%

8.235

. 260%

9.41720

4.85 %

Percentages


Tocalculatethepercentageofanumber:

Inmath,whenwetakeapercentofanumber,theword‘of’canbetranslatedasmultiplication,andtheword‘is’translatesasanequalssign.

30percentof120is40

30% ∗ 120 40

Tosolveaprobleminvolvingpercent,firsttranslatethesentencefromEnglishintomathematicalsymbols,thenworktheproblem.Sometimesyouwillbeabletomultiplytofindtheanswer,andothertimesyouwillneedtodivide.

Example

32%of55iswhatnumber?

Firsttranslatethesentenceintoamathequation 32% ∗ 55 _________?

Convertthepercentageintoeitherafractionoradecimal . 32 ∗ 55 _________?

Worktheproblem.Inthisexample,performthemultiplicationtofindtheanswer.

. 32 ∗ 55 17.6

Example

Ifyoudonotknowwhattomultiplyby,youshoulddivideinstead.Takealookatthenextproblem:

5%ofwhatnumberis24?

5% ∗ __________ 24

. 05 ∗ __________ 24

Inthisexample,oneofthefactorsisunknown,sowecannotmultiply.However,wedoknowtheanswertomultiplication,sowecanmakethisadivisionprobleminstead.

24 .05 __________

24 .05 480

Example

Inthethirdexample,theunknownnumberisthepercentage.Besuretoputyouranswerintherightform!Changetheanswertopercentageform.

Whatpercentof40is15?

__________?% ∗ 40 15

15 40 __________?

15 40 0.375

. 375 37.5%

Acommonmistakethatstudentsmakewhenperformingadivisionproblemisdividinginthewrongorder.Hereareafewpointerstohelpyougettheorderright‐

Dividebythesamenumberthatyoumultiplyby. Whendividingbyanumberlargerthan1 or100% ,youranswerwillgetsmaller. Whendividingbyanumbersmallerthan1 or100% ,youranswerwillgetlarger. Checkyourworkbyrewritingthesentence‐37.5%of40is15 thirdexample .

Doesthisseemreasonable?

Practice

1. 32%of46iswhatnumber?2. 125%of250iswhat?3. Whatnumberis45%of12?4. 15%ofwhatnumberis45?5. 150%ofwhatnumberis270?6. 540is90%ofwhatnumber?7. Whatpercentof25is30?8. Whatpercentof155is12?9. 4.2iswhatpercentof200?

Key

1 14.72 2 312.5 3 5.4 4 300 5 180

6 600 7 120% 8 7.74% 9 2.1%

AverageorMeanAverage

Thissheetisdesignedasareviewaid.Ifyou havenotpreviouslystudiedthisconcept,orifafterreviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.

Theaverage,themeanaverage,orthemeanallrefertothesameconcept.Ameanaverageisawaytotakealistofpossiblyverydifferentnumbersandtreatthemasthoughtheywereallthesame.Ifyouaregivenalistofnumbers,theaverageisthenumberthatallthevalueswouldbeiftheywereallthesame.

Considerthelistofnumbers:5,5,5

Inthiscaseallthreenumbersarethesamenumber,sotheaverageis5.Three5’shaveatotalof15.

Nowlookatthislistofnumbers:3,6,6

Thesethreenumbersalsohaveatotalof15,andtheaverageis5.

Tocalculateamean,addallthevaluesanddividebythetotalnumberofvalues.

Example

Findtheaverageofthefollowingnumbers:

1, 2, 5, 6, 3, 7

Addthevalues: 1 2 5 6 3 7 24

Dividebythenumberofvalues: 24 6 4

Theaverageis4

Example

Supposethatseveralemployeesinashopweremakingparts.Oneemployeecanmake5partsinanhourwhileanotheremployeemakes4partsinanhourandthethirdemployeemakes9partsinanhour.Theemployerwantstobeabletopredicthowmanypartswillbemadeeachhour.Whatistheaveragenumberofpartsthateachemployeeproduceseachhour?

Thelistis: 5, 4, 9

Addtofindthetotal: 5 4 9 18

Dividetofindtheaverage: 18 3 6

Theemployeesaverage6partsperhour.

WorkingBackwards

Youcanuseanaveragetofindamissingvaluebyworkingbackwards.Insteadofaddingthendividingtofindtheaverage,multiplyandsubtracttofindthemissingvalue.

Example

Theaverageof4numbersis12.Thefirstthreenumbersare10,15,and6.Whatisthemissingvalue?

Thelistis: 10, 15, 6, ? ?

Theaverageis: 12

Multiplytofindthetotal: 4 ∗ 12 48

Subtracttofindthemissingvalue:

10 15 6 _____? 48

31 _____? 48

48 31 _____?

48 31 17

17isthemissingvalue.

Practice

Findtheaverageofthefollowinglistsofnumbers:

1. 13, 15, 17, 18, 122. 8.2, 1.6, 3.5, 2.8, 7.4, 2.0

3. , , 1 , 2

4. 9.5,11 , 16.12, 3

Findthemissingvaluefromthelisttoobtainthegivenaverage:

5. 3,5,4,7,___?Average 5

6. 3 , 2 , 4 , 6 , ___?Average 4

7. 93.7,94.1,96.5,_____?Average 95.5

8. 12.12, 3 , 6.34, 7 , 5 , _____?Average 8

Solvethefollowingwordproblemsinvolvingaverages:

9. Maryannhastakenthreetestsforhermathclass,andearnedscoresof85%,91%and94%.Whatisheraverageintheclass?

10. Michaelneedsa70%averagetopasshishistoryclass.Ifhistestscoressofarare82%,65%,and74%,whatscoredoesheneedtogetonthefourthtesttopasstheclass?

Key

1.15 2.4.25 3. 1 1.2125 4. 10.03

5.6 6.7 7. 97.7 8. 13.84

9.Maryannhasa90%intheclass. 10. Michael needs a 59% onhislasttest.

Math Review Packet you answer all the questions? Does you answer make sense? Did you work the...

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