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)
EXERCISE 1.02 (page 6)
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
EXERCISE 1.03 (page 8)
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.
EXERCISE 1.04 (page 11)
2–3 –1 –4
0 3 1–2
0 –2 2–3 4 –4
–1 1 3
EXERCISE 1.05 (page 13)
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
EXERCISE 1.06 (page 14)
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.
EXERCISE 1.07 (page 16)
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
EXERCISE 1.08 (page 16)
EXERCISE 1.09 (page 17)
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
EXERCISE 1.10 (page 18)
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
EXERCISE 1.11 (page 21)
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
EXERCISE 1.12 (page 22)
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
EXERCISE 2.01 (page 25)
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
EXERCISE 2.02 (page 26)
1 12.7322 20.3723 710.80434 115.0535 0.1016
6 17.457 1.588 431.8149 0.67
10 0.0967
EXERCISE 2.03 (page 27)
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
EXERCISE 2.04 (page 28)
1 3.567082 102.343 1.24 0.725 66.672
6 1027 115.468028 1.12899 0.015
10 0.9376
EXERCISE 2.05 (page 28)
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
EXERCISE 2.06 (page 29)
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
EXERCISE 2.07 (page 30)
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
EXERCISE 3.01 (page 34)
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
EXERCISE 3.02 (page 35)
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
EXERCISE 3.03 (page 39)
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
EXERCISE 3.04 (page 40)
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
EXERCISE 3.05 (page 41)
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
EXERCISE 3.06 (page 43)
EXERCISE 3.07 (page 43)
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.
EXERCISE 3.08 (page 45)
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
EXERCISE 3.09 (page 45)
EXERCISE 3.10 (page 47)
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
EXERCISE 3.11 (page 48)
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
EXERCISE 3.12 (page 49)
EXERCISE 3.13 (page 51)
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
EXERCISE 3.14 (page 52)
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
EXERCISE 4.01 (page 54)
EXERCISE 4.02 (page 55)
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
EXERCISE 4.03 (page 55)
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%
EXERCISE 4.04 (page 57)
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%
EXERCISE 4.05 (page 58)
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%
INvEstIgatIoN (page 60)
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
EXERCISE 4.06 (page 58)
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
INvEstIgatIoN (page 62)
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%
EXERCISE 4.07 (page 61)
EXERCISE 4.08 (page 64)
INvEstIgatIoN (page 65)
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.
EXERCISE 4.09 (page 67)
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
EXERCISE 4.10 (page 69)
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
EXERCISE 4.11 (page 71)
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
EXERCISE 5.01 (page 73)
EXERCISE 5.02 (page 75)
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
EXERCISE 5.04 (page 79)
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
INvEstIgatIoN (page 80)
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.
EXERCISE 5.05 (page 83)
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
EXERCISE 6.01 (page 86)
EXERCISE 6.02 (page 88)
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
EXERCISE 6.03 (page 90)
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
EXERCISE 6.04 (page 92)
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
EXERCISE 6.05 (page 92)
1 6102 18003 934 4200005 73.36 5.657 80108 78000
9 16610 26.4511 3.15812 40900000013 5033014 11170015 6000
EXERCISE 6.06 (page 93)
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
EXERCISE 6.07 (page 94)
EXERCISE 6.08 (page 95)
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
EXERCISE 6.09 (page 95)
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
EXERCISE 6.10 (page 96)
1 0.0000000000000000000000272g2 1.0352×10-13light-years3 0.00016m4 0.000000001m5 1.6×10-3watts6 a 0.0000026kg b 0.0026g c 1000000
EXERCISE 6.11 (page 96)
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
EXERCISE 6.12 (page 97)
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
EXERCISE 6.13 (page 97)
EXERCISE 6.14 (page 98)
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
EXERCISE 7.01 (page 103)
5 2xy
6 2xy
7 6x
843
qp
9 6xy10 3xy
EXERCISE 7.02 (page 104)
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
EXERCISE 7.03 (page 107)
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
EXERCISE 7.04 (page 107)
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
INvEstIgatIoN (page 108)
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.
EXERCISE 7.05 (page 109)
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
INvEstIgatIoN (page 114)
Floor joists
1 (B),(A),(C)2 a 300mm b 467mm
3 D = 25 5x
3+ 2
EXERCISE 7.06 (page 112)
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
EXERCISE 8.01 (page 115)
14 210mnq15 -2r16 45fgh17 -12xy18 -30pq19 xy20 -12x21 -21d22 30ab23 9pq24 24xyz25 -12pqr
EXERCISE 8.02 (page 116)
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
EXERCISE 8.03 (page 116)
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
EXERCISE 8.04 (page 118)
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.
EXERCISE 8.05 (page 121)
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
EXERCISE 8.06 (page 122)
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
EXERCISE 8.07 (page 123)
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
EXERCISE 8.08 (page 124)
1 3y3
2 5x3 4y6
4 9x8
5 2x6 8y9
7 5x10
8 x8
9 10y50
10 xy3
11 7x3y2
12 6xy2
INvEstIgatIoN (page 124)
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)
EXERCISE 9.01 (page 126)
6 x2-7x7 10x2+15x8 18x2-6x9 -8x-2
10 -24x+8
EXERCISE 9.02 (page 127)
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
EXERCISE 9.03 (page 127)
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
EXERCISE 9.04 (page 128)
EXERCISE 9.05 (page 129)
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)
EXERCISE 9.06 (page 130)
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)
EXERCISE 9.07 (page 130)
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
EXERCISE 10.01 (page 133)
EXERCISE 10.02 (page 134)
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
EXERCISE 10.03 (page 135)
1 x=162 x=303 x=244 x=425 x=-3
6 x=2007 x=-158 x=-409 x=7
10 x=-100
EXERCISE 10.04 (page 135)
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
EXERCISE 10.05 (page 136)
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
EXERCISE 10.06 (page 137)
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
EXERCISE 10.07 (page 140)
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
EXERCISE 10.08 (page 141)
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.
EXERCISE 10.09 (page 142)
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
EXERCISE 10.10 (page 143)
EXERCISE 10.11 (page 144)
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
EXERCISE 10.12 (page 144)
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.
EXERCISE 10.13 (page 147)
EXERCISE 10.14 (page 147)
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
INvEstIgatIoN (page 156)
x-blocks1 x2+5x+62 (x+3)(x+2)3 x+3+x+2+x+3+x+2
=4x+104, 5
EXERCISE 11.01 (page 149)
9 x2-2x10 x2+3x11 x2-x12 x2+4x13 x2-5x14 x2+x15 x2-7x
EXERCISE 11.02 (page 150)
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
EXERCISE 11.03 (page 150)
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
EXERCISE 11.04 (page 152)
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
EXERCISE 11.05 (page 153)
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
EXERCISE 11.06 (page 153)
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.
EXERCISE 11.07 (page 154)
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
EXERCISE 11.08 (page 155)
(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
EXERCISE 11.09 (page 157)
EXERCISE 11.10 (page 158)
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!
INvEstIgatIoN (page 163)
the dimensions of the sand-pit5mby5m
EXERCISE 11.11 (page 158)
EXERCISE 11.12 (page 158)
6 (x-6)(x+5)7 (x-24)(x-1)8 Nofactors9 Nofactors
10 (x+7)2
EXERCISE 11.13 (page 159)
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).
EXERCISE 11.14 (page 161)
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
EXERCISE 11.15 (page 161)
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
EXERCISE 11.16 (page 162)
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
EXERCISE 11.17 (page 162)
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 $800 d
e 120m2.Thisistheareaforwhichthetwolinesintersectonthegraph.
8
9 aContainer Depth–time graph
Bowl A
Trough A
Cylinder B
Cuboid B
Cone A
b
10 a -5°C b 3hoursc Itreachedroomtemperatureat5am,thenlateronin
themorning(9am)theroomtemperatureincreased.
d1 1
2hours e 6:30pm
11 12
13
×A
Hei
ght
Weight
×
×C
B
Wei
ghti
nkg
Weightinlb
10
20
30
40
50
10 20 30 40 50 60 70 80
EXERCISE 12.01 (page 166)
20 40 60 80 100
200
400
600
800
1000
1200
Cos
tofp
aint
ing
($)
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
INvEstIgatIoN (page 170)
BMI graphs for boys and girls
1 Pink2 a Healthyweight b Overweight3 BMI=22.37;heisontheboundary
betweenhealthyweightandoverweight.
4 6yearsoldand8yearsold5 14yearsold(Year10).The50th
percentileforaBMIof19isroughlylevelwithanageof14forbothboysandgirls.
EXERCISE 12.02 (page 174)
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.
INvEstIgatIoN (page 178)
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
EXERCISE 13.03 (page 182)
a bc
d
e
f
EXERCISE 13.04 (page 183)
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
EXERCISE 13.05 (page 185)
1 a -1 b-14
c -4 d-15
e-34
2
ab
c
de
3 -6
4-13
5 a -1 b -2
EXERCISE 13.06 (page 186)
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
EXERCISE 13.07 (page 187)
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
EXERCISE 13.08 (page 188)
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.
EXERCISE 13.09 (page 190)
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
EXERCISE 13.10 (page 191)
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
INvEstIgatIoN (page 193)
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.
EXERCISE 13.11 (page 196)
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
EXERCISE 14.01 (page202)
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.
EXERCISE 14.02 (page204)
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)
EXERCISE 14.03 (page206)
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
EXERCISE 14.04 (page207)
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
EXERCISE 14.05 (page210)
EXERCISE 14.06 (page212)
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
EXERCISE 14.07 (page214)
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
EXERCISE 15.01 (page220)
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
EXERCISE 15.02 (page222)
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
InvEsTIgATIon (page224)
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
EXERCISE 15.03 (page226)
8 54 cm2
9 36 cm2
10 51 cm2
11 132 cm2
12 36 cm2
13 124 cm2
InvEsTIgATIon (page226)
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
EXERCISE 15.04 (page228)
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
EXERCISE 16.01 (page237)
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
EXERCISE 16.02 (page240)
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
EXERCISE 16.03 (page241)
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
EXERCISE 16.04 (page243)
1 0.4312 m2 6.685 m3 466.6 m4 30.83 m
5 32 cm6 318 mm7 170 m
EXERCISE 16.05 (page244)
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
EXERCISE 16.06 (page245)
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
InvEsTIgATIon (page248)
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
EXERCISE 17.01 (page247)
EXERCISE 17.02 (page248)
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
InvEsTIgATIon (page254)
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.
EXERCISE 17.03 (page250)
EXERCISE 17.04 (page251)
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)
EXERCISE 17.05 (page253)
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
EXERCISE 17.06 (page255)
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
EXERCISE 17.07 (page259)
Edgelength Surfacearea
1 6
2 24
3 54
4 96
…
8 384
…
x 6x2
2.6
4.2
2.5
EXERCISE 17.08 (page262)
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
EXERCISE 18.01 (page264)
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°
EXERCISE 18.02 (page266)
InvEsTIgATIon (page267)
security sensors1 a b
2 a
b 10 more (11 in total)
cd 10%
3 a 3 b
c d 80
4
EXERCISE 18.03 (page268)
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°
EXERCISE 18.04 (page271)
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°)
EXERCISE 18.05 (page273)
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
EXERCISE 18.06 (page275)
EXERCISE 18.07 (page277)
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
EXERCISE 18.08 (page280)
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.
EXERCISE 18.09 (page282)
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
EXERCISE 19.01 (page285)
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)
InvEsTIgATIon (page286)
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
EXERCISE 19.02 (page287)
1 360°2 360°3 50°4 85°5 a 108° b 38° c 33° d 62°
EXERCISE 19.03 (page288)
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
EXERCISE 19.04 (page289)
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
EXERCISE 20.01 (page293)
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.)
InvEsTIgATIon (page294)
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.
EXERCISE 20.02 (page296)
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
InvEsTIgATIon (page298)
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
EXERCISE 21.01 (page301)
Vertical cut
Vertical cut
Horizontal cut
EXERCISE 21.02 (page303)
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
EXERCISE 21.03 (page306)
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.
EXERCISE 21.04 (page309)
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
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
EXERCISE 22.01 (page313)
EXERCISE 22.02 (page315)
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
EXERCISE 22.03 (page317)
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
EXERCISE 22.04 (page317)
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
EXERCISE 22.05 (page320)
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.
InvEsTIgATIon (page323)
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.
EXERCISE 22.06 (page324)
EXERCISE 22.07 (page325)
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
InvEsTIgATIon (page326)
The ants and the sugar bowl 7.810 m
PUZZLE (page281)
What day is it? Today is yesterday tomorrow.
23 Trigonometry 1 − an introduction
EXERCISE 23.01 (page328)
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
InvEsTIgATIon (page329)
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
InvEsTIgATIon (page335)
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
EXERCISE 23.02 (page331)
EXERCISE 23.03 (page333)
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
EXERCISE 23.04 (page334)
1 0.70712 0.48483 0.1392
4 0.99255 0.24196 0.7986
7 0.63208 0.9995
EXERCISE 23.05 (page334)
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
EXERCISE 23.06 (page337)
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
EXERCISE 23.07 (page338)
1 0.70712 0.87463 0.99034 0.1219
5 0.97036 0.60187 0.77498 0.0332
EXERCISE 23.08 (page339)
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
EXERCISE 23.09 (page340)
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
EXERCISE 23.10 (page341)
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
EXERCISE 23.11 (page343)
1 0.17632 0.62493 14 0.14055 1.600
6 1.9977 6.3148 11.439 28.64
10 44.07
EXERCISE 23.12 (page344)
6 f = 16.63 m7 g = 13.49 m8 h = 123.1 m9 i = 5.412 km
10 j = 59.40 cm
EXERCISE 23.13 (page345)
1 sin2 tan3 tan4 sin
5 cos6 tan7 cos8 tan
EXERCISE 23.14 (page346)
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
EXERCISE 24.01 (page348)
EXERCISE 24.02 (page348)
EXERCISE 24.03 (page349)
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
EXERCISE 24.04 (page352)
EXERCISE 24.05 (page353)
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
InvEsTIgATIon (page355)
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
EXERCISE 25.01 (page358)
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°
InvEsTIgATIon (page358)
Inverse tan check2 It should be 35°.
3
710
4 0.75 34.99° (2 dp)
EXERCISE 25.02 (page359)
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°
EXERCISE 25.03 (page360)
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.
EXERCISE 25.04 (page361)
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
EXERCISE 26.01 (page365)
6 5.5 cm
InvEsTIgATIon (page369)
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.
InvEsTIgATIon (page371)
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.
EXERCISE 26.05 (page370)
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
EXERCISE 26.06 (page373)
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
EXERCISE 26.07 (page375)
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
EXERCISE 27.01 (page379)
EXERCISE 27.02 (page383)
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.
EXERCISE 27.03 (page384)
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
EXERCISE 27.04 (page385)
PUZZLE (page387)
Ringed trapezium20 cm
EXERCISE 27.05 (page388)
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’
EXERCISE 27.06 (page391)
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)
InvEsTIgATIon (page392)
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
EXERCISE 27.07 (page393)
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
EXERCISE 28.01 (page 397)
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
InvestIgAtIon (page 400)
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
EXERCISE 28.02 (page 401)
B’A
C
A’
C’D D’X
B
EXERCISE 28.03 (page 402)
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)
EXERCISE 28.04 (page 404)
O
O O
O
12345
O
554 Answers
O
O
O×O
6789
EXERCISE 28.05 (page 405)
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°
EXERCISE 28.06 (page 406)
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
EXERCISE 28.07 (page 408)
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’
EXERCISE 28.08 (page 411)
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
EXERCISE 29.01 (page 415)
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
EXERCISE 29.02 (page 420)
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
EXERCISE 30.01 (page 425)
200
400
800
600
1000
1200
Auc
klan
dW
ellin
gton
Chr
istc
hurc
hD
uned
in
Ave
rage
ann
ual r
ainf
all (
mm
)
ACTUnited Future
MaoriGreens
Progressives
New ZealandFirst
National
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.
EXERCISE 30.02 (page 431)
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.
EXERCISE 30.03 (page 435)
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
EXERCISE 31.03 (page 443)
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
EXERCISE 31.04 (page 446)
EXERCISE 31.05 (page 448)
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
EXERCISE 31.06 (page 450)
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.
20
40
60
80
100
120
140
160
Num
ber
of c
igar
ette
s sm
oked
Males Females
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
20
30
40
50
60
Min
utes
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
Pointsscored
Pointsagainst
Weekly cost of renting ($)100 200 300 400 500 600 700
4
3
2
1
Num
ber
ofbe
dro
oms
100 200 300 400 500Amount spent ($)
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
EXERCISE 31.07 (page 455)
2
4
6
8
10
12
0 1 2 3 4
Freq
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
EXERCISE 31.08 (page 457)
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
InvestIgAtIon (page 460)
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
EXERCISE 32.01 (page 466)
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
EXERCISE 32.02 (page 469)
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
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
Dry catfood
Dry dogfood
Tinnedcat food
Tinneddog food
Cos
t per
yea
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
EXERCISE 32.03 (page 477)
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
EXERCISE 33.01 (page 481)
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
EXERCISE 33.02 (page 484)
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
EXERCISE 33.03 (page 487)
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
InvestIgAtIon (page 490)
the crooked cricket captainThe probability that the crooked cricket
captain wins is
1325
= 0.52.
1 402 153 1504 325 96 757 5
InvestIgAtIon (page 491)
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.
EXERCISE 33.04 (page 491)
EXERCISE 33.05 (page 494)
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
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