Decimal Fractions. Decimal Fractions. Curriculum Ready.

Decimal Fractions

Curriculum Ready

Decimal

Fractions

Tomakedark-greencolouredpaint,youcanmixyellowandbluetogether,usingexactly0.5(half)asmuch

yellowasyoudoblue.

Howmuchdark-greenpaintwillyoumakeifyouuseallofthe12.5 mLofbluepaintyouhave?

Work through the book for a great way to do this

Give this a go! Give this a go! Decimalfractionsallowustobemoreaccuratewithourcalculationsandmeasurements. Becausemostofushavetenfingers,itisthoughtthatthisisthereasonthedecimalfractionsystem isbasedaroundthenumber10! Sowecanthinkofdecimalfractionsasbeingfractionswithpowersof10 inthedenominator. WriteinthisspaceEVERYTHINGyoualreadyknowaboutdecimalfractions. Q

Decimal Fractions

How

does it work?

1st_{decimalplace: 10} 10 1 # ' = =onetenth 2nd_{decimalplace: 10} 100 1 # ' = =onehundredth 3rd_{decimalplace: 10} 1000 1 # ' = =onethousandth 4th_{decimalplace: 10} 10 000 1 # ' = =onetenthousandth etc... Decimalpoint Add‘th’tothe namefordecimal placevalues or or or or ... ... ... ... 2 10 2 10 1 7 100 7 100 1 0 0 1000 0 1000 1 3 3 10 000 3 10 000 1 2 7 # # # # ' ' ' ' ` ` ` ` j j j j ... ... ... 4 100 6 10 5 1 4 6 5 # # # 10 2 100 7 1000 0 10 000 3 = = = = 400 60 5 = = = Multiplybymultiplesof 10 Dividebymultiplesof 10 1st_decimalplace 2nd_decimalplace 3rd_decimalplace 4thdecimalplace = 2tenths = 7hundredths = 0thousandths = 3tenthousandths = 4hundred = 6tens(orsixty) = 5ones(orfive) # 10 000 # 1000 # 100 # 10 # 1 ' 10 ' 100 ' 1000 ' 10 000 ' 100 000 ' 1 000 000 ' 10 000 000 • Tens ofthousands TenthsHundr edths

ThousandthsTenthousandthsHundr

edthousandths Million ths TenMillion ths ThousandsHundr eds Tens Ones

Place value of decimal fractions

Decimalfractionsrepresentpartsofawholenumberorobject.

W H O L E

D E C

I

M

A

L

Writetheplacevalueofeachdigitinthenumber465.2703

4 6 5 . 2 7 0 3

Expanded forms Place values

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Decimal Fractions

Place value of decimal fractions

Writethedecimalfractionthatrepresentsthese: b c a e f d Writethefractionthatrepresentsthese: Writetheplacevalueofthedigitwritteninsquarebracketsforeachofthesedecimalfractions:

2hundredths 9tenths 1tenthousandth

3thousandths 6hundredthousandths

b c

a

e f

d

3tenths 7thousandths 1hundredth

9tenthousandths 51hundredths 11tenthousandths

b c a e f d Circlethedigitfoundintheplacevaluegiveninsquarebrackets: . 3 1 325 6 @ 6 @1 0 231. 6 @4 1 .5 046 . 5 0 05043 6 @ 6 @6 0 79264. 6 @0 8 56309. b c a e f d [tenths]

8 . 1 7 1 6 1 5

[thousandths]

4 . 3 2 1 2 3 0

[millionths] [hundredths] [tenthousandths] [hundredthousandths]

1 0 0 . 1 0 0 1 0 0 1

Alwaysputazeroinfront (calledaleading zero)when therearenowholenumbers 0.02 8millionths 2 3 4 1

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Decimal Fractions

Writethedecimalfraction23.401inexpandedform

Place value of decimal fractions

Eachdigitismultipliedbytheplacevalueandthenaddedtogetherwhenwritinganumberinexpandedform.

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PLACE VALUE OF DECI

MAL FR ACT ION S P LACE VA LUE OF DE CIM AL F RA CTIO NS a b c d e f . . . . . . 4 19 29 281 40 2685 3 74932 0 2306 0 0085 = = = = = = 5 . 23 401 2 10 3 1 4 10 1 ₀ 100 1 ₁ 1000 1 2 10 3 1 4 10 1 ₁ 1000 1 # # # # # # # # # = + + + + = + + + Multiplyeachdigitbyitsplacevalue Zerodigitscanberemovedtosimplify 6 Writethesedecimalfractionsinexpandedform: Simplifythesenumberswritteninexpandedform:

Psst:Remembertoincludealeading zerofortheseones.

1 1 4 10 1 ₆ 100 1 4 10 9 1 0 10 1 ₇ 100 1 5 100 2 10 0 1 2 10 1 ₁ 100 1 ₈ 1000 1 6 1 8 10 1 ₅ 100 1 ₀ 1000 1 ₂ 10 000 1 ₉ 100 000 1 # # # # # # # # # # # # # # # # # # # + + = + + + = + + + + + = + + + + + = e a b c d f g 2 10 1 ₀ 100 1 ₃ 1000 1 6 100 1 ₇ 1000 1 ₀ 10 000 1 ₁ 100 000 1 3 10 1 ₄ 100 1 ₁ 1000 1 ₀ 10 000 1 ₈ 100 000 1 # # # # # # # # # # # # + + = + + + = + + + + =

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Decimal Fractions

Approximations through rounding numbers

Lookatthesetwostatementsmadeaboutateamofsnowboarders:

• Theyhaveattempted4937trickssincestarting= Accurate statement

• Theyhaveattemptednearly5000trickssincestarting= Rounded off approximation

Closertolowervalue,soround down

Leavetheplacevalueunchanged

Closertohighervalue,soround up

Add1totheplacevalue

Roundthesenumbers

Thedigit‘4’isinthehundredsposition

Thenextdigitisa6,soround upbyadding1to4

Changetheothersmallerplacevaluedigitsto0’s

Thedigit‘3’isinthefirstdecimalplace

Thenextdigitisa1,soround down

Writedecimalfractionwithonedecimalplaceonly Thedigit‘1’isinthefourthdecimalplace Thenextdigitisa9,soroundupbyadding1to1 Writedecimalfractionwithfourdecimalplacesonly Herearesomeexamplestoseehowweroundoffnumbers. (i) 2462tothenearesthundred (ii)0.3145toonedecimalplace(ortothenearesttenth) (iii)26.35819 tofourdecimalplaces(ortothenearesttenthousandth) 2462 2500 ` . roundedtothenearesthundred . . 0 3145 0 3 ` . roundedtoonedecimalplace .3 . 26 5819 26 3582 ` . roundedtofourdecimalplaces 2 6 . 3 5 8 1 9 2 6 . 3 5 8 1 9 2 6 . 3 5 8 2 0 . 3 1 4 5 0 . 3 1 4 5 0 . 3 2 4 6 2 2 4 6 2 2 5 0 0 0 1 2 3 4 5 6 7 8 9 Nextdigit Roundingoffvaluesisusedwhenagreatdealofaccuracyisnotneeded. Thenextdigitfollowingtheplacevaluewhereanumberisbeingroundedofftoistheimportantpart.

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Decimal Fractions

Approximations through rounding numbers

1 Roundthesewholenumberstotheplacevaluegiveninsquarebrackets. Roundthesedecimalfractionstothedecimalplacesgiveninthesquarebrackets. 2 Approximatethefollowingdistancemeasurements: 3 Agroupofpeopleforman8.82 mlonglinewhentheystandtogether.

(i) Howlongisthislinetothenearest10cm(i.e.1decimalplace)?

(ii)Whatistheapproximatelengthofthislinetothenearest10metres? Underamicroscopethelengthofadustmitewas0.000194 m (i) Approximatethelengthofthisdustmitetothenearesttenthousandth ofametre. (ii)Approximatethelengthofthisdustmitetothenearesthundredthofametre. IfLichenCityis3 458 532 mawayfromMossCity: (i) Whatisthisdistanceapproximatedtothenearestkm? (i.e.nearestthousand) (ii)Whatistheapproximatedistancebetweenthecitiestothenearest100 km?

(iii)Arethedigits2, 3oreven5importanttoincludewhendescribingthetotal

distancebetweenthetwocities?Brieflyexplainherewhy/whynot. 4 5 44 3 APPROXIMAT_IO N THR OUG H R OUN DIN G NU MBER S

.

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a a [nearestten] [nearesttenth] b b c c [nearesthundred] [nearesthundredth] [nearestthousand] [nearestthousandth] (i) (i) (ii) (ii) (iii) (iii) 536 . 8514 . 93025 . (i) (i) (ii) (ii) (iii) (iii) 14 302 . 4764 . 80 048 . (i) (i) (ii) (ii) (iii) (iii) 98 542 . 18 401 . 120 510 . . 0 73 . . 3 47 . . 11 85 . . 2 406 . . 0 007 . . 1 003 . . 10 4762 . . 0 3856 . . 0 048640 . . . . . . . a b c

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Decimal Fractions

Approximations through rounding numbers

4 Roundoffthesenumbersaccordingtothesquarebrackets.

Roundingupcanaffectmorethanonedigitwhenthenumber9isinvolved.

Round0.95toonedecimalplace Thedigit‘9’isinthetenthsposition

Thenextdigitisa5,soround upbyadding1to9

Changetheothersmallerplacevaluedigitsto0s

. .

0 95 1 0

` . roundedtoonedecimalplace 9roundsupto10,so

the 9becomes0and1

isaddedtothedigitinfront. a _{[onedecimalplace]} . 1 98 . d [nearestones] . 79 9 . g _{[nearestthousand]} 49798 . b [nearestten] 398 . e _{[threedecimalplaces]} . 0 1398 . h [nearestones] . 199 9 . c _{[twodecimalplaces]} . 11 899 . f _{[threedecimalplaces]} . 2 1995 . i [fourdecimalplaces] . 9 89999 . 5 Approximatethesevalues: a _{Acallcentrereceivesanaverageof2495.9callseachdayduringonemonth.} (i) Approximatethenumberofcallsreceivedtothenearesthundreds. (ii) Approximatelyhowmanythousandsofcallsdidtheyreceive? (iii) Estimatethenumberofcallsreceiveddailythroughoutthemonth. b Aswimmingpoolhadaslowleak,causingittoempty_9599.5896Linoneweek.

(i) Howmuchwaterwaslosttothenearest10litres?

(ii) HowmuchwaterwaslosttothenearestmLif1mL =

1000 1 _L?

(iii) Isthedigit6importantwhenapproximatingtothenearestwholelitre?

Brieflyexplainherewhy/whynot. . . . . . 0 . 9 5 0 . 9 5 1 . 0

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Decimal Fractions

Decimal fractions on the number line

Thesmallestplacevalueinadecimalfractionisusedtopositionpointsaccuratelyonanumberline. • Decimalfractionsarebasedonthenumber10,sotherearealwaystendivisionsbetweenvalues Eg:Hereisthevalue3.6onanumberline: Soitseightthousandthsof thewayfrom1.240to1.250 Sixtenthsof thewayfrom 3.0to4.0 Herearesomemoreexamplesinvolvingnumberlines: 1.240 1.248 1.250 8 6

• Themajorintervalsonthenumberlinearemarkedaccordingtothesecond lastdecimalplacevalue

3.0 3.6 4.0 0.1 0.2 4 a) 2.14 2.15 a) (i) Whatvaluedotheplottedpointsrepresentonthenumberlinesbelow? (ii) Roundthevalueoftheplottedpointsbelowtothenearesthundredth.

Pointisfourstepsfrom0.1towards0.2,sotheplottedpointis:0.14

Pointisninestepsfrom10.06towards10.07,sotheplottedpointis:10.069

Pointisthreestepsfrom2.14towards2.15,sotheplottedpointis2.143

` thevalueoftheplottedpointtothenearesthundredthis:2.14

10.06 10.07 9

b)

Pointisfivestepsfrom8.79towards8.80,sotheplottedpointis8.795

` thevalueoftheplottedpointtothenearesthundredthis:8.80

8.79 8.80 b)

3

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Decimal Fractions

Decimal fractions on the number line

1 Displaythesedecimalfractionsonthenumberlinesbelow: 2 Labelthesenumberlinesandthendisplaythegivendecimalfractiononthem: 3 Roundthevalueoftheplottedpointsbelowtothenearestplacevaluegiveninsquarebrackets. ` thevalue. ` thevalue. ` thevalue. ` thevalue. ` thevalue. ` thevalue. ` thevalue. ` thevalue. DECI MAL FRACTI_ON S ON T HE NUM BER LI NE

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4 0.2 0.3 0.8 0.9 1.994 1.995 2.902 2.903 a c e g a c e [tenth] [tenth] [thousandth] [thousandth] [hundredth] [hundredth] [thousandth] [thousandth] 1.6 0.94 2.053 b d f 4.2 7.07 9.538 1.03 1.04 0.08 0.09 8.103 8.104 0.989 0.990 b d f h 0.0 1.0 0.1 0.2 2.3 2.4 a c e 0.7 0.13 2.34 2.0 3.0 9.1 9.2 5.21 5.22 b d f 2.1 9.15 5.212

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Decimal Fractions

Multiplying and dividing by powers of ten

Movethedecimalpointdependingonthenumberofzeros (i) 5 1000# (ii) 8'100 (iii)1 25893. #10 000 (iv) 24 905. '10 000 0 (v) 260 15. 1 1000 # Calculatethesemultiplicationanddivisionquestionsinvolvingpowersof10: =decimalpointmovesright, =decimalpointmovesleft . . . 8 100 8 0 100 8 0 ' = ' = 0.08 = . . . . 1 25893 10 000 1 2 5 8 9 3 12 5 8 9 3 # = = . . . . 24 905 100 000 2 4 905 0 00024905 ' = = . . . . 260 15 1000 1 _{260 15 1000} 2 6 0 15 # _{= '} = Thewholenumberindecimalfractionform

'100has2zeros,somovedecimalpoint2spacesleft

Filltheemptybounceswith0sandputazeroinfront Movedecimalpoint4spacesright Noemptybouncestofill,sothisistheanswer Movedecimalpoint5spacesleft Fillemptybounceswith0sandputazeroinfront 1000 1 # isthesameas_'1000 Movedecimalpoint3spacesleft Placealeadingzeroinfrontofthedecimalpoint . . . 5 1000 5 0 1000 5 0 # = # = Thewholenumberindecimalfractionform Filltheemptybounceswith0s 0.2 6 015 = Remembertoinclude theleadingzero 5 000 = 1 2 3 2 1 1 2 3 4 5 4 3 2 1 3 2 1 Ifthedecimalpointisontheleftafterdividing,anextra0isplacedinfront. Wecansimplyaddthesame numberofzerostotheend ofthewholenumber

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Decimal Fractions

Multiplying and dividing by powers of ten

1 Calculatethesemultiplications.Remember,multiplymeansmovedecimalpointtotheright:

2 Calculatethesedivisions.Remember,dividemeansmovedecimalpointtotheleft:

3 Calculatethesemixedproblemswritteninexponentform:

Herearesomeofthepowersof10inexponentform.Thepower=thenumberofzeros.

10 10 10 10 000 1 4 = = 10 100 10 100 000 2 5 = = 10 1000 10 1000 000 3 6 = = a a a d d d b b b c c c e e e f f f 8 100# 3.4 10# 29 1000# 12.45 10 000# 0.512 100# 0.0000469 1000 000# 100 2' 4590'1000 0 014. '10 70. 08 '10 000 1367 512. '1000 421 900'100 000 000 31 10# 2 2400'105 0.0027 10# 6 90.008 10# 4 3 45 10. ' 3 2159 951 10' 7

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Decimal Fractions

Multiplying and dividing by powers of ten

4 Forthesecalculations: (i)Showwhereourcharacterneedstospraypaintanewdecimalpoint,and (ii)writedownthetwonumbersthenewdecimalpointisbetweentosolvethepuzzle

2 8 3 0 3 9 2 0

2 3 8 5 7

0 4 7 6 3 8 9 2

3 8 2 9 6 2

1 9 2 3 8 0 7

8 9 2 3 6 7 0 1

2 0 9 1 7 9 8 3

8 3 9 1 7

9 0 2 8 7 3 2 0 1

0 0 8 3 9 0

I 9 and 2 Thisisanothermathematicalnameforadecimalpoint:

0and9 8and9 8and7 9and2 0and7 3and9 8and2 0and8 3and8 6and7

MULTIPLYING AND D IV ID ING BY POW ERS O F TE N

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a b c d e f g h i j 2830.3920 100# 23 857'1000 0.4763892 10# 5 382 961 10 000' 19 238.07 10# 1 8.9236701 10 000# 20 917 983 1000 000 1 # 83 917'105 902 873.021 10 1 2 # 0.08390 10# 3 N A O X T R I D P

I

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Decimal Fractions

(i)0.25 07isjust 7 Writeeachofthesedecimalfractionasanequivalent(equal)fractioninsimplestform Write1.07asafraction: Lastdigitisinhundredthsposition Decimalfractiondigitsinthenumerator 0 . 2 5 100 25 = Equivalent,un-simplifiedfraction DividenumeratoranddenominatorbyHCF Equivalentfractioninsimplestform Equivalent,un-simplifiedmixednumber DividenumeratoranddenominatorbyHCF Equivalentmixednumberinsimplestform Lastdigitisinthousandthsposition 100 25 25 25 ' ' = 4 1 = 2 . 1 0 5 2 1000 105 = 100 105 5 5 2 0'' = 2 200 21 = 1 . 0 7 1 100 7 = 10 3 = Write0.3asafraction: Lastdigitisintenthsposition Decimalfractiondigitsinthenumerator 0 . 3 Decimalfraction Fraction Lastdigitisinhundredthsposition MULTIPLYING AND D IV ID ING BY POW ERS O F TE N

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Terminating decimal fractions to fractions

Thesehavedecimalfractionpartswhichstop(orterminate)ataparticularplacevalue. Theplacevalueofthelast digit on the righthelpsustowriteitasafraction.

Integersinfrontofthedecimalfractionvaluesaresimplywritteninfrontofthefraction.

Alwayssimplifythefractionpartsifpossible.Thesetwoexamplesshowyouhow.

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Decimal Fractions

Terminating decimal fractions to fractions

1 Writeeachofthesedecimalfractionsasequivalentfractions: c d a Writeeachofthesedecimalfractionsasequivalentfractionsandthensimplify: 2 Simplestform . 0 1 = 0 09 =. 0 03 =. b c a _{0.5 = = 0.6 = = 0.02 = =} b 0 7 =. g h e 0.00 =1 f _{0.007 =} 0 013 =. 0 049 =. k l i _{0.129 =} j _{0 081 =. 0 1007 =. 0 0601 =.} e f d _{0.08 = = 0.004 = = 0.005 = =} h i g _{0.12 = = 0.25 = =} k l j _{0.045 = = 0.0028 = = 0.0605 = =} Simplestform Simplestform Simplestform Simplestform Simplestform Simplestform Simplestform Simplestform Simplestform Simplestform Simplestform 0.022 = =

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Decimal Fractions

Terminating decimal fractions to fractions

3 Writeeachofthesedecimalfractionsasequivalentmixednumbers: b c a _{2 3 =. 1 1 =. 3 7 =.0} e f d _{1 3 =.0} 4 001 =. 2.009 = b c a _{2.8 = 1.4 = 4 6 =.0} e f d _{3 5 =.0} 2 75 =. 5.005 = h i g _{1 004 =. 2.025 = 3 144 =.} 4 Writeeachofthesedecimalfractionsasequivalentmixednumbersandthensimplify:

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0

.

5 =

1

2

TERM

INATING DECIMA_L

FR AC TION S T O F RAC TIO NS * Simplestform = Simplestform = = = = Simplestform = Simplestform = Simplestform = Simplestform =

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Decimal Fractions

Fractions to terminating decimal fractions

Writetheseasanequivalentdecimalfraction (i) 12 3 Sometimesitiseasiertofirstsimplifythefractionbeforechangingtoadecimalfraction. Wherepossible,justwriteasanequivalentfractionwithapowerof10inthedenominatorfirst. . 12 3 4 1 4 1 4 1 100 25 0 25 3 3 25 25 # # ` ' ' <sub>=</sub> = = = . 2 15 3 <sub>2</sub> 5 1 2 5 1 2 <sub>2</sub> 10 2 2 2 3 3 2 # # ' ' <sub>=</sub> = = . 5 3 5 3 10 6 0 6 2 2 # # ` = = = numerator denominator

Threetwelfths=onequarter=twentyfivehundredths=zeropointtwofive

Twoandthreefifteenths=twoandonefifth=twoandtwotenths=twopointtwo

Threefifths=sixtenths=zeropointsix

Multiplynumeratoranddenominatorbythesamevalue Equivalentfractionwithapowerof10inthedenominator (ii) 2 15 3 Simplifyfraction Equivalentfractionwithapowerof10inthedenominator Simplifyfractionpart Equivalentfractionwithapowerof10inthedenominator

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Decimal Fractions

includea leadingzero

Fractions to terminating decimal fractions

1 Writeeachofthesefractionsasequivalentdecimalfractions.

2 Writeeachoftheseasequivalentfractionswithapowerof₁₀inthedenominator.

3 (i) Writeeachoftheseasequivalentfractionswithapowerof10inthedenominator.

(ii) Changetoequivalentdecimalfractions. c d a 1 2 = b 52 = 43 = 209 = g h e 258 = 200 11 ₌ 125 2 ₌ f 250 3 ₌ c d a 109 = 10011 = 1000 7 ₌ b 100 3 ₌ i ₁ j k 5 4 = 3 251 = 6207 = h i g 2 259 = = 1 200 1 ₌ = 8 507 = = b c a 5 1 = = e f d 4 1 = = 25 11 = = 254 = = 200 1 ₌ = 125 6 ₌ =

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Decimal Fractions

Fractions to terminating decimal fractions

4 Changeeachofthesefractionstoequivalentdecimalfractionsafterfirstsimplifying.

Showallyourworking.

a 20 12 c 24 18 e 75 9 g ₁ 600 36 b 25 20 d 40 22 f 3 40 12 h 2 150 12 FRACTIONS TO TE RM IN AT ING DE CIM AL FR AC TI ONS

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1

2

=

5

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Decimal Fractions

Fractions to terminating decimal fractions

Whenchangingthedenominatortoapowerof10isnoteasy,youcanwritethenumeratorasadecimal fractionandthendivideitbythedenominator. 5 Completethesedivisionstofindtheequivalentdecimalfraction: a . b 5 2 _{= 2 000'5} d e f Writenumeratorasadecimalfractionanddividebythedenominator Completedivision,keepingthedecimalpointinthesameplace Fiveeighths=zeropointsixtwofive Ifyouneedmore decimalplace0s,you canaddtheminlater! .000 4 1 _{= 1 '4 c .000} 8 3 _{= 3 '8} .000 5 5 8 _{= 8 ' 1.000} 8 11 _{= 1 '8 .000} 4 27 _{= 27 '4}

.

5 2

0 0 0

=

=

g

.

5 8

0 0 0

=

=

g

=

8 1 1

.

0 0 0

=

g

=

4 2 7

.

0 0 0

=

g

.

4 1

0 0 0

=

=

g

=

8 3

.

0 0 0

=

g

. . . 8 5 _{5 000 8} 8 5 0 0 0 0 6 2 5 ' ` = = =

g

. . 0 6 2 5 8 5 0 0 05 2 4 =

g

Writethisfractionasanequivalentdecimalfraction

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Decimal Fractions

Fractions to terminating decimal fractions

6 Simplifythesefractionsandthenwriteasanequivalentdecimalfractionusingthedivisionmethod.

Showallyourworking.

a b c 15 12 12 9 56 49 _d e f 24 81 8 18 16 26

Decimal Fractions

(i) 24 105. +11 06. +6 5902. • Add2.45to6.31(i.e.2.45+ 6.31) Decimalpointslinedupvertically Decimalpointslinedupvertically Roundingdecimalfractionsbeforeaddingissometimesusedtoquicklyapproximatethesizeof theanswer. 24.105 11.06 6.5902 41.7552 ` + + =

• Subtract5.18from11.89(i.e.11.89-5.18)

(ii)Roundeachvalueinquestion(i)tothenearestwholenumberbeforeadding. (iii)80 09. -72 6081. . . . 24 105 11 06 6 5902 24 11 7 42 ` . . + + + +

Note:Roundingvaluesbeforeadding/subtractingisnotasaccurateasroundingafteradding/subtracting. Anyplacevaluespaces

aretreatedas0s

Fillplacevaluespaces inthetopnumberwith

a‘0’whensubtracting `80.09-72.6081 = 7.4819

Where

does it work

?

2 4 . 1 0 5 + 1 1 . 0 6 1 6 . 51 9 0 2 4 1 . 7 5 5 2 Decimalpointslinedupvertically Addmatchingplacevaluestogether Valuesroundedtonearestones Approximatevalueforaddition Decimalpointslinedupvertically Subtractmatchingplacevalues 2 . 4 5 + 6 . 3 1 8 . 7 6 1 1 . 8 9 5 . 1 8 6 . 7 1 Addmatchingplacevaluestogether Subtractmatchingplacevalues

Adding and subtracting decimal fractions

Justaddorsubtractthedigitsinthesameplacevalue. Todothis,lineupthedecimalpointsandmatchingplacevaluesverticallyfirst. Calculateeachofthesefurtheradditionsandsubtractions 8 10 . 10 9 10 10 -71 21 . 6 01 81 1 7 . 4 8 1 9

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Decimal Fractions

Adding and subtracting decimal fractions

1 Completetheseadditionsandsubtractions: 2 Calculatetheseadditionsandsubtractions,showingallworking: c d a b g h e f Add8.75to1.24 a b _{Subtract3.15from4.79} Add0.936to0.865 c Subtract0.9356from8.6012 e Add2.19, 5.6and0.13 d Add10.206, 4.64and8.0159 f ADDI NG AND SUBTRA CTI_NG DE CIM AL F RAC TIO NS + +

-.

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0 . 1 4 + 0 . 7 3 0 . 9 9 0 . 2 6 1 . 6 8 + 5 . 3 0 0 . 2 4 6 + 0 . 8 3 2 5 . 2 4 0 . 8 3 5 . 0 7 4 1 . 0 6 4 1 2 . 1 9 4 + 9 . 0 5 7 2 4 . 1 5 8 -1 3 . 6 9 4

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Decimal Fractions

Adding and subtracting decimal fractions

3 a Approximatethesecalculationsbyroundingeachvaluetothenearestwholenumberfirst.

b

c 4

a b

Calculateparts(v)and(vi)again,thistimeroundingafteraddingthenumberstogetamore accurateapproximatevalue. (i) 8 34. +1 61. +0 54. (ii)2 71. +3 80. +1 92. Calculatethesesubtractions,showingallyourworking: 7.8-2.56 13 09. -8 4621. 0 52. -0 12532. 5.7+6.2 . . + 8.3-1.9 . . -8.34 + 1.61 + 0.54 . . 2.71 + 3.80 + 1.92 . . + + + + 11.3-0.2 . . -0.9+9.4 . . + (i) (iii) (v) (vi) (iv) (ii)

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Decimal Fractions

Multiplying with decimal fractions

Howdoesthisworkwhenmultiplyingwithdecimalfractions?Excellentquestion!Verygladyouasked! Justwritethetermsaswholenumbersandmultiply.Putthedecimalpointbackinwhenfinished. Thenumberofdecimalplacesintheanswer=thenumberofdecimalplacesinthequestion! 1 Calculate 2 Calculate 4 1.2# 0.02 1.45# 4 12# =4 8 . . 4 8 4 1 2# 4 8 ` = 1 Multiplybothtermsaswholenumbers 1decimalplaceinquestion= 1decimalplaceinanswer Multiplybothtermsaswholenumbers 4decimalplacesinquestion= 4decimalplacesinanswer 2 145# =2 9 0 0.02 1.45 0 . 0 2 9 0 ` # = 2 9 0 4 3 2 1 Let’sdothesecondoneagainbutthistimechangethedecimalfractionstoequivalentfractionsfirst Changingthedecimalfractionstofractions Multiplynumeratorsanddenominatorstogether Numberofzerosindenominator=totalofdecimalplaces inquestion Dividingby10 000movesdecimalpointfourplacestotheleft `4decimalplacesinquestion= 4decimalplacesinanswer Trythismethodforyourselfonthefirstexampleabove,rememberingthat4 1 4 = asafraction. . . . . 0 02 1 45 100 2 100 145 100 100 2 145 10 000 290 290 10 000 0 2 9 0 0 0290 # # # # ' = = = = = = 4 3 2 1

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Decimal Fractions

Multiplying with decimal fractions

1 Calculatethesewholenumberanddecimalfractionmultiplications,showingallyouworking: a _{0.8 2}# b 5 1 5# . c 0.14 6# d _{0.62 4}# e 3#0 032. f 1.134 2# 2 Calculatethesedecimalfractionmultiplications,showingallyourworking: a 3.8#0.2 b 1 09 0 08. # . c 2 7 2 5. # . d 7 1 1. # .4 e 3.21 2 1# . f 17 2 9 3. # . MULT

IPLYING WITH DECIMAL FR ACT ION_S MU LTI PLYIN G W ITH DE CIM AL FR AC TI ON S

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Decimal Fractions

Calculate1 26. '0 8. • Calculate4 28. '4 Divisoralreadyawholenumbersonochangeneeded • Calculate0 0456. '0 006. 0.0456 0.006 7 6. ` ' = . . . . 0 0456 0 006 0 0 4 5 6 0 0 0 6 4 5 6 6 ' ' ' = =

dividend'divisor=quotient

. 2 . 4 4 8 1 0 7 2

g

4.28 4 1.07 ` ' = Movebothdecimalpointsrightuntildivisorisawholenumber 4 . . 6 5 6 0 7 6 4 3

g

Dropoffany0satthefrontoftheanswer Quotient2Dividend ifdivisor1 1 Here’sanotherexampleshowinghowtotreatremainders . . . 1 26 0 8 1 575 ` ' = 8 . . 1 2 6 0 0 0 1 5 7 5 1 4 6 4 =

g

Movebothdecimalpointsrightuntildivisorisawholenumber Add0sontheendofthedividendforeachnewremainder Dropoffany0satthefront

Dividing with decimal fractions

Oppositetomultiplying,wemovethedecimalpointbeforedividingifneeded. Tofindthequotientinvolvingdecimalfractions,thequestionmustbechangedsothedivisorisawhole number. 1.26 0.8 1.2 6 0.8 . 12 6 8 ' ' ' = =

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Decimal Fractions

Dividing with decimal fractions

1 Calculatethesedecimalfractionandwholenumberdivisions: a b c 16 2. '9 d 0.63'3 e 0 489. '5 f 10 976. '7 3.6 4 ` ' = `17.5'5= `16.2'9= 0.63 3 ` ' = `0.489'5= `10.976'7= 2 Calculatethesedecimalfractiondivisions,showingallyourworking: a _{5 2. '0.4} b 9 6. '0 6. c 0 56. '0 8. d _{1 58. '0 4.} e 0.8125'0 05. f 5 3682. '0 006. 5.2 0.4 ` ' = `9.6'0.6= `0.56'0.8= 1.58 0.4 ` ' `0.8125'0.05 `5.3682'0.006 DIVIDING WIT_{H D} ECI MAL FR ACT IONS D IVI DIN G WI TH DE CI MA L FR ACTI ONS

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÷

g

g

g

g

g

g

g

g

g

g

g

g

= = = 3.6'4 17.5'5

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Decimal Fractions

Recurring decimal fractions

Identifythestartandendoftherepeatingpattern Dotabovestartandendoftherepeatingpattern Identifythestartandendoftherepeatingpattern Dotabovestartandendoftherepeatingpattern Identifythestartandendoftherepeatingpattern Dotabovestartandendoftherepeatingpattern Write1asadecimalfractionwithafew0s Repeatsthesameremainderwhendividing Recurringdecimalfractioninsimplestnotation Ifthedecimalpartshavearepeatingnumberpattern,theyarecalledrecurringdecimalfractions. Non-terminatingdecimalfractionshavedecimalpartsthatdonotstop.Theykeepgoingonandon. . ... 0 3582942049 . ... 5 212121 Adotabovethestartandenddigitoftherepeatingpatternisusedtoshowitisarecurring decimalfraction. (i) Writetheserecurringdecimalfractionsusingthedotnotation . ... . 10 81818 10 81 = o o Threedotsmeansitkeepsgoing StartEnd . . . ... 1 0000 6 6 1 0 0 0 0 0 1 6 6 6 4 4 4 4 ' =

g

0.1 '0.6 = 1'6 1 6 0.1666 ... 0.16 ` ' = = o Abaroverthewhole patterncanalsobeused insteadofdots Thepattern21keepsrepeatinginthedecimalparts Herearesomeexamplesinvolvingrecurringdecimalfractions a) b) c) 10.81818... 0.2052052... 1.047777... . . 0 205o o=0 205 . ... . 0 2052052 0 205 = o o StartEnd or ... . 1047777 1 047 = o StartandEnd 1.047 = r (ii) Calculate0.1'0.6

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Decimal Fractions

Recurring decimal fractions

1 Whatisthenameofthehorizontallineabovetherepeatednumbersinarecurringdecimalfraction? Highlighttheboxesthatmatchtherecurringdecimalfractionsineachrowwiththecorrectsimplified notationineachcolumntofindtheanswer. Notallofthematchesformpartoftheanswer! 2 Calculatethesedivisionswhichhaverecurringdecimalfractionsasaresult. Writeanswersusingdotnotation. a 1'3 b 4'9 c 5'6 d _1.6'6 e 2 5. '9 f 0 34. '3 1 3 ` ' = ` '4 9 = ` '5 6 = . 1 6 6 ` ' = `2 5. '9 = `0 34. '3 = c z h m n a f b

g

g

g

g

g

g

. 0 14o 0 4.r 41 1.o 0 144. 0.141o o 0 41. o 4 14. 0 401.o o 4 1.o 0 41.o o 4.1414 ... C z F h Nd W c D b A a U n P t L f O m 0.144144 ... Yn A m Rf T t K z Eh Rd Ic U b S a 0.1444 ... L a D b A m Ih M t B f S c A d U z Q n 0.401401 ... Rh Z d A n Ez A c Nt 0 a M b A h Gf 4.111 ... A f T z P c H d T a Yn A t A h C m A b 0.4111... Id Yt A b U n H m Iz Ef S m It T a 0.4141 ... A b L a D t Ef A d Nc L m Ez O d Nh 41.111 ... W c J f B d A a X h M m A b U n A A z 0.444 ... P m Vc Ea F b A n B d T Yf Ec It 0.1411411 ... H t A n A m A m U f A b A h A a D d Rc

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Decimal Fractions

Recurring decimal fractions

3 (i) Completethefollowingdivisionstofivedecimalplaces. (ii) Determinewhethertheanswerisarecurringdecimalfractionornot. a _2'3 b 1'6 c 1'7 d _{1 6. '7} e _{2 9. '3} f 0 33. '0 8. 2 3 ` ' = ` '1 6 = ` '1 7 = . 1 6 7 ` ' = `2 9. '3 = `0.33'0.8 = Recurring

decimalfraction? Recurringdecimalfraction? Recurringdecimalfraction?

Yes No Yes No Yes No

Recurring decimalfraction? Recurring decimalfraction? Recurring decimalfraction?

Yes No Yes No Yes No

g 0.68'0.3 h 0.019'0 06. i 0.00644'0.002 0.68 0.3 ` ' `0.019'0.06 `0.00644'0.002 Recurring decimalfraction? Recurring decimalfraction? Recurring decimalfraction?

Yes No Yes No Yes No RECU

RRING DECIMAL FRACT

IO_NS ... REC URRI NG D ECI MAL FR AC TI ON S...

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g

g

g

g

g

g

g

g

g

= = =

Decimal Fractions

What

else can you do

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Simple recurring decimal fractions into single fractions

Onlyrecurring,non-terminatingdecimalfractionscanbewritteninfractionform. Hereisaquickwayforsimpledecimalfractionswiththepatternstartingrightafterthedecimalpoint. (i) 3 777. ... Threedigitsinrepeatingpattern,sothosethreedigitsover999 Onedigitinrepeatingpattern,sothatdigitover9 Twodigitsinrepeatingpattern,sothosetwodigitsover99 Onedigitinrepeatingpattern,sothatdigitover9 Digitsinfrontofdecimalpointformthewholenumberpart Threedigitsinrepeatingpattern,sothosedigitsover999 Digitsinfrontofdecimalpointformthewholenumber Simplifythefractionpart . ... . 3 7777 3 7 3 9 7 = = o 99 12 33 4 3 3 ' ' = = (ii)16 345345. ... 0.111... 0.1 . ... . 9 1 0 1212 0 12 99 12 = = = = o o o 0.301301... 0.301 999 301 = o o = Herearesomeotherexamplesincludingmixednumbers. . ... . 16 345345 16 345 16 999 345 16 999 345 16 333 115 3 3 ' ' = = = = o o Alwayssimplify fractions Writeeachoftheserecurringdecimalfractionsasmixednumbersinsimplestform

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Decimal Fractions

Simple recurring decimal fractions into single fractions

1 Usetheshortcutmethodtowriteeachoftheserecurringdecimalfractionsasafractioninsimplestform: a _{0 4.}o b 0 8.r c 0 6.o d 0 11.o o e 0 27.o o f 0 57.o o g _0.162 h _5.1485 i 0.4896o o Usetheshortcutmethodtowriteeachoftheserecurringdecimalfractionsasmixednumbersin simplestform. 2 a _{1 5.}o b _{2 7.}r c _{4 3.}r d 3 6.r e 5.12 f 0.117o o

3 (i) Write0 9.oasafractioninsimplestform.

(Ii) Doesanythingunusualseemtobehappeningwithyouranswer?Explain.

SIMP

LE RECURRING DECIMAL F RACT IO NS IN TO S ING LE FRA CT IO NS

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= 9 = 0. ... 0.

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Decimal Fractions

Combining decimal fraction techniques to solve problems

Allthetechniquesinthisbookletcanbeusedtosolveproblems.

(i) Theserainfallmeasurementsweretakenduringthreedaysofrainfromasmallweathergauge:

Addthedecimalfractionvaluestogether

(ii) Theresultsforfiverunnersina100mracewereplottedonthenumberlinebelow.

13.8 36.1 27.6 77.5 + 78 . mm Readoffallthetimes Answerwithastatement a) Whatwasthefastesttimerun(tothenearestthousandthofasecond)? Fastesttime=left-mostplottedpoint= 11.221seconds

b) Whattimedidtworunnersfinishtheracetogetheron?

Tworunnerswiththesametime=twodotsatthesamepoint= 11.223seconds

c) Whatwastheaveragetimeranbyallrunnersinthisrace? Averagetime=Thesumofallthetimesrandividedbythenumberofrunners Theaveragetimeranbyalltherunnersintherace= 11 2242. seconds ( . . . ) . 11 221 11 223 11 223 11 226 11 228 5 56 121 5 ' ' = + + + + = 5 5 6. 1 2 1 0 . 1 1 2 2 4 2 1 1 2 1 =

g

11.22 11.23 seconds Theseexamplesshowdifferentwaysdecimalfractionspopupinevery-daylife Whatwasthetotalrainfallforthethreedays,tothenearestwholemm? `Thetotalrainfalloverthethreedayswasapproximately78 mm 13.8 mm 36.1 mm 27.6 mm Roundtonearestwholemm Answerwithastatement Add,thendivideby5

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Decimal Fractions

Remember me?

Combining decimal fraction techniques to solve problems

1 To make dark-green coloured paint, you can mix yellow and blue together, using exactly 0.5 (half)

as much yellow as you do blue.

a Use multiplication to show how much yellow paint you will need if you use all

of the 12.46 mL of blue paint you have.

b How many millilitres of dark-green paint can you make with 18.45 mL of

yellow paint in the mix? Round your answer to the nearest tenth of a mL.

2 Derek types his essays at an average speed of 93.45 words every minute. How many words does he type in five minutes (to the nearest whole word)?

3 Nine people were trying out for a speed roller skating team around an oval flat track.

The shortest time to complete six full laps of the track for each person were recorded on

the number line below:

a What was the slowest time recorded to 3 decimal places?

b To make the team, a skater had to complete the six laps in less than 126.245 seconds.

How many skaters made it into the team?

c _{How many skaters missed out making the team by less than 0.01 seconds?}

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COMB

INING DECIMAL FRACT IO N TE CH NI QUE S T O S OLV E P RO BL EM S 126.22 126.23 126.24 126.25 126.26 126.27 seconds

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Decimal Fractions

Combining decimal fraction techniques to solve problems

ThewirelesstransmitterinLaura’shousereducesinsignalstrengthby0.024forevery1metreof

distanceshemoveshercomputerawayfromthetransmittersantenna.Hercomputerdisplayssignal strengthusingbarsasshownbelow: 4bars= 0 81. to1.0signalstrength 3bars= 0. 16 to0.8signalstrength 2bars= 0. 14 to0.6signalstrength 1bar= 0. 12 to0.4signalstrength 0bars= 0.2orbelowsignalstrength HowmanybarsofsignalstrengthwouldLaurahaveifusinghercomputer16.25mawayfrom theantenna? Ruofanisputtingtogetheravideoofarecentkaraokepartywithherfriends.Shewillbeusingfiveof herfavouritemusictracksforthevideo. Thelengthoftimeeachofthetracksplayforis:

3.55 min,5.14 min, 2.27 min, 3.18 minand 4.86 min

Ifsheusestheentirelengthofthetrackswitha0.15minbreakineachofthefourgapsbetween

songs,howlongwillhervideorunfor(tothenearestwholeminute)?Showallyourworking.

4

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Decimal Fractions

Combining decimal fraction techniques to solve problems

6 Afterarecentstudybyacitycouncil,theaveragenumberofpeopleineachhouseholdwas

determinedtobe3.4.Explainhowthisispossibleifahouseholdcannotactuallyhave0.4ofaperson?

psst:Checkexampleonpage33toseehowaveragecalculationsaremade. 7 AMexicanchefhassplitupamysteryingredient“Sal-X”intofourexactlyidenticalquantitiesin separatejars.Hethendistributes138 2.omLofthesecretingredient“Sa-Y”amongstthefourjars, producingintotal863 9.omLofthespecialsauce“SalSa-XY”.Howmuchofthemysteryingredient “Sal-X”isthereineachjar(tothenearestmL)?Showallyourworking. 8 Aftercompletelyflatwaterconditions(waveswithaheightof0.0m),theheightofthewavesatalocal

beachstartincreasingby0.2 mevery0 3.ohours.

Ifthewavesneedtobeatleast1.4metreshighbeforesurferswillridethematthisbeach,howlong

willitbeuntilpeoplestartsurfingtheretothenearestminute?Showallyourworking. psst:1.0hours= 60minutes

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Decimal Fractions

Reflection Time

Reflectingontheworkcoveredwithinthisbooklet: Whatusefulskillshaveyougainedbylearningaboutdecimalfractions? 2 Writeaboutoneortwowaysyouthinkyoucouldapplydecimalfractionstoareallifesituation. 3 _{Ifyoudiscoveredorlearntaboutanyshortcutstohelpwithdecimalfractionsorsomeothercoolfacts,} jotthemdownhere: 1

Cheat Sheet

Decimal Fractions

1.240 1.248 1.250 8 =decimalpointmovesright, =decimalpointmovesleft . . 5 1000 5 0 1000 5 0 # ₌ # = 5000 = . . . 8 100 8 0 100 8 0 ' = ' = . 0 08 = 1 2 3 2 1

Closertolowervalue,soround down

Leavetheplacevalueunchanged Closertohighervalue,soround up Add1totheplacevalue 0 1 2 3 4 5 6 7 8 9 Nextdigit • # 10 000 # 1000 # 100 # 10 # 1 Tens ofthousands ThousandsHundr eds Tens Ones

W H O L E

' 10 ' 100 ' 1000 ' 10 000 ' 100 000 ' 1 000 000 ' 10 000 000 TenthsHundr edths

ThousandthsTenthousandthsHundr

edthousandths Million ths TenMillion ths

D E C

I

M

A

L

Here is a summary of the things you need to remember for decimal fractions Place value of decimal fractions

Approximations through rounding numbers

Thenextdigitfollowingtheplacevaluewhereanumberisbeingroundedofftoistheimportantpart.

Decimal fractions on the number line

Thesmallestplacevalueinadecimalfractionisusedtopositionpointsaccuratelyonanumberline. 3.0 3.6 4.0 6 Sixtenthsoftheway from3.0to4.0 Eightthousandthsofthe wayfrom1.240to1.250

Multiplying and dividing by powers of ten

Cheat Sheet

Decimal Fractions

. 1 07 1 100 7 = Eg: . 5 3 5 3 10 6 0 6 2 2 # # ` = = =

Multiply numerator and denominator by the same value Equivalent fraction with a power of 10 in the denominator

Three fifths = six tenths = zero point six

Terminating decimal fractions to fractions

The place value of the last digit on the right helps us to write it as a fraction.

0.3

10 3 =

Write 0.3 as a fraction: Write 1.07 as a fraction:

Last digit is in tenths position Last digit is in hundredths position

Decimal fraction Fraction Decimal fraction Fraction

Fractions to terminating decimal fractions

Where possible, just write as an equivalent fraction with a power of 10 in the denominator first.

When this method is not easy, write the numerator as a decimal fraction and then divide it by the denominator. Adding and subtracting decimal fractions

Line up the decimal points and matching place values vertically before adding or subtracting. Multiplying and dividing decimal fractions

Write the terms as whole numbers and multiply. Put the decimal point back in when finished.

The number of decimal places in the answer = the number of decimal places in the question!

4 1.2# 4.8

: = : 0.02 1.45# = 0.0290

Eg:

Eg:

Recurring decimal fractions

These have decimal parts with a repeating number pattern. Dividing with decimal fractions

The question must be changed so the divisor is a whole number first. dividend ' divisor = quotient

13.5 0.4 135 4

: ' = ' : 89.25'0.003 = 89250'3

Always simplify fractions Simple recurring decimal fractions into single fractions

Only recurring, non-terminating decimal fractions can be written in fraction form. This is the method for simple decimal fractions with the pattern starting right after

the decimal point.

0.111... 0.1 9 1 : = o = 0.1212... 0.12 99 12 33 4 : = o o = =

One digit in repeating pattern, so that digit over 9

Two digits in repeating pattern,

so those two digits over 99

8.301301... 8.301 8 999 301

: = o o =

Three digits in repeating pattern, so

those three digits over 999, Keep

Eg: : 5.212121... = 5.21o o = 5.21 : 0.3698698... = 0.3698o o = 0.3698

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