(7) Onanumberline,findthecoordinateofapoint 1 5 ofthedistancefrom 3 to17.
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1 (1) Express( ) 1 asacommonfraction () IfA B= A B,whatisthevalueof(3 4) 5?Expressyouranswerasa common fraction. (3) Theratioofdogstocatsatthepoundis4:3.Howmanydogswereatthe poundifatotalof80dogsandcatswereatthepound? (4) Whatisthemeanofthefollowinglist? 4x+3,7,3x,9x (5) Sixtimesthereciprocalofanumberis 9 4.Whatisthenumber?Express youranswerasamixednumber. (6) What percent of the five-by-five square is shaded? (7) Onanumberline,findthecoordinateofapoint 1 5 ofthedistancefrom 3 to17. (8) 4%of50iswhatfractionalpartof36?
2 (9) Evaluate: 56 (10) Forwhatvalueofnwill(7!)(5!)(3!)(1!)=n!? (11) What fractional part of 7 is 16? (1) Solveforp:3p (p 4)=7p+6 (13) Evaluate:+4(3) 5. (14) Supposethatbyusingthesparetireonhercarasmuchassheusestheother fourtires,coleendrivesthecar80,000miles.whatisthewear,inmiles,oneachtire? (15) Howmanymilescanacartravelin0minutesifittravels 3 4 asfastasa traingoing80milesperhour? (16) Danipurchases5applesfor$0.85.Atthesamerate,howmuchwoulda dozen apples cost? (17) Atrapezoidhasonebaseequaltoitsheight,x,andtheotherbaseistwice aslong.writetheexpressionfortheareaofthetrapezoidasacommonfractioninterms ofx. (18) Thearithmeticmeanof1scoresis8.Whenthehighestandlowestscores areremoved,thenewmeanbecomes84.ifthehighestofthe1scoresis98,whatisthe lowest score? (19) Simplify3( )+ (0) Simplify:(4+8) +(4 +8 ).
3 (1) Theratioofboystogirlsinaschoolis4to5.Howmanygirlsareinthe schoolifatotalof945studentsareenrolled? () A basketball player made the following number of free throws in 8 successive games:6,18,15,14,19,1,19,and15.whatisthemediannumberofsuccessfulfree throws? (3) Expressasafractioninlowestterms: (4) Whatisthegeometricmeanof19and304? (5) Round to the nearest hundredth: (6) Dawnhas$1.0innickels,dimesandquarters.Foreachnickelshehasone dimeandforeachdimeshehasonequarter.howmanytotalcoinsdoesshehave? (7) Solvefork.k+5=13. (8) There exist non-negative integers a and b such that for any x satisfying a<x<b,itistruethat x>x.findtheorderedpair(a,b). (9) Findthesumofthemeanandthemedianofthegivenset: { 0, 1,, 3, 4, 5 } Express your answer as a mixed number. (30) If(a b)=(a b)+a b +b a,find3 5. Copyright MATHCOUNTS Inc. All rights reserved
4 Answer Sheet Number Answer Problem ID 1 0/9 4C1B 3 0BB dogs x+ 0B 5 /3 C41 6 5% 00C AB 8 1/3 0A3B 9 C1B /9 C /3 A3D C0B A03A miles 4A dollars B1BB 3x 17 A13B BAC A CB0C CB D1 3 4/33 1BB B coins 1AC A101 8 (0,1) ABB / AC B5BB Copyright MATHCOUNTS Inc. All rights reserved
5 Solutions (1) 0/9 ID:[4C1B] () 3 ID:[0BB41] (3)160dogs ID:[401] Let4xbethenumberofdogsand3xbethenumberofcats.Then4x+3x=80,or 7x=80.Solving,wegetx=40.Thus,thenumberofdogsis4x=4(40)= 160. (4)4x+ ID:[0B] (5)/3 ID:[C41] Lettingxbethedesirednumber,wehave 6 x = 9 4.Multiplyingbothsidesbyxandby4gives 4=9x.Dividingbothsidesby9givesx= 4 9 = 8 3 = 3. (6)5% ID:[00C0] Eachsmallsquarehasthesameareasoweneedonlydividethenumberofshadedsquares bythetotalnumberofsquares.thereare13oftheformerand5ofthelatter,sothe answeris 13 5 = 5%. (7)1 ID:[5AB] (8)1/3 ID:[0A3B]
6 (9) ID:[C1B] Wehave 56= 16= 4=. (10) 10 ID:[0551] (11) /9 ID:[C101] Thefractionalpartof7that16iscanbefoundbysimplifyingthefraction16/7.16and 7bothcontainafactorof8,sowehavethat 16 7 = = 9. (1) 1/3 ID:[A3D1] First,wesimplifytheleftside.Wehave 3p (p 4)=3p p+8=p+8, sowecanwritetheoriginalequationasp+8=7p+6.subtractingpfrombothsides gives8=6p+6,andsubtracting6frombothsidesgives=6p.finally,dividingby6 givesp=/6= 1/3. (13)9 ID:[4C0B] (14) ID:[A03A1] (15) 0 miles ID:[4A041] Thecartravelsataspeedof milesperhour=3 0=60milesperhour. In0minutes,thecartravels 60 miles 0minutes=1 0= 0miles. 60minutes (16).04 dollars ID:[B1BB]
7 (17) 3x ID:[A13B] Theareaofatrapezoidisequaltotheproductoftheheightandtheaverageofthelength ofthebases.inthiscase,sincethelengthofthetwobasesarexandxandthelengthof theheightisx,theareaisequalto x+x x= 3x x= 3x /. (18) 46 ID:[BAC41] Ifthemeanofthe1scoresis8,thenthesumofthe1scoresis8 1.Afterthetwo scoresareremoved,thesumofthe10remainingscoresis84 10=840.Thesumofthe two removed scores is =4( )=4(46 10)=4(36)=144. Sinceoneoftheremovedscoresis98,theotherremovedscoreis144 98= 46. (19)4 ID:[A451] (0) 4 ID:[CB0C] Followingtheorderofoperations(computethesumin(4+8) beforeapplyingthe exponentof(4+8) ),wehave (4+8) +(4 +8 )=(1) +(4 +8 )=144+(16+64)=144+80= 4. (1) 55 ID:[3CB0] Sincetheratioofboystogirlsis4:5,thefractionofthestudentswhoaregirlsis 5 Therefore,thenumberofgirlsis 9 945= = 5 9. () 15 ID:[53D1] To find the median, we first arrange the number of successful free throws in increasing numerical order: 6,1,14,15,15,18,19,19. Sincethereisanevennumberofterms,themediancanbefoundthebyaveragingthe middletwo(thefourthandthefifth)terms.boththefourthandthefifthtermsare15,so themediannumberofsuccessfulfreethrowsthatthebasketballplayermadeis 15.
8 (3) 4/33 ID:[1BB] Webeginbyrealizingthat0.1=0.11,so = =0.1.(Notethatthis can be done because there is no carrying involved.) Toexpressthenumber0.1asafraction,wecallitxandsubtractitfrom100x: 100x = x = x = 1 Thisshowsthat0.1= Butthatisn tinlowestterms,since1and99shareacommonfactorof3.wecan reduce 1 99 to 4 33,whichisinlowestterms. (4)76 ID:[01B] (5) ID:[001] Toround tothenearesthundredth,wemustlookatthehundredsandthe thousands digits of the number in question. Since the thousands digit(5) is greater than or equalto5,thehundredsdigit8roundsupto9.therefore, roundedtothenearest hundredthisequalto (6)9coins ID:[1AC41] IfDawnhasxnickels,thenshehasxdimesandxquarters.Incents,wesetupan equationforhowmuchmoneyshehas. 5x+10x+5x=10 40x=10 x=3 Shehas3ofeachcoinforatotalof 9 coins. (7)4 ID:[A101] Subtracting5frombothsidesgivesk=8anddividingbygivesk= 4. (8)(0,1) ID:[ABB41]
9 (9)161/ ID:[AC41] Themeanofthenumbersis: = = 63 6 = 1. Asthenumbersareordered,themedianisjusttheaverageofthemiddletwonumbers: + 3 = 1 =6. Thesumofthesenumbersis6+ 1 = = 161. (30) 383 ID:[B5BB] Copyright MATHCOUNTS Inc. All rights reserved
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