text answers index

1 Number properties and operations

1 02 a 10 b 100 c 10003 a 99 b 999 c 99994 05 20196 4and25

7 a 11 b 6 c 12 d 36 e 5 f 15 g 2 h 1 i 99 j 1 k 2 l 128 a 3 b 4 c 2

d 5 e 2 f 4 g 4 h 1 i 19 12cards,with$3leftover.

10 45000km

PUZZLE (page 5)

Nines and sevens 1

9× 9 + 9 + 9 + 9

9or

999 - 999

277

+ 77

+ (7 + 7)×7

3 7×7÷(.7×.7)

EXERCISE 1.01 (page 4)


1 a 1,2,5and10 b 1,2,3,6,9and18 c 1,3and9 d 1,2,3,6,7,14,21and42 e 1

f 1,2,3,4,5,6,10,12,15,20,30and60

g 1and13 h 1and292 a 2,4,6,8,10,12,…

b 10,20,30,40,50,60,… c 5,10,15,20,25,30,… d 1,2,3,4,5,6,… e 8,16,24,32,40,48,… f 11,22,33,44,55,66,… g 75,150,225,300,375,450,… h 41,82,123,164,205,246,…3 a 10 b 30 c 12 d 304 a 2 b 3 c 16 d 155 a False(1doesnot) b True c True d True e True f False

6 a 45°isafactorof360°,itgoesinexactly.

b No,because50°isnotafactorof360°.

7 True8 a Yes b No c No9 16

10 A$6stampwouldbebest.6isthehighestcommonfactorof18and48,andwouldinvolveusingfewerstampsthan$2or$3stamps.

11 252012 a Wednesday

b Monday.35isamultipleof35butisnotamultipleof10.

c Tuesday.Therecouldnothavebeenoneatlassoldbecausethenthemoneyleftover($25)isnotamultipleof$10.Therecouldnothavebeentwoatlasessoldbecausethenthecost($70)exceedsthetakingsof$60.

d Thursday

INvEstIgatIoN

(page 7)

Perfect numbers1 1isnotaperfectnumber.Ithasno

factorsexceptforitself.2 28.Factorsof28,excludingitself,

are1,2,4,7and14;1+2+4+7+14=28.

3 Factorsof496,excludingitself,are1,2,4,8,16,31,62,124and248;1+2+4+8+16+31+62+124+248=496.

PUZZLE (page 7)

What am I? 441


1 b,c,d,f,h,j 2 91isnotprimebecause7×13=913 114 85 116 17 1018 Yes

PUZZLE (page 9)

Prime Boeings 1 727,757,7872 707=7×101 717=3×239 737=11×67 747=3×3×83 767=13×59 777=3×7×37

498 answers

1 a -5 b 8 c -13 d 98 e 1 f 02 Twizel,Alexandra,Christchurch,

Nelson,Blenheim,Dunedin,Greymouth,MilfordSound

3 {-10,-8,-5,-3,0,6,7}4 a 6>3 b -9<4 c 8>-2 d -3<1 e 7>-8 f -2<1 g -4>-10 h -5<6 i -11<-8 j 0>-155 a False b True c False d True6 a=-5,b=-3,c=-9, d=0,e=27 a -21°C b 7°C

PUZZLE (page 11)

aD−BC 1 ‘BeforeChrist’2 ‘AnnoDomini’,whichisLatinfor

‘YearofourLord’.3 14ADifweassumethereisaYear

0,otherwise15AD.4 79yearsifweassumethereisa

Year0,otherwise78years.

PUZZLE (page 11)

Consecutive integersEight integers(Otheranswersarepossible.)

Nine integers(Otheranswersarepossible.)

PUZZLE (page 13)

Magic square

Otheranswersarepossible.


2–3 –1 –4

0 3 1–2

0 –2 2–3 4 –4

–1 1 3


1 a 1 b -9 c -1 d -9 e -5 f -18 g -2 h 7 i 3 j -8 k -6 l -9 m 15 n 3 o -10 p -2 q -3 r 79 s -4 t 42 a 139 b 585 c -858 d -4313 e 937 f 115 g 8431

1 a -5 b 10 c -1 d 2 e -7 f -4 g -5 h -5

2 a -4 b -20 c 5 d -9 e -3 f -9 g 1 h -31

PUZZLE (page 17)

the credit card quartetx=7

1 –3 –1 3–1 3 1 –3

3 –1 –3 1–3 1 3 –1


1 a -11 b 3 c 8 d 4 e -4 f -8 g -8 h -1 i 38 j 1 k -10 l 9 m -17 n 2 o 6 p -13 q 2 r -9 s 10 t -80

2 a -3895 b -1219 c -80804 d 13931 e -369 f 360 g 2561

PUZZLE (page 14)

the missing dollar

Thereisnomissingdollar.The$2stolenbytheportershouldbesubtractedfrom$27toget$25,notaddedtowhatthethreemenhavepaid.Themenhaveactuallyonlypaid$27aftertherefund-theporterhas$2ofthisamount,andtheother$25iswiththemanager.Asfarastheoriginal$30isconcerned,thethreemenhave$3,theporterhas$2andthemanagerhas$25.


1 a -40 b -14 c 24 d 200 e 16 f -40 g -36 h -282 a 24 b -10 c 160 d 8 e -30 f 64 g -1 h 24

PUZZLE (page 16)

How many, how much? 1 $742 373 $518



1 a -12 b 7 c 4 d 56 e -14 f -1 g 31 h 24 i -40 j 422 a -47 b 12 c -3 d -4 e 6 f 8 g 11 h -4 i 9 j 4


1 a Overdrawn b Anoverdraftof$24.

27-51=-242 4−−9=13;thatis,thetemperature

hasrisenby13°C.3 a -78 b -97 c 3 d -12 e 134 a 5 b U25 13°C6 a 57yearsold b 39yearsold c 32-and-a-halfyears

d 40ADifweassumethereisaYear0,otherwise41AD.

7 a 10×4+5×-7 b 58 a i 4 ii 70 b i 10 ii 66

PUZZLE (page 19)

Why did Rupert take some sausages to the hairdressers?

Seepage508foranswer.

499Answers

PUZZLE (page 20)

Jumbled integers

3 0 -4

-3 1

-1 -2 2

3 -1 -2

-4 0

1 -3 2

-4 2 0

-1 1

3 -2 -3

-3 -1 1

2 0

-2 3 -4


1 a 84 b 56 c 42

d 133 e (-1)3 f (-7)6

2 a 3×3×3×3 b 10×10×10 c -3×-3×-3×-3×-3×-3×-3 d -2×-2×-2×-2×-2×-2×-2×-23 a 9 b 196 c 512 d 512 e 1296 f 16384 g 9 h 4054 a 16 b -32 c 729 d 1 e 625 f -1610515 a 9.61 b 117.649 c 58.0644 d 244.140625 e 0.0625 f 0.01446 a 17 b 768 c 80 d 0 e 400 f 11147 a False b True c True d False e False f True8 a 32768 b 1024 c 81 d 8 e 4096 f 7299 a 312=311×3=177147×3=531441 b 310=311÷3=177147÷3=59049

10 6411 a 64 b 60

7 a 7 b 78 a 1000 b 4 c 5 d 101 e 79 a False b True

PUZZLE (page 22)

What happens at the end? 1


1 a 4 b 9 c 7 d 12 e 1 f 202 a 6 b 9 c 7 d 2 e 11 f 293 a 2.5 b 2.36 c 0.31 d 0.74 815 26 a 1.732 b 2.999824 c 3

PUZZLE (page 23)

Dominoes1 282 55


2 Decimals

1 a 47.5 b 0.8 c 0.24 d 0.096 e 0.012 a 8tenthsand5hundredths b 9tenths

c 1one,3tenthsand8hundredths

d 5hundredths e 7tenthsand3thousandths

3 a a=8.4,b=10.9,c=11.5 b a=3.25,b=3.47,c=3.11 c a=4.94,b=5.18,c=5.05 d a=8.85,b=9.075, c=8.9254 a 13.8 b 0.7 c 0.91 d 0.06 e 1.3 f 0.23865 0.037,0.04,0.082,0.403,0.86 a 1.24m b Kim,Cameron,Lee

c ChrisSmith,LeeBrown,TracyEvans,PatO’Sullivan


1 12.7322 20.3723 710.80434 115.0535 0.1016

6 17.457 1.588 431.8149 0.67

10 0.0967


1 a 9.4 b 2.8 c 7.51 d 10 e 7.93 f 48.7 g 0.583 h 518.86 i 2.019 j 0.39682 a 0.3 b 3.28 c 1.55 d 0.4 e 4.5 f 7.61 g 0.007 h 2.03 i 11.89 j 6.3

500 answers

3

+ 4.9 1.5 1.78 13.8

5.1 10 6.6 6.88 18.9

4.05 8.95 5.55 5.83 17.85

1.44 6.34 2.94 3.22 15.24

1 $22.502 $44.403 Three;paywitha$5noteandthe

changeisa$1coinandtwo20-centcoins.

4 $18.555 506 $1.757 268 $9.809 2.5litres

10 50.35m11 a $46.94 b $3.1012 $138.5013 $14.1514 $9.5515 $294416 0.56mor56cm17 a $125.90 b 18.9m


1 3.567082 102.343 1.24 0.725 66.672

6 1027 115.468028 1.12899 0.015

10 0.9376


1 0.62 0.333 11.24 0.15 0.726 5.5447 0.0088 0.034

9 0.07810 4.3211 0.0050512 0.0005413 0.0002814 307.3615 35.7555

16× 0.4 0.02 0.1

0.5 0.2 0.01 0.05

0.6 0.24 0.012 0.06

0.03 0.012 0.0006 0.003


1 22.32 80.63 2.74 135 46 7507 2.38 5.95

9 61.510 280011 50012 1500013 25014 4.0215 0.4

PUZZLE (page 29)

Correcting a wrong answer

812.5


18 9.02519 a $232.95 b $79.0520 $27.4521 $2.4022 a CompanyY b 10cents23 Profitof$152.24 Thetakings($1847)shouldbea

multipleof$7.50,buttheyarenot.

25 $66826 $1.1527 a Smaller b Larger

c Whenmultiplyingbyadecimalgreaterthan1,numbersbecomelarger.

Whenmultiplyingbyadecimallessthan1,numbersbecomesmaller.

3 Fractions


1 a34

b

18

c

512

2 a34

b

25

c12

d

730

e

8100

f

490

3 a One-quarter b Three-tenths c Four-fifths d Thirty-one-thousandths e One-eightieth f Eleven-fortieths

4

5

6 Ron

7

613

islessthan12

because

6islessthanhalfof13.

8

38

, 48

, 58

, 34

, 35

, 45

9

320

10 a

36

or 12

b

411

11 a

25

b

35

12

3264

or12

13

3081

or 1027

14 One-tenth

15 a

35

b47

16

37

, 12

, 59

, 35{ }

17

520

, 519

, 514

, 920

, 919{ }

18 HotelA

501Answers

PUZZLE (page 35)

the heaviest money boxVernon

PUZZLE (page 36)

a cross within a square

15

1 a32

b34

c

23

d

65

e107

f41

or4

g

18

h21

or2 i

115

j

1100

2 a Yes b No c Yes d No e Yes f No g Yes h Yes i No j No


1 a14

b

23

c

23

d32

e

45

f

45

g

23

h

13

i 3

j 1 k

129

l

13

m

539

n

736

o

337

2 a 4 b 25 c 8 d 25 e 2 f 13

3

45

4 One-quarterofaslabeach.

5 a

23

b

13

6

710

7

932

8

38

9 a

724

;7and24have

nocommonfactors.

b

13

10

13


1 a

13

b

815

c

110

d

815

2 a

58

b34

c

23

d

56

e32

3 a

572

b

845

c

320

PUZZLE (page 39) Number nine cake

or

or

01

2

3

4 56

7

8

9 01

2

3

4 56

7

8

9

01

2

3

4 56

7

8

9


1

120

2 23 $64 $2.405 a 15 b 24 c 35

6

314

7 73(Assumingitisnotaleap-year.)8 Colinbecausehesaves$37.50,

whileDebbieonlysaves$36.9 78

10 4300

11

715

12 413 327m2


INvEstIgatIoN (page 42)

Egyptian fractions

1 a

56

b

772

c1112

2 a

12

+ 16

b

13

+ 14

c

15

+ 18

3 a

12

+ 13

+ 18

+ 1168

or

12

+ 14

+ 18

+ 116

+ 156

+ 1112

b 12

14

116

164

1512

1601

19616

119 232

138 464

+ + + + + + + + ++ + +176 928

1153 856

1307 712

4 a

13

+ 15

b14

+ 17

502 answers

1 a

49

b2021

c

38

d

1825

e

932

f12

2 a 8 b

503

c

13

d

112

3 a

49

b

1615

c

109

d

19

4 50glasses

1 a

1320

b

720

2

920

3 a14

b34

4

920

5

415

6

320

7

310

8

56

9 48000km

12 2

3bottles

25 1

4hours

316 1

2chairs

418 2

11;thatis,19candlesbecause

18wouldnotbeenough.

51 3

10cups

63 1

12packets

7 140litres

87 7

10pounds

9

19

10 7

113 3

8tanks

12 a 2040 b 2300



1 a

23

b47

c

79

d

2931

e34

f

25

g14

h 1 i 1

j

1513

2 a12

b

631

c14

d

13

e

56

f

310

g 3 h

15

3 a

95

b 1

PUZZLE (page 44)

apples, oranges and peaches1 Oranges,peaches,apples2 21orangesweighthesameas

10apples.


1 a1112

b

1115

c1112

d

710

e

2330

f

133

2 a

56

b

463

c14

d

112

e

1120

f

524

3 a

3130

b6744



1 a1 2

3 b

3 3

4 c

3 5

11

d 7 e10 4

9 f

1 1

44

g 0 h8 11

12 i

2 2

31

j20 1

2

2 a72

b194

c

1810

or

95

d

395

e667

f

416

g

4120

h

198

i

413100

j

2075


1 a3 1

8 b

1 1

4 c

5 2

3

2 a 2 b

3554

3 a 4 b6 4

5 c 10

d4 1

6 e

5 11

20

4 a2 2

3 b

2324

c1 9

10

57 19

24

6 a 1 b2 1

5 c

4 4

15

d3 2

3 e

8 1

12 f

41 1

6



1 a 0.5 b 0.375 c 0. 6 d 0.7 2 e 0.6 f 0.85 g 0.16 h 0.015 i 0.48 j 0.11 6 k 0. 2 7 l 0.0 6 m 0.584 n 0.25 o

0.142 85 7

2 a 0.7 b 0.83 c 0.792 d 0.04 e 0.093 f 0.00465

3 a

35

b47

c

73100

4

37

, 49

, 511

, 47100

, 12

5 Yes,

711

= 0. 6 3 isbetween

35

= 0.6

and

913

= 0. 692 30 7.

6 Gerald7 CinemaAhasthebetterdeal.Its

ticketsaresellingfor

35

=0.6of

theusualprice,whereasCinema

B’sticketsaresellingfor

58

=0.625oftheusualprice.

8 a

811

= 0. 7 2

b 0.75

503Answers

112

2

110

5

29100

6

3950


3

45

414

7

925

8

1720

9

18

10

43125

1 a 25% b 50% c 80% d 37.5% e 85% f 47% g 68% h 31.6% i 225%

j266 2

3%

2 a 20% b 10% c 85% d 41% e 9% f 4% g 12.5% h 99% i 130% j 0.6%3 58. 3%

15 a 156.25% b Lessthan100%16 a 16.2 b 12.5

c Toexpressthevalueasapercentage.

17 a

AnitaAdams 56.25%

NeilArmstrong 52.5%

RebeccaBarton 88.75%

SimonBolivar 81.25%

b Dividingby80andthenmultiplyingby100isthesame

asmultiplyingby

10080

,whichis1.25.

4 Percentages



1 a12

b

15

c

35

d

120

e

1920

f

925

g

225

h

1725

i

65

j

38

2 a 0.4 b 0.49 c 0.06 d 0.53 e 0.01 f 1.5 g 0.319 h 0.125 i 0.028 j 0.0006

3

130


1 a 55% b 25%2 a 63% b 1% c 12.5%3 60%4 82%5 20%6 50%7 75%8 91%9 a 84.8% b 15.2%

10 30%11 57.3%12 Whiteware13 KowhaiSchool14 68.8%


1 16.6%2 2.2%3 OriginalRecipeis10.8%fatby

weight,ExtraCrispyis12.4%fatbyweight.ExtraCrispyisfattier.

4 FrenchFries5 Filet-o-Fish6 Yes,aQuarterPounderwith

cheeseis15.0%fatbyweight;withoutcheeseitisonly12.8%fatbyweight.

7 39%


1 962 273 9km4 3755 Yes,Brucegetsabonusbecause

6.7%(1dp)ofhiscallsresultinapurchase,whichishigherthan5%.

6 18days7 490008 a 8 b 429 a $75.00 b $15.50 c $40.94

10 16.5m2

11 $1057.5012 $55000013 ForZap,15%of1125mL=

168.75mLisactive. ForSharp,22%of750mL=

165mLisactive. ThecostpermLofactive

ingredientforZapis$3.50÷168.75=2.074cents.

ThecostpermLofactiveingredientforSharpis$3.00÷165=1.818cents.

SharpisthebestbuybecauseeachmLofactiveingredientcostsless.

14 a $2600 b $7300 c $17500

1513 1

3%


spaceship Earth

1 509104200km2

2 147640200km2

3 3.48%4 5.16%

1 a 15 b 36 c 19 d 2356 e 22 a $367.50 b $194.99 c 16cents d 24cents e $5.133 a 90litres b 22.8g c $400 d 6.346kmor6346m e 620g


504 answers

1 a $46 b 8125litres c 900m2 d 532.48kg2 a $450 b 22litres c 6208ha d 61320kg3 $14.58perhour4 a 87 b 4935 a $37.39 b $336.536 3857 a $96.75 b $51.75 c $17.258 $496.08


Depreciation

1 20%2 20%3 a $408 b $768 c $2102.404 $67.505 $31.476 30%7 14%8 3.3%9 20%

10 19%11 37.5%12 15%




the house market

1 Ittakes10years.SeethespreadsheetThe house market Answers.xls.ThisisavailableontheBeta Mathematics WorkbookcompanionCD,orcanbedownloadedfromwww.mathematics.co.nz.

2 No.Regardlessofthestartingvalue,itwilltake10yearstodoubleinvalueifthepricesincreaseby8%eachyear.ThiscanbeseenbyputtingdifferentvaluesincellA3inthespreadsheet.

13 $1357.50ifpayinginnotesandcoins;otherwise$1357.55.

14 $19.1715 30%16 6.1%17 30%18 Iona19 $80727220 a 25% b 20%

c Thefirst(ororiginal)numberisdifferentfromoneyeartothenext.


1 a 30cents b $90 c $40.06 d $2.09 e $442.502 a $270 b $13.50

c $259.88(or$259.90,roundingforcash)

d $6.30 e $3.99(or$4,roundingforcash)3 a $18 b $1624 $11.705 $106 $506.257 $11.658 17.4cents9 a $2.40 b $19.20

10 a $184.80 b $23.1011 $550

PUZZLE (page 68)

the coloured casino tokens1 52 Max3 Max4 7


1 a $1000 b $60 c $129600 d $450 e $11252 $162003 $604 a $2400 b $64005 $465756 a $80 b $800 c 6% d 1.5years


1 a 60 b 80 c 35002 a 900m b $400 c 550g3 2404 120

5

Diningroomfurniture

$750

TVset $399

Washingmachine $600

6 65007 12m2

8 60009 a 80 b 28

10 40011 27

505Answers

1 a 2:5 b 18:11 c 4:75 d 6:5 e 8:52 a 2:5 b 9:8 c 1:4 d 3:4 e 2:1 f 5:73 a 3:2 b 3:8 c 1:3 d 3:4 e 2:3 f 3:54 a 3:5 b 4:5 c 1:2 d 8:75 a 7:3 b 5:1 c 5:4 d 3:1 e 3:8 f 1:56 a i F ii H iii J ivC b F7 Lesssweet8 Theratiosaredifferent.Yellow

volume:orangevolume=1:8.Yellowsurfacearea:orangesurfacearea=1:4.

9 3:410 4:1111 7:312 4:113 a 2:3 b 2:114 3:715 a 7:17 b 2 c More d 64:25

e Theratiointeriorcabins:balconycabinscannotbesimplifiedbecause375and748havenocommonfactor.

16 1:1917 Theratiowouldchangeto

15blue:8yellow.18 a 10kumaras

b 180carrots;300onions;90carrotsand150onions(Otheranswersarepossible.)

1 a $20,$30 b 15kg,25kg c 22cm,66cm d $8,$4 e 500people,2000people f $40.80,$27.202 a 5hours b 20hours3 284 $805 240mLofoil,960mLofpetrol6 1957 258 $72009 190

10 $200011 Hemi$100andIan$50.12 a Jane:Sarah=2:3 b Jane$24,Sarah$3613 Deirdreshouldget$2880,Eva

shouldget$4320.14 Ngashouldget$105000,Martin

shouldget$63000.15 52

5 Ratios and rates



1

49

227

3

89

4 30%

5 4%

6

512

7 73%8 1021mL(to

thenearestmL)

EXERCISE 5.03 (page 77) PUZZLE (page 78)

Bath temperature

1

38

2 30°C


1 a $50 b $2502 a 18litres b 400km3 a 75km b 450km c 2hours d 75km/h4 a 75 b 16minutes c 900 d 900piecesperhour5 a $60 b $165 c 7.5m6 a 3000litres/h b 0.8 3 litres/second c 3000000mL/h

d 833mL/second(tothenearestwholenumber)

7 1hour15minutes8 a $75/m2 b $67500 c 640m2

9 a 281 b 6.14runsperover

10 a 5minutes b 420litres11 6hours40minutes12 a Henry:48m2/h;

Rose:54m2/h b Rose13

8 1

3ha


Colour-mixing (RgB)1 0:0:02 a 1:0:1 b 0:1:13 Ingreythethreecolourscontribute

equally-itwillbesomewherebetweenabout200:200:200,whichisaverylightgrey(almostwhite),and50:50:50,whichisaverydarkgrey(almostblack).

4 a Lilac,mauve,lavender,purple(Thedescriptioncanvary.)

b 135:108:153;lilac,mauve,lavender,purple(Thedescriptioncanvary.)

c Thecoloursarethesame‘tint’butthesecondoneisdarker.

d 15:12:17;thecolourbecomesverydark-almostblack.

5 a Aflamingopinkcolour. b Alightgreen. c Arichpurple.6 Answerswillvary.


1 $19.902 21minutes3 $1.604 8days5 25minutes6 $135007 75skeins8 84minutes

921 1

3bags

10 1second11 12days

12 a 24jars b 10513 a 50

minutes b 72014 $4.8015 16hours16 10.8minutes17 72minutes18 9.6kg19 113minutes20 2.4days

PUZZLE (page 84)

thinking RatIonally

1 hxy

hours

2 xzk

workers

506 answers

6 Approximations, standard form and estimation

1 a 4 b 6 c 5 d 2 e 3 f 4 g 6 h 6 i 5 j 52 a 1 b 2 c 3 d 6 e 5 f 53 a 2 b 6 c 3 d 4 e 4 f 4 g 6 h 7

4 a 2 b 4

c Toshowtheexactbalanceindollarsandcents.

5 a 3b Torecorditwiththesame

degreeofaccuracyastheotherreadings.

PUZZLE (page 87)

a century of Po Boxes21



1 a 7 b 7 c 9 d 30 e 30 f 500002 a 7.9 b 5.6 c 58 d 350 e 57000 f 0.613 a 6.33 b 17.9 c 45300 d 14.1 e 0.128 f 5594 a 8.358 b 74660 c 63.73 d 0.004107 e 86.68 f 7.4835 a 13.9 b 1.45 c 1.445 d 2.999 e 3 f 0.67 g 0.1556 h 49

6

Number 1 sf 2 sf 3 sf 4 sf

a 49.285 50 49 49.3 49.29

b0.18537 0.2 0.19 0.185 0.1854

c 311.92 300 310 312 311.9

d673800 700000 670000 674000 673800

e 498905 500000 500000 499000 498900

7 a 3.7m b 2.1m c 4.7m d 6.9m8 66.59 a i $73.40 ii $5.40 iii $7.00 iv $6.90

b i Lowestprice$67.36,highestprice$67.45

ii Lowestprice$5.06,highestprice$5.15

iii Lowestprice$7.96,highestprice$8.05

iv Lowestprice$226.76,highestprice$226.85


1 a 63 b 630 c 6300 d 630002 a 18.11 b 181.1 c 1811 d 181103 a 4.95 b 49.5 c 495 d 49504 a 0.073 b 0.73 c 7.3 d 735 a 740 b 89120 c 180 d 1445.6 e 13 f 923000


1 5.07×101

2 5.748×102

3 6.310×102

4 8.92×102

5 4.9×104

6 6.822×101

7 9.1×100

8 8.32×105

9 1.2142×101

10 8.711×100

11 1.5×106

12 8×103

13 1.8×100

14 9.2×101

15 3.133×101

16 5.11186×102

17 2.567×107

18 4.5×1011

19 2.6×100

20 3.43×105


1 6102 18003 934 4200005 73.36 5.657 80108 78000

9 16610 26.4511 3.15812 40900000013 5033014 11170015 6000


1 a 1.0×106

b 1.0×109

c 1.0×1012

2 a $8972000000000=8.972×1012

b 302800000=3.028×108

c $29640=2.964×104

3 1.29533×109people;9.56098×106km2

4 384000km5 1516500000000000km36 4.6×109years7 2.6×104lightyears8 2200000lightyears9 a 5.974×1024kg b 5.974×1021tonnes

10 a 6.671×1021;3.546×1012

b 1881169920.Thisgivesthenumberofwaysinwhichapuzzlecanbereflected,rotated,andsoon,toessentiallygivethesamepuzzle.

507Answers

PUZZLE (page 93)

an awfully long time

1 60secondsinaminute,60minutesinanhour,24hoursinaday,100yearsinacentury

2 Everyfourthyearisaleapyear.365.25istheaverage(mean)of365,365,365,366.

i.e.365 + 365 + 365 + 366

4= 1461

4= 365.25

3 3.15576×109

4 Noteveryfourthyearisaleapyear-forexample2100,2200and2300willnotbe.

InsomeyearsanextrasecondisaddedtothetimegloballytoallowforvariationintheEarth’srotationaroundthesun.

1 a 0.01 b 0.0001 c 0.1 d 0.00001 e 0.001 f 0.00000012 a 0.000001 b 0.00000001

1 3×4=122 100×5=5003 1000×4=40004 21÷3=75 a 56=7×8 b 80=2×40 c 42=6×7 d 100=20×5 e 45=3×15 f 320=40×8

6 a80 = 80

1

b4 = 200

50

c50 = 100

2

7 a9 = 12×3

4

b80 = 40× 4

28 a 159.29 b 581.378 c 153.419306 d 8.206198812 e 20.903184 f 4.59815239



1 a 0.071 b 0.0058 c 0.63 d 0.0000422 a 0.143 b 0.0368 c 0.00801 d 0.00068 e 0.0244 f 0.016453 a 0.000000008171 b 0.0000000205 c 0.00074033 d 0.00604 a 0.000000000000512 b 0.000000000006 c 0.000000091135 a 0.000000000042 b 0.00000000000128 c 0.000000000064 d 0.000000000000000064


1 3.5×10-3

2 1.8×10-2

3 7.1×10-1

4 5.6×10-6

5 1.4×10-3

6 7.5×10−8

7 1.1×10-12

8 1.013×10-1

9 9.8×10-1

10 6.639×10-1

11 5.611×10-3

12 6.8609×10-7

13 2×10-1

14 8×10-7

15 1.38×10-4


1 0.0000000000000000000000272g2 1.0352×10-13light-years3 0.00016m4 0.000000001m5 1.6×10-3watts6 a 0.0000026kg b 0.0026g c 1000000


1 a 4600 b 0.0046 c 189000 d 0.0000189 e 0.03552 f 99100 g 7034000 h 0.00008116 i 6.667 j 0.53012 a 1.8×104

b 1.8×10-4

c 6.73×102

d 6.73×10-1

e 5.44×10-6

f 5.44×101

g 1.92×100

h 9.3×10-2

i 9.3×109

j 2.8×10-5


1 a 6 b 10 c 20 d 35 e 1 f 502 a 500 b 600 c 200 d 1200 e 900 f 10003 a 10 b 20 c 50 d 70 e 120 f 90



1 Itisunlikelytobeexact-itwouldbea‘guesstimate’.Itwouldbealmostimpossibletocountthatnumberofpeopleexactly.

2 a 40×20=$800 b $800×50=$400003 Each‘square’shapedblock(except

fortheonesattheback)has10rowsandabout15seatsperrow,soabout10×15=150seats.Thereare16blocks,buttoallowforthem‘taperingoff’towardthebackabetterestimatewouldbe14.Thetotalnumberofseatsatstagelevel≈150×14=2100.Theareaofseatsintheslopingsectionlooksaboutthesameastheareaatthestagelevel,orabitmore,sothetotalnumberofseatscouldbeabout4500.

4≈ 600

50= 12

5 39000÷300=130(Otheranswersarepossible.)

6 300×$5=$1500

508 answers

7 Thelahartook2hours8minutestoreachtheTangiwairoadbridge.Thisisabout2hours.Thedistancefromthecratertothebridgeisabout40km.Theaveragespeed=distance÷time=40km÷2h=20km/h.Beforerounding,theaveragespeed=39.4÷ 2.1 3 =18.46875km/h.

8 a 7.92/100×804≈0.08×800=64litres b Cost≈60×$1.60=$96≈$1009 a 300×0.2=60litres

b 60×$8=$480=$500,tothenearesthundred10 Oneapproachwouldbetolookata1cmby1cm

squareinthephoto.Eachsquarehasabout9×9≈80tiles.Thephotoisrectangularandmeasuresabout12cmby9cm=108cm2≈100cm2.Thereforethephotoshowsapproximately80×100=8000tiles.

11 400000×$1.60=$640000.12 Bothofthefirsttwojobsused1bottleper50m2,

approximately(

77916

≈ 80016

= 50 and

45910

≈ 50010

= 50).

Therefore,thenextjobmightneed22bottles

(

112850

≈ 110050

= 22).

13 a

3000295

≈ 3000300

= 10 and

2100295

≈ 2100300

= 7

b No,becauseeachestimateistoolow.

PUZZLE (page 19)

Why did Rupert take some sausages to the hairdressers?Hethoughthewasgoingtoabarberqueue.

1 4y2 3x

3 x2

4 abc

7 Formulae and substitution


5 2xy

6 2xy

7 6x

843

qp

9 6xy10 3xy


1 a x+5 b x-7 c x-20 d x-4 e 7x f 8x g 2x h x+18

i

x5

j x-3

k 6x+8 l 2x-4

2 a

x8

b x+8

c 8×x d x-8

e

8x

f x+x

g x×x h 8-x3 a Multiplyxby3,or3timesx.

b Add7tox,or7morethanx.c Subtract13fromy,or13less

thany.d Multiplyxby4andaddtwice

ytotheresult.e Multiplyxby3,andthentake

away1.f Dividexby6,orxdividedby

6.4 a x+2 b x+y c x-3 d x-t

5 a 2 b 9 c 6+x d 6-y6 a x-20 b x+15

c 2x d 12

xor x2

7 a 3d b d-6 c 100d8 a d−70 b

d2

c d+a9 a 5x b 20−3x

c x4

d 16x+7

10 55-4ycm

11 a V=x3

b I = PRT

100

c A h a b= +2

( )

d x = x1 + x2

2 e y=180-2x

f A bh= 12

g T=(n-2)×180


1 a 11 b 1 c 16 d 20 e 9 f 21 g 0 h 16 i 20 j 48 k 1 l 9 m 10 n 22 a 25 b 17 c 35 d 10 e 36 f 2


1 a 10 b 24 c 12 d 8 e 3 f 37 g 2 h 0 i 7.5 j 52 a 12 b -2 c -2 d 8 e -5 f -13 g -2 h -40 i 2 j 163 a 7 b -2 c 23 d -72 e 6 f -16 g -9 h 36 i -8 j 289


Hot cross buns1 242 pq3

4 85 p+q−2

509Answers

1 a

b 13c Wayne.Thecorrectpatternis

M=2T+1.d 161

2 a

b (C)P=5S+1 c 1013 15cm2

4 a Asuitcaseweighsabout20kg. b 660kg5 a $44 b $111

SeethespreadsheetExercise 7-06 Answers.xls.ThisisavailableontheBeta Mathematics WorkbookcompanionCD,orcanbedownloadedfromwww.mathematics.co.nz.1 a SeeWorksheet

‘Question1a’onthespreadsheet.

b SeeWorksheet‘Question1b’onthespreadsheet.

2 SeeWorksheet‘Question2’onthespreadsheet.

3 a h=71+2.9l b =71+2.9*A3

c SeeWorksheet‘Question3’onthespreadsheet.


6 a Thenumberofcups. b Thenumberofteabags. c 127 a $4275 b C=2n+35h+360 c C=3n

d Smartdrives:2×1000+35×6+360=2570

Cobbleco:3×1000=3000 Thecheapercompanyis

Smartdrives,andtheamountsavedis$430.

8 380g9 a i $5 ii $3.50 iii $3

b Thegraphapproachesalowestpossiblepriceof$2perbook.

10 16km11 a 37° b 45° c A3-iron. d n=1212 a 62.6kg b 76.6kg c 43.8kg d 48.7kgto60.8kg13 a 540000joules b 135000joules c four


Floor joists

1 (B),(A),(C)2 a 300mm b 467mm

3 D = 25 5x

3+ 2


8 Simplifying algebraic expressions

1 8cd2 6fg3 2ap4 8ab5 6pqr6 10def7 6pqr8 8cde9 6a

10 10q11 16pqr12 8abc13 pqr

1 6x2 4x3 9x4 7p5 x6 -2x7 30p8 7x9 4x

10 x11 6x+6y12 8x+313 4x+714 14x+3y15 2x+4y16 2x+3y17 -2x+3y18 3x-2y

1 a 10xb 20xc 3x-1d 7x+20

2 a 3p,5p(Otheranswerspossible.)

b x+y,5x+2y(Otheranswerspossible.)

c 3x-2,2x-6(Otheranswerspossible.)

3 a 8x+3y b 18c+38p4 a

b 12

a2 or a2

2

c 3a2

2


14 210mnq15 -2r16 45fgh17 -12xy18 -30pq19 xy20 -12x21 -21d22 30ab23 9pq24 24xyz25 -12pqr


19 16x-y20 3x-1921 5x-y22 11x+123 7x-224 -4x+725 -3x+1226 -7x+427 -6x-728 9p+5q29 14x-5y30 8x-9y31 11x+5y-1232 -8x-633 -x2+8x34 7x2-9x

35 -x2+6x-12a

a

a


510 answers

PUZZLE (page 117)

the 3x triangle

Otheranswersarepossible.

1 a

b

Number of squares (x)

Number of dots (d)

1 4

2 7

3 10

4 13

5 16

c 1 d 612 a

b (A)Numberofdots=2×numberofcircles-2 c d=2c-2 d 223 a b28

c

Height of shape (x)

Number of matchsticks (n)

1 12

2 20

3 28

4 36

5 44

d 8 e n=8x+44 a

Step (n) 1 2 3 4 5

Number of cubes (c) 1 3 5 7 9

b 2 c c=2n-15 a

Step (n) 1 2 3 4 5

Number of matchsticks (m) 6 9 12 15 18 b 3 c m=3n+3


x x–1 x+1

x–2

x+2

x–3

511Answers

6 a

Number of lamp-posts (l)

Number of flags (f)

1 0

2 4

3 8

4 12

5 16

6 20

b Thisruleonlyworksforl=1andl=5.Forexampleifyouusel=2,yougetf=1,notf=4.

c f=4(l−1)orf=4l−4 d 112 e 52

PUZZLE (page 120)

Fooled again1 102 n+(n-1)(n-2)(n-3)(n−4)

7 an 1 2 3 4 5 6

Number of squares (s) 1 4 7 10 13 16

b 3 c s=3n-28 a 8 b 12 c p=4s−4orp=4(s−1) d s=2 e 2889 a $250 b b=20w+250

10 Ateachstepsevenmatchsticksareaddedon. Atthebeginningtherewereninematchsticks,sotherearetwo

moreatthatstage. Theruleism=7h+2. Whenn=85,n=7×85+2=597.


1 r3

2 y4

3 p2

4 h6

5 6x2

6 2y3

7 6q2

8 3p2

9 8a2

10 6cd2

11 16qr2

12 8r3

13 4x3

14 16x2y15 6x4


1 a x9 b r3 c q5

d x6 e x8

2 a x3 b x3 c x2

d x e x5

3 a x8 b y15 c x8y12

d x2y2 e x18y6 f 1 g 14 a x4 b 1 c 2x2

d 05 p=66 k=9


1 a 12x7 b 18x3 c 16x3

d 10x5 e 20x4 f 96x10

g 30x7 h 72x9

2 a 2x4 b 2x c 7x3

d 2x2 e 2x3 f 5x2

2 g 4xy3 a 25x2 b 16x6 c 36x8

d 16x4y2 e 8x12

4 (E)simplifiesto4x.5 p=3,r=156 2xand10x7;4x3and5x5

(Otheranswersarepossible.)7 20y15

8 a=15,p=11


1 3y3

2 5x3 4y6

4 9x8

5 2x6 8y9

7 5x10

8 x8

9 10y50

10 xy3

11 7x3y2

12 6xy2


skid marks

1 Speed= 24× x

32 98.0km/h(1dp)3,5 SeethespreadsheetSkid mark Answers.xls.Thisisavailable

ontheBeta Mathematics WorkbookcompanionCD,orcanbedownloadedfromwww.mathematics.co.nz.

4

512 answers

9 Expanding and factorising

1 5x+152 5x-153 pq+4p4 pq-4p5 x2+7x

1 6(a+b)2 2(p-q)3 12(x+y)4 10(d-e)5 2(p+q-r)6 7(c-d+e)7 3(c-g-d)8 4(x+2y)9 3(a+2b)

10 12(x+2y)11 2(2c+3d)12 4(4x-5y)13 3(2x+3y+7z)14 4(2p+q-3r)15 8(2a-3b+c-d)

1 p(q+r)2 a(c+f)3 f(g-h)4 a(b+2)5 b(c-3)6 x(6-a)7 p(3+q)8 x(4-y)9 a(cg-2)

10 p(q-r+2)11 pq(r+t)12 xy(w-z)13 x(3y-4)14 p(6q-5r)15 x(3+y-z)


6 x2-7x7 10x2+15x8 18x2-6x9 -8x-2

10 -24x+8


1 7x+7y2 10p-10q3 5x-104 6x+125 2x-66 -4x-207 -3x+188 x2+5x9 x2-6x

10 pq+pr11 ab-ac12 xy+5x13 xy-4x14 12x+615 12x-1016 -20x+417 -10x-5

18 -21x+1419 30x+2420 10+20x21 -18+12x or

12x-1822 10-30xor

-30x+1023 6xy+6xz24 2pq-2pr25 3x2+2x26 -x2-3x27 6x2-3x28 4x2+28x29 14x2-21x30 12pq+30pr31 12cd -32c32 -2x2-x


1 a 6x+6 b 6x+22 c 4x-2 d 4x+16 e 6x+20 f -2x+13 g 12x-21 h 6x+8 i -x+2 j x-3 k 28x-8 l -x+8 m 2x-6 n 12x-7 o -15x p 16x+62 a 6x+10 b 7x+27 c -3x-13 d 4x+13 e 2x+14 f -5x-20 g 4x-18 h 2x-10 i -2x-18 j -3x-10 k 7x+23 l -29x+46 m 5x+3 n -26x-93 8(x-1)+5(x+1)=13x-34 2(x+7)+3(x+4)+4×4=5x+42



1 3(x+2)2 4(x+2)3 2(3x+4)4 4(2x+3)5 4(3x-2)6 3(x+10)7 2(2x+3)8 7(3x+2)9 2(2x+9)

10 3(2x+3)11 5(x-3)12 8(3x-2)13 5(x+1)14 7(x-1)


PUZZLE (page 131)

the tenz family

AdamlivesinNelson.BernicelivesinInvercargill.ColinlivesinNewPlymouth.DeniselivesinAuckland.EvanlivesinGreymouth.FleurlivesinNapier.GarylivesinDunedin.HannahlivesinWellington.IanlivesinWanganui.JilllivesinChristchurch.

15 2(2x+1)16 3(5x-7)17 7(2x+5)18 4(4x-1)19 5(3x-y)20 23(2x+1)21 30(2x-3)22 5(6x-1)23 15(3x+2)24 3(2x-3y+4z)25 6(4p-3q+5r)26 3(a+2b+6)27 4(x+y-1)28 4(10x+2y+1)

16 2x(2-3y)17 3x(y+2p)18 ab(4c-5d)19 7xy(3z+5p)20 x(y+1)21 ac(d+1)22 qr(2p-1)23 y(6x-1)24 6a(2x+y)25 3e(d+20f)26 6x(4y+3)27 6px(7-3q)28 3px(2y+z)


1 x(3x+5)2 3x(2-x)3 x2(1+x)4 x2(2x+5)5 3x2(2+3x)6 2(2x2+1)7 x(3x2-1)8 x(x2+x-1)9 12x(2x-1)

10 2x3(2x2+3)

513Answers

10 Solving equations

1 a x=3 b x=9 c x=15 d x=17 e x=63 f x=18 g x=78 h x=202 i x=0 j x=152 a x=-4 b x=4 c x=-9 d x=-5 e x=-2 f x=-3 g x=-14 h x=-1 i x=2 j x=-24

PUZZLE (page 135) Upside-down equation

n=8



1 x=42 x=33 x=-54 x=-45 x=-76 x=47 x=38 x=-129 x=9

10 x=-5

11 x=12

12 x=-213 x=-7

14 x=

13

15 x=34

16 x=154

or3 34

17 x = 8

3or2 2

3

18 x =-47

19 x=-1820 x=6

21 x = 3

2or1 1

222 x=3

23 x =-32

or

-1 1

224 x=1125 x=-5


1 x=162 x=303 x=244 x=425 x=-3

6 x=2007 x=-158 x=-409 x=7

10 x=-100


1 x=42 x=33 x=144 x=65 x=156 x=9

7 x=-18 x=-59 x=0

10 x=-811 x=512 x=15


1 a x=4 b x=3 c x=1 d x=2 e x=-2 f x=3 g x=3 h x=-1 i x=4 j x=-32 a x=-4 b x=2

c x = 4

5 d

x = 7

2or3 1

2

e x =-43

or

-1 13

f x=5 g x=-5

h x =-45

i x = 3

2or1 1

2 j x=-19

3 184 8


1 a x+6=14;x=8 b x-8=17;x=25 c 4x=28;x=7

d x2

= - 4 ;x=-8

e 10-x=13;x=-32 a 3x+4=22 b x=63 a 2x-8=12 b x=104 a x+10 b x+x+10=70 c x=30cm d 40cm5 a (B)3x+6=45 b x=136 a 2x-5=11 b x=87 a 2x-10=100 b x=55kg8 a 43and45 b Onefollowingaftertheother. c x+x+1+x+2=24 3x+3=24 d x=79 a 3x+120=360 b x=80°

10 90x+40=175;x=1.5,i.e.anhourandahalf.

11 a

b x+x+12+x+12=42 3x+24=42 c x=6 d 18cm12 a n=14

b Itisthenumberofdaysalarge-sizecarcanbehired.

c 17p+60=1250;p=70,thatis,therateforhiringamedium-sizecaris$70perday.

13 35n+90=1000;n=2614 a 6x+22=100 b x=$1315 8x−30=240;x=$33.7516 a $416 b 6x+152=344;x=32 c Morethan$13.60.

x+12

x+12

x

==

1 x=-32 x=-63 x=-64 x=45 x=46 x=27 x=88 x=4

9 x=-210 x=611 x=-3

12 x = 5

2or2 1

213 x=-7

14 x = 1

2


15 x = 15

2or7 1

216 x=-917 x=-118 x=1

19 x = 7

5or1 2

520 x=2

514 answers

PUZZLE (page 140)

the cool sunglasses Seepage526foranswer. 1 a 7x+8 b 16x+6

c 7x-1 d 8x+6 e -5x+2 f 11 g 7x+4 h x-1 i 8x+30 j -2x+4 k -6x+6 l 10x-13 m -5x-2 n -2x+182 a 5x+10 b 5x-12 c -4x-9 d 12x+3 e -x+4 f 9x-10 g 5x+10 h x-8 i x-23 j -5x-15

1 a x=8 b x=6 c x=15 d x=20 e x=62 a x=2 b x=27 c x=9 d x=6

e x = 15

2or7 1

23 a x=11 b x=134 a x=2

b x = 5

2or2 1

2

5 a x = 1

2

b x = 21

2or10 1

26 a x=5

b x =-

-152

7 12

or

c x=4

d x =-

-2310

2 310

or

7 a x=-3 b x=-22 c x=16

d x = 17

4or4 1

4 e x=168 a Whenmultiplyingby4,

Ashleyshouldhaveonlymultiplied8by4,butnot7.

b

3 74

8

3 7 32

3 39

13

x

x

x

x

- =

- =

=

=

PUZZLE (page 148)

granddad’s family

90


1 a Correspondinganglesonparallellinesareequal.

b 3x=x+40 c x=20° d Eachmarkedangleis60°.2 5x=x+8;x=23 3x-12=x;x=64 a Oppositesidesinarectangle

arethesamelength. b 2x+1=x+5 c x=4cm d Eachsideis9cm.5 3x+6=5x-10;x=86 a No,becauseonlywhole

numbersarepossibleforthe‘numberofeggs’.

b 5.5or5 1

2minutes.Youcould

checkbyseeingifthepoint

(5, 5 1

2) liesonaline

throughtheotherpoints. c n=8

d Anomelettemadewitheighteggstakes7minutestocook.

7 x+x+12=70;x=29 T-shirtcosts$29,sweatshirtcosts

$418 Meter1:29;meter2:66;

meter3:1329 6x=x+6;1.2

10 5x+40=3x+120;x=$4011 a 3hours

b DJ1wouldworklonger,andthesocialcouldrunforanother17minutes.


1 2x+142 3x+123 2x-64 7x+425 6x+86 15x-67 -3x-128 -2x-109 -5x+30

10 -x-611 -x+412 -x+1713 6x-314 -15x-1015 12x-3216 -2x+1417 -x-318 -4x+8



1 x=112 x=23 x=34 x=-15 x=26 x=-27 x=18 x=-4

9 x =-12

10 x=811 x=-26

12 x=013 x=12

14 x = 7

2or3 1

2

15 x = 5

2or2 1

216 x=1617 x=0

18 x = 21

4or5 1

4 19 x=320 x=0


1 a 2(n-5)=48 2n-10=48 b 292 10.25km3 6(3c+1)=204;c=114 a x=7;thisisthepricepaidto

hireeachmovielastmonth.b i a=11,b=3,c=297 ii $24

5 13(x−2)=143;x=136 a Thetotalcostoffencingthe

sidenexttotheroad. b 15(3x+60) c 80mby140m7 14ononesideand70ontheother.8 0.39(x−41000)+14670

=32129.91;x=$857699 3(x-5)+2=29;x=14,thatis,

Ashleighis14yearsold.



1

x + 35

= 2;x=7

2

x - 86

= 5;x=38

3 3x4

= 12;x=16

4

2x5

= 14;x=35

5 x + 8

2= 11;x=14

6

x + 153

= x - 1;x=9

515Answers

11 Two pairs of brackets

1 3x-122 -2x-23 4x-124 -5x+105 7x+76 2x-27 -4x+88 -10x-20

1 x2-42 x2-163 x2-814 x2-645 x2-36


x-blocks1 x2+5x+62 (x+3)(x+2)3 x+3+x+2+x+3+x+2

=4x+104, 5


9 x2-2x10 x2+3x11 x2-x12 x2+4x13 x2-5x14 x2+x15 x2-7x


1 x2+13x+302 x2+4x-43 x2+3x-104 x2+x-425 x2+2x-15

6 x2-6x+87 x2-4x-58 x2+4x+39 x2-2x-15

10 x2-12x+20


1 x2+6x+82 x2+x-123 x2+6x+54 x2+2x-85 x2-7x+12

6 x2-3x+27 x2+10x+218 x2-3x-49 x2+3x-10

10 x2-12x+32


1 x2+5x+62 x2+6x+53 x2+12x+364 x2+12x+325 x2+13x+426 x2+5x+47 x2+x-68 x2+3x-409 x2-6x-40

10 x2+9x-3611 x2-8x+1512 x2-x-7213 x2-5x+4

14 x2+11x+1815 x2+x-4216 x2+10x+917 x2-9x-2218 x2+10x+2519 x2-17x+7220 x2+9x-3621 x2-5x-3622 x2-9x+1823 x2-3x-1024 x2-16x+6025 x2-3x-88


1 x2+4x+42 x2+6x+93 x2-10x+254 x2-4x+45 x2+2x+16 x2-12x+36

7 x2+14x+498 x2+20x+1009 x2-24x+144

10 x2+18x+8111 x2-30x+22512 x2+40x+400


6 x2-17 x2-1008 x2-1219 x2-225

10 x2 14

-

PUZZLE (page 154)

Dollar days1 Caitlinispaid$1morethan

Dennis.2 SupposeCaitlinworksxhoursfor

$xperhour.Caitlinispaidx×x=$x2intotal.

Dennisworks(x+1)hoursfor$(x−1).Dennisispaid(x+1)(x−1)=$x2−1intotal.


1 2x2-x-32 2x2+3x-23 3x2-20x+124 15x2-x-25 2x2-15x+286 12x2-13x+37 4x2+20x+258 9x2-6x+19 100x2−140x+49

10 64x2−9


(Note:thetwopairsofbracketscanbeineitherorder.)

1 (x+3)(x+4)2 (x+3)(x+5)3 (x+1)(x+2)4 (x+2)(x+5)5 (x+2)(x+3)6 (x+2)(x+7)7 (x+1)(x+14)8 (x+3)(x+6)9 (x+1)(x+18)

10 (x+3)(x+3)

(Note:thetwopairsofbracketscanbeineitherorder.)

1 (x-7)(x+2)2 (x+3)(x-1)3 (x-6)(x+1)4 (x-9)(x-1)5 (x-2)(x-9)6 (x+3)(x-6)7 (x+6)(x-2)8 (x+10)(x-2)9 (x-20)(x+1)

10 (x+4)(x-3)11 (x-3)(x-4)12 (x+1)(x-17)13 (x-3)(x+2)14 (x+9)(x-5)15 (x-13)(x-3)

PUZZLE (page 157)

Which swimmer was the winner?Quentin

x+4

x+

5



1 a x(x+2) b x(x-8) c x(x+10) d x(x-7)2 a (x+3)(x-3) b (x-10)(x+10) c (x+6)(x-6) d (x-2)(x+2) e (x+8)(x-8) f (x-9)(x+9)

516 answers

1 a (x+2)(x+3) b (x-3)(x-6) c (x+3)(x-2) d (x+1)(x-12) e (x-4)(x+4) f (x+6)(x+2) g (x+10)(x-2) h (x-8)(x+3) i (x-1)(x-5) j (x+3)(x-11)2 a (x+4)(x+3) b (x+7)(x-7) c (x-10)2 d (x+6)(x-1) e (x+1)(x+2) f (x-1)(x+1) g (x-9)(x+3) h (x+7)(x-5) i (x-1)2 j (x-6)(x-8)3 (x+3)4 (x+120)5 (x−25)

1 (x+10)(x+1)2 (x-3)(x+3)3 Nofactors4 (x-6)(x-3)5 x(x+7)

PUZZLE (page 159)

the age of augustus1 AugustusdeMorganwasbornin1806(hewas43years

oldintheyear1849).2 Itisveryunlikelythatanyonealivetodaywasyyears

oldintheyeary2.Therearetwocasestolookat:y=44andy=45.(i) y=44.Someonebornin1892wouldhavebeen44

yearsoldintheyear1936,sowouldbeolderthan116now.Theoldestlivingpersonatthetimeofwriting(November2007)isEdnaParker,bornin1893,soisage114.Nooneelsecurrentlyalivewasbornbeforeher.

(ii) y=45.Anyonethatwillbe45yearsoldintheyear2025wouldhavebeenbornin1980.Theycanmakethatclaimin2025,butnotyet!


the dimensions of the sand-pit5mby5m



6 (x-6)(x+5)7 (x-24)(x-1)8 Nofactors9 Nofactors

10 (x+7)2


1 2(x+2)(x+5)2 5(x-3)(x+2)3 3(x-7)(x-3)4 2(x+2)(x-2)5 3x(x-3)

6 6x(x+4)7 10(x+4)(x+1)8 4(x-5)(x+5)9 2(x+24)(x-1)

10 4(x-2)(x+9)

PUZZLE (page 159)

Dog Leg ParkTheareaisc2−d2or(c−d)(c+d).


1 a x=2orx=−2 b x=9orx=−9 c x=10orx=−10 d x=1orx=−12 a x=3orx=−3 b x=5orx=−5 c x=8orx=−8 d x=4orx=−43 a x=2orx=−2 b x=3orx=−3 c x=4orx=−4 d x=5orx=−5


1 x=−2orx=32 x=−1orx=43 x=5orx=−84 x=2orx=85 x=−9orx=−16 x=−20orx=77 x=4orx=−28 x=6orx=89 x=−12orx=−3

10 x=7orx=−111 x=4orx=2

12 x=−6orx=−513 x=2orx=314 x=−4orx=1515 x=1orx=−3016 x=−8orx=−1917 x=17orx=14

18 x=12

orx=4

19 x =-34 orx=5

20 x=0orx=6


1 x=3orx=52 x=1orx=63 x=10orx=24 x=−3orx=−75 x=−2orx=−16 x=4orx=−37 x=2orx=−78 x=8orx=−19 x=6orx=5

10 x=12orx=−2

11 x=5orx=212 x=−6orx=−113 x=1orx=314 x=−8orx=−915 x=−2orx=1516 x=−30orx=217 x=17orx=−118 x=10orx=2019 x=0orx=−320 x=0orx=4


1 a x=−3orx=−2 b x=3orx=5 c x=4orx=−12 a x=2orx=5 b x=3orx=−2 c x=15orx=−33 a x=−1orx=−10 b x=1orx=104 a x=5orx=0 b x=2orx=65 x=3

517Answers

12 Two-dimensional graphs

1 a A=minibus;B=ship;C=bicycle;D=trainb Eitherextendtheverticalaxisorchangethescale

ontheverticalaxis.2 a Nga b Chris c Nguyen d Sue3

4 Ingeneral,thetallerastudentis,thehighertheycanclearthehigh-jump.

5 a 48km/h(or50km/hwhenrounded) b 80km/h c 62mph d i 64.4km/h ii 55.9mph6 a i 45kg ii 90lb b

7 a $950 b 80m2 c $800 d

e 120m2.Thisistheareaforwhichthetwolinesintersectonthegraph.

8

9 aContainer Depth–time graph

Bowl A

Trough A

Cylinder B

Cuboid B

Cone A

b

10 a -5°C b 3hoursc Itreachedroomtemperatureat5am,thenlateronin

themorning(9am)theroomtemperatureincreased.

d1 1

2hours e 6:30pm

11 12

13

×A

Hei

ght

Weight

×

×C

B

Wei

ghti

nkg

Weightinlb

10

20

30

40

50

10 20 30 40 50 60 70 80


20 40 60 80 100

200

400

600

800

1000

1200

Cos

tofp

aint

ing

($)

Areatobepainted(m2)

3 6 9 15

Hei

ghto

ffla

g

Time(seconds)12 18 21 24

Tem

pera

ture

(°C

)

Time(minutes)

20

40

60

80

100

10 20 30 40 50 60 70 80

A

Dep

th

Time

A

BC

C

Hei

ghta

bove

gro

und

Time

518 answers


BMI graphs for boys and girls

1 Pink2 a Healthyweight b Overweight3 BMI=22.37;heisontheboundary

betweenhealthyweightandoverweight.

4 6yearsoldand8yearsold5 14yearsold(Year10).The50th

percentileforaBMIof19isroughlylevelwithanageof14forbothboysandgirls.


6 Theboyis11yearsoldormore.7 TheBMIisbetween21and24.8 Severalreasonsarepossible:

(i)growthinheightisfasterthangrowthinweightbetween2yearsoldand4yearsold,(ii)childrenlearntowalkatthisage,and(iii)childrenlose‘puppyfat’atthisage.

9 a Heweighslessthan50kg.b Sheweighsbetween45kgand

65kg.

10 Thegraphsshouldkeeprisingbecauseadultshavestoppedgrowinginheightbyage20butcontinuetoputonweight;however,thegraphswill‘flattenout’tosomeextent.

11 BoyshaveahigherBMIforunder7andover15;girlshaveahigherBMIforbetween7and15.

1 a 10km b 2hours15minutes c 1km d 15minutes

e Thelineshowingthecarjourneyissteeperthanthelineshowingthebusjourney.

2

3 a 1340hours b 2hours40minutes c 8.5km d 45minutes

e Thetwogroupsarewalkingtogetheratthesamespeed.

4 (B) Studentwalkstoafriend’splace,hasarest,continueswalkingslightlyfastertoschoolinordertogetthereontime.

(C)Studentwalkssteadilytoschool.(D)Studentcyclestoschool,issenthomebecauseisin

wronguniform,cycleshome,changes,cyclesbacktoschool.

(E) Studenttravelstoschoolbybus,whichstopsatregularintervals.

5 a Red b Orangec Bothgroupsstoppedforlunchbetween12:30pm

and1:30pm;thelunchstopswere6kmapart. d Justbefore3:30pm. e 15km f 1hour g 60km

h Theredgroup;between1:30pmand4pm.i theredgroupdrifteddownstreamwiththecurrent

for2.5hoursandcovered6km.Thecurrentspeed

is

distancetime

= 33 - 272.5

= 6km2.5h

= 2.4km/h.

j Theorangegroupwerepaddlingagainstthecurrent.

6 a 1500hours b 1hour c 3 d Christchurch e KeruruinChristchurch,KeainAuckland

f No,becausethehorizontalpartsofthegraphdonotoverlap.

7

8 a 4:15pm b 370km c, d

20 40 60 80 100 120 140

2

1

Time(minutes)

Dis

tanc

efr

omho

me

(km

)

Dis

tanc

e(k

m)

2 4 6

4

2

Time(minutes)

150

230

370

11 12 1 2 3 4 5 6Timeofday

Dis

tanc

efr

omC

hris

tchu

rch

(km

)

Motorist

Truck-driver

e 140km

519Answers

9 (A)ApassengertrainleavingfromDunedinat3pmandarrivingatInvercargillat5:50pm.

(B) Atrainthathasbrokendown80kmfromDunedin.

(C)AgoodstrainwhichhasalreadyleftDunedinandarrivesatInvercargillatabout6:10pm.

(D)ApassengertrainleavingfromInvercargillat3pmandarrivingatDunedinat5:50pm.

13 Graphs using rules from algebra

1

2 A=(1,3) B=(3,1) C=(-4,-2) D=(-3,5) E=(-2,0) F=(5,-3) G=(0,6) H=(0,0)3 a {A,B,C} b {G,H,I} c {D,E,F}4 Astar5 (11,5)6 a (3,1) b Aninfinitenumber. c Thex-co-ordinateis3. d (3,10)or(3,-10)

PUZZLE (page 178)

Which vegetable is most environmentally friendly?Seepage526foranswer.


Wholly equilateral1 False2 Thisisnotpossible.

1 a Co-ordinatesare(2,6),(1,3),(0,0),(-1,-3)and(-2,-6).

b Co-ordinatesare(2,3),(1,2)(0,1),(-1,0)and(-2,-1).

c Co-ordinatesare(2,0),(1,-1)

(0,-2),(-1,-3)and(-2,-4).

y

x2 4 6–2–4

–2–4

2

4

6

A

E

BC

D

F

EXERCISE 13.01 (page 177) EXERCISE 13.02 (page 180)

2 4 6–2–4–2–4

2

4

6

y

x

–6

8y=3x

–2

–4

2

4

2 4–2

–4

y = x +1

y

x

–4

2

4

2 4–2–4

y = x –2

y

x

–2

d Co-ordinatesare(2,5),(1,3)(0,1),(-1,-1)and(-2,-3).

2 a

b

c

2 4–2–4–2

–4

2

4

6

y

x

y = 2x +1

y

y=2x+3

–2 –1

3

x

x

2

y

y=x+4

–4 –2

4

x3

–3

y

y =x –3

520 answers

d

1 a 1 b

13

c 2

d14

e12

f 0

2

3 4

412

5 (B),(G),(C),(E),(F),(A),(D)

6 a 3 b

25

2 4–2–4–2

2

y

xy=x1

2


a bc

d

e

f


1 a 5 b14

c

35

d12

e 2 f 3

g32

h1 1

4

2 a

b

c

y

x

y =3x

y

x

y=x12

d

e

f

g

y

x

y=x32

y

x

y=2x

y

x

y=x13

y

x

y=x53

y

x

y=x35

h

i

j

k

y

xy = 1x

y

x

y =5x

y

x

y=x23

y

x

y= x14

521Answers

l INvEstIgatIoN (page 184)

the water-pipe

1

Length of farm on plan

1 2 3 4 5 6

1 1 2 3 4 5 6

Width of 2 2 4 4 6 6 8

farm on 3 3 4 7 6 7 10

plan 4 4 6 6 10 8 10

5 5 6 7 8 13 10

6 6 8 10 10 10 16

y

xy = x

2 j3 m+n-14 3p-2


1 a -1 b-14

c -4 d-15

e-34

2

ab

c

de

3 -6

4-13

5 a -1 b -2


1

2

y

x

y=–2x

y

x

y=–x23

y

x

y=–x32

y

x

y=– x53

y

x

y=–x

y

xy=2x

y

x

y=x25

y

x

y=–4x

y

xy=–x1

3

y

xy = x

3

4

5

6

7

8

9

10

522 answers

1 a 3 b -1 c 5 d -10 e 125 f -7 g 0 h 0

2 a b c d


y

x

y=x+4

y

xy=x+1

y

x

y=x+5

y

x

y=x–1

e f g

y

x

y = x–6

y

x

y = x–3

y

xy = x


1 a 2 b 4 c -3 d

23

e-14

f 1 g 1 h -8 i -1

2 a 3 b -1 c 2 d -5 e 1 f 0 g 4 h 0 i 23 a b c d ey

x

y = 2x +11

x

y = 3x+2

y

2

y

x

y = 2x–3–3

y

xy = x+11

21

y

x

–4 y = x–425

4 a b c d y

x

y=–2x+55

y

xy=–2x–1

–1

y

x

4y=–3x+4

y

x

y=–x+1

1

y

x

y = – x+212

2

y

x

y = – x+453

4

e f

523Answers

5 a Yes b 26 a -2 b 5 c

8 y=2x+39 a y=2x-1 b y=x+1 c y=-x+2

d y x= --12

1

10 C=(2,7)

1112

× 42 - 4 = 21- 4 = 17

12 No,because6≠36-5×5.

13

(1,3)

y

x

y=5–2x5

7 a b c d

ef

y

x

y = x+212

y

x

y =–5x+1

1

y

x

3y = x+34

3

y

x

y = – x–132

–1

y

x

y = x–123

–1

y

x

y =6–3x6

y

x

y =–x+4

y =2x+1

14 y

x

y =2x+1y =2x–3

y = x+212

a Thegradientswillbethesame.b Steeperlineshavebiggergradients.c Itcutsthey-axisabove(0,0)ifthe

y-interceptispositive,itcutsthey-axisbelow(0,0)ifthey-interceptisnegative.


1 a b c dy

x

y =1

y

x

x=5

y

x

y=–3

y

x

x=–2

2 a x=5 b y=2 c x=-2 d y=-4

3 a False b False c Trued True e True f False

524 answers

4 a, b c y=-2

1 a 1.5litres b

c32

or1.5 d y = 3

2x

e About2 1

2hours(actually

2hours40minutes).

2 a 10m2 b 3 c 4 d y=3x+4

e Weneed4m2forstoringequipment,andeachpersonintheclassneeds3m2offloorarea.

3 a

4 a

y

x

–2

2

y=–2

y=2

5 Theimageisinthesameplace.6 a, b c y=4y

x

x=–4

y=4


3

6

9

12

2 4 6 8x

Fuel

use

d(l

itre

s)

Timeused(hours)

y

3

3 6 9 12

y

x

Temperature(°C)

Num

ber

ofsc

arve

sso

ld 6

x

y

Are

am

own

(m2 )

Time(minutes)

b 5 c y=5x-4 d 46m2

e Valueslessthan0.8minutes.

b-12

c y x= +-12

6

d 7 e 12°Cf Itwouldbecomehorizontal,

andcontinuetotherightalongthex-axis.

5 a

23

b Allow

23

ofanhour(i.e.40minutes)perkilogramtocooktheturkey.

c t = 2

3w + 1

2d

e 3.75kgf Thetimescaleissplitupinto10-minuteintervalsbecausethegiventime

informationisinmultiplesof10,andalsoitmakesiteasytosplitupthehours.Theweightscaleischosentomatchthetimescalesothatitiseasytorelatethegradientandy-interceptwiththeequation.

5

4

3

2

1

01 2 3 4 5 6 7 8 9

Weightofturkey(kg)

Coo

king

tim

e(h

ours

)

x

y

525Answers

6 a Possibleanswersarefivetokensand16notes,10tokensand12notes,15tokensandeightnotes,20tokensandfournotes.

b

Number of $4 game tokens (x) 5 10 15 20

Number of $5 notes (y) 16 12 8 4

cd y x= +- 45

20

24

20

16

12

8

4

0 4 8 12 16 20 24Numberof$4tokens

Num

ber

of$

5no

tes

y

x


glove-sizing1

14

12

10

8

6

4

2

03 6 7 8 9 10

Con

tine

ntal

glo

ves

ize

y

x

–6

Englishglovesize

2 y=2x-63 64 Thewidthofmostpeople’sknucklesisbetween

7cmand14cm.


1 a Co-ordinatesare(-2,5),(-1,2),(0,1),(1,2)and(2,5).

b Co-ordinatesare(-2,2),

(-1,-1),(0,-2),(1,-1)and(2,2).

y

x

y=x2+11

y

xy=x2–2

–2

c Co-ordinatesare(-2,9),(-1,4),(0,1),(1,0),(2,1),(3,4)and(4,9).

d Co-ordinatesare(-2,9),(-1,3),(0,0),(1,-1),(2,0),(3,3)and(4,8).

y

x

y=(x–1)2

1

1

x

y=x2–2x

y

2

e Co-ordinatesare(-2,9),(-1,5),(0,3),(1,3),(2,5)and(3,9).

2 SeethespreadsheetEx 13-11 Qn

2 Answers.xls.ThisisavailableontheBeta Mathematics Workbook companionCD,orcanbedownloadedfromwww.mathematics.co.nz

x

y=x2–x+3

y

3

526 Answers

a b

c

EXERCISE 13.12 (page197)

1 a

b 80 m c 4.5 seconds

d The graph becomes steeper and steeper, showing the computer is accelerating towards the ground.

100

80

60

40

20

y

x1 2 3 4 5

2 a 2400 b

c As the number of workers increases, the number of instant messages increases faster and faster.

120

100

80

60

40

20

y

x1 2 3 4 5 6

3 a

b $200 c 9 m

900

700

500

300

100

2 4 6 8 10x

y

PUZZLE (page140)

The cool sunglasses

They are made up of the letters cool.

InvEsTIgATIon

(page178)

Which vegetable is most environmentally friendly

Green peas.

527Answers

14 The metric system, scales and tables


5 You could measure the height of a stack of 10, say; then multiply by 100. The height would be about 2.6 m

10 a 6700 m b 4000 m c 1.3 m d 1.1 m e 11.25 m f 600 m

11 3.764 km 12 6400 m13 a 25.8 cm b Multiply by 10.14 2 km15 42 195 m16 154717 a 7600 m b 7.6 km18 27 minutes

PUZZLE (page205)

How thick is photocopy paper?

0.1 mm

1 a 6000 mL b 45 000 mL c 850 mL d 4.9 mL e 15 mL f 20 441 mL2 a 5 litres b 0.75 litres c 88 litres d 0.0683 litres e 0.002 litres f 100 litres3 a 4.5 litres b 1.55 litres4 a 3850 mL b 5550 mL5 156 3.3 litres or 3300 mL7 4.7 litres or 4700 mL8 $2.949 14

10 Cans Volume: 6 × 333 mL =

1998 mL = 1.998 litres. Cost of cans is 0.45 × 6 = $2.70.

Cost per litre is

2.701.998

= $1.35/litre= $1.35/litre.

Plasticbottle Cost per litre is

2.501.5

= $1.67/litre.

It is cheaper to buy the six-pack of cans.

11 a 20 mL b 6 years old c 13 years old d 4 e 5 days12 a 40 g b 600 g

InvEsTIgATIon (page209)

Decimal time in France

1 Longer. There would only be 1000 of these seconds in a day, instead of 86 400 of our seconds.

2 If there were 10 days in a month then a month would no longer correspond roughly to the periods of the moon. If in addition there were 10 months in a year then there would only be 100 days in a year, instead of 365, and so the calendar would go too fast in relation to the seasons.


1 a m b km c cm d mm2 a 12 m b 1.9 m c 18 mm d 1.3 km e 590 km f 4 mm3 a 1.2 m b 0.492 m c 18.5 m4 a 3500 mm b 78 200 mm c 400 mm5 a 1.5 km b 0.75 km c 46.83 km6 a 2000 m b 4700 m c 350 m7 a 534 cm b 7 cm c 3.2 cm d 49 cm8 a cm b m c km d mm e cm

9

m cm mm

a 6.3 630 6300

b 5 500 5000

c 0.12 12 120

d 0.08 8 80

e 0.497 49.7 497

f 12.8 1280 12 800

PUZZLE (page203)

Can you fathom this?

1 A foot was the length of a typical adult human foot.

2 3 feet = 1 yard (and is about 1 arm’s length)

3 1 fathom = 6 feet (and is the height of a tall person)


1 a 5000 g b 5480 g c 60 000 g d 700 g2 a 8 kg b 7.36 kg c 0.45 kg d 0.0112 kg3 a 53 000 mg b 800 mg4 a 50 g b 8.6 g5 a 6 tonnes b 13.255 tonnes c 0.439 tonnes6 a 11 000 kg b 6420 kg c 75 kg7 13.36 kg

8 a $48.46 b $6.609 a i $7.90 ii $63.20 b $39.50/kg

10 $43.7611 1112 a i $5.10 ii $2.51 b 73 cents c 13 or 1413 53


528 Answers

1 a 0630 b 2145 c 1200 d 1340 e 0255 f 2330 g 0000 h 1030 i 2315 j 16452 a 7:30 am b Midnight c 9 am d 9 pm e midday f 3:55 pm g 7:20 pm h 8:40 pm i 11:20 pm j 12:10 am3 12004 a 1940 b 2100 c 1810 d 19565 8 is incorrect. The third digit in

24-hour time must be between 0 and 5 inclusive, because there are only 60 minutes in an hour.

6 0700 or 7 am7 a 8 minutes b 25 minutes c 10218 a 3.45 b 1005 c 17479 a 1 hour 15 minutes b 1 hour 55 minutes

10 a 3 hours 50 minutes b 142011 3 minutes 20 seconds12 1 hour 15 minutes

PUZZLE (page212)

The time in Letterland(C) and (E)

PUZZLE (page213)

Best before when?1 8 July 20022 231/093 Advantages: It may serve as identification for a particular batch in case there

are complaints about the product. Another reason is to avoid confusion between the DD/MM/YYYY used in some countries and the MM/DD/YYYY format used in North America.

Disadvantages: Both customers and retailers may find it confusing because it requires relatively complicated calculations both of today’s date and the date on the packet before deciding whether stock is past its use-by date.

1 a 842 km b Blenheim c Picton and Blenheim d Takaka and Milford Sound e 10 km

f Milford−Queenstown = 296 km.

Queenstown−Mt Cook = 272 km.

When these are added the result is 568 km, which is 12 km more than the direct route from the table: Milford−

Mt Cook = 556 km.2 a 75 minutes or 1 hour

15 minutesb 75 minutes or 1 hour

15 minutes c Rare3 a 7 minutes b 11 minutes c Well-done4 a 7367 units b 8505 units c



1 a 333 mL b 45 litres c 600 mL d 2.2 m e 105 kg f 1600 litres g 350 mL h 2 litres i 10 mL j 38 m k 3.5 kg l 32 cm m 22 mm n 364 km2 a 8 g b 333 mL c 65 mm d 1.2 kg e 90 L f 15 cm g 2151 km h 4.8 m3 a Nearest day b Nearest cm c Nearest mm d Nearest kg e Nearest tonne

f Nearest

1100

of a second

g Nearest 5 minutes


5 a 6.8 cm b 9.05 cm c 2.83 d

a = 7

3c

6 a 1.8 m3

b 3.5 m3

c 20 m2

d i 20 bags ii 1.5 m3

e i It should be twice the amount needed for 20 m2 − i.e. 2 × 1.4 m3 = 2.8 m3.

ii About 26 bags of cement, 1.3 m3 of sand and 5 m3 of builder’s mix.

7 a A = 1250 mL, B = 600 mL, C = 150 mL

b 20 mL8 a 2.4 m b 2.9 m

c Size 14, because 70 cm is close to the 71 cm waist measurement for size 14.

d Cushla will need 3.10 m for the jacket and 2.95 m for the skirt; this is 6.05 m altogether, so 6 m is not quite enough.

e $43.66

0123

4 5 678

9

1000

0 123

45678

9

100

0123

4 5 678

9

10

0 123

45678

9

kWh per div

15 Area of polygons


1 a 18 cm2 b 12 cm2

c 16 cm2 d 20 cm2

e 3000 cm2 f 20 cm2

g 60 cm2 h 108 cm2

2 62 370 mm2

3 Eight bags4 208 cm2


1 a 4 cm b 8 cm c 8 cm d 15 cm2 a 28 cm b 36 cm c 26 cm d 20 cm3 128 cm4 a 2240 m b 21 760 m5 56 cm6 21 cm

529Answers

7 No, the perimeter of a 3 cm × 4 cm rectangle is 14 cm, while the perimeter of a 2 cm × 6 cm rectangle is 16 cm.

8 a Sidelength Area Perimeter

2 cm 4 cm2 8 cm

5 cm 25 cm2 20 cm

6 cm 36 cm2 24 cm

7 cm 49 cm2 28 cm

8.4 cm 70.56 cm2 33.6 cm

10 cm 100 cm2 40 cm

12 cm 144 cm2 48 cm

b 4 cm9 a 28 m b 22.361 cm c 8.185 cm

10 144 m11 a 180 m

b The mesh may not touch the edges of the driveway, the mesh might overlap in places, the mesh might have to be cut before being placed in position so more would be required at first, the surface may not be flat so that the mesh bends, etc.

1 8 cm2

2 42 cm2

3 20 cm2

4 30 cm2

5 60 cm2

6 42 cm2

7 72 cm2

PUZZLE (page224)

Prisoner in the middle


The soccer field

The perimeter of the field is 2 × 100 + 2 × 50 = 300 m.The centre line measures 50 m.The extra lines needed for the penalty areas measure 2 × (10 + 15 + 10) = 70 m.The total length of all the lines is 300 + 50 + 70 = 420 m.The width of the lines is 0.04 m.The area of the lines is 420 × 0.04 = 16.8 m2.The number of litres of paint required is 16.8 ÷ 2 = 8.4 litres. (Note: the overlapping on the corners is ignored in this answer.)

Y

X


8 54 cm2

9 36 cm2

10 51 cm2

11 132 cm2

12 36 cm2

13 124 cm2


Halving triangles

1 Draw a line from one corner to the middle of the opposite side. The two triangles formed each have the same height, and their bases are equal.

2 Repeat the process above. Take each triangle, and from one corner, draw a line to the middle of the opposite side.

h


1 a 40 cm2 b 48 cm2

c 35 cm2 d 72 cm2

e 50 cm2 f 30 cm2

g 104 cm2 h 21.2 cm2

i 120 cm2 j 30 cm2

2 a 96 cm2 b 9.6 cm

PUZZLE (page229)

Honey, I’ve shrunk the area!

In the ‘rectangle’ drawing, the trapezium pieces overlap the triangle pieces.

530 Answers

1 a 78 cm2 b 68 cm2

c 360 cm2 d 108 cm2

e 240 cm2 f 124 cm2

g 68 cm2 h 67.5 cm2

i 144 cm2 j 532 cm2

2 4300 cm2

3 a

2449

b

2549

PUZZLE (page231)

The four blocks1 400 cm2

2 Between 0 and 400 cm2.

1 a 60 cm2 b 480 cm2

2 24 m2

3 a 5409 cm2 b 46.8%4 55 m2

5 1433.5 cm2

6 a 150 cm2 b 607 a 16 500 m2 b 660 m2

8 (D) because the envelope has a front and a back, and also needs overlapping edges when glued together.

9 4050 cm2

10 50 m11 290 m2

12 25 m2

13 64 m14 a No, because 250 mm divides

exactly into both 3 m and 5 m. b 24015 130016 a 50 m2

b 5 cm, because it is the depth. c 1700

1 a 400 ha b 3900 ha c 832 ha d 10 040 ha2 a 6.3 ha b 0.4 ha c 13.76 ha d 3.95 ha3 a 550 000 m2

b 64 000 m2

c 3500 m2

d 490 m2

4 a 2 km2 b 68.2 km2

c 550 km2 d 0.803 km2

5 5 790 000 m2 = 579 ha (dividing by 10 000).

579 ha = 5.79 km2 (dividing by 100).

6 300 000 m2

7 a Abel Tasman b 4000.36 km2

c Mt Cook (Aorangi) d 3 085 036 ha = 30 850.36 km2

e 11.4%8 Golf courses require 3 × 7 = 21 ha. Polo field requires 200 × 500 =

100 000 m2 = 10 ha. Equestrian course requires

0.2 km2 = 20 ha. Total land required is at least

51 ha = 0.51 km2. 0.5 km2 is not enough.9 154 500 m2 or 15.45 ha

EXERCISE 15.05 (page230) EXERCISE 15.06 (page231) EXERCISE 15.07 (page234)

16 Circles − circumference and area


1 a 62.83 cm b 15.83 m2 a 12.57 cm b 1131 mm3 39 990 km4 471 cm5 True6 235 7 1339 mm or 133.9 cm8 a 46.27 m b 20.14 cm c 41.71 cm d 96.82 cm

9 Both paths are the same length, 12.57 cm.10 607.1 cm11 a 94.25 mm b 98.96 mm12 a The Earth takes 1 year = 365.25 × 24 = 8766

hours to orbit the sun. b The distance travelled is the circumference

of a circle with radius 149 000 000 km. This is 936 195 000 km (6 sf).

The speed is 936 195 000 km ÷ 8766 h = 106 800 km/h (4 sf).

1 a 271.72 m2 b 12 707.62 cm2

c 88.25 m2 d 26.06 cm2

e 6939.78 m2 f 59.94 cm2

2 1018 m2

3 113.1 km2

4 34 636 cm2

5 a

b 10 936 mm2

6 25%

3 mm

3 mm

3 mm 3 mm124 mm

124 mm


531Answers

PUZZLE (page240)

Pizza please470 g 1 a 11.43 m2

b 102.5 cm2

c 1122 cm2

2 Both designs need the same amount − i.e. area is 2146 mm2.

3 a 616 cm2

b 292 cm2

4 a

b 1579 m2


1 a 14

b 8 cm2

2 a

13

b 37.70 cm2

3 65.97 cm2

4 a 736.3 cm2 b 61.96 cm2

5 351.9 cm2

6 a 201.1 cm2 b 32 cm2

c True d 18.27 cm2

PUZZLE (page242)

Circular leftoversAll of the patterns have the same shaded area.

30 m

30 m

5 m

40 m

40 m


1 0.4312 m2 6.685 m3 466.6 m4 30.83 m

5 32 cm6 318 mm7 170 m


1 a 3.868 m b 18.70 cm c 1.376 cm d 0.7979 km2 a 3.545 m b 7.089 m c 22.27 m3 3.2 m4 29 m5 a 78.54 cm2 b 157.1 cm2

PUZZLE (page244)

The first and last ever school Certificate Mathematics exam question

283.7 m2


17 Volume and surface area

1 a 24 cm3 b 36 cm3

2 a 72 cm3 b 36 m3

c 1020 m3

3 a 90 cm3 b 72 cm3

c 73.44 m3 d 10 240 cm3

4 6 cm5 6.2 m6 72 cm3

7 a 216 m3 b 117.649 cm3

8 a 8 cm b 11 m c 4.291 cm d 79.370 m


volume conversions1 True2 Yes3 1000

4

Volumeincm3 Volumeinmm3

512 512 000

8 8 000

89 000 89 000 000

9 9 000

71 000 71 000 000

5 64 m3; 64 000 000 cm3. These volumes are equivalent, so there are 64 000 000 ÷ 64 = 1 000 000 cm3 in 1 m3.

6Volumeinm3 Volumeincm3

64 64 000 000

2 2 000 000

500 500 000 000

800 800 000 000

0.05 50 000



1 a 300 cm3 b 240 m3

c 400 mm3

2 240 000 cm3

3 a 0.12 m b 1.2 m3

4 155 144 000 cm3

6 204 288 cm3

7 a 4.5 m3

b 4050 kg c 8.1 tonnes8 a 213 m2 b 0.15 m c 31.95 m3

9 0.251 m or 25.1 cm10 1 m

532 Answers

1 a 90 cm3 b 800 cm3 c 225 m3

2 a 60 cm3 b 80 cm3 c 80 cm3

d 140 cm3 e 330 cm3

3

Areaofcross-section Height Volume

a 12 cm2 2 cm 24 cm3

b 45 m2 10 m 450 m3

c 31 cm2 17.4 cm 539.4 cm3

d 8 m2 2.45 m 19.6 m3


The Arch of Constantine

1 The archways are the same depth as the Arch itself.2 a 519.6 m3

b 3.25 is the radius of the semi-circular part at the top of the archway. It is half of the width of the archway.

c 8.25 is the height of the archway walls before they start curving. It is the overall height (11.5 m) minus the radius of the semi-circular part (3.25 m).

3 175.0 m3

4 Volume = V(cuboid) − V(central archway) − V(side archways)

= 25.7 × 21 × 7.4 − 519.6 − 2 × 175.0 = 3124 m3 5 The Arch does not have smooth faces − there are

protrusions and indentations in many places.



1 96 cm3

2 a x = 0.6, y = 4.8, z = 1.2 b 1.728 m3

3 108 m3

4 48 000 cm3

5 787 cm2

6 4264 cm3

7 963 m3 (rounded from 963.144 m3)


1 a 226.2 cm3 b 3633 m3

2 a 50.27 m3 b 2601 cm3

3 62.83 cm3

4 36 290 m3

5 0.8906 m3

6 a 5500 cm b 276 500 cm3

7 0.015 90 m3

8 a Adding on the thickness of concrete to the radius of a large tunnel you get 1.5 + 3.8 = 5.3 m, so the diameter including the concrete part is 2 × 5.3 = 10.6 m. A greater volume of rock than the finished volume would have been excavated before the tunnel was lined.

b 7.8 m c 11 315 000 m3

d Rubble is less compacted than the rock where it is removed from, there would be places where excavations needed to go beyond the minimum diameter of 7.6 m, there would be places where access would be needed between the service tunnel and each main tunnel.

9 21110 2262 seconds = 37 minutes 42 seconds11 43 980 cm3


1 a 4 mL b 500 litres c 300 mL d 2 litres e 45 litres2 a 40 b 6 c 8000 d 49 700 e 800 f 453 a 3 kg b 46.8 kg c 0.6 kg4 a 50 g b 8000 g c 788 g5 a 160 litres b 160 kg6 34.56 cm3

8 3456 mL9 a 38.23 m3 b 38 230 litres

10 a 3 b 8 c 10011 a 804.2 cm3

b The label of 750 mL is likely to be correct when allowing for the thickness of the glass. To calculate the volume of wine exactly we would use interior measurements and would also have to assume the bottom of the bottle was completely flat.

12 a 4247 m3

b 4 247 000 litres c 212 400 litres13 $4514 121.5 kg15 a 13 to 14 days b 75 g

c A cylinder is only an approximate model because a tube of toothpaste is not perfectly round and flattens out towards the base. At the other end the nozzle protrudes, so at both ends the tube lacks the flat, round base that cylinders have.

PUZZLE (page257)

I have suctionSee page 538 for answer.

533Answers

PUZZLE (page257)

Tug of war

40 kg

1 a 52 cm2 b 74 cm2

2 a 126 cm2 b 36 cm2

3 56 cm2

4 180 cm2

5 204 cm2

6 a Isosceles trapezium b 1952 cm2

7 64 m2

8 a Rectangle b Because the cut exposes wood which originally was not on a surface. c 2000 cm2

9 a Volume of cuboid = 18 × 10 × 3 = 540 m3.

Volume of triangular prism = 12

×10×12×9 = 540 m3.

b The cuboid has the greater surface area (528 m2). It exceeds the surface area of the prism (444 m2) by 84 m2.

10 a b

11 4.64 m2

12 a 8 m2 b 20 00013

Area of front and back = 2 × 2.6 × 2.5 = 13 m2

Area of both sides = 2 × 4.2 × 2.5 = 21 m2 Area of floor and ceiling = 2 × 2.6 × 4.2 = 21.84 m2

Total area to be painted: (13 + 21 + 21.84) m2 = 55.84 m2

Number of litres needed =

55.8415

= 3.723 litres, that is, 3.7 litres to the nearest 100 mL.

PUZZLE (page261)

Wine and cheese

6 cm by 8 cm by 12 cm


Edgelength Surfacearea

1 6

2 24

3 54

4 96

…

8 384

…

x 6x2

2.6

4.2

2.5


1 a 502.7 cm2

b 85.77 m2

c 177.3 cm2

2 a 1407 cm2

b 144.5 m2

3 728.8 cm2

4 37 960 mm2

5 0.7118 m2

6 Area of bottom: πr2 = π × 2.42 = 18.096 m2.

Area of top: πr2 = π × 2.42 = 18.096 m2.

Area of curved surface = 2πrh = 2 × π × 2.4 × 3 = 45.239 m2

Total surface area = 18.096 + 18.096 + 45.239 = 81.431 m2.

Area to be coated inside and outside = 2 × 81.432 = 162.86 m2.

Total cost = 162.86 × $8.50 = $1384.33, i.e. $1400, approximately.

PUZZLE (page263)

The three cubes1 704 cm2

2 800 cm2

18 Angles 1 − intersecting and parallel lines


1 a ABC b EDF

2 a EFD b ∠STR

3 a BAC b EFD

4 a = ∠PRQ b = ∠RSQ c = ∠QPR d = ∠PTS e= ∠QTR f= ∠PQR5 a 4 b 1 c 6 d 3 e 5 f 2

534 Answers

1 a 54° b 59° c 118° d 92° e 21°2 a 83° b 105° c 101° d 71°3 a 131° b 34° c 15°



security sensors1 a b

2 a

b 10 more (11 in total)

cd 10%

3 a 3 b

c d 80

4


1 a 111° b 039° c 322° d 239°2 a b

c d

3 a 090° b 270° c 045° d 225°4 a South b North c North-west d South-east

N N

N N

5 a Whakatane b Mt Ruapehu c East Cape d New Plymouth e Auckland f 020° g 073° h 331° i 129° j 205°

(Note that answers for parts f−j are based on using the full-page version of the North Island map in the blackline master. If students use the diagram in the book it is not quite so accurate.)

k Gisborne l Bay of Islands6 a 6.2 km b 338°7 230°


1 a = 110°, b = 58°, c = 90°, d = 210°, e = 90°, f = 47°, g = 129°, h = 50°, i = 73°,

j = 47°, k = 86°, l = 94°, m = 20°, n = 91°, o = 89°2 a ∠’s at a pt add to 360° b vert. opp. ∠’s = c ∠’s on line add to 180°

3 a = 135° (∠’s on line add to 180°) b = 102° (vert. opp. ∠’s =) c = 135° (∠’s at a pt add to 360°) d = 38° (∠’s on line add to 180°) e = 142° (vert. opp. ∠’s =) f= 148° (∠’s at a pt add to 360°)


1 a x = 70° (∠ sum of is 180°) b x = 28° (∠ sum of is 180°) c x = 134° (∠ sum of is 180°) d x = 66° (∠ sum of is 180°) e x = 60° (∠ in equilat. ) f x = 73° (isos. , base ∠’s =) g x = 36° (isos. , base ∠’s =) h x = 52° (isos. , base ∠’s =) y = 76° (∠ sum of is 180°) i x = 75° (∠ sum of is 180°) y = 75° (isos , base ∠’s =)

j x = 34° (∠ sum of is 180°) y = 34° (isos. , base ∠’s =) k x = 128° (isos. , base ∠’s =,

then ∠ sum of is 180°) l x = 81° (∠ sum of is 180°,

then isos. , base ∠’s =)2 112.5°3 a 60° b 150° c 15° d 75°

535Answers

1 a x = 130° (ext. ∠ of = sum of int. opp. ∠’s ) b x = 122° (ext. ∠ of = sum of int. opp. ∠’s) c x = 31° (ext. ∠ of = sum of int. opp. ∠’s)

d x = 110° (isos , base ∠’s =, then ext. ∠ of = sum of int. opp. ∠’s)

e x = 120° (∠’s at a pt add to 360°, then ext. ∠ of = sum of int. opp. ∠’s)

f x = 139° (vert. opp. ∠’s =, then ext. ∠ of = sum of int. opp. ∠’s)

2 a x = 68°, y = 44° b x = 111° c x = 130°, y = 50° d x = 62°3 x = 19° (vert. opp. ∠ ‘s =, then ∠ sum of is 180°)4 α = 50°

PUZZLE (page279)

Parallel framework 18



1 a Corresponding b Co-interior c Alternate d Alternate e Corresponding f Co-interior2 a b b c

c a d e3 a x = 70° (corresp. ∠’s =, || lines) b x = 60° (co-int. ∠’s add to 180°, || lines) c x = 73° (corresp. ∠’s =, || lines) d x = 119° (co-int. ∠’s add to 180°, || lines) e x = 100° (corresp. ∠’s =, || lines) f x = 95° (alt. ∠’s =, || lines)4 a x = 50°, y = 50° b x = 139°, y = 139° c x = 65°, y = 115°, z = 65° d x = 68°, y = 68°, z = 68° e x = 68° f x = 70° g x = 66°, y = 110° h x = 50° i x = 50° j x = 79°, y = 36°, z = 65° k x = 80° l x = 31°, y = 74°, z = 75° m x = 122°5 Translation6 Rotation7 Yes, because the two alternate angles are equal.8 No, because the two co-interior angles add to 182°, not

180°.9 Yes

10 q and r11 a and d; and c and f12 a 8 b 16 c 8


1 a 3x + 7x = 180° (∠’s on line add to 180°); x = 18°b 5x + 3x + x = 360° (∠’s at a pt add to 360°); x = 40°c x + 2x + 36° = 180° (∠’s on line add to 180°);

x = 48°d 2x + x + 60° = 180° (∠ sum of is 180°); x = 40°e 2x = x + 15° (alt. ∠’s =, || lines); x = 15°f 3x + x + 40° = 180° (co-int. ∠’s add to 180°,

|| lines); x = 35°g 2x + x = x + 40° (ext. ∠ of = sum of int. opp. ∠’s);

x = 20°h 5x = 3x + 62° (ext. ∠ of = sum of int. opp. ∠’s);

x = 31°i 3x + 6x = 180° (corresp. ∠’s =, || lines and ∠’s on

line add to 180°); x = 20°2 a x = 15°; the three angles are 90°, 60° and 30°. b x = 10°; the three angles are 70°, 60°and 50°.

PUZZLE (page281)

What day is it? See page 543 for answer.


DAILY

OBTUSE

C U

X

EH A L V E S

E

R

D

GREES

COINTERIOR

PAST

PARALLEL

REFLEX

V R T X

MOON

C O

MINUS

ANT

PERPENDICULAR

BEARING

SUBTRACT

ALTERNATE

EQUILATERAL

HAKA

L U

E X T

P L E M E N T R Y

N GDS P ORC O

M U T I P L Y

P O R C O

E ET S

O C K

O V EI O S E L S

R E

536 Answers

19 Angles 2 − polygons


1 a = 62°, b = 80°, c = 63°, d = 83°, e = 72°; 360°

2 a = 74°, b = 93°, c = 94°, d = 99°; 360°

3 a = 135°, b = 50°, c = 100°, d = 75°4 a = 130°, b = 110°, c = 120°,

d = 90°, e = 90°5 a Concave b Convex c Convex d Concave e Concave f Convex6 Yes

7 No, because a and b are equal (vertically opposite angles).

8 (A)


Chessboard squares

1

Sizeofsquare Number

1 × 1 64

2 × 2 49

3 × 3 36

4 × 4 25

5 × 5 16

6 × 6 9

7 × 7 4

8 × 8 1

Total 204

2

717


1 360°2 360°3 50°4 85°5 a 108° b 38° c 33° d 62°


1 540°2 900°3 a 3 b 180° c Quadrilateral d 360° e 360° f 5 g 360° h Hexagon i 720° j Octagon k 1080° l 360° m (n− 2)180°

4 a 50° b 120° c 90° d 102° e 210° f 85° g 120° h 124°5 a 4x − 20 = 360; x = 95° b 3x + 210 = 540; x = 110° c 6x + 30 = 720; x = 115°6 1800°7 32


1

Nameofpolygon Numberofsides

Sumofexteriorangles

Eachexteriorangle

Equilateral triangle 3 360° 120°

Square 4 360° 90°

Pentagon 5 360° 72°

Hexagon 6 360° 60°

Octagon 8 360° 45°

Decagon 10 360° 36°

2

Nameofpolygon Numberofsides

Sumofinteriorangles

Eachinteriorangle

Equilateral triangle 3 180° 60°

Square 4 360° 90°

Pentagon 5 540° 108°

Hexagon 6 720° 120°

Octagon 8 1080° 135°

Decagon 10 1440° 144°

3 a 30° b 150°4 a 160 b 185 366 247 The angles are not all equal to each other.8 a x = 120° b No, the interior angles are not all equal to each other.

9 a (7 − 2)×180

7

b 128 47

°

10 a Yes, because 15° is a factor of 360°.b No, because if the interior angle was 155°, the exterior angle would have

to be 25°, and 25° does not divide exactly into 360°.11 x = 72°, y = 54°12 x = 36°, y = 72°, z = 36°13 30 cm14 12 cm

537Answers

5 Yes, all quadrilaterals tessellate.6

7 a 6 b

PUZZLE (page291)

Irregular polygons1

2 The interior angles are not all equal, some are 90° and others are 270°.

20 Tessellation

1

2

The cross does tessellate.3 Yes

4 a

b Yes


9 Yes10 Yes

11 a 120° b 108°c They would need to be placed

together at one point, and the interior angles (108° each) are not a factor of 360° (angles at a point).

12

13 a The ‘Z’ shaped tetromino. b

8

538 Answers

PUZZLE (page295)

Chopping up trominoes123 (Other answers are possible.)


Hexagonal cobblestones

1

Numberofblackcobblestones

Numberofwhitecobblestones

1 6

2 11

3 16

4 21

5 26

6 31

2 Number of white = 5n + 1.

3Numberofblack

intoprowNumberofblack

inbottomrowNumberof

white

1 2 15

2 3 23

3 4 31

4 5 39

5 6 47

4 Number of white = 8n + 7.


1

2

3

4 a No b No c

5

6 a Equilateral triangle, square and hexagon.

b

PUZZLE (page297)

Hearts, diamonds, clubs and spades

PUZZLE (page257)

I have suctionThis machine sucks.

539Answers


Tessellating jigsaw pieces1

2

3 a The red and yellow pieces (the two on the right).b i The red piece tessellates. ii

c

d It must have the same number of indentations as protrusions; it must have matching (same size and shape) pairs of indentations and protrusions.

e Traditional jigsaw puzzles have straight edges − the pieces at the outside cannot be the same shape and size as ones in the middle.

21 Three dimensions

1 a Square b Yes2 a 8 b 12 c 6 d Rectangle e Yes3 a 4 b 6 c 4 d Yes4 Equilateral triangle5

6

7 a 5 b 8 c 5 d Yes8 a Yes b Isosceles9 a No b No c EH, EB, EC

10 a AE, BF, DH b DCGH c H11 a 10 vertices,

15 edges, 7 faces

b Equilateral12 a 45° b 45° c 45° d 90° e 60°

PUZZLE (page303)

Cheesy cylinder

U

S W

R V

T

Q

P

H

Rewi

F

E

GB

AD

C


Vertical cut

Vertical cut

Horizontal cut


1 302 283 a b c

540 Answers

4 a

b

5

6

2

3 2

51

4

C

F

B

G

D

H

E

EH

DC

B

FA

7

8

9 a

b c

10 a 8 b 24 c 24 d 811 a 8 b

12 a Spade b Club c Anchor


1 A cuboid.2

3 a B b

4 a E and F b

5

or

6 a b F

7

8

Six different answers are possible.9 a B and E b C and D

c Because they each have two sides that join up to a 3 m edge.

A D

CB

R

P

S

Q

A B

CD

H

F

G

E d

10

11 a 3 b

12

4 m

3 m

3 m

2 m

4 cm

2 cm2 cm

2 cm2 cm

2 cm

2 cm 2 cm

2 cm

541Answers

13 aA B

C

D

b There are four possible answers − with the extra square in positions A, B, C and D.


1 abc

2 a

b

c

3 a

b

4 a Clockwiseb Travel up in a straight line to

the top then turn right and drop down quickly. Go up again to the right, drop suddenly then go through a tight right-hand turn and go up again in a straight line. Go down while turning to the left, then go up while turning to the right. Go down and return to the start.

1 111

3 1122

111

3 333

332

333

1 111

111

111

CD

A B

A D

5

6 47 a b

8 a b

c

Rugby posts

Top Front Side

TV set(Cathode ray)

Bed

Netball hoop

Submarine

a

b

c

d

e

Side (right)Top Front Side (right)Top Front

Side (right)Top Front



542 Answers

9 a b1 112

2 111

1 211

10 a b

PUZZLE (page311)

The exposed surfaceA pentagon with one axis of symmetry.

22 Pythagoras

1 a i Hypotenuse is f. ii f2 = e2 + g2

b i Hypotenuse is r. ii r2 = p2 + q2

c i Hypotenuse is x. ii x2 = y2 + z2

d i Hypotenuse is h. ii h2 = g2 + i2

e i Hypotenuse is c. ii c2 = b2 + d2

2 a 10 b 15 c 50 d 9.220 e 11.31 f 37.01

PUZZLE (page319)

sides of the diamond148 cm

1 a b 3.425 m

2 a b 3.622 m

3 11.66 m4 a

b 4.472 m5 a

b 25 km c 50 km6 a 125 m b 45 m



1 15 cm2 7.810 cm3 9.220 cm4 25 cm5 11.31 cm

6 13.45 cm7 34 cm8 101 cm9 11.77 cm

10 778.2 cm


1 3 m2 6 m3 5 m4 24 m5 12 m

6 4.359 m7 23.52 m8 7.937 m9 3.439 m

10 488.5 m


1 4.34 m (3 sf)2 112 mm3 2.21 m (3 sf)4 58.83 m5 38 m6 4367 mm

7 6.3 m (2 sf)8 105 mm9 479 m

10 a 6.15 m b 0.10 m


7

x2 = (2.8)2 − (1.6)2 = 7.84 − 2.56 = 5.28

x = 5.28 = 2.3 m (2 sf)

8 40 m9 The distance from the farm to the showgrounds = 2× 692 + 822 . The return distance is twice this

− i.e. 214.3 km, which exceeds the range. It would not be safe to make this flight.

10 275 mm (to the nearest mm)11 a

b 84 km12 a Opening b

Using Pythagoras to calculate the widest diagonal measurement of the aperture:

x2 = 202 + 52 = 425 x = 425 = 20.6 The two largest measurements

of the parcel are both larger than this, so the parcel will not fit through.

Stay3.7 m

1.4 m

Polex

� �

Wall3.6 m

Polex

0.4 m

4 m

2 mx

Bracing

24 km

7 km

Yacht

Base

x

N

Heightx

1.6 m

Tape = 2.8 m

A

B

C

12 km31 km42 km

29 km

20 cm

5 cm x

543Answers

PUZZLE (page322)

The wooden lamp-post16.88 cm

PUZZLE (page322)

The Jurassic Park puzzleSee page 556 for answer.


The tractor and the gate(i) When driven as close as possible to the

left: 2.83 m.(ii) When driven along the middle: 2.96 m.

1 422 + 292 = 1764 + 841 = 2605 512 = 2601 422 + 292 ≠ 512, so the triangle is not right-angled.2 182 + 38.52 = 324 + 1482.25 = 1806.25 42.52 = 1806.25 182 + 38.52 = 42.52, so the triangle is right-angled.3 (D)4 652 + 722 = 4225 + 5184 = 9409 972 = 9409 There is a pair of opposite angles that are each 90° and the other pair are

equal because of alternate angles, so all four angles are 90°, and therefore the parallelogram is a rectangle.



1 Teresa has forgotten to square the side lengths. The correct answer is 17.

2 a It is impossible for the third side of a triangle to measure more than the sum of the other two sides (4 cm in this example).

b The square root key − i.e. x

c 2.828 cm (4 sf)3 a x is one of the two shorter sides and must be less

than the hypotenuse, which is 6 cm.b He used the + key instead of the − key.

4 a x = 5 cm, y = 13 cm b x = 12 cm, y = 5 cm c x = 25 cm d x = 7 cm, y = 25 cm e x = 17 cm

5 a

b 15.65 units c 3.91 m6 The height of triangle P is 4 cm (from x2 = 52 − 32). The height of triangle Q is 3 cm (from x2 = 52 − 42).

Area of triangle P = 12

×6× 4 = 12 cm2 .

Area of triangle Q = 12

×8×3 = 12 cm2.

Both triangles have the same area.

P


The ants and the sugar bowl 7.810 m

PUZZLE (page281)

What day is it? Today is yesterday tomorrow.

23 Trigonometry 1 − an introduction


1 a Hypotenuse b Adjacent c Opposite2 a Opposite b Adjacent c Hypotenuse

3 a Adjacent b Hypotenuse c Opposite4 a Opposite b Hypotenuse c Adjacent

5

Triangle Hypotenuse Oppositeside Adjacentside

∆PQR PR QR PQ

∆STU TU SU ST

∆VWX VX VW WX

∆ABC BC AB AC

544 Answers


40° right-angled triangles

1

Triangle o hoh

(to2dp)

a 45 mm 70 mm

4570

= 0.64

b 37 mm 57 mm 3757

0 65= .

c 34 mm 54 mm 3454

0 63= .

d 38 mm 59 mm 3859

0 64= .

e 44 mm 68 mm 4468

0 65= .

f 88 mm 140 mm 88140

0 63= .

g 22 mm 35 mm 2235

0 63= .

2 0.642 787 610

1Givenangle 10° 20° 30° 40° 50° 60° 70° 80°Oppositeside 13 mm 26 mm 50 mm 59 mm 67 mm 72 mm 76 mmHypotenuse 76 mm 76 mm 76 mm 76 mm 76 mm 76 mm 77 mmRatioofoppositesidetohypotenuse,asadecimal

0.17 0.34 0.5 0.66 0.78 0.88 0.95 0.99

2

Givenangle sin(to9dp) sin(to2dp)

10° 0.173 648 178 0.17

20° 0.342 020 143 0.34

30° 0.5 (exactly) 0.5 (exactly)

40° 0.642 787 609 0.64

50° 0.766 044 443 0.77

60° 0.866 025 403 0.87

70° 0.939 692 620 0.94

80° 0.984 807 753 0.98


The cos ratio

1

Triangle a h ah

(to2dp)

a 22 mm 52 mm 2252

0 42= .

b 29 mm 68 mm 2968

0 43= .

c 35 mm 81 mm 3581

0 43= .

d 38 mm 89 mm 3889

0 43= .

e 51 mm 120 mm 51120

0 43= .

f 23 mm 29 mm 2329

0 79= .

g 64 mm 80 mm 6480

0 80= .

h 42 mm 52 mm 4252

0 81= .

i 81 mm 101 mm 81101

0 80= .

2 cos (65°) = 0.422 618 2623 cos (37°) = 0.798 635 510



1 187.94 m2 500 m3 273.60 m4 76.60 m

5 a 2.12 m b 3.90 km c 0.47 m d 5.81 m e 163.16 km f 34.79 cm


1 0.70712 0.48483 0.1392

4 0.99255 0.24196 0.7986

7 0.63208 0.9995


1 2.270 cm2 1.035 cm3 7.552 m

4 22.07 cm5 11.47 cm6 6.676 m

7 3.719 km8 2.560 cm


1 68.4 m2 866 m3 751.76 m4 64.28 m

5 a 2.12 m b 2.25 km c 0.17 m d 15.97 m e 136.92 km f 34.79 cm

545Answers

1 a = 11.92 m2 b = 8.660 m3 c = 17.39 m4 d = 881.6 m5 e = 21.45 m


1 0.70712 0.87463 0.99034 0.1219

5 0.97036 0.60187 0.77498 0.0332


1 4.455 cm2 3.864 cm3 9.326 m4 11.74 cm

5 8.030 cm6 2.697 m7 1.208 km8 3.942 cm


1 a = 8.988 m b = 4.384 m 2 c = 7.314 m d = 6.820 m3 e = 18.47 cm f= 23.64 cm4 g = 42.40 cm h = 26.50 cm5 i= 0.6180 km j = 1.902 km6 k = 0.1040 m m = 0.4891 m

7 n = 5.346 m p = 2.724 m8 q = 5.286 m r = 5.871 m9 s = 20.08 cm t = 80.53 cm

10 u = 17.10 m v = 13.85 m11 w = 14.34 m x = 4.386 m12 y = 8.211 cm z = 3.485 cm


1 1.2 m2 3.346 m3 1.145 m4 5.955 m5 14.16 m6 9.235 cm7 2.783 m8 0.726 m9 The two walls measure 5.92 m and

12.69 m. The total length is 19 m (to the nearest metre).

10 a 7.55 m b 4.90 m11 a 7.654 m b 36.96 m12 a 4.774 m b 3.538 m c 8.924 m


1 0.17632 0.62493 14 0.14055 1.600

6 1.9977 6.3148 11.439 28.64

10 44.07


6 f = 16.63 m7 g = 13.49 m8 h = 123.1 m9 i = 5.412 km

10 j = 59.40 cm


1 sin2 tan3 tan4 sin

5 cos6 tan7 cos8 tan


1 2.12 m2 2.62 m3 6.55 m4 2.51 m5 10.5 m

6 8.91 m7 4.73 m8 1.34 m9 2.14 m

10 4.19 m

24 Trigonometry 2 − calculating any side length

Hypotenuse BC DE GH

Oppositeside AB EF GI

Adjacentside AC DF HI

1 a = 5.88 cm2 b = 4.46 cm3 c = 4.79 cm4 d = 3.93 cm5 e = 5.25 cm6 f = 4.60 cm7 g = 6.34 cm8 h = 29.54 cm9 i = 13.49 cm

10 j = 9.37 cm

1 2.14 m2 40 mm3 52 mm4 2.18 m5 27.5 m6 a b 12 cm

7 a

b 5.3 km c 2.8 km




40°14 cm

x

28°

6 kmx

y

8 55 m9 942 m

10 17 m11 32 m12 3.3 m13

The end of the nail will be 12.9 mm from the surface of the gib-board so will not go through all of it.

40°20 mm

x

Width of gib-board thatnail will go through

546 Answers

1 a 7.832 m b 13.86 cm c 4.801 cm d 2.996 cm e 17.92 m f 11.41 cm g 13.82 m h 123.2 cm i 8.115 cm j 7.132 cm2 a 29.71 cm b 75.40 m

1 6 km2 a

b 15 m3 176 mm4 2.98 m5 a

b 14.68 m



15°4 m

x

8.42 mx

35°

6 a b 79 km

7 1662 m8 a b 3.84 m

20°

74 km x

19°

1.25 m

x

9 13 m10 a 9.47 m

b You have to assume the rope is straight, which is unrealistic unless there is a very strong current, and even then it is likely to sag a bit. You have to assume the seafloor is flat, which is unlikely if it is rocky. You have to assume the end of the anchor is touching the seafloor, which is probably realistic because of its weight.

11 711 m 12 11.86 m


The cuboctahedron1 Tetrahedron2 No3 64 Equilateral triangles; 85 24; yes6 127 473.2 cm2

PUZZLE (page356)

Trig decoding See page 556 for answer.

25 Trigonometry 3 − calculating angles


1 6.2°2 52.5°3 29.2°4 24.0°5 43.2°6 53.9°7 30.0°8 29.6°

9 0.5°10 12.9°11 48.9°12 80.4°13 90°14 30°15 45°


Inverse tan check2 It should be 35°.

3

710

4 0.75 34.99° (2 dp)


1 48.6°2 29.0°3 67.4°4 59.0°5 54.0°6 32.0°

7 62.7°8 60°9 9.3°

10 57.7°11 31.2°12 49.7°


1 45.6°2 51.3°3 36.9°4 40.3°5 55.7°

6 44.9°7 43.0°8 77.0°9 30.4°

10 47.9°

547Answers

1 a

b 6.9°2 a b 23.0°

3 28.4°4 26.6°5 38.7°

6 a

b

cos(A) = 1.35

= 0.26

A = 74.9°

This angle is in between 74° and 78° so the ladder has been positioned safely.


10 m1.2 m

A

1.8 m

4.6 m A

WallLadder

5 m

1.3 m

A

7 039°8 a

b 16.1°9 a 75.5°

b co-int. ∠’s add to 180°, || lines, or ∠ sum of quadrilateral is 360°

10 24.0°11 82.7°

PUZZLE (page362)

The slipping ladder2.4°

Finish

AStart

90 25

26 Construction and loci


6 5.5 cm


The circumcircle of a triangle

4 The lines should intersect at the same point. When three lines pass through one point they are said to be concurrent.

5 Draw a circle with its centre at the point of intersection of the three perpendicular bisectors, and radius set to be the distance between this centre and any one of the three vertices A, B or C.


The dead centre

The perpendicular bisector of a chord in a circle is always a diameter of that circle. Two different diameters of a circle must intersect at the centre of the circle.


1 c 5 cm2 c 6 cm5 c Yes6 d Yes

8 d ABC = °60 (∠ of equilat. ∆) ABE = °30 (∠ of equilat. ∆ has been bisected)

CBE ABC ABE = +

= ° + °

= °

60 30

90

e BE


1 a b

c d

2

A

B

CD

SR

3

4

5 a

b

P

Q

HG

A

B

AA

B

548 Answers

9

10

11

PUZZLE (page375)

Loci in three dimensions

1 A sphere.2 A cylinder, infinitely long.3 A plane (flat surface),

perpendicular to, and bisecting, the line joining the two points.

c

6 ab c 4

A

B

P

Q

S

R

P

Q

S

R

7 abc

de f

g

8 abc

P Q R P Q R P Q R

P Q R P Q R

P Q R

P Q R

C DC D

C D

A

l1

l2

A

l

D

E

F


1 a (E) b (H) c (B) d (F) e (A) f (C) g (D) h (G)2

3

4

E F

DC

B

A

S1

S2

5

3 m

Scale: 1 unit = 1 m

549Answers

6

0 100 200 300

Scale

N

E

S

W

25

27

7 a The places where both buoys can be seen from the sea surface.

b The places where B1 can be seen and B2 cannot be seen.

8

9

W B P

10 a

b

0 2 4 6

Scale

B N

m

0 2 4 6

Scale

B N

m

27 Transformations 1 – symmetry and congruence

1 a 1 b 2 c 0 d 4 e 22 a 3 b 4 c 2 d 6 e 13 a 2 b 6 c 6 d 2 e 6 f 8 g 4 h 14 a 1 b 2 c 8 d 4 e 10 f Infinite g 8 h 32 i 1 j 15 a (A)

b For (B) the total order of symmetry is 10, for (C) it is 38.

c Rotational symmetry helps with the balancing of the tyres.

6

7

8

a b c d e f g h

Orderofrotationalsymmetry

1 2 4 2 1 1 2 1

Numberofaxesofsymmetry 1 0 4 2 1 1 2 0

Totalorderofsymmetry 2 2 8 4 2 2 4 1

9

10 One possible answer is shown.

11 Yes



1 a Rhombus b Isosceles trapezium c Square d Arrowhead e Rectangle f Parallelogram g Kite

550 Answers

2

Orderofrotationalsymmetry

Numberofaxesofsymmetry

Totalorderofsymmetry

Parallelogram 2 0 2

Kite 1 1 2

Rectangle 2 2 4

Square 4 4 8

Arrowhead 1 1 2

Isoscelestrapezium 1 1 2

Rhombus 2 2 4

1 a x = 45° b x = 103°, y = 77° c a = 8 m, b = 12 m d a = 12 cm, b = 10 cm, x = 90° e x = 70°, y = 42°, z = 68° f a = 5 cm, b = 9 cm, c = 6 cm g x = 15°, y = 49°, z = 28° h x = 23°, y = 23°, z = 53°2 a x = 85° b x = 76°, y = 68°

c x = 5 cm, y = 4 cm, z = 8 cmd w = 90°, x= 70°, y= 30°,

z = 60°e w = 43°, x= 80°, y = 47°,

z = 10°f x = 90°, y = 37°, z = 47°

g x = 28°, y = 34°, z = 118° h x = 26°, y = 110° i x = 42°, y = 28°3 x = 38°, y = 52°, z = 52°4 x = 45°, y = 90°5 a x = 34°, y = 34°, z = 112° b a = 12 m, b = 9 m, c= 6 m c x = 66°, y = 30°6 a x = 75° b x = 33° c x = 80° 7 a 10 cm b 8 cm c 4 cm d 8 cm

e Yes, the parallelogram has half-turn symmetry so the opposite angles must be equal.


1 a Isosceles trapeziumb Kite, arrowhead, isosceles

trapeziumc Square, rhombusd Parallelogram, rectangle,

rhombus, square2 No3 Yes4 No5 No6 a Kite (or arrowhead) b Isosceles trapezium c Rectangle d Rhombus e Square f Parallelogram 7 a the same size b sides c diagonals

d bisects (and is perpendicular to)

8 Rhombus9 Square

10 Rhombus11 Arrowhead


PUZZLE (page387)

Ringed trapezium20 cm


1

2

3 a b

c

4 a R b P c D d C5 a P b Q c 270°6 a E b AJ c B and G d 4 cm e 2 cm f 10 cm7 a F b G c E8 a

b Reflection in the line y = x (a diagonal line through the intersection of the x- and y-axes).

m

OC

PPP S

RQ

Q’

R’ S’

y

x

551Answers

9 a The sloping sides (e.g. BC) are longer than the horizontal and vertical sides (e.g. AB).

b

c i D ii H iii FE

m

C D

E

F

GH

A

B

d

10 a 180° b 90° c 270°

m

C D

E

F

GH

A

B

D’B’

A’

E’

F’

H’


1

2 a

23

b

21−

c

25

d −

−

22

e

05

f

−

20

g −

41 h

31

3 a b

c d

e

4 a

23

b R c PS

5 (6, 3)6 (–3, 3)


The knight swaps corners

1 12

12

12−

− −

−

, ,

−

−, , ,2

121

21

−

−, 21

2 a 6

b 12

21

12

12

, , ,

−, ,

12

12

Other answers are possible.

a b c

d f g

h

e


1 a,b

c Reflection in the x-axis.2 a,b,c

b F′ = (–5, 1) c F′′ = (–3, 5)

A

B C

D

B’

D’ A’

C’ B”

D”A”

C”

R

Q

P

S

R’P’

S’

S”

Q’

Q”

P”

R”

3 a,b

c Rotation of 180° about (0, 0).4 a, b,c

b R′ = (7, 5) c R′′ = (–7, 5)5 Yes

D

E

FF’ D’

E’

E”

F” D”

P

Q

R

P’ R’

Q’Q”

R” P”

552 Answers

InvestIgAtIon (page 396)

Frieze patterns 2 a 5 Half-turn rotation

b 6 Half-turn rotation and reflection in a perpendicular mirror line

c 4 Reflection in both a centre and a perpendicular mirror line

d 1 Translation onlye 7 Simultaneous reflection in a centre mirror line

and a translationf 3 Reflection in a centre lineg 2 Reflection in a perpendicular mirror line

1 a 1 Translation onlyb 4 Reflection in both a centre and a perpendicular

mirror linec 2 Reflection in a perpendicular mirror lined 3 Reflection in a centre linee 5 Half-turn rotationf 7 Simultaneous reflection in a centre mirror line and

a translationg 6 Half-turn rotation and reflection in a perpendicular

mirror line

28 Transformations 2 - enlargement


1 O, U, S2 A3 a Yes b Yes c No d Yes4 a 2 b 4 c 2.55 a 4 b 1.5 c 1.5 d 46 a 3 b 2.5 c 1.2


typographical type sizes and line weights

1 48 point text is about 12 mm high, so 1 mm = 48 ÷ 12 = 4 points.

2 25 mm3 3.6 times larger4 0.75 mm5 4.5 point6 No, it would not fit, because as well as the height

increasing by 60% (a factor of 1.6), the width will increase too by the same factor, and the spaces between the lines.

It would take up almost 2 1

2 times as much space.

6 a,b

c Translation by the vector

55

.

d By adding the two vectors -

13

and

62

.

A

C

D

D’A’

C”

D”

B”

B

B’

A”

C’

NM

LM’

L’N’

M”

N” L”

7 a,b

c Rotation of 270° about (0, 0).d One rotation followed by

another rotation about the same point is equivalent to a rotation through the sum of the two angles.

8 a

m2

m1y = –3

y = 1

Y

ZX

Y’X’

Z’

Z”X”

Y”

A D

B C

D

CB

A

9 b

Reflection in m4

c

Rotation of 90°10 a B (rotation) b Yes

J

K

7 a 2 b 3 c 2.4 d 1.5 8 a 2 b 1.259 a 90 mm b 105 mm c 12 cm d 112 mm by 76 mm

b 08

c t = 2r

553Answers


B’A

C

A’

C’D D’X

B


1 a bcd

B’

A

B

A’

C’

XC

1

2

3

4 5

6

B’

A

CB

C’

EF

F’

E’

X

D

D’

A’

Q’

PSP’

R’

Q

S’

XR

B’A’

C’D’

B

C

A

DX

B’

C

BA’

C’D’

XA

D

H’

I

H

G’ I’

G

XB’

A

C

A’

C’

X

B

efg h

2 a (4, 5) b (2, 0) c (7, -2)3 (7, 1)4 (3, 1)5 (6, 3)


O

O O

O

12345

O

554 Answers

O

O

O×O

6789


1 a E b D c T d EF e EDF f 32 a R b S c L d SR e 2

3 a 2

b AB = 1.5 cm, DE = 3 cm c No d No

e BAC = 45°, EDF = 45° f Yes g Yes h Clockwise i Clockwise j Yes

4 ‘Under enlargements, lengths of lines are not invariant, but sizes of angles are invariant. This means that when figures are enlarged, they change in size but remain the same shape’.

5 30°


1 a x = 10 b x = 44 c x = 27, y = 45 d x = 9

e x = 75, y = 45 f x = 35 5

9, y =

42 2

3

2 a Scale factor = 2, a = 40°, b = 90°, x = 14b Scale factor = 1.5, a = 80°, x = 13.5c Scale factor = 1.25, a = 85°, b = 95°, x = 37.5, y = 75,

z = 62.5 d Scale factor = 1.25, a = 100°, b = 130°, x = 40, y = 42.5

1 a bc

Scale factor =

12

Scale factor =

13

Scale factor =

35

2 abcd


A B

C C’

B’O

A’

D

F

E

F’

D’

O

E’

R

P

R’

P’

S

S’

O

S’

O

Q’

P’ R’OA’

C’B’E’D’

F’ O

G’

H’

G’

I’

O

555Answers

2 a b

Scale factor = -2

Scale factor = -3 c

Scale factor = -1

d

Scale factor = -23

3 a -1 b Rotation of 180° about O.4 A′ = (10, 4), B′ = (10, 0), C = (6, 4)

efg

3 a Scale factor = 34

, x = 18

b Scale factor = 12

, x = 9

c Scale factor =

23

, x = 18, y = 24

A’

B’

O

C’

D’

Q’

O

P’ R’

S’E’ F’O

�

D’


1 ab

cd

e f

g

O O

O

O

O O

O

B’C’

A’B C

A D

D’

O

S’ P’

Q’R’

Q

P

R

SO

G’F’

E’H’

E

F

H

G

O

L

M

NM’

N’ L’

O

556 Answers

6

a Rhombus b G′ = (6, 0)

c (6, 3) d -12

7 a Rectangle b (3, 1) c -1 d C′ = (-2, -1), D′ = (-2, 3)

5

a -2 b (3, 1)

y

x

R’

Q’

P’

P R

Q

y

x

E

F

G

H

G’

H’E’

F’

PUZZLe (page 322)

the Jurassic Park puzzle Do you think he saw us?

PUZZLe (page 356)

trig decodingWhen the going gets tough the tough get going.

29 Statistical literacy - interpreting graphs and reports


1 a Life expectancy has been increasing for all groups from 1951 to 2001.

b Maori males 1966-71 and 1991−96; Maori females 1991−96.

c It has remained about the same - approximately 5 years apart.

2 a To provide a standard way of comparing charges; it is easy to find the US$ exchange rate in most countries.

b $400 c Korea d United Kingdom

e The fixed charge is extremely high compared with the usage charges.

3 a 71%b No information.c As the fee for dumping waste

increases, the amount of waste dumped per person decreases; or the lower the charge, the more rubbish is dumped.

d Waitakere - not much waste is being dumped (there is a high fee for this) and it is likely they are encouraging residents to recycle.

4 a 13% b 1995

c That is where the graph is steepest.

d In order to estimate the total number of computers in New Zealand households, information about the number in each individual household would be needed − many households have more than one computer.

e The graph would start levelling out and become almost horizontal. The percentage cannot be any higher than 100%.

5 a Stratosphere; 20-30 km.b More in the spring. In spring

the maximum level exceeds 15 mPA; in autumn it is less than 15 mPa.

6 a US$2300 b United States and Luxembourg c Korea

d Japan and Finland are closest; Spain and Italy could also be mentioned.

e The higher GDP per person, the more that is spent on health per person - ‘rich countries spend more on health per person’.

557Answers


1 a The exact number (as an average rate per day) is 71 646 ÷ 365 = 196 (to the nearest whole number).

b There is no information given about the numbers of learner drivers caught for any of the years 2003, 2004, 2005 and 2006. There could have been decreases from year to year in that period, whereas if numbers increase steadily that means the number each year is higher than for the previous year.

2 a If the survey was held in partnership with an exhibitor at the Ideal Home Show, it may have been taken from people who have been looking at the anti-snore bedroom.

b To get people thinking about their partner’s snoring, and hence be receptive to purchasing anti-snoring products.

c No, the survey says nothing about the actual numbers of people who snore. For all we know, only 100 of the 2000 people surveyed snore, and 80 of their partners gave that response.

3 a 52.7%b The Aucklanders are opposed to a new fuel tax

because they would be the only ones who would have to pay it; the people outside Auckland are not affected financially, but may in principle approve of better public transport for environmental reasons, which benefit everybody. Some may think it is a good idea because they do not like Aucklanders!

c If the numbers in each group surveyed (Aucklanders and ‘Rest of NZ’) were equal you would expect the overall results for those agreeing, for example, to be exactly half-way between the two. The average of 26.8% and 64.5% is 45.65%, but the overall result for ‘Agree’ is 52.7%, which is much closer to the ‘Rest of NZ’ result, showing more respondents were in this group than were Aucklanders.

d Given that the percentages are ‘exactly’ correct to 1 dp - e.g. ‘exactly 52.7%’ - then there must have been at least 1000 people surveyed. A possible example is 527 people out of 1000. There is no other number less than 1000 that when placed in a denominator will give exactly 52.7%.

4 a 20−24b About 15−20%.c Not many older people ride motorbikes compared

with younger people, so you would expect the proportion, and the total number, of accidents to be much higher for younger people. However, for the few older people who do ride motorbikes it could be very risky.

5 a The period of decrease is from 1985 to 2001.b There may be fewer motorbikes on the road now

than there were 20 years ago; the student is using the data for 1985 to 2001 to make that statement − since 2001 the number of these accidents has started to increase again, and also the 20-year period ends now, not in 2001.

c i The graph shows about 1000 fatal or injury accidents for motorcyclists in 2006. In the same year there were 14 907 of these accidents for car and van drivers. 14 907 ÷ 1000 = 14.907 ≈ 15.

ii No, it is much riskier to ride a motorbike than drive a car because there are comparatively fewer motorbikes on the road compared with cars. In fact, the New Zealand Travel Survey indicates that, on average, the risk of being involved in a fatal or injury crash is more than 14 times higher for a motorcyclist than for a car driver over the same distance travelled.

558 Answers

2 a 2b 2 goals per game - this is

shown by the column with the most dots.

c 48 d 6−0, 5−1, 4−2 e 1173 a 360° ÷ 120 = 3° b 18°

c

30 Displaying statistical information

1


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ACTUnited Future

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Labour

4 a

b Paragliding cannot be represented by a bar graph because there is no ‘range’. It could either be represented by a single point, or a small range like $135−$145 could be used instead.

5

6 a 13 b 55 c 64 or 657 Over this period, people were eating more chicken and less

sheep meat as a proportion of their total meat consumption.8

559Answers

9 See the spreadsheet NZRegionalCouncilpopulation2006Answers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.a West Coast Region. b

The graph shows high concentrations of population in the Auckland, Wellington and Canterbury regions in particular; most other Regional Councils have a much smaller population.

c 149 750d 258 725e The mean is considerably higher than the median because of the one extreme value of the Auckland

Region pulling it up.f 4 139 600. The individual regions have been rounded to the nearest 100, and there has been more

rounding up than rounding down, which is why the total of the South Island and North Island regions has gained an extra 100. The 16 regions do not include other New Zealand Territories or Dependencies, such as the Chatham Islands and the Ross Dependency in Antarctica.

10 See the spreadsheet PassengerArrivalsatAucklandAirportbycountryAnswers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.

a

560 Answers

b

c India, by 17.1%. See the spreadsheet for the calculations.d The data for New Zealand should be excluded.

f 29%g Europe and Asia would be under-represented because

there are countries not listed, for example France in Europe, Thailand in Asia, and at present these would be included under ‘Other’.

11 a Natural forest: 86.4° Planted production forest: 18° Total pasture and arable land: 187.2°

Other land: 68.4°b It is unlikely that forests would be cut down for housing, so

the proportion of pasture and arable land would decrease, while other land (which includes urban areas) would increase.

e

561Answers

12 See the spreadsheet Digitalcameras-weightvspriceAnswers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.a

b There is a weak positive relationship - as weight increases so does price, so in general heavier cameras tend to be more expensive. Some of the light cameras are very expensive, and so are heavy ones, but there is so much scatter that other factors must be involved. Moana, Nigel and Olinda are all (partially) correct.


1 a Heading is wrong (Honolulu is not in Australia); scale on vertical axis is uneven.

b No indication of size of angles, not circular.

c Scale on vertical axis does not start at 0; there is no scale on the horizontal axis so cannot tell the period of time.

d Difficult to read off scale on vertical axis because cannot judge where the top of these symbols is; misleading size of oil barrels - they should be the same width.

2 Some months are missing on the horizontal axis - e.g. April; the scale on the vertical axis is incorrect (130 is missing); the heading mentions ‘weekly’ spending while the graph implies it is ‘monthly’; the heading mentions TV4 while the graph is labelled TV6.

3 The sectors are not proportional to the number of stores; the heading should not occupy a sector in the pie graph, because the proportions have to fill the entire circle.

4 a The graph at the bottom-left of the screenshot is the most accurate, because it clearly shows that most drivers do wear seatbelts. However, the top-right graph is the most useful as far as reading the actual percentages concerned.

b For the graph at the top-right the vertical scale starts at 88% instead of 0%, and this exaggerates the increase; for the graph at the bottom-right a curve has been applied to the data and this implies there was a decrease from 2001 to 2003, and also the graph reaches the top in 2005, which could imply no more improvement in the seatbelt wearing rate is possible.

5 a The graph has been rotated so that it tilts downwards; most of the vertical scale is missing.

b 97 or 98 km/hc Place a ruler on the graph so

that it touches the top-front of the 2002 column and is parallel to the Year axis. Read off the mean speed where the ruler crosses the vertical axis.

6 ‘Most’ means more than half.7 a The ‘Yes’ column is about

twice the height of the ‘No’ column.

b ‘Little interest in a new Aquatic Centre’. Other answers are possible.

8 a It does not say more than ‘what’. More than previously, more than other brands, etc.

b A product can only be 100% pure.

c It does not say what the ‘lifetime’ is of. The charger, the batteries, the user?

562 Answers

9 a 1b Data for fatalities and serious injuries for several years before 1989.

10 50% have to be in the bottom half, just as the other 50% have to be in the top half. It does not matter how high or low the educational standard is, these percentages always apply.


1 a August b September c $20 d May e Decreasing2 a

b Both fatalities and serious injuries are decreasing in the long term.

5

10

15

1989 1990 1991 1992 1993 1994 1995

Number ofinjuries

Number offatalities

AccidentsonNorthernMotorway

3 See the spreadsheet Newcarregistrations1987-2006Answers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.

a

Year Totalcars Carspreviouslyregisteredoverseas

Newcars

1987 90 000 12 000 77 000

1988 89 000 17 000 72 000

1989 135 000 51 000 84 000

1990 160 000 8 5000 75 000

1991 103 000 47 000 56 000

1992 92 000 39 000 53 000

1993 98 000 44 000 54 000

1994 124 000 62 000 62 000

1995 147 000 81 000 66 000

1996 176 000 112 000 64 000

1997 156 000 97 000 59 000

1998 154 000 100 000 54 000

1999 189 000 131 000 58 000

2000 174 000 116 000 58 000

2001 187 000 129 000 58 000

2002 201 000 136 000 65 000

2003 227 000 157 000 70 000

2004 229 000 154 000 75 000

2005 230 000 152 000 78 000

2006 200 000 123 000 77 0003 b

c The number of new cars registered each year has remained about the same. The steady increase in the number of cars previously registered overseas is responsible for the overall increase in the total number of cars registered.

d You would need to know the number of cars registered in the years before 1987, and you would also need to know the number of cars de-registered each year.

563Answers

4 a

b Steady long-term trend; seasonal (weekly) variation.c Tuesday - fewest sales so plenty of seats available.

31 Working with data

1 a 4 b 15 c 358 d 913 g e 39.375 f $31.902 a 40 b 8.7 c 92 d 23 e 53 m f 6173 a 6 b 8 c No mode d Two modes - $2 and 20 cents4 2 hours 5 minutes5 $46176 32°7 a $2456 b $30.328 a 6058 km b 4093 km c Perth d Papeete

EXERCISE 31.01 (page 439) EXERCISE 31.02 (page 441)

1 a 26 b 0 c 23 d The median

2 Mean = median = mode = 63 For example, 5, 8, 8, 8, 114 Mode5 a 1.6 litres b 1.65 litres

c The mode - this would be the size of which there is most stock, and the wrecker does not sell 1.65 litre engines.

6 a 10 b The median is most typical,

and is not influenced by extreme values such as 23 in this example. It is probably a coincidence that the mode is 17.

7 a 2 b 3 c 3. 3d The mean, because the total

number of people to cook for can be worked out from the mean.

e The mode, this is the size table that will be most useful.

8 (A)9 26

10 18011 45 kg12 a 319 (to the nearest whole

number) b 331 c The mean d 17 00013 614 56 kg15 61.2 points


1 See the spreadsheet BuildingconsentfeesAnswers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.a-c See the first Worksheet in the spreadsheet.b Mean = $695.72, median = $704.53,

mode = $726.75.d See the second Worksheet in the spreadsheet.

2 See the spreadsheet DivingscoresAnswers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.a,b See the spreadsheet.c Annetted Annette, Nga, Helga, Charlotte, Teresa, Moana,

May-Li, Denise.e Helga, Teresa, Moana, Nga, Charlotte/May-Li

equal, Denise.f The winner would now be Nga.

564 Answers

1 a 14 b 15 c 10 d 54 e 1182 a LQ = 9, UQ = 20, interquartile range = 11

b LQ = 3.5, UQ = 10.5, interquartile range = 7c LQ = 40, UQ = 56, interquartile range = 16d LQ = 2.5, UQ = 8, interquartile range = 5.5e LQ = 25, UQ = 91, interquartile range = 66

3 a 13 secondsb LQ = 60 seconds, UQ = 66 seconds

c 6 seconds4 a LQ = $77, UQ = $120

b Range = $255, interquartile range = $43c He will look at those between $120 and $77 in price.

1 a

b 42 c The 1960s; the leaf for the 1960s is longer than the

others.2 a Keith - it is easier to read information this way, and

Kevin’s diagram would have a very long stem and very short leaves.

b



190 0 2 5191 1 1 8192 4 5 7 9 9193 1 2 4 9194 1 5 6 7 8 8195 3 6 9196 1 3 3 4 5 5 9197 4198 2 4 6 7 8 9199 0 2 4 8

6 05 30 457 00 12 24 36 488 00 12 24 36 489 00 15 30 45

10 00 20 4011 00 20 40

3 a

b The Mathematics marks are more spread out than the English marks.

4 a 135 b 125.5

c i 86 ii 106 iii 135 iv 29

English Mathematics1 4 8 8

8 2 5 59 7 3 9

9 7 2 1 4 4 6 6 79 9 8 5 4 4 4 3 0 5 3 8

8 3 2 0 6 0 1 2 88 5 4 0 7 3 4 5 5 6

2 8 7 99 2 8


1 a

b Males - most (three-quarters) smoked less than 45, whereas more than half the females smoked more than 45. That is, the males’ UQ < females’ median.

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2 a Median = 18, LQ = 10, UQ = 30 b

3 a Action b Others4 See the spreadsheet Creditcardfees

Answers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.a Median = $60, LQ = $28, UQ = $80. Note: if the

formula =QUARTILE(D2:D75,1) is used in the spreadsheet it gives $28.25.

10

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Timetotraveltoschool

565Answers

b 0 is an outlier - it represents a credit card with no yearly fees; the Platinum card fees are also outliers, they represent very expensive credit cards for ‘high net worth’ individuals.

c

d There is a high concentration of fees between the $25 to $80 range (approximately), whereas the high-fee cards seem to be spread out. The box and whisker diagram ‘over-simplifies’ the data to some extent. It does not show that there are hardly any fees over $105, and does not show the clumping of low-fee and medium-fee cards, whereas the dot plot shows these features very clearly.

e If the term ‘on average’ refers to the mean credit card fee then it would be higher than $60 because the few cards that have extremely high fees would pull the mean upwards from $60. If ‘average’ is being used to mean ‘typical’, then this is also incorrect because most fees are around $25 or around $80.

5 a

b No. You cannot tell whether the high scores for went with low scores against, for example.

6 a $193 b $53 c Adults d $15 e $88

f The adult box plot has no bottom whisker because the lower quartile and the bottom value are the same - they are both $0.

g This shows that at least one-quarter of adults in the survey either have no cell-phone or use it very infrequently - only for emergency purposes.

h Teenagers are more consistent in their use of cell-phones, while some adults are big spenders on cell-phones, probably due to business use, and higher incomes.

i

j The top number.

7 See the spreadsheet WeeklyrentalpricesinAucklandAnswers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.

a

Numberofbedrooms 1 2 3 4

Topvalue $330 $400 $550 $655

Upperquartile* $256 $344 $450 $597

Medianvalue $236 $300 $396 $495

Lowerquartile* $212 $280 $343 $427

Bottomvalue $190 $242 $300 $350

* Calculated by spreadsheet

b

c For the less expensive areas, it costs about $60 more for each additional bedroom. There is less spread for rental prices in the cheaper areas than in the more expensive areas - this is shown by the graphs not being symmetrical - the boxes and whiskers on the right are longer than the ones on the left.

100 200 300Annual fee ($)

20

40

60

80

100

120

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Pointsagainst

Weekly cost of renting ($)100 200 300 400 500 600 700

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566 Answers

c

4 a 4 b 5 c 22d Because you cannot

distinguish how many in the 2−3 minute interval were

more/less than 2 1

2minutes.

1 a 2 b 36 c 52 a 8 b 26 c


2

4

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0 1 2 3 4

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uenc

y

Number of goalsper game

3 a

Totalofdocket Frequency,f

$0-$9.99 7

$10-$19.99 9

$20-$29.99 5

$30-$39.99 2

$40-$49.99 1

$50-$59.99 1

b 16

10 20 30 40 50 60

Freq

uenc

y

Total of docket ($)

108642


1 a

Numberofstrokes,x

Frequency,f

x×f

1 0 0

2 1 2

3 1 3

4 5 20

5 7 35

6 2 12

7 1 7

8 1 8

9 0 0

Total 18 87

b 18 c 87 d

8718

= 4.8 32 a 20.10 cm

b Maria is correct - the mode is 20 - a hand-span of 20 cm has the highest frequency − it occurred 59 times, which is more than any other measurement.

3 $151.674 a 84 b 1.57 (2 dp) c 132

5 a 15 b

Numberoftoheroaperm2,x

Frequency,f

x×f

5 1 5

6 6 36

7 3 21

8 2 16

9 1 9

10 0 0

11 0 0

12 1 12

13 0 0

14 0 0

15 1 15

Total 15 114

c 115 d i 6 ii 7

iii

11415

= 7.6

e The mean - if the median or mode was used these ignore the fact that in some places there are high concentrations of toheroa.

6 a 9 °C b 9 times c 28 °C d 29 °C e 27.1 °C

f The mode - they want air-conditioning so should give the highest of the three temperatures.

567Answers


the old dunga

See the spreadsheet TheolddungaAnswers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.1

2 At about 20-25 years. At 20 years there is a sharp drop-off in the number of cars, and after about 25 years the number that are that particular age ‘settles down’ to around 5000 each year.

3 That is the age at which used-car imports from Japan come into the country.4

5 There is a fairly steady decrease in ages until about 16 years old. Then the number is fairly stable until about 34 years old. In contrast, the car ages graph shows a bulge for cars between about 6 and 18 years old. This is because there are a lot of used cars imported into the country, and no trailers.

6 The ages are continuous - although they are given in a whole number of years, they would be older than that. For example, if a car is less than 1 year old it does not mean it is brand-new (0 years old). 0.5 is a good ‘average estimate’ for the cars in this group.

7 12.04 years8 There is no information given

about the exact ages of the cars that are over 40 years old.

32 The statistical enquiry cycle


1 a ‘I wonder what proportion of students in Years 9 and 10 arrived late at school so far this term.’

b Students late yesterday might be absent today. Students may not remember, students may not give the exact time in case they get into trouble. The times may not be synchronised. Students may refuse to answer. Asking all students would take a very long time in a large school. It may not be possible to easily find all students.

c You could measure the number of students outside the school grounds after a particular time.

568 Answers

4 Choose a group of 30 adults at random and another group of 30 teenagers at random. Give each person a pedometer and record their total number of steps over a week. (This is a suggested answer − several different approaches are possible.)

5 a Gender: 17; ethnic group: 18b Student 3 should be recorded as 171 months, student

7 should be recorded as 168 months, or possibly 174 months if we are not sure whether he is closer to his 14th or 15th birthday.

c Student 12 is significantly younger than all the others.

d Student 5: probably read the wrong scale off a 150-cm tape-measure, and it could be recorded as 36 cm. Student 16: neck measure might have been recorded in inches instead of centimetres, and therefore could be 17 × 2.54 = 43 cm.

e Student 4: maybe reading the wrong scale on a 150-cm tape-measure, so should be 29 cm. Student 6: either ‘1 foot’ is an attempt at a joke and should be ignored, or it is an exact measurement and then 1 foot = 12 inches = 12 × 2.54 cm = 30 cm.

6 a The question is about investigating whether the speed limit is being observed, and this is expressed in km/h, so the data should also be expressed in km/h.

b

Speed DistanceTime

ms

km/h

=

= × =12014 8

3 6 29.

.

c In the same order as given, the time data converts to this speed data (whole number km/h): 29, 26, 32, 30, 29, 29, 26, 29, 24, 27, 28, 25. 23, 35, 28, 27, 28, 25, 28, 26.

d Lateness needs to be defined carefully(e.g.define being late to school as being still outside the school grounds 1 minute before the starting bell). You need to make sure watches are synchronised with ‘official’ school time. You need to make sure late students are not confused with ‘absent’ students. If students knew about the survey in advance they may come early - just for that day.

2 a ‘I wonder how much money each student in this class spends on cell-phone use each week.’

b Call records may have been deleted. The number of calls made is not necessarily a measure of how much is spent, due to special deals and differing rates. The ‘last month’ needs to be defined more clearly - is it the previous 30 days, or the month before the current one? The student may not have their cell-phone with them. Some students have more than one cell-phone.

c You could measure the amount of prepay credit each student had on a particular day, and then 30 days later measure this again, taking into account any ‘top-ups’. Some students might be on a billing plan and the payments for this could be measured separately.

d It would require the co-operation of all students involved to measure these amounts at the same date and on the same day. If the student is on a calling plan instead of prepay you would need to adjust to allow for different time periods. Students may not remember what they had spent on top-ups.

3 ‘I wonder if there is a relationship between wrist circumference and neck size?’

‘I wonder what proportion of students in Years 9 and 10 have a part-time job?’

‘I wonder if Year 10 students have a faster reaction time than Year 9 students?’

(to the nearest whole number)

6 d See the spreadsheet Localstreetcarspeedinvestigation.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.

e Most of the speeds except for two are on or under the speed limit. There is some clumping immediately below the speed limit, which probably shows drivers are aware of the limit but otherwise driving as fast as allowed.

7 a The data may have been collected by requiring students to ‘sign in’ at some central location. In the future students may be scanned as they enter school grounds!

Possible problems include the fact that students have a wide number of reasons for different arrival times, such as before-school rehearsals, practices, etc. There is no information given about whether this data was collected on the same day for each group, or not.

569Answers

b

c The arrival times for the two groups are fairly similar, as shown by the LQ, median and UQ for each group being within 3 minutes of each other. The outliers for Year 13 are easily explained by the free period for some of those students.

d

Year10 Year13

Mean 8:37 am 8:46 am

Median 8:38 am 8:36 am

The median gives more useful information because the times after 9:00 am for the Year 13 students pull their mean upwards so that it is slightly higher than the upper quartile.

e If students are required to be present for the first period of the school day there is no significant difference between the arrival times for Year 10 students and Year 13 students.

8:00 8:10 8:20 8:30 8:40 8:50 9:00 9:10 9:20 9:30 9:40 9:50

Arrival time at school

Year 13

Year 10


1 For parts a–e the suggested answers refer to investigating what the difference is between the cost of vaccinating a dog and vaccinating a cat.a ‘I wonder if it costs about $5 more to vaccinate a dog

compared with a cat.’ The problem is to summarise the prices given and

make a comparison. I will compare the prices given in the table for the two types of animal.

b I will use the vaccination prices for cats for all six regions, and the vaccination prices for dogs for all six regions.

c The data is a summary of prices collected from many vets in New Zealand and the individual data is not given. You could assume that if published in a reputable magazine like Consumer it is likely to be accurate. There are no obvious items of data that do not seem to fit.

d

The dog vaccination prices are higher than each corresponding cat vaccination price, and are more spread out. The graph shows this by generally being shifted to the right. This table summarises the data.

Cats Dogs

Mean $45 $51

Median $45 $50

e The values of the mean and median support the suggestion that it does cost about $5 more to vaccinate a dog than a cat. However, the actual difference varies more in Auckland and Wellington than it does elsewhere in New Zealand.

f South Island Provincialg It is over $100 more to spay a dog compared with

a cat in each one of these regions, and it is at least $60 more to neuter a dog than it is to neuter a cat in each region. Both kinds of desexing procedures are definitely more expensive for a dog than a cat.

h 120 vets. This is the total of the number surveyed for each of the six regions.

i The mean cost of microchipping a dog would be less than $43. The classmate has averaged the six figures given, without realising that only 32 vets are in Auckland and Wellington (with a price over $43), while 88 are not in those centres (with average prices no more than $40). When the mean is calculated for all 120 vets, the 88 vets outside Auckland and Wellington will be more influential.

2 The suggested answers refer to investigating what the difference is between the cost of dry food and the cost of tinned food.a ‘I wonder if it is cheaper to feed an animal dry food

or tinned food.’ The problem is to use the data to make a comparison, bearing in mind that you should probably keep the cat data separate from the dog data rather than combining it. The prices for various types of food brand are given. However, there is no information about how popular each brand is.

b The prices for all the dry food and all the tinned food products, keeping the dog and cat groups separate so that your conclusions are not muddied if there are differences between the products for each animal.

c The data is a summary of retail prices, and you could assume that if published in a reputable magazine like Consumer it is likely to be accurate. There are no obvious items of data that do not seem to fit.

40 50 60

Cats Dogs

Cost of vaccination ($)

570 Answers

d

The medians are: dry cat food: $139; dry dog food: $450; tinned cat food: $321; tinned dog food: $1107.

e The graph and the summary statistics show that tinned food is obviously much more expensive than dry food. The outliers for the very expensive products reinforce this observation. The separation of the products into cat and dog categories was useful, because if only ‘dry’ and ‘tinned’ were analysed then the overlap between dry dog and tinned cat prices would have ‘muddied’ this conclusion.

3 The suggested answers explore whether the size of a mobile phone is related to how long it takes to recharge.a ‘I wonder if small mobile phones take longer to

recharge than medium ones.’ I will need to take the size of a phone and how long

it takes to recharge.b The recharge time for all phones that fit into the

‘small’ and ‘medium’ size category.c The data has been collected by testing mobile

phones on a controlled basis in a laboratory, and you could assume that if published in a reputable magazine like Consumer it is likely to be accurate. There are no obvious items of data that do not seem to fit.

d

e It is not obvious which size of mobile phone recharges fastest in general. There is more variation in the times for the medium-size phones, whereas small phones are close to 2 hours recharging time.

4 a ‘I wonder if the typical teenager in New Zealand consumes 15 kg of sugar in drinks each year?’

b i Recall of students, getting students to record their drink consumption over an extended period, measuring uncompleted drinks, etc.

ii You would need to survey a wide range of ages and backgrounds in the target group - i.e. New Zealand teenagers.

c The carbonated drinks figure for student number 31 is obviously wrong. It represents consumption of more than 20 drinks per day. If it is discarded it will make this student’s consumption of sugar appear too low, so it could be replaced by the mean number of carbonated drinks per month for all the other students.

d Differences in how the sugar content in drinks is measured, defining a standard drink, variation between brands, not all types of drink are included, drinks may not be finished off by students.

e There appears to be a lot of substitution between drinks - that is, if they drink a lot of water they may not drink so much of other drinks, and vice-versa. There is no variation for any individual student in the amount of sugar they have in drinks like tea or coffee - they have a preferred sweetness and stick to it. The consumption of energy drinks was either high (about one per day) or very low.

f 16 × 21 + 32 × 19 + 7 × 18 = 1070g See the spreadsheet Sugarconsumptionindrinks

(monthly)Answers.xls. This is available on the Beta Mathematics Workbook companion CD, or can be downloaded from www.mathematics.co.nz.

h The mean amount consumed per month is 1247 g.

2800

2600

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Dry catfood

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Cos

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r ($

)

571Answers

i j The mean consumption per month is 1247 g, which is equivalent to 14.96 kg per year. Given that the graph also shows that consumption between 1000 g and 1500 g per month is much more common that other consumption levels, the claim seems reasonable.

k The main problem is the difficulty in measuring sugar consumption accurately. The list given does not include all possible alternatives (e.g. other drinks, like sweetened milk or alcoholic drinks, also contain sugar), and there is no information given about how the 40 students were chosen - they may not have been representative of all New Zealand teenagers. This kind of survey is difficult to run under controlled conditions (keeping people under watch while measuring their drink consumption very accurately). Another possible approach would be to focus on the supply of drinks in co-operation with sugar and soft-drink manufacturers - they would have precise figures on sales and sugar content of their products. However, they would not have data on the age range of people who consume their products.

16

14

12

10

8

6

4

2

500

1000

1500

2000

2500

3000

3500

Grams per month

Sugarconsumption(fromsweeteneddrinks)permonth

Freq

uenc

y


1 a Discrete b Continuous c Continuous d Discrete e Discrete f Continuous g Continuous h Discrete2 a Census b Sample c Sample d Census e Sample f Census3 a It may be difficult for someone to remember this

information. ‘Movies’ is not defined - does it include videos, or movies on television?

b Invasion of privacy.c ‘Exercise’ is not defined, ‘enough’ is not defined.

4 a The sample is biased because it is only chosen from Foodcity shoppers - therefore the claim only applies to Foodcity shoppers.

b People may own an answering machine but answered the phone themselves. Not everyone has a telephone. Answering machines are owned by households, not individuals.

c It is not a question that would always be answered honestly, particularly when asked ‘face to face’.

d The sample surveyed is not large enough to justify an estimate of 100%. If even one person had changed their mind the estimate would only be 75%.

5 a Students who borrow books are probably less likely to watch television.

b Girls are not included in the sample.c Students may not be Year 9.

6 (C). (A) is not suitable because there might be a fault that just affects the last item, and it also involves too much work. (B) is not suitable because the fault may not be at this particular check-out but elsewhere.

7 a Men are excluded from the survey, and may have different opinions to women, because their life expectancy is lower.

b Only those watching the show can participate. Only those who care about this issue and who are prepared to spend money can be surveyed.

8 a Unsuitable, because the sample is not large enough. The principals at a conference in Queenstown may not be typical of all schools.

b Suitable, because it gives every school the chance to see the product. However, some schools may not have the time to complete the questionnaire and it might be too expensive and time-consuming to do this.

c Unsuitable, because secondary schools are excluded.d Unsuitable, because only schools interested would

respond, and also e-mail surveys have a very low response rate. There may be a connection between use of stationery software (which is what the survey is about) and usage of the internet.

e Only suitable if the schools concerned got the chance to see the product.

572 Answers

33 Probability


1 a Unlikely b Unlikely c Certain d Unlikely e Likely f Likely g Certain2 a (A) and (C) b (D)

3 (Suggested answers.)a A gold ring will sink when

you throw it into water.b The phone will ring sometime

in the next week.c New Zealand will host the

Olympic Games sometime this century.

d You will get a total of 13 when you throw two six-sided dice each numbered from 1 to 6.

4 (D), (E) and (F)5 a The top face will be pink. b The top face will be green. c The top face will be blue.

6

7 a will never b is certain to c is unlikely to8 a unlikely b impossible c certain d even chance e likely


1 a H, T, H, H, T, T b It is getting closer to 65%. c 50%

d No − percentage of heads is not approaching 50% in the long run.

2 a i

1640

= 25

= 0.4 ii

2440

= 35

= 0.6 b 1

3 a

45100

= 920

= 0.45 b

16100

= 425

= 0.16

c

84100

= 2125

= 0.84

4 a

1150

b

3750

5 a 2114 b

2851057

c 0.40 d 3%

6

111500

= 0.007 3


1 a Green b

15

c

710

2 a

15

b 0

3 14

4

15

5 a

13

b

712

6 a The probability of getting a 4 when a fair six-sided die is tossed once.

b The probability of getting a red ball when choosing a ball at random from a bag containing four red and six blue balls.

7 a

112

b 12

c

712

d 14

e

56

8

119

9 There must be other colour tickets because the given probabilities only add up to 0.88. If they added up to 1 you could be sure there were no other colours.

10 a

35

b

23

c Box B, because a probability of

23

= 0. 6 is higher

than a probability of

35

= 0.6.

11 a 19 b 59

12 a

113

b

152

c

113

d 14

e 34

f 12

g

313

h

413

13 a Taupo → New Plymouth Taupo → Wanganui → New Plymouth Taupo → Rotorua → New Plymouth Taupo → Rotorua → Hamilton → New Plymouth

b 12

14 Remove 10 black cards.

15 a 17

b 27

c

16

d

2584

Red Red Red Green

Blue

Blue

573Answers

PUZZLe (page 489)

Lost your marbles? 16


the crooked cricket captainThe probability that the crooked cricket

captain wins is

1325

= 0.52.

1 402 153 1504 325 96 757 5


Have I won a prize yet? 1 The bottom graph shows what happens to the proportion of winning

tickets as more are purchased.

2

320

= 0.15

3 The graph ‘settles down’ (i.e. not much variation up or down) to a value close to 0.15.

4 No5 The prizes are high enough to reward people when they eventually get

a winning ticket.6 Unlikely. Both of us are likely to win somewhere fairly close to 14 or 15

times, but not to get exactly the same number.7 Unlikely. There will be considerable variation, and other outcomes,

such as 13, 15 and 16 winning tickets, are also quite likely. Although 14 wins is the most likely, the chances of 11, 12, 13, 15, 16, 17, etc. will add to more.

8 10–20 winning tickets out of 100.9 (C), (A), (B), (D)

10 Increase the number of runs to significantly more than 100.



1 a

b {PP, PN, NP, NN}

c 14

d 14

2 a

b 8 c

18

d

38

Prize

No prize

Prize P P

Prize

No prize

No prize

P N

N P

N N

H

T

H

H

T

H

H

T

T

{HHH}

{HHT}

T

H

T

H

T

{HTH}

{HTT}{THH}

{THT}

{TTH}

{TTT}

1stcoin

2ndcoin3rdcoin

3 a

b

16

c

23

4 a

b 12 c

16

d 12

S

BC

M

B

C

MBCS

B

S

M

C

SM

BC

BMBSCB

CM

CSMBMCMSSBSC

SM

PurpleOrangeGreenGreen

Orange

Green

Green

GreenGreen

Purple

OrangeGreen

Purple

OrangeGreen

Purple

5 a

b 16 c 4 d

38

6 a 20 b

110

c

310

d

15

7 a

b

16

c 12

SG

S

G

S

G

S

G

S

G

S

G

S

G

S

G

SGSGSGSGSGSGSG

1stset

2ndset

3rdset

4thset

T

M Midday

10 am

2 pm

10 am

Midday

2 pm

Index

π—2363-D shapes—30024-hour clock—20960° angles (construction)—367

AD dates—11adding

decimals—25fractions—43, 44integers—12

adjacent side (of triangle)—327algebraic expressions—103alternate angles—276angle—264angle bisectors—368, 372angles

on a line—270measuring—265at a point—270of a triangle—271

arcs—364area—219

of a circle—239of a parallelogram—227of a rectangle—220of a square—220of a trapezium—228of a triangle—225

arrowhead—382average—438axis of symmetry—378

bar graph—423, 454, 462base—20, 121

of a triangle—225BC dates—11B-DM-AS mnemonic—3bearings—268biased sample—475bisector, perpendicular—364BMI (body-mass index)—170body-mass index (BMI) —170box and whisker diagram—449, 462boxplot—449, 462brackets, implied—4

capacity—207, 254ceiling—350census—414, 475CensusAtSchool database—463

centreof enlargement—401of rotation—388

chance—480change sides, change operations—139circle, area—239circumcircle, of a triangle—369circumference—237coefficients—125co-interior angles—276column vectors—390combined transformations—393common denominator—44common factor—33, 129, 130comparison question—461compass—364composite areas—229, 245compound interest—68concave polygons—158congruence transformations—387, 388congruent shapes—292constant term—157construction—363, 364continuous data—476converse of Pythagoras—323convex polygons—158co-ordinates—177corresponding angles—276cos ratio—335counting numbers—3cross-multiplying—146cross-section—250cube—247cube root—22cubic units—246cuboid—246cyclic quadrilateral—382cylinder—252

net for—261surface area—261volume—252

data—414, 415dates, AD–BC—11decagon—283decimal form—24decimal point—24decimals

adding—25converting to fractions—52converting to percentages—54

575Index

dividing—29multiplying—27recurring—51subtracting—25

degree of dominance—37degrees—265denominator—29, 32

same—43depreciation—62diameter—237difference of two squares—153, 157direct proportion—82discounts—63discrete data—476distance–time graphs—172distributive law—126dividend—29dividing

decimals—29fractions—41integers—16

divisor—29Domesday Book—414dominance, degree of—37dominoes—23dot plot—423, 462

edge—301enlargement—397

properties of—404equally likely outcomes—482, 487equations—132

quadratic—160that link fractions—146

equilateral triangle—272Escher—292estimation—97expanding brackets—126, 142, 149expected number—490expressions—103exterior angles

of a polygon—284, 286of a triangle—274

face—301factorising—128, 154

quadratic expressions—154two-stage—159

factors—6five number summary—449flow chart—135, 136fractional scale factor—407fractions—32

adding—43, 44

converting to decimals—50converting to percentages—54dividing—41multiplying—38reciprocals—41simplifying—33subtracting—43, 44

frequency—423frequency table—454frieze patterns—396front views—309

glove-sizing—193gradient—181

negative—185GST (Goods and Services Tax)—66

hectares—223hexagon—283histogram—455, 462horizontal axis—165horizontal lines—190hypotenuse—313, 327

calculating using Pythagoras—314

imperial units—201implied brackets—4impossible triangle—300improper fractions—46index—20index form—121integers—10

adding—12dividing—16multiplying—15subtracting—12

intercepts—187interest—68interior angles of a polygon—284, 287interquartile range—446invariant points—388inverse proportion—82inverse (trig) keys—357isometric drawing—300isometric paper or grid—303isosceles trapezium—382isosceles triangle—272

jigsaw pieces—298

kite—382Klein bottle—300

land area—223

576 Index

leaf—447Leaning Tower of Pisa—362left-handedness—37length

conversions—203units—203

level staircase—300like terms—115, 143, 150line, definition of—264liquid volume—207, 254loci, in three dimensions—375locus—372long-run relative frequency—482, 483lower quartile—445, 446

magic square—13mean—438

calculated from a frequency table—456median—438metric units—201midpoint—364mirror line—378, 387misleading graphs—430mixed numbers—46mode—438multiples—6multiplying

decimals—27fractions—38integers—15by powers of 10—89

National Parks—234natural numbers—3negative gradients—185negative scale factor—409nets—306

of a cylinder—261number line, integer—10number rules—3numerator—29, 32

oblique drawing—300octagon—283, 284opposite, of an integer—10opposite side (of a triangle)—327order of operations—3orientation—404origin—182original quantity, working it out—70outliers—447

parabola—194paradox—496

parallel lines—276parallelogram—382

area—227pentagon—283percentage—53

change—61percentages

converting to decimals—54converting to fractions—54increasing and decreasing—60of quantities—58

perfect cube—21perfect numbers—7perfect square—21, 152perimeter—221perpendicular bisectors—364, 372perpendiculars—365

at a point on the line—366from a point not on the line—365

pie graph—424, 462pipe—262placeholders—86plan views—308point, definition of—264point sizes—400polygon—283

exterior angles—284, 286interior angles—284, 287

powers—20, 121of 10, multiplying by—89of powers—122of zero—122

PPDAC—461prefixes, metric—201prime numbers—8principal—68priority of operations—17prism—250probability—480probability trees—493product—38proof—315properties of enlargement—404proportion—82protractor—265Pyramids of Giza—363Pythagoras

converse of—323proof—315Theorem of—313

quadratic equations—160quadratic expressions—149, 154quadratic relationships—194

577Index

quadrilateral—283, 382quartiles—445quotient—16

radius—237random choice—475random numbers—476random variation—434range—445rate of interest—68rates—78ratio—72reading tables/scales—214reciprocals—41rectangle—382

area—220recurring decimals—51reflection—387regular polygons—289relationship question—461relative frequency—482, 483repeated unknowns—139rhombus—382roots—22rotation—388rounding—87

same denominator—43sample—475scale factor—397

fractional—407negative—409

scalene triangle—272scales—214scatter plot—424School Certificate Mathematics—244seasonal variation—434second, definition of—208sense—404side views—309significant figures—86, 88similar figures—397simple interest—68simplifying

expressions—115fractions—33

sin ratio—330SOH-CAH-TOA mnemonic—347solution of an equation—132solving equations—132splitting in a given ratio—76spread—445square—382

area—220

square root—22, 124square units—219standard form—90

converting to ordinary form—92statistical enquiry cycle—461statistical graphs—415, 423statistical investigations—475statistical reports—418statistics—414Statistics New Zealand (Tatauranga Aotearoa)—414stem and leaf diagram—447, 462stopping distances—164straight-line rules—191substitution—107subtracting

decimals—25fractions—43, 44integers—12

summary question—461surface area—258surveys—475symmetry—378

tables—214tan ratio—342tessellation—292Theorem of Pythagoras—313three-dimensional shapes—300time—208time series—434

top views—309total order of symmetry—378translation—390transversal—276trapezium—382

area—228trends—434triangle

area—225triangles of facts—347trigonometry—327tromino—295two-dimensional graphs—165two-stage factorising—159typographical point sizes—400

uniform rates—83unitary method, in proportion problems—83upper quartile—445, 446

vectors—390vertex—301

of an angle—264

578 Index

vertical axis—165vertical lines—190vertically opposite angles—270views—308volume—246

liquid—254conversions—248of a cuboid—246of a cylinder—252of a prism—250

weight, of a line—400weight units—206whole numbers—3

y = mx + c rule—187

text answers index

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