Page 1
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.
Page 2
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.
Page 3
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.
Page 4
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).
Page 5
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.
Page 6
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.
Page 7
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.
Page 8
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.
Page 9
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.
Page 10
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@
Page 11
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.
Page 12
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
Page 13
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
Page 14
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
Page 15
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
Page 16
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
Page 17
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
Page 18
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.
Page 19
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°)
Page 20
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