ACTTest67C

1. Answer:B.AdditionalCommaUsesandMisusesTheoriginaltextisincorrectbecauseitincludesacommainaplaceusedtoseparatearticlesinalist.Listsmustbecomprisedof2ormorearticles.Forexample,wereit“asnakecurving,winding,andbending,”itwouldbenecessary.Dsimilarlybaitsthismistake,however“snake”isnotanarticleinalist.Imagineanarratorreadingthisdramatically.Wherewouldhepause?

2. Answer:J.JoiningandSeparating

Sentences:Comma+FANBOYSNotethe“NOT”inthisquestion.Jistheonlyoptionwhichdoesnotofferanyformofconjunctionforthedoublepredicateinthissentence.“Clearedthispath,”and“Paveditwithpackedgravel,”arebothpredicates,independentofoneanother,whichshareasubject.Inordertocompoundthem,theymustbeseparatedwithacomma,conjunction,orsimilar,asprovidedinoptionsF-H.

3. Answer:D.Pronouns:Agreementand

CaseThemosttemptinganswerhereisC,howeveritisincorrectbecauseaddinga“that”doesnotactuallyfixtheproblemwiththeword“they”inthissentence.Inthiscontext,theantecedenttothepronoun“they”wouldbe“thecounty,”whichwouldmeanthegoverningbodyitselfwaspeacefullyhikingandbiking.Joffersamoregeneralanswer,howevergeneralitytrumpsspecificitywhenthespecificsarewrong.

4. Answer:F.Dashes

Emdashesaretricky,butdon’tbeintimidated.Ifyouaren’tsurewhetherit’sbeingusedcorrectly,insteadlookattheoptionsandseewhichyouknowarewrong.Gcanberuledoutbecauseasemicoloncanonlyleadintoanindependentclause,howeverherewehaveafragment.Hshouldbeafairlyclearrun-onsentencewithnopunctuationwhatsoever.JiswrongforthesamereasonasGyetwithasuperfluouscommathrownin.Evenifyoudon’tquiteunderstandtheemdash,ifyou

understandsemicolonsyoucansolvethisquestion.

5. Answer:B.IsitRelevant:Deleting

Itmaybeagoodideatoreadaheadalittlebithere;it’shardtosaywhattheessaycouldloseifyoudon’tknowwhatithastobeginwith.Thenconsiderthesentenceinquestion.Itdoesn’tprovideareasonforthenarratorbeingintheforest,soAisout.Itcertainlydoesn’tcontrastthelightheartedness;infactitreinforcesit.Disonethatcan’tberuledoutbecauseitdoesn’tmakeadirectclaim,howeverwhencomparedtoBit’sfairlyclearwhichmakessense—otherwisehemayaswellberidingagreenplumber.

6. Answer:J.JoiningandSeparating

Sentences:Comma+FANBOYSHereitisimportanttoincludeaconjunctionforthecompoundsentence.OptionsG&Hbothhaveatransition;however,theywouldonlybecorrectatthestartofanewsentence.Iftherewasaperiodintheplaceofthecommatheymaybefeasible,howeveronlyJ(whichdoesnotrefertotheprevioussentencebymeansofapronoun)isconciseandwithoutredundancy.

7. Answer:D.WordPairsand

ComparisonsThisquestioncanbefrustratingbecauseit’snotanobjectiveone,butratherdependsonyourabilitytoreadtheconnotationofthewordsused.Thetextaswrittenisentirelyneutralintone,sowecanbesureitbegsimprovement.BandCarebothsomewhereonascalefromneutraltofond,howeverDmatchestheaffectionatetonesoughtinthequestion.

ACTTest67C

8. Answer:F.Pronouns:AgreementandCaseG&Haregrammaticallycorrectoptions,howeverinthecontextofthepassagetheydonotmakesense.“You”and“one”donotmatchthefirst-personnarration,socanbeeliminated.Unliketheformertwo,Jisnotgrammaticallycorrectbecausetheadditionof“while”wouldsuggestanewindependentclausetobeaddedattheend.

9. Answer:A.JoiningandSeparating

Sentences:Comma+FANBOYSThisisacompoundsentence,withtwoindependentclausesjoinedbyacommaandtheconjunction“and.”B&Dbothcausethefirstclausetobecomedependent,thusbreakingthesentence.WhileCisgrammaticallycorrect,inthecontextofthesentenceitisfarlessreasonablethantheoriginaltext.

10. Answer:H.JoiningandSeparating

Sentences:Comma+FANBOYS&10,AdjectivesandAdverbs“Ibouncegentlyalonginmypaddedchair”isanindependentclause,acompletethought,soitneedsnocommawhatsoever.Ifyoureadthesentencetoyourself,youwilllikelynoticethatthere’snonaturalpauseintheunderlinedportion.Ifthecommasconfuseyou,youcanstillnotethatonlyoptionsH&Fcorrectlyuseanadverbinsteadofanadjective.

11. Answer:A.IsitRelevant:Replacing

Inordertoanswerthisquestion,youhavetofirstfindthesubjectofthenextparagraph.Notingtheemphasisonthenarrator’sleisurelypace,itmakessensetobeginwithareferencetospeed.Theword“their”abitfurtheronalsoprovidesahintinthatitisapluralpronoun;“Nature”and“Thesun”aresingular,sotheycannotbecorrectanswers.Giventhechoicebetweencyclistsanddays,itmakesfarmorecontextualsensetopicktheformer.

12. Answer:H.JoiningandSeparatingSentences:PeriodHerewemustdeterminewhethertheportionaftertheunderlineisanindependentclauseoradependentclause.Todothis,allyouneeddoisreaditonitsown.“Idon’tunderstandtheirhurry.”isacompletesentence,soitisanindependentclause.Assuch,theansweristomakeitanewsentence.Iftherewereanoptionforasemicolonthatcouldalsobecorrect,howevertheotheroptionsarenot.

13. Answer:D.ParallelStructure:Lists&5,

AdditionalCommaUsesandMisusesHere,again,wemustfindthepotentialclauses.While“Ispendanhour,”isanindependentclause,“lookingandlisteningandlearning,”isnot.Itcannotcomeafterasemicolon,then,soweknowrightawaytheremustbeachange.Cisnobetter,sowemustlooktoB&D.ThecommasinoptionBaresuperfluousandincorrect,soforegoingpunctuationtocreateonelongclauseistheanswer.

14. Answer:G.Apostrophes:Possessivevs

PluralSingularpossessive.Thereisonetrailwhichhasoneend.Aswrittenitisapluralwithoutpossession,Hisapluralpossessive,andJisn’tevenathing.Gisthecorrectsingularpossessive.

15. Answer:B.SupposetheWriter’sGoal

Thetrickwiththisquestionisthatthegoaldescribedinthequestionisnotactuallyagreatfitfortheessay.However,whenwelookatthe“No”optionsit’sclearthatneitherisappropriate.Cwronglyassertsafocusonthecity,whichisblatantlyirrelevant,whileDsuggestsatechnicalfocusonthewheelchair,Luigi.Bothoftheseareflawedreasoning,sowemustdiscernwhich“Yes”ismoreappropriate.Aalsooffersaninvalidexplanation,whileBgivesageneralyetcorrectevidence.Ifyoureadthepassage,itshouldbeeasytodeterminewhichanswersarenotrelevanttothetext.

ACTTest67C

16. Answer:G.Non-EssentialTransitions,Words,andClausesIfwerecognizetheportionofthesentenceleadinguptothesemicolonisnotanindependentclause,wecanruleoutbothanswersF&H.ThedifferencebetweenG&Jisasubtleone,giventheheavily-punctuatedsentence:thefinalcomma.Itdoesnotmakesensetohaveacommaafter“thousands,”asitispartofthesubject“thousandsofNewYorkCityresidents,”soitcanbedeterminedinvalid.Gistheremaininganswer.

17. Answer:C.Colons

Thecolonisaninterestingcreature.Itisusedtopresentalist,example,orothersubject.Inthiscase,wearebeingintroducedto“anincredibleengineeringfeat.”Asemicolonisnotcorrectbecausewedonothaveanindependentclause,norcanweforegopunctuationasinDbecauseweareclearlynotinthesameidea.Thepreposition“over”inoptionBwouldgivethesentenceanewmeaning,suggestingthatthecitywascelebratingliterallyabovethecompletedsubway.

18. Answer:G.IsitRelevant:Deleting

Thisquestionreliesonyourcomprehensionskills.Whenweconsiderwhatexplicitlycanbedeterminedfromthephraseinquestion,onlyGisareasonableanswer.Itcouldbearguedthatonemightinfertheotheranswers,howeverthetestwon’tlistentoyourarguments:onlyoneanswerisdefinitelycorrect.

19. Answer:D.ShorterisBetter:

RedundancyandWordinessWhendeletionisananswer,alwaysgiveitspecialattention;itisoftencorrect.Inthiscaseitis,howeverwestillmustdeterminewhy.Thebeginningofthesentencealreadystatesthattheroutetook26minutes,soaddingthatitwasunderahalfanhourisredundant.Thetesthatesredundancy,soyoucanbesureitisbetteroffomitted.

20. Answer:F.IsitRelevant:ReplacingTheanswerhereliesinthequestion.WhichchoiceCLEARLYandEFFECTIVELYdescribesthesolution?OptionsG-Jareallextremelyvague,andthoughnotinbreachofanygrammarrulestheyarefarfromwhatthequestionsought.

21. Answer:B.Pronouns:Agreementand

CaseInordertodeterminethecorrectpronoun,wemustconsidertheantecedent(theimpliednountowhichthepronounrefers).Theword“that,”asusedinthetext,canonlybereferringto“proposals.”Thisdoesnotmakesenseinthecontextofthesentence.“Those”asinChasthesameproblem.WecannotdeleteasinD,becausethen“took”wouldhavenosubject.“It,”thoughvagueandwithnoclearimpliedantecedent,iscorrectbecauseitreferstotheprocessofresolutionmentionedlaterinthesentence.Thisisadifficultquestionfromatechnicalstandpoint,butifyoureadaloudtoyourselfyouwilllikelybeabletodeterminewhichjustsoundsright!

22. Answer:J.Non-EssentialandEssential

ClausesThissentencehasafairlysimplesubjectandpredicate.“TheengineerWilliamBarclayParsons,”isthesubject,and“acceptedresponsibilityforoverseeingthisproject,”isthepredicate.Thereisoneotherwaytopunctuatethissentence,butitisnotoneoftheoptions.IftheemdashinGenclosedBOTHsidesofthename,itwouldbeacorrectnotationforadescriptiveaside.Similarly,commascouldbeusedoneitherside.HOWEVERaswehavenooptionforthis,weknowtokeepthesubjectasawhole,andthesimplesentenceisinnoneedofpunctuation.

ACTTest67C

23. Answer:C.AdjectivesandAdverbsThisquestiontestsyourunderstandingofpartsofspeech.Whatweneedhereisanadjectiveandanoun.Thiswouldfittheprecedingsentencemostsimply.“Innovation,”isanoun,however,soasiswehavenoun-noun.Boffersverb-noun.Disverb-noun.OnlyChasanadjectivedescribinganounforacorrectcompaniontotheverb“using.”

24. Answer:F.ShorterisBetter:

RedundancyandWordinessMorecomplicateddoesnotmeanmorecorrect!OptionsGandHaresimplymessesofwordsputthereonlytobeconfusingandmakeyoudoubtthecorrect,simpleanswer.NotethatinJomissionisnotcorrect,becauseinthiscaseweloseanimportantdetail.Whilethesentencecouldendhere,itisnotthesameasspecifyingwhattheyweredigging.

25. Answer:C.Colons

Whilethestartofthissentence(upto“created”)isanindependentclause,thefactthatwemustincludetheword“with”inallanswersdoesruleoutmanyoptions.Thetextasisincorrectlyusesacolonwithoutintroducingasubject,whileDtemptsonewithanappearanceofbeginningalist.Inreality,however,thelistdoesnotbeginwiththeword“layers.”Asitisdescribingallofthematerialsasbeinglayers,acoloncouldonlybeusedaftertheword“of.”Bincorrectlyplacesacommabetween“created”and“with,”whicharemore-or-lessconjoinedatthehip.Nopunctuationisneededheretocompletethesentence.

26. Answer:G.RelativePronouns:Who(se),

Whom,Which,Where,andThatManydonotunderstandthecorrectuseoftheword“whom.”Asageneralruleofthumb,itwillalmostalwaysfollowtheword“of.”Youwillneversee“ofwho,”exceptinveryspecificcornercases.“Them”isincorrectbecausetheantecedentwouldbethesubjectofthesentence,“stations.”Jmaybetemptingbecauseofthepotentialof“many”tobeanounmeaning“manypeople,”howevera

commawouldnotbecorrectifwesoughttouseitassuch.Beginninganewindependentclauseispossible,butnotanavailableanswer.

27. Answer:A.Non-EssentialTransitions,

Words,andPhrases“Therefore”means“becauseofwhatwasjustsaid.”“Forinstance”isspecifyingsomethingasanexample.“Thatis”isaclarification.“However,”however,isaddingthatsomethingiscontrarytowhatwasjustpresented.Herewearetoldthatitmanywereskeptical,butthen,contrarytothisstatement,wearetoldtheyadaptedquickly.“However”isthebest,mostlogicalanswer.

28. Answer:G.IsitRelevant:Inserting

Whichanswerclearlyandeffectivelyportraysthescaleinthepresentday?Foffersnoreferencetoscalethoughitdoesrefer;tothequoteHrefersmoretoafeatofmanagementthanofmagnitude,butisacloseroptionthantheothers;itdoesnot,however,flowfromtheparagraph.Jhasasimilarprobleminthat,whiledirectlyaboutsize,ithasnothingtodowiththenewspaperquote.OnlyGsuccessfullyintegratesthequotetomodernstatisticstosupporttheparagraph.

29. Answer:C.ParagraphOrder&9,

Pronouns:AgreementandCaseThecluehereshouldbethepronoun“this.”Whereistherealogicalantecedentjustbeforethatthiscouldfollow?OnlyCdescribesatechnique,soitmustlogicallygothere.

30. Answer:J.WordPairsandComparisons

Thisvocabquestiondirectlyasksforyoutointerprettheconnotationofthewordsprovided.TheexistingwordandoptionGarebothveryneutral,andinnowaysuggestthepreviousdecline.While“influenced”isamorevividword,italsodoesnotreferencethedeclineaddressedinthequestion.“Revived,”however,impliesithadbeendying(thus,inneedofrevival).

ACTTest67C

31. Answer:B.Verbs:AgreementWecaninferthatthe“elitegalleries”andthe“upperclass”arethesameorsimilargroupofpeople,sosomeanswersdonotmakesense.A,C,andDallsuggesthebroughthisartoutoftheupperclassbyorwhilecateringtoit,whichhardlymakessense.Instead,thepronouninBchangestheverb“cater”tobedonebythegalleriesinsteadofRivera,makingalogicalsentence.

32. Answer:J.SentenceOrder

Thisisacomplicatedquestionandisveryintimidatingatfirstglance.However,ifyoureadeachanswerallthewaythroughyouwillfindit’smucheasierthanitlooks.Gisimmediatelyeliminatedbecauseitisinthepresenttensewiththeactiveverbbeing“should.”Nowthatwehaveconfirmedtheroot“Riveraattracted,”wecanask:whatdidheattract?Controversy,ofcourse.Andwhy?Forhisbelief.Toexpand,hisbeliefthattheworkingclassshouldwieldmorepoliticalpower.ThisallfitstogetherinanswerH.

33. Answer:D.ShorterisBetter:

RedundancyandWordinessAsalways,whendeletionisanoptionmakesureyouconsideritstrongly.Hereitiscorrect,becauseitisalreadyspecifiedtobehis“artisticsubject,”soaddingthatitisinhisartisredundant.Alwayscutredundancywhenyoucan!

34. Answer:H.ParallelStructure:

Prepositions“As”isacomparativewordwhichimpliesequality.It’susebeforetheword“expansive”meansweareabouttobetoldsomethingequallyexpansive.Becausetwoequalthingscannotbemoreorlessequalthantheother,“than,”whichimpliesthisinequality,isnotcorrect.“If”mandatesacondition,andthereisnoconditioninthiscomparison.“Then”denotestimeorsequence,whichisalsoirrelevant.Another“as”properlynotestheequalityofthecomparison.

35. Answer:B.Verbs:Agreement

Theactionverb“depict”hereisourclue.“Depict”musthaveapluralsubject,while

“depicts”wouldbethesingularform.ThisalsorulesoutoptionD.Thoughhardertotellatfirstlook,Cisalsosingular.“Eachofhisfrescoes”wouldmeananyonespecificfrescooutofalargegrouping.“Many”istheonlypluraloption,andthustheonlyonewhichaccuratelypairswiththeverbinthesentence.

36. Answer:H.Non-EssentialClause,“,...,”

Thecoresubjectandpredicateherearerespectively“Thesameworkers”and“standtall.”Therestisdescriptive,andinordertodeterminehowtheyfittogetherinthegivenorderwehavetorecognizewhichpartsareandaren’tpartofthesubject.“Ontheright...throughhistory”isadescriptionoftheworkers.Itisnotanindependentclause,andthuscannotretaintheperiodnoruseasemicolon.Emdashesalsofollowthisrule.Whileitisclumsyandloadedwithcommas,usingonetoseparatethedescriptionfromthenouniscorrecthere.

37. Answer:C.Verbs:AgreementandTense

Theplural“were”immediatelyrulesoutleavingthesentenceunchanged.Theadditionof“if”inbothB&Dactuallycausesthepredicatetobecomepartofthesubjectbyturningitintoadescriptor,causingthesentencetobecomeafragment.ThesimpleCiscorrectandconcise.

38. Answer:F.AdditionalCommaUsesand

MisusesBecausethereareonlytwoadjectiveshere,notthethreerequiredtomakealist,theredoesnotneedtobeanypunctuation.Iftherewasonemoreadjectiveitwouldbenecessarytopunctuate,butasisthesentenceiscorrect.

ACTTest67C

39. Answer:C.IsitRelevant:DeletionWhatdetailisprovidedintheunderlinedportion?Isitelsewhereinthesentence?Thesearethetwoquestionstoconsiderwhengivenanomissionquestion.Becauseitisnotelsewhere,weimmediatelyknowAtobeuntrue.Bisnottruebecausethesentenceisstillcompletewithouttheknowledgeofwhatisbeingpainted.Disincorrectbecause,whileonecouldargueitisunnecessary,itisinnowayambiguous.Ccorrectlypointsoutthatthedetailisnewandrelevant.

40. Answer:J.JoiningandSeparating

Sentences:Comma+FANBOYSChoosingthecorrectconjunctionisentirelydependentoncontext.“When”isnotcorrectbecausethereisneveratimewhentheSouthcentralpanelshowssomethingdifferent.“Since”isalsoincorrectbecausetheSouthcentralpaneldidnotcausetheNorthtoshowwhatitdoes.“Thus”alsoincorrectlyimpliescausation.“And”correctlyjoinsthetwoideaswithoutimplyingone’sinfluencingtheother.

41. Answer:B.Verbs:Agreement

Wemusthaveapredicateforthissentencetobecomplete.Here,“depicting”servesnotasanactionverbbutsimplypartofadescription.CandDsimilarlyfailtocreateaction.OnlyBcreatesapredicate,andthusacomplete,independentclause.

42. Answer:F.IsitRelevant:Inserting

Whatmattershereisnotwhetheritshouldorshouldnotbeadded,butratherthereasonprovided.Evenifyouthinkitshouldnot,H&Jarenotsufficientreasons.Jisclose,howevertheexamplesarenotatruedigression.Gsimilarlyoffersincorrectevidence.

43. Answer:B.Non-EssentialClause,“,...,”

Theportionbetweentheunderlineand“celebrates”isadistractor.Thedescriptionofferedcannotbechanged,andneedn’timpactyouranswer.Insteadimaginethesentencemovingdirectlyfrom“because”into“celebrates.”Theonlyanswerthatmakessenseis“Thefrescois

adynamicworkthatcelebratesallworkingmenandwomen.”

44. Answer:J.Non-EssentialTransitions,

Words,andPhrasesAsisoftenthecase,thedeletioniscorrecthere.Onehinttothisasidefromthefactthatnoneoftheotheranswersmakemuchsenseinunderlineisthattheyallmeanmoreorlessthesamething;theyallleadintoanexceptionorsomethingcontrarytowhatwasjustsaid.Thisdoesn’tfithere,andevenifitdidtheycouldn’tallbecorrect.Thus,delete.

45. Answer:A.SentencesandFragments

Youhopefullynoticewhenfirstreadingthesentencethattherearenoglaringerrors;thatis,itsoundscorrect.Oftenthisgutfeelingisright.Notethat“eventhoughweattendeddifferenthighschoolsandcolleges”isnotanindependentclause.Itcan’tbeseparatedbyaperiodorsemicolonasinC&D.Theadditionof“yet”canbedeemedunnecessarybecause“eventhough”alreadyservesthepurposeofconnectingtheseideas.“Eventhough”couldbereplacedwith“yet,”howeverhavingbothisnotcorrect.

46. Answer:J.JoiningandSeparating

SentencesBecareful!Thisisa“NOT”question.Ifyoudon’treadcarefullyyoumayseethatFisgrammaticallycorrectandimmediatelyfillitin.HopefullyyouwouldreadtheotheranswersandnoticethatallexceptJare,becauseJistheonlyonewhichdoesnotinanywayjointhecompoundsentence.

ACTTest67C

47. Answer:C.AdditionalCommaUsesandMisusesThiscanbeadifficultcommaquestion.Whenstuckoncommaquestions,readingaloudwithlongpausesforcommashelpstodeterminewhatiscorrect.WecanruleoutBandDfairlyquicklybecausetheydonotofferadverbs,and“enjoyed”isalreadytheactiveverb.BetweenBandA,youcanhopefullytellthattwocommassoundbetterthannone.Grammatically,theredoesnotneedtobea“though,”inthissentenceatall:that’showyoucantellitshouldbeseparatedwithnestedcommas.

48. Answer:F.WordPairsand

ComparisonsWeneedaphraseherethatexpresses“fondness”andshowsa“positivereaction.”GandHareveryneutral,withalmostnoexpression.Jexpressessurprise,whichthoughnotexclusivetofondnessisnowherenearasdirectasthewrittensentence.

49. Answer:B.Pronouns:Agreementand

Case“Whom”isusedwithprepositions.Herewedonothaveone,so“who”isappropriate.“Whose”ispossessive,whichclearlyisnotright.“Which”doesnotrefertopeople,even,soasimple“Who”isthecorrectanswer.

50. Answer:F.Pronouns:Agreementand

CaseHerewehavetodeterminewhichpronounimpliesthemostappropriateantecedent.“We,”wouldmeanwho,thespeakerandhisparents?Thatdoesn’tmakesense,seeingastheytoldhimabouther.“They”wouldimplyhisparentslearnedthroughthemselves.“He”makeslesssenseyet,asthere’snosingularmalereferredtohere.“I,”wecanassume,iscorrect,andissimilarlyconsistentwiththefirstpersonnarrative.

51. Answer:B.SentenceOrder

Thebiggesthinthereshouldbeinsentence2,wherethereferencetoJoanisunprompted.Itwouldmakemoresensefollowingsentence3explaininghowhe

foundher,whichitselfmakesmoresenseonceFairbanksisintroducedinsentence1.Thesequestionscanbetrickylittlepuzzles,butifyoulookoutforsignalwordstheybecomemuchsimpler.

52. Answer:J.ShorterisBetter:

RedundancyandWordinessInthisquestion,deletionisagaincorrect.Themostcommonreasonfordeletionisredundancy,andthissentencehasit.“Tokeepthebatteryfromfreezing”alreadyimplies“sothebatterywouldstaywarm.”

53. Answer:D.WordPairsand

ComparisonsAlloftheseareviablepast-tensenotationsfor“shine”except“shoned,”whichisnotaword.Thehybridof“Shined”and“Shone”ismeanttoseemlikeanotherpossibility.

54. Answer:H.IsitRelevant:Deleting

Thedetailinquestion,wecansee,isadescriptiveonewhichoffersasettingandsomesimpleimagery.Itdoesnotincludearesponsetotheweather,nordoesitanalyzetheplugging.Whileyoumayfindthedetailunnecessary,itdoesnotrepeatanything,andyoucan’tchoosehalfofananswer.Thus,Histhemostcorrectoption.

55. Answer:A.WordPairsand

ComparisonsAllanswersherearegrammaticallycorrect,soyoumustdeterminewhichmakesthemostsenseandconveystheauthor’sthoughtsmostaccurately.Bimpliestheyfoundthehabitsbychance,Cthattheypaidthehabitsavisit,andDthattheyleftwithoutthehabitsandcamebackwiththem.“Slippedinto,”correctlysuggeststheyunknowinglyrecommencedtheiroldhabits.

56. Answer:H.SentenceOrder

Asis,thesentencestatesJoan’shousewasontopoftherabbithutch.Gsaysthefieldisonthehutch.Jalsoplacesthefieldatopthehutch.OnlyHaccuratelyplacesJoanandthenarratoronthehutch.

ACTTest67C

57. Answer:B.WordPairsandComparisonsAlloftheoptionsexceptBshowJoan’sbeingextremelyoccupiedbyherwork.Bseemstosayherworkgainedpossessionofher.

58. Answer:H.ParallelStructure:

Prepositions“Along”suggestspositioning,whichisnotapplicablehere.Theotherwordsallshowthatherdecisionswerethesubjectofherfirmness,which“Along”merelyplacesherfirmnesssomewheretherein.

59. Answer:A.IsitRelevant:Replacing

Whilethechangeslistedarecorrectsentences,theydonotaptlysumupthepassage.Ifyouread,you’llknowtherewasnohintofthe“sadly”inB,norafocusonamutualdevotiontowork,norindeedanyrealexplanationofthetimespenttogetherwhentheywereyoung.

60. Answer:G.ParagraphOrder&18.1,IsitRelevant:InsertingThebestanswerisGbecausethesuggestedsentenceisatransitionfromthedescriptionoftheirbondtotheireventualreunion.Wordslike“yet”and“such”areexcellentindicatorsofwheretoplacesentencesinthesequestions.

61. Answer:A.ShorterisBetter:

RedundancyandWordinessTheword“attaching”needstwonouns;onetobeattached,andonetowhichtoattach.Becausewealreadyhave“beganattaching,”wedonotneedtheadditionalactionsof“whichtheyconnected”and“whichtheyrelated”asofferedintheanswers.AnswerDdoesnotofferthesecondnountowhichtoattachthestories.Thus,onlyAisavalidanswer.

62. Answer:H.JoiningandSeparating

Sentences:Comma+FANBOYSThisisactuallytwoindependentclauses,bestwrittenasacompoundsentence.“Thesunsets,”isanentireidea,and“Theimagesof...appearinlightsoverhead,”issimilarlyso.Theycanbeconnectedwithasimplecommaandconjunction.Jcouldbe

agrammaticallycorrectsentence,howeveritwouldmeanthesunwassettingtheimages;thesun,notholdingthemup,couldhardlysetthem.

63. Answer:A.Non-EssentialClause,“,...,”

Inthiscase,“orconstellation”isofferinganalternatewording.Asitisnotafunctionalpartofthesubject-predicaterelationship,itisbestnestedbetweentwocommas.Ifyouareunsurewhenaphraseisonesuchextradetail,seeifthesentencemakessensewithoutit.Ifitdoes,youarelikelysafetoputitinsomecommas(oremdashes,ifthey’reanoption).

64. Answer:H.IsitRelevant:Replacing

Readahead!It’sokaytoskipaquestionlikethisandcomebacktoitwhenyouhavethecontext.Neartheend,awholeparagraphisdevotedtothemythandconstellationofOrion,sothisquestionbecomessimpleifyouhavealittlepatience.

65. Answer:D.WordPairsand

Comparisons&IsitRelevant:ReplacingDon’tbefooledbythespecification“inmoderntimes!”Thequestionislookingforvisualinformation,whichisonlyprovidedinoptionD.Whilenotexplicitlyreferencingtimeperiods,itisstillthecorrectanswertothequestion.

66. Answer:G.WordPairsand

ComparisonsThisisaquestionofvocab.“Overseers”arepeopleinamanagerialpositionwhomonitortheactionsofothers.Eachotheroptionheredescribessomeonewholooksatthesky.

67. Answer:A.WordPairsand

ComparisonsThesignificancecomesfromthedetailoftheKarasukirepresentingaplow.Plowsbeingatraditionalagriculturaltool,it’srelevantthattheJapanesecommunitiesmentionedareagriculturalones.Theirdistance,population,andhistorical-nesshavenopertinencetotheplowimagery.

ACTTest67C

68. Answer:J.DictionandRegisterForaformal,informativeessay,it’snotappropriatetousesuchcasualphrasesasinFandG.H,whilenotsuchabreachofetiquette,iswhollyunnecessary,anditisafaux-pastousesecond-personpronounsinaninformativeessay.

69. Answer:A.Non-EssentialTransitions,

Words,andPhrasesBissimplyfalse,whileC&Dbothleaveoutimportantdetails.ItissignificantthatthisinterpretationislocaltootherpartsofJapan,andnotthosealreadymentioned.

70. Answer:G.Verbs:Agreement

“Stars”istherootofthesubjecthere,andisplural.“Has”issingular;“couldof”iswrongnotonlybecauseitdoesn’tfit,butalsobecauseitwouldbe“couldhave”ifitwasevenapt;“hasbeen”isjustassingularasF.OnlyGhasapluralverbandactuallymakessense.

71. Answer:D.Non-EssentialTransitions,

Words,andPhrasesThisonesoundsreallyweird,andyoumaybetemptedtothrowcommasintherejustbecauseitseemslikearun-on.Butdon’tletyourearsfoolyou;thereisnoneedforacommauntiltheend.“ThemythologyoftheTswanapeopleofSouthAfrica”isonelongsubject,and“In”isnecessaryfortheword“represent”laterinthesentence.Imagineitflippedthisway:“ThesesamestarsrepresentthreepigsinthemythologyoftheTswanapeopleofSouthAfrica.”Theonlycommaweneedistoshowwhereweflipped.

72. Answer:J.RelativePronouns:Who(se),

Whom,Which,Where,andThatGandHarenotgrammaticallycorrect,howeverJandFare.Todeterminewhichiscorrect,wehavetouselogictoanalyzewhateachissaying.Fseemstosaythatthescorpion’sinabilitytostingisthecauseofthedistance,whileJsaysthedistancepreventsthestinging.Jmakesmoresenseinthiscontext.

73. Answer:C.Apostrophes:They’re,Their,ThereTheir,there,they’re:thebanesofEnglish-speakers.

Their=Belongingtothem.They’re=Theyare.There=Thatplace.

74. Answer:G.JoiningandSeparating

Sentences:Comma+FANBOYSThisisacompoundsentence!Aswithalltheothersinthistest,itcanbejoinedwithsimplyacommaandan“and.”Notethat“span”isnot“spanning,”whichwouldvalidateotheroptions.

75. Answer:D.ParagraphOrder

Theimportantparthereis“freeofdirectreferencestoaspecificculture’sviewofthethreestars.”Dividinganywhereearlierthansentence7wouldleavesomereferencesintheconclusion,andthusbeincorrect.

ACTTest67C

1.A.LinearFunction:Rate$20perv,vehicles;$10perp,persons

A.𝟐𝟎𝒗 + 𝟏𝟎𝒑2.F.SolvingEquations(9 + 5 − (−6))(5 + −6 )20 −1

F.−𝟐𝟎Tip:payattentiontoNEGATIVEsigns.WhensubstitutinganddistributingnegativenumbersALWAYSuseparentheses.3.E.Rate&Proportion1st:60𝑝𝑒𝑟𝑚𝑖𝑛. ∗ 80𝑚𝑖𝑛 = 4802nd:80𝑝𝑒𝑟𝑚𝑖𝑛.∗ 60𝑚𝑖𝑛 = 480Total=480+480=960

E.9604.J.Mean,AverageYouknowhowtodoanaverage.Whenaskedtokeeptheaveragethesame,thenextvalueaddedmustsimplybeequaltotheaverage!210 + 225 + 254 + 231 + 280

5= 240

J.240

5.C.Rate&ProportionTheACTlovestoaskabouthourlywages(payperhour).Hereshemakes$7.50perhourfor40hours.$7.50 ∗ 40 = $300And1.5timesformorethan40hours.$7.50 ∗ 1.5 ∗ 2 = $22.50$300+$22.50=$322.50

C.$322.50

6.K.WordProblems:Translation&Vocabulary“Anumbersquared”is𝑥@“is39more”is“=39+”“productof10andx”is10x

K.𝒙𝟐 = 𝟑𝟗 + 𝟏𝟎𝒙7.E.SolvingEquations9 𝑥 − 9 = −11*Distribute99𝑥 − 81 = −11*Add819𝑥 = 70*Divideby9

𝑥 =709

E:DE

F

8.H.Rate&ProportionGiven:$4.00isdiscountprice.Spent$6060/4=15ticketspurchasedDiscountis$37.50less($37.50 + $60.00) 15 = $6.50

H.$6.50Trap:Fisonlythediscountperticket.9.A.Factoring&FOILFOIL!(onlyifyouhaveto)

A.𝟗𝒙𝟐 − 𝟏𝟔𝒚𝟒TheACTlovestheDifferenceoftheSquares.Askilledtest-takershouldbeabletorecognizethesebackwardsandforwards.Tip:Memorize

• 𝑎 − 𝑏 𝑎 + 𝑏 = 𝑎@ − 𝑏@• (𝑎 + 𝑏)@ = 𝑎@ + 2𝑎𝑏 + 𝑏@• (𝑎 − 𝑏)@ = 𝑎@ − 2𝑎𝑏 + 𝑏@

ACTTest67C

10.J.QuadrilateralsArea.𝑙 ∗ 𝑤 = 32Perimeter2𝑙 + 2𝑤 = 24AlgebrasolutionSolvefor𝑤,bysubstitution.2𝑙 + 2𝑤 = 24Subtract2wfrombothsides2𝑙 + 2𝑤 − 2𝑤 = 24 − 2𝑤2𝑙 = 24 − 2𝑤Dividebothsidesby22𝑙2=24 − 2𝑤

2

𝑙 = 12 − 𝑤Substitute12 − 𝑤for𝑙in𝑙 ∗ 𝑤 = 32(12 − 𝑤) ∗ 𝑤 = 3212𝑤 − 𝑤@ = 32𝑤@ − 12𝑤 − 32 = 0Factor𝑤 − 8 𝑤 − 4 = 0𝑤 = 4𝑜𝑟8

J.4That’saheckofalotofalgebra.Guessandcheckisbetter.Becausetheareaandtheperimeterareintegers,thenthelengthandthewidthbothhavetobeintegers,since:IntegerxInteger=IntegerInteger+Integer=IntegerSowhatarethefactorsof32?8x4&16x2.Guessandcheck.Trap:K:8,don’tmakethemistakeofputtingthelongerside.

11.D.TrianglesTheSUMoftheanglesis47°,thentheotherissimply180° − 47° = 133°

D.𝟏𝟑𝟑°Trap:B.ThemeasureofAandBaddedis47,ifyoumakeeachone47,thentheresumis94,makingC86.12.K.Counting,Permutations,&CombinationsThisisstraightupcounting.Tomakealunchyouneedoneofeach,sojustmultiplythenumberofoptionstogether.3×3×4×2 = 72

K.72Traps:F:2istheaverageofthenumbersandH:12isthesum.Don’tmakethesemistakes.13.B.PropertiesofIntegersConsecutiveintegerscanberepresentedas(𝑛, 𝑛 + 1)𝑛 + 3 𝑛 + 1 = 79;solveforn.4𝑛 + 3 = 79𝑛 = 19

B.19,20Trap:WhosaystheACTisnottricky?Eisatrapoftwoconsecutivenumberswhosesumis79.14.F.Functionsf(x)Thisisaverybasicquestionthatisoneverytest:evaluateafunctionforagivenvalueofx.𝑓 −3 = −8(−3)@𝑓 −3 = −8 9 𝑓 −3 = −72

F:−𝟕𝟐Trap:WATCHOUTFORNEGATIVESIGNS!OneveryACT,therearesimplesubstitutionsandevaluationsthatincludeoperationswithnegativenumbers.Theylovesubtractinganddistributingnegativenumbers,becauseyoudonot.

ACTTest67C

15.C.Exponents&Roots3T = 54LookattheanswersMUSTbetrue.Youhavememorizedthepowersof3(becauseItoldyouto).3@ = 9;3V = 27;3W = 81Since54isbetween27and81,xMUSTbeanumberbetween3and4.

C.𝟑 < 𝒙 < 𝟒Trap:E.Anumberbetween3and4isalsolessthan5.Butanumberlessthan5,suchas1or2doesnotHAVEtobebetween3and4,sowhileE.COULDbetrust,itisn’talways.16.J.Numbers:LCMLeastCommonMultipleThat’srightclass,youarenowbackinfourthgradebeingtestedonmultiples.Iliketobreakthesenumbersintofactors:

70 = 7 ∗ 1060 = 6 ∗ 1050 = 5 ∗ 10

Ouranswerhastohavecommonparts,weonlyneedone10,becauseeachofthemhasa10.

(7 ∗ 6 ∗ 5) ∗ 10 = 2,100

J:2,10017.B.SolidsVolumeofaboxis𝑙 ∗ 𝑤 ∗ ℎGivenvolume,andtwodimensions,findthethird.

81,000 = 45 ∗ 30 ∗ ℎ

B.60

18.J.CirclesDRAW!Ifyoudon’tdrawthisoneout,youaremakingamistake.Clockwise:movetotherightCounterclockwise:movetotheleft.

J.A,C,D,BTip:Whenyoucandrawsomethingforaproblem,DRAW!Physicallymovingyourpencilusesadifferentpartofyourbrain19.D.Exponents&RootsSubstitute𝑡 = 5into𝑦 = 16(2)\

𝑦 = 16(2)] = 512D.512

Trick:Setupyourequationandthentakeapeekattheanswerchoices.DistheonlypossibleanswersinceNOpowerof2endsinZERO!16isapoweroftwo,thereforetheanswerMUSTbeapowerof2.Youknowthat512isapowerof2.Thiscouldbea10secondproblem–youcansolveitfasterthanmoststudentscanpickupandentertheequationintheircalculator.20.J.QuadrilateralsDRAW!

SmallerRectangle𝑙 ∗ 𝑤 = 𝐴LargerRectangle:3𝑙 ∗ 𝑤 = 𝑘𝐴𝑘 = 3

J.3

ACTTest67C

21.E.AlgebraicOperations𝑎 + 2𝑏 + 3𝑐 − (4𝑎 + 6𝑏 − 5𝑐)Distributethenegativesign𝑎 + 2𝑏 + 3𝑐 − 4𝑎 − 6𝑏 + 5𝑐CLT:Combineliketerms

E.−𝟑𝒂 − 𝟒𝒃 + 𝟖𝒄Trap:Beverycarefuldistributingnegativesigns.Thisproblemdoesnottakelongtodo,sogetitrightbybeingmindfulandexecuting.22.G.Trigonometrysin 𝜃 = 𝑂𝑝𝑝/𝐻𝑦𝑝sin 𝜃 =

𝑎𝑐

G:

lm

Tip:TheTrigontheACTisusuallythiseasy.Practicethisproblemandgainspeedbutdoitwithaccuracy.23.B.COUNTING,PERMUTATIONS,&COMBINATIONS

LabelthefiveplayersA,B,C,D,Eandputtheminacircle.

1. AcanonlypasstoCorD,pickC2. CcanonlypasstoE3. EcanonlypasstoB4. BcanonlypasstoD5. DcanonlypasstoA-andthusittakes

fivepassesfortheballtogetbacktoA.

B.5th

24.H.LinearFunctions:y=mx+b𝑦 = 0.12𝑥 + 3000theslopeis0.12add0.1to0.12

H.0.2225.A.Exponents&Roots−8𝑥V(7𝑥n − 3𝑥])Distribute,becarefulofthenegativesigns.Whenmultiplyingvariableswithsamebasebutdifferentexponents,addthem.

(−8𝑥V) ∗ (7𝑥n) + (−8𝑥V) ∗ (−3𝑥])AlwaysuseparenthesiswhenmanipulatingtermsinAlgebraicexpressionsandequations.Tip:Lookattheanswerchoices.Theanswermustbetwoterms,firstnegative,secondpositive.ThatleavesonlyAandC.Youcanbedownto50/50justbylookingatthesigns. −8 ∗ 7 ∗ 𝑥Von + (−8 ∗ −3 ∗ 𝑥Vo])

A.−56𝑥F + 24𝑥pEisjustwrong;itisforsomeonewhoclearlydoesnotunderstandexponents.26.G.ABSAbsoluteValueThisoneyoucantyperightintoyourcalculator

−3 −6 + 8 −3 2

G.– 𝟔

27.B.TrianglesSolvefor𝐵𝐶,a3-4-5righttriangle.AECisasimilartriangle𝐴𝐶 =𝐵𝐶 ∗ 4so𝐴𝐸 =𝐵𝐷 ∗ 4 = 3 ∗ 4 = 12

B.12

ACTTest67C

28.H.Functionsf(x)Thisisalinearfunctionbecauseitsays“constantrate.”Theeasiestthingtodoissubstitutet=0;y=14intoeachanswer.EliminateG,J,KThensubstitutet=1;y=19𝐹: 19 = (1) + 14–𝑤𝑟𝑜𝑛𝑔𝐻: 19 = 5 1 + 14 − 𝑐𝑜𝑟𝑟𝑒𝑐𝑡

H.𝒚 = 𝟓𝒕 + 𝟏𝟒29.E.Inequalities6𝑥 + 12 > 7𝑥 − 35*Simplify12 + 35 > 7𝑥 − 6𝑥47 > 𝑥

E.𝒙 < 𝟒𝟕30.K.Coordinate

OnlyKis3unitsawayfrom(2,0)

K.(5,0)

31.F.TrianglesItdoesnotgetmucheasierthanthis:𝐴@ + 𝐵@ = 𝐶@𝑦@ = 𝑥@ + 4@𝑦 = 𝑥@ + 16

E. 𝒙𝟐 + 𝟏𝟔32.G.ProbabilityChange|@

V@toV

]

1.BACKSOLVE:Addanswerstoboththenumeratoranddenominatorandreduce.Tip:thenewfraction’sdenominatoris5,thereforetherightcombinationMUSTbeamultipleof5;endingin5or10.J&Kcanbeeliminated.F:|@o|V

V@o|V= @]

W]= ]

F

G:|@o|p

V@o|p= VE

]E= V

]correct!

2.AlgebraicSolution:12 + 𝑥32 + 𝑥

=35

5 12 + 𝑥 = 3 32 + 𝑥 60 + 5𝑥 = 96 + 3𝑥2𝑥 = 36𝑥 = 18

G.18

ACTTest67C

33.D.CoordinateGeometry&XY-Plane4𝑥 − 2𝑦 = 8Thelineisinstandardform,toplotonxy-plane,justfindtheintercepts(zero’s).4 0 − 2𝑦 = 8𝑦 = −4; 0, −4 4𝑥 − 2 0 = 8𝑥 = 2; 2,0 Plot:

D.I,III,IVonly34.F.LinearFunctionsSubstituteandsolve𝑦 = −5𝑥@ + 9Substitute(1,2𝑎)2a = −5 1 @ + 92𝑎 = 4; 𝑎 = 2

F.235.D.Ratio12𝑆 +

13𝑆 + 𝑥 𝑆 = 𝑆

𝑥 =16

Converttoratio|@∶ |V∶ |nisequalto3:2:1

D.3:2:1

36.F.EquationofCircleTheequationofacircleinthestandard(𝑥, 𝑦)planeis:𝑥 − ℎ @ + (𝑦 − 𝑘)@ = 𝑟@Where(ℎ, 𝑘)isthecenterofthecircleandristheradius.(𝑥 − 5)@ + 𝑦@ = 38𝑟@ = 38; 𝑟 = 38𝑎 = 5, 𝑏 = 0centeris(5.0)Tip:Becarefulofnegativesignsforthecenterofacircle.Theymovelikefunctions(x-5)movestotheright.

F.Radius= 38Center=(5,0)37.B.MultipleFigures

Youareaskedfortheoutsideperimeteroftheofthefigure.2semicirclesmakeonecircle,findcircumference𝐶 = 𝜋𝑑; 𝐶 = 8𝜋Addonlytwosidesofthesquare,16

B.𝟏𝟔 + 𝟖𝝅

ACTTest67C

38.G.MultipleFiguresTwo“Midpoints”cutarectangleinhalf,formingtwocongruentsmallerrectangles.

𝐴𝐵𝐸𝐹 ≅ 𝐸𝐹𝐶𝐷

Adiagonalofarectanglecutstherectangleintoequaltriangles.

∆𝐴𝐵𝐹 ≈ ∆𝐴𝐸𝐹&∆𝐸𝐹𝐶 ≈ 𝐸𝐷𝐶

Thediagonalsofarectangleintersectatthecenteroftherectangleandformfourcongruenttriangles.

Soeachhalfoftherectangleissplitintofourequalpieces,onenon-shadedpieceandthreeshadedpieces,makingaratioof1:3.

G.3:1Tip:“Midpoints”ALWAYSmeanscutthethinginhalf!

39.C.LinearFunctions:y=mx+bTrick:Readthequestioncarefully.ItisONLYaskingaboutxcoordinates.Theycoordinatesareirrelevant.

Midpoint=(|W� �W )

@

−4 + 9 = 514 − 9 = 5

C.540.G.SolidsAreaofonefaceofacube,side8=8@ = 64Sixsidesofacube6 ∗ 64 = 384

G.38441.B.SystemofEquations

𝑎𝑦 + 𝑏𝑥 = 𝑐𝑎𝑦 − 𝑏𝑥 = 𝑐

Putbothequationsintoslope-interceptform:𝑦 = − �

l𝑥 + m

l *slope=− �

l

𝑦 = �

l𝑥 + m

l *slope=�

l

I. ParallelLines–No.Parallellineshavethesameslope.

II. IntersectingLines–Yes.Intersectinglineshavedifferentslopes(sameyinterceptmeansthattheyintersectonthex-axis).

III. SingleLine–No.Singlelineshavethesameslopeandthesameyintercept.

B.IIonly

ACTTest67C

42.F.401;Trigonometry

Fromtheangleof52,theoppositesideisxandtheadjacentsideis30.𝑡𝑎𝑛𝜃 = ������\�

l��lm��\𝑡𝑎𝑛52 = T

VE*solveforx

𝑥 = 30 𝑡𝑎𝑛 52

F.𝟑𝟎 𝒕𝒂𝒏𝟓𝟐

43.D.Charts&GraphsOdds–inage:notinage(25–35)=42%(allothers)=58%42:58=21:29

D.21:2944.H.MultipleFigures“Linesofsymmetry”arelinesthatarewhenfoldedeachsideisamirrorreflection.Thereare8waysto“fold”thisfigure.

H.8

45.A.CirclesDiameter=2,soradius=1Area=𝜋𝑟@ = 3.14A.3.146.J.PercentDiameter=2,75%longer

2 ∗ 1.75 = 3.5

J.3.5047.C.GeometryLines∠𝐵𝐴𝐶&∠𝐴𝐶𝐷arecomplimentaryangles(oppositeinterior)andaddupto180.∠𝐵𝐴𝐶 = 82∠𝐴𝐶𝐷 = 180 − 82 = 98Eacharebisected

∠𝐸𝐴𝐶 =822= 41

∠𝐴𝐶𝐸 =982= 49

∠𝐴𝐸𝐶 = 180 − 41 + 49 = 90

C.90

ACTTest67C

48.H.CirclesTip:Whenatriangleisinacircleandonevertexisthecenterandtheothertwoareonthecircle,radiiformtwosides,makingitanisoscelestriangle–thisisastandardizedtestfavorite.Arcangleistwicetheangleofthetwochords.

H.60

49.B.210;SystemofEquationsInfinitesolutionsmeans“sameline.”1.Usingslopeinterceptform𝑦 = 2𝑥 − 8

𝑦 = 2𝑥 −43𝑎

Thetwolineshavethesameslopetheyneedtohavethesameintercept.Therefore:− WV𝑎 = −8*solvefora

B.6

50-52.SystemsofEquationsThisisararetopic,butonethatusessimplelinearfunctionsandcanbeunderstoodeasily.50.F.SystemsofEquationsThegraphshowsahorizontallineat𝑦 = 2,asthebottomconstraint.Becausethisisasystemofinequalities,theshadedareaabovetheline𝑦 = 2representsthenumberoflargeframesthatshecanmake.Whereas,thexaxisrepresentsnumberofsmallframes.Forexample:Thepoint(2,3),sheCANmake2smalland3large.But(3,1)willnotbemadesinceitisnotshaded.

Theyaxisisthenumberoflargeframes.Therefore,shemakesaminimumof2andamaximumof8.

F.251.C.Rate&Proportion,Percent“Foreveryhour”=$3Findthenumberofhoursworked.Large=(3hrs)(4L)=12hrsSmall=(2hrs)(2S)=4hrsTotal=12+4=16hrs($3/hr)(16hrs)=$48donatedNowfindprofit30(2)+70(4)=340Percentdonated:48/340=.1411

C.14%

52.J.SystemsofEquationsAmaximumcomesatthevertexofthepolygonformedbytheinequalitiesinlinearprogramming(asystemoflinearinequalities).Themaximumprofitiswhenshemakesonly8largeframes.(8frames)*($70)=$560

J.$56053.E.MatrixDon’tsweatmatrixproblems,theACTisveryniceandtellsyouexactlywhattodo.𝑎𝑏 − 𝑐𝑏𝑥 ∗ 𝑥 − 8𝑥 = −16*solveforx𝑥@ − 8𝑥 + 16 = 0*perfectsquare=factor!𝑥 − 4 𝑥 − 4 𝑜𝑟 𝑥 − 4 @ = 0𝑥 = 4

E.4.

ACTTest67C

54.K.AlgebraicOperationsThisproblemhasnothingtodowithinterest!Itissimplysolvingforonevariableintermsofanother.TheACTisjusttryingtoscaresomestudentsoff,iftheygetthisfar.𝐴 = 𝑃(1 + 0.01𝑖)�*tosolveforP,simplydividebywhatisnotP

𝐴(1 + 0.01𝑖)�

=𝑃(1 + 0.01𝑖)�(1 + 0.01𝑖)�

𝑃 =𝐴

(1 + 0.01𝑖)�

K. 𝑨

(𝟏o𝟎.𝟎𝟏𝒊)𝒏

55.C.SolvingInequalities“Mustbetrue”–meansthatyouonlyneedtofindoneexamplethatprovesitwrong.Picksomenumbersandgettowork.A.T

�> 1

Nevertrue:Apositivenumberdividedbyanegativenumberisalwaysnegative.B.|𝑥|@ > |𝑦| Thisstatementcouldbetrue,butisnotalwaystrue.Make𝑥 = 2&𝑦 = −200.4 ≯ 200C.T

V− 5 > �

V− 5

Simplify:add5tobothsidesandthenmultiplyby3𝑥 > 𝑦Apositivenumberisalwaysgreaterthananegativenumber.

Thismustbetrue.D.𝑥@ + 1 > 𝑦@ + 1UsethesamelogicasB.Make𝑥 = 2&𝑦 = −200.E.𝑥�@ > 𝑦�@UsethesamelogicasB.Make𝑥 = 2&𝑦 = −200

56.J.TrianglesFunlittleproblem.Hereyouhavetofindtheheightofthetriangles(alsocalledaltitude).Andthencomparetriangles.Setxasthebaseofbothtriangles:

Thenthetrickistoseethattheheightisthesame.Theheight,h,mustbecongruentusingAngle-Side-Angle(yup,thoseoldlovelyGeometrytriangleproofs).

J.3057.E.TrigonometryTip:MemorizetheLawofSinesandtheLawofCosines.LookattheexamplesofwheneachareusedontheACT.Generally,theACTonlyasksyoutosetuptheequation,notsolveit.HereitistheLawofCosinesthatmustbeusedbecausewehavethelengthoftwosidesandtheangleinbetweenthem.

𝐵𝐶@ = 12@ + 18@ − 2 12 18 𝐶𝑜𝑠(40°)E.𝑩𝑪 = 𝟏𝟐𝟐 + 𝟏𝟖𝟐 − 𝟐 𝟏𝟐 𝟏𝟖 𝑪𝒐𝒔(𝟒𝟎°)

ACTTest67C

58.G.SequenceFormostsequences,Iliketodrawapictureofthenumberblanks.

From8to13moving4spacesis |V�p

W.Eachspace

is]W

Nowfindthefirst4terms.

Addthemup:1.75 + 3 + 4.25 + 5.5 = 14.5

G.14.559.C.Quadratics&ParabolasTip:Whenaquadraticequationisequaltozero,youaresolvingforthe“roots”–otherwiseknownas“factoring.”Whenthereisonlyonesolutionorrootthequadraticisa“perfectsquare”withitsvertexonthexaxis.Iftheonlysolutionis𝑥 = −3thenthequadraticis(𝑥 + 3)@ = 0Expand:𝑥@ + 6𝑥 + 9 = 0;𝑚 = 6

C.6

60.F.ABSAbsoluteValueTip:Absolutevalueisusedtomeasuredistancefromzero.“5unitsfrom-3”meansnumbersthatareadistanceof5.

Thesolutionsetis(-8,2);Eitherputthesolutionintotheequationsorsolvetheequations:𝑥 + 3 = 5𝑥 + 3 = −5& 𝑥 + 3 = 5𝑥 = −8&𝑥 = 2

F. 𝒙 + 𝟑 = 𝟓