piqosity

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WhattoExpectontheISEEERB’sOfficialPracticeTestAnswerExplanationsByStephenHayesforPiqosity.com®ISEEisaregisteredtrademarkoftheEducationalRecordsBureau,whichisneitheraffiliatedwithnorendorsesPiqosity.

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Contents

VerbalReasoning page3QuantitativeReasoning 10ReadingComprehension 20MathematicsAchievement 27

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VerbalReasoning-Synonyms1. INITIAL=existingoroccurringatthebeginningA. first B. mutual C. orderly D. propercomingbeforeallothersintimeororder

heldincommonbytwoormoreparties

neatlyandmethodicallyarranged

suitableorappropriate

2. MANNEQUIN=adummyusedtodisplayclothesinastorewindowA. actor B. aide C. leader D. modelonewhoperformsaroleinTV,movies,oronstage

onewhoassistssomeoneofimportance

onewhocommandsagroupororganization

oneemployedtodisplayclothing

3. AGENDA=aplanofthingstobedoneorproblemstobeaddressedA. accident B. composition C. duty D. programeventthathappensbychance

thewayinwhichsomethingismadeup

moralorlegalobligation;responsibility

aplannedseriesoffutureevents

4. ADVERSARY=one'sopponentinacontest,conflict,ordisputeA. agent B. coward C. opponent D. rascalonewhoactsonbehalfofanother

onewholackscouragetoendureunpleasantthings

onewhocompetesorfightsanother

mischievousorcheekyperson

5. PERSONIFY=representorembodya(usuallyhuman)quality,concept,orthinginphysicalformA. argue B. fulfill C. replace D. representexchangeorexpressopposingviews

bringtocompletionorreality;carryout

taketheplaceofsomething

beasymbolorembodimentofathing

6. EQUITY=thequalityofbeingfairandimpartialA. fairness B. harshness C. humor D. knowledgequalityofbeingfreefrombiasorinjustice

qualityofbeingdisagreeabletosenses

qualityofbeingamusingorcomic

facts/skillsgainedbyexperienceoreducation

7. ANTHOLOGY=apublishedcollectionofpoemsorotherpiecesofwritingA. agreement B. collection C. disease D. extensionharmonyinopinionorfeeling

assemblyofitems,suchaswrittenworks

conditionthatimpairsnormalfunctioning

continuation;partthatisaddedtoprolong

8. OPAQUE=notabletobeseenthrough;nottransparentA. antique B. clouded C. exhausted D. pretentiousbelongingtoancienttimes

madeunclearorlesstransparent

verytired;completelyusedup

actinggreaterthanoneistoimpressothers

9. PALPABLE=abletobetouchedorfelt;cleartothemindorplaintoseeA. docile B. political C. sluggish D. tangiblesubmissive;readytoacceptcontrol

interestedinoractiveinpolitics

slow-movingorinactive perceptiblebytouch;clearanddefinite;real

10. FATHOM=understand(adifficultproblemoranenigmaticperson)aftermuchthoughtA. comprehend B. hasten C. question D. trickgraspmentally;understand

causetohappensoonerthannormal

feelorexpressdoubtaboutsomething

deceiveoroutwitthroughcunning/skill

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11. DIMINISH=makeorbecomeless;makeseemlessimpressiveorvaluableA. eliminate B. evade C. examine D. reducecompletelyremoveorgetridofsomething

escapeoravoid;avoidgivingadirectanswerto

inspectindetail;investigatethoroughly

makesmallerorlessinamount,degree,orsize

12. PERPETUATE=makeathing(undesirablesituationorunfoundedbelief)continueindefinitelyA. continue B. convince C. enclose D. introducepersistinanactivityorprocess

causeonetobelievefirmlyinatruth

surroundorcloseoffonallsides

bringintouse/operationforfirsttime

13. ADMONISH=warnorreprimandsomeonefirmlyA. delay B. organize C. suffer D. warnmakesomethinglateorslow

arrangeintoastructuredwhole

experiencesomethingbadorunpleasant

givesomeoneforcefulorcautionaryadvice

14. DEPICT=showorrepresentbyadrawing,painting,orotherartformA. describe B. discard C. include D. reversegiveanaccountinwordsofsomething

getridofsomethingasnolongeruseful/wanted

makepartofawholeorset

movebackward;makeoppositeofwhatis

15. EPITOME=apersonorthingthatisaperfectexampleofaparticularqualityortypeA. embodiment B. equilibrium C. resilience D. viewpointatangibleformofanidea,quality,orfeeling

astateofphysicalormentalbalance

capacitytorecoverquicklyfromdifficulties

wayofconsidering/one’spositiononamatter

16. TRANSITORY=lastingonlyforashorttimeA. active B. essential C. fleeting D. immediatereadytoengageinenergeticpursuits

absolutelynecessary;extremelyimportant

lastingforaveryshorttime

occurringordoneatonce;instant

17. INCITE=encourageorstirup(usuallyviolentorunlawfulbehavior)A. explain B. investigate C. provoke D. requestmakesomethingmoreclearthroughdescription

carryoutinquirytoestablishtruth

stimulateareactionoremotion(unwelcome)

politelyorformallyaskforsomething

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VerbalReasoning-SentenceCompletions18. Poet-novelistRitaDove,formerUnitedStatesPoetLaureate,wastherecipientof

the1966HeinzAwardinthecategoryofartsandhumanities. D

Explanation

Workingwiththeinformationprovidedbythesentence,Dovewritespoetryandnovels(bothworksofarts/humanities)andwastheUnitedStatesPoetLaureate.YoudonothavetoknowwhataPoetLaureateis,butyoucanrecognizethefactthetitleincludes“UnitedStates,”whichindicatesthatDovewasaveryimportantpoetthroughoutthewholeoftheUnitedStates.Thus,itmakessensethatsuchanimportantpoetwouldreceiveanawardforherefforts.Plus,wewouldneedmoreinformationfortheotheranswerchoicestobetrue(didshemakeorbringabouttheaward?)

19. AlfredJarry’sfirstplay,UbiRoi,isconsideredthefirstworkofthetheaterofthe

absurd;althoughitcausedascandalwhenitopenedin1896,todayitisacclaimedforitsinnovativeplot.

A

Explanation

Focusingonthepartsofthesentencethatfollowthesemicolon,thecontextualstructureshiftsfromnegativetopositive.While“scandal”putstheplayinanegativelight,the“although”marksashiftfrom“scandal,”and“innovative”suggeststheplaywaswellreceived.Thus,“acclaimed”(praisedenthusiasticallyandpublically)makesthemostsense.

20. Manypeopleraisetheirvoicesinanargument,asthoughhighervolumeprovidesa

greaterabilitytopersuade. C

Explanation

“asthough”indicatesthatsecondpartofthesentenceisthereasonforthefirstpart—raisedvoiceshelpwithpersuasioninanargument.Thus,“provides”makesthemostsense.Thesentencedoesnotdiscussasecondaspectofvolumeorargumentfor“balances”tomakesense(lowvolumeorfighting,forexample,isnotmentioned).“necessitates”doesnotworkbecause“agreaterabilitytopersuade”isthereasonfor“highervolume”andnotitsconsequence.

21. Inthesecondhalfofthenineteenthcentury,thenumberofAmericanbison,which

wereonceabundant,begantodeclineasthebisonbecameasourceoffoodforwestward-movingpioneersandrailroadworkers.

A

Explanation

Boileddown,thesentencestatesthenumberofbisondeclined.However,“whichwereonce”indicatesthenumberofbisonweresomethingelsebeforethedecline.Wewantananswerchoicethatmeansthebisonweregreatinnumberorweresomethinggreaterthanbeforetheirdecline.Thus,“abundant”(existingoravailableinlargequantities)makesthemostsense.“vibrant”means“fullofenergyandenthusiasm,”whichtellsusnothingaboutthenumberofbisonbeforetheirdecline(justthattheyhadgreatpersonalities).

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22. Unlikeothergreatapes,whicharesocial,orangutansaresolitarycreaturesexceptforplayfuljuvenilesandmotherswithbabies. D

Explanation

Thesentencedefinesgreatapesassocialthings,andthe“Unlike”indicatesthatorangutansaretheoppositeofsocial.Wewantananswerchoicethatmeanstheoppositeofwantingtointeractwithothers,likethejuvenilesandmotherswithbabies.Thus,“solitary”(wantingtoactalone)makesthemostsense.Wedonothaveenoughinformationinthesentencetosaywhetherornotothergreatapesareunhappy(“contented”)orstrange(“mysterious”),and“friendly”hasasimilarmeaningto“social.”

23. Thearticleongenesplicingwassoesotericthatonlyahandfulofthestudents

wereabletounderstandit. B

Explanation

Ifonlyasmallportionofagroupofpeopleunderstandsomething,thenthatsomethingisdifficultorabovetheheadsofmost.Whileyoumaynotknowthedefinitionof“esoteric,”youlikelyknowthedefinitionoftheotheranswerchoices.“contrite”meanstoshoworfeelremorse(feelingguilty).Ifthearticlewere“functional,”thenmorepeoplewouldunderstandit.Thereisnotenoughinformationinthearticletodetermineifthetruth(“genuine”)ofthearticlewouldhelpmakeitmoreunderstandable.Thus,ouronlypossibleansweris“esoteric”(onlyunderstoodbyafewpeople).

24. ThefirstAfricanAmericanactortoattaininternationalrenownwasIraAldridge,

oneoftheleadingShakespeareanperformersofthe1800s. D

Explanation

IraAldridgeisaleadingperformer,whichindicatesthatsheisimportant(apositiveconnotation).WewantananswerchoicethatmeansIraattainssomethinggoodregardingtheinternationalcommunity.Thus,“renown”(fame)makesthemostsense,since“rejection”isnegative.International“permanence”doesnotmakesense(whatisbecomingpermanent?Herfameorinfamy?),andwewouldneedtoknowwhatIrais“provoking”internationally(goodorbadthings?).

25. Ancientcavepaintingsofthesun,themoon,andwildanimalstestifytothe

inherenthumandesireandabilitytoportraytheenvironment. D

Explanation

Thesentencestatesthathumanswantandareabletodraworportraytheirenvironment,asevidencedorshownbythepaintings.Youmightthinkthat“cater”makesthemostsense,sinceitreferstotryingtosatisfyaparticularneedordemand.However,theblankisthesubject’sverb—thepaintings’verb.Thehumanscreatethepaintingstosatisfytheirdesire,butthatisnottheactiontakingplaceinthesentence.Instead,theactionistheexistenceofthepaintingsshowor“testify”tothisinherentdesire.

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26. Eachafternoontheshepherdwoulddrivehisflockalongthenarrowroad,effectivelyobstructingthewayforanhour. A

Explanation

Thesentencestipulatesthattheroadis“narrow,”whichisenoughinformationforustodeterminethataflockofsheepwouldblock“theway”foranhour.Thus,“obstructing”(blocking)makesthemostsense.Wewouldneedmoreinformationtodetermineifthesheep/shepherdwere“plundering”(stealing)itemsfromtheroad.Iftheflockwere“renouncing”(declaringabandonmentofsomething)theroadforonlyanhour,it’snotmuchofarenouncement.Wewouldneedmuchmoreinformationfor“transplanting”(moveortransfersomethingtoanotherplace)tomakeanysortofsense.

27. Thecitycouncillookedattheproposalforanewlibrarywithanindifferencethat

borderedonscornfulness. C

Explanation

Theimportantwordinthissentenceis“indifference”(lackofconcern,interest,orcare).Becausethecitycouncildoesnotcareaboutorhaveanyinterestintheproposal,thedirectionthatindifference“borders”onmustmakesense.Nothinginthesentencesuggestsanythingthatwouldmakethecouncilinterestedintheproposal.Thus,“scornfulness”(deepcontemptforsomething)makesthemostsense.

28. TheartofFridaKahlowasstronglyinfluencedbyherlifelonginterestinand

fascinationwithMexicanfolkloreandculture. B

Explanation

Theconjunction“and”between“lifelonginterest”andthesecondblankindicatesthatthesecondblankmimicsthepositivecontextof“lifelonginterest.”IfKahlo’srelationshipwithMexicanfolkloreandcultureispositive,thentheaffectithasonKahlo’sartisalsopositive.“irritation,”“repelled,”andallofanswerchoiceDarenegative.Thus,“influenced”and“fascination”makesthemostsense.

29. Likemostotherchronicmedicalconditions,arthritisisnotcurable;physiciansdo

theirbest,however,toameliorateitssymptoms. A

Explanation

“notcurable”showsthatarthritisdoesnotend,andthefirstblankisawordthatmeansthisfact.“temporary”isthecompleteoppositeofwhatwewant,and“complicated”hasnobearingonwhetherornotsomethingwilleverend.Ifamedicalconditionis“imaginary,”itneverexistedinthefirstplace.Thus,“chronic”istheonlyanswerthatmakessenseforthefirstblank.IfyouwanttomakesureAisthebestanswer,youcanlooktotherelationshipbetween“physician”(ahealer)andthesecondblank.Evenifsomethingisn’tcurable,aphysicianwillwanttohelpasickpersonasmuchaspossible.Aphysicianwouldnotwantto“mimic”(imitate)or“extend”(causetolastlonger)arthritis’ssymptoms.Aphysicianwouldwantto“minimize”(reducetothesmallestamountordegree)thesymptoms,butwealreadyestablishedthat“imaginary”doesn’twork.

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30. AlthoughmuchoftheworstpollutionhasbeeneliminatedintheUnitedStates,tracesofmanytoxicchemicalsstillpersist. B

Explanation

“Although”indicatesthatthesecondblankwillbetheoppositeinsomewayofthefirstblank.Plus,thecombinationof“traces”(smallamounts)and“still”showsthatthepollutioniscontinuingsomethinginsomeway.Thus,ifthepollutioniscontinuing,thefirstblankisawordforstoppingorending.Only“eliminated”fitsthisdescription,and“persists”means“tostubbornlycontinueagainstallodds.”Also,AnswerchoiceBistheonlyoptionwithopposingmeanings.

31. QueenVictoriahadmixedopinionsontheemancipationofwomen;whileshe

fosterededucationforwomen,sheopposedtheirrighttovote. C

Explanation

“mixedopinions”indicatesthatthetwoblankswillbeopposinginmeaning—whatevershewantsforwomen’seducation,itwillbetheoppositeforwomen’svotingrights.AnswerchoicesA,B,andDarecomplimentaryinmeaningsomewayandcontext,butanswerchoiceChasopposingwords—“fostered”(promotethedevelopmentof)and“opposed”(disapproved).

32. Thecasualobserverofalichengrowingonarockwouldneversuspectthatitwasa

compositeoflife-formsinteractingwithoneanother. A

Explanation

Althoughthisisatwo-blanksentence,wereallyonlyneedthewordforthefirstblank.Anobserverthatwouldneversuspectsomethingisonewhoisnotlookingcarefullyatthings.An“inquiring”(showinganinterestinlearningnewthings)wouldwanttothoroughlyobservethelichenontherock,whilean“expert”or“knowledgeable”observerwouldalreadyknowthesecondpartofthesentenceorwhattolookfor.Thus,ouronlyoptionisa“casual”(relaxedorunconcerned)observerinanswerchoiceA.Youcancheckyouranswerwith“composite”(madeupofvariouspartsandelements),butitalsoistheonlywordthatworksfor“interactingwithoneanother.”

33. Iftheauthorshadwrittenwithmorerestraintandavoidedsuchsentimental

language,theirarticleswouldhavehadmorepower. D

Explanation

Theconjunction“and”betweenthefirstblankand“avoided”indicatesthatthefirstblankmimicstherelativemeaningof“avoided.”Plus,“wouldhavehadmorepower”indicatesthatthesecondblankisanegativeadjectivefor“language.”BothanswerchoicesAandBhavepositivewordsanddonotmatchwhatweneed.While“excess”matchesthenegativecontextthatweneed,itdoesnotmimicthemeaningof“avoided.”Thus,“restraint”(self-control)and“sentimental”(sadinanexaggeratedway)makethemostsense—iftheauthorshadrestrainedtheirlanguage,theywouldhaveavoidedlosingpower.

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34. DespitetheapprehensionIfeltatthethoughtofmeetingLuisa,ourbusinesswastransactedinanatmospherethatwasclearlycongenial. A

Explanation

“Despite”indicatesthatthefirstblankisopposinginmeaningandcontexttothesecondblank—ifthefirstblankisnegative,thenthesecondblankmustbepositive.OnlyanswerchoiceAhaswordsthatopposeinmeaningandcontext.AnswerchoicesBandChavewordsthatarebothpositive,whileanswerchoiceDhaswordsthatarenegative.Thus,“apprehension”(fearthatsomethingbadwillhappen)and“congenial”(pleasantoragreeable)makethemostsense.

35. Becausethecaretakerhadledafrugallifestyleformostofhislife,hismilliondollar

bequesttothesettlementhouseamazedthetrustees. A

Explanation

“Because”and“amazed”indicatethatthefirstblankwillopposeinmeaningandcontexttothesecondblankinsomeway.Ifthecaretakerleda“lavish”(verygenerousorextravagant)lifestyle,thenamilliondollar“generosity”wouldnotamazethetrustees—nottomentionthefactthiswouldbeanimproperuseofgenerosityasanactualobjectandnotaquality.Ifthecaretakerleda“generous”lifestyle,anysortof“legacy”(amountofmoneyorpropertylefttosomeone)wouldnotamazethetrustees.Wewouldneedmoreinformationforwhata“unique”lifestyleisandwhya“milliondollarentreaty”(“entreaty”=humblerequest)wouldamazethetrusteesforanswerchoiceDtowork.Thus,“frugal”(sparingoreconomicalwithfoodormoney)and“bequest”(legacyorendowment)makesthemostsense.

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QuantitativeReasoning1. 35 A

Tools: functionnotationSteps: (1) 𝑛∗isjustlike𝑓(𝑥),inthatyouinputaspecificvaluefornforthefunction

(equation)𝑛∗ = 4𝑛 + 3(theinputismultipliedby4andthenaddedto3)(2) Inthiscase,8isourspecificvalueà𝑛∗ = 4𝑛 + 3 → 4 8 + 3(3) 4 8 + 3 → 32 + 3 = 35

QuickTips: • Functionnotationcancomeinmanyformsandwithmultiplevariables• Example:𝑛∎𝑜∎𝑝 = 𝑛 + 𝑜 + 𝑝 → 4∎5∎6 = 4 + 5 + 6

2. 𝑥 − 3 B

Tools: balancingequationsSteps: (1) Inordertofindwhichexpressionisequaltoy,wemustgetybyitselfinthe

givenequation(2) 𝑥 − 𝑦 = 3 → −𝑦 = 3 − 𝑥 → −𝑦 = −𝑥 + 3(3) Since–yisnotthesamethingasy,weneedtomake–ypositivebymultiplying

bothsidesby–1(4) −1 −𝑦 = −1 −𝑥 + 3 → 𝑦 = 𝑥 − 3

QuickTips: • Paycloseattentiontoyoursignsasyourbalancetheequation• Remember,youmustperformthesameoperationtobothsidesofanequation

tokeepitbalanced3. 𝑥 − 1,999 A

Tools: algebrawordproblem,integersSteps: (1) Wearegiventhesumofallintegersfrom1to1000asx,whichcanbewritten

as1 + 2 + 3…+ 998 + 999 + 1000 = 𝑥(2) Youcanalsowriteoutthesumofallintegersfrom1to998inasimilar

manner:1 + 2 + 3…+ 996 + 997 + 998 =?(3) Noticethedifferencebetweenthetwoequations:thefirstequationincludesall

integersfrom1to998,andthenadds999and1000toachievex(4) Thus,thesecondequationissimplymissingthevalues999and1000(missing

canbeatranslationofsubtracting)(5) Ifthesumofthefirstequationisx,thenwecansubtract999and1000fromx

toachievethesumofallintegersfrom1to998(6) 1 + 2 + 3…+ 998 + 999 + 1000 = 𝑥 → 1 + 2 + 3…+ 998 = 𝑥 − 1000 − 999(7) 𝑥 − 1000 − 999 → 𝑥 − 1999

QuickTips: • Writeoutallinformationtothesideofthewordproblemsothatitisseparatefromthetext

4. 12% D

Tools: percentofchange,trianglesSteps: (1) Whilewearenotgivenanyvalues,wecanstillfindtheanswer

(2) Chooseavalueforthetriangle’sheightandbasethatiseasytomanipulatewithpercentages(suchas10or100forboth)

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(3) Using10forbothmeasurements,theareaofthetriangleis50(=>×=>@

= 50)(4) Increasetheheightby10%:10 + 10×0.1 = 10 + 1 = 11(5) Decreasethebaseby20%:10 − 10×0.2 = 10 − 2 = 8(6) Findthenewtriangle’sarea: 11×8 ÷ 2 = 88 ÷ 2 = 44(7) Usethepercentofchangeformula:𝑃 = |EFGHGIJKLMNO|

EFGHGIJK

(8) 𝑃 = |P>LQQ|P>

= RP>→ =@

=>>→ 12%

QuickTips: • Percentageswillworkforwhatevervaluesyouchoosefortheoriginaltriangle• Itiseasiertoconvertfractionswithadenominatorof100topercentages(in

thiscasewemultipliedthedenominatorandnumeratorby2)5. 14 B

Tools: multiplyingpolynomials,balancingequationsSteps: (1) Startwiththeleftsideoftheequationbysquaring𝑥 + 7andfollowFOIL

(2) 𝑥 + 7 @ → 𝑥 + 7 𝑥 + 7 (3) First:(𝑥)(𝑥) = 𝑥@(4) Outside: 𝑥 7 = 7𝑥(5) Inside: 7 𝑥 = 7𝑥(6) Last: 7 7 = 49(7) 𝑥@ + 7𝑥 + 7𝑥 + 49 → 𝑥@ + 14𝑥 + 49(8) Now,lookattheplacementof14andmin𝑥@ + 14𝑥 + 49 = 𝑥@ + 𝑚𝑥 + 49(9) Ifyouremovesimilarelementsfrombothsides,14 = 𝑚remains

QuickTips: • IfyouareunfamiliarwithFOIL,youcanalsousethedistributiveproperty• 𝑥 + 7 𝑥 + 7 → 𝑥 𝑥 + 7 + 7 𝑥 + 7 → 𝑥@ + 7𝑥 + 7𝑥 + 49 → 𝑥@ + 14𝑥 + 49

6. 91.00 C

Tools: mean,multiplesSteps: (1) Findthesumof370and85,andthendividebythenumberoftests(5)

(2) 370 + 85 = 455(3) 455 ÷ 5 = 91

QuickTips: • Since370and85aremultiplesof5,theirsumwillalsobeamultipleof5• Theresultofdividingamultipleof5by5canonlybeaninteger,soanswer

choicesBandDcannotbetrue7. 42inches C

Tools: perimeter,wholenumbersSteps: (1) Sincethemeasurementsareinwholenumbers,wecanlooktothefactorsof

110tofindourmeasurements(2) 110 → 1, 2, 5, 10, 11, 22, 55, 110(3) Wecanquicklyeliminatetheperimetersof1and110and2and55(4) Ifweuse5and22,theperimeteris54inches,answerchoiceD(5) However,ifweuse10and11,theperimeteris42

QuickTips: • Sincethemeasurementsarewholenumbers,answerchoicesAandBcannotbetruesincethesumofall4sidesofarectanglewillbeanevennumber

• Thesumoftwoevennumbersisanevennumber

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• Thesumoftwooddnumbersisanevennumber8. AnswerchoiceA’sgraph A

Tools: analyzingchartsandgraphs,temperatureSteps: (1) Sincethepotatostartsoutcool,answerchoicesBandCdonotworkbecause

theyplacethepotatoabove300°beforeanytimehaselapsed(2) Sincetheovenissetto350°,thepotatocannotgoabovethattemperatureasin

answerchoiceD(3) OnlyanswerchoiceAaccuratelyportraysthecookingofthepotatoovertime

QuickTips: • Paycloseattentiontotheinformationprovidedbythewordproblem• “coolpotato”and“hot(350°)oven”giveusourparametersfortheproblem

9. 6cm B

Tools: similartrianglesSteps: (1) Becausethetrianglesaresimilar,theysharethesamedegreevaluesand

proportionalsidelengths(2) IfyoulooktosidelengthsQRandTU,you’llnoticethatQRisincreasedbyV

@to

achieveV@𝑥;thus,ourproportionisV

@forthetriangles

(3) MultiplythesidelengthofQSbyV@tofindthelengthofTV

(4) 4× V@= =@

@= 6cm

QuickTips: • Similartriangleshavethesamedegreevaluesandproportionalsidelengths• Congruenttriangleshavethesamedegreevaluesandsidelengths

10. 1 B

Tools: exponents,distributivepropertySteps: (1) Thefastestwaytosolvethisquestionisbychangingallvaluesintotermsof3

(2) 9 → 3@;therefore,V(VWXVY)

VW(VXVW)

(3) Followingthedistributiveproperty,you’llnoticethatthesamethingishappeninginthenumeratorandthedenominatorV(V

WXVY)VW(VXVW)

→ VYXVZ

VYXVZ= 1

(4) Theresultofanumberdividedbyitselfisalways1(5) Youcanalsomultiplyeverythingout(reducewhereyoucantomakeiteasier)(6) V(V

WXVY)[(VX[)

→ =([X@\)V(=@)

→ = VRV =@

→ =(V)V(=)

→ 1QuickTips: • AnswerchoiceAcanbequicklyeliminatedbecausenopartoftheexpression

willresultin0• Inmanycases,youdonotneedtomultiplyeverythingouttoarriveatyour

answer11. 0.25miles A

Tools: analyzingchartsandgraphsSteps: (1) SincewearelookingforthepointduringJane’swalkwhereshewaitedforher

friend,weneedtofindaspotonthegraphwherethedistancefromJane’s

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homeremainsthesamebutthetimestillincreases(2) Thevaluesonthex-axisandy-axisarethesame,butthex-axisisclearly

markedastimewhilethey-axisismarkedasdistance(3) Atthefirsthalf-hourmark,Jane’sdistanceis0.25milesfromherhome(4) Noticethatthisdistancestaysat0.25milesfrom0.50hourstoalittlebefore

the0.75hourmark—thisistheonlytimesomethinglikethishappens(5) Thus,theonlypossibleconclusionisJanewas0.25milesfromherhomewhen

shewaitedforherfriendQuickTips: • Sincethequestionislookingforapointwhereonevaluedoesnotincrease,

youcanquicklyobservethatthisonlyoccursatonepointforthewholegraph• Paycloseattentiontothedesignationsfortheaxes

12. range D

Tools: mean,median,mode,rangeSteps: (1) Muchoftheinformationinthisproblemissuperfluous(unnecessary)

(2) Thinkabouthowthemean,median,mode,andrangewillbeaffectedbyadding6pointstoeveryscore

(3) Themean,median,andmodewouldchangeinsomeway,butfocusontherange(thedifferencebetweenthelowestandhighestvalueinadataset)

(4) Ifthehighestscoreisa100,thenthelowestscoreisa36(rangeof64)(5) Ifyouincreasebothofthosescoresby6points,thehighestscorewouldbe

106andthelowestwouldbe42(6) Therangewillstillbe64withthepointincrease;thus,therangechangesthe

leastQuickTips: • Confirmwhatthequestionislookingforbeforewritingoutinformationtothe

sideoftheproblem(you’dsaveyourselfsometime)• KeepinmindthedefinitionsofconceptsintheQuantitativeReasoningsection

sincesomeofthequestionsaretestingthatknowledge13. Maud B

Tools: probabilitySteps: (1) Forprobability,itisimportanttounderstandtheoperationthatoccursfor

eventswithwordslike“or”(2) The“or”meansthattheprobabilityofrollingasumof6pointsisaddedtothe

probabilityofrollingasumof4pointsforMaud,givingherthegreaterprobabilityofreceivingapoint

(3) Jimonlyhasthefirstprobability,whileMaudhasboth(givinghertheedge)(4) Theprobabilityofrollingasumof6pointsis P

VRandtheprobabilityofrollinga

sumof4pointsis VVR; PVR+ V

VR= ]

VR

(5) Jim: PVR< ]

VR:Maud

QuickTips: • Youdonotneedtoknowtheactualprobabilitiestoanswerthisquestion• ThefactthatJimandMaudreceiveapointforthefirstprobabilitybutMaud

alsoreceivesapointforthesecondprobabilityisalltheinformationweneed

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14. 14 DTools: mean,median,range,symmetry,analyzingchartsandgraphsSteps: (1) Themedian(ormiddle)ofthedatais7(thetallestbaronthegraph)

(2) Themode(orvaluethatappearsthemost)ofthedataisalso7(3) Ifthedataissymmetrical(onehalfisthesameastheotherhalf)aboutthe

valueof7,thenthefrequencyofthevaluesontheleftsideofthevalueof7willbemirroredforthevaluesontherightsideofthevalueof7(thebarswilllooklikeatriangle)

(4) Therange(differencebetweenthelowestandhighestvalue)ofthedatais8andthemaximumvalueis11,whichmeansthelowestvalueis3

(5) Wewantthenumberofdatapoints(totalnumberoffrequency)thatfallabovethevalueof7,whicharethedatapointsforthevaluesof8,9,10,and11

(6) Sincetheleftof7ismirroredontheright,thebarfor6isthesamefor8andthebarfor5isthesamefor9

(7) Countingeachpointoffrequency,8has7points,9has4points,10has2points,and11has1point

(8) 7 + 4 + 2 + 1 = 14QuickTips: • Thisquestionisthemostconvolutedquestiononthispracticetest,andmost

studentshavetroublewithit• YoucanwritealloveryourtestbookletontherealISEE(doso!)

15. 9 C

Tools: functions,compoundinequalitiesSteps: (1) Thecompoundinequalityisstatingthatthesmallestvalueforxis–2andthe

largestvalueforxis1(2) Inputthesetwovaluestofindy(moststartwith1sinceitisthelargest)(3) 𝑦 = 2𝑥@ + 1 → 2 1 @ + 1 → 2 + 1 → 𝑦 = 3(4) 𝑦 = 2𝑥@ + 1 → 2 −2 @ + 1 → 2 4 + 1 → 8 + 1 → 𝑦 = 9

QuickTips: • Youcaninputtheanswerchoicesforyandsolveforxtofindtheanswer• AnswerchoiceDisachievedbyinputting 8,whichisnotapossiblevalueofx,

since 8isapproximately±2.8• AnswerchoiceAisachievedbyinputting0

16. 𝑔 0.9 < 𝑓 0.9 < 𝑓 1.1 < 𝑔(1.1) B

Tools: radicals,exponentsSteps: (1) Whileyouarenotallowedacalculatoronthistest,youdon’tneedtoknowthe

exactresultsforeachoftheinputsof0.9and1.1intofunctions𝑓(𝑥)and𝑔(𝑥),whichtakethesquarerootoftheinputandsquaretheinputrespectively

(2) Ifyousquareapositivedecimalnumberthatislessthan1,suchas0.9,thentheresultwillbelessthanbefore(0.9@ = 0.81)

(3) Ifyoutakethesquarerootofapositivedecimalnumberthatislessthan1,suchas0.9,thentheresultwillbegreaterthanbefore( 0.9 ≈ 0.95)

(4) Theoppositeistrueforadecimalnumberthatisgreaterthan1,suchas1.1,where1.1@ = 1.21and 1.1 ≈ 1.05

(5) 0.9@ < 0.9 < 1.1 < 1.1@QuickTips: • Consider 1and1@torecognizehow0.9and1.1willplayoutwith 𝑥and𝑥@

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17. thedifferenceinJohn’sandErin’sspeeds DTools: distance,rate,andtimeSteps: (1) NotethatJohnandErinrunataconstantrateandonthesamepath

(2) Johnis500metersaheadofErinwhenshestartstorun,andJohnwillmaintainhisconstantratethewholetimeErinisattemptingtocatchup

(3) IfJohnandErinarebothrunningat20metersperminute,thenErinwillforeverremain500metersbehind

(4) IfJohnisrunning20metersperminuteandErinisrunning25metersperminute,Erininwillbeabletocatchupin100minutes(𝑡 = c

F→ 𝑡 = P>>

P= 100)

(5) IfweknowthedifferenceinJohnandErin’sspeeds,thenwecandeterminehowlongitwilltakeErintocatchuptoJohn

QuickTips: • Inputeasyvaluesforeachvariableinawordproblemtotestoutinformationtoseewhatwouldbetrue

• Multiplesof5or10workwellforinput18. answerchoiceA’sfigure A

Tools: geometricpatternsSteps: (1) Notethatthecubehasacirclewithonetrianglepointingawayfromitandthe

othertrianglepointingtoit(2) Foldedover,answerchoiceBwouldhavethepatternr¡r(3) Foldedover,answerchoiceCwouldhavethepatterns¡s(4) Foldedover,answerchoiceDwouldhavebothtrianglespointedawayfrom

thecircle(5) OnlyanswerchoiceAwouldhavethepatternofonetrianglepointingaway

fromthecircleandtheothertrianglepointingtothecirclewhenfoldedoverQuickTips: • AnswerchoiceAistheonlypatternwithatrianglepointingtowardsthecircle,

sotheotheranswerchoicesdonotwork19. Thetwoquantitiesareequal C

Tools: orderofoperationsSteps: (1) AllweneedtodoisfindthevalueofthequantityunderColumnA(ColumnBis

alreadysimplified)(2) Followtheorderofoperations:5 + 2× 4 + 3 → 5 + 2×(7)(3) 5 + 2× 7 → 5 + 14 → 19(4) 19 = 19

QuickTips: • Manystudentsmissthisquestionbecausetheyadd5and2firstandthenmultiplythatproducttothesumof4and3—don’tforgettheproperorder!

20. yisthegreatervalue B

Tools: area,perimeter,algebraicequationsSteps: (1) SincetheareaofrectangleQis18cm2,wecanfindxbyusingtheformulafor

theareaofarectangle(𝐴 = 𝑙𝑤)(2) 2𝑥 𝑥 = 18 → 2𝑥@ = 18 → 𝑥@ = 9 → 𝑥 = 3(3) SincetheperimeterofrectangleRis30cm,wecanfindybyusingtheformula

fortheperimeterofarectangle(𝑃 = 2𝑙 + 2𝑤)

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(4) 2 2𝑦 + 2 𝑦 = 30 → 4𝑦 + 2𝑦 = 30 → 6𝑦 = 30 → 𝑦 = 5(5) 3 < 5

QuickTips: • Assoonasyouseeinformationregarding“area”or“perimeter,”quicklyjotdowntheappropriateformulasforthoseconcepts

21. Thetwoquantitiesareequal C

Tools: multiplyingpolynomialsSteps: (1) ThereisnothingtobedonewithColumnB,sowecanfocusonColumnA

(2) Usethedistributivepropertytocorrectlymultiplythepolynomials(3) 𝑥 − 𝑦 𝑥@ + 𝑥𝑦 + 𝑦@ → 𝑥 𝑥@ + 𝑥𝑦 + 𝑦@ − 𝑦 𝑥@ + 𝑥𝑦 + 𝑦@ (4) 𝑥 𝑥@ + 𝑥𝑦 + 𝑦@ − 𝑦 𝑥@ + 𝑥𝑦 + 𝑦@ → 𝑥V + 𝑥@𝑦 + 𝑥𝑦@ − 𝑥@𝑦 − 𝑥𝑦@ − 𝑦V(5) 𝑥V + 𝑥@𝑦 + 𝑥𝑦@ − 𝑥@𝑦 − 𝑥𝑦@ − 𝑦V → 𝑥V + 𝑥@𝑦 + 𝑥𝑦@ − 𝑥@𝑦 − 𝑥𝑦@ − 𝑦V →

𝑥V − 𝑦V(6) 𝑥V − 𝑦V = 𝑥V − 𝑦V

QuickTips: • Youcouldalsoinput1,0,and–1asvaluesforbothxandytotestthem,butdoingsomightactuallytakeyoulongerthansimplymultiplyingColumnA

22. $3.00isthegreatervalue B

Tools: substitutioninlinearequationsSteps: (1) Createanequationforthesumofthevalueofdimesandquartersinthe

parkingmeter:0.10𝑑 + 0.25𝑞 = $4.50(2) Iftherearetwiceasmanydimesastherearequartersintheparkingmeter,

thenwecanshowdas2q(𝑑 = 2𝑞)(3) Now,substitute2qfordinthefirstequation:0.10𝑑 + 0.25𝑞 = 4.50 →

0.10 2𝑞 + 0.25𝑞 = 4.50(4) Solveforq:0.10 2𝑞 + 0.25𝑞 = 4.50 → 0.20𝑞 + 0.25𝑞 → 0.45𝑞 = 4.50(5) 0.45𝑞 = 4.50 → 𝑞 = 10quarters(6) Thetotalvalueofthequartersinthemeteris$2.50(or0.25×10)(7) $2.50 < $3.00

QuickTips: • Inthefirstequation,youcannotsimplyadddtoqandsetitequalto4.50—thenumberofitemsdoesnotequateto$4.50,butthevalueoftheitemstimesthenumberoftheitemsdoes(10quartersat$0.25eachplus20dimesat$0.10eachmakes$4.50)

23. Theslopeoflinekisthegreatervalue A

Tools: parallellines,slopeoflinearequationsSteps: (1) Ifalineisparalleltoanotherline,thentheirslopesareequal

(2) Theslopeoflinejis3(𝑦 = 𝑚𝑥 + 𝑏;wheremistheslopeand𝑚 = 3)(3) 3 > −3

QuickTips: • Noticethatthelinesareatanincline;thus,theirslopesmustbepositive• Anypositivevalueisgreaterthananegativevalue—evenifyouweren’tgiven

theequationforlinejyoucouldfindtheanswerwiththisknowledgealone

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24. Therelationshipcannotbedeterminedfromtheinformationgiven DTools: perimeter,areaSteps: (1) Whilewearenotgivenanyvaluesforthelengthandwidthoftherectangle,we

cantestsomepossibilitiesbyusingtheformulasforperimeter(𝑃 = 2𝑙 + 2𝑤)andarea(𝐴 = 𝑙𝑤)

(2) Ifthelengthis10,thenthewidthwouldbe15:2 10 + 2𝑤 = 50 → 20 + 2𝑤 =50 → 2𝑤 = 30 → 𝑤 = 15

(3) Theareaofthesedimensionswouldbe150: 10 15 = 150(4) Inthiscase,ColumnAisgreater(5) Ifthelengthis5,thenthewidthwouldbe20:2 5 + 2𝑤 = 50 → 10 + 2𝑤 =

50 → 2𝑤 = 40 → 𝑤 = 20(6) Theareaofthesedimensionswouldbe100: 5 20 = 100(7) Inthiscase,ColumnBisgreater(8) Becauseouranswerschange,theonlypossiblesolutionisanswerchoiceD

QuickTips: • Wheninputtingvalues,ifyouranswerchangesfromonecasetothenext,thentherelationshipcannotbedetermined(moreconcreteinformationisneeded)

25. Areaoftheshadedregionisthegreatervalue A

Tools: area,operationsonalgebraicexpressionsSteps: (1) Findtheareaofthelargersquareasifthesmallcutoutisnotthere

(2) 𝑥 𝑥 = 𝑥@(3) Findtheareaofthecutout(4) 𝑦 𝑦 = 𝑦@(5) Subtracttheareaofthecutoutfromtheareaofthelargersquare(6) 𝑥@ − 𝑦@(7) If𝑥 > 0and𝑦 > 0,thenxandyarepositivevalues(8) Input2forxand1forytotestColumnAandB(9) 𝑥@ − 𝑦@ → 2 @ + 1 @ = 4 + 1 = 5vs.𝑥@ − 𝑥𝑦 − 𝑦@ → 2 @ − 2 1 − 1 @ →

4 − 2 − 1 = 1(thiswillbetrueforallinputs)QuickTips: • NotethatthequantityunderColumnBisgenerallythesameasColumnA

exceptthatanadditionaltermissubtractedfrom𝑥@• Sinceweknowbothxandyarepositivevalues,wecansafelyconcludethatan

additionalpositivevalueissubtractedfromthequantityunderColumnB,makingColumnAlargerinallinstances

26. 10isthegreatervalue B

Tools: integers,consecutivenumbersSteps: (1) Althoughwedon’tneedtodoanythingwithColumnB,wecanstilluseit’s

quantitytodeterminethevalueofthequantityinColumnA(2) Ifweassumethatgreatestofthe3consecutiveintegersis10,wecanfindthe

productofthe3integersandcomparethattheproductoftheactual3consecutiveintegers

(3) If10isthegreatestinteger,thenitwouldbe8×9×10 = 720(4) 720ismuchlargerthan210,whichmeansthatthegreatestintegerforColumn

Acannotbegreaterorequalto10(itmustbeasmallerinteger)(5) Incidentally,theconsecutiveintegersare5,6,and7(5×6×7 = 210)

QuickTips: • Itwouldbeverychallengingtotrytofindthesolutionalgebraically

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• 𝑥 𝑥 + 1 𝑥 + 2 → 𝑥@ + 𝑥 𝑥 + 2 → 𝑥@ 𝑥 + 2 + 𝑥 𝑥 + 2 → 𝑥V + 2𝑥@ +𝑥@ + 2𝑥 → 𝑥V + 3𝑥@ + 2𝑥 = 210(Goodlucksolvingthisinatimelymanner!)

27. 25𝑛 − 1isthegreatervalue A

Tools: orderofoperationsSteps: (1) Althoughwearenotgivenanyvaluesforn,wecannotassumetheanswerisD

(2) Instead,testyourowninputsforn(1,0,and–1tostart)(3) If𝑛 = 1,thenAis25 1 − 1 = 24andBis25 1 − 1 = 0;24 > 0(4) If𝑛 = 0,thenAis25 0 − 1 = −1andBis25 0 − 1 = −25;−1 > −25(5) If𝑛 = −1,thenAis25 −1 − 1 = −26andBis25 −1 − 1 = −50;−26 >

−50(6) Inallcases,ColumnAisgreaterthanColumnBandwillremaintrueforall

otherinputsQuickTips: • Remember,ifyouaren’tgivenanyvaluesforvariables,inputsimpleto

calculatevalues(2,1,0,-1,and-2)28. Therelationshipcannotbedeterminedfromtheinformationgiven D

Tools: perimeterSteps: (1) Itmayseemlikewe’regivenadecentamountofinformation,butyou’llquickly

seewereallyneedtohavesomeparametersforxandytosolvethisone(2) If𝑥 = 1and𝑦 = 2,thenAis3andBis6(3) If𝑥 = 2and𝑦 = 1,thenAis6andBis3(4) Becauseouranswerschanged,theonlypossiblesolutionisD

QuickTips: • Wheninputtingvalues,ifyouranswerchangesfromonecasetothenext,thentherelationshipcannotbedetermined(moreconcreteinformationisneeded)

29. Thetwoquantitiesareequal C

Tools: probabilitySteps: (1) WhileitmayseemlikethefirstpartofeachstatementunderColumnAand

ColumnBmatter,therollofanumbercubehasnobearingontheprobabilityofacoinlandingheadsortailsup(theseeventsareindependent)

(2) Ifaneventisindependentofanotherevent,thenthereisnocorrelation(connection)totheevents’probabilities

(3) Thus,therealquantityunderColumnAistheprobabilityofthecoinlandingtailsupor=

@

(4) TherealquantityunderColumnBistheprobabilityofthecoinlandingheadsupor=

@

(5) =@= =

@

QuickTips: • Independenteventsdonotinfluenceeachother’sprobabilities(adieiscastandacoinflipped)

• Dependenteventsdoinfluenceeachother’sprobabilities(2marblesareremovedfromabag—theprobabilityofbothbeingred)

• Mutuallyexclusiveeventscannothappenatthesametime(coinlandingheadsandtailsup)

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30. Themedianscoreisthegreatervalue ATools: median,range,analyzingchartsandgraphsSteps: (1) Youcouldspendagreatdealoftimefindingtheactualmedianforthegraph,

butitisfastertofindtherangeandcomparethattoapossiblemedian(2) Thehighestpossiblevalueofthisgraphis100andthelowestpossiblevalueis

51(3) Therangeisthedifferencebetweenthesetwovalues:100 − 51 = 49(4) Therangeofthisgraphis2lessthanthelowestpossibleexamscoreonthis

graph;thus,therangecannotbegreaterthanthemedian(middlevalue)ofthisgraph

QuickTips: • Ifaquestionseemslikeitwilltakealongtimetodeterminesomething,likethemedianofthisgraph,thenthereislikelyamoresimplewaytosolvingtheproblem

• Paycloseattentiontowhatthequestionislookingforandkeepinmindthedefinitionsforrange,median,mode,andmean

31. Theprobabilitythatthefirstcandyselectedisgreenisthegreatervalue B

Tools: probabilitySteps: (1) Sincethereare5candies,wecandeterminethatforasingleselectionthe

probabilityofselectinganorangecandyis@PandagreencandyisV

P

(2) ColumnBisprobabilityofselectingagreencandyonthefirstdraworVP

(3) ColumnAistheprobabilityofselectingagreencandy,puttingitback,andthenselectinganothergreencandy(dependentevents)

(4) ForColumnA,wemultiplytheprobabilityofthefirstdraw(VP)tothe

probabilityoftheseconddraw(VP)

(5) VP× VP= [

@P

(6) [@P< V

P

QuickTips: • Iftheeventsaredependent,thenyoumultiplytheprobabilityofeacheventtoeachother

• Ifthefirstdrawisnotreplaced,thenyoumustreducethenumeratoranddenominatoroftheprobabilityaccordingly(V

P→ @

Q)

32. $1.50isthegreatervalue B

Tools: algebrawordproblemsSteps: (1) FindtheincreasedpriceofapplesinMarch:𝐴 + 𝐴×% = 𝐵

(2) 1.50 + 1.50×0.10 = 1.50 + 0.15 = 1.65(3) FindthedecreasedpriceofapplesinAprilfromMarch’sprice(4) 1.65 − 1.65×0.10 = 1.65 − 0.165 = 1.485(5) $1.485 < $1.50

QuickTips: • Increasingavaluebyapercentageandthendecreasingtheresultingvaluebythesamepercentagewillresultinasmallervaluethantheoriginalvalue

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ReadingComprehension-Passage11. Theprimarypurposeofthepassageistodescribeadiscoverythatexcitedthe

author’sinterest. B

Explanation

Thewholeofthepassagediscussestheauthor’sexperimentwithcaddislarvae,fromcollectingcreaturesinjarstoshowingoffhisexperiment’sresultstohisfriends.Whilethepassagestartsoffbymentioningtheauthor’sinterestinfreshwaterbiology,thepassagedoesnotdiscusswhyheisinterestedinit(answerchoiceA).Theauthordoesnotcomparethecaddislarvaetoanyothercreature(answerchoiceC)anddoesnotdiscussthecompletelifecycleofthecaddislarvae(answerchoiceD).

2. Inline4,“minute”mostnearlymeanstiny.

C

Explanation

Ifacreaturecanfitintoajar,evenalargeone,thenitissmallinsize.Thus,theauthoruses“minute”(mı̄̍ n(y)o͞otormy-noot)tomean“tiny”or“extremelysmall.”Evenifyoudonotknowthisdefinitionof“minute,”youdoknowthemeasurementoftimeforwhich“minute”stands.Youknowthataminuteisaveryshortorsmallamountoftime,soyoucanstillthinkofcreaturesinthatgeneralsense.Nothinginthesentenceoritssurroundingcontentsuggeststhatthecreaturesare“timely”(punctualoron-time),“timorous”(nervousorfearful),or“tireless”(nevertiring).

3. Theauthorcausedthelarvaetodecoratetheircocoonswithstripesbychanging

theirenvironmentatvariousstagesofcocoondevelopment. C

Explanation

Looktothesecondandthirdparagraphs.Theauthorfirststateshisfriendsaid,“…thatifyouremoveacaddislarvafromitscocoonandplaceitinajarofclearwater,itwouldspinitselfanewcocoonanddecoratetheoutsidewithwhatevermaterialsyousupplied.”Then,inthethirdparagraph,theauthorstates,“Idiscoveredthatbymovingthelarvaetoadifferentjarwithanewsubstance,theywouldproducenewmulticoloredcocoons.”Theauthormovesthelarvaefromjartojar(answerchoiceC).Hedoesnotkeepthelarvaeinasinglejarthroughoutthewholeprocess(answerchoicesA,B,andD).

4. Inline8,theauthordescribesthecaddislarvaeas“ratherdull”becausetheyhad

beenlivinginastagnantpool. C

Explanation

Looktoline7-11,“ThecaddisIhadcollectedlookedratherdull,forIhadcollectedthemfromastagnantpool…”Theauthordoesnotsaythatthelarvaewerestillinthecaterpillarstageorthatthelarvaewereremovedfromthepoolbeforefinishingtheircocoons(AandD).Whiletheauthordoesplacethelarvaeintoajar,hedoesnotsaythat’swhythelarvaeweredull(answerchoiceB).

5. Inthefinalsentence(lines33-36),theauthorsuggeststhatthecaddislarvaewere

annoyedbytheauthor’sexperiments. B

Explanation

Theauthorcontinuallymakesthelarvaecreatenewcocoonsdecoratedwithnewmaterialsagainandagainwithhisexperiments.Inlines33-36,theauthorstatesthe“poorcreatureswerereallyratherrelieved”whentheywereallowedtohatchandflyaway,insteadofconstantlybuildingcocoons.“relieved,”“forget,”and“problems”allindicatethatthelarvaewereinanegativesituationfromwhichtheywereeventuallyfreed.AnswerchoicesAandCreflectpositivereactionsthat

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wouldnotresultfromnegativesituations,whileweneedmoreinformation(moreofareactionfromthecaddis)foranswerchoiceDtowork.

6. Aconversationwiththeauthor’sfriendledtheauthortoexperimentwithcaddis

larvae. B

Explanation

Looktolines12-18,“Ihadbeentoldbymyfriend…Decidingtoexperiment…”Theauthorhasaconversationwithhisfriendaboutthefriend’sobservationofcocoondevelopmentincaddislarvae,andtheauthorthendecidestoexperimentwithhisfriend’sobservation.Nobookorfamousnaturalistismentionedinthepassage(AandC).Whiletheauthordoesspendmostofhistimecollectingcreaturesfrompondsandstreams(answerchoiceD),itistheconversationwithhisfriendthatleadshimtoexperimentonthelarvae.WewouldneedproofthattheauthorexperimentsonhiscollectionbeforetheconversationforDtobetrue.

ReadingComprehension-Passage27. “NorthwestCoastIndiansarefamousforlarge,beautifultotempoles”best

expressesthemainideaofthispassage. D

Explanation

ThepassagediscussesthetotempolescreatedbytheNorthwestCoastIndians—thepoles’varyingstyles,howthepolesaremade,andthepoles’history,decline,andreturn.AnswerchoicesAandConlycoveronepartofthepassage,whileanswerchoiceBisnotmentionedinthepassage.

8. Theauthorimpliesthattotempolecarvingwasabandonedforalongperiod.

A

Explanation

Whilethepassagedoesmentionthe1800sandthe1950s,thereisnodirectconnectionbetweenthesedatesandtheabandonmentoftotempolecarving.However,looktolines23-26,“Inthe1950s,thefewremainingcarvers…reproduceoldanddecayingKwakiutlpoles.”Ifsomethingisoldanddecayingandnomentionofanynewerpolesismadeinthepassage,thenwehaveourevidenceforanswerchoiceA.Plus,theauthordoesnotmentionanythingaboutmakingalivingorrespectfortotempolecarvinganddoesnotmentionthepopulationofcedartreesinthepassage.

9. “Aprocessisdescribedinchronologicalorder”bestdescribestheorganizationof

lines8-17. B

Explanation

Lines8-17discusstheprocessofplanning,designing,carving,and(insomecases)paintingtotempoles(inthatorder).Plus,“before”and“After”areindicatorsoftime.Theauthordoesnotcontrastdifferenttotempoledesignsandisnotexpressinganopinion(AandC),whileanswerchoiceDismoreaboutthepassageasawholethanlines8-17.

10. Accordingtothepassage,totempoleswerecarvedbyNorthwestCoastIndian

tribes. D

ExplanationLooktolines1-3,“Totempoles…areatrademarkoftheNorthwestCoastIndians.”AnswerchoiceAisincontradictionwiththepassage’sfinalparagraph,whileanswerchoicesBandCareincontradictionwiththefirstparagraph.

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11. TheauthorofthepassageappearstocaremostdeeplyaboutthefactthattheartisticheritageofNorthwestCoastIndianswassaved. D

Explanation

Sincethepassageisabouttheuniquenessoftotempoles(“Eachpoleisdifferent,andeachpoletellsitsownstory”),theauthorisdemonstratinghisinterestintheartoftotempolecarving.Plus,theauthordevotesafullparagraphtothenearextinctionandrevivaloftotempolecarving.AnswerchoicesAandCarenotgivennearlyasmuchemphasisasanswerchoiceD.AnswerchoiceBisnotevenmentionedinthepassage.

12. Accordingtothepassage,amuseumhelpedpreservetheartoftotempolecarving

bycommissioningcarverstoduplicateexistingtotempoles. B

Explanation

Looktolines23-26,“…fewremainingcarverswerehiredbyUniversityofBritishColumbiaMuseumofAnthropologytoreproduceoldanddecayingKwakiutlpoles.”“commissioning”isthesamethingas“hired”and“reproduce”isthesamethingas“duplicate.”“reproduce”isnotthesamethingas“preserve”(answerchoiceA),andanswerchoicesCandDarenotmentionedinthepassage.

ReadingComprehension-Passage313. Theprimarypurposeofthispassageistosuggestthatreportsexpressingconcern

overthestateofeducationalpreparednessintheUnitedStatesmaybeunnecessarilyalarming.

A

Explanation

Thepassageopenswithastatementabouthow“newsmediaseemtobefilledwithalarmingeditorials”aboutthestateofeducationalpreparedness.Theauthorthendiscussestwostudiesthatcountertheclaimsofthenewsmedia.Finally,theauthorcloseswithstatingthatthenewsmediamisunderstandsjobsandtheirgrowthrate.AnswerchoiceBiscountertowhatisstatedinthepassage,whiletheauthordoesnotexpressanylamentationforhighschooldropouts(answerchoiceD).AnswerchoiceCdoesnotappearanywhereinthepassageeither.

14. Inline6,“pundits”mostnearlymeansexperts.

B

Explanation

The“pundits”inlines5-11aredeclaringthat“lasertechnology,robotics,andcomputer-controlledequipment”willbesoinfluentialinourlivesthatstudentsmusthave“advancedtrainingorevencollegedegrees.”Inotherwords,the“pundits”inthenewsmedia“know”somethingtobetrueandaremakingrecommendationsbasedonthatknowledge.Ifyouwantadviceaboutsomething,yougenerallyseekan“expert”onthematter,orapersonwhohasacomprehensiveorauthoritativeknowledgeofsomething.Thus,answerchoiceBbestmatches“pundit”(anexpertinaparticularsubjectorfieldwhoisfrequentlycalledontogiveopinionsaboutittothepublic).AnswerchoiceAwouldonlyworkifthepeopleinthenewsmediawerestatingafactinsteadofanopinion(“editorials”areopinionpieces).

15. TheauthorofthepassagedoesallofthefollowingEXCEPTcomparetrendsin

differentcountries. C

Explanation Theauthorprovidesdata(answerchoiceA)throughoutthesecondparagraphandtheendofthethirdparagraphanddescribestheresearchoftwostudiesinthe

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secondparagraph(answerchoiceB).Theauthorcitesthestatementsofcommentatorsfromthenewsmediainthefirstparagraph(answerchoiceD).NowhereinthepassagedoestheauthorcomparetheUnitedStatestoanothercountry.

16. Theworkforcebeingpreparedbyourschoolstodaymatchesfairlycloselythe

workforcelikelytobeneededbyoursocietyinthenearfuture. B

Explanation

AnswerchoiceAandDcontradictthefindingsofthetwostudies,whichstatethattheupcomingworkforcewillmatchfutureworkforceneeds.Whilethestudiesdoincludedataabouthighschooldropouts,nowhereinthepassagedoestheauthordiscusstheneedforthepublicandeducatorstobandtogethertopreventthedropouts.OnlyanswerchoiceBistheonlyconclusionthatcanbedrawnfromthestudiesinthesecondparagraphofthepassage.

17. Theauthor’stonewhendiscussingthenewsmediaisbestdescribedascritical.

B

Explanation

Theauthorpresentsevidencethatcontradictsthecommentsmadebythenewsmediaandsuggeststhatthenewsmediaisconfusedaboutthedata.Thus,theauthorisplacingtheblameforthe“discrepantconclusions”uponthemediaandnottheresearchers.Theauthorwouldhavepresenteddatathatconfirmedthestatementsmadebythenewsmediaifhe“admired”themorhewas“worried”abouttheirstatements(answerchoicesAandD).Althoughhumorissubjective,nowhereinthepassagedoestheauthormakelightofthenewsmedia’scommentsorcrackjokesatthenewsmedia’sexpense(answerchoiceB).

18. Thepurposeofthelastparagraph(lines40-55)istoprovideanexplanationforthe

differingpointsofview. A

Explanation

Looktolines43-47,“Onepossibleexplanationforthediscrepantconclusions…”Theauthorisattemptingtoexplainthedifferingpointsofviewbystatingthatthenewsmediaispossiblyconfusedaboutthe“ratesofgrowthwithactualnumbersofjobs.”Nowhereinthethirdparagraphdoestheauthorexpressconcernforthefuturewelfareoftheeconomy(answerchoiceB)orproposethatadditionalresearchisneeded(answerchoiceC).Theauthoronlycriticizestheshortcomingsofthenewsmedia’sargumentandnottheresearchers’(answerchoiceD).

ReadingComprehension-Passage419. Thepassageisprimarilyconcernedwithprovidingbackgroundinformationfora

discussionofthemedievalhomeanditscomforts. C

Explanation

Theauthoropenswiththestatement,“Anydiscussionaboutdomesticlife…cannotrefertomostofthepopulation,whowerepoor.”Theauthorlaysthegroundworkfordiscussingthemedievalhomeanditscomfortsbyfirstdiscussingwhythepoor,aristocracy,andtheclergycannotbeincludedinthediscussion.Theauthorisleavingouttherichfromthediscussion,soanswerchoiceAdoesn’twork.Medievalartismentionednotasimportantforthe“bourgeoisandtherich”butasanantidoteforthepoor,soanswerchoiceBdoesn’twork.Thepoliticalviewsintownsareonlybrieflymentionedinthethirdparagraph(answerchoiceD).

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20. Accordingtothepassage,medievalpageantsandfestivalsforthepoorwereappealingbecausetheyprovidedrelieffromahard,bleakexistence. D

Explanation

Looktolines19-23,“Theextravagantpageantsandreligiousfestivals…butalsoasantidotestothemiseriesofeverydaylife.”Thepageantsandfestivalsactedasanescapefromthe“wretchedconditionsunderwhichtheylived.”Thepassagedoesnotmentionthatthepageantsandfestivalswerefree,hadmuchreligiousimportance,orthattheycreatedanexcusetocelebrate.Instead,thefocusofthepassageisontheterriblelivingconditionsofthepoorandhowthesepageantsandfestivalsprovidedrelieffromthoseconditions.

21. Theauthorsuggeststhatwedonotunderstandthe“keenness”(line10)ofcertain

pleasuresenjoyedbymedievalpeoplebecauseweenjoythepleasuresmentionedfairlyfrequently.

D

Explanation

Inlines9-12,theauthorcitesaprominent(importantorfamous)historian,“We,atthepresentday…wereformerlyenjoyed.”Thehistorianisstatingthatpresentdaypeoplecannotappreciatethelevelofjoymedievalpeoplefeltfortheeverydaycomfortswenowenjoy.Plus,thehistorianstatesthat“health,wealth,andgoodfortune”werea“rarity”duringthattime.Whilethisquestionrequiresalittlemorereadingbetweenthelines,youcanalsosafelyeliminateanswerchoicesA,B,andCbecausetheyaren’tevenmentionedinthepassage.

22. Inthesecondparagraph(lines24-34),theauthorstatesthattheconceptof

“family”didnotexistbecausechildrenweresentawayassoonastheywereoldenoughtowork.

D

Explanation

Looktolines28-31,“Therewasroomonlyfortheinfants—theolderchildrenwereseparatedfromtheparentsandsenttoworkasapprenticesorservants.”Becausetheirhomesweretoosmalltohousethem,thechildrenweresentofftoworkassoonaspossible—thesechildrenwereneverabletogrowupwiththeirparentsandsiblingsasa“family.”Thus,theideaof“family”and“home”hadnomeaningforthepoor.Again,answerchoicesA,B,andCarenotmentionedinthepassage.

23. Theauthormostlikelyusessimilartermsfromdifferentlanguages(lines38-42)in

ordertoemphasizethewidespreadnatureofasimilarconcept. B

Explanation

Looktolines36-38,“Thefreetown…wasuniquelyEuropean.”Tofurtherprovethisstatement,theauthorpresentsthenamesfortheinhabitantsofthefreetownsfromvariouspartsofEurope(France,Germany,Italy,andEngland)toshowhowwidespreadtheconceptwas.Thesetermsareallrelatedtoasingleideaanddonotrepresentarangeorbreadthofideas(answerchoiceA).AnswerchoiceCdoesn’tworkbecausetheauthoristryingtoshowhowsimilarinstructureandmeaningthedifferenttermsaretooneanother.Theoriginofmedievallanguagesisn’tevendiscussedinthepassage(answerchoiceD).

24. Thepassagesuggeststhatloyaltytoakingratherthanalordhastheadvantageof

morepotentialforself-government. C

Explanation

Looktolines44-48,“Itdescribed…electedcouncils…allegiancedirectlytothekinginsteadofalord.”Sincethe“bourgeois”inmostcasesalliedthemselvesdirectlytothekingandnotalord,theywereabletogovernthemselveswithelectedcouncils.AnswerchoicesA,B,andDaren’tevenmentionedinthepassage.

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ReadingComprehension-Passage525. TheprimarypurposeofthepassageistoshowhowWrightmethisbusiness

partner. B

Explanation

Honestly,thisquestionisansweredbytheitalicizedtextabovethepassage,“…describesanincidentfromhisyouththatwastoleadtoabusinesspartnershipinlaterlife.”Plus,thepassagetalksaboutWrightmeetingLampandtheirmanyactivitiestogether.AnswerchoiceAdoesn’tworkbecauseonlyLamp’scourageisshown,andwedon’tknowifthebulliesbackeddownornot(answerchoiceD).Plus,answerchoiceCisn’tevendiscussedinthepassage.

26. Themoodofthefirstparagraph(lines1-10)canbestbedescribedasoneof

youthfulenthusiasm. C

Explanation

Thefirstparagraphisfilledwithpositivewordsandphrases—quick,“couldworkalmostashardasaman,”“wasn’tafraidofanything,”buoyantly,exclaimed.Theauthorisillustratinghowmuchheaccomplishedandgrewonhisuncle’sfarm.Heisnotdemonstrating“overbearingpride”becausehecountershis“wasn’tafraidofanything”statementwith“—well,maybealittleafraidofstormsandofpeople”(answerchoiceA).Heisnotshowinghowhis“adolescentshyness”kepthimfromperformingtasksathisuncle’sfarm(answerchoiceB),andhislongingforSeptemberisonlybrieflymentionedatthebeginningoftheparagraph(answerchoiceD).

27. ItcanbeinferredthatWrightandLamprequiredCharlieDoyontogivethem

moneybeforejoiningtheirbusinessbecausetheythoughtthatthebusinesswouldbenefitfromalargermodelpress.

D

Explanation

WrightandLampcreatedthingstogetherinlines26-28,includingthejointeffortofsettingtypeontheirsmallprintingpressinlines28-29.SincebothWrightandDoyonputineffortintotheirprojects,itcanbeinferredthatDoyonwouldneedtodothesame—hemustprovidealargerpress.ThefactthatWrightusedtheword“more”indescribingthelargerpressalsoindicatesWrightandLampthoughtthelargerpresswouldhelpmorethantheirsmallerpress.AnswerchoiceCisaveryenticingchoice,butwedonothaveanyevidencefromthepassagethatsuggestsWrightandLampwerecommittedtotheirbusiness.WealsodonothaveanyevidenceinthepassageforanswerchoicesAandB.

28. Thephrase“myforayintotheunknown”(line15)referstoWright’sentranceinto

anewschool. A

Explanation

Looktolines11-12,thesentencebefore“myforayintotheunknown,”“OnthedayIapproachedtheforbiddingSecondWardSchool,Iwaslesssureofmyself.”Wrightisunsureofhimselfasheentersthisnewschoolbecausehehadn’tmadeanyfriendsduringthesummeronhisuncle’sfarm.HisencounterwithRobieLamp(answerchoiceB)occursinaseparateparagraph,andWright’sexperiencesonhisuncle’sfarmswereknowninthefirstparagraph(answerchoiceC).AnswerchoiceDisnotevenmentionedinthepassage.

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29. Thesentence“IsoadmiredRobie’scourageandresourcefulnessthatwebecamefriendsoftheheart”(lines24-26)isincludedinordertoexplainwhyWrightandLamp’sfriendshipwasalastingone.

C

Explanation

Thephrase“friendsoftheheart”indicatesthatWrightsawakindredspiritinLampandWright’sgreatadmirationforLampcreatedalinkbetweenthemasimportantasthelinkbetweentheheartandthebody.WedonotknowifWrightwantedtotormentLamp(answerchoiceA)orifLampwasolderthanWright(answerchoiceD).Plus,answerchoiceBdoesn’tworkbecauseLampdemonstratesuttercouragewhileWrightis“lesssureofhimself”—whywouldLampbeluckytowinWright’sfriendship?

30. Inline31,“inveigle”mostnearlymeansacquire.

A

Explanation

Theword“inveigle”meanstopersuadesomethingtodosomething,usuallythroughflatteryordeception.Whilewedon’tknowwhatDoyondidtoobtaintwohundreddollarsfromhisfather,wedoknowthathedidobtainitandtheboyswereabletobuythelargerpress.Thus,“acquire”bestmatcheswhathappensinthepassage.“dismiss”and“return”havetheoppositeeffectofobtainingthetwohundreddollars,andDoyonwouldnot“purchase”twohundreddollarsfromhisfather.

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MathematicsAchievement1. 40cm2 A

Tools: area,evaluatingshapesSteps: (1) Sinceeachshadedgridsquare’sareais5cm2,wesimplyneedtocounthow

manyshadedgridsquaresthereareandmultiplythatamountby5(2) 8shadedgridsquares×5cm2=40cm2

QuickTips: • YouareallowedtowriteonyouractualISEEtestbooklet• Drawlinesthroughtheshadedregiontohelpaccuratelycountthenumberof

shadedgridsquares2. P

=[× P=[ DTools: probabilitySteps: (1) Determinetheprobabilityofchoosingaredballfromthejarforthefirst

drawing(IpqrNFstusvGwGxNspwysqNvtsFwzNNxNIwwswJKIpqrNFstusvvGrKNspwysqNv

)

(2) 5redballsoutofatotalof19balls= P=[

(3) Sincetheballisreturned,theprobabilityforchoosingaredballforthesecondeventisalso P

=[

(4) Todeterminetheprobabilityofbotheventsresultinginaredball,wemustmultiplythetwoprobabilities: P

=[× P=[

QuickTips: • Beforestartingyourcalculations,seehowtheanswerchoicesareformatted—sometimestheanswersarepresentedasastepintheprocessandnotthefinalresult

3. 3.241×10\ C

Tools: scientificnotationSteps: (1) Convertthevaluesintostandardnotation

(2) 3.2×10\ = 32,000,000and4.1×10P = 410,000(3) Findthesumofthesevalues:32,000,000 + 410,000 = 32,410,000(4) Convertthissumintoscientificnotation(5) 32,410,000 = 3.241×10\

QuickTips: • Inconvertingvaluesinstandardnotationtoscientificnotation,bringthedecimaltotherightofthelargestplacevalue

• Inaddinganumberwithmoreplacevaluestoanumberwithfewerplacevalues,suchas10\and10P,thenotationwillmostlikelynotchange(10\willremain10\)

4. 0.6666667 A

Tools: fractions,decimalsSteps: (1) Compareeachanswerchoiceto@

V

(2) D)@.QV.R= @

V;2.4and3.6aremultiplesof1.2—ifyoudividebothvaluesby1.2,

youget@V

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(3) C) ==.P= @

V;ifyoumultiply1and1.5by2,youget@

V

(4) B)0. 6 = @V;ifyouconvert@

Vtoafraction,youget0.666666repeating

(5) A)0.6666667 > @V;answerchoiceAis@

Vroundeduptothetenmillionthsplace

value,whichisnotthesamethingas@V

QuickTips: • Comparefractionsbymultiplyingthedenominatorofonetothenumeratoroftheother

5. 0 D

Tools: zeroproductpropertySteps: (1) Noticethattheequationisavaluetimestisequaltot,or10𝑡 = 𝑡

(2) Ifyousubtracttfrombothsides,yournewequationis9𝑡 = 0(3) Thezeroproductpropertystatesthatiftheproductoftwovaluesiszero,one

ofthevaluesiszero—inthiscase,tmustbezeroQuickTips: • Whateveractionyoutaketobalanceanequationmustmakesense

• Ifyoudividebothsidesbyt,youget10 = 1(whichisnottrue)6. Therearenovaluesforxthatwouldmaketheequationtrue. D

Tools: divisionproperties,commutativepropertySteps: (1) Whenaddingvalues,thecommutativepropertystatesthatyoucanchangethe

orderwithoutchangingtheresult(2) Ifyouswitchtheorderofthevaluesinthedenominator,you’llnoticethatitis

thesameastheexpressioninthenumerator:{XVVX{

→ {XV{XV

(3) Wheneveranumberisdividedbyitself,thequotientis1(exceptfor0);thus,

theresultfor{XV{XV

= 1andnot0(4) Novaluesforxwillmaketheoriginalequationtrue

QuickTips: • Keepallmathematicalpropertiesinmindwhendealingwiththeseproblems• Ifyouinputvalues,you’llnoticethatDistheonlypossibleansweraswell

7. 13 B

Tools: orderofoperationsSteps: (1) Youmustfirstaddthevaluestogetherundertheradicalin 25 + 144before

takingthesquareroot(2) 25 + 144 = 169 → 169 = 13(3) Youcannottakethesquarerootofeachvaluefirstandthenaddtheresulting

values: 25 + 144 ≠ 25 + 144 → 5 + 12 → 17QuickTips: • Treatthevaluesundertheradicalasiftheyaresurroundedbyparentheses:

(25 + 144)—youmustcompleteoperationsintheparenthesesbeforetakingthesquareroot

• However,inmultiplicationofvaluesunderaradical,youcantakethesquarerootfirst: (25)(144) = 3600 = 60 → 25 144 = 5 12 = 60

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8. 185 CTools: median,analyzingchartsandgraphsSteps: (1) Arrangethedogs’scoresinorderfromleasttogreatest

(2) 155,160,175,175,180,190,195,195,195,200(3) Themedianisthemiddlevalueofarangeofvalues,butwhentwovaluesshare

themiddle(180and190)youmustfindtheaverageofthosevalues(4) =]>X=[>

@= V\>

@= 185

QuickTips: • Ifarangeofvalueshasanoddnumberofvalues,aspecificvaluefromthatrangewillbethemedian

• Ifarangeofvalueshasanevennumberofvalues,thentwovalueswillsharethemiddleandtheaverageofthesetwovaluesistheactualmedian

9. 2 A

Tools: systemoflinearequations(orproportion)Steps: (1) Createanequationthatrepresentstherelationbetweenthenumberof

defectivepartseachmachinemakes:𝐴 = 2𝐵(2) Createanotherequationtherepresentsthetotalnumberofdefectiveparts

madeyesterday:𝐴 + 𝐵 = 6(3) SincewearelookingforthenumberofdefectivepartsmadebyMachineB,

substituteAwith2BsothatthesecondequationonlyhastheBvariable(4) 𝐴 + 𝐵 = 6 → 2𝐵 + 𝐵 = 6 → 3𝐵 = 6 → 𝐵 = 2

QuickTips: • Youcouldalsosetupaproportionforthefirstequation:2to1or2:1(whereMachineAmakes2defectsforevery1defectmadebyMachineB)

• Fromthisproportion,youcanevaluatehowmanydefectivepartscreatedbyeachmachinewouldaddupto6andstillkeeptheproportiontrue

• 2: 1 = 3parts;thus, 2 2 : 1 2 = 3 2 → 4: 2 = 6parts10. 94 B

Tools: meanSteps: (1) Wedonotknowtheexactfinalexamscore,butwecansetupameanequation

withthefinalexamscoreas2𝑥(thefinalexamismultipliedby2becauseitiscountedtwiceinhermean)setequaltothedesiredmeanof93

(2) [VX][X[PX@{P

= 93 → @{X@\\P

= 93 → 2𝑥 + 277 = 93 5 → 2𝑥 + 277 = 465(3) 2𝑥 + 277 = 465 → 2𝑥 = 188 → 𝑥 = 94

QuickTips: • AlthoughLisaisonlytaking4tests,wedividethesumofherscoresby5becausethefinalexamisbeingcountedtwicetowardshermean

11. 1 A

Tools: dataanalysis,modeSteps: (1) Modeisthevaluethatappearsthemostfrequentlyinarangeofvalues,butwe

mustmakesurewearechoosingthemodefromthecorrectrangeofvalues(theoutputs)

(2) Whiletheleftcolumnofthedatatabledoeshavevalues,thosevaluessimplyrepresentthenumberoftimestherightcolumn’svaluesoccur—theleftcolumnisourinputswhiletherightcolumnisouroutputs

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(3) Ifyouwriteoutallofthevaluesfromtherightcolumnaccordingtotheirfrequencyfromtheleftcolumn,thenyouhavethedataset{0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,4,4}

(4) Thevalue1(1pet)occursthemostinthissetofdataandisourmodeQuickTips: • AnswerchoicesCandDcanbequicklyeliminatedbecausenostudenthas

morethan4pets• Ifyouweretoconvertthisdatatableintoagraph,thex-axiswouldbethe

NumberofPets(ourinputs)whilethey-axiswouldbetheNumberofStudentsOwningThatNumberofPets(ouroutputs)

12. 24𝑛@𝑚 D

Tools: primenumbers,multiplesSteps: (1) Weneedavaluethateachofthealgebraicexpressions8𝑛, 6𝑛𝑚, and4𝑛@can

divideintowithoutanyremainders(2) While6𝑛𝑚candivideintoanswerchoiceA,theothertwoexpressionscannot(3) While6𝑛𝑚candivideintoanswerchoiceB,theothertwoexpressionscannot(4) While6𝑛𝑚and8𝑛candivideintoanswerchoiceC,4𝑛@cannot(5) AllthreeexpressionscandivideintoanswerchoiceD

QuickTips: • AnswerchoicesAandBcanbequicklyeliminatedbecause6isnotamultipleof4or8

• AnswerchoiceCcanbequicklyeliminatedbecause𝑛isnotamultipleof𝑛@—instead,𝑛isafactorof𝑛@

13. 3 D

Tools: balancingequationsSteps: (1) Lookattheplacementofthe3sontheleftsideandtheysontherightsideof

theequation—theysharethesameplacesineachterm(2) 3𝑥 − 3 = 𝑥𝑦 − 𝑦 → 3𝑥 − 3 = 𝑦𝑥 − 𝑦;thus,𝑦 = 3becausetheequationis

balanced(3) Youcanalsoinputsomepossiblevaluesforx,suchas0(4) 3𝑥 − 3 = 𝑥𝑦 − 𝑦 → 3 0 − 3 = 0 𝑦 − 𝑦 → −3 = −𝑦 → 3 = 𝑦

QuickTips: • Youcanalsoinputtheanswerchoicesinforytoseeiftheequationremainsbalanced,butbemindfulofsubtractionsignscoupledwithnegativeinputs

14. 5𝑥@𝑦Q − 𝑥Q𝑦@ C

Tools: operationsonalgebraicexpressionsSteps: (1) Addandsubtractliketermsintheexpression—“liketerms”referstoterms

withthesamevariablessettothesameexponents,suchas3𝑥Q𝑦@ − 4𝑥Q𝑦@(2) 2𝑥@𝑦Q + 3𝑥Q𝑦@ − 4𝑥Q𝑦@ − 3𝑥@𝑦Q → 2𝑥@𝑦Q + 3𝑥Q𝑦@ − 4𝑥Q𝑦@ + 3𝑥@𝑦Q(3) 2𝑥@𝑦Q + 3𝑥@𝑦Q + 3𝑥Q𝑦@ − 4𝑥Q𝑦@ → 5𝑥@𝑦Q − 𝑥Q𝑦@

QuickTips: • Besuretodistributethenegativesigntothetermsinsideoftheparentheses• 𝑥@𝑦Qand𝑥Q𝑦@arenotliketermsbecauseeachxandyinthetermsareraised

todifferentexponents

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15. 𝑥 = −5and𝑥 = 5 CTools: factoringquadraticequations,definitionofzero,zeroproductpropertySteps: (1) Noticethattherationalequationissetequaltozero—since0dividedbyany

othernumber(except0)isequaltozero,theresultoftheexpressioninthenumeratormustbe0

(2) Setthenumeratorequaltozeroandsolveforx:𝑥@ − 25 = 0 → 𝑥@ = 25 →𝑥 = 5

(3) However,thesquareof5and–5bothresultin+25,so𝑥 = ±5(4) Ifthedenominatorwereequalto0,theequationwouldbeundefined—we

mustmakesurethat5and–5won’tmakethedenominatorequalto0(5) Seteachterminthedenominatorequaltozeroandfollowthezeroproduct

property(iftheproductoftwotermsisequaltozero,thenoneofthetermsiszero)

(6) 𝑥 + 2 = 0 → 𝑥 = −2(7) 𝑥 − 3 = 0 → 𝑥 = 3(8) Thus,𝑥 ≠ −2or3(our5and–5remaintrue)

QuickTips: • Itmightbefastertosimplyinputtheanswerchoicesintotheequationandseewhathappens

• Ifyouseeacomplicatedrationalequation,thequestionislikelytestingasimplepropertythatsolvesthequestionquickly

16. 𝑥@ + 𝑥 − 6 D

Tools: multiplyingpolynomials(FOIL/distributiveproperty)Steps: (1) FollowFOILtosolvetheexpression(𝑥 − 2)(𝑥 + 3)

(2) First: 𝑥 𝑥 = 𝑥@(3) Outside: 3 𝑥 = 3𝑥(4) Inside: −2 𝑥 = −2𝑥(5) Last: −2 3 = −6(6) Addalltermstogether:𝑥@ + 3𝑥 − 2𝑥 − 6 → 𝑥@ + 𝑥 − 6

QuickTips: • Youcanalsousethedistributivepropertyifyouaremorecomfortablewithit• 𝑥 − 2 𝑥 + 3 → 𝑥 𝑥 + 3 − 2 𝑥 + 3 → 𝑥@ + 3𝑥 − 2𝑥 − 6 → 𝑥@ + 𝑥 − 6

17. –2 B

Tools: graphsoflinearequations(slope)Steps: (1) Graphsoflinearequationswithadownwardslope,liketheonepresentedin

thequestion,representanegativeslope(2) SlopeisthesamethingasFGvN

FpIor yzJIHNGI�

yzJIHNGI{or �WL��

{WL{�

(3) Selecttwopointsonthegraph,suchasthey-intercept(0,5)and(2,1)(4) =LP

@L>= LQ

@= −2;thus,theslopeis–2forthisline

QuickTips: • Ifyoustartwiththey-intercept,youcanseethatthelineslopesdown2(–2)andtotheright1(+1)(aslopeofL@

== −2)

18. 5gridunits C

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Tools: distanceformula(graphs),circleSteps: (1) Youaregiventwopoints—oneinthecenterofthecircleandoneontheedge

ofthecircle(2) Thedistancebetweenthesetwopointsisthesamethingastheradiusofthe

circle(thelengthfromthecentertotheedgeofthecircle)(3) Usethedistanceformulatosolve:𝑑 = 𝑥= − 𝑥@ @ + 𝑦= − 𝑦@ @(4) 𝑑 = 1 − (−2) @ + 8 − 4 @ → 1 + 2 @ + 4 @ → 3 @ + 4 @(5) 3 @ + 4 @ → 9 + 16 → 25 = 5gridunits

QuickTips: • Ifyouforgetthedistanceformula,youcandrawoutthegraphofthiscircleandapproximatethedistancebetweenthetwopoints

• Youcanevenusethepreviousquestion’sgraphtodrawyourcircle19. arandomsampleofallthestudentsintheschool B

Tools: dataanalysisSteps: (1) Considerthebias(one-sidedness)ofeachanswerchoiceandtheprobabilityof

hoursspentwatchingTVbythegroups(2) ForanswerchoiceA,Terri’sfriendswilllikelywatchthesameshowsasTerri

andforthesamenumberofhours,sothisgroupwillnotaccuratelyreflectthestudentbody

(3) ForanswerchoiceB,arandomselectionassurestheleastamountofbiasandanunknownprobabilityofhoursspentwatchingTV,sothisgroupwillbestreflectthestudentbody

(4) ForanswerchoiceC,ifthewholegrouphasshownuptowatchafootballgame,thentheyshareabiasandschedulethatonlyreflectstheirgroup,sothisgroupdoesnotaccuratelyreflectthestudentbody

(5) ForanswerchoiceD,ifthewholegroupisinthelibrarybeforeschool,thentheyshareabiasandschedulethatonlyreflectstheirgroup,sothisgroupdoesnotaccuratelyreflectthestudentbody

QuickTips: • Arandomsampleofthepopulation(freefrombiasandcategorization)providesthebestrepresentationofthatpopulation

• Specificgroupswillsharebiasesthatwarpthecollecteddatatowardstheirinterestsandwillnotreflectthevarietyanddiversityofapopulation

20. 45° A

Tools: anglesinaquadrilateralSteps: (1) Aquadrilateralisapolygonwith4straightsides—squaresandrectanglesare

quadrilaterals(2) Ifthesumoftheinterioranglesofasquareorrectangleequal360°,thenso

mustotherquadrilaterals(3) Findthesumoftheinterioranglesofthisquadrilateralandsetitequalto360°(4) 110° + 75° + 130°+? ° = 360° → 315°+? ° = 360° →? ° = 45°

QuickTips: • Thesumofanyquadrilateral’sinterioranglesisalways360°nomatterhowitlooks

21. 20 B

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Tools: combinationsSteps: (1) WhileyoumightbetemptedtousetheFundamentalCountingPrinciplefor

thisproblem,weareonlydealingwithonesetofdata—thedebateteammembers

(2) Wemustusethecombinationformulatosolvethisproblem,sincetheorderoftheselectiondoesn’tmatter:nCr=

I!F! ILF !

(3) nisthenumberofitemsinaset(6debateteammembers)andristhenumber

ofselectionsbeingmadefromthatset(3)—the!symbolistheproductofthenandallpositiveintegersbelowit(indescendingorder)

(4) I!F! ILF !

→ R!V! RLV !

→ R×P×Q×V×@×=(V×@×=)(V×@×=)

→ R×P×Q×V×@×=(V×@×=)(V×@×=)

→ R×P×QV×@×=

→ =@>R= 20

QuickTips: • Combinationshasasecondformulathatyoucanuseiffactorialsseemtooconfusing:I IL= IL@ …

F FL= FL@ …wherethenumberofvaluesmultipliedinthe

numeratormatchersthenumberofvaluesinthedenominator• Ifyouhave3choicesorselectionsinacombinationsquestion,thenyouwill

have3values(ex.6×5×4)inthenumeratorand3values(ex.3×2×1)inthedenominator

22. (7,2) D

Tools: parallelogram,xy-coordinateplaneSteps: (1) Aparallelogramisapolygonwithparallelsides,suchasarhombus

(2) Thefastestwaytodeterminewhichofthefollowinganswerchoicesinthefourthvertex,orpoint,oftheparallelogramisbyplottingthemonthegraph

(3) Ifyoulookatpoints(–2,3)and(1,2),you’llnoticethat(1,2)is4totherightandtwodownfrom(–2,3);thus,thefourthvertexshouldmatchthispattern

(4) (7,2)is4totherightand2downfrompoint(3,4)QuickTips: • Afteryouplotthepoints,youcandrawfaintlinesbetweenthepointsto

visuallyevaluatewhichoftheanswerchoicesisthefourthvertex23. 𝑥 ≤ − =

@or𝑥 ≥ 4 D

Tools: absolutevalue,algebraicinequalitiesSteps: (1) Tosolveforabsolutevaluewithalgebraicinequalities,yousolvetheequation

twice—1)withtheinequalityasisbutwithoutthebarsand2)againwithoutthebarsbytherightsideoftheinequalitymultipliedby–1

(2) 4𝑥 − 7 ≥ 9 → 4𝑥 − 7 ≥ 9 → 4𝑥 ≥ 16 → 𝑥 ≥ 4(3) 4𝑥 − 7 ≥ 9 → 4𝑥 − 7 ≤ −9 → 4𝑥 ≤ −2 → 𝑥 ≤ − @

Q→ 𝑥 ≤ − =

@

(4) Youcanchecktheseanswersbyinputtingthembackintotheinequalitywiththeabsolutevaluebars

QuickTips: • Whenyoumultiplyordivideasideoftheinequalitybyanegativenumber,theinequalitysignmustswitchdirections

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24. complexnumber CTools: numbertypesSteps: (1) Thisquestionrequiresthatyouknowthedefinitionsofintegers,rational

numbers,complexnumbers,andirrationalnumbers(2) Irrationalnumbersarevaluesthatcannotberepresentedasfractions(ex.𝜋)(3) Anintegerisavaluethatcanbeapositiveornegativenumberorzero,butit

cannotbeafractionordecimal(4) Arationalnumberisavaluethatcanberepresentedasafraction(5) Acomplexnumbercontainsarealnumberandanimaginarynumber—an

imaginarynumberisavaluethatdoesnotexist,suchas −1(6) Thedifferencebetweentwoirrationalnumberscouldresultinaninteger,

rationalnumber,orirrationalnumberbecauseallthreeofthesenumbertypesarerealnumbers

(7) Becauseirrationalnumbersarerealnumbersandhavenoimaginarypart,thedifferencebetweentwoirrationalnumberscannotresultinacomplexnumber—𝑎 + 𝑏𝑖isacomplexnumberwhereaisarealnumberwhilebiisanimaginarynumber(bisarealnumber,but𝑖 = −1)

QuickTips: • Whileyoumaynotknowthedefinitionofacomplexnumber,youshouldknowthedefinitionsoftheotherthreenumbertypesandcanreasonouttheresultofsubtractingtwoirrationalnumbers

25. 3.25 B

Tools: mean,analyzingchartsandgraphsSteps: (1) Identifythenumberofbooksreadfromthedata

(2) 5studentseachread1book(5books),4studentseachread2books(8books),2studentseachread3books(6books),1studentread4books(4books),6studentseachread5books(30books),and2studentseachread6books(12books)

(3) Findthesumofthebooksreadanddividethatsumbythetotalnumberofstudents

(4) 5 + 8 + 6 + 4 + 30 + 12 = 65books(5) 5 + 4 + 2 + 1 + 6 + 2 = 20students(6) RP

@>→ =V

Q→ 3 =

Q→ 3.25booksperstudent

QuickTips: • AnswerchoicesAandDcanbequicklyeliminatedbecausedividing65by20willnotresultinaninteger

• Sincemultiplicationcanbeeasierforsomestudentsthandividing,youcanmultiplyanswerchoicesBandCby20toseeiftheyequal65(totalbooks)

26. 𝑥 + 3 < 4 D

Tools: absolutevalue,inequalities,numberlinesSteps: (1) Theopencirclesindicatethatthepossibleinputsforxarelessthan1but

greaterthan–7,suchas0or–6,buttheinputscannotbe1or–7(2) Input0foreachanswerchoice(3) A) 0 − 4 < 3 → −4 < 3 → 4 < 3(nottrue,soeliminateit)(4) B) 0 + 4 < 3 → 4 < 3 → 4 < 3(nottrue,soeliminateit)(5) C) 0 − 3 < 4 → −3 < 4 → 3 < 4(true,sowemusttestitagain)(6) D) 0 + 3 < 4 → 3 < 4 → 3 < 4(true,sowemusttestitagain)

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(7) Input–6foreachremaininganswerchoice(8) C) −6 − 3 < 4 → −9 < 4 → 9 < 4(nottrue,soeliminateit)(9) D) −6 + 3 < 4 → −3 < 4 → 3 < 4(true,thisisouranswer)

QuickTips: • Usesimpleintegersfromarangeofinputswhenpossible• Youcanalsosolveeachanswerchoice,butthismaytakesomestudentslonger

thansimplytestinginputs• 𝑥 + 3 < 4 → 𝑥 + 3 < 4 → 𝑥 < 1• 𝑥 + 3 < 4 → 𝑥 + 3 > −4 → 𝑥 > −7;thus,Dis−7 < 𝑥 < 1

27. V

@ B

Tools: probabilitySteps: (1) Eachtossofthecoinisanindependentevent,whichmeansthattheoutcome

ofonetossdoesnotaffecttheoutcomeoftheothertosses(2) Theprobabilityofcointosslandingheadsupis=

@;thus,theprobabilityof

tossing3headsinarowis=@× =@× =@= =

](thesameistruefornoheadstossed)

(3) Therearethreepossibilitiesforcointosslandingheadsup2times—thefirsttwotossesorthefirstandthirdtossorthesecondandthirdtoss(the“or”isveryimportant)

(4) The“or”signifiesthatwewilladdeachpossibilities’probabilitytogethertofindtheactualprobabilityofacoinlandingheadsuptwice:=

]+ =

]+ =

]= V

](the

sameistrueforthecoinlandingheadsuponce)(5) Hereisavisualrepresentationofallthepossibleoutcomes:HHH,HHT,HTH,

THH,HTT,THT,TTH,TTT(8possibleoutcomes)(6) Thereareatotalof12Hsoutofthe8possibleoutcomes,whichcanbe

representedas=@]→ V

@

(7) Thequestioncanalsobereadtosay“Whatistheprobabilityofthefirstcointosslandingheadsup,thesecondtosslandingheadsup,ORthethirdtosslandingheadsup?”Inthiscase,youfindthesumoftheprobabilityofeachevent:=

@+ =

@+ =

@= V

@

QuickTips: • Youcanalsomultiplythenumberofheadstotheirrespectiveprobabilities• 3 =

]+ 2 V

]+ 1 V

]+ 0 =

]= V

]+ R

]+ V

]+ 0 = =@

]→ V

@

• Paycloseattentiontohowprobabilityeventsinfluenceoneanother(ifatall)28. Q>>×R>

>.V>P×P,@]> A

Tools: convertingmeasurementsSteps: (1) Sincethequestionisaskingforthehorse’sspeedinmilesperhour,weneedto

writeoutratiosforeachmeasurementprovided(2) >.V>PqNwNFv

=tssw,P,@]>tNNw

=qGKN,R>qGIpwNv

=zspF,Q>>qNwNFv=qGIpwN

(theratioscanbeflippedtoaccuratelyconvertmeasurements)

(3) OurfinalratioshouldbeqGKNvzspF

—startwiththehorse’srate

(4) Convertmeterstofeet:Q>>qNwNFv=qGIpwN

→ Q>>qNwNFv=qGIpwN

× =tssw>.V>PqNwNFv

(weflippedtheratiotoeliminatemetersfromtheconversion

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(5) Convertfeettomiles: Q>>=qGIpwN

× =tssw>.V>P

× =qGKNP,@]>tNNw

(weflippedtheratiotoeliminatefoot/feetfromtheconversion)

(6) Convertminutestohours: Q>>=qGIpwN

× R>qGIpwNv=zspF

× =>.V>P

× =qGKNP,@]>

(eliminateminutes)

(7) AllthatremainsisqGKNvzspF

:Q>>=× R>=zspF

× =>.V>P

× =qGKNP,@]>

→ Q>>×R>>.V>P×P,@]>

QuickTips: • Lookattheanswerchoicestoseehowfaryouneedtogoinyour

calculations—thisproblemjustwantstoseeifyouknowhowtosetupconversions

29. 12feet B

Tools: proportionSteps: (1) Thequestionisstatingtheheightofanobjectisproportionaltothelengthof

itsshadow—theratioofthepole’sheighttoitsshadowisVP(thenumeratoris

theheightandthedenominatoristheshadowlength)(2) Wecancreatethesameratioforthetree’sheightandshadowlength: {

@>(the

numeratoristheheightandthedenominatoristheshadowlength)(3) Setthetworatiosequaltooneanothertocreatetheproportionandsolvefor

x:VP= {

@>

(4) Crossmultiply:VP= {

@>→ 5𝑥 = 60 → 𝑥 = 12feet

QuickTips: • Thetypeofvalueinthenumeratorofoneratiomustmatchthetypeofvalueinthenumeratoroftheotherratioinaproportion(thesameistrueforthedenominators)—inthiscase,bothnumeratorsaretheheightoftheobjectswhilebothdenominatorsaretheshadowlengths

30. centimeters A

Tools: measurementsSteps: (1) AnswerchoicesBandCaremeasurementsforweightandwillnotmeasure

thelengthoftheleaf(2) AnswerchoicesAandCaremeasurementsforlength,butameterisagreater

degreeofmeasurementthanacentimeter(100centimetersfor1meter)(3) Sincethequestionisaskingforareasonablemeasurementforaleaf,imagine

ordrawastandardsizedleaf—whiletherearesomeleavesthatcouldbemeasuredusingmeters,theseleaveswouldlikelybenogreaterthanameter

(4) Thus,themostreasonablemeasurementformeasuringthelengthofaleafiscentimeters

QuickTips: • TheISEErequiresthatyouunderstandmetricmeasurements(conversionsandapplicationintherealworld)

31. 25 − 4 C

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Tools: radicals,numbertypesSteps: (1) Anintegerisapositiveornegativenumber(includingzero)thatcannot

includeafractionordecimal(2) AnswerchoiceA: 4 − 25 → 2 − 5 = −3(integer)(3) AnswerchoiceB: 4× 25 → 2×5 = 10(integer)(4) AnswerchoiceC: 25 − 4 → 21 = 4.582…(notaninteger)(5) AnswerchoiceD: 4×25 → 100 = 10(integer)

QuickTips: • AnswerchoicesBandDcanimmediatelybeeliminatedbecausetheyaretheexactsame,justpresentedinaslightlydifferentway: 4× 25 = 4×25

32. (100 − 25𝜋)cm2 C

Tools: circle(area),square(area)Steps: (1) Findtheareaofthesquare(𝑠@ = 𝐴v)

(2) 10@ = 100(3) Findtheareaofthecircle(𝜋𝑟@ = 𝐴y)(4) Theradiusishalfthediameterofthecircle,andthediameteristhesame

lengthasthewidthofthesquare(10);thus,𝑟 = c@→ 𝑟 = =>

@→ 𝑟 = 5

(5) 𝜋𝑟@ → 𝜋 5 @ → 25𝜋(6) Theareaoftheshadedregionistheareaofthecirclesubtractedfromthearea

ofthesquare:100 − 25𝜋QuickTips: • Looktotheanswerchoicestoseehowfaryouneedtogowithyour

calculations—thisquestiondoesnotneedyoutofinishsubtractingtheareaofthesquarefromtheareaofthecircle

33. 108𝜋 B

Tools: volume(cylinder)Steps: (1) Findtheheightofthecylinder,whichistwotimesthecylinder’sdiameter

(2) ℎ = 2𝑑 → ℎ = 2 6 → ℎ = 12(3) Usethegivenformulaforthevolumeofacylindertosolve(risthecylinder’s

radius,orhalfofthecylinder’sdiameter(3inches))(4) 𝑉 = 𝑟@ℎ𝜋 → 3 @ 12 𝜋 → 9 12 𝜋 → 108𝜋

QuickTips: • Looktotheanswerchoicestoseehowfaryouneedtogowithyourcalculations—wedonotneedtomultiplythevaluestotheactualvalueofpi

34. 55 B

Tools: box-and-whiskerplot,numberlines,rangeSteps: (1) Whileyoumayormaynotknowwhatabox-and-whiskerplotis,youshould

knowrange(thedifferencebetweenthesmallestandlargestvalueinasetofdata)andnumberlines

(2) Thesetofthedatafallsbetween25degreesand80degrees,statingthatatsomepointover50years,thesamedayinthemonthofMaywas25degreesatleastonceand80degreesatleastonce

(3) Thus,25degreesisoursmallestvaluewhile80degreesisourlargestvaluefromthesetofdata

(4) 80 − 25 = 55

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QuickTips: • Abox-and-whiskerplotrepresentsthemedian,lowerandupperquartiles,andtheextremesofasetofdataonanumberlinetodemonstratethedistributionofthesevalues

• Thefarleftandrightpointsofthethinlineindicatetheextremevalues(lowestandhighestamount)—whatweneededforthisproblem

• Theboxrepresentsthequartiles,wherethefarleftlineindicatesthefirstquartile,the“middle”lineindicatesthemedian,andthefarrightlineindicatesthethirdquartile

• Quartilesarethreepointsthatdividethesetofdataintofourgroups,witheachgroupcomprisingaquarterofthedata(thedataisalsoranked)

35. Q

=Q A

Tools: probabilitySteps: (1) BecauseKateremovedayellowmarblefromthebag,theprobabilityof

Joanne’sselectionisinfluenced;however,weareonlylookingfortheprobabilityofJoanne’sselectionforthisproblem( #stusvGwxNspwysqNv

wswJKusvvGrKNspwysqNv)

(2) Thebagcontains15marbles—4green,5blue,2yellow,and4orange(3) SinceKatehasremovedayellowmarble,thebagnowcontains14marbles(4) Thebagnowcontains4greenmarblesoutof14marbles,andtheprobability

ofJoannedrawingagreenmarbleis Q=Q

QuickTips: • Looktotheanswerchoicestoseehowfaryouneedtogowithyourcalculations—wedonotneedtoreduce Q

=Q→ @

\

• AnswerchoicesCandDrepresenttheprobabilityofKateANDJoanne’sselection,whileanswerchoiceBdoesnotreflectKatekeepinghermarble

36. 4𝑥] B

Tools: radicals,operationsonalgebraicexpressions,commutativepropertySteps: (1) 16and𝑥=Rarecommutative;thus,youcanplaceeachvalueunderitsown

radical: 16𝑥=R → ( 16 𝑥=R → ( 16)( 𝑥=R)(2) 16 𝑥=R → (4)( 𝑥=R)(3) Whenyoumultiplyvalueswiththesamebase,suchas𝑥Pand𝑥](xisthe

base),thenyousimplyaddtheexponentstogether((𝑥P) 𝑥] → 𝑥=V)(4) Inthiscase,youneedtwoofthesamevariablewhoseproductis𝑥=R—inother

words,twoofthesamevariablewiththesameexponentsthataddupto16(5) Ifyoudivide16inhalf,youfindthatyouneedxraisedtothepowerof8(6) 𝑥=R → 𝑥](ifyoumultiply𝑥]and𝑥],youget𝑥=R)(7) 4 𝑥] → 4𝑥]

QuickTips: • Thesquarerootofavaluecanalsobewrittenasthevaluetakentothe½power: 𝑥 = 𝑥

�W(numeratoristhepowerwhilethedenominatoristheroot)

• Youcanconvert 𝑥=Rthesameway: 𝑥=R → 𝑥��W → 𝑥]

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37. @��� @>°

ATools: trigonometrySteps: (1) Sine(sine),cosine(cos),andtangent(tan)aretrigonometricfunctionsof

anglesinthecontextofrighttriangles.Thesefunctionsrepresentaparticularratiooftwosidelengthsofthetriangleinrelationtoaknownangle(notthe90°angle)

(2) Sinrepresentstheratioofthesidelengthoppositeoftheknownangleandtherighttriangle’shypotenuse( suusvGwN

z�uswNIpvN)

(3) Tanrepresentstheratioofthesidelengthoppositeoftheknownangleandthesidelengthadjacent(nextto)theknownangle(suusvGwN

Jc�JyNIw)

(4) Inthisrighttriangle,wearegiventheangle20°,thesidelengthof2cm(whichisoppositetheknownangle),andthefactwearelookingforthelengthofthetriangle’shypotenuse(x)

(5) Sincetheinformationincludestheoppositelengthandthehypotenuse,weneedtousesine’sratioof suusvGwN

z�uswNIpvN

(6) sin 20° = @{→ 𝑥 sin 20° = 𝑥 @

{→ 𝑥 sin 20° = 2 → ({)(��� @>°)

��� @>°= @

��� @>°

(7) 𝑥 = @��� @>°

QuickTips: • Youcanremembertheratiosofsine,cosine,andtangentusingthemnemonic

deviceSOH-CAH-TOA• sin 𝜃 = suusvGwN

z�uswNIpvN;cos 𝜃 = Jc�JyNIw

z�uswNIpvN;andtan 𝜃 = suusvGwN

Jc�JyNIw

38. AnswerchoiceA’snumberlinegraph A

Tools: compoundinequalities,numberlinesSteps: (1) Simplify41 ≤ 2𝑥 − 1 ≤ 51sothatxisbyitself

(2) Startwiththeleftpartofthecompoundinequality:41 ≤ 2𝑥 − 1(3) 41 ≤ 2𝑥 − 1 → 42 ≤ 2𝑥 → 21 ≤ 𝑥(4) Solveagainfortherightpartofthecompoundinequality:2𝑥 − 1 ≤ 51(5) 2𝑥 − 1 ≤ 51 → 2𝑥 ≤ 52 → 𝑥 ≤ 26(6) Rewrite21 ≤ 𝑥and𝑥 ≤ 26asacompoundinequality:21 ≤ 𝑥 ≤ 26(7) OnlyanswerchoiceA’snumberlinegraphrepresentsthesmallestpossible

valueforxas21andthelargestpossiblevalueforxas26QuickTips: • AnswerchoiceBrepresentsanestimationof41 ≤ 2𝑥 − 1 ≤ 51,where41and

51aredividedby2.Theproblemisthat51dividedbytwofallsbelow30,sothisestimationistoolargeofarangeofvalues

• AnswerchoiceCrepresents41 ≤ 𝑥 ≤ 51,butnot41 ≤ 2𝑥 − 1 ≤ 51• AnswerchoiceDrepresentsanestimationof41 ≤ 2𝑥 − 1 ≤ 51,where41is

dividedby2and51isdoubled.Theproblemisthat51shouldbedividedby2justas41is,sothisestimationdoesn’tfollowproperoperationprocedures

39. 78 C

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Tools: median,stem-and-leafplotSteps: (1) Ifyoudonotknowhowastem-and-leafplotworks,thenlooktotheanswer

choices—eachvalueisdoubledigit(2) Youcanthenreasonthattheleafistheone’splacevalueofasinglevalueand

thestemisthehigherplacevaluesofthatsamevalue(3) Ifthestemis5andtheleafis7,thenthevalueis57.Ifthestemis10andthe

leafis0,thenthevalueis100(4) Writtenout,thedatapointsare55,57,58,62,62,62,64,66,67,68,74,76,76,

76,77,78,78,79,79,83,83,83,84,84,85,91,92,92,93,97,98,99,and100(5) Inall,thereare33datapoints,whichmeansthatthemedianwillbeoneof

thesepoints—ifthedatasetcontainsanoddnumberofvalues,thenoneofthevalueswillbethemedian;ifthedatasetcontainsanevennumber,thentheaverageofthetwomiddlenumbersisthemedian

(6) Whileyoucanspendyourtimecrossingoutnumbersuntilyoureachthemiddlenumber,youcanalsorecognizethat70,75,and80arenotdatapointsthatexistinthisdataset

(7) Thus,onlyanswerchoiceCcanbecorrectsinceitistheonlydatapointthatexistsintheset

QuickTips: • Ifaproblemseemslikeitwilltakealongtimetofigureout,thereislikelyaneasierwaytosolvetheproblem

• Inthiscase,weknowthatthemedianwillbeoneoftheexistingdatapointsinasetofdatathatcontainsanoddnumberofdatapointsorvalues

40. ±7𝑖 D

Tools: imaginarynumbers,commutativeproperty,associativepropertySteps: (1) Ataquickglance,youcanrecognizethattheonlywayfor49tobecome0isby

subtracting49(oradding–49)toit;however,youalsoknowthatnosquaredrealnumbercanresultinanegativenumber

(2) Somehow,weneedthesquareof7or–7tomake–49forthisequationtobetrue—entertheimaginarynumber −1or𝑖

(3) Setxbyitselfintheequation:𝑥@ + 49 = 0 → 𝑥@ = −49 → 𝑥 = −49(4) −49isalsoimaginary,butwecanwriteitinadifferentwayusingthe

commutativeandassociativeproperties(5) −49 → −1 49 → −1 49 → −1 7 → 7 −1(6) Sincethesquareofboth7and–7make49,wemustincludebothnumbersas

squarerootsof49:±7 −1(7) Ifweuse𝑖for −1,then±7 −1is ± 7𝑖

QuickTips: • Ifyousquare −1,thenyouget–1(justlikesquaring 8: 8@= 8)

• −1@→ −1 −1 → −1

• Convertingtheradicaltoapowerdemonstrateswhytheradicalsarecanceled

out: −1@→ −1

�W@→ −1

WW → −1

�� → (−1)

41. 7 6

2 5 A

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Tools: matrices(addition)Steps: (1) Whenaddingmatrices,theseboxeswithcolumnsandrowsofvalues,you

simplyaddthevalueslocatedinthesamecolumnandrowinonematrixastheothermatrix

(2) Inthiscase,2inthefirstmatrixand5inthesecondmatrixarelocatedinthefirstrowandfirstcolumn;thus,wefindthesumofthesetwovaluesandplaceitinthesamespotintheresultingmatrix(andsoonfortheothervalues)

(3) 2 30 4 + 5 3

2 1 = 2 + 5 3 + 30 + 2 4 + 1 = 7 6

2 5 QuickTips: • AnswerchoicesCandDdonothave6inthefirstrowandsecondcolumnspot,

sowecaneliminatetheseanswers• AnswerchoiceBdoesnothave2inthesecondrowandfirstcolumnspot,so

wecaneliminateitasananswer• Inadditionofmatrices,thetwoormorematricesmusthavethesamenumber

ofrowsandcolumns—youcannotaddtogetheramatrixwith2rowsand2columnstoamatrixwith3rowsand3columns

42. 2cm A

Tools: surfacearea(sphere)Steps: (1) Wearegiventheformulaforthesurfaceareaofasphere(𝑆𝐴 = 4𝜋𝑟@)andthe

actualsurfaceareaofaparticularsphere(16𝜋cm2)(2) Inputtheknownsurfaceareaintotheformulaandsolvefortheradius(3) 16𝜋 = 4𝜋𝑟@ → 16 = 4𝑟@ → 4 = 𝑟@ → 𝑟 = 2

QuickTips: • Sincebothsidesoftheequationcontain𝜋,wecancancelitoutandtheequationwillremainbalanced

• Youcanestimatethattheradiuswillbeamuchsmallernumberthan16becauseitwillbedividedby4andthentakentoitssquareroot(wecanquicklyeliminateanswerchoicesCandD)