Year 11Mathematics
Contents
uLake Ltdu a e tduLake LtdInnovative Publisher of Mathematics Texts
IAS 1.1Robert Lakeland & Carl Nugent
Numeric Reasoning
• AchievementStandard .................................................. 2• PrimeNumbers ....................................................... 3• FactorsandMultiples ................................................... 4• RoundingandEstimation................................................ 7• StandardForm......................................................... 12• OrderofOperation..................................................... 15• Integers(+,x,÷,–)....................................................... 17• Fractions(+,x,÷,–)...................................................... 20• Percentages............................................................ 24• Ratio................................................................... 32• Proportion............................................................. 37• Rates.................................................................. 41• Powers................................................................. 44• CompoundingRates..................................................... 49• PracticeInternalAssessment1 ............................................ 55• PracticeInternalAssessment2............................................ 56• PracticeInternalAssessment3............................................ 57• Answers............................................................... 58
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
2 IAS 1.1 – Numeric Reasoning
NCEA 1 Internal Achievement Standard 1.1 – Numeric ReasoningThisachievementstandardinvolvesapplyingnumericreasoninginsolvingproblems.
◆ ThisachievementstandardisderivedfromLevel6ofTheNewZealandCurriculum,Learning Media.Thefollowingachievementobjectives,takenfromtheNumberStrategiesandKnowledge threadoftheMathematicsandStatisticslearningarea,arerelatedtothisachievementstandard: ❖ reasonwithlinearproportions ❖ useprimenumbers,commonfactorsandmultiples,andpowers(includingsquareroots) ❖ understandoperationsonfractions,decimals,percentages,andintegers ❖ useratesandratios ❖ knowcommonlyusedfraction,decimal,andpercentageconversions ❖ knowandapplystandardform,significantfigures,rounding,anddecimalplacevalue ❖ applydirectandinverserelationshipswithlinearproportion ❖ extendpowerstoincludeintegersandfractions ❖ applyeverydaycompoundingrates.
◆ Applynumericreasoninginvolves: ❖ selectingandusingarangeofmethodsinsolvingproblems ❖ demonstratingknowledgeofnumberconceptsandterms ❖ communicatingsolutionswhichwouldusuallyrequireonlyoneortwosteps. Relationalthinkinginvolvesoneormoreof: ❖ selectingandcarryingoutalogicalsequenceofsteps ❖ connectingdifferentconceptsandrepresentations ❖ demonstratingunderstandingofconcepts ❖ formingandusingamodel;
andalsorelatingfindingstoacontext,orcommunicatingthinkingusingappropriatemathematical statements. Extendedabstractthinkinginvolvesoneormoreof: ❖ devisingastrategytoinvestigateorsolveaproblem ❖ identifyingrelevantconceptsincontext ❖ developingachainoflogicalreasoning,orproof ❖ formingageneralisation;
andalsousingcorrectmathematicalstatements,orcommunicatingmathematicalinsight.
◆ Problemsaresituationsthatprovideopportunitiestoapplyknowledgeorunderstandingof mathematicalconceptsandmethods.Thesituationwillbesetinareal-lifeormathematicalcontext.
◆ Thephrase‘arangeofmethods’indicatesthatevidenceoftheapplicationofatleastthreedifferent methodsisrequired.
◆ Studentsneedtobefamiliarwithmethodsrelatedto: ❖ ratioandproportion ❖ factors,multiples,powersandroots ❖ integerandfractionalpowersappliedtonumbers ❖ fractions,decimalsandpercentages ❖ rates ❖ roundingwithdecimalplacesandsignificantfigures ❖ standardform.
Achievement Achievement with Merit Achievement with Excellence• Applynumericreasoningin
solvingproblems.• Applynumericreasoning,
usingrelationalthinking,insolvingproblems.
• Applynumericreasoning,usingextendedabstractthinking,insolvingproblems.
3IAS 1.1 – Numeric Reasoning
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
Prime Numbers
Prime NumbersAprimenumberisanumbergreaterthan1thathasnopositivedivisorsotherthan1anditself.Aprimenumberhasexactlytwofactors1anditself.Forexample17isaprimebecauseithasonlytwofactors1and17.Thesmallestprimenumberaswellastheonlyevenprimenumberis2,becauseitisdivisibleby1and2.Anumberngreaterthan1isdefinedasaprimenumberifitisonlydivisibleby1andn.Positivenumbersotherthan1thatarenotprimenumbersarecalledcompositenumbers.
A factor is a number that divides into another number without remainder. For example 2 is a factor of 6 because 2 divides into 6 without remainder.
Product of PrimesItispossibletowriteanypositivenumbergreaterthanoneasaproductofprimenumbers.
Thebestwaytodothisistouseafactortree.
Atthetopofthetreeyoustartwiththenumberyouwishtowriteasaproductofprimenumbers.
Youthenfindtwonumbersthatmultiplytogivethenumber.Onceoneofthebranchesofthetreehasaprimenumberatitsbranchendyoustopsimplifyingthatbranch.
Youcontinueworkingoneachbranchuntilonlyaprimenumberremains.
Ifyoumultiplyalltheprimenumbersattheendofeachofthebranchesyoushouldgetthenumberyoustartedwith.Aprimefactortreefor120isdrawnbelow.
80
20 4x
4 5 2 2xx
2 2x
So80=2x2x 5x 2x 2.Itdoesnotmatterwhattwonumbersyoufindtomultiplytogive80(i.e.40x2or8x5or10x8)youwillalwaysendupwiththesameprimefactorsattheend.
Example
a) Copythenumbers5,19,32,37,39,52andcircle thosethatareprime.b) Listthenexttwoprimenumbersafter61.c) Drawaprimefactortreefor150.
a) 5,19,32,37,39,52
b) Thenexttwoprimenumbersafter61are67 and71.c) Primefactortreefor150isasfollows.
Primefactorsare2x3x5x5.
150
15 10x
3 5 2 5xx
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IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
12 IAS 1.1 – Numeric Reasoning
To write any number in standard form first move the decimal point, so that the number has a value between 1 and 10
then multiply by an appropriate power of 10, so the number has the same value (found by counting the number of decimal places the decimal point has moved).
Writethefollowinginstandardform.a) 1320000 b) 0.00056 c) 2.1
a) 1.32x106 b) 5.6x10–4 c) 2.1x100as100=1
Writethefollowingasanordinarynumber.
a) 2.4x10–3 b) 8.647x102 c) 3.91x105
a) 0.0024 b) 864.7 c) 391000
Example
Example
To convert a number into standard form on your calculator use the scientific mode. Each calculator is a little different.
On the TI-84 Plus press MODE then choose SCI and then the number of decimal places you want to display. On the Casio 9750GII press SHIFT MENU then scroll down and select Display and choose SCI and then the number of decimal places you want to display.
Standard Form
Standard FormWeusestandard formasaconcisewayofwritingverylargeandverysmallnumbers.Anumberinstandardformiswrittenasanumberbetween1and10,multipliedbyapowerof10.Consider 3127686.2Youfirstmovethedecimalpointsothenumberhasavaluebetween1and10,i.e.3.1276862Younowmultiplyby1000000or106tomakethenumberequal3127686.2Instandardform3127686.2=3.1276862x106
To enter a number in standard form on a graphics calculator we use the EXP button (Casio 9750GII) or the EE button (TI-84 Plus),
e.g. for 3.127 686 2 x 106 we enter
3.1276862 Casio 9750GII
3.1276862 TI-84 Plus
EXP 6
2nd , 6EE
Withverysmallnumbersweworksimilarly.Consider 0.0045Movethedecimalpoint,sothenumberhasavaluebetween1and10,i.e.4.5.Younowneedtodividethisnumberby1000ormultiplyby0.001=10–3tomake4.5equal0.0045.Instandardform0.0045=4.5x10–3
Thedecimalpointneedstomove3placesleftfrom4.5sothepoweris–3.
0.0045=4.5x10–3
Thisnumbermustalwaysbebetween1and10.
65. 41500 66. 591 67. 12.75 68. 0.045
69. 0.592 70. 7 71. 12700000 72. 0.00000956
Merit–Writethefollowinginstandardform,calculatingtheanswerfirstifrequired.
A number displayed as 8.89E+05 means 8.89 x 105. A number displayed as 8.89E–05
means 8.89 x 10–5.
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IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
14 IAS 1.1 – Numeric Reasoning
100. Acountry’snationaldebtis$1.5x1012.Ifthepopulationofthecountryis285million,howmuchisowedperindividual?
99. Abladeofgrassgrowsonaverageatarateof2.0x10–8metrespersecond.Howmuchwillabladeofgrassgrowinoneweekinmillimetres?
101.Usestandardformtocalculatethefollowinginformationabout75yearoldMartin.Roundallyouranswersto3significantfigures.
Oneestimateofthenumberofcellsintheaveragehumanbodyisseventy-threepointeightmillionmillion.
a) Writethisasanordinarynumber.
b) Writethisnumberinstandardform.
c) ThepopulationoftheEarthisapproximately 6150000000people. Calculatethetotalnumberofhumancellsfor
theentirepopulationofEarth.
d) Theaveragehumanweighs65.5kg.Whatistheaveragemassofonecellingrams?(1000g=1kg).
e) Themassofasinglehydrogenatomis1.66x10–24g.Calculatehowmanytimesheavierasinglehumancellisthanahydrogenatom.
Aperson’sheartbeats,onaverage,85beatsperminuteforeveryminuteoftheirlife.f) HowmanyminuteshasMartinlivedfor? Ignoreleapyearsandgiveyouranswerin standardform.
g) HowmanytimeshasMartin’sheartbeaten?
h) Eachbeatoftheheartpumpsabout67mL. Howmanylitreshastheheartpumpedin Martin’slife?
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IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
24 IAS 1.1 – Numeric Reasoning
211. Keithisausedcarsalesmanandatthe beginningofamonthhas120vehiclesinstock.
a) Keithsells 245 ofhisstockinthemonth.How
manyvehiclesdoeshehaveleftattheendofthemonth?
b) 31 ofthevehiclesarecommercialvehicles,suchasvansandutes,and 8
5 ofthesecostlessthan$10000.HowmanyofKeith’scommercialvehiclescostlessthan$10000?
c) 32 ofthevehiclesarecarsandofthese 4
3 haveanenginecapacitylessthan2000cc.HowmanycarsdoesKeithhavewithanenginecapacitygreaterthan2000cc?
d) Keithsellsoneofhiscarsfor$27000.Thepurchaserputsdownadepositof 5
1 andpaysthebalanceoffover24monthsat0%interest.Howmuchdoesthepurchaserpaypermonth?
e) 45 ofthecustomerswhoboughtvehiclesonemonthpaidcashforthem.If20customerspaidcash,howmanyvehiclesintotaldidKeithsellduringthemonth?
f) Keithsells 245 ofhisstockinamonthand 5
2 oftheremainingstockthefollowingmonth.WhatfractionoftheoriginalstockofvehiclesremainafterthetwomonthsandhowmanycarsdoesKeithhaveleft?
To convert a decimal to a percentage move the decimal point two places to the right. On a Casio 9750GII to convert a fraction to a percentage on the calculator enter
On the TI-84 Plus to convert a fraction to a percentage enter
To simplify a fraction on the Casio 9750GII enter it into your calculator as a fraction and then press EXE. The calculator will automatically reduce
it down to its simplest form. If the simplified fraction is a mixed numeral press to convert it to an improper fraction. On the
TI-84 Plus to simplify a fraction like 35100
we enter
100. Percentagethereforemeans‘outof100’.Byrepresentingfiguresasapercentagewecaneasilymakeacomparisonbetweentwoormoresetsoffigures.
Thewordpercentageismadeupoftheprefix‘per’meaningoutofand‘centage’fromthesamerootas‘century’meaning
SHIFT
3 a b/c 5 x 1
EXE
0 0
ENTER
MATH
1Frac
3 ÷ 15 0 0
3 ÷ 5 x 1
ENTER
0 0
Percentages to Fractions
Percentages
Toconvertapercentagetoafractionwewritethepercentageasafractionoutof100andthensimplifythefractionifpossible.
Consider 35%
Asafraction = 35100
Simplify = 720
Decimal to a PercentageToconvertadecimaltoapercentagewemultiplyby100%.Consider 0.05Mult.by100% = 0.05x 100% = 5%
Fraction to a PercentageToconvertafractiontoapercentagewemultiplyby100%.
Consider 35
Mult.by100% = 35x100%
= 3005%
Simplify = 60%
F D
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28 IAS 1.1 – Numeric Reasoning
ExampleCalculatethefollowing.
247. Increase$210by23% 248. Decrease194by35%
Achievement–Answerthefollowingpercentagequestions.
a) Decrease$95by15%. b) Increase$255by16.5%
a) Startx(100±Change)% =Result 95x(100–15)%=Result 95x85% =Result Result =$80.75
b) Startx(100±Change)% =Result 255x(100+16.5)%=Result 255x116.5% =Result Result =$297.08(2dp)
249. Decrease$47.50by12.5% 250. Decreaseby30%ashirtthatcosts$67.50
Examplea) Anarticleisboughtfor$58andsoldfor$70.
Whatisthepercentageincrease?b) Findthepre-GSTamountwhenanarticle
sellingfor$250includesGSTof15%.
a) Startx(100±Change)%=Result
58x(100±Change)%=70
(100±Change)%=1.2069
(100±Change)=120.69 asa%
Change=20.7%(1dp)
Startx(100±Change)% =Result
Startx(100+15.0)% =250
Startx115.0% =250
Start =
250115.0%
=$217.39
251. Alechastopaya15%surchargeonameal costing$96.50.Howmuchwillhepay altogether?
252. Aplasmascreenusuallyretailsfor$3999.If apurchaserpayscashtheyareeligiblefora 12.5%discount.Howmuchdoestheplasma screencostwiththecashdiscount?
253. Duetoafluepidemic12%ofthepupilsina schoolareabsentoneday.Iftheschoolroll isnormally575pupils,howmanypupils arepresent?
254. Ahousesellsfor34%aboveitsgovernment valuationof$485000.Howmuchdoesitsell for?
255. Aschool’srollhasincreasedby6.5%overthe last10years.Iftenyearsagoithadarollof 1450,whatisitsrollnow?
256. Thevalueofacarhasdepreciatedinvalueby 55%.Ifitwasinitiallypurchasedfor$42500 whatisitworthnow?
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37IAS 1.1 – Numeric Reasoning
IAS 1.1 – Year 11 Mathematics and Statistics – Published by NuLake Ltd New Zealand © Robert Lakeland & Carl Nugent
Proportion
ProportionAproportionisapartconsideredinrelationtoawholeorastatementofequalitybetweentwoormoreratios.
i.e. ab
cd
==ab
cd
=
Directly Proportional Problems Directlyproportionalproblemsareproblemswhereachangeinonequantitycausesaproportionalchangeinanotherquantity.Twoquantitiesyandxareindirectproportionifbywhateverychanges,xchangesbythesameproportionormultiplier.Wewritey∝x,whichisreadasyisdirectlyproportionaltox,thismeansy=kx,wherekisaconstant.
E.g.Thecostofpensisdirectlyproportionaltothenumberofpensyoubuy.Iftwopenscost$1.50,howmanypenscanyoubuyfor$10.50? Firstwefindk,1.50=2k k=0.75(eachpencosts75cents)Tocalculatehowmanypenswecanbuyfor$10.50wedivide10.50bythecostofasinglepen0.75whichequals14pens.
Inversely Proportional Problems Inverselyproportionalproblemsareproblemsthataresimilartodirectlyproportionalproblemsexceptthatwhenxincreasesywilldecreaseandviceversa.Twoquantitiesyandxareinverselyproportionaliftheirproductalwaysremainconstant,i.e.xy=kor
y= kxwherekisaconstant.
E.g.Ifittakes4men6hourstodigadrain,howlongwillittake7mentodothesamejob? Firstwefindk,whichis4x6=24(totalnumberofmanhours).Tofindhowlongitwilltake7mentodigthedrainwedivide24(totalnumberofmanhours)
by7= 3 3
7hours.
The ratio of the number of men = the inverse ratio of the number of hours. i.e. 4 : 7 = x : 6
47=
x6
7x = 24
x = 3 3
7 hours
Directly proportional problems can also be set up as ratios, but make sure that the two ratios are written in the correct order.In the problem on the left,
if x = the number of pens then 1.52
= 10.5x
= 1.52
= 10.5x
Solving this equation gives 1.5x = 21 and then x = 14
Two quantities x and y are said to be inversely proportional if their product xy always remains constant.In the problem on the left, if x = the number of hours then 4 men x 6 hours = 7 men x x 24 = 7x
x = 3 3
7 hours
The more men the less time to complete the job, hence this is an inversely proportional problem. When x increases, y decreases.
A good test of an inverse proportional problem is to ask yourself,“If one quantity doubles, will the other half”, i.e. if x increases by a multiplier, y will decrease by the same divisor.
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58 IAS 1.1 – Numeric Reasoning
Page 9
30. 6.0(1dp)
31. 9(1sf)
32. 3.22(2dp)
33. 39(2sf)
34. 2.62(3sf)
35. 12(2sf)
36. 110(2sf)
37. 0.13(2sf)
38. 22.8m(1dp)
39. 47cm3(2sf)
40. 5.4m(1dp)
41. 5.6L(2sf) asonlythe45litresis measured(the8partsare counted).
42. 43cm2(2sf)
43. $353.50(2sf)
44. a) 21m2 b) 4 c) 55cm3(2sf) d) $170.95 e) $3897 f) Noshouldberoundedto 3sfi.e52.8m3. g) 18hours.
Page 10
45. 120(accept102)
46. 70(accept77)
47. 500
48. 9
49. 200
50. 60
51. 100
52. 250
53. $150
54. $10000
55. $100
56. $3
57. $12000
58. 1500cm3
59. $300
60. 1250km
AnswersPage 5
1. a) 1,2,4,7,14,28
b) 1,2,3,6,7,14,21,42
c) 1,19
2. a) 17,34,51,68,...
b) 21,42,63,84,...
c) 62,124,186,248,...
3. a) 1,3,7,21 and1,2,4,7,8,14,28,56
HCF=7
b) 1,3,5,9,15,45 and1,3,9,13,39,117
HCF=9
c) 1,5,19,95 and1,2,3,6,19,38,57,114
HCF=19
4. a) 12,24,36,48,60,72,...
20,40,60,...
LCM=60
b) 6,12,18,24,30,36,42,48, 54,60,66,...
11,22,33,44,55,66,..
LCM=66
c) 9,18,27,36,45,54,...
15,30,45,...
LCM=45
5. a) primenumber
b)
2x3x 23
c)
2x5x 2x 7
Page 6
6. 8,16,24,32,40,48,56,...
14,28,42,56,...
56seconds
138
2 69x
3 23x
140
10 14x
2 5 2 7xx
Page 6 cont...
7.
8. 4=2+2,5=2+3,6=3+3, 7=2+5,8=5+3, 9=2+7,10=5+5, 12=5+7,13=11+2, 14=11+3,15=2+13 16=13+3,18=7+11 19=2+17allcan.
14numberslessthan20.
9. HCFof448and616is 56sogreatestpossiblelength is56cm.
10. LCMof28and24whichis 168.So168seconds
11. 48=2x2x 2x2x 3
108=2x2x 3x3x 3
HCF=2x2x 3=12
12. LCMof40,48and60=240, so240minutes(4hours)
13. HCFof96,144and224, so16piecesofchicken.Page 8
14. 789.88
15. 0.0024
16. 50.0
17. 67500
18. 0.0098
19. 2000
20. 655.0
21. 0.0480
22. 27000
23. 44
24. 0.9
25. 5.23
26. 480
27. 8.4
28. 60
29. 7220
47
113
17
29
59
89
101
5
71
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