isa ~~b~~ o ~~k~~ emicah@gmail.com MATHEMATICSNOTES FORM ~~1TUTORIAL~~ ##### CHAPTERONE SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Identify,readandwritenaturalnumbersinsymbolsandwords; b.)Roundoffnumberstothenearesttens,hundreds,thousands,millionsandbillions; c.)Classifynaturalnumbersaseven,oddorprime; d.)Solvewordproblemsinvolvingnaturalnumbers. Content a.)Placevaluesofnumbers b.)Roundingoffnumberstothenearesttens,hundreds,thousands,millionsandbillions c.)Oddnumbers d.)Evennumbers e.)Primenumbers f.) Wordproblemsinvolvingnaturalnumbers ##### Introduction Placevalue Adigithaveadifferentvalueinanumberbecauseofitspositioninanumber.Thepositionofadigitina numberiscalleditsplacevalue. ##### Totalvalue Thisistheproductofthedigitanditsplacevalue. ##### Exam p le Number Hundred 345,678,901 |Col1|Col2|Col3|Col4|Col5|Col6|Col7|Col8|Col9| |---|---|---|---|---|---|---|---|---| |||||||||| Novels, UpdatedCPA,CIFA,KASNEB,ICT,College,Hig hSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com |Col1|Col2|Col3|Col4|Col5|Col6|Col7|Col8| |---|---|---|---|---|---|---|---| ||||||||| Aplacevaluechartcanbeusedtoidentifybothplacevalueand totalvalueofadigitinanumber.Theplace valuechartisalsousedinwritingnumbersinwords. ##### Example  Threehundredandfortyfivemillion,sixhundredandseventyeightthousand,ninehundredandone.  Sevenhundredandsixtyninemillion,Threehundredandonethousand,eighthundredandfiftyfour. ##### Billions Abillionisonethousandsmillion,writtenas1,000,000,000.Therearetenplacesinabillion. ##### Example Whatistheplacevalueandtotalvalueofthedigitsbelow? a.)47,397,263,402(placevalue7and8). b.)389,410,000,245(place3and9) ##### Solution a.)Theplacevaluefor6istenthousands.Itstotalvalueis60,000. b.)Theplacevalueof3ishundredbillions.Itstotalvalueis300,000,000,000. ##### Roundingoff Whenroundingofftothenearestten,theonesdigitdeterminestheteni.e.iftheonesdigitis1,2,3,or4the nearesttenisthetennumberbeingconsidered.Iftheonesdigitis5ormorethenearesttenisthenexttenor roundedup. Thus641tothenearesttenis640,3189tothenearestis3190. Whenroundingofftothenearest100,thenthelasttwodigitsornumbersendwith1to49roundoff downwards.Numberendingwith50to99areroundedup. Thus641tothenearesthundredis600,3189is3200. ##### Example Roundingoffeachofthefollowingnumberstothenearestnumberindicatedinthebracket: a.)473,678(100) b.)524,239(1000) c.)2,499(10) ~~Novels,UpdatedCPA~~, ~~C~~ ~~IFA~~, ~~KASNEB~~, ~~ICT~~, ~~C~~ ~~oll~~ ege ~~,High~~ ~~School&P~~ rimaryNotes0714 ~~497530~~ ----- ~~isabokemicah@g~~ ma ~~il~~ .co ~~m~~ ##### Solution a.)473,678is473,700tothenearest100. b.)524,239is524,000tothenearest1000 c.)2,499is2500tothenearest10. ##### OperationsonwholeNumbers Addition Example Findout: a.)98+6734+348 b.)6349+259+7954 ##### Solution Arrangethenumbersinverticalforms 98 6734 + 348 7180 6349 259 +79542 86150 ##### Subtracting Example Find:734698971 ##### Solution 73469 - 8971 64498 ##### Multiplication Theproductistheresultoftwoormorenumbers. ~~Novels~~, ~~Upd~~ ate ~~dCPA~~, ~~CIFA~~, ~~KASNEB~~, ~~ICT~~, ~~College~~, ~~HighSchool&Primar~~ ~~y~~ ~~N~~ ~~o~~ tes ~~0714~~ ~~497530~~ ----- ~~is~~ ~~abokemic~~ ~~ah@gmail~~ . ~~com~~ ##### Example Workout:469x63 ##### Solution 469 X63 1407469x3=1407 +28140 469x60=28140 29547 ##### Division Whenanumberisdividedbytheresultiscalledthequotient.Thenumberbeingdividedisthedividedandthe numberdividingisthedivisor. ##### Example Find:649314 ##### Solution Weget463and11istheremainder ##### Note: 6493=(463x14)+11 Ingeneral,dividend=quotientxdivision+remainder. |Col1|Col2|Col3|Col4|Col5| |---|---|---|---|---| |||||| |||||| |||||| |||||| |||||| ----- isabokem ~~i~~ ca ~~h@~~ gmail.com ##### Wordproblem Inworkingthewordproblems,theinformationgivenmustbereadandunderstoodwellbeforeattemptingthe question. Theproblemshouldbebrokendownintostepsandidentifyeachotheroperationsrequired. ##### Example Otegohad3469bagsofmaize,eachweighing90kg.Hesold2654ofthem. a.)Howmanykilogramofmaizewasheleftwith? b.)Ifheadded468morebagsofmaize,howmanybagsdidheendupwith? ##### Solution a.)Onebagweighs90kg. 3469bagsweigh3469x90=312,210kg 2654bagsweigh2654x90=238,860kg Novels,U p dat edCPA,CIFA,KA SNEB,ICT, ~~Colleg~~ ~~e,Hi~~ ~~ghSchool&P~~ ~~rimaryNotes0714497530~~ ----- ~~i~~ sa ~~b~~ okemicah@gma ~~il~~ . ~~com~~ Amountofmaizeleft =312,210238,860 =73,350kg. b.)Numberofbags=815+468 =1283 ##### EvenNumber Anumberwhichcanbedividedby2withoutaremainderE.g.0,2,4,60or8 3600,7800,806,78346 ##### OddNumber Anynumberthatwhendividedby2givesaremainder.E.g.471,123,1197,7129.Thenumbersendswiththe followingdigits1,3,5,7or9. ##### PrimeNumber Aprimenumberisanumberthathasonlytwofactorsoneandthenumberitself. Forexample,2,3,5,7,11,13,17and19. ##### Note: i.) 1isnotaprimenumber. ii.) 2istheonlyevennumberwhichisaprimenumber. Endoftopic ##### PastKCSEQuestionsonthetopic 1.)Write27707807inwords 2.)Allprimenumberslessthantenarearrangedindescendingordertoformanumber a.)Writedownthenumberformed b.)Whatisthetotalvalueoftheseconddigit? c.)Writethenumberformedinwords. ##### CHAPTERTWO ~~Nove~~ ~~ls,Upda~~ ~~tedCPA,CIFA,KASNEB,ICT,~~ ~~Colle~~ ~~ge~~, ~~HighSch~~ ~~oo~~ ~~l&PrimaryN~~ ~~otes0714497530~~ ----- ~~isabokemicah@gm~~ ~~ail.com~~ SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Expresscompositenumbersinfactorform; b.)Expresscompositenumbersasproductofprimefactors; c.)Expressfactorsinpowerform. Content a.)Factorsofcompositenumbers. b.)Primefactors. c.)Factorsinpowerform d.) ##### Introduction Definition Afactorisanumberthatdividesanothernumberwithoutaremainder. Anaturalnumberwithonlytwofactors,oneanditselfisaprimenumber.Oranynumberthatonlycanbe dividedby1anditself.Primenumbershaveexact ly2 differentfactors. ##### PrimeNumbersupto100 . |Col1|Col2|Col3|Col4| |---|---|---|---| ||||| ||||| ----- isabokemicah@gmail.com ##### Composite Anynumberthathas |Col1|Col2|Col3|Col4|Col5| |---|---|---|---|---| |||||| |||||| morefactorsthanjustitselfand1.Theycanbesaidtobenaturalnumberotherthan1whicharenotprime numbers.Theycanbeexpressedasaproductoftwoormoreprimefactors. 9=3x3 12=2x2x3 105=3x5x7 Thesamenumbercanberepeatedseveraltimesinsomesituations. 32=2x2x2x2x2=2 5 3 2 x 3 72=2x2x2x3x3=2 x 3 2 Toexpressanumberintermsofprimefactors,itisbesttotakethenumbersfrom thesmallestanddivideby eachofthemasmanytimesaspossiblebeforegoingtothenext. ##### Example Expressthefollowingnumbersintermsoftheirprimefactors a.)300 b.)196 ##### Solution a.) |Col1|Col2| |---|---| ||| ||| ||| ||| ||| 300=2x2x3x5x5 x3x 2 5 2 =2 x3x 2 b.) No v ~~els,Upda~~ ~~tedCPA~~, ~~CIFA,KA~~ ~~S~~ NEB, ~~ICT,Coll~~ ~~e~~ ~~ge,Hi~~ ghScho ol&P ri ~~maryN~~ ~~otes0714~~ ~~49753~~ 0 ----- ~~isabokemicah@gm~~ ~~ail.com~~ |Col1|Col2| |---|---| ||| ||| ||| ||| 196=2x2x7x7 2 2 x 7 =2 x 7 2 ##### Exceptions Thenumbers1and0areneitherprimenorcomposite.1cannotbeprimeorcompositebecauseitonlyhas onefactor,itself.0isneitheraprimenoracompositenumberbecauseithasinfinitefactors.Allother numbers,whetherprimeorcomposite,haveasetnumberoffactors.0doesnotfollowtherules. Endoftopic ##### PastKCSEQuestionsonthetopic 1.)Expressthenumbers1470and7056,eachasaproductofitsprimefactors. Henceevaluate: 1470 [2] Leavingtheanswerinprimefactorform 2.)Allprimenumberslessthantenarearrangedindescendingordertoformanumber (a)Writedownthenumberformed (b)Whatisthetotalvalueoftheseconddigit? ##### CHAPTERTHREE Novels,U p da tedCPA,CIFA,KASNEB,I ~~CT,College,H~~ ~~ighSchool&Primar~~ ~~yNotes07~~ ~~14497530~~ ----- isabokemicah@gma ~~il~~ .com SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: Thelearnershouldbeabletotestthedivisibilityofnumbersby2,3,4,5,6,8,9,10and11. Content Divisibilitytestofnumbersby2,3,4,5,6,8,9,10and11 ##### Introduction Divisibilitytestmakescomputationonnumberseasier.Thefollowingisatablefordivisibilitytest. |Col1|Col2| |---|---| ||| ||| ||| ||| ||| ||| ||| ~~N~~ ~~ovel~~ s,Up datedCPA,CIFA, ~~KASNEB~~, ~~ICT~~ ~~,College,Hig~~ h Sc ~~h~~ oo ~~l~~ & ~~P~~ r ~~i~~ m ~~aryNote~~ ~~s071~~ 4497530 ----- isabokemicah@gmail.com |Col1|Col2| |---|---| |st rd th intheoddpositonslike1 3 5 7 nd th th th positionlike2,4,6,8|| Endoftopic ##### PastKCSEQuestionsonthetopic CHAPTERFOUR SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)FindtheGCD/HCFofasetofnumbers. b.)ApplyGCDtoreallifesituations. Content a.)GCDofasetofnumbers b.)ApplicationofGCD/HCFtoreallifesituations ### Introduction AGreatestCommonDivisoristhelargestnumberthatisafactoroftwoormorenumbers. WhenlookingfortheGreatestCommonFactor,youareonlylookingfortheCOMMONfactorscontainedin ~~No~~ ~~vels~~, ~~Updated~~ ~~CPA~~, ~~CIFA~~, ~~KASNE~~ ~~B,ICT~~ ~~,Col~~ lege,HighSchool&Prim ~~a~~ ~~ryNotes07144975~~ ~~3~~ 0 ----- isabokemicah@gmail.com bothnumbers.TofindtheG.C.Doftwoormorenumbers,youfirstlistthefactorsofthegivennumbers, identifycommonfactorsandstatethegreatestamongthem. TheG.C.Dcanalsobeobtainedbyfirstexpressingeachnumberasaproductofitsprimefactors.Thefactors whicharecommonaredeterminedandtheirproductobtained. ##### Example FindtheGreatestCommonFactor/GCDfor36and54is18. ##### Solution Theprimefactorizationfor36is2x2x3x3. Theprimefactorizationfor54is2x3x3x3. Theybothhaveincommonthefactors2,3,3andtheirproductis18. Thatiswhythegreatestcommonfactorfor36and54is18. ##### Example FindtheG.C.Dof72,96,and300 ##### Solution |Col1|Col2|Col3| |---|---|---| |||| |||| |||| Endoftopic ##### PastKCSEQuestionsonthetopic 2 4 1.)Findthegreatestcommondivisoroftheterm.144x [3] yand81xy 2 4 b)Hencefactorizecompletelythisexpression144x [3] y-81xy (2marks) 2.)TheGCDoftwonumbersis7andtheirLCMis140.ifoneofthenumbersis20,findtheothernumber ~~Nove~~ ~~ls~~, ~~UpdatedC~~ P ~~A~~, ~~CIFA~~, ~~K~~ ~~ASNEB~~, ~~I~~ CT,College,HighSchool&P ~~rimary~~ ~~Notes0714497530~~ ----- isabokemicah@gmail.com 3.)TheLCM ofthreenumbersis7920andtheirGCDis12.Twoofthenumbersare48and264.Using factornotationfindthethirdnumberifoneofitsfactorsis9 ##### CHAPTERFIVE SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Listmultiplesofnumbers. b.)FindtheLCMofasetofnumbers. c.)ApplyknowledgeofLCMinreallifesituations. Content a.)Multiplesofanumber b.)LCMofasetofnumbers c.)ApplicationofLCMinreallifesituations. ##### Introduction Definition LCM orLCFisthesmallestmultiplethattwoormorenumbersdivideintoevenlyi.e.withoutaremainder. A multipleofanumberistheproductoftheoriginalnumberwithanothernumber. Somemultiplesof4are4,8,12,16,20,24,28,32,36,40,44,48,52,56 Somemultiplesof7are7,14,21,28,35,42,49,56 ACommonMultipleisanumberthatisdivisiblebytwoormorenumbers.Somecommonmultiplesof4and7 are28,56,84,and112. WhenlookingfortheLeastCommonMultiple,youarelookingforthesmallestmultiplethattheybothdivide intoevenly.Theleastcommonmultipleof4and7is28. ##### Example FindtheL.C.Mof8,12,18and20. ( usingtables ) ##### Solution |Col1|Col2|Col3|Col4|Col5| |---|---|---|---|---| |||||| ----- isabokemicah@gmail.com |Col1|Col2|Col3|Col4| |---|---|---|---| ||||| ||||| ||||| ||||| ||||| TheL.C.Mistheproductofallthedivisionsused. Therefore,L.C.M.of8,12,18and20=2x2x2x3x3x5 3 2 x 3 2 =2x 3 x5 =360 ##### Note; UnliketheG.C.Dtables,ifthedivisor/factordoesnotdivideanumberexactly,thenthenumberisretained,e.g., 2doesnotdivide9exactly,therefore9isretained.Thelastrowmusthaveallvalues1. Endoftopic ##### PastKCSEQuestionsonthetopic 4.)FindtheL.C.Mofx [2] +x,x [2] 1andx [2] x. 5.)Findtheleastnumberofsweetsthatcanbepackedintopolythenebagswhichcontaineither9or15or 20or24sweetswithnoneleftover. 6.)Anumbernissuchthatwhenitisdividedby27,30,or45,theremainderisalways3.Findthesmallest valueofn. 7.)Apieceoflandistobedividedinto20acresor24acresor28acresforfarmingand Leave7acresforgrazing.Determinethesmallestsizeofsuchland. 8.)Whenacertainnumberxisdividedby30,45or54,thereisalwaysaremainderof21.Findtheleast valueofthenumberx. N ovels,Updat ~~edCPA,CIF~~ ~~A,KASNEB,ICT,College~~, ~~HighSchool&~~ ~~Primary~~ ~~Notes0714497530~~ ----- isabokemicah@gmail.com Anumberm issuchthatwhenitisdividedby30,36,and45,theremainderisalways7.Findthesmallest possiblevalueofm. ##### CHAPTERSIX SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Defineintegers b.)Identifyintegersonanumberline c.)Performthefourbasicoperationsonintegersusingthenumberline. d.)Workoutcombinedoperationsonintegersinthecorrectorder e.)Applyknowledgeofintegerstoreallifesituations. Content a.)Integers b.)Thenumberline c.)Operationonintegers d.)Orderofoperations e.)Applicationtoreallifesituations ### Introduction ##### TheNumberLine Integersarewholenumbers,negativewholenumbersandzero.Integersarealwaysrepresentedonthenumber lineatequalintervalswhichareequaltooneunit. ##### OperationsonIntegers AdditionofIntegers Additionofintegerscanberepresentedonanumberline.Forexample,toadd +3to0,webeginat0andmove3unitstotherightasshownbelowinredtoget+3,Alsotoadd+4to+3we move4unitstotherightasshowninbluetoget+7. Novels ~~,Upd~~ ~~atedCP~~ ~~A,CI~~ FA,KAS NEB,IC T,Col ~~lege~~ ~~,HighSchool&Primar~~ ~~y~~ ~~Notes0714~~ ~~497530~~ ----- ~~isab~~ o ~~k~~ emicah@gmail.com Toadd-3tozerowemove3unitstotheleftasshowninredbelowtoget-3whiletoadd-2to-3wemove2 stepstotheleftasshowninbluetoget-5. ##### Note;  When adding positive numbers we movetotheright.  Whendealingwithnegativewemove totheleft. ##### Subtractionofintegers. ##### Example (+7)(0)=(+7) Tosubtract+7from 0,wefindanumbernwhichwhenaddedtoget0weget+7andinthiscasen=+7as shownaboveinred. ##### Example (+2)(+7)=(-5) Startat+7andmoveto+2.5stepswillbemadetowardstheleft.Theansweristherefore-5. ##### Example -3(+6)=-9 |__|__|__|__|__|__|__|__|__|__|__|__|__|__| -4-3-2 -1 0 1 2 34 5 6 7 8 9 10 Westartat+6andmovesto-3.9stepstotheleft,theansweris-9. ##### Note:  Ingeneralpositivessignscanbeignoredwhenwritingpositivenumbersi.e.+2canbewrittenas2but negativesignscannotbeignoredwhenwritingnegativenumbers-4canonlybewrittens-4. 4(+3)=4-3 =1 -3-(+6)=36 Nov els,Upda ~~tedCPA~~ ~~,~~ ~~CIFA~~ ~~,~~ ~~KASNEB~~, ~~ICT~~, ~~Colle~~ ~~ge~~, ~~H~~ ighSchoo ~~l&Prim~~ aryNotes0 ~~71449753~~ 0 ----- ~~i~~ sa ~~b~~ o ~~k~~ emicah@gmail.com =-3  Positiveintegersarealsoreferredtoasnaturalnumbers.Theresultofsubtractingthenegativeofa numberisthesameasaddingthatnumber. 2(-4)=2+4 =6 (-5)(-1)=-5+2 =-3  Inmathematicsitisassumedthatthatthenumberwithnosignbeforeithasappositivesign. ##### Multiplication Ingeneral i.) (anegativenumber)x(appositivenumber)=(anegativenumber) ii.) (apositivenumber)x(anegativenumber)=(anegativenumber) iii.) (anegativenumber)x(anegativenumber)=(apositivenumber) ##### Examples -6x5=-30 7x-4=-28 -3x-3=9 -2x-9=18 ##### Division Divisionistheinverseofmultiplication.Ingeneral  i.) (apositivenumber) (apositivenumber)=(apositivenumber)  ii.) (apositivenumber) (anegativenumber)=(anegativenumber)  iii.) (anegativenumber) (anegativenumber)=(apositivenumber)  iv.) (anegativenumber) (appositivenumber)=(anegativenumber) Note; Formultiplicationanddivisionofinteger:  Twolikesignsgivespositivesign.  Twounlikesignsgivesnegativesign  Multiplicationbyzeroisalwayszeroanddivisionbyzeroisalwayszero. N ~~ovels~~, ~~Updat~~ ~~edCPA~~, ~~CIFA~~, ~~KASNEB~~, ~~ICT,Co~~ ~~llege,H~~ ighSc ~~hool&PrimaryNotes~~ ~~071449~~ ~~7530~~ ----- ~~isab~~ ~~okemi~~ cah@gmail.com ##### Orderofoperations BODMASisalwaysusedtoshowastheorderofoperations. BBracketfirst. OOfissecond. DDivisionisthird. MMultiplicationisfourth. AAdditionisfifth. SSubtractionisconsideredlast. ##### Example 6x342+5+ ( 2-1 ) = ##### Solution UseBODMAS (21)=1wesolvebracketsfirst (42)=2wethensolvedivision (6x3)=18nextismultiplication Bringthemtogether 182+5+1=22wesolveadditionfirstandlastlysubtraction 18+62=22 Endoftopic ##### PastKCSEQuestionsonthetopic 1.)Thesum oftwonumbersexceedstheirproductbyone.Theirdifferenceisequaltotheirproductlessfive. Findthetwonumbers. (3mks) 2.) 3x1>-4 2x+17 - -  3.) Evaluate 12(3)x4 (15) -5x62+(-5) 4.)Withoutusingacalculator/mathematicaltables,evaluateleavingyouranswerasasimplefraction Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&Primar y Notes07144 97530 ----- ~~isabokemicah@gmail.com~~ ( - 4)( - 2)+( - 12)  (+3) + - 20+(+4)+ - 6) -9(15) 46-(8+2)-3 5.) Evaluat e - 8  2+12x9  4x6 56  7x2 6.)Evaluatewithoutusingmathematicaltablesorthecalculator 1.9x0.032 20x0.0038 ##### CHAPTERSEVEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Identifyproperandimproperfractionsandmixednumber. b.)Convertmixednumberstoimproperfractionsandviceversa. c.)Comparefractions; d.)Performthefourbasicoperationsonfractions. e.)Carryoutcombinedoperationsonfractionsinthecorrectorder. f.)Applytheknowledgeoffractionstoreallifesituations. Content a.)Fractions b.)Proper,improperfractionsandmixednumbers. c.)Conversionofimproperfractionstomixednumbersandviceversa. d.)Comparingfractions. e.)Operationsonfractions. f.)Orderofoperationsonfractions g.)Wordproblemsinvolvingfractionsinreallifesituations. ~~Nove~~ ~~ls,UpdatedCPA~~, ~~CIFA~~, ~~KAS~~ ~~NEB~~, ~~ICT~~, ~~College~~, ~~Hi~~ g ~~hS~~ c ~~h~~ oo ~~l~~ & ~~Primary~~ ~~No~~ ~~tes0~~ 7144 97530 ----- ~~isabokemicah@gmai~~ ~~l.com~~ ### Introduction Afractioniswrittenintheform awhereaandbarenumbersandbisnotequalto0.Theuppernumberis b calledthenumeratorandthelowernumberisthedenominator. ##### anumerator bdenominator Properfraction Inproperfractionthenumeratorissmallerthanthedenominator.E.g. 2 1 3,4 ##### Improperfraction Thenumeratorisbiggerthanorequaltodenominator.E.g. ##### 15 ##### 9 2 ##### 7 15 9 3,,6 2 ##### 6 ##### Mixedfraction Animproperfractionwrittenasthesumofanintegerandaproperfraction.Forexample 7 1 =2 ~~+~~ 3 3 = ~~2~~ [1] 3 ##### ChangingaMixedNumbertoanImproperFraction Novels,U p datedCPA,CIFA, K ~~ASNEB,ICT~~, ~~College~~, ~~Hi~~ ~~g~~ ~~hSchool&~~ ~~PrimaryNot~~ ~~es0714497530~~ ----- isabokemicah@gmail.com Mixednumber 4 (containsawholenumberandafraction) Improperfraction- (numeratorislargerthandenominator) Step1Multiplythedenominatorandthewholenumber Step2Addthisanswertothenumerator;thisbecomesthenewnumerator Step3Carrytheoriginaldenominatorover ##### Example 3 = 38+1=25 = ##### Example 4 = 49+4=40 = ##### ChanginganImproperFractiontoaMixedNumber Step1Dividethenumeratorbythedenominator Step2Theanswerfromstep1becomesthewholenumber Step3Theremainderbecomesthenewnumerator Step4Theoriginaldenominatorcarriesover Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example = 475 or 5 = 5 = 9 2 ##### Example = 2 = 2 =4  ##### ComparingFractions Whencomparingfractions,theyarefirstconvertedintotheirequivalentformsusingthesamedenominator. ##### EquivalentFractions Togettheequivalentfractions,wemultiplyordividethenumeratoranddenominatorofagivenfractionbythe samenumber.Whenthefractionhasnofactorincommonotherthan1,thefractionissaidtobeinitssimplest form. ##### Example Arrangethefollowingfractionsinascendingorder(fromthesmallesttothebiggest):   /3 Step1:Changeallthefractionstothesamedenominator. Step2:Inthiscasewewilluse12because2,4,6,and3allgointoi.e.Weget12byfindingtheL.C.M ofthe denominators.TogettheequivalentfractionsdividethedenominatorbytheL.C.M andthenmultiplyboth thenumeratoranddenominatorbytheanswer, Forwedivide122=6,thenmultiplyboththenumeratoranddenominatorby6asshownbelow. 1 [x6] 1 [x3] 5 [x2] 2 [x4] 2 x6 4 x3 6 x2 3 x4 Step3:Thefractionswillnowbe:     Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Step4:Nowputyourfractionsinorder(smallesttobiggest.)     Step5:Changeback,keepingtheminorder.     Youcanalsousepercentagestocomparefractionsasshownbelow. ##### Example Arrangethefollowingindescendingorder(fromthebiggest)     ##### Solution 5 X100=41.67% 12 7 X100=233.3% 3 11 X100=220% 5 9 X100=225% 4 ,,, OperationonFractions AdditionandSubtraction Thenumeratorsoffractionswhosedenominatorsareequalcanbeaddedorsubtracteddirectly. ##### Example += = Whenaddingorsubtractingnumberswithdifferentdenominatorslike: +=? =? Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Step1Findacommondenominator(anumberthatbothdenominatorswillgointoorL.C.M) Step2DividethedenominatorofeachfractionbythecommondenominatororL.C.M andthenmultiplythe answersbythenumeratorofeachfraction Step3Addorsubtractthenumeratorsasindicatedbytheoperationsign Step4Changetheanswertolowestterms ##### Example + = Commondenominatoris8becauseboth2and8willgointo8 ##### + = [4+7] 8 Whichsimplifiesto 1 ##### Example 4  = Commondenominatoris20becauseboth4and5willgointo20 4 = 4  = 4 Or ### 12-5 4  = 4 = 4 ### 20 ### 7 20 Mixednumberscanbeaddedorsubtractedeasilybyfirstexpressingthemasimproperfractions. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Examples 2 4 3 5 ##### Solution 2 4 2 ~~5~~ + ~~1~~ =5 ~~+~~ +1 ~~+~~ 3 5 3 4 5 4 5 2 3 = ( 5+1 ) ~~+~~ [2] ~~+~~ 3 4 5 ##### =6 + [10+12] 15 =6 + [22] 15 7 7 =6+ 1 = 7 15 15 ##### Example -2 -1 Evaluate ~~+~~ 3 ##### Solution 5 -2 -1 -16-3 ~~+~~ = ~~=~~ -16-3 -19 3 -1 5 24 24 ##### MultiplyingSimpleFractions Step1Multiplythenumerators Step2Multiplythedenominators Step3Reducetheanswertolowesttermsbydividingbycommondivisors ##### Example  = whichreducesto ##### MultiplyingMixedNumbers Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Step1Convertthemixednumberstoimproperfractionsfirst Step2Multiplythenumerators Step3Multiplythedenominators Step4Reducetheanswertolowestterms ##### Example 2 1 =  = Whichthenreducesto3 ##### Note: Whenopposingnumeratorsanddenominatorsaredivisiblebyacommonnumber,youmayreducethe numeratoranddenominatorbeforemultiplying.Intheaboveexample,afterconvertingthemixednumbersto improperfractions,youwillseethatthe3inthenumeratorandtheopposing3inthedenominatorcouldhave beenreducedbydividingbothnumbersby3,resultinginthefollowingreducedfraction:  = = 3 ##### DividingSimpleFractions Step1Changedivisionsigntomultiplication Step2Changethefractionfollowingthemultiplicationsigntoitsreciprocal(rotatethefractionaroundsothe olddenominatoristhenewnumeratorandtheoldnumeratoristhenewdenominator) Step3-Multiplythenumerators Step4Multiplythedenominators Step5simplifytheanswertolowestterms ##### Example  = becomes  whichwhensolvedis ##### DividingMixedNumbers Step1Convertthemixednumberornumberstoimproperfraction. Step2Changethedivisionsigntomultiplication. Step3Changethefractionfollowingthemultiplicationsigntoitsreciprocal(flipthefractionaroundsothe olddenominatoristhenewnumeratorandtheoldnumeratoristhenewdenominator) Step4-Multiplythenumerators. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Step5Multiplythedenominators. Step6Simplifytheanswertolowestform. ##### Example 3 2 = becomes  becomes  = Whichwhensolvedis  = whichsimplifiesto1 ##### OrderofoperationsonFractions Thesamerulesthatapplyonintegersarethesameforfractions BODMAS ##### Example 1 1 15 ~~~~ of12=15 [x12] (westartwithofthendivision) ### 4 ( 4 ) =153 =5 ##### Example 1 3 11 x ### 2 [{] 8 [+] [(] 3 [-] 4 1 = ### [-] 4 [)] [}] 1 1 3 11 ~~+~~ x = ### 6 2 [{] 8 [+] [(] 3 [-] 4 [)] [}] ##### Solution 4-1 1  ~~=~~ ~~=~~ (westartwithbracket) 12 1 12 3 1 11 ~~=~~ (Wethenworkouttheouterbracket) ### { 8 [+] 12 [}] 21 1 ### 12 [}] 11 21 1 1 11 1 11 ~~+~~ ~~x~~ ~~=~~ ~~+~~ (Wethenworkoutthemultiplication) 6 2 48 6 48 1 2 11 48 1 6 11 48 1 11 19 ~~+~~ ~~=~~ (Additioncomeslasthere) 6 48 48 11 48 19 48 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example ## 1 1 ~~+~~ 2 3 Evaluate + ## 1 21 o f(- ) 7 56 ## 2 ~~f(-~~ 5 ## 1 6 ##### Solution Wefirstworkoutthisfirst ## 1 1 ~~+~~ 2 3 ## 1 21 o f(- ) 7 56 ## 2 ~~f(-~~ 5 ## 1 6 1 1 3+2 ~~+~~ ~~=~~ ~~=~~ 2 3 6 1 3 3+2 6 5 6 1 21 1 7 o ~~f~~ ~~=~~ ~~x~~ ~~=~~ ## 7 [(] 5 [-] 6 [)] 7 30 1 ## [-] 6 [)] 1 7 7 30 1 30 5 30=25 6 ## 1 1 ~~+~~ 2 3 Therefo re +=25+ ## 1 21 o f(- ) 7 56 ## 2 ~~f(-~~ 5 ## 1 6 ##### =25  Note: Operationsonfractionsareperformedinthefollowingorder.  Performtheoperationenclosedwithinthebracketfirst.  If(of)appears,performthatoperationbeforeanyother. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example 13 173 1 Evaluate ~~:~~ of ~~1~~ 5= ### 2 [{] 5 [+] 4 [(] 3 [-] 7 [)] 2 [}] 173 ### 4 [(] 3 [-] 7 3 ### [-] 7 [)] 1 5= ### 2 [}] ##### Solution 13 140 1 of ~~1~~ 5 ### 2 [{] 5 [+] 4 [(] 21 [)] 2 1 ### 4 [(] 40 ### [(] 21 [)] [1] 3 1 ~~[x]~~ 40 ~~x~~ 3 5 ### 2 [{] 5 [+] 4 21 2 [}] ~~=~~ [1] 1 4 40 21 1 ~~1~~ 5 ### 2 [}] 3 5 ### 2 [}] ##### [1] 3 10 x 3 5 # 2 [(] 5 [+] 21 2 ##### = [1] ##### 10 21 ##### 3 5 # 2 [)] ##### [1] 3 # 2 [(] 5 [+] ##### = [1] ##### 5 # 35 [)] ##### 121+5 1 26 ~~= = x =~~ # 2 [(] 35 [)] 2 35 ##### 21+5 # 35 [)] ##### 21+5 ##### 1 2 ##### 26 ~~x~~ 35 ##### 13 35 ##### Example TwopipesAandBcanfillanemptytankin3hrsand5hrsrespectively.PipeCcanemptythetankin 4hrs.IfthethreepipesA,BandCareopenedatthesametimefindhowlongitwilltakeforthetank tobefull. ##### Solution +-= 20+1215 60 = =1hr 1=1x60/17 =3.5294118 =3.529hrs. Endoftopic Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### PastKCSEQuestionsonthetopic 1. Evaluatewithoutusingacalculator. 2. Atwodigitnumberissuchthatthesumoftheonesandthetensdigitisten.Ifthedigitsarereversed,the newnumberformedexceedstheoriginalnumberby54. Findthenumber. 3. Evaluate 4. Converttherecurringdecimall intofraction 5. Simplify withoutusingtablesorcalculator 6. Evaluatewithoutusingtablesorcalculators 7. Mr.Saidikeepsturkeysandchickens.Thenumberofturkeysexceedsthenumberofchickensby6. Duringanoutbreakofadisease,ofthechickenand [1] / 3 oftheturkeysdied.Ifhelosttotalof30birds, howmanybirdsdidhehavealtogether? 9. Workout 10. Evaluate -4of(-4+-5  15)+-34  2) 84  -7+3--5 11.Writetherecurringdecimal0.CanasFraction Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 13. Withoutusingtablesorcalculatorsevaluate. 14. Withoutusingtablesorcalculator,evaluatethefollowing. - -  8+(13)x3 (5) - 1+( - 6)2x2 15. Express asasinglefraction 16. Simplify of 3+1(2- [2] / 3 ) of2   17. Evaluate: 2 / 5  of [4/] 9 1 [1] / 10 1  [1] / 8 / 6 of3/ 8 18. Withoutusingacalculatorortable,workoutthefollowingleavingtheanswerasamixednumberinits simplestform: +1 [2] / 7  [3] / 7 of2 [1] / 3 (9/ 7 - 3 / 8 )x [2] / 3 19. Workoutthefollowing,givingtheanswerasamixednumberinitssimplestform. ##### 2 / 5   of 4/ 9  1 [1] / 10 1  [1] / 8 / 16 x3/ 8 20. Evaluate; 23. Withoutusingacalculator,evaluate: Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 1 [4] / 5 of25/ 18  1 [2] / 3 x24 2 [1] / 3 of12  [5] / 3 leavingtheanswerasafractioninitssimplestform 24. Therewasafund-raisinginMatissehighschool.Oneseventhofthemoneythatwasraisedwasusedto constructateachershouseandtwothirdsoftheremainingmoneywasusedtoconstructclassrooms.If shs.300,000remained,howmuchmoneywasraised ##### CHAPTEREIGHT SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Convertfractionsintodecimalsandviceversa b.)Identifyrecurringdecimals c.)Convertrecurringdecimalsintofractions d.)Roundoffadecimalnumbertotherequirednumberofdecimalplaces e.)Writenumbersinstandardform f.)Performthefourbasicoperationsondecimals g.)Carryouroperationsinthecorrectorder h.)Applytheknowledgeofdecimalstoreallifesituations. Content a.)Fractionsanddecimals b.)Recurringdecimals c.)Recurringdecimalsandfractions d.)Decimalplaces e.)Standardform Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com f.) Operationsondecimals g.)Orderofoperations h.)Reallifeproblemsinvolvingdecimals. ### Introduction Afractionwhosedenominatorcanbewrittenasthepowerof10iscalledadecimalfractionoradecimal.E.g. ,,. Adecimalisalwayswrittenasfollows1/10iswrittenas0.1while5/100iswrittenas0.05.Thedotiscalledthe decimalpoint. Numbersafterthedecimalpointsarereadassingledigitse.g.5.875isreadasfivepointeightsevenfive.A decimalfractionsuch8.3means8+.Adecimalfractionwhichrepresentsthesumofawholenumberand aproperfractioniscalledamixedfraction. Placevaluechart |Col1|Col2|Col3|Col4|Col5|Col6|Col7|Col8|Col9|Col10|Col11| |---|---|---|---|---|---|---|---|---|---|---| |||||||||||| ##### DecimaltoFractions Toconvertanumberfrom fractionform todecimalform,simplydividethenumerator(thetopnumber)bythe denominator(thebottomnumber)ofthefraction. ##### Example:  Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 8  Addasmanyzerosasneeded. 48 20 16 40 40 0 ##### Convertingadecimaltoafraction Tochangeadecimaltoafraction,determinetheplacevalueofthelastnumberinthedecimal.Thisbecomes thedenominator.Thedecimalnumberbecomesthenumerator.Thenreduceyouranswer. Example: .625-the5isinthethousandthscolumn,therefore, .625= =reducesto ##### Note: Yourdenominatorwillhavethesamenumberofzerosastherearedecimaldigitsinthedecimalnumberyou startedwith-.625hasthreedecimaldigitssothedenominatorwillhavethreezero. ##### RecurringDecimals Thesearedecimalfractionsinwhichadigitoragroupofdigitsrepeatcontinuouslywithoutending. 1 =0.333333 3 5 =0.454545454 11 Wecannotwriteallthenumbers,wethereforeplaceadotaboveadigitthatisrecurring.Ifmorethanonedigit recursinapattern,weplaceadotabovethefirstandthelastdigitinthepattern. E.g. 0.3333.iswrittenas0.3 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 0.4545iswrittenas0.45 0.324324.iswrittenas0.324 Anydivisionwhosedivisorhasprimefactorsotherthan2or5formsarecurringdecimalornon-terminating decimal. ##### Example Expresseachasafraction a.)0.6 b.)0.73 c.)0.15 ##### Solution a.)Letr=0.66666-------(I) 10r=6.6666---------------(II) SubtractingIfromII 9r=6 r ~~=~~ [6] 9 ~~=~~ [2] 3 b.)Letr=0.73333---------(I) 10r=7.3333333---------(II) 100r=73.33333---------(III) Subtracting(II)from(III) 90r=66 r ~~=~~ [66] 90 ~~=~~ [11] 15 c.)Letr=0.151515---------(I) 100r=15.1515-----------(II) 99r=15 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com r ~~=~~ [15] 99 ~~=~~ [5] 33 ##### Decimalplaces Whentheprocessofcarryingoutdivisiongoesoverandoveragainwithoutendingwemayroundoffthedigits toanynumberofrequireddigitstotherightofdecimalpointswhicharecalleddecimalplaces. ##### Example Round2.832tothenearesthundredth. ##### Solution Step1Determinetheplacetowhichthenumberistoberoundedis. 2 .832 Step2Ifthedigittotherightofthenumbertoberoundedislessthan5,replaceitandallthedigitstothe rightofitbyzeros.Ifthedigittotherightoftheunderlinednumberis5orhigher,increasetheunderlined numberby1andreplaceallnumberstotherightbyzeros.Ifthezerosaredecimaldigits,youmayeliminate them. 2 .832 = 2.830 = 2.83 ##### Example Round43.5648tothenearestthousandth. ##### Solution 43.5 648 = 43.5650= 43.565 ##### Example Round5,897,000tothenearesthundredthousand. ##### Solution 5,89 7,000 =5,900,000 ##### StandardForm Anumberissaidtobeinstandardformifitisexpressedinform AX10 n,Where1<A<10andnisaninteger. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example Writethefollowingnumbersinstandardform. a.)36 b.)576c.)0.052 ##### Solution a.)36/10x10=3.6x10 1 b.)576/100x100=5.76x10 2 c.)0.052=0.052x100/100 ##### OperationonDecimals 5.2 ~~~~ [1] 100 5.2 ~~(~~ [1] 2 ~~(~~ 100 [)] 5.210 -2 ##### AdditionandSubtraction Thekeypointwithadditionandsubtractionistolineupthedecimalpoints! ##### Example 2.64+11.2= 2.64 +11.20  inthiscase,ithelpstowrite11.2as11.20 13.84 ##### Example 14.7312.155= 14.730  againaddingthis0helps - 12.155 2.575 ##### Example 127.5+0.127= 327.500 + 0.127 327.627 ##### Multiplication Whenmultiplyingdecimals,dothesum asifthedecimalpointswerenotthere,andthencalculatehowmany numbersweretotherightofthedecimalpointinboththeoriginalnumbers-next,placethedecimalpointin youranswersothattherearethisnumberofdigitstotherightofyourdecimalpoint? Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example 2.1x1.2. Calculate21x12=252.Thereisonenumbertotherightofthedecimalineachoftheoriginalnumbers, makingatotaloftwo.Wethereforeplaceourdecimalsothattherearetwodigitstotherightofthedecimal pointinouranswer. Hence2.1x1.2=2.52. Alwayslookatyouranswertoseeifitissensible.2x1=2,soouranswershouldbecloseto2ratherthan20 or0.2whichcouldbetheanswersobtainedbyputtingthedecimalinthewrongplace. ##### Example 1.4x6 Calculate14x6=84.Thereisonedigittotherightofthedecimalinouroriginalnumberssoouransweris8.4 Check1x6=6soouranswershouldbecloserto6than60or0.6 ##### Division Whendividingdecimals,thefirststepistowriteyournumbersasafraction.Notethatthesymbol/isusedto denotedivisioninthesenotes. Hence2..2= 2.14 1.2 Next,movethedecimalpointtotherightuntilbothnumbersarenolongerdecimals.Dothisthesamenumber ofplacesonthetopandbottom,puttinginzerosasrequired. ##### 2.14 Henc ~~e~~ become ~~s~~ ##### 1.2 ##### 214 120 Thiscanthenbecalculatedasanormaldivision. Alwayscheckyouranswerfrom theoriginaltomakesurethatthingshaventgonewrongalongtheway.You wouldexpect2..2tobesomewherebetween1and2.Infact,theansweris1.78. Ifthismethodseemsstrange,tryusingacalculatortocalculate2..2,21.,214/120and . Theanswershouldalwaysbethesame. ##### Example 4..14= 4.36 = 436 =31.14 1.14 14 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example 27..2= 27.93 = 2793 =23.28 1.2 120 ##### RoundingUp Somedecimalnumbersgoonforever!Tosimplifytheiruse,wedecideonacutoffpointandroundthemupor down. Ifwewanttoround2.734216totwodecimalplaces,welookatthenumberinthethirdplaceafterthedecimal, inthiscase,4.Ifthenumberis0,1,2,3or4,weleavethelastfigurebeforethecutoffasitis.Ifthenumberis 5,6,7,8or9weroundupthelastfigurebeforethecutoffbyone.2.734216thereforebecomes2.73when roundedto2decimalplaces. Ifweareroundingto2decimalplaces,weleave2numberstotherightofthedecimal. Ifweareroundingto2significantfigures,weleavetwonumbers,whethertheyaredecimalsornot. ##### Example 243.7684=243.77(2decimalplaces) =240(2significantfigures) 1973.285=1973.29(2decimalplaces) =2000(2significantfigures) 2.4689=2.47(2decimalplaces) =2.5(2significantfigures) 0.99879=1.00(2decimalplaces) =1.0(2significantfigures) ##### Orderofoperation Thesamerulesonoperationsisalwaysthesameevenfordecimals. ##### Examples Evaluate 0.02+3.5x2.60.1(6.2-3.4) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution 0.02+3.5x2.60.1x2.8=0.02+0.91-0.28 Endofto p ic ##### PastKCSEQuestionsonthetopic 1.)Withoutusinglogarithmtablesoracalculatorevaluate. 384.16x0.0625 96.04 2.)Evaluatewithoutusingmathematicaltable 3.)Expressthenumbers1470and7056,eachasaproductofitsprimefactors. Henceevaluate: 1470 [2] 7056 Leavingtheanswerinprimefactorform 4.)Withoutusingmathematicaltablesorcalculators,evaluate |Col1|Col2| |---|---| ||| 5.)Evaluatewithoutusingmathematicaltablesorthecalculator 0.0625x2.56 0.25x0.08 CHAPTERNINE Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Findsquaresofnumbersbymultiplication b.)Findsquaresfromtables c.)Findsquarerootbyfactormethod d.)Findsquarerootfromtables. Content a.)Squaresbymultiplication b.)Squaresfromtables c.)Squarerootsbyfactorization d.)Squarerootsfromtables. ### Introduction ##### Squares Thesquareofanumberissimplytheumbermultipliedbyitselfonce.Forexamplethesquareof15is225.That is15x15=225. Squarefromtables Thesquaresofnumberscanbereaddirectlyfrom tableofsquares.Thistablesgiveonlyapproximatevalues ofthesquaresto4figures.Thesquaresofnumbersfrom1.000to9.999canbereaddirectlyfromthetables. Theuseoftablesisillustratedbelow ##### Example Findthesquareof: a.)4.25 b.)42.5 c.)0.425 ##### Tables a.)Toreadthesquareof4.25,lookfor4.2downthecolumnheadedx.Movetotherightalongthisrow,up towhereitintersectswiththecolumnheaded5.Thenumberinthispositionisthesquareof4.25 So4.2 =18.06to4figures5 2 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 0 2 0 2 2 b.)Thesquareof4.25liesbetween4 and50 2 0 2 between1600and2500. .5 =( 2 4.25x10 1 2 42.5 =( 2 1 ) 2 2 2 x 10 =4.25 x 2 =18.06x100 2 1 2 =(4.25 ~~x~~ 10 [)] c.)0.425 =( 2 4.25 ~~x~~ 2 2 2 1 x ~~(~~ ) 10 =4.25 x ~~(~~ =18.06x1/100 =0.1806 Thesquaretableshaveextracolumnslabeled1to9totherightofthethickline.Thenumbersunderthese columnsarecalledmeandifferences.Tofind3.162,read3.16toget9.986.Thenreadthenumberinthe positionwheretherowcontaining9.986intersectswiththedifferencescolumnheaded2.Thedifferenceis13 andthisshouldbeaddedtothelastdigitsof9.986 9.986 + 13 9.999 56.129has5significantfiguresandinordertouse4figurestables,wemustfirstrounditofftofourfigures. 56.129=56.13to4figures 2 =(5.613x10 1 2 56.13 2 =(5.613x 1 ) 2 =31.50x10 2 =3150 ##### SquareRoots Squarerootsaretheoppositeofsquares.Forexample5x5=25,wesaythat5isasquarerootof25. Anypositivenumberhastwosquareroots,onepositiveandtheothernegative.Thesymbolforthesquareroot ofanumberis. Anumberwhosesquarerootisanintegeriscalledaperfectsquare.Forexample1,4,9,25and36areperfect squares. ##### SquarerootsbyFactorization. Thesquarerootofanumbercanalsobeobtainedusingfactorizationmethod. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example Findthesquarerootof81byfactorizationmethod. ##### Solution 81=3x33x3 (Findtheprimefactorof81) =(3x3)(3x3)(Grouptheprimefactorsintotwoidenticalnumbers) =3x3 (Outofthetwoidenticalprimefactors,chooseoneandfindtheirproduct) =9 ##### Note: Pairtheprimefactorsintotwoidenticalnumbers.Foreverypair,pickonlyonenumberthenobtaintheproduct. ##### Example Find1764byfactorization. ##### Solution 1764= 2x2x3x3x7x7 =2x3x7 =42 ##### Example Find441byfactorization ##### Solution 441= 3x3x7x7 =3x7 =21 ##### SquareRootfromtables Squarerootsofnumbersfrom1.0to99.99aregiveninthetablesandcanbereaddirectly. ##### Examples Usetablestofindthesquarerootof: a.)1.86 b.)42.57 c.)359 d.)0.8236 ##### Solution a.)Toreadthesquarerootof1.86,lookfor1.8inthecolumnheadedx,movetotherightalongthisrowto whereitintersectswiththecolumnheaded6.Thenumberinthispositionisthesquarerootof 1.86.Thus1.86=1.364to4figures. b.) 42.57Lookfor42inthecolumnheadedxandmovealongtherowcontaining42towhereitintersects Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com withthecolumnheaded5.Readthenumberinthisposition,whichis6.519.Thedifferencefor7from thedifferencecolumnalongthisrowis6.Thedifferenceisaddedto6.519asshownbelow: 6.519 +0.006 Thus,42.57=6.525to4figures. Foranynumberoutsidethisrange,itisnecessarytofirstexpressitastheproductofanumberinthisrange andanevenpowerof10. c.)359=3.59x10 2 =1.895x10 =18.95(fourfigures) 359= ( 3.59x1000 ) d.)0.8236=82.36x ### ~~(~~ 1 2 ### ( 10 [)] 2 =(9.072+0.004) ~~x~~ [1] 10 =0.9076(4figures) Endoftopic ##### PastKCSEQuestionsonthetopic 1.)Evaluatewithoutusingtablesorcalculators 2.)Evaluateusingreciprocals,squareandsquareroottablesonly. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 3.)Usingacalculator,evaluate (Showyourworkingateachstage) 4.)Usetablesofreciprocalsandsquarerootstoevaluate 5.)Usetablestofind; a)I)4.978 [2] ii)Thereciprocalof31.65 b)Henceevaluateto4.S.Fthevalueof 4.978 [2]  [1] / 31.65 6.)Usetablesofsquares,squarerootsandreciprocalstoevaluatecorrectto4S. 3 2 7. Withoutusingmathematicaltablesorcalculator,evaluate :153x1.8 givingyouranswerinstandard form 0.68x0.32 CHAPTERTEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Useletterstorepresentnumbers b.)Writestatementsinalgebraicform c.)Simplifyalgebraicexpressions d.)Factorizeanalgebraicexpressionsbygrouping e.)Removebracketsfromalgebraicexpressions |Col1|Col2| |---|---| ||| f.)Evaluatealgebraicexpressionsbysubstitutingnumericalvalues g.)Applyalgebrainreallifesituations. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Content a.)Lettersfornumbers b.)Algebraicfractions c.)Simplificationofalgebraicexpressions d.)Factorizationbygrouping e.)Removalofbrackets f.) Substitutionandevaluation g.)Problemsolvinginreallifesituations. ### Introduction Analgebraicexpressionisamathematicalexpressionthatconsistsofvariables,numbersandoperations.The valueofthisexpressioncanchangeClarifythedefinitionsandhavestudentstakenotesontheirgraphic organizer. ##### Note: o AlgebraicExpressioncontainsatleastonevariable,onenumberandoneoperation. An exampleofanalgebraicexpressionisn+9. o Variablealetterthatisusedinplaceofanumber.Sometimes,thevariablewillbegivena value. Thisvaluewillreplacethevariableinordertosolvetheequation. Othertimes,the variableisnotassignedavalueandthestudentistosolvetheequationtodeterminethevalue ofthevariable. o Constantanumberthatstandsbyitself.The9inourpreviousvocabularyisanexampleofa constant. o Coefficientanumberinfrontofandattachedtoavariable.Forexample,intheexpression5x+ 3,the5isthecoefficient. o Termeachpartofanexpressionthatisseparatedbyanoperation.Forinstance,inourearlier examplen+9,thetermsarenand7. ##### Examples Writeeachphraseasanalgebraicexpression. Nineincreasedbyanumberr 9+r Fourteendecreasedbyanumberx  14-x Sixlessthananumbert t-6 Theproductof5andanumbern5Xn or5n Thirty-twodividedbyanumbery 31x or [31] x ##### Example Anelectricianchargessh450perhourandspendssh200adayongasoline.Writeanalgebraicexpressionto representhisearningsforoneday. Solution: Letxrepresentthenumberofhourstheelectricianworksinoneday.Theelectrician'searningscan Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com berepresentedbythefollowingalgebraicexpression: ##### Solution 450x-200 ##### SimplificationofAlgebraicExpressions Note: Basicstepstofollowwhensimplifyanalgebraicexpression:  Removeparenthesesbymultiplyingfactors.  Useexponentrulestoremoveparenthesesintermswithexponents.  Combineliketermsbyaddingcoefficients.  Combinetheconstants. ##### Likeandunliketerms Liketermshavethesamevariable/lettersraisedtothesamepoweri.e.3b+2b=5bora+5a=6aandthey 2 2 canbesimplifiedfurtherinto5band6arespectively( anda 3a arealsoliketerms).Whileunliketermshave 2 2 ( anda 3a 2 differentvariablesi.e.3b+2cor4b+2xandtheycannotbesimplifiedfurther. ##### Example 3a+12b+4a-2b = 7a+10b (collectthelickterms) 2x-5y+3x-7y+3w =5x-12y+3w ##### Example Simplify:2x6y4x+5zy ##### Solution 2x6y4x+5zy=2x4x6yy+5z =(2x4x)(6y+y)+5z =-2x7y+5z ##### Note: -6yy=-(6y+y) ##### Example 1 1 1 Simplify ~~:~~ a ~~-~~ b ~~+~~ a 1 a 4 2 1 3 ##### Solution TheL.C.Mof2,3and4is12. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ~~=~~ [6a+3a-4b] 12 = [9a-4b] 12 1 1 1 Therefor ~~e~~ a ~~-~~ b ~~+~~ a ~~=~~ 2 3 4 1 3 1 a 4 6a-4b+3a 12 ##### Example a+b Simplify ~~:~~ ~~-~~ 2a-b 3 2 ##### Solution Example 5x-2x=3x 4abc-2abc=2abc a+b 2a-b ~~-~~ ~~=~~ 2a-b 3 ( a+b ) -2 ( 2a-b ) 2 3 6 3a+3b-4a+2b 6 3a-4a+3b+2b 6 ~~=~~ [-a+5b] 6 ab2 c+3ab+ c=4ab c 3 b 3 b 3 b 3 3 ab2 c+3ab+ c=4ab cb 3 b 3 b 3 ##### Note: Capitalletterandsmalllettersarenotliketerms. ##### Brackets Bracketsservethesamepurposeastheydoinarithmetic. ##### Example Removethebracketsandsimplify: a.)3(a+b)2(ab) b.)1/3a+3(5a+bc) c.)2b+3(32(a5) ##### Solution a.)3(a+b)2(ab)=3a+3b2a+2b Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com =3a-2a+3b+2b =a+5b b.)1/3a+3(5a+bc)=1/3a+15a+3b3c 1 =1 ~~5~~ a+3b-3c 3 c.)2b+a{32(a5)}=2b+a{32a+10} 2 =2b+3a2 +10aa 2 =2b+3a+10a-2a 2 =2b+13a-2a Theprocessofremainingthebracketsiscalledexpansionwhilethereverseprocessofinsertingthebrackets iscalledfactorization. ##### Example Factorizethefollowing: a.)3m+3n=3(m+n) (thecommontermis3soweputitoutsidethebracket) 3 4 +ar 4 5 +ar 3 4 b.)ar+ar+ 5 2 y+20x 4 2 3 - 36x 4 2 y 3 2 4 2 3 c.)4xy+20xy- 36xy ##### Solution 3 4 +ar 5 3 ar(ar 4 5 +ar 3 3 4 5 3 b.)ar+ar+ar(ariscommon) ###### 3 ( 1+r+r 2 ar ###### 2 ) 2 y+20x 4 2 3 - 36x 4 2 y 3 2 4 2 3 c.)4xy+20xy- 36xy 4x 2 y(Iscommon) 2 y(1+5x 2 2 =4xy(1+5xy-9x ##### Factorizationbygrouping Whenthetermsofanexpressionwhichdonothaveacommonfactoraretakenpairwise,acommonfactorcan befound.Thismethodisknownasfactorizationbygrouping. ##### Example Factorize: a.)3ab+2b+3ca+2c b.)ab+bxa-x Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution a.)3ab+2b+3ca+2c=b(3a+2)+c(3a+2) =(3a+2)(b+c) b.)ab+bxax=b(a+x)1(a+x) =(a+x)(b1) ##### Algebraicfractions Inalgebra,fractionscanbeaddedandsubtractedbyfindingtheL.C.Mofthedenominators. ##### Examples Expresseachofthefollowingasasinglefraction: x-1 x+2 a. ~~)~~ ~~+~~ ~~+~~ x+2 x 5 2 4 a+b b. ~~)~~ ~~-~~ b-a b a 1 3 c. ~~)~~ ~~+~~ ~~+~~ 3 ( a+b ) 8 ( a+b ) 3 8 ( a+b ) 5 12a Solution #### a. ) x-1 + x+2 + x = 10 ( x-1 ) +5 ( x+2 ) +4x (10x-10+5x+10+4x) ##### x+2 ##### x 5 #### 10 ( x-1 ) +5 ( x+2 ) +4x ##### 12a ##### 2 ##### 4 ##### = [19x] 20 ##### a+b b-a b. ) - = ##### b-a #### b ( a+b ) -a ( b-a ) ##### ab ##### b ##### a 2 2 ##### a [+] b 2 ##### a b = [+] ##### ab 1 3 c. ~~)~~ ~~+~~ ~~+~~ 3 ( a+b ) 8 ( a+b ) 3 ~~+~~ 8 ( a+b ) 5 12a [(] [findtheL.C.M.of3,8and12whichis24,] [)] ( findtheL.C.M.ofaand ( a+b ) isa(a+b) ) ( TheL.C.M.of3 ( a+b ) (,8 a+b ) and12is24a(a+b) ) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### 8a+9a+10a+10b 1 3 5 ~~+~~ ~~+~~ ~~=~~ 3 ( a+b ) 8 ( a+b ) 12a 3 8 ( a+b ) 5 12a ##### 24a(a+b) ##### Simplificationbyfactorization Factorizationisusedtosimplifyexpressions ##### Examples Simplify p [2] 2pq+q [2] 2p [2] -3pq+q [2] ##### Solution Numeratorissolvedfirst. ~~=~~ [27a+10b] 24a(a+b) p(p-q)-q(p+q) 2 2 -Pq-pq+q p-Pq-pq+ 2 (p-q) 2 Thensolvethedenominator 2p [2] -2pq-pq-q [2] (2p-q)(p-q) 2 ### - (pq) ~~=~~ ### (2p-q)(p-q) ##### Example Simplify ##### Solution Num.(4m3n)(4m+3n) ### - pq 2p-q 2 (p-q)=(p-q)(p-q)henceitcancelwiththedenominator Den.4m [2] 4mn+3mn3n [2] (4m+3n)(mn)  (4m 3n)(4m+3n) (4m+3n)(mn) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com  4m 3n mn ##### Example Simplifytheexpression. ##### Solution Numerator 18x(yr) Denominator 9x(ry) ##### Therefore [18x] [(y] [r] [)] 9x(r-y) = [(y] [r] [)] (r-y) Example 2 2 ##### 12 +ax-6x a 2 ##### 12 +ax-6x Simplify 2 2 2 ##### 9 -4x a 2 ##### 9 -4x a ##### Solution = ##### Example Simplifytheexpressioncompletely. ##### a y -ax bx-by Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution ##### a (y -x ) = a (y -x ) = a = b(x-y) -b(x-y) -b ##### a (y -x ) -b(x-y) ##### a -b ##### -a b x-y=-(y-x) ##### Substitution Thisistheprocessofgivingvariablesspecificvaluesinanexpression ##### Example 2 2 x+ y y+2 2 x+ y Evaluatetheexpressio ~~n~~ ifx=2andy=1 ##### Solution 2 2 x+ y y+2 2 2+ 1 2 2 2+ 1 1+2 4+1 3 2 x+ y = 2+ 1 ~~=~~ ##### 5 2 = = 1 3 3 Endoftopic ### PastKCSEQuestionsonthetopic  1.Giventhaty= 2x z x+3z expressxintermsofyandz 2.Simplifytheexpression x  1 - 2x+1 x 3x Hencesolvetheequation x  1 - 2x+1 = 2 x 3x 3 3.Factorizea [2] b [2] Hencefindtheexactvalueof2557 [2] -2547 [2] 4.Simplify p [2]  2pq+q [2] Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com P [3] -pq [2] +p [2] qq [3] 5.Giventhaty=2xz,expressxintermsofyandz. Fourfarmerstooktheirgoatstoamarket.MohammedhadtwomoregoatsasKoechhad3timesas manygoatsasMohammed,whereasOdupoyhad10goatslessthanbothMohammedandKoech. (i) Writeasimplifiedalgebraicexpressionwithonevariable,representingthetotalnumberofgoats. (ii) Threebutchersboughtallthegoatsandsharedthem equally.Ifeachbutchergot17goats,how manydidodupoyselltothebutchers? 6.Solvetheequation 1 = 5 -7 4x 6x 7.Simplify a + b 2(a+b) 2(a-b) 8.Threeyearsago,Jumawasthreetimesasold.AsAliintwoyearstime,thesum oftheirageswillbe 62.Determinetheirages CHAPTERELEVEN Bytheendofthetopicthelearnershouldbeableto: a.)Definerates b.)Solveproblemsinvolvingrates c.)Defineratio d.)Comparetwoormorequantitiesusingratios e.)Changequantitiesinagivenratio f.) Comparetwoormoreratios g.)Representandinterpretproportionalparts h.)Recognizedirectandinverseproportions i.) Solveproblemsinvolvingdirectandinverseproportions j.) Convertfractionsanddecimalstopercentagesandvice-versa Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com k.)Calculatepercentagechangeinagivenquantity l.) Applyrates,ratios,percentagestoreallifesituationsandproportion. Content a.)Rates b.)Solvingproblemsinvolvingrates c.)Ratio d.)Comparingquantitiesusingratio e.)Increaseanddecreaseinagivenratio f.) Comparingratios g.)Proportion:directandinverse h.)Solveproblemsondirectandinverseproportions i.) Fractionsanddecimalsaspercentages j.) Percentageincreaseanddecrease k.)Applicationofrates,ratios,percentagesandproportiontoreallifesituations. ### Introduction ##### Rates Arateisameasureofquantity,andcomparingonequantitywithanotherofdifferentkind. ##### Example Ifacartakestwohourstotraveladistanceof160km.thenwewillsaythatitistravellingatanaveragerateof 80km perhour.Iftwokilogramsofmaizemealissoldforsh.38.00,thenwesaythatmaizemealissellingat therateofsh.19.00perkilogram. ##### Example Whatistherateofconsumptionperdayiftwelvebagsofbeansareconsumedin120days? ##### Solution. Rateofconsumption=numberofbags/numberofdays ~~=~~ [12] 120 =1/10bagsperday ##### Example Alaborerswageissh.240pereighthoursworkingday.Whatistherateofpaymentperhour? Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution Rate=amountofmoneypaid/numberofhours ~~=~~ [240] 8 =sh.30perhour ##### Ratio Aratioisawayofcomparingtwosimilarquantities.Forexample,ifaliasis10yearsoldandhisbrotherbasher is14yearsold.Thenaliasageis10/14ofBashirsage,andtheiragesaresaidtobeintheratioof10to14. Written,10:14. Aliasage:Bashirsage=10:14 Bashirsage:aliasage=14:10 Instatingaratio,theunitsmustbethesame.Ifonamap2cm rep5km ontheactualground,thentheratioof mapdistancetomapdistanceis2cm:5x100000cm,whichis2:500000. Aratioisexpressedinitssimplestforminthesamewayasafraction, E.g.=,hence10:14=5:7. Similarly,2:500000=1:250000, Aproportionisacomparisonoftwoormoreratios.If,example,a,bandcarethreenumberssuchthata:b: c=2:3:5,thena,b,caresaidtobeproportionalto2,3,5andtherelationshipshouldbeinterpretedtomeana/2 =b/3=c/5. Similarly,wecansaythata:b=2:3,b:c=3:5a:c=2:5 Example3 Ifa:b=3:4andb:c=5:7finda:c Solution a:b=3:4(i) b:c=5:7.(ii) Considertherighthandside; Multiply(i)by5and(ii)by4toget,a:b=15:20andb:c=20:28 Thus,a:b:c=15:20:28anda:c=15:28 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Increaseanddecreaseinagivenratio Toincreaseordecreaseaquantityinagivenratio,weexpresstheratioasafractionandmultiplyitbythe quantity. ##### Example Increase20intheratio4:5 ##### Solution Newvalue=5/4x20 =5x5 =25 ##### Example Decrease45intheratio7:9 ##### Solution Newvalue =7/9x45 =7x5 =35 ##### Example Thepriceofapenisadjustedintheratio6:5.Iftheoriginalpricewassh.50.Whatisthenewprice? ##### Solution Newprice:oldprice=6:5 Newprice/oldprice= Newprice =6/5x50 =sh.60 ##### Note: Whenaratioexpressesachangeinaquantityanincreaseordecrease,itisusuallyputintheform ofnew value:oldvalue ##### Comparingratios Inordertocompareratios,theyhavetobeexpressedasfractionsfirst,ie.,a:b=a/b.theresultantfractioncan thenbecompared. ##### Example Whichratioisgreater,2:3or4:5? Solution2:3=,4:5= =,==> Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Thus,4:5>2:3 ##### Distributingaquantityinagivenratio Ifaquantityistobedividedintheratioa:b:c,thefractionofthequantityrepresentedby: (i) Awillb ~~e~~ [a] a+b+c a (ii) Bwillbe a+b+c a (iii) C willbe a+b+c ##### Example A72-hactarefarm istobesharedamongthreesonsintheratio2:3:4.Whatwillbethesizesinhectaresofthe threeshares? ##### Solution Totalnumberofpartsis2+3+4=9 Theshesharesare:2/9x72ha=16ha 3/9x72ha=24ha 4/9x72ha=36ha ##### Directandinverseproportion Directproportion Thetablebelowshowsthecostofvariousnumbersofcupsatsh.20percup. |Col1|Col2|Col3|Col4|Col5|Col6| |---|---|---|---|---|---| ||||||| Theratioofthenumbersofcupsinthefourthcolumntothenumberofcupsinthesecondcolumnis4:2=2:1. Theratioofthecorrespondingcostsis80:40=2:1.Byconsideringtheratioofcostsinanytwocolumnsandthe correspondingratioandthenumberofcups,youshouldnoticethattheyarealwaysthesame. Iftwoquantitiesaresuchthatwhentheoneincreases(decreases)inparticularratio,theotheronealso increases(decreases)intheratio, ##### Example Acartravels40kmon5litresofpetrol.Howfardoesittravelon12litresofpetrol? ##### Solution Petrolisincreasedintheratio12:5 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Distance=40x12/5km ##### Example Atraintakes3hourstotravelbetweentwostationsatanaveragespeedof40km perhour.Atwhataverage speedwoulditneedtotraveltocoverthesamedistancein2hours? ##### Solution Timeisdecreasedintheratio2:3Speedmustbeincreasedintheratio3:2averagespeedis40x3/2km =60 km/h ##### Example Tenmenworkingsixhoursadaytake12daystocompleteajob.Howlongwillittakeeightmenworking12 hoursadaytocompletethesamejob? ##### Solution Numberofmendecreasesintheratio8:10 Therefore,thenumberofdaystakenincreasesintheratio10:8. Numberofhoursincreasedintheratio12:6. Therefore,numberofdaysdecreasesintheratio6:12. 10 Numberofdaystaken=12 ~~x~~ ~~x~~ 8 6 12 =7days ##### Percentages Apercentage(%)isafractionwhosedenominatoris100.Forexample,27%means27/100. ##### Convertingfractionsanddecimalsintopercentages Towriteadecimalorfractionasa%:multiplyby100.Forexample 0.125 =0.125  100=12.5% ### 2 2 2 =  100 (i.e. Of100%)=40% ### 5 5 5 ### 2  ### 5 ### 2 5 ### 2 Or =2  5  100=40% ### 5 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example Change2/5intopercentage. ##### Solution 2 x ~~=~~ 5 100 2 x ~~=~~ x100 5 =40% ##### Example Convert0.67intoapercentage:solution 0.67 ~~=~~ [67] 100 67 Asapercentage,0.67 ~~=~~ x100 100 =67% ##### Percentageincreaseanddecrease Aquantitycanbeexpressedasapercentageofanotherbyfirstwritingitasafractionofthegivenquantity. ##### Example Afarmerharvested250bagsofmaizeinaseason.Ifhesold200bags,whatpercentageofhiscropsdoesthis represent? Letxbethepercentagesold. Then,x/100= [200] 250 200 So,x == x100 250 =80% ##### Example Amanearningsh.4800permonthwasgivena25%payrise.Whatwashisnewsalary? ##### Solution Newsalary=25/100x4800+4800 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com =1200+4800 =sh.6000 ##### Example Adresswhichwascostingsh.1200nowgoesforsh.960.Whatisthepercentagedecrease? ##### Solution Decreaseincostis1200-960=sh.240 Percentagedecrease=240/1200x100 =20% ##### Example Theratioofjohnsearningstomusesearningsis5:3.Ifjohnsearningsincreaseby12%,hisnew figure becomessh.5600.Findthecorrespondingpercentagechangeinmusesearningsifthesum oftheirnew earningsissh.9600 ##### Solution Johnsearningsbeforetheincreaseis100/112x5600=sh.5000 Johnsearnings/musesearnings= Musasearningsbeforetheincrease=3/5x5000 =sh.3000 Musasnewearnings =9600-5600 =sh.4000 Musaschangeinearnings =4000-1000 =sh.3000 Percentagechangeinmusesearnings=1000/3000x100 1 =3 ~~3~~ % 3 Endoftopic Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ### PastKCSEQuestionsonthetopic 1.Akinyiboughtandbeansfrom awholesaler.Shethenmixedthemaizeandbeansintheratio4:3she boughtthemaizeatKshs21perkgandthebeans42perkg.Ifshewastomakeaprofitof30%.What shouldbethesellingpriceof1kgofthemixture? 2.Waterflowsfrom atapattherateof27cm [3] persecondintoarectangularcontaineroflength60cm, breadth30cm andheight40cm.Ifat6.00PM thecontainerwashalffull,whatwillbetheheightof waterat6.04pm? 3.Twobusinessmenjointlyboughtaminibuswhichcouldferry25payingpassengerswhenfull.Thefare betweentwotownsAandBwasKshs80perpassengerforoneway.Theminibusmadethreeround tripsbetweenthetwotownsdaily.ThecostoffuelwasKshs1500perday.Thedriverandthe conductorwerepaiddailyallowancesofKshs200andKshs150respectively. AfurtherKshs4000perdaywassetasideformaintenance,insuranceandloanrepayment. (a) (i) Howmuchmoneywascollectedfromthepassengersthatday? (ii) Howmuchwasthenetprofit? (b) Onanotherday,theminibuswas80%fullontheaverageforthethreeroundtrips,howmuchdid eachbusinessmangetifthedaysprofitwassharedintheratio2:3? 4.Wainainahastwodairyfarms,AandB.FarmAproducesmilkwith3percentfatandfarmBproduces milkwith4percentfat. (a) Determine (i) Thetotalmassofmilkfatin50kgofmilkfromfarmAand30kgofmilkfromfarmB (ii) Thepercentageoffatinamixtureof50kgofmilkAand30kgofmilkfromB (b) Therangeofvaluesofmassofmilkfrom farm Bthatmustbeused ina50kgmixturesothat themixturemayhaveatleast4percentfat. 5.Intheyear2001,thepriceofasofasetinashopwasKshs12,000 (a) Calculatetheamountofmoneyreceivedfromthesalesof240sofasetsthatyear. (b) (i) Intheyear2002thepriceofeachsofasetincreasedby25%while thenumberofsetssolddecreasedby10%.Calculatethepercentageincreaseinthe amountreceivedfromthesales (ii) Iftheendofyear2002,thepriceofeachsofasetchangedintheratio16:15,calculate thepriceofeachsofasetintheyear2003. (c) Thenumberofsofasetssoldintheyear2003wasP%lessthanthenumbersoldintheyear 2001. CalculatethevalueofP,giventhattheamountsreceivedfromsalesifthetwoyearswereequal. 6.Asolutionwhosevolumeis80litresismadeupof40%ofwaterand60%ofalcohol.Whenxlitresof waterisadded,thepercentageofalcoholdropsto40%. (a) Findthevalueofx (b) Thirtylitresofwaterisaddedtothenewsolution.Calculatethepercentageofalcoholinthe Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com resultingsolution (c) If5litresofthesolutionin(b)aboveisaddedto2litresoftheoriginalsolution,calculateinthe simplestform,theratioofwatertothatofalcoholintheresultingsolution. 7.Threebusinesspartners,Asha,NangilaandCheropcontributedKshs60,000,Kshs85,000andKshs105, 000respectively.Theyagreedtoput25%oftheprofitbackintobusinesseachyear.Theyalsoagreedto putaside40%oftheremainingprofittocaterfortaxesandinsurance.Therestoftheprofitwouldthen besharedamongthepartnersintheratiooftheircontributions.Attheendofthefirstyear,the businessrealizedagrossprofitofKshs225,000. (a) CalculatetheamountofmoneyCheropreceivedmorethanAshaattheendofthefirstyear. (b) NangilafurtherinvestedKshs25,000intothebusinessatthebeginningofthesecondyear. Giventhatthegrossprofitattheendofthesecondyearincreasedintheratio10:9,calculate Nangilasshareoftheprofitattheendofthesecondyear. 8.Kipketercancultivateapieceoflandin7hrswhileWanjikucandothesameworkin5hours.Findthe timetheywouldtaketocultivatethepieceoflandwhenworkingtogether. 9.MogakaandOndisoworkingtogethercandoapieceofworkin6days.Mogakaworkingalone,takes5 dayslongerthanOnduso.HowmanydaysdoesittakeOndusotodotheworkalone. 10.Acertainamountofmoneywassharedamong3childrenintheratio7:5:3thelargestsharewasKshs 91.Findthe (a) Totalamountofmoney (b) Differenceinthemoneyreceivedasthelargestshareandthesmallestshare. CHAPTERTWELVE SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Statetheunitsofmeasuringlength b.)Convertunitsoflengthfromoneformtoanother c.)Expressnumberstorequirednumberofsignificantfigures d.)Findtheperimeterofaplanefigureandcircumferenceofacircle. Content a.)Unitsoflength(mm,cm,m,km) b.)Conversionofunitsoflengthfromoneformtoanother Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com c.)Significantfigures d.)Perimeter e.)Circumference(includelengthofarcs) ### Introduction Lengthisthedistancebetweentwopoints.TheSIunitoflengthismetres.Conversionofunitsoflength. 1kilometer(km)=1000metres 1hectometer(hm)=100metres 1decameter(Dm)=10metres 1decimeter(dm)=1/10metres 1centimeter(cm)=1/100metres 1millimeter(mm)=1/1000metres Thefollowingprefixesareoftenusedwhenreferringtolength: Mega1000000 Kilo1000 Hecto100 Deca-10 Deci- Centi- Milli- Micro- ##### Significantfigures Theaccuracywithwhichwestateorwriteameasurementmaydependonitsrelativesize.Itwouldbe unrealistictostatethedistancebetweentownsAandBas158.27km.amorereasonablefigureis158 km.158.27kmisthedistanceexpressedto5significantfiguresand158kmto3significantfigures. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example Expresseachofthefollowingnumbersto5,4,3,2,and1significantfigures: (a)906315 (b)0.08564 (c)40.0089 (d)156000 ##### Solution |Col1|Col2|Col3|Col4|Col5|Col6|Col7| |---|---|---|---|---|---|---| |||||||| |||||||| |||||||| |||||||| Theaboveexampleshowhowwewouldroundoffameasurementtoagivennumberofsignificantfigures Zeromaynotbeasignificant.Forexample: (i) 0.085 -zeroisnotsignificanttherefore,0.085isatwo-significantfigure. (ii) 2.30zeroissignificant.Therefore2.30isathree-significantfigure. (iii) 5000zeromayormaynotbesignificantfigure.Therefore,5000tothreesignificantfigureis5 00(zeroafter5issignificant).Toonesignificantfigureis5000.Zeroafter5isnotsignificant. (iv) 31.805Or305zeroissignificant,therefore31.805isfivesignificantfigure.305isthree significantfigure. ##### Perimeter Theperimeterofaplaneisthetotallengthofitsboundaries.Perimeterisalengthandisthereforeexpressed inthesameunitsaslength. ##### Squareshapes 5cm Itsperimeteris5+5+5+5=2(5+5) =2(10) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com =20cm Hence 5x4=20 Soperimeterofasquare=Sidesx4 ##### Rectangularshapes Figure12.2isarectangleoflength5cmandbreadth3cm. 5cm 3cm Itsperimeteris5+3+5+3=2(5+3)cm =2x8 =16cm Hence perimeterofarectangle p=2(L+W) ##### Triangularshapes Tofindtheperimeterofatriangleaddallthethreesides. c a b Perimeter=(a+b+c)units,wherea,bandcarethelengthsofthesidesofthetriangle. ##### Thecircle Thecircumferenceofacircle=2rorD ##### Example (a)Findthecircumferenceofacircleofaradius7cm. (b)Thecircumferenceofabicyclewheelis140cm.finditsradius. ##### Solution (a)C=d Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com =22/7x7 (b)C=d =2x22/7xr =140 =22.27cm ##### Lengthofanarc Anarcofacircleispartofitscircumference.Figure12.10(a)showstwoarcsAMBandANB.ArcAMB,which islessthanhalfthecircumferenceofthecircle,iscalledtheminorarc,whilearcANB,whichisgreaterhalfof thecircumferenceiscalledthemajorarc.Anarcwhichishalfthecircumferenceofthecircleiscalleda semicircle. ##### Example Anarcofacirclesubtendsanangle60atthecentreofthecircle.Findthelengthofthearciftheradiusofthe circleis42cm.(=). Solution Thelength,l,ofthearcisgivenby: L=/360x2r. =60,r=42cm Therefore,l=60/360x2x22/7x42 =44cm ##### Example Thelengthofanarcofacircleis62.8cm.findtheradiusofthecircleifthearcsubtendsanangle144atthe centre,(take=3.142). Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution L=/360x2r=62.8and =144 144 Therefore, x2x3.142xr=62.8 360 360 R=62.8 ~~x~~ x2x3.142 144 =24.98cm ##### Example Findtheanglesubtendedatthecentreofacirclebyanarcoflength11cmiftheradiusofthecircleis21cm. ##### Solution L=/360x2xr=11cmandr=21m L=/360x2x22/7x21=11 Thus,=11x360x7/2x22x21 =30 0 Endoftopic ### PastKCSEQuestionsonthetopic 1.) Twocoilswhicharemadebywindingcopperwireofdifferentgaugesandlengthhavethesamemass. Thefirstcoilismadebywinding270metresofwirewithcrosssectionaldiameter2.8mm whilethe secondcoilismadebywindingacertainlengthofwirewithcross-sectionaldiameter2.1mm.Findthe lengthofwireinthesecondcoil. 2. ThefigurebelowrepresentsamodelofahutwithHG=GF=10cm andFB=6cm.Thefourslanting edgesoftheroofareeach12cmlong. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Calculate LengthDF. AngleVHF ThelengthoftheprojectionoflineVHontheplaneEFGH. Theheightofthemodelhut. ThelengthVH. TheangleDFmakeswiththeplaneABCD. 3. Asquarefloorisfittedwithrectangulartilesofperimeters220cm.eachrow(tilelengthwise)carries 20lesstilesthaneachcolumn(tilesbreadthwise).Ifthelengthoftheflooris9.6m. Calculate: a.Thedimensionsofthetiles b.Thenumberoftilesneeded c.Thecostoffittingthetiles,iftilesaresoldindozensatsh.1500perdozenandthelabourcost issh.3000 CHAPTERTHIRTEEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Stateunitsofarea b.)Convertunitsofareafromoneformtoanother c.)Calculatetheareaofaregularplanefigureincludingcircles d.)Estimatetheareaofirregularplanefiguresbycountingsquares e.)Calculatethesurfaceareaofcubes,cuboidsandcylinders. Content a.)Unitsofarea(cm 2,m 2,km 2,Ares,ha) b.)Conversionofunitsofarea Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com c.)Areaofregularplanefigures d.)Areaofirregularplaneshapes e.)Surfaceareaofcubes,cuboidsandcylinders ##### Introduction UnitsofAreas Theareaofaplaneshapeistheamountofthesurfaceenclosedwithinitsboundaries.Itisnormallymeasured insquareunits.Forexample,asquareofsides5cmhasanareaof 5x5=25cm Asquareofsides1mhasanareaof1m,whileasquareofside1kmhasanareaof1km Conversionofunitsofarea 1m=1mx1m =100cmx100cm =10000cm 1km=1kmx1km =1000mx1000m =1000000m 1are =10m x10m =100m 1hectare(ha)=100Ares =10000m ##### Areaofaregularplanefigures Areasofrectangle 5cm 3cm Area,A=5x3cm 2 =15m Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Hence,theareaoftherectangle,A=LXWsquareunits,wherelisthelengthandbbreadth. ##### Areaofatriangle H Base Areaofatriangle A=1/2bhsquareunits ##### Areaofparallelogram Area=1/2bh+1/2bh =bhsquareunits ##### Note: Thisformulaeisalsousedforarhombus ##### Areaofatrapezium Thefigurebelowshowsatrapezium inwhichtheparallelsidesareaunitsandbunits,long.Theperpendicular distancebetweenthetwoparallelsidesishunits. AreaofatriangleABD=1/2ahsquareunits AreaoftriangleDBC=bhsquareunits ThereforeareaoftrapeziumABCD=1/2ah+1/2bh =1/2h(a+b)squareunits. Thus,theareaofatrapezium isgivenbyahalfthesum ofthelengthofparallelsidesmultipliedbythe perpendiculardistancebetweenthem. 1 Thatis,areaoftrapezium ~~=~~ h 2 [(] [a+b] [)] ##### Areaofacircle TheareaAofacircleofradiusrisgivenby:A=r 2 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Theareaofasector Asectorisaregionboundedbytworadiiandanarc. Supposewewanttofindtheareaoftheshadedpartinthefigurebelow Theareaofthewholecircleisr Thewholecirclesubtends360atthecentre. Therefore,360 correspondstor 1 correspondsto1/360xr 60 correspondsto60/360 xr Hence,theareaofasectorsubtendinganangleatthecentreofthecircleisgivenby ###  A = x 0 ### 360 2 ### xr 0 ##### Example Findtheareaofthesectorofacircleofradius3cmiftheanglesubtendedatthecentreis140(take=) ##### Solution AreaAofasectorisgivenby  A= xr 2 360 [0] Here,r=3cmand=140 0 140 22 Therefore,A ~~=~~ ~~x~~ x3x3 360 [0] 7 =11cm ##### Example Theareaofasectorofacircleis38.5cm.Findtheradiusofthecircleiftheanglesubtendedatthecentreis 90 0 (Take=) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution Fromtheformulaa=/360xr,weget90/360x22/7xr=38.5 ##### = [38.5x360x7] Therefore,r ##### 90x22 Thus,r=7 ##### Example Theareaofacircleradius63cmis4158cm.Calculatetheanglesubtendedatthecentreofthecircle.(Take =) Usinga=/360xr, #####  = [4158x7x360] 22x63x63 0 =120 ##### Surfaceareaofsolids ConsideracuboidABCDEFGHshowninthefigurebelow.Ifthecuboidiscutthroughaplaneparalleltothe ends,thecutsurfacehasthesameshapeandsizeastheendfaces.PQRSisaplane.Theplaneiscalledthe cross-sectionofthecuboid Asolidwithuniform cross-sectioniscalledaprism.Thefollowingaresomeoftheprisms.Thefollowingare someoftheprisms. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Thesurfaceareaofaprismisgivenbythesumoftheareaofthesurfaces. Thefigurebelowshowsacuboidoflengthl,breathbandheighth.itsareaisgivenby; A=2lb+2bh+2hl =2(lb.+bh+hl) Foracubeoffside2cm; A=2(3x2) =24cm ##### Example Findthesurfaceareaofatriangularprismshownbelow. Areaofthetriangularsurfaces=x5x12x2cm Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com =60cm Areaoftherectangularsurfaces=20x13+5x20+12x20 =260+100+240=600cm Therefore,thetotalsurfacearea=(60+600)cm =660cm ##### Cylinder Aprismwithacircularcross-sectioniscalledacylinder,seethefigurebelow. Ifyourollapieceofpaperaroundthecurvedsurfaceofacylinderandopenitout,youwillgetarectangle whosebreathisthecircumferenceandlengthistheheightofthecylinder.Theendsaretwocircles.The surfaceareaSofacylinderwithbaseandheighthisthereforegivenby; S=2rh+2r ##### Example Findthesurfaceareaofacylinderwhoseradiusis7.7cmandheight12cm. Solution S=2(7.7)x12+2(7.7)cm =2(7.7)x12+(7.7)cm =2x7.7(12+7.7)cm =2x7.7x(19.7)cm =15.4(19.7)cm =953.48cm ##### Areaofirregularshapes Theareaofirregularshapecannotbefoundaccurately,butitcanbeestimated.Asfollows; (i) Drawagridofunitsquaresonthefigureorcopythefigureonsuchagrid,seethefigurebelow Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (ii) Countalltheunitsquaresfullyenclosedwithinthefigure. (iii) Countallpartiallyenclosedunitsquaresanddividethetotalbytwo,i.e..,treateachoneofthem as halfofaunitsquare. (iv) Thesumofthenumbersin(ii)and(ii)givesanestimateoftheareasofthefigure. Fromthefigure,thenumberoffullsquaresis9 Numberofpartialsquares=18 Totalnumberofsquares=9+ =18 Approximatearea=18sq.units. Endoftopic ### PastKCSEQuestionsonthetopic 1.)Calculatetheareaoftheshadedregionbelow,giventhatACisanarcofacirclecentreB.AB=BC=14cm CD=8cmandangleABD=75 [0] (4mks) A 14cm B 75 [0] 14cm C B ----- isabokemicah@gmail.com 2.) Thescaleofamapis1:50000.Alakeonthemapis6.16cm [2] .findtheactualareaofthelakeinhactares. (3mks) 3.) ThefigurebelowisarhombusABCDofsides4cm.BDisanarcofcirclecentreC.Giventhat  ABC= 138 [0] .Findtheareaofshadedregion. (3mks) 4.) ThefigurebelowsowstheshapeofKamausfarmwithdimensionsshowninmeters 140 100 80m FindtheareaofKamausfarminhectares (3mks) 5.) InthefigurebelowABandACaretangentstothecirclecentreOatBandCrespectively, theangleAOC=60 [0] Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Calculate (a)ThelengthofAC 6.) Thefigurebelowshowsthefloorofahall.Apartofthisfloorisintheshapeofarectangleof length 20mandwidth16mandtherestisasegmentofacircleofradius12m.Usethefigureto find: (a)ThesizeofangleCOD (2mks) (b)TheareaoffigureDABCO (4mks) (c)AreaofsectorODC (2mks) (d)Areaofthefloorofthehouse. (2mks) 7.) Thecirclebelowwhoseareais18.05cm [2] circumscribesatriangleABCwhereAB=6.3cm,BC=5.7cm andAC=4.8cm.Findtheareaoftheshadedpart 8.) Inthefigurebelow,PQRSisarectangleinwhichPS=10kcmandPQ=6kcm.MandNaremidpointsof QRandRSrespectively Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com a) Findtheareoftheshadedregion (4marks) b) GiventhattheareaofthetriangleMNR=30cm [2] .findthedimensionsoftherectangle (2marks) c) Calculatethesizesofangles and givingyouranswerto2decimalplaces (4marks) 9.) Thefigurebelowshowstwocircleseachofradius10.5cmwithcentresAandB.thecirclestoucheach otheratT GiventhatangleXAD=angleYBC=160 [0] andlinesXY,ATBandDCareparallel,calculatetheareaof: d)TheminorsectorAXTD (2marks) e)FigureAXYBCD (6marks) f) Theshadedregion (2marks) 10.) Thefloorofaroomisintheshapeofarectangle10.5mlongby6mwide.Squaretilesof length30cmaretobefittedontothefloor. (a)Calculatethenumberoftilesneededforthefloor. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (b)Adealerwishestobuyenoughtilesforfifteensuchrooms.Thetilesarepackedincartons eachcontaining20tiles.ThecostofeachcartonisKshs.800.Calculate (i)thetotalcostofthetiles. (ii)Ifinaddition,thedealerspendsKshs.2,000andKshs.600ontransportandsubsistence respectively,atwhatpriceshouldheselleachcartoninordertomakeaprofitof12.5% (GiveyouranswertothenearestKshs.) 11.) Thefigurebelowisacircleofradius5cm.PointsA,BandCaretheverticesofthetriangle ABCinwhich  ABC=60 [o] and  ACB=50 [o] whichisinthecircle.Calculatetheareaof  ABC) B 12.) Mr.Wanyamahasaplotthatisinatriangularform.Theplotmeasures170m,190m and210m,butthealtitudesoftheplotaswellastheanglesarenotknown.Findthearea oftheplotinhectares 13.) Threesirenswailatintervalsofthirtyminutes,fiftyminutesandthirtyfiveminutes. Iftheywailtogetherat7.18a.monMonday,whattimeanddaywilltheynextwailtogether? 14.) Afarmerdecidestoputtwo-thirdsofhisfarmundercrops.Ofthis,heputaquarterundermaizeand four-fifthsoftheremainderunderbeans.Therestisplantedwithcarrots. If0.9acresareundercarrots,findthetotalareaofthefarm Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- CHAPTERFOURTEEN SpecificObjectives isabokemicah@gmail.com Bytheendofthetopicthelearnershouldbeableto: a.)Stateunitsofvolume b.)Convertunitsofvolumefromoneformtoanother c.)Calculatevolumeofcubes,cuboidsandcylinders d.)Stateunitsofcapacity e.)Convertunitsofcapacityfromoneformtoanother f.) Relatevolumetocapacity g.)Solveproblemsinvolvingvolumeandcapacity. Content a.)Unitsofvolume b.)Conversionofunitsofvolume c.)Volumeofcubes,cuboidsandcylinders d.)Unitsofcapacity e.)Conversionofunitsofcapacity f.) Relationshipbetweenvolumeandcapacity g.)Solvingproblemsinvolvingvolumeandcapacity ### Introduction Volumeistheamountofspaceoccupiedbyasolidobject.Theunitofvolumeiscubicunits. Acubeofedge1cmhasavolumeof1cmx1cmx1cm=1cm. ##### Conversionofunitsofvolume Acubeofside1mhasavolumeof1m But1m=100cm 1mx1mx1m=100cmx100cmx100cm Thus,1m=(0.01x0.01x0.01)m =0.000001m =1x10m Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Acubeside1cmhasavolumeof1cm. But1cm=10mm 1cmx1cmx1cm=10mmx10mmx10mm Thus,1cm=1000mm ##### Volumeofcubes,cuboidsandcylinders Cube Acubeisasolidhavingsixplanesquarefacesinwhichtheanglebetweentwoadjacentfacesisaright-angle. Volumeofacube=areaofbasexheight =lxl =l ##### Cuboid Acuboidisasolidwithsixfaceswhicharenotnecessarilysquare. Volumeofacuboid=lengthxwidthxheight =asq.unitsxh =ahcubicunits. ##### Cylinder Thisisasolidwithacircularbase. Volumeofacylinder=areaofbasexheight =rxh =rhcubicunits ##### Example Findthevolumeofacuboidoflength5cm,breadth3cmandheight4cm. ##### Solution Areaofitsbase=5x4cm Volume =5x4x3cm =60cm ##### Example Findthevolumeofasolidwhosecross-sectionisaright-angledtriangleofbase4cm,height5cm andlength 12cm. ##### Solution Areaofcross-section=1/2x4x5 =10cm Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Thereforevolume =10x12 =120cm ##### Example Findthevolumeofacylinderwithradius1.4mandheight13m. ##### Solution Areaofcross-section=22/7x1.4x1.4 =6.16m Volume=6.16x13 =80.08m Ingeneral,volumevofacylinderofradiusrandlengthlgivenbyv=rl ##### Capacity Capacityistheabilityofacontainertoholdfluids.TheSIunitofcapacityislitre(l) Conversionofunitstocapacity 1centiliter(cl)=10millilitre(ml) 1decilitredl=10centilitre(cl) 1litre(l)=10decilitres(dl) 1Decalitre(Dl)=10litres(l) 1hectolitre(Hl)=10decalitre(Dl) 1kilolitre(kl)=10hectolitres(Hl) 1kilolitre(kl)=1000litres(l) 1litre(l)=1000millilitres(ml) ##### Relationshipbetweenvolumeandcapacity Acubedofanedge10cmholds1litreofliquid. 1litre=10cmx10cmx10cm =1000cm 1m=10cm 1m=10litres. Endoftopic Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ### PastKCSEQuestionsonthetopic 1. )Allthewaterispouredintoacylindricalcontainerofcircularradius12cm.Ifthecylinderhasheight45cm, calculatethesurfaceareaofthecylinderwhichisnotincontactwithwater. 2.)TheBritishgovernmenthiredtwoplanestoairliftfootballfanstoSouthAfricafortheWorldcup tournament.Eachplanetook10hourstoreachthedestination. Boeng747hascarryingcapacityof300peopleandconsumesfuelat120litresperminute.Itmakes5 tripsatfullcapacity.Boeng740hascarryingcapacityof140peopleandconsumesfuelat200litresper minute.Itmakes8tripsatfullcapacity.Ifthegovernmentsponsoredthefansonewayatthecostof 800dollarsperfan,calculate: (a)ThetotalnumberoffansairliftedtoSouthAfrica. (2mks) (b)Thetotalcostoffuelusedifonelitrecosts0.3dollars. (4mks) (c)Thetotalcollectionindollarsmadebyeachplane. (2mks) (d)Thenetprofitmadebyeachplane. (2mks) 3.)Arectangularwatertankmeasures2.6mby4.8matthebaseandhaswatertoaheight of3.2m.Findthevolumeofwaterinlitresthatisinthetank 4.)Threelitresofwater(density1g/cm [3] )isaddedtotwelvelitresofalcohol(density0.8g/cm [3] . Whatisthedensityofthemixture? 5.)Arectangulartankwhoseinternaldimensionsare2.2mby1.4mby1.7misthreefifthfull ofmilk. (a)Calculatethevolumeofmilkinlitres (b)Themilkispackedinsmallpacketsintheshapeofarightpyramidwithanequilateralbasetriangle ofsides10cm.Theverticalheightofeachpacketis13.6cm.Fullpacketsobtainedaresoldatshs.30per packet.Calculate: (i)Thevolumeincm [3] ofeachpackettothenearestwholenumber (ii)Thenumberoffullpacketsofmilk Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (iii)Theamountofmoneyrealizedfromthesaleofmilk 6.)An890kgculvertismadeofahollowcylindricalmaterialwithouterradiusof76cmandaninnerradius of64cm.Itcrossesaroadofwidth3m,determinethedensityofthematerialssusedinitsconstruction inKg/m [3] correctto1decimalplace. CHAPTERFIFTEEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Definemass b.)Stateunitsofmass c.)Convertunitsofmassfromoneformtoanother d.)Defineweight e.)Stateunitsofweight f.) Distinguishmassandweight g.)Relatevolume,massanddensity. Content a.)Massandunitsofmass b.)Weightandunitsofweight c.)Density d.)Problemsolvinginvolvingreallifeexperiencesonmass,volume,densityandweight. ### Introduction ##### Mass Themassofanobjectisthequantityofmatterinit.Massisconstantquantity,wherevertheobjectis,and matterisanythingthatoccupiesspace.Thethreestatesofmatteraresolid,liquidandgas. TheSIunitofmassisthekilogram.Othercommonunitsaretone,gramandmilligram. Thefollowingtableshowsunitsofmassandtheirequivalentinkilograms. ##### Weight Theweightofanobjectonearthisthepulloftheearthonit.Theweightofanyobjectvariesfromoneplaceon theearthssurfacetotheother.Thisisbecausetheclosuretheobjectistothecentreoftheearth,themore thegravitationalpull,hencethemoreitsweight.Forexample,anobjectweighsmoreatsealevelthenontopof Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com amountain. ##### Unitsofweight TheSIunitofweightisnewton.Thepulloftheearth,sunandthemoononanobjectiscalledtheforceof gravityduetotheearth,sunandmoonrespectively.Theforceofgravityduetotheearthonanobjectofmass 1kgisapproximatelyequalto9.8N.Thestrengthoftheearthsgravitationalpull(symbolg)onanobjecton thesurfaceoftheearthisabout9.8N/Kg. Weightofanobject=massofanobjectxgravitation WeightN=masskgxgN/kg ##### Density Thedensityofasubstanceisthemassofaunitcubeofthesubstance.Abodyofmass(m)kgandvolume(v) mhas: (i) Density(d)=mass(m)/density(d) (ii) Mass(m)=density(d)xvolume(v) (iii) Volume(v)=mass(m)/density(d) ##### Unitsofdensity TheSIunitsofdensityiskg/m.theothercommonunitisg/cm 1g/cm=1000kg/m ##### Example Findthemassofanicecubeofside6cm,ifthedensityoftheiceis0.92g/cm. ##### Solution Volumeofcube=6x6x6=216cm Mass=densityxvolume =216x0.92 =198.72g ##### Example Findthevolumeofcorkofmass48g.giventhatdensityofcorkis0.24g/cm ##### Solution Volume=mass/density =.24 =200cm ##### Example Thedensityofironis7.9g/cm.whatisthisdensityinkg/m Solution Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 1g/cm=1000kg/m 7.9g/cm=7.9x1000/1 =7900kg/m ##### Example Arectangularslabofglassmeasures8cm by2cm by14cm andhasamassof610g.calculatethedensityof theglassinkg/m ##### Solution Volumeoftheslab=8x2x14 =224cm Massoftheslab=610g Density = =2.5x1000kg=25000kg/m Endoftopic ### PastKCSEQuestionsonthetopic 1.)Asquaredbrassplateis2mmthickandhasamassof1.05kg.Thedensityofbrassis8.4g/cm.Calculate thelengthoftheplateincentimeters. (3mks) 2.)Aspherehasasurfacearea18cm [2] .Finditsdensityifthespherehasamassof100g.(3mks) 3.)NyahururuMunicipalCouncilistoconstructafloorofanopenwholesalemarketwhoseareais800m [2] .The flooristobecoveredwithaslabofuniformthicknessof200mm.Inordertomaketheslab,sand,cementand ballastaretobemixedsuchthattheirmassesareintheratio3:2:3.The massofdryslabofvolume1m [3] is 2000kg.Calculate (a)(i)Thevolumeoftheslab (2mks) (ii)Themassofthedryslab. (2mks) (iii)Themassofcementtobeused. (2mks) (b)Ifonebagofthecementis50kg,findthenumberofbagstobepurchased. (1mk) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (c)Ifalorrycarries10tonnesofballast,calculatethenumberoflorriesofballasttobe purchased. (3mks) 4.)Aspherehasasurfaceareaof18.0cm [2] .Finditsdensityifthespherehasamassof100grammes. (3mks) 5.)Apieceofmetalhasavolumeof20cm [3] andamassof300g.Calculatethedensityofthemetal inkg/m [3] . 6.)2.5litresofwaterdensity1g/cm [3] isaddedto8litresofalcoholdensity0.8g/cm [3] .Calculatethedensity ofthemixture CHAPTERTEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Convertunitsoftimefromoneformtoanother b.)Relatethe12hourand24hourclocksystems c.)Readandinterprettraveltime-tables d.)Solveproblemsinvolvingtraveltimetables. Content a.)Unitsoftime b.)12hourand24hourclocksystems c.)Traveltime-tables d.)Problemsinvolvingtraveltimetables Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ### Introduction ##### Unitsoftime 1week=7days 1day=24hours 1hour=60minutes 1minutes=60seconds ##### Example Howmanyhoursarethereinoneweek? ##### Solution 1week=7days 1day=24hours 1week=(7x24)hours =168hours ##### Example Covert3h45minintominutes ##### Solution 1h=60min 3h=(3x60)min 3h45min=((60x3)+45)min =(180+45)min =225min ##### Example Express4h15mininsec ##### Solution 1hour=60min 1min=60sec 4h15min=(4x60+15)min =240+15min Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com =255min =255x60sec =15300sec. ##### The12andthe24hoursystems Inthe12hoursystem,timeiscountedfrom midnight.Thetimefrom midnighttomiddayiswrittenasam . whilethatfrommiddaytomidnightiswrittenaspm. Inthe24hoursystem,timeiscountedfrommidnightandexpressedinhours. ##### Traveltimetable Traveltimetablesshowstheexpectedarrivalanddeparturetimeforvehicles.Ships,aeroplanes,trains. ##### Example ThetablebelowshowsatimetableforapublicservicevehicleplyingbetweentwotownsAandDviatownsB andC. |Col1|Col2|Col3| |---|---|---| |||| (a)WhattimedoesthevehicleleavetownA? (b)AtwhattimedoesitarriveintownD? (c)HowlongdoesittaketotravelfromtownAtoD. (d)WhattimedoesthevehicletakestotravelfromtownCtoD? ##### Solution (a)8.20A.M (b)4.00P.M (c)ArrivaltimeintownDwas4.00p.m.itdeparturefromtownAwas8.20a.m. Timetaken=(12.00-8.20+4h) =3h40min+4h =7h40min (d)ThevehiclearrivedintownDat4.00p.m.itdepartedfromtownCat2.50p.m. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Timetaken=4.00-2.50 =1h10min Endoftopic ### PastKCSEQuestionsonthetopic 1.) AvantravelledfromKitaletoKisumuadistanceof160km.Theaveragespeedofthevanforthefirst 100kmwas40km/handtheremainingpartofthejourneyitsaveragespeedwas30km/h.Calculatethe averagespeedforthewholejourney. (3mks) 2.) Awatchwhichloosesahalf-minuteeveryhourwassettoreadthecorrecttimeat0545honMonday. Determinethetime,inthe12hoursystem,thewatchwillshowonthefollowingFridayat1945h. 3.) Thetimetablebelowshowsthedepartureandarrivaltimeforabusplyingbetweentwotowns MandR, 300kmapart 0710982617 |Col1|Col2|Col3| |---|---|---| |||| |||| |||| |||| |||| (a)HowlongdoesthebustaketotravelfromtownMtoR? Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (b)Whatistheaveragespeedforthewholejourney? CHAPTERSEVENTEEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)solvelinearequationsinoneunknown b.)solvesimultaneouslinearequationsbysubstitutionandelimination c.)Linearequationsinoneandtwounknown. Content a.)Linearequationsinoneunknown b.)Simultaneouslinearequations c.)Linearequationsinoneandtwounknownsfromgivenreallifesituations ### Introduction Linearequationsarestraightlineequationsinvolvingoneortwounknowns.Inthischapter,wewilldealwith theformationandsolvingofsuchequationsconsiderthefollowingcases. ##### Example Solvefortheunknownsineachofthefollowingequations 3x+4=10 ##### x 2=4 ##### 3 ##### p +5 = ##### 3 Solution 3x+4=10 3x+44=104 (tomakexthesubjectsubtract4onbothsides) 3x=6 X=2 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 2=4 2+2=4+2 (tomakexthesubjectadd2tobothsides) ##### x =6 ##### 3 X=18 ##### p +5 = ##### 3 ##### 5 4 ##### 3x ( p +5) = 5 x3 ##### 3 ##### 5 4 ##### 5 P+5 = x3 4 5 4(p+5) = x4 4 4p+20=15 4p=-5 P ~~=~~ [-5] 4 =-1 Solvinganequationwithfractionsordecimals,thereisanoptionofclearingthefractionsordecimalsinorder tocreateasimplerequationinvolvingwholenumbers. 1. Toclearfractions,multiplybothsidesoftheequation(distributingtoallterms)bytheLCDofallthe fractions. 2. Tocleardecimals,multiplybothsidesoftheequation(distributingtoallterms)bythelowestpowerof 10thatwillmakealldecimalswholenumbers. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### StepsforSolvingaLinearEquationinOneVariable: 1. Simplifybothsidesoftheequation. 2. Usetheadditionorsubtractionpropertiesofequalitytocollectthevariabletermsononesideofthe equationandtheconstanttermsontheother. 3. Usethemultiplicationordivisionpropertiesofequalitytomakethecoefficientofthevariableterm equalto1. 4. Checkyouranswerbysubstitutingyoursolutionintotheoriginalequation. ##### Note: Allotherlinearequationswhichhaveonlyonesolutionarecalledconditional. ##### Example Solvefortheunknownineachinofthefollowingequations ##### x+1 x-1 a. ) - = ##### x-1 ##### 2 ##### 3 ##### b. ) 3 y - 14 y -3 = y ##### 14 y -3 ##### y -1 ##### 2 ##### 5 ##### 4 ##### c.) 1 - y -2 y -3 =0 ### 2 [(] 2 [)] y -3 ### 2 [)] y -3 ##### Solution ##### x+1 x-1 a.) x24 - x24 =1/8x24(multiplybothsidesbytheL.C.Mof2,3and8) # ( 2 [)] ( 3 [)] ##### x+1 x24 # 2 [)] ( ##### x+1 ##### x-1 ##### x-1 ## 3 [)] #### 12 ( x+1 ) 8(x2)=3 ##### 12x+12 8x16=3 4x=-25 ##### 4x+28=3 x= - ~~6~~ [1] 4 #### ( 14 y -3 ) ##### y -1 # 4 [)] #### b. ) 3 y x20 - ( 14 y -3 ) x20= y -1x20 (multiplybothsidesbytheL.C.Mof25and4) # ( [)] ##### y -1 ##### 2 #### y -3 ) x20= y # 5 ( 30y-4(14y-3)=5(y4) -26+12=5y-20 31y=32 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com c.) ( 1x2 ) ~~-~~ y x2-2 y -3x2=0x2 ### ( 2 [) ] [(] 2 [)] y -3 ### 2 [)] 32 1 y ~~=~~ = ~~1~~ 31 31 2-y-2 ( y-3 ) =0 2-y-2y+6=0 8-3y=0 y -3 8 2 3y ~~=~~ = ~~2~~ 3 3 ##### ProblemsleadingtoLinearequations Equationsareveryusefulinsolvingproblems.Thebasictechniqueistodeterminewhatquantityitisthatwe aretryingtofindandmakethattheunknown.Wethentranslatetheproblem intoanequationandsolveit.You shouldalwaystrytominimizethenumberofunknowns.Forexample,ifwearetoldthatapieceofrope8 metreslongiscutintwoandonepieceisxmetres,thenwecanwritetheremainingpieceas(8x)metres, ratherthanintroducingasecondunknown. ##### Wordproblems Equationsariseineverydaylife.ForexampleMaryboughtanumberoforangesfrom Anitaskiosk.Shethen wenttoMarkskioskandboughtthesamenumberoforanges.Markthem gaveherthreemoreoranges.The orangesfrom thetwokioskswerewrappedindifferentpaperbags.Onreachingherhouse,shefoundthata quarterofthefirstLotorangesandafifthofthesecondwerebad.Ifintotalsixorangeswerebad,findhow manyorangessheboughtfromAnitaskiosk. ##### Solution LetthenumberoforangesboughtatAnitaskioskbex. Then,thenumberoforangesobtainedfromMarkskioskwillbex+3. x+3 NumberofBadorangesfromMarkskioskwa ~~s~~ . 5 x TotalnumberofBadorangesisequalto ~~=~~ + 4 x+3 5 x+3 x x+3 Thus, ~~=~~ ~~+~~ =6 4 5 5 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Multiplyeachtermoftheequationby20(L.C.Mof4and5)togetridofthedenominator. x x+3 =20 ~~x~~ +20 =6x20 ## 4 ( 5 [)] x+3 ## 5 [)] x+3 5x+4(x+3)=120 5x+4x+12=120(Removingbrackets) Subtracting12frombothsides. 9x=108 X=12 Thus,thenumberoforangesboughtfromAnitaskioskwas12. ##### Note: Ifanyoperationisperformedononesideofanequation,itmustalsobeperformedontheotherside. ##### Example x+3 x-4 Solveforxintheequation ~~:~~ ~~-~~ =4 x-4 2 3 ##### Solution Eliminatethefractionsbymultiplyingeachtermby6(L.C.M,of2and3). x+3 x-4 6x ~~(~~ )-6 ~~(~~ )=4x6 x-4 2 3 3(x+3)2(x4)=24 3x+9-2x+8=24(notethechangeinsighwhenthebracketareremoved) x+17=24 x=7 ##### LinearEquationsinTwoUnknowns Manyproblemsinvolvefindingvaluesoftwoormoreunknowns.Theseareoftenlinkedviaanumberoflinear equations.Forexample,ifItellyouthatthesumoftwonumbersis89andtheirdifferenceis33,wecanletthe largernumberbexandthesmalleroneyandwritethegiveninformationasapairofequations: x+y=89(1) xY=33.(2) Thesearecalledsimultaneousequationssinceweseekvaluesofxandythatmakesbothequationstrue simultaneously.Inthiscase,ifweaddtheequationsweobtain2x=122,sox=61.Wecanthensubstitutethis valuebackintoeitherequation,saythefirst,then61+y=89givingy=28. ##### Example Thecostoftwoskirtsandthreeblousesissh600.Ifthecostofoneskirtandtwoblousesofthesamequality sh350,findthecostofeachitem. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Solution Letthecostofoneskirtbexshillingsandthatofoneblousebeyshillings.Thecostoftwoskirtsandthree blousesis2x+3yshillings. Thecostofoneskirtandtwoblousesisx+2yshillings. So,2x+3y=600..(I) X+2y=350..(II) Multiplyingequation(II)by2togetequation(III). 2x+4y=700(III). 2x+3y=600(I) Subtractingequation(I)from(II),y=100. Fromequation(II), X+2y=350buty=100 X+200=350 X=150 Thusthecostofoneskirtis150shillingsandthatofablouseis100shillings. Insolvingtheproblem above,wereducedtheequationsfrom twounknownstoasingleunknowninyby eliminating.Thisistheeliminationmethodofsolvingsimultaneousequations. ##### Examples a.)a+b=7 a-b=5 b.)3a+5b=20 6a-5b=12 c.)3x+4y=18 5x+6y=28 ##### Solutions a.) a+b=7-------(i) a-b=5-------(ii) AddingItoII 2a=12a=6 SubtractingIIfromI; 2b=2 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com b=1 b.)3a+5b=20-------(i) 6a-5b=12-------(ii) Toeliminateb,wesimplyaddthetwoequations. 9a=32 a ~~=~~ [32] 9 Findthevalueofb c.)3x+4y=18-----------(i) 5x+6y=28-----------(ii) Hereitisnoteasytoknowtheobviousunknowntobeeliminated Toeliminate(I)by5and(II)by3toget(III)and(IV)respectivelyandsubtracting(IV)from(III); d.)15x+20y=90-----------(iii) 15x+18y=84-----------(iv) 2y=6 Thereforey=3 Substitutingy=3in(I); 3x+12=18 Thereforex=2 NotethattheL.C.Mof3and5is15. Toeliminatey; Multiplying(I)by3,(II)by2toget(V)and(VI)andSubtracting(V)from(VI); 9x+12y=54-----------(v) 10x+12y=56-----------(vi) Subtractingx=2in(ii); 10+6y=28 6y=18 Thereforey=3. ##### Note;  Itisadvisabletostudytheequationanddecidewhichvariableiseasiertoeliminate. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com  Itisnecessarytocheckyoursolutionbysubstitutingintotheoriginalequations. ##### Solutionbysubstitution 2x+3y=600--------- ( i ) x+2y=350-----------(ii) Takingequation(II)alone; x+2y=350 Subtracting2yfrombothsides; x=350-2y----------(iii) Substitutingthisvalueofxinequation ; ( i ) 2(3502y)+3y=600 7004y+3y=600 Y=100 Substitutingthisvalueofyinequation ( iii ) ; x=350-2y =350-200 x=150 Thismethodofsolvingsimultaneousequationsiscalledthesubstitutionmethod Endoftopic ### PastKCSEQuestionsonthetopic 1.Aclothdealersold3shirtsand2trousersforKshs840and4shirtsand5trousersforKshs1680 findthecostof1shirtandthecostof1trouser 2.Solvethesimultaneousequations 2xy=3 x [2] xy=-4 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 3.Thecostof5skirtsandblousesisKshs1750.Mueniboughtthreeoftheskirtsandoneofthe blousesforKshs850.Findthecostofeachitem. 4.AkinyiboughtthreecupsandfourspoonsforKshs324.WanjiruboughtfivecupsandFatuma boughttwospoonsofthesametypeasthoseboughtbyAkinyi,WanjikupaidKshs228morethan Fatuma.Findthepriceofeachcupandeachspoon. 5.Maryhas21coinswhosetotalvalueisKshs.72.Therearetwiceasmanyfiveshillingscoinsas therearetenshillingcoins.Therestoneshillingscoins.Findthenumberoftenshillingscoinsthat Maryhas. (4mks) 6.Themassof6similarartbooksand4similarbiologybooksis7.2kg.Themassof2suchartbooks and3suchbiologybooksis3.4kg.Findthemassofoneartbookandthemassofonebiology book 7.Karanibought4pencilsand6birospensforKshs66andTachorabought2pencilsand5biro pensforKshs51. (a) Findthepriceofeachitem (b) MusomaspentKshs.228tobuythesametypeofpencilsandbiropensifthenumberofbiro pensheboughtwere4morethanthenumberofpencils,findthenumberofpencilsbought. 8.Solvethesimultaneousequationsbelow 2x3y=5 -x+2y=-3 9.Thelengthofaroomis4metreslongerthanitswidth.Findthelengthoftheroomifitsareais32m [2] 10.HadijaandKagendoboughtthesametypesofpensandexercisebooksfromthesametypesof pensandexercisebooksfromthesameshop.Hadijabought2pensand3exercisebooksforKshs 78.Kagendobought3pensand4exercisebooksforKshs108. Calculatethecostofeachitem 11.Infourteenyearstime,amotherwillbetwiceasoldasherson.Fouryearsago,thesumoftheir ageswas30years.Findhowoldthemotherwas,whenthesonwasborn. 12.ThreeyearsagoJumawasthreetimesasoldasAli.Intwoyearstimethesumoftheirageswillbe 62.Determinetheirages. 13.TwopairsoftrousersandthreeshirtscostsatotalofKshs390.Fivesuchpairsoftrousersandtwo shirtscostatotalofKshs810.Findthepriceofapairoftrousersandashirt. 14.Ashopkeepersellstwo-typesofpangastypexandtypey.Twelvexpangasandfivetypeypangas costKshs1260,whileninetypexpangasandfifteentypeypangascost1620.Mugalabought eighteentypeypangas.Howmuchdidhepayforthem? CHAPTEREIGHTEEN ##### COMMERCIALARITHMETIC SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Statethecurrenciesofdifferentcountries b.)Convertcurrencyfromoneformintoanothergiventheexchangerates c.)Calculateprofitandloss d.)Expressprofitandlossaspercentages Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com e.)Calculatediscountandcommission f.)Expressdiscountandcommissionaspercentage. Content a.)Currency b.)Currentcurrencyexchangerates c.)Currencyconversion d.)Profitandloss e.)Percentageprofitandloss f.)Discountsandcommissions g.)Percentagediscountsandcommissions ### Introduction Incommercialarithmeticwedealwithcalculationsinvolvingbusinesstransaction.Themedium ofany businesstransactionsisusuallycalledthecurrency.TheKenyacurrencyconsistofabasicunitcalleda shilling.100centsareequivalenttooneKenyanshillings,whileaKenyanpoundisequivalenttotwentyKenya shillings. ##### CurrencyExchangeRates TheKenyancurrencycannotbeusedforbusinesstransactionsinothercountries.Tofacilitateinternational trade,manycurrencieshavebeengivendifferentvaluesrelativetoanother.Theseareknownasexchange rates. Thetablebelow showstheexchangeratesofmajorinternationalcurrenciesatthecloseofbusinessona certaindayintheyear2015.Thebuyingandsellingcolumnrepresentstheratesatwhichbanksbuyandsell thesecurrencies. ##### Note Theratesarenotalwaysfixedandtheykeeponcharging.WhenchangingtheKenyancurrencytoforeign currency,thebanksellstoyou.Therefore,weusethesellingcolumnrate.Converselywhenchangingforeign currencytoKenyanCurrency,thebankbuysfromyou,soweusethebuyingcolumnrate. Currency Buying Selling DOLLAR 102.1472 102.3324 STGPOUND 154.0278 154.3617 EURO 109.6072 109.8522 SARAND 7.3332 7.3486 KES/USHS 33.0785 33.2363 KES/TSHS 20.9123 21.0481 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com KES/RWF 7.2313 7.3423 AEDIRHAM 27.8073 27.8653 CAN$ 77.6018 77.7661 JAPNESEYEN 84.0234 84.1964 SAUDIRIYAL 27.2284 27.2959 CHINESEYUAN 16.0778 16.1082 AUSTRALIAN$ 71.8606 72.0420 ##### Example Converteachofthefollowingcurrenciestoitsstatedequivalent a.)Us$305toKsh b.)530Dirhamtoeuro ##### Solution a.)ThebankbuysUs1atKsh 102.1472 ThereforeUS$305=Ksh(102.1472x305) =Ksh31,154.896 =Ksh31,154.00(Tothenearestshillings) Thebankbuys1DirhamatKsh27.8073 Therefore530Dirham=Ksh(21.8073x530) =Ksh11,557.00 (Tothenearestshillings) Thebanksells1Euroat109.8522 Therefore530Dirham=11,.8522 =105.170Euros ##### Example Duringacertainmonth,theexchangeratesinabankwereasfollows; |Col1|Col2|Col3| |---|---|---| |||| |||| Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com AtouristleftKenyatotheUnitedStateswithKsh.1000,000.Ontheairportheexchangedallthemoneyto dollarsandspent190dollarsonairticket.WhileinUShespent4500dollarsforupkeepandproceededto Europe.WhileinEuropehespentatotalof2000Euros.HowmanyEurosdidheremainwith? (3marks) ##### Solution ProfitandLoss Thedifferencebetweenthecostpriceandthesellingpriceiseitherprofitorloss.Ifthesellingpriceisgreater thanthecostprice,thedifferenceisaprofitandifthesellingpriceislessthanthetotalcostprice,the differenceisaloss. ##### Note Sellingprice-costprice=profit ##### Profit Percentageprofit = x100 Costprice Costprice-sellingprice=loss ##### Profit Percentageloss ~~=~~ x100 ##### Costprice Example Ollieboughtacowatsh18000andsolditatsh21000.Whatpercentageprofitdidhemake? ##### Solution Sellingprice=sh21000 Costprice=sh18000 Profit=sh(21000-18,000) =sh3000 ##### 3000 Percentageprofit ~~=~~ x100 ##### 18000 2 =1 6 % 3 Example Johnnyboughtadressat3500andlatersolditatsh.2800.whatpercentagelossdidheincurs? Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Costprice=sh3500 Sellingprice=sh2800 Loss=sh(3500-2800) =Sh700 ##### 700 Percentageloss= x100=20% ##### 3500 ##### Discount Ashopkeepermaydecidetosellanarticleatreducedprice.Thedifferencebetweenthemarkedpriceandthe reducedpriceisreferredtoasthediscount.Thediscountisusuallyexpressedasapercentageoftheactual price. ##### Example Thepriceofanarticleismarkedatsh120.Adiscountisallowedandthearticlesoldatsh96.Calculatethe percentagediscount. ##### Solution Actualprice=sh120.00 Reducedprice=sh96.00 Discount=sh(120.0096.00) =sh24 Percentagediscount=24/120x100 =sh20% ##### Commission Acommissionisanagreedrateofpayment,usuallyexpressedasapercentage,toanagentforhisservices. ##### Example Mr.Neasa,asalesmaninasoapindustry,sold250piecesoftoiletsoapatsh45.00and215packetsof determiningatsh75.00perpacket.Ifhegota5%commissiononthesales,howmuchmoneydidhegetas commission? ##### Solution Salesforthetoiletsoapwas250x45=sh11250 Salesforthedetergentwas215x75=sh16125 ##### Commission = [5] #### 100 [(] [11250+16125] [)] ##### 5 x27375=sh1368 100 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Example Asalesmanearnsabasicsalaryofsh.9,000permonth.Inadditionheisalsopaidacommissionof5%for salesabovesh.15,000.Inacertainmonthhesoldgoodsworthsh.120,000atadiscountof2%.Calculatehis totalearningsthatmonth. {3marks} ##### Solution salessh.120,000 97.5 netafterdiscou ~~nt~~ 120,000=17,000 100 salesabovesh.15,000=117,000-15,000 =kshs.102,000 5 commisiio ~~n~~ 102,000=5,100 100 totlearnings=9,000+5,100 =kshs14,100 Endoftopic ### PastKCSEQuestionsonthetopic 1.ThecashprizeofatelevisionsetisKshs25000.AcustomerpaidadepositofKshs3750.Herepaid theamountowingin24equalmonthlyinstallments.Ifhewaschargedsimpleinterestattherateof 40%p.ahowmuchwaseachinstallment? 2.MrNgenyborrowedKshs560,000from abanktobuyapieceofland.Hewasrequiredtorepaythe loanwithsimpleinterestforaperiodof48months.TherepaymentamountedtoKshs21,000per month. Calculate (a)Theinterestpaidtothebank Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (b)Therateperannumofthesimpleinterest 3.Acardealercharges5%commissionforsellingacar.HereceivedacommissionofKshs17,500for sellingcar.Howmuchmoneydidtheownerreceivefromthesaleofhiscar? 4.AcompanysalesladysoldgoodsworthKshs240,000from thissalesheearneda commissionof Kshs4,000 (a)Calculatetherateofcommission (b)IfshesoldgoodwhosetotalmarkedpricewasKshs360,000andallowed adiscountof2%calculatetheamountofcommissionshereceived. 5.Abusinesswomanboughttwobagsofmaizeatthesamepriceperbag.Shediscoveredthatone bagwasofhighqualityandtheotheroflowquality.Onthehighqualitybagshemadeaprofitby sellingatKshs1,040,whereasonthelowqualitybagshemadealossbysellingatKshs880.Ifthe profitwasthreetimestheloss,calculatethebuyingpriceperbag. 6.Asalesmangetsacommissionof2.4% onsalesuptoKshs100,000.Hegetsanadditional commissionof1.5%onsalesabovethis.CalculatethecommissionhegetsonsalesworthKshs 280,000. 7.ThreepeopleKoris,WangareandHassancontributedmoneytostartabusiness.Korircontributeda quarterofthetotalamountandWangaretwofifthsoftheremainder. HassanscontributionwasoneandahalftimesthatofKoris.Theyborrowedtherestofthemoney from thebankwhichwasKshs60,000lessthanHassanscontribution.Findthetotalamount requiredtostartthebusiness. 8.AKenyantouristleftGermanyforKenyathroughSwitzerland.WhileinSwitzerlandheboughta watchworth52deutscheMarks.Findthevalueofthewatchin: (a) SwissFrancs. (b) KenyaShillings Usetheexchangertesbelow: 1SwissFranc=1.28DeutscheMarks. 1SwissFranc=45.21KenyaShillings 9. AsalesmanearnsabasicsalaryofKshs.9000permonth Inadditionheisalsopaidacommissionof5%forsalesaboveKshs15000 InacertainmonthhesoldgoodsworthKshs.120,000atadiscountof2%.Calculatehistotal earningsthatmonth 10.Inthisquestion,mathematicaltableshouldnotbeused AKenyanbankbuysandsellsforeigncurrenciesasshownbelow Buying Selling (InKenyashillings) InKenyaShillings 1HongKongdollar 9.74 9.77 1SouthAfricanrand 12.03 12.11 AtouristsarrivedinKenyawith105000HongKongdollarsandchangedthewholeamountto Kenyanshillings.WhileinKenya,shepentKshs403897andchangedthebalancetoSouthAfrican randbeforeleavingforSouthAfrica.Calculatetheamount,inSouthAfricanrandthatshereceived. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 11.AKenyanbusinessmanboughtgoodsfrom Japanworth2,950000Japaneseyen.Onarrivalin Kenyacustomdutyof20%waschargedonthevalueofthegoods. Iftheexchangerateswereasfollows 1USdollar=118JapaneseYen 1USdollar=76Kenyashillings CalculatethedutypaidinKenyashillings 12.Twobusinessmenjointlyboughtaminibuswhichcouldferry25payingpassengerswhenfull.The farebetweentwotownsAandBwasKshs.80perpassengerforoneway.Theminibusmadethree roundtripsbetweenthetwotownsdaily.ThecostoffuelwasKshs1500perday.Thedriverand theconductorwerepaiddailyallowancesofKshs200andKshs150respectively. AfurtherKshs4000perdaywassetasideformaintenance. (a)Onedaytheminibuswasfulloneverytrip. (i) Howmuchmoneywascollectedfromthepassengersthatday? (ii) Howmuchwasthenetprofit? (b)Onanotherday,theminibuswas80%ontheaverageforthethreeround trips.Howmuchdideachbusinessgetifthedaysprofitwassharedintheratio2:3? 13.Atravelerhadsterlingpounds918withwhichheboughtKenyashillingsattherateofKshs84per sterlingpound.Hedidnotspendthemoneyasintended.Later,heusedtheKenyanshillingstobuy sterlingpoundattherateofKshs.85persterlingpound.Calculatetheamountofmoneyinsterling poundslostinthewholetransaction. 14.AcommercialbankbuysandsellsJapaneseYeninKenyashillingsattheratesshownbelow Buying 0.5024 Selling 0.5446 AJapanesetouristattheendofhistourofKenyawasleftwithKshs.30000whichheconvertedto JapaneseYenthroughthecommercialbank.HowmanyJapaneseYendidheget? 15.InthemonthofJanuary,aninsurancesalesmanearnedKshs.6750whichwascommissionof4.5% ofthepremiumspaidtothecompany. (a)Calculatethepremiumpaidtothecompany. (b)InFebruarytherateofcommissionwasreducedby66 [2] / 3 %andthe premiumsreducedby10% calculatetheamountearnedbythesalesmaninthemonthof February 16.Akinyi,Bundi,Cura and Diba invested somemoneyin a businessin theratio of7:9:10:14 respectively.ThebusinessrealizedaprofitofKshs46800.Theyshared12%oftheprofitequally Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com andtheremainderintheratiooftheircontributions.Calculatethetotalamountofmoneyreceived byDiba. 17.AtelephonebillincludesKshs4320foralocalcallsKshs3260fortrankcallsandrentalcharge Kshs2080.Avalueaddedtax(V.A.T)isthenchargedat15%,Findthetotalbill. 18.Duringacertainperiod.Theexchangerateswereasfollows 1sterlingpound=Kshs102.0 1sterlingpound=1.7usdollar 1U.Sdollar=Kshs60.6 AschoolmanagementintendedtoimporttextbooksworthKshs500,000from UK.Itchangedthe moneytosterlingpounds.Laterthemanagementfoundoutthatthebooksthesterlingpoundsto dollars.UnfortunatelyafinancialcrisisaroseandthemoneyhadtobeconvertedtoKenyashillings. Calculatethetotalamountofmoneythemanagementendedupwith. 19.Afruitererbought144pineapplesatKshs100foreverysixpineapples.Shesoldsomeofthem at Kshs72foreverythreeandtherestatKshs60foreverytwo. Ifshemadea65%profit,calculatethenumberofpineapplessoldatKshs72foreverythree. CHAPTERTEN SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)DrawandlabelthecompleteCartesianplane b.)LocateandplotpointsontheCartesianplane c.)Chooseanduseappropriatescaleforagivendata d.)Makeatableofvaluesforagivenlinearrelation e.)Usevaluestodrawalineargraph f.)Solvesimultaneouslinearequationsgraphically g.)Draw,readandinterpretgraphs. Content a.)Cartesianplane b.)Cartesianco-ordinate c.)PointsontheCartesianplane d.)Choiceofappropriatescale e.)Tableofvaluesforagivenlinearrelation Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com f.)Lineargraphs g.)Graphicalsolutionsofsimultaneouslinearequations h.)Interpretationofgraphs. ### Introduction Thepositionofapointinaplanislocatedusinganorderedpairofnumberscalledco-ordinatesandwrittenin theform (x,y).Thefirstnumberrepresentsthedistancealongthexaxisandiscalledthexco-ordinates.The secondnumberrepresentsdistancealongtheyaxisanditscalledtheycoordinates. Thexandycoordinatesintersectsat(0,0)apointcalledtheorigin.Thesystem oflocatingpointsusingtwo axesatrightanglesiscalledCartesianplansystem. TolocateapointontheCartesianplane,movealongthex-axistothenumberindicatedbythex-coordinateand thenalongthey-axistothenumberindicatedbythey-coordinate.Forexample,tolocatethepointwith coordinates(1,2),move1unittotherightoftheoriginandthen2unitsup ##### TheCartesianplan ### The Graph ofastraight line ConsidertheLinear equationy=2x+ 1.Some corresponding valuesofxand y are given in the table below.Ifwe plotthepointswe noticethattheyall lieinastraightline. ##### Solution Step1writetherule y=2x+1 Step2Drawatableandchoosesimplexvalues Step3Usetheruletofindeachyvalueandentertheninthetable. E.g.whenx=-2,y=2x-2+1=-3. whenx=-1,y=2x-1+1=-1 step4DrawaCartesianplanandplotthepoints. Step5Jointhepointstoformastraightlineandlabelthegraph Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Note:  Twopointsaresufficienttodetermineastraightline,butweusethethirdpointasacheck.  Itisadvisabletochoosepointswhichcanbeplottedeasily. ##### Graphingsolutionsofsimultaneouslinearequation Thegraphsoftheformax+by=crepresentsastraightline.Whentwolinearequationsarerepresentedonthe sameCartesianplan,theirgraphsmayormaynotintersect.Forexample,insolvingthesimultaneous equationsx+3y=5and5x+7y=9graphically,thegraphsofthetwoequationaredrawn. Thetwolinesintersectatp(-1,2).Thesolutiontothesimultaneousequationsis,therefore,x=-1andy=2. ##### Generalgraphs Graphsareappliedwidelyinscienceandmanyotherfields.Thegraphsshouldtheirfoebedrawninawaythat conveyinformationeasilyandaccurately.Themostofimportanttechniqueofdrawinggraphsisthechoiceof Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com appropriatescale. Agoodscaleisonewhichusesmostofthegraphpageandenablesustoplotpointsandreadoffvalueseasily andaccurately. Avoidscaleswhich:  Givetinygraphs.  Cannotaccommodateallthedatainthetable. Itisgoodpracticeto:  Labeltheaxesclearly.  Givethetitleofthegraph. Endoftopic ### RevisionQuestionsonthetopic 1.)Copyandcompletethetableandhencedrawthecorrespondinggraph. Y=4x+3 |Col1|Col2|Col3|Col4|Col5|Col6| |---|---|---|---|---|---| ||||||| 2.)Drawthegraphofthefollowing: a.)Y+2x=5 b.)y/2+2x=5 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- CHAPTERTWENTY SpecificObjectives isabokemicah@gmail.com Bytheendofthetopicthelearnershouldbeableto: a.)Nameandidentifytypesofangles b.)Solveproblemsinvolvinganglesonastraightline c.)Solveproblemsinvolvinganglesatapoint d.)Solveproblemsinvolvinganglesonatransversalcuttingparallellines e.)Stateanglepropertiesofpolygons f.)Solveproblemsinvolvinganglepropertiesofpolygons g.)Applytheknowledgeofanglepropertiestoreallifesituations. Content a.)Typesofangles b.)Anglesonastraightline c.)Anglesatapoint d.)Anglesonatransversal(corresponding,alternateandalliedangles) e.)Anglepropertiesofpolygons f.)Applicationtoreallifesituations. ### Introduction Aflatsurfacesuchasthetopofatableiscalledaplane.Theintersectionofanytwostraightlinesisapoint. ##### Representationofpointsandlinesonaplane Apointisrepresentedonaplanebyamarklabelledbyacapitalletter.Throughanytwogivenpointsonaplane, onlyonestraightlinecanbedrawn. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ThelinepassesthroughpointsAandBandhencecanbelabelledlineAB. ##### TypesofAngles Whentwolinesmeet,theyform anangleatapoint.Thepointwheretheangleisformediscalledthevertexof theangle.Thesymbol<isusedtodenoteanangle. Acuteangle. Reflexangle. Obtuseangle Rightangle Toobtainthesizeofareflexanglewhichcannotbereaddirectlyfrom aprotractor,thecorrespondingacuteor obtuseangleissubtractedfrom 360 0 .IfanytwoanglesXandYaresuchthat: i.) AngleX+angleY=90 0,theanglesaresaidtobecomplementaryangles.Eachangleisthensaidto bethecomplementoftheother. ii.) AngleX+angleY=180 0,theanglesaresaidtobesupplementaryangles.Eachangleisthensaidto bethesupplementoftheother. Inthefigurebelow<POQand<ROQareapairofcomplementaryangles. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Inthefigurebelow<DOFand<FOEareapairofsupplementaryangles. ##### Anglesonastraightline. Thebelowshowsanumberofangleswithacommonvertex0.AOEisastraightline. Twoanglesoneithersideofastraightlineandhavingacommonvertexarereferredtoasadjacentangles. Inthefigureabove: <AOBisadjacentto<BOC <BOCisadjacentto<COD <CODisadjacentto<DOE 0 ##### Anglesonastraightlineaddupto180. Anglesatapoint Twointersectingstraightlinesform fourangleshavingacommonvertex.Theangleswhichareonopposite sidesofthevertexarecalledverticallyoppositeangles.Considerthefollowing: Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Inthefigureabove<COBand<AOCareadjacentanglesonastraightline.Wecannowshowthata=cas follows: a+b=180 0 (Anglesonastraightline) c+d=180 0 (Anglesonastraightline) 0 0 + 180 0 0 = 360 0 0 So,a+b+c+d=180+ 180 = 0 Thisshowsthatanglesatapointaddupto360 0 ##### Anglesonatransversal Atransversalisalinethatcutsacrosstwoparallellines. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com IntheabovefigurePQandSTareparallellinesandRUcutsthroughthem.RUisatransversal. Name: i.) CorrespondinganglesareAnglesbande,candh,aandf,dandg. ii.) Alternateanglesaandc,fandh,bandd,eandg. iii.) Co-interiororalliedanglesarefandd,cande. ##### Anglepropertiesofpolygons Apolygonisaplanfigureborderedbythreeororestraightlines ##### Triangles Atriangleisathreesidedplanefigure.Thesum ofthethreeanglesofatriangleaddupto18 .trianglesare0 0 classifiedonthebasisofeitheranglessides. i.) Atriangleinwhichoneoftheanglesis90 0 iscalledarightangledtriangle. ii.) Ascalenetriangleisoneinwhichallthesidesandanglesarenotequal. iii.) Anisoscelestriangleisoneinwhichtwosidesareequalandtheequalsidesmakeequalangles withthethirdside. iv.) Anequilateraltriangleisoneinwhichallthesideareequalandalltheanglesareequal Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Exteriorpropertiesofatriangle AngleDAB=p+q. Similarly,AngleEBC=r+qandangleFCA=r+p. Butp+q+r=180 0 Butp+q+r=180 0 ThereforeangleDAB+angleEBC+angleFCA=2p+2q+2r =2(p+q+r) =2x180 0 =360 0 0 Ingeneralthesumofalltheexterioranglesofatriangleis360. ##### Quadrilaterals Aquadrilateralisafoursidedplanfigure.Theinterioranglesofaquadrilateraladdput360 0 .Quadrilaterals arealsoclassifiedintermsofsidesandangles. ##### PROPERTIESOFQUADRILATERALS PropertiesofParallelograms Inaparallelogram, 1. Theparallelsidesareparallelbydefinition. 2. Theoppositesidesarecongruent. 3. Theoppositeanglesarecongruent. 4. Thediagonalsbisecteachother. 5. Anypairofconsecutiveanglesaresupplementary. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### PropertiesofRectangles Inarectangle, 1. Allthepropertiesofaparallelogramapplybydefinition. 2. Allanglesarerightangles. 3. Thediagonalsarecongruent. ##### Propertiesofakite 1. Two disjointpairs ofconsecutive sides are congruentby definition. 2. Thediagonalsareperpendicular. 3. Onediagonalistheperpendicularbisectoroftheother. 4. Oneofthediagonalsbisectsapairofoppositeangles. 5. Onepairofoppositeanglesarecongruent. ##### PropertiesofRhombuses Inarhombus, 1. Allthepropertiesofaparallelogramapplybydefinition. 2. Twoconsecutivesidesarecongruentbydefinition. 3. Allsidesarecongruent. 4. Thediagonalsbisecttheangles. 5. Thediagonalsareperpendicularbisectorsofeachother. 6. Thediagonalsdividetherhombusintofourcongruentright triangles. ##### PropertiesofSquares Inasquare, 1. Allthepropertiesofarectangleapplybydefinition. 2. Allthepropertiesofarhombusapplybydefinition. 3. Thediagonalsformfourisoscelesrighttriangles. ##### PropertiesofIsoscelesTrapezoids Inanisoscelestrapezoid, 1. Thelegsarecongruentbydefinition. 2. Thebasesareparallelbydefinition. 3. Thelowerbaseanglesarecongruent. 4. Theupperbaseanglesarecongruent. 5. Thediagonalsarecongruent. 6. Anylowerbaseangleissupplementarytoanyupperbaseangle. ##### ProvingThataQuadrilateralisaParallelogram Anyoneofthefollowingmethodsmightbeusedtoprovethataquadrilateralisaparallelogram. 1. Ifbothpairsofoppositesidesofaquadrilateralareparallel,thenitisaparallelogram(definition). 2. Ifbothpairsofoppositesidesofaquadrilateralarecongruent,thenitisaparallelogram. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 3. Ifonepairofoppositesidesofaquadrilateralarebothparallelandcongruent,thenitisaparallelogram. 4. Ifthediagonalsofaquadrilateralbisecteachother,thentheitisaparallelogram. 5. Ifbothpairsofoppositeanglesofaquadrilateralarecongruent,thenitisaparallelogram. ##### ProvingThataQuadrilateralisaRectangle Onecanprovethataquadrilateralisarectanglebyfirstshowingthatitisaparallelogramandthenusingeither ofthefollowingmethodstocompletetheproof. 1. Ifaparallelogramcontainsatleastonerightangle,thenitisarectangle(definition). 2. Ifthediagonalsofaparallelogramarecongruent,thenitisarectangle. Onecanalsoshowthataquadrilateralisarectanglewithoutfirstshowingthatitisaparallelogram. 3. Ifallfouranglesofaquadrilateralarerightangles,thenitisarectangle. ##### ProvingThataQuadrilateralisaKite Toprovethataquadrilateralisakite,eitherofthefollowingmethodscanbeused. 1. Iftwodisjointpairsofconsecutivesidesofaquadrilateralarecongruent,thenitisakite(definition). 2. Ifoneofthediagonalsofaquadrilateralistheperpendicularbisectoroftheotherdiagonal,thenitisa kite. ##### ProvingThataQuadrilateralisaRhombus Toprovethataquadrilateralisarhombus,onemayshowthatitisaparallelogram andthenapplyeitherofthe followingmethods. 1. Ifaparallelogram containsapairofconsecutivesidesthatarecongruent,thenitisarhombus (definition). 2. Ifeitherdiagonalofaparallelogrambisectstwoanglesoftheparallelogram,thenitisarhombus. Onecanalsoprovethataquadrilateralisarhombuswithoutfirstshowingthatitisaparallelogram. 3. Ifthediagonalsofaquadrilateralareperpendicularbisectorsofeachother,thenitisarhombus. ##### ProvingThataQuadrilateralisaSquare Thefollowingmethodcanbeusedtoprovethataquadrilateralisasquare: Ifaquadrilateralisbotharectangleandarhombus,thenitisasquare. ##### ProvingThataTrapezoidisanIsoscelesTrapezoid Anyoneofthefollowingmethodscanbeusedtoprovethatatrapezoidisisosceles. 1. Ifthenonparallelsidesofatrapezoidarecongruent,thenitisisosceles(definition). 2. Ifthelowerorupperbaseanglesofatrapezoidarecongruent,thenitisisosceles. 3. Ifthediagonalsofatrapezoidarecongruent,thenitisisosceles. ##### Note:  Ifapolygonhasnsides,thenthesumofinteriorangles(2n-4)rightangles. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com  Thesumofexterioranglesofanypolygonis360 0 .  Atriangleissaidtoberegularifallitssidesandallitsinterioranglesareequal. Thefigurebelowisahexagonwithinterioranglesg,h,I,kandIandexterioranglesa,b,c,d,e,andf. Endoftopic ### PastKCSEQuestionsonthetopic Inthefigurebelow,linesABandLMareparallel. A B Findthevaluesoftheanglesmarkedx,yandz (3mks) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com CHAPTERONE SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Usearulerandcompassesonlyto:  constructaperpendicularbisectorofaline  constructananglebisector  constructaperpendiculartoalinefromagivenapoint  constructaperpendiculartoalinethroughagivenpointontheline  constructangleswhosevaluesaremultiplesof7  constructparallellines  dividealineproportionally b.)Usearulerandasetsquaretoconstructparallellines,dividealineproportionally,andtoconstruct perpendicularlines c.)Constructaregularpolygonusingrulerandcompassesonly,andruler,compassesandprotractor d.)Constructirregularpolygonsusingaruler,compassesandprotractor. Content a.)Constructionoflinesandanglesusingarulerandcompassesonly b.)Constructionofperpendicularandparallellinesusingarulerandasetsquareonly c.)Proportionaldivisionofaline d.)Constructionofregularpolygons(uptoahexagon) e.)Constructionofirregularpolygons(uptoahexagon). ### Introduction ##### ConstructionInstruments Thefollowingminimumsetofinstrumentsisrequiredinordertoconstructgoodqualitydrawings: o Twosetsquares. o Aprotractor. o A 15cmor150mmruler o Compass o Protractor o Divider o Aneraser/rubber o Twopencils-a2HandanHB,togetherwithsomesharpeningdeviceRazorbladeorshaper. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### ConstructionofPerpendicularLines Perpendicularlines ThefigurebelowshowsPQasaperpendicularbisectorofagivenlineAB. ##### ToobtaintheperpendicularbisectorPQ  WithAandBascentre,andusingthesameradius,drawarcsoneithersideofAB tointersectatPand Q.  JoinPtoQ. ThefigurebelowshowsPE,aperpendicularfromapointPtoagivenlineAB. ##### Toconstructaperpendicularlinefromapoint  TodropaperpendicularlinefrompointPtoAB.  SetthecompasspointatPandstrikeanarcintersectingABatCandD.  WithCandDascentresandanyradiuslargerthanone-halfofCD, Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com  StrikearcsintersectingatE.  AlinefromPthroughEisperpendiculartoAB. ##### Toconstructaperpendicularlinefromapoint  UsingPascentreandanyconvenientradius,drawarcstointersectthelinesatAandB.  UsingAascentreandaradiuswhosemeasureisgreaterthanAP,drawanarcabovetheline.  UsingBasthecentreandthesameradius,drawanarctointeracttheonein(ii)atpointQ.  Usingaruler,drawPQ. ### FIGUR21.4 ##### Constructionofperpendicularlinesusingasetsquare. Twoedgesofasetsquareareperpendicular.Theycanbeusedtodrawperpendicularlines.Whenoneofthe edgesisputalongaline,alinedrawnalongtheotheroneisperpendiculartothegivenline. ##### Toconstructaperpendicularfromapointptoaline  Placearuleralongtheline.  Placeoneoftheedgesofasetsquarewhichformarightanglealongtheruler.  Slidethesetsquarealongtheruleruntiltheotheredgereachesp.  HoldthesetsquarefirmlyanddrawthelinethroughPtomeetthelineperpendicularly. ##### ConstructionofAnglesusingaRulerandapairofcompassonly Thebasicanglefromwhichalltheotherscanbederivedfromisthe60 0,45 0 and90 0 0,45 0 0 and90 0 0 0 ##### ConstructionofanAngleof60  LetAbetheapexoftheangle  WithcentreAdrawanarcBCusingasuitableradius.  WithbasthecentredrawanotherarctointersectarcBCatD. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com  DrawalineAEthroughD.TheangleEACis60 0 0 ##### ConstructionofanAngleof90  LetAbetheapexoftheangle.  WithcentreAdrawanarcBCoflargeradius.  DrawanarconBCusingasuitableradiusandmarkitD.  UsingthesameradiusandpointDasthecentredrawanarcE.  BDandDEareofthesameradius.  WithcentreDdrawanyarcF.  WithcentreEdrawanarcequalinradiustoDF.  JoinAFwithastraightline.AngleBAFis90 0 0 ##### ConstructionofanAngleof45 DrawABandACatrightangles45 0 toeachother.WithcentreAandwithlargeradius,drawanarctocutABat Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com D andandACatE.WithcentresEandDdrawarcsofequalradiustointersectatDrawastraightlinefrom A throughAngleBAFis45 0 0 ##### 1 0 Constructionofanglesofmultipleof 7 30 [0] 15 [0] .. 2 ##### 30 [0] 15 [0] 0 angleproduces30 0 0 andthesuccessivebisectionofthisangleproduces15 0 0 Thebisectionof 60 0 angleproduces30 0 andthesuccessivebisectionofthisangleproduces15 0 whichis 0 bisectedtoproduce ~~7~~ [1] asshownbelow. 2  DrawABandACat60 0 toeachotherasshownabove.  WithcentreA,andalargeradius,drawanarctocutABatEandACatD.  WithcentresEandDdrawarcsofequalradiustointersectatF.  DrawalinefromAthroughF. 0 half .60 0 0  AngleCAFis30 0 half .60 0 0 ##### 0 1 ~~7~~ ##### Toconstruct15 ##### 2  DrawACandAFat30 0 toeachotherasdescribedabove.  WithcentresGandDdrawarcsofequalradiustointersectatH  DrawalinefromAthroughH  AngleCAHis15 0 . Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com 0  WithcentresJandDafurtherbisectioncanbemadetogive ~~7~~ [1] 2 ### Constructionofparallellines Toconstructalinethroughagivenpointandparalleltoagivenline,wemayusearulerandapairofcompass only,orarulerandasetsquare. ##### Usingarulerandapairofcompassonly Parallelogrammethod ThelineEPparalleltoACisconstructedasfollows:  WithXasthecentreandradiusPQ,drawanarc.  WithQasthecentreandradiusPX,drawanotherarctocutthefirstarcatR.  JoinXtoR. ##### ProportionalDivisionoflines Linescanbeproportionatelydividedintoagivennumberofequalpartsbyuseofparallellines. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com TodividelineAB  DividelineABintotenequalparts.  Throughb,drawalineCBofanyconvenientlengthatasuitableanglewithAB.  Usingapairofcompasses,markoff,alongBC,tenequalintervalsasshownabove.  JoinCtoA.Byusingasetsquareandaruler,drawlinesparalleltoCA.  Thelineisthereforedividedtenequalpartsorintervals. ##### ConstructionofRegularPolygons Apolygonisregularifallitssidesandanglesareequal,otherwiseitisirregular. ##### Note: Forapolygonofnsides,thesum ofinterioranglesis(2n-4)rightangles.Thesizeofeachinteriorangleofthe 2n-4 0 regularpolygonisthereforeequalto ~~(~~ ) [90] . 0 ) n [90] Thesum ofexterioranglesofanypolygonis360 0 .Eachexteriorangleofaregularpolygonisthereforeequal 360 0 ~~to~~ . n ##### ConstructionofaregularTriangle Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### ConstructionofaregularQuadrilateral Constructionofaregularpentagon. ToconstructaregularpentagonABCDofsides4cm. ~~=~~ ( 10-4 ) Eachoftheinteriorangles rightangles= 0 rightangles=108 5  DrawalineAB=4cmlong.  DrawangleABC=108 0 andBC=4cm  UsethesamemethodtolocatepointsDand Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Note; Usethesameproceduretoconstructotherpoints. ##### Constructionofirregularpolygons Constructionoftriangles Toconstructatrianglegiventhelengthofitssides  DrawalineandmarkapointAonit.  OnthelinemarkoffwithapairofcompassapointB,3cmfromA.  WithBasthecentreandradius5cm,drawanarc  WithAasthecentreandradius7cm,drawanotherarctointersectthearcin(iii)atC.JoinAtoCandB toC. Toconstructatriangle,giventhesizoftwoanglesandlengthofoneside. 0,<ABC==50 0 0 ConstructatriangleABCinwhich<BAC=60 0,<ABC==50 0 andBC=4cm.Thesketchisshownbelow.  DrawalineandmarkapointBonit.  MarkoffapointContheline,4cmfromB. 0 and=70 0 0  Usingaprotractor,measureanangleof=50 0 and=70 0 atBandCrespectively. Toconstructatrianglegiventwosidesandoneangle. Giventhelengthsoftwosidesandthesizeoftheincludedangle.ConstructatriangleABC,inwhichAB=4 cm,BC=5cmand<ABC=60 0 .Draw asketchasshownbelow.  DrawalineBC=5cmalong  Measureanangleof=60 0 atBandmarkoffapointA,4cmfromB.  JoinAtoC. Toconstructatrapezium. TheconstructionofatrapeziumABCDwithAB=8cm,BC=5cm,CD=4cmandangleABC=60 0 andAB=8 cm Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com  DrawalineAB=8cm.  ConstructanAngleof=60 0 atB.  UsingBasthecentreandradiusof5cm,markanarctoinsectthelinein(ii)atC.  ThroughC,drawalineparalleltoAB  UsingCasthecentreandradiusof4cm,Markanarctointersectthelinein(iv)atD.  JoinDtoAtoformthetrapezium. Endofto p ic ### PastKCSEQuestionsonthetopic 1.Usingarulerandapairofcompassesonly, a) ConstructatriangleABCinwhichAB=9cm,AC=6cmandangleBAC=37 [0] a)DropaperpendicularfromCtomeetABatD.MeasureCDandhencefindtheareaofthetriangle 2. ABC b)PointEdividesBCintheratio2:3.UsingarulerandSetSquareonly,determinepointE.MeasureAE. B Onthediagram,constructacircletotouchlineABatXandpassesthroughthepointC. (3mks) 3. Usingrulerandpairofcompassesonlyforconstructionsinthisquestion. (a)ConstructtriangleABCsuchthatAB=AC=5.4cmandangleABC=30 [0] .MeasureBC (4mks) (b)Onthediagramabove,apointPisalwaysonthesamesideofBCasA.Drawthe locusofPsuchthatangleBACistwiceangleBPC (2mks) (c)DropaperpendicularfromAtomeetBCatD.MeasureAD (2mks) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (d)DeterminethelocusQonthesamesideofBCasAsuchthattheareaoftriangle BQC=9.4cm [2] (2mks) 4. (a)Withoutusingaprotractororsetsquare,constructatriangleABCinwhichAB=4cm, BC= 6cmand  ABC=67 [0] .TakeABasthebase. (3mks) MeasureAC. (b)DrawatriangleA [1] BC [1] whichisindirectlycongruenttotriangleABC. (3mks) 5. ConstructtriangleABCinwhichAB=4.4cm,BC=6.4cmandAC=7.4cm.Constructanescribedcircle oppositeangleACB (5mks) (a)Measuretheradiusofthecircle (1mk) (b)MeasuretheacuteanglesubtendedatthecentreofthecirclebyAB (1mk) (c)ApointPmovessuchthatitisalwaysoutsidethecirclebutwithintriangleAOB,whereOisthe centreoftheescribedcircle.ShowbyshadingtheregionwithinwhichPlies. (3mks) 6. (a)Usingarulerandapairofcompassesonly,constructaparallelogramPQRSinwhichPQ=8cm,QR =6cmand PQR=150 [0] (3mks) (b)DropaperpendicularfromStomeetPQatB.MeasureSBandhencecalculatetheareaofthe parallelogram. (5mks) (c)MarkapointAonBSproducedsuchthattheareaoftriangleAPQisequaltothreequartersthearea oftheparallelogram (1mk) (d)Determinetheheightofthetriangle. (1mk) 7. Usingarulerandapairofcompassesonly,constructtriangleABCinwhichAB=6cm,BC=8cmand angleABC=45 [o] .DropaperpendicularfromAtoBCatM.MeasureAMandAC (4mks) 8. a)UsingarulerandapairofcompassesonlytoconstructatrapeziumABCD suchthat,, and (5mks) b)FromthepointDdropaperpendiculartothelineABtomeetthelineatE.measureDEhence calculatetheareaofthetrapezium (5mks) 9. Usingapairofcompassesandruleronly; (a)ConstructtriangleABCsuchthatAB=8cm,BC=6cmandangleABC=30 [0] .(3marks) (b)MeasurethelengthofAC (1mark) (c)DrawacirclethattouchestheverticesA,BandC. (2marks) (d)Measuretheradiusofthecircle (1mark) (e)Henceorotherwise,calculatetheareaofthecircleoutsidethetriangle. (3marks) 10. Usingarulerandapairofcompassesonly,constructthelocusofapointPsuchthatangleAPB=60 [0] onthelineAB=5cm. (4mks) Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com A B 11. Usingasetsquare,rulerandpairofcompasesdividethegivenlineinto5equalportions.(3mks) 12. Usingarulerandapairofcompassesonly,drawaparallelogramABCD,suchthatangleDAB=75 [0] . LengthAB=6.0cmandBC=4.0cmfrompointD,dropaperpendiculartomeetlineABatN a)MeasurelengthDN b)Findtheareaoftheparallelogram (10mks) 13. Usingarulerandapairofcompassesonly,drawaparallelogramABCDinwhichAB=6cm,BC=4cm andangleBAD=60 [o] .Byconstruction,determinetheperpendiculardistancebetweenthelinesABandCD 14. Withoutusingaprotractor,drawatriangleABCwhere  CAB=30 [o],AC=3.5cmand AB=6cm.measureBC 15. (a)Usingarulerandapairofcompassonly,constructatriangleABCinwhich angleABC=37.5 [o],BC=7cmandBA=14cm (b)DropaperpendicularfromAtoBCproducedandmeasureitsheight (c)Useyourheightin(b)tofindtheareaofthetriangleABC (d)UseconstructiontofindtheradiusofaninscribedcircleoftriangleABC 16. Inthisquestionuseapairofcompassesandaruleronly a)ConstructtrianglePQRsuchthatPQ=6cm,QR=8cmand<PQR=135 b)ConstructtheheightoftrianglePQRin(a)above,takingQRasthebase 17. OnthelineACshownbelow,pointBliesabovethelinesuchthat  BAC=52.5 [o] and] AB=4.2cm. (Usearulerandapairofcompassesforthisquestion) C A (a)Construct  BACandmarkpointB (b)DropaperpendicularfromBtomeetthelineACatpointF.MeasureBF CHAPTERTWENTYTW0 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Interpretagivenscale b.)Chooseanduseanappropriatescale c.)Drawsuitablesketchesfromgiveninformation d.)Statethebearingofonepointfromanother e.)Locateapointusingbearinganddistance f.) Determineanglesofelevationanddepression g.)Solveproblemsinvolvingbearingselevationsandscaledrawing h.)Applyscaledrawinginsimplesurveying. Content a.)Typesofscales b.)Choiceofscales c.)Sketchingfromgiveninformationandscaledrawing d.)Bearings e.)Bearings,distanceandlocatingpoints f.)Anglesofelevationanddepression g.)Problemsinvolvingbearings,scaledrawing,anglesofelevationanddepression h.)Simplesurveyingtechniques. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### Introduction Thescale Theratioofthedistanceonamaptotheactualdistanceonthegroundiscalledthescaleofthemap.Theratio canbeinstatementforme.g.50cmrepresents50,000cmorasarepresentative fraction(R.F),1:5,000,000is ##### 1 writtena ~~s~~ . ##### 5,000,000 Example Thescaleofamapisgiveninastatementas1cm represents4km.convertthistoarepresentativefraction (R.F). ##### Solution Onecmrepresents4x100,000cm.1cmrepresents400,000 ##### 1 Therefore,theratiois1:400,000andtheR.Fi ~~s~~ ##### 400,000 ##### Example Thescaleofamapisgivenas1:250,000.Writethisasastatement. ##### Solution 1:250,000means1cm onthemaprepresents250,000cm ontheground.Therefore,1cm represents ##### 250, 000km. 100,000 I.e.1cmrepresents2.5km. ##### ScaleDiagram Whenduringusingscale,oneshouldbecarefulinchoosingtherightscale,sothatthedrawingfitsonthe paperwithoutmuchdetailsbeingleft. ##### BearingandDistances Directionisalwaysfoundusingacompasspoint. Acompasshaseightpointsasshowabove.ThefourmainpointsofthecompassareNorth,South,East,and West.TheotherpointsaresecondarypointsandtheyincludeNorthEast(NE),SouthEast(SE),SouthWest (SW)andNorthWest(NW).Eachangleformedatthecentreofthecompassis45 0 theanglebetweenNandE 0 is90. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com ##### CompassBearing Whenthedirectionofaplacefrom anotherisgivenindegreesandintermsoffourmainpointsofacompass. E.g.N45 0 W,thenthedirectionissaidtobegivenincompassbearing.Compassbearingismeasuredeither clockwiseoranticlockwisefromNorthorsouthandtheangleisacute. ##### Truebearing NorthEastdirection,writtenasN45 0 Ecanbegiveninthreefiguresas045 0 measuredclockwisefrom True 0 Ecanbegiveninthreefiguresas045 0 0 North.Thisthree-figurebearingiscalledthetruebearing. 0 .DuesouthEastas135 0 0 anddueNorthWestas315 0 0 Thetruebearingsduenorthisgivenas000 0 .DuesouthEastas135 0 anddueNorthWestas315 . 0 ##### Example From townP,atownQis60km awayonabearingSouth80east.AthirdtownRis100km from Ponthe bearingSouth40west.Acyclisttravellingat20km/hleavesPforQ.HestaysatQforonehourand thencontinuestoR.HestaysatRfor1hrs.andthenreturnsdirectlytoP. (a) CalculatethedistanceofQfromR. P=100+60-2(100)(60)Cos120 P=1360012000Cos120 P=19600 P =140km (b) CalculatethebearingofRfromQ. M1 M1 =38.2 A1 Bearing27038.2=241.8 B1 Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com (c) Whatisthetimetakenforthewholeroundtrip? TimefromPtoR TimefromQtoR B1forallthree correct FromRtoP Takentravelling=3+7+5 M1 =15hrs ##### Example AportBisonabearing080 [o] fromaportAandadistanceof95km.ASubmarineisstationedataportD,which isonabearingof200 [o] from A,andadistanceof124km from B.AshipleavesBandmovesdirectly SouthwardstoanIslandP,whichisonabearingof140 [o] from A.TheSubmarineatDonrealizingthattheship washeadingtotheIslandP,decidestoheadstraightfortheIslandtointercepttheship.Usingascaleof1cm torepresent10km,makeascaledrawingshowingtherelativepositionsofA,B,DandP. {4marks} Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- isabokemicah@gmail.com Hencefind: b) ThedistancefromAtoD. {2marks} 4.60.110=46KM 1KM c) ThebearingotheSubmarinefromtheshipwhentheshipwassettingofffromB. {1mark} 2401ORS60W 2 d) ThebearingoftheIslandPfromD. {1mark} 122ORS58E1 e) ThedistancetheSubmarinehadtocovertoreachtheIsland P1270.110=1271 KM {2marks} ##### AngleofElevationandDepression AngleofElevation: Theangleabovethehorizontalthatanobservermustlooktoseean object thatishigherthantheobserver.Example,amanlookingatabird. Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- ~~isabokemica~~ ~~h@gmail~~ ~~.com~~ ##### AngleofDepression: Theanglebelowhorizontalthatanobservermustlooktoseean object thatislowerthantheobserver.Example,abirdlookingdownata bug. Anglesofdepressionandelevationcanbemeasuredusingan instrumentcalledclinometer Tofindtheheightsorthelengthswecanusescaledrawing. ##### Simplesurveymethods Thisinvolvestakingfieldmeasurementsoftheareasothatamapoftheareacanbedrawntoscale.Piecesof landareusuallysurveyedinorderto:  Fixboundaries  Fortownplanning  Roadconstruction  Watersupplies  Mineraldevelopment ##### Areasofirregularshapes Areasofirregularshapecanbefoundbysubdividingthem intoconvenientgeometricalshapese.g.triangles, rectanglesortrapezia. ##### Example Theareainhectaresofthefieldcanbefoundbythehelpofabaselineandoffsetsasshown. Fig22.26 XYisthebaseline360m.SM,RPandQNaretheoffsets. TakingXasthestartingpointofthesurvey,theinformationcanbeentredinafieldbookasfollows. ~~Novels~~, ~~UpdatedCPA~~, ~~CIFA~~, ~~KASN~~ ~~EB~~, ~~ICT,Colle~~ ge,HighScho ~~ol&P~~ ~~rimaryNotes071449~~ 7530 ----- isabokemica ~~h@~~ gma ~~il~~ .com Thesketchisasfollows: Usingasuitablescale. Theareaoftheseparatepartsisfoundthencombined. Areaof: 2 TriangleXPRisx180x90=8100m 2 TrianglePRYisx180x90=8100m 2 TriangleXSMisx120x60=3600m 2 TriangleQNYisx120x180=10800m 2 TrapeziumSQNM=(QN+SM)xSQm 2 (180+60)x120m 2 =14400m 2 Totalarea=8100+8100+3600+10800+14400=45000m ~~No~~ ~~vels,Up~~ d atedCPA, CIFA, KASNEB, IC T,Col ~~lege,Hig~~ ~~hScho~~ o ~~l&Prim~~ ~~aryN~~ otes0714497530 ----- ~~isabokemicah@~~ gmail.com Endofto p ic ### PastKCSEQuestionsonthetopic 1.ApointBisonabearingof080 [0] fromaportAandatadistanceof95km.Asubmarineisstationedata portD,whichisonabearingof200 [0] fromAMandadistanceof124kmfromB. AshipleavesBandmovesdirectlysouthwardstoanislandP,whichisonabearingof140fromA.The submarineatDonrealizingthattheshipwasheadingfrotheislandP,decidestoheadstraightforthe islandtointercepttheship Usingascale0f1cmtorepresent10km,makeascaledrawingshowingtherelativepositionsofA,B,D, P. Hencefind (i)ThedistancefromAtoD (ii)ThebearingofthesubmarinefromtheshipwassettingofffromB (iii)ThebearingoftheislandPfromD (iv)ThedistancethesubmarinehadtocovertoreachtheislandP 2.FourtownsR,T,KandGaresuchthatTis84km directlytothenorthR,andKisonabearingof295 [0] fromRatadistanceof60km.Gisonabearingof340 [0] fromKandadistanceof30km.Usingascaleof1 cm torepresent10km,makeanaccuratescaledrawingtoshow therelativepositionsofthetown. Find (a)ThedistanceandthebearingofTfromK (b)ThedistanceandthebearingGfromT (c)ThebearingofRfromG 3. Twoaeroplanes,SandTleaveairportsAatthesametime.Sfliesonabearingof060at750km/h whileTfliesonabearingof210 [0] at900km/h. (a) Usingasuitablescale,drawadiagramtoshowthepositionsoftheaeroplaneaftertwohours. (b) Useyourdiagramtodetermine (i)Theactualdistancebetweenthetwoaeroplanes (ii)ThebearingofTfromS (iii)ThebearingofSfromT N ~~ov~~ ~~els,UpdatedCPA,CIF~~ A,KASNEB,ICT, Colle g e,HighSchool&Pr ~~imar~~ ~~yNotes0714497530~~ ----- ~~i~~ sa ~~b~~ o ~~k~~ em ~~i~~ ca ~~h~~ @gmail.com 4. ApointAisdirectlybelowawindow.AnotherpointBis15mfromAandatthesamehorizontallevel. FromBangleofelevationofthetopofthebottomofthewindowis300andtheangleofelevationof thetopofthewindowis350.Calculatetheverticaldistance. (a) FromAtothebottomofthewindow (b) Fromthebottomtotopofthewindow 4. FindbycalculationthesumofalltheinterioranglesinthefigureABCDEFGHIbelow 6. ShoppingcentersX,YandZaresuchthatYis12kmsouthofXandZis15kmfromX.Zisona bearingof330 [0] fromY.FindthebearingofZfromX. 7. Anelectricpylonis30mhigh.ApointSonthetopofthepylonisverticallyaboveanotherpointRonthe ground.PointsAandBareonthesamehorizontalgroundasR.PointAduesouthofthepylonandthe angleofelevationofSfromAis26 [0] .PointBisduewestofthepylonandtheangleofelevationofS fromBis32 [0] Findthe (a) DistancefromAandB (b) BearingofBfromA 8. ThefigurebelowisapolygoninwhichAB=CD=FA=12cmBC=EF=4cm andBAF=-CDE=120 [0] .ADisalineofsymmetry. ~~N~~ ove ls, U p datedCPA,CIFA ~~,KASNEB,IC~~ T,Coll e g e,Hi g hSchool&Primar yNotes ~~07144~~ ~~97530~~ ----- isa bok ~~emic~~ ~~ah@gmail~~ . ~~com~~ Findtheareaofthepolygon. 9. ThefigurebelowshowsatriangleABC a) Usingarulerandapairofcompasses,determineapointDonthelineBC suchthatBD:DC=1:2. b) FindtheareaoftriangleABD,giventhatAB=AC. 10. Aboatatpointxis200mtothesouthofpointY.TheboatsailsXtoanother pointZ.PointZis200monabearingof310 [0] from X,YandZareonthesamehorizontalplane. (a)CalculatethebearingandthedistanceofZfromY (b)WisthepointonthepathoftheboatnearesttoY. CalculatethedistanceWY (c)AverticaltowerstandsatpointY.TheangleofpointXfromthetopofthetoweris6 [0] calculatethe angleofelevationofthetopofthetowerfromW. 11. ThefigurebelowshowsaquadrilateralABCDinwhichAB=8cm,DC=12cm,  BAD=45 [0],  CBD= 90 [0] andBCD=30 [0] . Novels,Update dCPA, CIFA, KASNEB,ICT,College,HighSchool&Primar y Notes0714497530 ----- isabokem ~~icah@gm~~ ai ~~l~~ .com Find: (a)ThelengthofBD (b)ThesizeoftheangleADB 12. Inthefigurebelow,ABCDEisaregularpentagonandABFisanequilateraltriangle Findthesizeof a)  ADE b)  AEF c)  DAF 13. Inthisquestionuseapairofcompassesandaruleronly (a)constructtriangleABCsuchthatAB=6cm,BC=8cmand  ABC135 [0] (2marks) (b)ConstructtheheightoftriangleABCina)abovetakingBCasthebase (1mark) 14. Thesizeofaninteriorangleofaregularpolygonis3x [0] whileitsexteriorangleis(x-20) 0 .Findthe numberofsidesofthepolygon Novels,UpdatedCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryNotes 071449753 0 ----- isabokemicah@gmail.com 15. PointsLandMareequidistantfromanotherpointK.ThebearingofLfromKis330 [0] .ThebearingofM fromKis220 [0] . CalculatethebearingofMfromL 16. FourpointsB,C,QandDlieonthesameplanepointBisthe42kmduesouth-westoftownQ.PointCis 50kmonabearingof560 [0] fromQ.PointD isequidistantfromB,QandC. (a) Usingthescale1cmrepresents10km,constructadiagramshowingthepositionofB,C,Qand D (b) Determinethe (i) DistancebetweenBandC (ii) BearingDfromB 17. TwoaeroplanesPandQ,leaveanairportatthesametimefliesonabearingof240 [0] at900km/hrwhile QfliesdueEastat750km/hr (a) Usingascaleof1vcmdrawingtoshowthepositionsoftheaeroplanesafter40minutes. (b) Usethescaledrawingtofindthedistancebetweenthetwoaeroplaneafter40minutes (c) Determinethebearingof (i) PfromQans254 [0] (ii) QfromPans74 [0] 18. AportBisnoabearingof080fromaportAandatadistanceof95km.Asubmarineisstationedport Dwhichisonabearingof200 [0] fromA,andadistanceof124kmfromB. AshipleavesBandmovesdirectlysouthwardstoanislandP,whichisonabearingof140 [0] fromA.The submarineatDonrealizingthattheshipwasheadingfortheislandPdecidestoheadstraightforthe islandtointercepttheship. Usingascaleof1cmtorepresent10km,makeascaledrawingshowingtherelativepositionofA,BD andP. Hencefind: (i) ThedistancefromAandD (ii) ThebearingofthesubmarinefromtheshipwhentheshipwassettingofffromB (iii) ThebaringoftheislandPfromD (iv) Thedistancethesubmarinehadtocovertoreachtheisland 19. FourtownsR,T,KandGaresuchthatTis84kmdirectlytothenorthRandKisonabearingof295 [0] fromRatadistanceof60km.Gisonabearingof340 [0] fromKandadistanceof30km.Usingascale of1cmtorepresent10km,makeanacutescaledrawingtoshowtherelativepositionsofthetowns. Find Nove ~~ls,UpdatedC~~ ~~PA~~, ~~CIFA~~, ~~KASNEB~~, ~~ICT~~, ~~Colle~~ ge,H ighScho ol ~~&PrimaryNot~~ ~~es07~~ 14497530 ----- isabokemicah@gma ~~il~~ .c ~~om~~ (a)ThedistanceandbearingofTfromK (b)ThebearingofRfromG 20. Inthefigurebelow,ABCDEisaregularpentagonandMisthemidpointofAB.DMintersectsEBatN. (T7) Findthesizeof (a)  BAE (b)  BED (c)  BNM 21. Usearulerandcompassesinthisquestion.DrawaparallelogramABCDinwhichAB=8cm,BC=6cm andBAD=75.Byconstruction,determinetheperpendiculardistancebetweenABandCD. 22. Theinterioranglesofthehexagonare2x [0],x [0],x+40 [0],110 [0],130 [0] and160 [0] .Findthevalueofthe smallestangle. 23. Thesizeofaninteriorangleofaregularpolygonis156 [0] .Findthenumberofsidesofthepolygon. CHAPTERTWENTYTHREE ~~Novels,Up~~ d atedCPA,CIFA,K ~~ASNEB,ICT,Colle~~ ge,Hig ~~hSch~~ ~~ool&Pri~~ m ar yN ~~otes0~~ ~~7144~~ ~~97530~~ ----- ~~isabok~~ e ~~m~~ ~~icah@gmail.com~~ SpecificObjectives Bytheendofthetopicthelearnershouldbeableto: a.)Identifyandsketchcommonsolids b.)Identifyvertices,edgesandfacesofcommonsolids c.)Statethegeometricpropertiesofcommonsolids d.)Drawnetsofsolidsaccurately e.)Makemodelsofsolidsfromnets f.)Calculatesurfaceareaofsolidsfromnets Content a.)Commonsolids(cubes,cuboids,pyramids,prisms,cones,spheres,cylindersetc) b.)Vertices,edgesandfacesofcommonsolids. c.)Geometricpropertiesofcommonsolids. d.)Netsofsolids. e.)Modelsofsolidsfromnets. f.) Surfaceareaofsolidsfromnets(includecubes,cuboids,cones,pyramidsprisms) ##### Introduction Asolidisanobjectwhichoccupiesspaceandhasadefiniteorfixedshape.Solidsareeitherregularorirregular. Nov els,UpdatedCPA,CIFA ~~,KAS~~ ~~NEB~~, ~~ICT~~, ~~C~~ o ~~ll~~ e g e, ~~Hi~~ g ~~h~~ ~~S~~ c ~~hool&PrimaryN~~ ~~otes0714497530~~ ----- isabokemicah@gmail.com ##### Note:  Intersectionsoffacesarecallededges.  Thepointwherethreeormore edges meetiscalledavertex. ##### Sketchingsolids Todrawareasonablesketchofasolidonaplanpaper,thefollowingideasarehepful: ##### Useofisometricprojections Inthismethodthefollowingpointsshouldbeobtained:  Eachedgeshouldbedrawntothecorrectlength.  Allrectangularfacesmustbedrawnasparallelograms.  Horizontalandverticaledgesmustbedrawnaccuratelytoscale. Novels, U p datedCPA, CIFA, KASNEB,ICT,College,HighSchool&PrimaryNotes0714497530 ----- ~~i~~ sa ~~b~~ o ~~k~~ em ~~i~~ ca ~~h@~~ gma ~~il~~ .com  Thebaseedgesaredrawnatanangle30 0 withthehorizontallines.  Parallellinesaredrawnparallel. ##### Examples a.) Cubenet b.) Cuboidnet |Col1|Col2|Col3|Col4|Col5|Col6|Col7|Col8|Col9|Col10|Col11|Col12|Col13|Col14|Col15|Col16|Col17|Col18|Col19|Col20|Col21|Col22|Col23|Col24|Col25|Col26| |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| ||||||||||||||||||||||||||| ----- isabo ~~kemicah~~ ~~@gmail~~ . ~~com~~ ##### c.) Triangularprismnet d.) Pentagonalprism ~~Novel~~ s, ~~U~~ p ~~d~~ a ~~t~~ e ~~d~~ CPA, ~~CIFA,KA~~ ~~SNEB~~ ~~,ICT,College,HighSchool&PrimaryNotes07~~ ~~14497530~~ ----- ~~isabokemicah@gmail~~ . ~~com~~ ##### e.) Squarebasepyramid f.) Cylindernet g.) Conenet ~~Novels~~, ~~UpdatedCPA~~, ~~CIFA~~, ~~KASNEB~~, ~~IC~~ T,C ollege, ~~HighSchool&PrimaryN~~ ~~otes0714497530~~ ----- ~~isabokemicah@gm~~ ~~ail.com~~ ##### Conenet h.) Hexagonalprism i.) Tetrahedron ~~Novels,Update~~ ~~dCPA,CIFA,KASNEB,ICT,College,HighSchool&PrimaryN~~ ~~ote~~ s07 14497530 ----- ~~isabokemicah@gm~~ ~~ail.com~~ ##### j.) Octahedron k.) Dodecagon Example AnantmovedfromYtoXthemidpointofRSthroughPintherightpyramidbelow Y Q R Drawthenetofthepyramidsho ~~wing~~ thepathoftheanthencefindthedistanceitmoved. ~~Novels~~, ~~UpdatedC~~ ~~PA,CIFA~~, ~~KA~~ ~~SNEB~~, ~~ICT~~, ~~Co~~ ~~llege~~, ~~Hi~~ gh ~~School~~ ~~&~~ ~~PrimaryNote~~ ~~s0714497530~~ ----- ##### Solution Distance=15+ =27.649cm ##### Example Drawthenetofthesolidbelow. ~~i~~ sa ~~b~~ o ~~k~~ em ~~icah@gm~~ ~~ail.com~~ H 1c I A E ##### Solution Novels,UpdatedCPA,CIFA,KASNEB,ICT,Colle g e,Hi g hSchool&Primar y Notes0714497530 |Col1|Col2| |---|---| ||| ----- ~~isab~~ ~~okemicah@gm~~ ~~ail.com~~ Endofto p ic ### PastKCSEQuestionsonthetopic 1.Thefigurebelowshowsanetofaprismwhosecrosssectionisanequilateraltriangle. ----- ~~i~~ sabokemica ~~h@gma~~ ~~il.com~~ a) Sketchtheprism b)Statethenumberofplanesofsymmetryoftheprism. 2.Thefigurebelowrepresentsasquarebasedsolidwithapathmarkedonit. Sketchandlabelthenetofthesolid. 3.Thefigurebelowrepresentsbelowrepresentsaprismoflength7cm AB=AE=CD=2cmandBCED=1cm ~~Novels~~, ~~UpdatedCPA~~, ~~CIFA~~, ~~KASNEB~~, ~~ICT~~,Co ~~ll~~ e ~~ge~~, ~~HighSchool&Prim~~ aryNotes ~~0714497530~~ ----- ~~isabokemicah@gmail~~ . ~~com~~ Drawthenetoftheprism (3marks) 4.Thediagrambelowrepresentsarightpyramidonasquarebaseofside3cm.Theslantofthepyramidis4 cm. (a)Drawanetofthepyramid (2marks) (b) Onthenetdrawn,measuretheheightofatriangularfacefromthetopof thePyramid (1mark) 5. (a) Drawaregularpentagonofside4cm (1mark) (b) Onthediagramdrawn,constructacirclewhichtouchesallthesidesofthepentagon (2marks) 6.Thefigurebelowshowsasolidregulartetrapackofsides6cm (a) Drawanetofthesolid (b) Findthesurfaceareaofthesolid 7.Thefigurebelowshowsasolidmadebypastingtwoequalregulartetrahedral ~~Novels~~, ~~U~~ pdatedCPA,CIFA,K ~~ASNEB,ICT,College,HighSchool&PrimaryNotes0714497530~~ ----- ~~i~~ sa ~~b~~ o ~~k~~ em ~~i~~ ca ~~h@~~ gma ~~il~~ . ~~com~~ (a) Drawanetofthesolid (b) Ifeachfaceisanequilateraltriangleofside5cm,findthesurfaceareaofthesolid. 8(a)Sketchthenetoftheprismshownbelow (b)Findthesurfaceareaofthesolid ~~Nov~~ els,Update ~~dCPA,CIFA,KA~~ SN EB,ICT,College,Hig ~~h~~ Sc ~~h~~ oo ~~l&Prim~~ ~~aryNo~~ tes 0714497530 -----