Surname ………………………………………… Candidate number ……….

First name …………………………………….

Entrance examination 2012 Arithmetic Paper 1

30 minutes Do not open this booklet until told to do so

Calculators may not be used

Write your names and candidate number in the spaces provided at the top of the page.

You have 30 minutes for this paper which is worth 20 marks.

Answer all the questions, attempting them in order and writing your answers clearly. If you find that you cannot answer a question straight away leave it blank and return to it later if you have time. Do not leave blank answer spaces, make the best attempt at an answer that you can.

If you need to change an answer cross it out neatly and write the new answer alongside the box. ____________________________________________________ Marker 1 Marker 2 Agreed mark

Number correct

Number wrong

Entrance Examination 2013

Arithmetic Paper 2

1 Hour

Do not open this booklet until told to do so

Calculators may not be used

Surname..............................................................Candidate number..............................

First name ...........................................................

Current school....................................................

Write your names, school and candidate number in the spacesprovided at the top of this page. Foreachquestion,showallyourworkinginfull,asthiswillbemarked,andthenwriteyouranswerclearlyinthespaceprovided.

Youhave1hourforthispaperwhichisworth80marks.

The ManchesterGrammar School

Page2

1. Completethefollowingsentenceswiththemostappropriatemetricunit

(a)Andyis183_______________tall

(b)Ourfootballpitchis80_______________long

(c)Mybirthdaycakeweighs700________________

(d)Theareaofapieceoflinedwritingpaperis

600____________

(e)Mywaterbottlecontains1.5__________ofwater

[5 marks]

1a

1b

1c

1d

1e

2. Agameofcountershassixrounds.Ineachroundofthegame,theplayersscorepointswhich

arethenaddedtotheirtotalscore.

ThegraphbelowshowsClaire’stotalscoreaftereachroundofthegame.

(a) HowmanypointsdidClairescoreinround1?

(b) HowmanypointsdidClairescoreinround4?

(c) WhichwasClaire’smostsuccessfulroundin

whichshescoredmostpoints?

Nigelalsoplaysthegameandhisscoresarerecorded.Whenthepointsshowinghistotal

scoresareconnected,thegraphofhistotalscoresaftereachroundisasinglestraightline.

(d) Whatcanyousayaboutthenumberofpoints

Nigelscoredineachround?

[5 marks]

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2a

2b

2c

2d

Round

Totalscoreaftereachround

0123456

70

60

50

40

30

20

10

0

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3. Apieceinagamecanmoveinfourdifferentways:

Lown-onespacetotheleftandonespacedown

Rown–onespacetotherightandonespacedown

Lup–onespacetotheleftandonespaceup

Rup–onespacetotherightandonespaceup

Rown: Lup:

Thesemovescanbecombinedtoreachdifferentpositionsonaboard.Forexampleona

3x3board,twoRownswouldmovethepiecefromthetopleftcornertothebottomright

corner:

Start: After1Rown: After2Rowns:

Inparts(a)and(b),onthediagramalongsidethequestion,showwherethecounterwillbe

afterthemovesgiven.Ineachcasethestartingpositionisshownbythegreycircle.

(a) aRupfollowedbyaLup diagramfor(a)

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(b) 2LownsfollowedbyaLup

Inparts(c)and(d)statethemovewhichistheoppositeofthegivenmove.

(c) Lup

(d) Rown

(e) ListasequenceofTWOmoveswhichwouldtakeyoufromthelightcounterto

thedarkcounter.

[5 marks]

3e

3c

3d

diagramfor(b)

then

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4. Oneofthewaystocheckifanumberisdivisibleby11isbyalternatelysubtractingandadding

itsdigitsasfollows

Consider the digits in 121. Since 1 - 2 + 1 = 0 we can say that 121 is divisible by 11

For the number 16918, the digits combine as 1 - 6 + 9 - 1 + 8 = 11, which is a

multiple of 11 so 16918 is divisible by 11

However for 147, 1 - 4 + 7 = 4, which is not in the 11 times table, so 147 is NOT

divisible by 11.

Neither is 3276 as 3 - 2 + 7 - 6 = 2, which is also not a multiple of 11

Byapplyingthisprocessofsubtractingandadding,answerthefollowingquestions

(a) Showthat62557isdivisibleby11

(b) Can138251bedividedby11andgivenoremainder?

(GiveyouransweraseitherYesorNo)

(c) Inthenumber74n130whatvaluedoesthedigitn

havetobe,ifthenumberisdivisibleby11?

(d) Inthenumber83a4b9,thedigitbisthreetimes

digita.Findvaluesforaandbifthenumberis

divisibleby11.

[5 marks]

4a

4b

4d

4c n=

b=

a=

5. Theancient“Viva”tribeofCentralAmericacombinednumbersinaparticularway.Their

methodusedthefollowingsymbol(.)andworkedlikethis

a ( . ) b = 5 times a plus b ( ie 5 x a + b )

s o 2 ( . ) 7 = 5 x 2 + 7 = 1 7 and 9 ( . ) 3 = 5 x 9 + 3 = 4 8

WorkoutwhattheanswerwouldbefortheseVivacalculations

(a) 3(.)4=

(b) 5(.)3=

Workoutwhatvaluetheletterwouldhavetohavetogivethefollowingresultsinthese

Vivaproblems

(c) Ifm(.)2=37,whatnumbermustmbe?

(d) Ifn(.)4=8(.)n,whatnumbermustnbe?

[5 marks]

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5d

5c

5a

5b

m=

n=

6. Whenaballisdropped,therelationshipbetweenthespeed(S)oftheballinm/sand

thedistance(d)inmithasfallenis

S 2 = 2 0 x d where S 2 = S x S

So after dropping 5m, S 2 = 20 x 5 = 100 and therefore the speed would be 10m/s

as 10 x 10 = 100.

(a) Findthespeedwhentheballhasdropped45m

(b) Whenthespeedis40m/s,howfarhasthe

ballfallen?

(c) Howmuchfurtherwouldtheballhavetofall

tobetravellingattwicethespeedthattheball

hadinpart(b)?

[5 marks]

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6a

6b

6c

m/s

m

m

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7. Ina“prod sequence”,anumberinthesequenceistheproductofsomeoftheprevious

terms.So in a “2 prod sequence” we start with 2 numbers and then each number

that follows after is the product of the previous 2 numbers.

For example 1, 2, 2, 4, 8 ……..

Similarly, where we start with the three numbers 1, 2 and 3 then a “3 prod sequence”

would be

1, 2, 3, 6, 36, 648 ……

Fillinthemissingnumbersinthefollowingsequences

(a)2prodsequence2,3,_____,_____

(b)3prodsequence2,3,3,_____,_____

(c)4prodsequence2,3,5,_____,210

(d)2prodsequence

2,_____,_____,_____,500

(e)4prodsequence

2,2,_____,7,_____,3528

[10 marks]

7a

7b

7c

7d

7e

then then

and

,

,

8. Followingindependence,thenewcountryofOldhalliaistryingtodesignitsnationalflag.

Thegovernmentdecidethattheirflagwillconsistoftwocolours.

Forexample:

becauseinternationalrulesstatethattheflagmaynotbeallonecolourandtherulesalso

statethattheyareonlyallowedtopickfromacertainnumberofcolours.

The country always chooses what colour to put in the left section first. When

calculating how many possible choices that they have for their flag; the following

system is adopted. So with seven possible colours the choices would be

Which gives a total number of possible flags = 7 x 6 = 42

(a) Howmanypossibleflagsarethereiftheycan

choosefrom10colours?

(b) Ifthereareatotalof12flags,howmany

coloursdidtheyhavetochoosefrom?

8a

8b

Blue Green

Any of the 7 coloursAny of the 6

remaining colours

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(c) Iftheinternationalruleswererelaxedtoallowflags

ofallonecolouraswell,howmanypossibleflags

wouldtherebechoosingfrom6colours?

If the country wants to make a flag with three colours, the rules state that

they may not have two sections of the same colour next to each other.

If there were 4 choices of colour the calculation would be:

which gives a total number of possible flags = 4 x 3 x 3 = 36

(d) Howmanyflagsarethereiftheycanchoose

from10colours?

(e) Ifthereareatotalof150possibleflags,how

manycoloursdidtheyhavetochoosefrom?

[10 marks]

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8c

8e

8d

Any of the 4 coloursAny of the 3

remaining colours

Any of the 3 colours

which are different

from the middle

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9. HereisadiagramofarectanglewithcornerslabelledABCD.

A B

D C

Whentherectangleisreflectedabouttheverticallineshown,thecornersendupinthe

positionsindicatedonthepicturebelow.We will call this change M1.

B A

C D

Iftheoriginalrectangleisreflectedaboutthehorizontallineshown,thecornersendupinthe

positionsonthepictureunderneath.This change will be called M2.

D C

A B

Iftheorginalrectangleisrotatedthrough180°aboutitscentreOthenthecornersendupin

thepositionsonthepicturebelow.This change will be known as R.

C D

B A

M1, M2 and R are known as transformations of the original rectangle

(a) Inthespacebelow,drawandlabeltherectangleafterperforming

transformationRfollowedbyM1on the original rectangle.

O

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(b) Inthespacebelow,drawandlabeltherectangleafterperformingtransformation

M1followedbyM2on the original rectangle.

(c) Usingyouranswerstoparts(a)and(b)andanyotherdiagramsyoumaywish

todraw,complete the table below,showinghowthe original rectangle is

changedwhenapairoftransformationsareperformed.

RM1M2

RI

M1 I

M2 I

whereIindicatesthatapairoftransformationshavetakenyoubacktotheoriginalrectangle

(d) Eitherbyusingyourtableorotherwise,find

thesingletransformationwhichwouldtakeyou

backtothestarthavingperformedM1thenM2?

(e) Eitherbyusingyourtableorotherwise,find

thesingletransformationwhichwouldtakeyou

backtothestartafterhavingperformedM1thenR?

[10 marks]

9d

9e

2ndtransformation

1stt

rans

form

atio

n

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10.Olliethesportscoachwantedtotakegroupsofchildrenandtheirparentstoseearangeof

sportingfixtures.Whenhemadeenquiriesabouttheticketstherewerealwaystwotypesof

ticketavailable,oneforadultsandoneforchildren.

(a) ForfootballatRoversgroundanadultticket

costs£15andachild’sticketcosts£4.

Findthetotalcostfor20adultsand12children.

(b) Forarugbymatch,30adultsand16children

paid£408toseethegame.Achild’sticketcost£3.

Workoutthecostofoneadultticket.

(c) 50adultsand40childrenwenttothevelodrome

forthecyclingchampionships.Thetotalbillforthe

ticketswas£760.Anadultticketcostthreetimes

asmuchasachild’sticket.Workoutthecostof

anadultticket.

(d) Attheswimmingtournamentachild’sticket

cost£6lessthananadultticket.Thetotalcost

oftheticketsfor25adultsand30childrenwas

£370.Workoutthecostofonechildticket.

[10 marks]

10b

10c

10d

10a £

£

£

£

11. WhenIthrowagreendiceandareddicetogetherthereare36numberoutcomes.Note that

green1, red2 and green2, red1 are different outcomes.

Youmayfindithelpfultocompletethegridsbelowtohelpyou,althoughcompletingthegrids

doesnotgainextramarks.Thegridswillbeusefulinansweringthequestionsbelow

(a)Howmanysumsofthetwonumbersarefactorsof24?

(b)Howmanyoftheproductsareeven?

(c)Howmanyofthedifferencesareprimenumbers (remember1isnotaprimenumber)

(d)Forwhatfractionofthethrowsisthegreenscore greaterthantheredscore?

(e)Whatistheprobabilitythatthesumofthetwonumbers ismorethansix(6)?

[10 marks]

Page15

The End

11e

11d

11c

11b

11a

Red

Sum (+) 1 2 3 4 5 6

1 2 4 7

24 7

Green3 4 6 7 9

4 6 9

5 7 9 11

6 7 8 9 11

Red

Difference (-) 1 2 3 4 5 6

1 0 1 4 5

20 3

Green3 0 1 3

4 0

5 43 1 0 1

6 4 1 0

Red

Product (x) 1 2 3 4 5 6

1 1 2 3 5 6

24 6 10

Green3 3 91518

4 4 16

5 10 15

66 36

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