Download Math Review Packet

WordProblems
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
SimplifyingWordProblems
Mostofusareintimidatedbywordproblems.Butifwetakeawaytheextrawords,alotof
theconfusiongoeswiththem.Usethesefourstepsasanoutlinetoguideyouthroughthe
problem:
Step1‐UnderstandtheProblem.Readthroughtheproblemtogetageneralideaof
thesituation.Whatishappeninginthestory?Whatquestion s doyouneedtoanswer?
Whatinformationcanbedisregarded?
Step2‐DeviseaPlan.Readthroughtheproblemagain,lookingfordetails.Decide
whatoperationstouse addition,multiplication,etc. .Setuprelationships,translatefrom
Englishtomath,makelists,drawandlabelpictures.Useanystrategyyouknowto
organizetheinformationintoaworkableformat.
Step3‐WorkthePlan.Nowthatyouhaveaplan,workit.Dothemath,crunchthe
numbers.Someproblemsrequireseveralsteps,sobesuretoworkthemall.
Step4‐LookBack.Nowthatyouaredone,areyoureallydone?Didyouanswerall
thequestions?Doesyouanswermakesense?Didyouworktheproblemcorrectly?
Someextraadvice


Manywordproblemshavemorethanonepartthatneedstobeworkedout.Break
downtheproblemtoitsindividualpartsandworkoneatatime.Ifsomething
seemstobemissinginordertoworkasub‐problem,setitasideandworkanother
part.Youmayfindthemissinginformationasyouworkothersub‐problems.
Problemswithlistsofthesametypeofthingtendtobeadditionproblems.“John
worked5hoursonMonday,12hoursonTuesday,6hoursofWednesdayand7
hoursand30minutesonThursday.Howmanyhourshasheworkedthisweek?”Be
surethattheunitsmatchbeforeadding.
o Units:hours,hours,hours,hoursandminutes.

Problemswithmixedunitstendtobemultiplicationordivisionproblems.“An
airplanetravelsat350milesperhour.Howfardoestheairplanetravelin6hours?”
o Units:milesperhour,hours,miles.
Vocabulary
TherearemanykeywordsinEnglishthatcanhelpusdecidewhatmathematicaloperation
orrelationshiptouse.Herearesomeexamples.
Words,Phrases
Symboloroperation
plus,sumof,addedto,joinedwith,increasedby,more,more
than,and,total
minus,differencebetween,subtractedfrom,decreasedby,
reducedby,less,lessthan,exceeds,takeaway,remove,purchase
times,timesmorethan,multipliedby,product,equalamounts
of,of,at,each,every,total
∗
dividedby,divides,quotient,ratio,compared to,separatedinto
equalparts,per,outof,goesinto,over
equals,isequalto,is,isthesameas,was,were,willbe,resultsin,
makes,gives,leaves
twotime,double,twice
2 ∗
half,halfasmuch,halves,onehalfof
1
∗
2
What number,part,percent,amount,price,etc. ,anumber,the
number,how many,much,often,far,few,etc. , , ? , _______
Thisisasymbol placeholder forthenumberyouaretryingto
findasyouconstructyourformula.
Example
BestBakeryishavingasaleonpumpkinpies.Eachpieiscutinto8equal
slices.Theyhave2 piesleft.Jimwantstopurchase1 piesfordinner.
Howmanypieswillbeleft?
At$0.75perslice,howmuchwillthepieJimisbuyingcost?
Step1‐UnderstandtheProblem.Youcandothisbycrossingoutunnecessary
information,organizingtheinformationintocharts,andwritingoutrelationships.
Bestbakeryishavingasaleonpumpkinpies.Eachpieiscutinto8equal
slices.Theyhave2 piesleft.Jimwantstopurchase1 piesfordinner.
Howmanypieswillbeleft?
At$0.75perslice,howmuchwillthepieJimisbuyingcost?
Step2‐DeviseaPlan.Translatekeywordsandphrasesintotheirproper
mathematicalsymbols,breakdifficultproblemsintosubproblems,andwritedownany
pertinentformulasorcautions.
Question1‐Howmanypieswillbeleft?
Theproblemisaskingustocomparethenumberatthebeginningtothe
numberaftersomepiesaretakenaway byJim .Thisisasubtractionproblem.
Numberof
piesinthe
store
Numberofpies
resultsin
bought
takeaway
1
2 4
1
3
8
1
3
8
Numberof
piesleft.
? ?
Step3‐WorkthePlan.
2
1
4
7
8
Step4‐LookBack.Justbecauseyouhaveananswer,doesn’tmeanthatitistheright
answer.Sometimesstoryproblemstakeseveralstepstocomplete,sobesurethatyou
haveansweredallthequestionsthatwereasked.Workbackwardsoruseestimationto
checkyouranswer,makesurethatyouranswerisreasonable.Mostimportantly,answer
thequestionthatwasasked.
7
8 ofapieisleft.
Question2‐Howmuchwillthepiecost?
Theproblemisaskingforatotalcost,butweknowthecostforeachslice.
Thisisamultiplicationproblem.
Costofaslice foreach
$. 75
numberof
slices
∗
resultsin
??
TotalCost
? ?
Thisproblemhasasubproblem,becausewedon’tactuallyknowhowmany
slicesofpieJimbought.Jimbought1 ofapie,andeachpiewascutinto8slices.
Thisisalsoamultiplicationproblem.
Numberof
slices
8
foreach
pie
∗
resultsin
1
3
8
totalslices
? ?
Step3‐WorkthePlan.
8∗1
3
8
11
Jimbought11slices,sousethisanswerintheoriginalquestion.
Costofaslice foreach
$. 75
∗
numberof
slices
resultsin
11
TotalCost
? ?
$. 75 ∗ 11 $8.25
Step4‐LookBack.Didyouanswertheoriginalquestion?Didyouincludethe
appropriateunits?Doesyouranswerseemlogical?
Howmuchwillthepiecost?Thepiewillcost$8.25.
Thepiecosts$.75foronesliceofpie.Wecanestimatethateachslicecostsalittle
lessthan$1.Usingthisestimate,weknowthatifhebought11slicesfor$1each,hewould
pay$11.Becauseouractualpriceforeachsliceislessthanourestimate,ourtotalisless
thantheestimatedtotal.$8.25islessthan$11,sotheanswermakessense.
Practice
1. AplanetakesofffromapointonthefloorofDeathValleywhichis200feetbelow
sealevel.Theplanemustclearan800ftmountainby300ft.Howmanyfeetmust
theplaneclimbaftertakeofftosafelyclearthemountain?
a. 900ft
b. 1300ft
c. 1100ft
d. 500ft
2. Johnpaid$45.00forashirt.Thisrepresentsadiscountof25%.Whatwasthe
originalpriceoftheshirt?
a. $60.00
b. $180.00
c. $11.25
d. $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. 87.5
d. 114.3%
6. Aplanetravelsatanaveragespeedof600milesperhourfor4hours.Howfardid
theplanetravel,inmiles?
a. 150miles
b. 240miles
c. 2400miles
d. 1500miles
7. Mariaismakingascaledrawingofherbasement,whichis20ftlong.Ifherscaleis
½in 1ft,howmanyincheslongshouldshemakeherdrawing?
a. 20inches
b. 6inches
c. 10inches
d. 15inches
8. Theweightsof4childrenare80,85,90,and75lbs.Whatistheaverageweight,in
pounds,ofthefourchildren?
a. 90lbs
b. 75lbs
c. 80.4lbs
d. 82.5lbs
9. Allofthefollowingarewaystowrite20percentofnEXCEPT
a. . 2 b.
c.
d. 20 10. Whichofthefollowingisclosestto√10.5?
a. 3
b. 4
c. 5
d. 8
11. Threepeoplewhoworkpart‐timearetoworktogetheronaproject,buttheirtotal
timeontheprojectistobeequivalenttoonlyonepersonworkingfulltime.Ifoneof
thepeoplebudgeted½ofhistimefortheprojectandthesecondpersonbudgeted
1/3ofhertime,whatfractionofhistimeshouldthethirdpersonputintothe
project?
a. 1/3
b. ¼
c. 1/6
d. 1/8
Key
1. B
2. A
3. D
4. A
5. B
6. C
7. C
8. D
9. D
10. A
11. C
Integers
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
Integersarepositiveandnegativewholenumbers.Applicationswithintegersinvolve
OrderofOperations.Operationsarewaystocombinetwonumbers.Theoperationswe
usemostoftenareaddition,subtraction,multiplicationanddivision.
OrderofOperations
1. Dooperationsinsidegroupingsymbolsbeforetheoperationsoutsideofthe
groupingsymbols.
a. Workgroupingsymbolsfromtheinsidetotheoutside.
3 12 5 3 7 21
b. Groupingsymbolsinclude parentheses , brackets , braces ,aswellas
operationssuchas
,|absolutevalues|,and√roots.
5
7
3 5 √4
5 2
7
c. Onceanoperationsuchasarootoranabsolutevaluehasbeenperformed,it
mayhelptoreplaceitwithparenthesestoavoidmistakes.
3| 6|
3 6 18
2. Evaluateexponentsandrootsbeforeotheroperations.Followtheappropriaterules
ofexponents.
12 3 12 9
3
3. Performmultiplicationanddivisionbeforeadditionandsubtraction.Workleftto
right,andworkmultiplicationanddivisiontogether.DONOTperformallthe
multiplicationandthenallthedivision.
4 20 2 ∗ 5
4 10 ∗ 5
4 50
54
4. Additionandsubtractionisthelaststepintheorderofoperations.Rememberthat
numberscanbeaddedinanyorder.
a. Itisoftenhelpfultochangetheordertomakesumsthatareeasytowork
with,suchasmultiplesof10.
53 14 7
53 7 4
60 14
74
b. Tochangetheorderforsubtraction,keepthenegativeorthesubtraction
signwiththenumberthatfollowsit.
10 15 12
10 12 15
22 15
7
5. Rememberspecialproperties,suchastheDistributiveProperty.Theseallowyouto
workinaslightlydifferentorderwithoutchangingtheanswer.
4 25 3
12
4 25
4 3
12
100 12 12
100
RulesofIntegers
1. Whenaddingtwointegers,thelargersignwins:
a. Ifthesignsarethesame,addandkeepthesign.
3 5
8
3 5
8
b. Ifthesignsaredifferent,subtractandkeepthesignofthelargernumber.
7 4
3
7 4
3
2. Whensubtractingtwointegers,changesubtractiontoadditionoftheopposite.
13
4 13
4 17
3. Whenmultiplyingintegers,twonegativesmakeapositive:
a. Whenmultiplyingtwopositives,theanswerispositive.
3
8
24
b. Whenmultiplyingtwonegatives,theanswerispositive.
3
8
24
c. Whenmultiplyingapositiveandanegative,theanswerisnegative.
3
8
24
3
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| 5
b. | 3| 3
c. |0| 0
Practice
1. 27
3 ∗ 2
2. 17
15
13
7. 2|5
9|
8. √9
16
3. 3 10
2
7
9. 3
4. ‐4 5
3
12
10. 10
5. 5
6.
Key
1.
2.
3.
4.
5
18
15
31
‐28
4 ∗ 3
4
11. 4
4 ∗ 3
12.
5.
6.
7.
8.
17
27
8
5
6
6
7
2 15
9. 1
10. 0
11. 8
12. 9
15
5
5 ∗ 3
‐1 8
AddingandSubtractingDecimals
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass, youshouldenrollintheappropriatemathclass.
AddingandSubtractingDecimals
1.
2.
3.
4.
Writetheproblemincolumnform.
Lineupthedecimals.
Addzerostotherighttofillinmissingplacevalues.
Addorsubtractjustlikeawholenumber.Carrythedecimalstraightdownintothe
answer.
:24.62 3.4
Line up the decimal :24.62
3.40
Fill in zeros to the right 21.22
Bring the decimal straight down Practice
1.
2.
3.
4.
24.62 3.4
61.4 0.34 1.2
. 4 .023 .06
1.6 0.7
2.42
5. .3
Key
1. 21.22
2. 62.94
6.
3. .483
4. 2.3
5.1
. 32 1.456
7. 2.12
.32
5. 2.12
6. 6.876
. 41
8. . 023 .68
7. 1.8
8. 1.113
MultiplyingandDividingDecimals
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
MultiplyingDecimals
Multiplyingdecimalsisverysimilartomultiplyingwholenumbers,exceptthatyouneedto
determinewheretoputthedecimalintheanswer.Followthesestepstomultiplydecimal
numbers:
1. Multiplythenumberasthoughyouwereworkingwithwholenumbers.
2. Countthetotalnumberofdigitsafter totherightof thedecimal.
3. Beginningatthefarright,countthesamenumberofdigitstotheleft,sothatthe
answerhasthesamenumberofdigitsafterthedecimalasthetotalofthenumberof
digitsintheproblem.
Example
:1.03 ∗ 2.5
: 103 ∗ 25
1.03 ∗ 2.5
2575
2.575
2 digits after the decimal 3 digits after the decimal 1 digit after the decimal
Practice
1. 1.03 ∗ .25
5. 3 ∗ 1.42
2. 2.4 ∗ 1.2
6. 12.3 ∗ .005
3. . 06 ∗ .63
7. 300 ∗ .04
4. 0.4 ∗ 1.5
8. 0.0007 ∗ .002
Key
1.
2.
3.
4.
0.2575
2.88
.0378
0.6
5.
6.
7.
8.
4.26
0.0615
12
0.0000014
DividingDecimals
Dividingdecimalssimilartodividingwholenumbers.Againtheconcerniswheretoplace
thedecimal.Followthesestepstodividedecimals.
1. Writetheprobleminlongdivisionformat.Besuretoidentifytheplacementofthe
decimalinboththedivisor outside andthedividend inside .
2. Inthedivisor,movethedecimaltotherightuntilallnon‐zerodigitsareinfrontof
thedecimal.Movethedecimalinthedividendthesamenumberofplacestothe
right.Thedecimalsshouldbemovedthesamedistanceandthesamedirection.
3. Addzerostotheend afterthedecimal ifneeded.
4. Divide.Bringthedecimalinthedividendstraightupintothequotient answer .
5. Rememberthatdivisiondoesnotalwayscomeouteven.Asaruleofthumb,you
shouldroundyouranswertofourdecimalplaces.Youmayneedtoaddzerostothe
endofthedividendsothatyouhaveenoughdigits.
Example
38.8
1.2isthesameas1.238.8
32.333333…
12.388.0000
Move the decimal so that the divisor is a whole number Add zeros if needed Move this decimal the same distance and direction, and then carry it straight up. 38.8
1.2
32.3333
Round your answer to 4 decimal places. Practice
1. 37.5
2. 2.123
0.2
3. 471.012
4. 1.159
Key
1.
2.
3.
4.
5. 0.055
4
2.3
4.8
9.375
10.615
204.7878
0.2415
6. 725
0.005
7. 1.25
50
8. 453
0.7
5.
6.
7.
8.
11
0.005
145000
0.025
647.1429
AddingandSubtractingFractions
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
TheLowestCommonDenominator LCD Whenaddingandsubtractingfractions,thedenominatorsofthefractionsmustbeequal,or
like.Followthesestepstofindacommondenominator:
1. Identifythelargestdenominator.TheLCDcannotbesmallerthanthisnumber,and
mustbeamultipleofthisnumber.
2. TherearepossibilitieswhenlookingforanLCD:
a. Thesmallernumberdividesevenlyintothelargernumber.Inthiscase,the
largernumberistheLCD.
Example: and .
8
4
2.8dividesevenlyby4
8istheLCD
b. Thenumbersmaynotdivideevely,buttheyhaveacommonfactor.Divide
thesmallernumberbythecommonfactorandmultiplytheresulttothe
largernumber.ThisistheLCD.
Example: and 9
6
?, butboth6and9divideevelyby3
6
3
2, 2 ∗ 9
18.
18istheLCD
c. Thenumbersdonotdivideevenlyandhavenofactorincommon otherthan
1 .TofindtheLCD,multiplythetwonumbers.
Example: and 4
3
4∗3
? andtheyhavenofactorsincommon
12
12istheLCD
AddingandSubtractingFractions
1. FirstidentifytheLCD.Multiplyeachfractionby1 1 2/2or3/3or4/4etc. to
buildacommondenominator.
2. Add orsubtract thenumerators tops andkeepthedenominator bottom .
3. Simplify.Thefractionmustbereducedtolowestterms.
Example
1
3

2
5
TheLCDof3and5is15.Multiplyeachfractionbytheappropriateversionof1.
1 5
∗
3 5
2 3
∗ 5 3
5
15

6
15
Nowthatwehavecommondenominators,addthetopsandkeepthebottom.
5
6
15
11
15

Reduceifpossible.Inthiscase,thefractiondoesnotreduce.
AddingandSubtractingMixedNumbers
1. Writetheadditionorsubtractionproblemincolumnform.Makesurethefraction
partshaveacommondenominator.
2. Addandsubtractthefractionpartsfirst.
a. Foraddition,ifthefractionsadduptoanumber1orlarger,youwillneedto
changetheanswertoamixednumberandcarrythewholenumberpartto
thewholenumbercolumn.
b. Forsubtraction,ifthefirstfractionissmallerthanthesecondfraction,you
willneedtoborrow1fromthewholenumbercolumn.Rememberthat
1 1/1or2/2or3/3,etc.
3. Addorsubtractthewholenumbercolumn.
4. Besurethatyouranswerisinlowestterms.Allfractionanswersmustbereduced
tolowestterms.
Example
3

1
3
1
1 2
Writetheproblemincolumnformat.Findacommondenominatorforthefraction
parts.
1 2
2
3 ∗ → 3 3 2
6
3
1 3
1 ∗ → 1 6
2 3

Because2/6issmallerthan3/6wewillneedtoregrouporborrowfromthewhole
number.
21
2
8
→ 2 6
6
3
1 6


Performthesubtraction.
5
1 6
Reducethefractionpart,ifpossible.Inthiscase,thefractionisalreadyinlowest
terms.
Practice
1.
4.
2.
5. 1
2 3.
6. 5
3 Key
1.
4.
1 2. 5. 4 3. 6. 1 MultiplyingandDividingFractions
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
Whenmultiplyinganddividingfractions,itisnotnecessarytogetacommondenominator.
However,multiplyingworksmuchbetterifyouworkwitheitherproperorimproper
fractions.Donotattempttomultiplymixednumberswithoutfirstchangingthemto
improperfractions!
MultiplyingFractions
1‐ Changeanymixednumberstoimproperfractions
2‐ Multiplystraightacross,numeratortonumeratoranddenominatortodenominator.
3‐ Reducethefraction.
Example
4
2
∗1 5
3
Firstchangeanymixednumbersto
improperfractions
4 5
∗ 5 3
Multiplystraightacross.Becausewewillbe
reducingthefraction,don’tactuallycarry
outthemultiplication.Instead,lookfor
commonfactors.
4∗5
5∗3
4
3
4∗5 5
5 5∗3
Dividethenumerator top and
denominator bottom bythesamenumber.
Youcanalsoreducebeforeyoumultiplyandmanypeoplepreferthismethod.Becautious
toonlyreducethiswayformultiplication.Thisdoesnotworkwithanyotheroperation.
5
5
1 ∗1 7
9
Changemixednumberstoimproper
fractions
12 14
∗ 7 9
Reducebeforemultiplying.Besuretodivide
anumeratorandadenominatorbythesame
number.
12 3 14 7
∗
7 7 9 3
4 2
∗
1 3
8
3
Multiplystraightacross.Checktoseeifyour
answerreduces‐ theremayhavebeena
commonfactorthatwasmissed.
DividingFractions
1‐ Changeanymixednumberstoimproperfractions.
2‐ Changedivisiontomultiplicationbymultiplyingbythereciprocal
3‐ Followtherulesformultiplyingfractions.
Example
5
6
2
2 3
First, changemixednumberstoimproper
fractions.
8
3
5
6
Second,changedivisionbyafractionto
multiplicationbythereciprocal.
5 3
∗ 6 8
5
6
3
∗
DONOTREDUCEFRACTIONSTHATARE
BEINGDIVIDED.
3
5 1
∗
2 8
8
3
Nowthatwehavetwofractionsbeing
multiplied,wecanreduceandmultiply.
5
16
Practice
1.
5 3
∗ 12 4
2.
3
1 ∗2
4
3.
1
2
2 ∗3 3
3
4.
4
1
∗1 3
5
5.
1
2
2 ∗3
4
3
6.
1
2
2
2
3
8.
9.
3
4
15
16
7.
2
1
4
10. 3
4
9
16
11.
3
2
5
6
1
5
8
8
1
5
12.
2
1
3
3
1 7
Key
1. 5
16
7.
3
3 8
1
3 2
3.
8. 17
30
9.
2.
8
4
5
5
9
4.
10.
3
5
1
1
3
5.
1
8 4
6.
11.
65
328
12.
1
4
1
19
30
Conversions
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
ConvertingFractionstoDecimals
Everyfractionisreallyadivisionproblem.Tochangeafractiontoadecimal,dothe
division.
0.8
4
4
5 4.0
isthesameas4 5
0.8
5
5
ConvertingDecimalstoFractions
Therearetwotypesofdecimalsthatcanbeconvertedintofractions,thedecimalsthat
terminate stop ,andthedecimalsthatrepeat.
Terminatingdecimals‐tochangeaterminatingdecimaltoafractionsayitsname.Besure
toreducethefractiontolowestterms.
. 25
25
100
1
4
Noticethatthedecimalformofthenumberhastwodigitsafterthedecimal,andthe
denominatoroftheoriginalfractionhastwozerosaftertheone.
Twodecimaldigits twozeros.
Repeatingdecimals‐tochangearepeatingdecimaltoafraction,firstdeterminewhatis
repeating.Thisisthenumeratorofthefraction.
27
?
. 27272727 …
Becausetherearetwodigitsthatrepeat,thedenominatorwillbe99.Besuretoreduce
yourfractiontolowestterms.
Twodigitsrepeating twonines.
. 27272727 …
27
99
3
11
Thisruleappliesregardlessofhowmanydigitsarerepeating.
. 2222222 …
2
9
Onedigitrepeating onenineinthedenominator.
. 358358358 …
358
999
Threedigitsrepeating threeninesinthedenominator.
ConvertingPercentagestoDecimals
Apercentisaspecialfractioninwhichthedenominatorisalways100.Percentliterally
meansdivideby100.Tochangeapercenttodecimal,justdivideby100.Theeasiestway
todothisistomovethedecimaltwoplacestotheleftanddropthepercentsign.
325%
325. 100
3.25
Rememberthatwearedividing by100,sothe
decimalshouldappearsmaller thanthepercent.
ConvertingaDecimaltoaPercent
Tochangeadecimalintoapercent,reversetheprocessofchangingapercentintoa
decimal.Thismeansthatinsteadofdividingby100andremovingthepercentsign,you
willneedtomultiplyby100andaddthepercentsign.Todothis,movethedecimaltwo
placestotherightandputapercentsignontheendofthenumber.
0.357
.357 ∗ 100%
Herewearemultiplyingby100,sothepercent
shouldappearlargerthanthedecimal.
35.7%
ConvertingaPercenttoaFraction
Rememberthatpercentmeansdivideby100.Whenworkingwithfractions,wechange
divisionintomultiplicationbythereciprocal,somultiplyby1/100.
35%
35
35 1
∗
1 100
100
35
100
7
20
Simplifyyouranswer.Ifthereisadecimalinthenumerator,youwillneedtomoveboth
decimalsthesamedistanceandthesamedirectionuntiltherearenodigitsafterthe
decimal. sameruleasdecimaldivision .
2.4
100
2.4%
24
1000
3
125
ConvertingaFractiontoaPercent
Ifyoucanmakethedenominatorequalto100,thenumeratoristhepercent
3
10
30
100
30%
Ifyoucannot,firstchangethefractionintoadecimal,thenconvertthedecimalintoa
percent.
1
7
.1429
14.29%
Practice
Completethechartbyconvertingeachnumbertotheothertwoforms.
Fraction
Decimal
Percent
Ex
3
1 4
1.75
175%
1.
1
7
2.
3.
0.125
12%
5
8
4.
5.
6.
0.33333 …
3.5%
2
5 5
7.
8.
9.
2.6
485%
Key
Ex
Fraction
Decimal
Percent
3
4
1.75
175%
. 142857 …
14.29%
1
1.
2.
1
8
.
3.
3
25
0.12
4.
1
3
6.
7
200
7.
8.
9.
.
62.5%
…
. 035
33.33%
. %
5.4
540%
3
5
.
260%
17
20
4.85
2
4
%
0.625
5.
12.5%
%
Percentages
Thissheetisdesignedasareviewaid.Ifyouhavenotpreviouslystudiedthisconcept,orifafter
reviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemathclass.
Tocalculatethepercentageofanumber:
Inmath,whenwetakeapercentofanumber,theword‘of’canbetranslatedas
multiplication,andtheword‘is’translatesasanequalssign.
30percentof120is40
30% ∗ 120
40
Tosolveaprobleminvolvingpercent,firsttranslatethesentencefromEnglishinto
mathematicalsymbols,thenworktheproblem.Sometimesyouwillbeabletomultiplyto
findtheanswer,andothertimesyouwillneedtodivide.
Example
32%of55iswhatnumber?
Firsttranslatethesentenceintoamathequation
32% ∗ 55
Convertthepercentageintoeitherafractionoradecimal
. 32 ∗ 55
_________?
Worktheproblem.Inthisexample,performthe
multiplicationtofindtheanswer.
. 32 ∗ 55
17.6
_________?
Example
Ifyoudonotknowwhattomultiplyby,youshoulddivideinstead.Takealookatthenext
problem:
5%ofwhatnumberis24?
5% ∗ __________
24
. 05 ∗ __________
24
Inthisexample,oneofthefactorsisunknown,sowecannotmultiply.However,wedo
knowtheanswertomultiplication,sowecanmakethisadivisionprobleminstead.
24
24
.05
.05
__________
480
Example
Inthethirdexample,theunknownnumberisthepercentage.Besuretoputyouranswer
intherightform!Changetheanswertopercentageform.
Whatpercentof40is15?
__________? % ∗ 40
15
40
15
40
. 375
15
__________?
0.375
37.5%
Acommonmistakethatstudentsmakewhenperformingadivisionproblemisdividingin
thewrongorder.Hereareafewpointerstohelpyougettheorderright‐




Dividebythesamenumberthatyoumultiplyby.
Whendividingbyanumberlargerthan1 or100% ,youranswerwillgetsmaller.
Whendividingbyanumbersmallerthan1 or100% ,youranswerwillgetlarger.
Checkyourworkbyrewritingthesentence‐37.5%of40is15 thirdexample .
Doesthisseemreasonable?
Practice
1.
2.
3.
4.
5.
6.
7.
8.
9.
32%of46iswhatnumber?
125%of250iswhat?
Whatnumberis45%of12?
15%ofwhatnumberis45?
150%ofwhatnumberis270?
540is90%ofwhatnumber?
Whatpercentof25is30?
Whatpercentof155is12?
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,orif
afterreviewingthecontentsyoustilldon’tpass,youshouldenrollintheappropriatemath
class.
Theaverage,themeanaverage,orthemeanallrefertothesameconcept.Ameanaverage
isawaytotakealistofpossiblyverydifferentnumbersandtreatthemasthoughthey
wereallthesame.Ifyouaregivenalistofnumbers,theaverageisthenumberthatallthe
valueswouldbeiftheywereallthesame.
Considerthelistofnumbers:5,5,5
Inthiscaseallthreenumbersarethesamenumber,sotheaverageis5.Three5’s
haveatotalof15.
Nowlookatthislistofnumbers:3,6,6
Thesethreenumbersalsohaveatotalof15,andtheaverageis5.
Tocalculateamean,addallthevaluesanddividebythetotalnumberofvalues.
Example
Findtheaverageofthefollowingnumbers:
1, 2, 5, 6, 3, 7
Addthevalues:
Dividebythenumberofvalues:
1
24
2
5
6
Theaverageis4
6
4
3
7
24
Example
Supposethatseveralemployeesinashopweremakingparts.Oneemployeecanmake5
partsinanhourwhileanotheremployeemakes4partsinanhourandthethirdemployee
makes9partsinanhour.Theemployerwantstobeabletopredicthowmanypartswillbe
madeeachhour.Whatistheaveragenumberofpartsthateachemployeeproduceseach
hour?
Thelistis:
5, 4, 9
Addtofindthetotal:
5
Dividetofindtheaverage: 4
18
9
3
18
6
Theemployeesaverage6partsperhour.
WorkingBackwards
Youcanuseanaveragetofindamissingvaluebyworkingbackwards.Insteadofadding
thendividingtofindtheaverage,multiplyandsubtracttofindthemissingvalue.
Example
Theaverageof4numbersis12.Thefirstthreenumbersare10,15,and6.Whatisthe
missingvalue?
Thelistis:
Theaverageis:
10, 15, 6, ? ?
12
Multiplytofindthetotal: 4 ∗ 12
48
Subtracttofindthemissingvalue:
10
15
6
31
_____?
48
31
48
31
_____?
48
48
_____?
17
17isthemissingvalue.
Practice
Findtheaverageofthefollowinglistsofnumbers:
1. 13, 15, 17, 18, 12
2. 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.Ifhistestscoressofarare
82%,65%,and74%,whatscoredoesheneedtogetonthefourthtesttopassthe
class?
Key
1.15
2.4.25
3. 1
5.6
6.7
7. 97.7
9.Maryannhasa90%intheclass.
1.2125
4.
10.03
8. 13.84
10. Michael needs a 59% onhislasttest.