Decimals. Decimals. Curriculum Ready.

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1 Curriculum Redy

2

3 llow us to be more ccurte with our clcultions nd mesurements. Becuse most of us hve ten finers, it is thouht tht this is the reson the deciml system is bsed round the number 0! So we cn think of decimls s bein frctions with powers of 0 in the denomintor. Write in this spce EVERYTHING you lredy know bout decimls. Give this o! Q To mke drk-reen coloured pint, you cn mix yellow nd blue toether, usin exctly 0.5 (hlf) s much yellow s you do blue. How much drk-reen pint will you mke if you use ll of the 2.5 ml of blue pint you hve? Work throuh the book for ret wy to do this H SERIES TOPIC

4 How does it work? Plce vlue of decimls represent prts of whole number or object. # Tens of thousnds # 000 # 00 Thousnds Hundreds Tens Ones # 0 Deciml point st deciml plce: ' 0 # 0 one tenth 2 nd deciml plce: ' 0 # 00 one hundredth 3 rd deciml plce: ' 0 # 000 one thousndth 4 th deciml plce: ' 0 # one ten thousndth etc... # ' 0 ' 00 Tenths Hundredths Thousndths Ten thousndths W H O L E D E C I M A ' 000 ' ' Hundred thousndths Millionths ' L ' Add th to the nme for deciml plce vlues Ten Millionths Write the plce vlue of ech diit in the number Multiply by multiples of 0 Divide by multiples of 0 Expnded forms Plce vlues # hundred... # 0 0 tens (or sixty) Inteer prts # 5 5 ones (or five) ' 0 or 2 ` # 0 j ' 00 or 7 ` # 00 j ' 000 or 0 ` # 000 j ' or 3 ` # j tenths 7 hundredths 0 thousndths 3 ten thousndths st deciml plce 2 nd deciml plce 3 rd deciml plce 4 th deciml plce 2 H SERIES TOPIC

5 How does it work? Your Turn Plce vlue of decimls Write the deciml tht represents these: d 2 hundredths b 9 tenths c ten thousndth 0.02 Alwys put zero in front (clled ledin zero) when there re no whole numbers 3 thousndths e hundred thousndths f 8 millionths 2 Write the frction tht represents these: 3 tenths b 7 thousndths c hundredth d 9 ten thousndths e 5 hundredths f ten thousndths 3 Write the plce vlue of the diit written in squre brckets for ech of these decimls: 3@ d 5@ Circle the diit found in the plce vlue iven in squre brckets: [tenths] b [thousndths] c [hundred thousndths] d [hundredths] e [ten thousndths] f [millionths] H SERIES TOPIC 3

6 How does it work? Your Turn Plce vlue of decimls Ech diit is multiplied by the plce vlue nd then dded toether when writin number in expnded form. Write the deciml in expnded form # + # + # + # + # # + # + # + # Multiply ech diit by its plce vlue Zero diits cn be removed to simplify 5 Write these decimls in expnded form: 49. b c d e f Simplify these numbers written in expnded form: b 4 # + # + # # + # + # + # /.../20... LACE VALUE OF DECIMALS PPLACE VALUE OF DECIMALS c d # + # + # + # + # + # # + # + # + # + 2 # + 9 # Psst: Remember to include ledin zero for these ones. e f # # # 7 0 # + # + # + # # # # # # 4 H SERIES TOPIC

7 How does it work? Approximtions throuh roundin numbers Look t these two sttements mde bout tem of snowborders: They hve ttempted 4937 tricks since strtin Accurte sttement They hve ttempted nerly 5000 tricks since strtin Rounded off pproximtion Roundin off vlues is used when ret del of ccurcy is not needed. The next diit followin the plce vlue where number is bein rounded off to is the importnt prt. Next diit Closer to lower vlue, so round down Leve the plce vlue unchned Closer to hiher vlue, so round up Add to the plce vlue Here re some exmples to see how we round off numbers. Round these numbers (i) 242 to the nerest hundred The diit 4 is in the hundreds position The next diit is, so round up by ddin to 4 Chne the other smller plce vlue diits to 0 s ` rounded to the nerest hundred (ii) to one deciml plce (or to the nerest tenth) The diit 3 is in the first deciml plce The next diit is, so round down Write deciml with one deciml plce only ` rounded to one deciml plce (iii) to four deciml plces (or to the nerest ten thousndth) The diit is in the fourth deciml plce The next diit is 9, so round up by ddin to Write deciml with four deciml plces only ` rounded to four deciml plces H SERIES TOPIC 5

8 How does it work? Your Turn APPROXIMATION THROUGH ROUNDING NUMBERS. Approximtions throuh roundin numbers Round these whole numbers to the plce vlue iven in squre brckets /.../20... [nerest ten] b [nerest hundred] c [nerest thousnd] (i) 53. (i) (i) (ii) 854. (ii) 474. (ii) (iii) (iii) (iii) Round these decimls to the deciml plces iven in the squre brckets. [nerest tenth] b [nerest hundredth] c [nerest thousndth] (i) (i) (i) (ii) (ii) (ii) (iii). 85. (iii) (iii) Approximte the followin distnce mesurements: A roup of people form n 8.82 m lon line when they stnd toether. (i) How lon is this line to the nerest 0 cm (i.e. deciml plce)?. (ii) Wht is the pproximte lenth of this line to the nerest 0 metres?. b Under microscope the lenth of dust mite ws m (i) Approximte the lenth of this dust mite to the nerest ten thousndth of metre.. (ii) Approximte the lenth of this dust mite to the nerest hundredth of metre.. c If Lichen City is m wy from Moss City: (i) Wht is this distnce pproximted to the nerest km? (i.e. nerest thousnd). (ii) Wht is the pproximte distnce between the cities to the nerest 00 km?. (iii) Are the diits 2, 3 or even 5 importnt to include when describin the totl distnce between the two cities? Briefly explin here why/why not. H SERIES TOPIC

9 How does it work? Your Turn Approximtions throuh roundin numbers Roundin up cn ffect more thn one diit when the number 9 is involved. Round 0.95 to one deciml plce The diit 9 is in the tenths position 9 rounds up to 0, so the 9 becomes 0 nd is dded to the diit in front The next diit is 5, so round up by ddin to 9 Chne the other smller plce vlue diits to 0s ` rounded to one deciml plce 4 Round off these numbers ccordin to the squre brckets. [one deciml plce] 98.. d [nerest ones] [nerest thousnd] b [nerest ten] 398. e [three deciml plces] h [nerest ones] c [two deciml plces] f [three deciml plces] i [four deciml plces] Approximte these vlues: A cll centre receives n vere of clls ech dy durin one month. (i) Approximte the number of clls received to the nerest hundreds. (ii) Approximtely how mny thousnds of clls did they receive? (iii) Estimte the number of clls received dily throuhout the month.... b A swimmin pool hd slow lek, cusin it to empty L in one week. (i) How much wter ws lost to the nerest 0 litres? (ii) How much wter ws lost to the nerest ml if ml 000 L?.. (iii) Is the diit importnt when pproximtin to the nerest whole litre? Briefly explin here why/why not. H SERIES TOPIC 7

10 How does it work? on the number line The smllest plce vlue in deciml is used to position points ccurtely on number line. re bsed on the number 0, so there re lwys ten divisions between vlues E: Here is the vlue 3. on number line: Six tenths of the wy from 3.0 to 4.0 The mjor intervls on the number line re mrked ccordin to the second lst deciml plce vlue 8 So its eiht thousndths of the wy from.240 to Here re some more exmples involvin number lines: (i) Wht vlue do the plotted points represent on the number lines below? ) Point is four steps from 0. towrds 0.2, so the plotted point is: 0.4 b) Point is nine steps from 0.0 towrds 0.07, so the plotted point is: 0.09 (ii) Round the vlue of the plotted points below to the nerest hundredth. ) Point is three steps from 2.4 towrds 2.5, so the plotted point is 2.43 ` the vlue of the plotted point to the nerest hundredth is: 2.4 b) Point is five steps from 8.79 towrds 8.80, so the plotted point is ` the vlue of the plotted point to the nerest hundredth is: H SERIES TOPIC

11 How does it work? Your Turn on the number line Disply these decimls on the number lines below: 0.7 b 2. 4 ECIMALS ON THE NUMBER LINE.../.../20... DDECIMALS ON THE NUMBER LINE c 0.3 d e 2.34 f Lbel these number lines nd then disply the iven deciml on them:. b 4.2 c 0.94 d 7.07 e f Round the vlue of the plotted points below to the nerest plce vlue iven in squre brckets. [tenth] b [hundredth] ` the vlue. ` the vlue. c [tenth] d [hundredth] ` the vlue. ` the vlue. e [thousndth] f [thousndth] ` the vlue. ` the vlue. [thousndth] h [thousndth] ` the vlue. ` the vlue. H SERIES TOPIC 9

12 How does it work? Multiplyin nd dividin by powers of ten Move the deciml point dependin on the number of zeros deciml point moves riht, deciml point moves left Clculte these multipliction nd division questions involvin powers of 0: (i) 5 # 000 We cn simply dd the sme number of zeros to the end of the whole number 5 # # The whole number in deciml form Fill the empty bounces with 0s If the deciml point is on the left fter dividin, n extr 0 is plced in front. (ii) 8 00 ' 8' ' 00 The whole number in deciml form Remember to include the ledin zero '00 hs 2 zeros, so move deciml point 2 spces left 0.08 Fill the empty bounces with 0s nd put zero in front (iii) # # Move deciml point 4 spces riht No empty bounces to fill, so this is the nswer (iv) ' ' Move deciml point 5 spces left Fill empty bounces with 0s nd put zero in front (v) 205. # # 205. ' # is the sme s ' Move deciml point 3 spces left Plce ledin zero in front of the deciml point 0 H SERIES TOPIC

13 How does it work? Your Turn Multiplyin nd dividin by powers of ten Clculte these multiplictions. Remember, multiply mens move deciml point to the riht: 8 00 # b 3.4 # 0 c 29 # 000 d # e 0.52 # 00 f # Clculte these divisions. Remember, divide mens move deciml point to the left: 2 ' 00 b 4590 ' 000 c. 004 ' 0 d ' e ' f ' Here re some of the powers of 0 usin exponent nottion. The power the number of zeros Clculte these mixed problems written usin exponent nottion: # b 2400 ' 0 5 c # 0 d # e ' f ' 0 7 H SERIES TOPIC

14 MULTIPLYING AND DIVIDING BY POWERS OF TEN How does it work? Your Turn Multiplyin nd dividin by powers of ten 4 For these clcultions: (i) Show where our chrcter needs to spry pint new deciml point, nd (ii) write down the two numbers the new deciml point is between to solve the puzzle.../.../ # I 9 nd 2 b ' N c # A d ' O e # X f # T # R h ' I i # D j # P This is nother mthemticl nme for deciml point: I 0 nd 9 8 nd 9 8 nd 7 9 nd 2 0 nd 7 3 nd 9 8 nd 2 0 nd 8 3 nd 8 nd 7 2 H SERIES TOPIC

15 How does it work? Termintin decimls to frctions These hve deciml prts which stop (or terminte) t prticulr plce vlue. The plce vlue of the lst diit on the riht helps us to write it s frction. Write 0.3 s frction: Deciml 0.3 Frction 3 0 Deciml diits in the numertor Lst diit is in tenths position Inteers in front of the deciml vlues re simply written in front of the frction. Write.07 s frction: Deciml diits in the numertor Lst diit is in hundredths position 07 is just 7 Alwys simplify the frction prts if possible. These two exmples show you how. Write ech of these decimls s n equivlent (equl) frction in simplest form (i) Equivlent, un-simplified frction 00 Lst diit is in hundredths position 25 ' ' 25 Divide numertor nd denomintor by HCF 4 Equivlent frction in simplest form (ii) Equivlent, un-simplified mixed number Lst diit is in thousndths position 2 05 ' ' 5 Divide numertor nd denomintor by HCF Equivlent mixed number in simplest form H SERIES TOPIC 3

16 How does it work? Your Turn Termintin decimls to frctions Write ech of these decimls s equivlent frctions: 0. b. c 009. d f h. e i 0.29 j. k l Write ech of these decimls s equivlent frctions nd then simplify: 0.5 b 0. c 0.02 Simplest form Simplest form Simplest form d 0.08 e f Simplest form Simplest form Simplest form 0.2 h 0.25 i Simplest form Simplest form Simplest form j k l Simplest form Simplest form Simplest form 4 H SERIES TOPIC

17 TERMINATING DECIMALS TO FRACTIONS * Where does it work? Your Turn Termintin decimls to frctions 3 Write ech of these decimls s equivlent mixed numbers: /.../ b. c e f.00 d Write ech of these decimls s equivlent mixed numbers nd then simplify: 2.8 b.4 c.0 4 Simplest form Simplest form Simplest form 3 5 e 275. f d.0 Simplest form Simplest form Simplest form. 004 h i. 344 Simplest form Simplest form Simplest form H SERIES TOPIC 5

18 How does it work? Frctions to termintin decimls Where possible, just write s n equivlent frction with power of 0 in the denomintor first. numertor denomintor 3 3 # # 2 0 ` 0. Multiply numertor nd denomintor by the sme vlue Equivlent frction with power of 0 in the denomintor Three fifths six tenths zero point six Sometimes it is esier to first simplify the frction before chnin to deciml. Write these s n equivlent deciml (i) ' 3 2 ' 3 4 Simplify frction 4 4 # # Equivlent frction with power of 0 in the denomintor ` 025. Three twelfths one qurter twenty five hundredths zero point two five (ii) ' 3 5 ' Simplify frction prt 2 # 2 5 # Equivlent frction with power of 0 in the denomintor Two nd three fifteenths two nd one fifth two nd two tenths two point two H SERIES TOPIC

19 How does it work? Your Turn Frctions to termintin decimls include ledin zero Write ech of these frctions s equivlent decimls. 9 b 3 c d Write ech of these s equivlent frctions with power of 0 in the denomintor. c 2 b d 20 9 e 8 f 3 h i 4 j 3 k (i) Write ech of these s equivlent frctions with power of 0 in the denomintor. (ii) Chne to equivlent decimls. 5 b 4 c 25 d 4 25 e 200 f h 200 i H SERIES TOPIC 7

20 How does it work? Your Turn Frctions to termintin decimls 4 Chne ech of these frctions to equivlent decimls fter first simplifyin. Show ll your workin b c d 40 FRACTIONS TO TERMINATING DECIMALS /.../20... e 9 75 f h H SERIES TOPIC

21 How does it work? Your Turn Frctions to termintin decimls When chnin the denomintor to power of 0 is not esy, you cn write the numertor s deciml nd then divide it by the denomintor. Write this frction s n equivlent deciml ' 8 Write numertor s deciml nd divide by the denomintor If you need more deciml plce 0s, you cn dd them in lter! Complete division, keepin the deciml point in the sme plce ` Five eihths zero point six two five 5 Complete these divisions to find the equivlent deciml: ' 5 b ' 4 3 c ' ' ' ' d e f H SERIES TOPIC 9

22 How does it work? Your Turn Frctions to termintin decimls Simplify these frctions nd then write s n equivlent deciml usin the division method. Show ll your workin. 2 5 b 9 2 c 49 d e 8 24 f 2 20 H SERIES TOPIC

23 Where does it work? Addin nd subtrctin decimls Just dd or subtrct the diits in the sme plce vlue. To do this, line up the deciml points nd mtchin plce vlues verticlly first. Add 2.45 to.3 (i.e ) Deciml points lined up verticlly Add mtchin plce vlues toether Subtrct 5.8 from.89 (i.e ) Deciml points lined up verticlly Subtrct mtchin plce vlues Clculte ech of these further dditions nd subtrctions (i) Deciml points lined up verticlly Add mtchin plce vlues toether Any plce vlue spces re treted s 0s ` Roundin deciml vlues before ddin is sometimes used to quickly pproximte the size of the nswer. (ii) Round ech vlue in question (i) to the nerest whole number before ddin. ` Vlues rounded to nerest ones Approximte vlue for ddition Note: Roundin vlues before ddin/subtrctin is not s ccurte s roundin fter ddin/subtrctin. (iii) Fill plce vlue spces in the top number with 0 when subtrctin ` Deciml points lined up verticlly Subtrct mtchin plce vlues H SERIES TOPIC 2

24 ADDING AND SUBTRACTING DECIMALS Where does it work? Your Turn Addin nd subtrctin decimls Complete these dditions nd subtrctions: b. 8 + c d e f h Clculte these dditions nd subtrctions, showin ll workin: Add 8.75 to.24 b Subtrct 3.5 from /.../20... c Add 0.93 to 0.85 d Add 2.9, 5. nd 0.3 e Subtrct from 8.02 f Add 0.20, 4.4 nd H SERIES TOPIC

25 Where does it work? Your Turn Addin nd subtrctin decimls 3 Approximte these clcultions by roundin ech vlue to the nerest whole number first. (i) (ii) (iii) (iv) (v) (vi) b Clculte prts (v) nd (vi) in, this time roundin fter ddin the numbers to et more ccurte pproximte vlue. (i) (ii) Clculte these subtrctions, showin ll your workin: - b c H SERIES TOPIC 23

26 Where does it work? Multiplyin with decimls Just write the terms s whole numbers nd multiply. Put the deciml point bck in when finished. The number of deciml plces in the nswer the number of deciml plces in the question! Clculte 4 # # Multiply both terms s whole numbers 48 deciml plce in question deciml plce in nswer ` 4 # Clculte 0.02 # # Multiply both terms s whole numbers ` 0.02 # deciml plces in question 4 deciml plces in nswer How does this work when multiplyin with decimls? Excellent question! Very ld you sked! Let s do the second one in but this time chne the decimls to equivlent frctions first 002. # # # 45 00# Chnin the decimls to frctions Multiply numertors nd denomintors toether Number of zeros in denomintor totl of deciml plces in question 290' Dividin by moves deciml point four plces to the left ` 4 deciml plces in question 4 deciml plces in nswer Try this method for yourself on the first exmple bove, rememberin tht 4 4 s frction. 24 H SERIES TOPIC

27 Where does it work? Your Turn Multiplyin with decimls Clculte these whole number nd deciml multiplictions, showin ll you workin: # b 5 5. c # 0.4 # 0.2 # 4 3 # # 2 d e f 2 Clculte these deciml multiplictions, showin ll your workin: 3.8 # # # 25. b c ULTIPLYING WITH DECIMALS.../.../20... MMULTIPLYING WITH DECIMALS 7. # # # 93. d e f H SERIES TOPIC 25

28 Where does it work? Dividin with decimls Opposite to multiplyin, we move the deciml point before dividin if needed. To find the quotient involvin decimls, the question must be chned so the divisor is whole number. dividend ' divisor quotient Clculte 428. ' Divisor lredy whole number so no chne needed ` 4.28 ' 4.07 Clculte ' ' ' 000. Move both deciml points riht until divisor is whole number 45. ' Quotient 2 Dividend if divisor ` ' Drop off ny 0s t the front of the nswer Here s nother exmple showin how to tret reminders Clculte 2. ' ' ' 0.8 Move both deciml points riht until divisor is whole number 2. ' Add 0s on the end of the dividend for ech new reminder ` 2. ' Drop off ny 0s t the front 2 H SERIES TOPIC

29 Where does it work? Your Turn Dividin with decimls Clculte these deciml nd whole number divisions: 3.' ' 5. 2' 9 b c.../.../20... IVIDING WITH DECIMALS DDIVIDING WITH DECIMALS ` 3.' 4 ` 7.5 ' 5 `.2 ' ' ' ' 7 d e f ` 0.3 ' 3 ` 0.489' 5 ` 0.97 ' 7 2 Clculte these deciml divisions, showin ll your workin: 52. ' ' ' 08. b c ` 5.2' 0.4 ` 9.' 0. ` 0.5 ' ' ' ' d e f `.58 ' 0.4 ` ' 0.05 ` ' 0.00 H SERIES TOPIC 27

30 Where does it work? Repetin decimls Non-termintin decimls hve deciml prts tht do not stop. They keep oin on nd on Three dots mens it keeps oin If the deciml prts hve repetin number pttern, they re clled repetin decimls Here re some exmples involvin repetin decimls A dot bove the strt nd end diit of the repetin pttern is used to show it is repetin deciml. (i) Write these repetin decimls usin the dot nottion The pttern 2 keeps repetin in the deciml prts ) b) c) o o A br over the whole pttern cn lso be used insted of dots Strt End 0. 8 o o Strt End o o Strt nd End 047. o or.047r Identify the strt nd end of the repetin pttern Dot bove strt nd end of the repetin pttern Identify the strt nd end of the repetin pttern Dot bove strt nd end of the repetin pttern Identify the strt nd end of the repetin pttern Dot bove strt nd end of the repetin pttern (ii) Clculte ' 0. ' 0. ' ' Write s deciml with few 0s ` ' o Repets the sme reminder when dividin Repetin deciml in simplest nottion 28 H SERIES TOPIC

31 Where does it work? Your Turn Repetin decimls Wht is the nme of the horizontl line bove the repeted numbers in repetin deciml? Hihliht the boxes tht mtch the repetin decimls in ech row with the correct simplified nottion in ech column to find the nswer. Not ll of the mtches form prt of the nswer! r. 4. o o o o o C z F h N d W c D b A U n P t L f O m Y n A m R f T t K z E h R d I c U b S L D b A m I h M t B f S c A d U z Q n R h Z d A n E z A c N t 0 M b A h G f A f T z P c H d T Y n A t A h C m A b I d Y t A b U n H m I z E f S m I t T A b L D t E f A d N c L m E z O d N h W c J f B d A X h M m A b U n A A z P m V c E F b A n B d T Y f E c I t H t A n A m A m U f A b A h A D d R c c z h m n f b 2 Clculte these divisions which hve repetin decimls s result. Write nswers usin dot nottion. ' 3 4' 9 5' b c ` ' 3 ` 4' 9 ` 5'.' 25. ' ' 3 d e f `. ' ` 25. ' 9 ` 034. ' 3 H SERIES TOPIC 29

32 ...REPEATING DECIMALS...REPEATING DECIMALS...REPEATING DECIMALS Where does it work? Your Turn Repetin decimls 3 (i) Complete the followin divisions to five deciml plces. (ii) Determine whether the nswer is repetin deciml or not. 2' 3 ' ' 7 b c.../.../20... ` 2' 3 ` ' ` ' 7 Repetin deciml? Repetin deciml? Repetin deciml? Yes No Yes No Yes No. ' ' ' 08. d e f `. ' 7 ` 29. ' 3 ` 0.33 ' 0.8 Repetin deciml? Repetin deciml? Repetin deciml? Yes No Yes No Yes No 0.8' ' ' h i ` 0.8 ' 0.3 ` 0.09' 0.0 ` ' Repetin deciml? Repetin deciml? Repetin deciml? Yes No Yes No Yes No 30 H SERIES TOPIC

33 Wht else cn you do? Simple repetin decimls into sinle frctions Only repetin, non-termintin decimls cn be written in frction form. Here is quick wy for simple decimls with the pttern strtin riht fter the deciml point o 9 One diit in repetin pttern, so tht diit over o o ' 3 99 ' Two diits in repetin pttern, so those two diits over 99 Alwys simplify frctions o o Three diits in repetin pttern, so those three diits over 999 Here re some other exmples includin mixed numerls. Write ech of these repetin decimls s mixed numerls in simplest form (i) o One diit in repetin deciml pttern, so tht diit over Diits in front of deciml point form the whole number prt (ii) o o Three diits in repetin deciml pttern, so those diits over Diits in front of deciml point form the whole number 345 ' ' Simplify the frction prt H SERIES TOPIC 3

34 Wht else cn you do? Your Turn Simple repetin decimls into sinle frctions Use the shortcut method to write ech of these repetin decimls s frction in simplest form: 04. o 08r. 0. o b c 0. oo 027. o o 057. o o d e f 2 Use the shortcut method to write ech of these repetin decimls s mixed numbers in simplest form. 5. o 27r. 43r. b c 3r o o d. e f o o h i 3 (i) Write 09. o s frction in simplest form. (Ii) Does nythin unusul seem to be hppenin with your nswer? Explin. SIMPLE REPEATING DECIMALS INTO SINGLE FRACTIONS /.../ H SERIES TOPIC

35 Wht else cn you do? Combinin deciml techniques to solve problems All the techniques in this booklet cn be used to solve problems. These exmples show different wys decimls pop up in every-dy life (i) These rinfll mesurements were tken durin three dys of rin from smll wether ue: 3.8 mm 3. mm 27. mm Wht ws the totl rinfll for the three dys, to the nerest whole mm? mm ` The totl rinfll over the three dys ws pproximtely 78 mm Add the deciml vlues toether Round to nerest whole mm Answer with sttement (ii) The results for five runners in 00 m rce were plotted on the number line below seconds ) Wht ws the fstest time run (to the nerest thousndth of second)? Fstest time left-most plotted point.22 seconds b) Wht time did two runners finish the rce toether on? Two runners with the sme time two dots t the sme point.223 seconds c) Wht ws the vere time rn by ll runners in this rce? Avere time The sum of ll the times rn divided by the number of runners ( ) ' ' The vere time rn by ll the runners in the rce. 2242seconds Red off ll the times Add, then divide by 5 Answer with sttement H SERIES TOPIC 33

36 Wht else cn you do? Your Turn Combinin deciml techniques to solve problems To mke drk-reen coloured pint, you cn mix yellow nd blue toether, usin exctly 0.5 (hlf) s much yellow s you do blue. Use multipliction to show how much yellow pint you will need if you use ll of the 2.4 ml of blue pint you hve. Remember me? b How mny millilitres of drk-reen pint cn you mke with 8.45 ml of yellow pint in the mix? Round your nswer to the nerest tenth of ml. 2 Derek types his essys t n vere speed of words every minute. How mny words does he type in five minutes (to the nerest whole word)? COMBINING DECIMAL TECHNIQUES TO SOLVE PROBLEMS 3 Nine people were tryin out for speed roller sktin tem round n ovl flt trck. The shortest time to complete six full lps of the trck for ech person were recorded on the number line below:.../.../ seconds Wht ws the slowest time recorded to 3 deciml plces? b To mke the tem, skter hd to complete the six lps in less thn seconds. How mny skters mde it into the tem? c How mny skters missed out mkin the tem by less thn 0.0 seconds? 34 H SERIES TOPIC

37 Wht else cn you do? Your Turn Combinin deciml techniques to solve problems 4 The wireless trnsmitter in Lur s house reduces in sinl strenth by for every metre of distnce she moves her computer wy from the trnsmitters ntenn. Her computer displys sinl strenth usin brs s shown below: 4 brs 08. to.0 sinl strenth 3 brs 0. to 0.8 sinl strenth 2 brs 0. 4 to 0. sinl strenth br 0. 2 to 0.4 sinl strenth 0 brs 0.2 or below sinl strenth How mny brs of sinl strenth would Lur hve if usin her computer.25m wy from the ntenn? 5 Ruofn is puttin toether video of recent kroke prty with her friends. She will be usin five of her fvourite music trcks for the video. The lenth of time ech of the trcks ply for is: 3.55 min, 5.4 min, 2.27 min, 3.8 min nd 4.8 min If she uses the entire lenth of the trcks with 0.5 min brek in ech of the four ps between sons, how lon will her video run for (to the nerest whole minute)? Show ll your workin. H SERIES TOPIC 35

38 Wht else cn you do? Your Turn Combinin deciml techniques to solve problems After recent study by city council, the vere number of people in ech household ws determined.to be 3.4. Explin how this is possible if household cnnot ctully hve 0.4 of person? psst: Check exmple on pe 33 to see how vere clcultions re mde. 7 A Mexicn chef hs split up mystery inredient Sl-X into four exctly identicl quntities in seprte jrs. He then distributes 382. o ml of the secret inredient S-Y monst the four jrs, producin in totl 839. o ml of the specil suce SlS-XY. How much of the mystery inredient Sl-X is there in ech jr (to the nerest ml)? Show ll your workin. 8 After completely flt wter conditions (wves with heiht of 0.0m), the heiht of the wves t locl bech strt incresin by 0.2 m every 03. o hours. If the wves need to be t lest.4 metres hih before surfers will ride them t this bech, how lon will it be until people strt surfin there to the nerest minute? Show ll your workin. psst:.0 hours 0 minutes 3 H SERIES TOPIC

39 Wht else cn you do? Your Turn Reflection Time Reflectin on the work covered within this booklet: Wht useful skills hve you ined by lernin bout decimls? 2 Write bout one or two wys you think you could pply decimls to rel life sitution. 3 If you discovered or lernt bout ny shortcuts to help with decimls or some other cool fcts, jot them down here: H SERIES TOPIC 37

40 Chet Sheet Here is summry of the thins you need to remember for decimls Plce vlue of decimls # # 000 # 00 Tens of thousnds Thousnds Hundreds Tens Ones W H O L E # 0 # ' 0 Tenths ' 00 ' 000 Hundredths Thousndths Ten thousndths D E C I M A ' ' Hundred thousndths Millionths ' L ' Ten Millionths Approximtions throuh roundin numbers The next diit followin the plce vlue where number is bein rounded off to is the importnt prt. Next diit Closer to lower vlue, so round down Leve the plce vlue unchned Closer to hiher vlue, so round up Add to the plce vlue on the number line The smllest plce vlue in deciml is used to position points ccurtely on number line Six tenths of the wy from 3.0 to 4.0 Eiht thousndths of the wy from.240 to.250 Multiplyin nd dividin by powers of ten Move the deciml point dependin on the number of zeros deciml point moves riht, deciml point moves left 5 # # 000 8' ' H SERIES TOPIC

41 Chet Sheet Termintin decimls to frctions The plce vlue of the lst diit on the riht helps us to write it s frction. Write 0.3 s frction: Lst diit is in tenths position Deciml 03. Frction 3 0 Deciml Frction Write.07 s frction: Lst diit is in hundredths position Frctions to termintin decimls Where possible, just write s n equivlent frction with power of 0 in the denomintor first. E: 3 3 # 2 Multiply numertor nd denomintor by the sme vlue 5 5 # 2 Equivlent frction with power of 0 in the denomintor 0 ` 0. Three fifths six tenths zero point six When this method is not esy, write the numertor s deciml nd then divide it by the denomintor. Addin nd subtrctin decimls Line up the deciml points nd mtchin plce vlues verticlly before ddin or subtrctin. Multiplyin nd dividin decimls Write the terms s whole numbers nd multiply. Put the deciml point bck in when finished. The number of deciml plces in the nswer the number of deciml plces in the question! E: : 4 # : 0.02 # Dividin with decimls The question must be chned so the divisor is whole number first. E: : 3.5 ' ' 4 : ' ' 3 dividend ' divisor quotient Repetin decimls These hve deciml prts with repetin number pttern. E: : o o 5.2 : o o Strt End Simple repetin decimls into sinle frctions Only repetin, non-termintin decimls cn be written in frction form. This is the method for simple decimls with the pttern strtin riht fter the deciml point. Strt End Alwys simplify frctions : : o o One diit in repetin Two diits in repetin pttern, pttern, so tht diit over 9 so those two diits over : o o Three diits in repetin pttern, so those three diits over 999, Keep whole number out the front. H SERIES TOPIC 39

42 Notes 40 H SERIES TOPIC

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