Chapter33. Real numbers

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1 Chpter Rel numers Contents: A Frtions B Opertions with frtions C Deiml numers D Opertions with deiml numers E Rtionl numers F Irrtionl numers

2 46 REAL NUMBERS (Chpter ) Opening prolem Qi-Zheng hs irulr pond with rdius m. He wnts to know its surfe re, so he sks his fmily for help. Their responses re given in the tle elow: Are (m 2 ) Dd :4 Mum 22 Brother Qi-Ren ¼ : :::: Things to think out: Cn ll of the nswers e illustrted on numer line? Use your lultor to write 22 s deiml. How n you write this deiml so it is extly 22? Consider the three nswers given s deimls. In whih nswer does the deiml: i keep repeting in pttern ii stop or terminte iii go on forever without repeting? The set of rel numers inludes ll numers whih n e pled on the numer line. It inludes the set of whole numers or integers, s well s the frtions nd deimls etween them. In this hpter we first revise frtions nd deimls. We will then see how the set of rel numers n lso e divided into two groups: rtionl nd irrtionl. A FRACTIONS In previous yers we hve seen how frtions re otined when we divide whole into equl portions. In generl, the division n e written s the frtion. mens we divide whole into equl portions, nd then onsider of them. the numertor is the numer of portions onsidered the r indites division the denomintor is the numer of portions we divide whole into A frtion written in this form is lled ommon frtion. Et is shded

3 REAL NUMBERS (Chpter ) 4 TYPES OF FRACTIONS 2 4 is lled proper frtion s the numertor is less thn the denomintor. is lled n improper frtion s the numertor is greter thn the denomintor. 2 2 is mixed numer s it is relly When we perform lultions involving mixed numers, it is often useful to first onvert the mixed numer to n improper frtion. Exmple Convert 2 2 to n improper frtion. 2 2 =2+ 2 = fwriting with equl denomintorsg = 2 NEGATIVE FRACTIONS In Chpter we sw tht whenever we divide positive y negtive, or negtive y positive, the result is negtive. Sine the r of frtion indites division, the frtion 2 mens ( ) 2 = 2 negtive positive negtive Also, 2 mens ( 2) = 2 positive negtive negtive So, 2 = 2 = 2, nd in generl = =

4 48 REAL NUMBERS (Chpter ) PLACING FRACTIONS ON A NUMBER LINE We n represent frtions on numer line y dividing eh whole into the numer of prts in the denomintor. By extending the numer line either side of zero, we n represent positive nd negtive frtions. -It -Wt Qt Rt EXERCISE A Desrie the following s proper frtions, improper frtions, or mixed numers: 2 8 d e 0 f Convert these mixed numers to improper frtions: d 2 8 e f Plot these frtions on numer line:, 2,, 4 Write s division nd hene evlute: 6 6, 6, 2, 4, 0 6 EQUAL FRACTIONS When we multiply or divide the numertor nd denomintor of frtion y the sme non-zero numer, we otin n equl or equivlent frtion. For exmple, *2 * = 2 6 = 6 8, nd 20 = 4. *2 * / / Exmple 2 Express: 4 with denomintor 2 24 with denomintor To onvert the denomintor to 2 we need to multiply y 8. We must therefore multiply the numertor y 8 lso. To onvert the denomintor to we need to divide y. We must lso divide the numertor y. *8 4 = 24 2 *8 / 24 = 8 /

5 REAL NUMBERS (Chpter ) 49 SIMPLIFYING FRACTIONS We n simplify frtion y dividing the numertor nd denomintor y their highest ommon ftor. When we hve simplified frtion in this mnner, we sy it is in lowest terms. Exmple Express in lowest terms: = 0 2 fhcf =g = 24 fhcf =g = 2 = 8 EXERCISE A.2 Write with denomintor 20: d 00 2 Write the frtions 0,, nd 2 with denomintor 0. Hene write the frtions in order from smllest to lrgest. Express in lowest terms: d 2 e 2 2 f g 6 48 h 8 20 i 22 2 j An enlosure t the zoo ontins 6 meerkts. 6 re sleep nd 0 re wke. Find, in lowest terms, the frtion of meerkts tht re: sleep wke. 2 eginners, 2 mteurs, nd 8 professionls ompeted in poker tournment. How mny plyers ompeted in the tournment? istokphoto.om/pjmlsury Find, in lowest terms, the frtion of ompetitors who were: i mteur ii professionl iii either mteur or professionl.

6 0 REAL NUMBERS (Chpter ) B OPERATIONS WITH FRACTIONS ADDING AND SUBTRACTING FRACTIONS To dd or sutrt frtions, we onvert them to frtions with the lowest ommon denomintor. This is the lowest ommon multiple of the originl denomintors. We then dd or sutrt the new numertors. Exmple 4 Find: = = = 8 flcd =8g fdding numertorsg = = flcd =2g 2 = = 2 Mixed numers should e written s improper frtions efore the ddition or sutrtion is performed. Exmple Find: = 2 fwrite s improper frtionsg = = = 0 or 0 flcd =0g EXERCISE B. Find: d 4 e f 6 4 g h 2 9

7 REAL NUMBERS (Chpter ) 2 Find: d + 4 Find: d + e f 6 8 g h Crter red of ook during flight from Melourne to Brisne, nd nother 8 ook during the flight k. Wht frtion of the ook hs Crter red? Wht frtion of the ook remins for Crter to red? of the A reipe requires 2 up of self-rising flour, nd 2 ups of plin flour. In totl, how muh flour is used in the reipe? How muh more plin flour is used thn self-rising flour? 6 Use your lultor to evlute: MULTIPLYING AND DIVIDING FRACTIONS To multiply two frtions, we multiply the numertors together nd multiply the denomintors together. d = d Exmple 6 Find: 2 2 = 2 = 6 = 4 = 4 We nel ny ommon ftors in the numertor nd denomintor efore ompleting the multiplition.

8 2 REAL NUMBERS (Chpter ) For ny frtion, we notie tht =. nd re lled reiprols euse their produt is one. The reiprol of ny frtion is otined y swpping its numertor nd denomintor. To divide y frtion, we multiply y the reiprol of tht frtion. d = d Exmple Find: =4 =4 =2 = = 4 2 = 0 = EXERCISE B.2 Find: d 4 e 2 2 f 8 0 g ( 2 )2 h ( 2 ) i (2 )2 j ( )2 6 k 8 ( 2 ) l Evlute: d 6 e 8 4 f 4 g 2 h Find: ( ) 6 d 2 4

9 REAL NUMBERS (Chpter ) Exmple 8 During seson, Joe hit 2 of the home runs for his tem. How mny home runs did he hit if there were 40 sored in totl? Rememer tht of mens. Joe hit 2 of 40 = 2 40 = =6home runs. 4 Find: 2 of 0 8 of 24 d 4 of $60 e 6 of 0 m f 0 of of 80 kg Lee is trvelling 00 km from Rokhmpton to Brisne. He stops t Gympie, hving trvelled of the wy. How fr hs Lee trvelled? of the money rised t hrity event is given to the lol hospitl. The hospitl spends of their money on new X-ry mhine. Wht frtion of the totl money rised y the 4 hrity ws spent on the X-ry mhine? Tin uses 2 tlespoon of utter to mke n priot lof. How mny loves n she mke with 4 tlespoons of utter? 8 Trevor te 9 of lsgne. Elenor te of wht 6 remined. Wht frtion of the lsgne did Elenor et? Wht frtion of the lsgne now remins? 9 Emm uys two identil ottles of shmpoo. She uses 8 of one of them t home, nd uses of ll of the remining shmpoo while she is on holidys. In totl, how muh shmpoo 0 remins? 0 Use your lultor to evlute: µ 4 2

10 4 REAL NUMBERS (Chpter ) C DECIMAL NUMBERS A deiml numer is numer whih ontins deiml point. We n use deiml numers to disply frtions of whole numers. For exmple: ² 4:6 is quik wy of writing This numer n lso e 00 written s the improper frtion ² 4:062 is quik wy of writing or s the mixed numer Expnsions of deimls like those ove re referred to s expnded frtionl form. Exmple 9 Write :04 in expnded frtionl form. Write in deiml form. Stte the vlue of the digit 6 in 0:06 24 :04= =:240 In 0:06 24, the 6 stnds for CONVERTING DECIMALS TO FRACTIONS We n use our knowledge of deiml ple vlues to onvert deimls to frtions. Exmple 0 Write s frtions: 0:6 0:04 Rememer to write your nswer in lowest terms. 0:6 = 6 0 = = 0:04 = = = PLACING DECIMALS ON A NUMBER LINE To represent deiml numers on numer line, we hoose the sle ording to the lowest deiml ple vlue.

11 REAL NUMBERS (Chpter ) For exmple, when representing f0:2, 0:9, :, :8g, the lowest ple vlue is tenths. We divide eh whole into tenths Extending the numer line in oth diretions llows us to represent negtive deiml numers: ROUNDING DECIMAL NUMBERS We often need to round numers to ertin numer of deiml ples. For exmple, if you wish to reord your weight in kilogrms, you will proly round the result to the nerest whole numer, or to one deiml ple. RULES FOR ROUNDING ² If the digit fter the one eing rounded is less thn, we round down. ² If the digit fter the one eing rounded is or more, we round up. Exmple Round 9:48 to: the nerest whole numer one deiml ple two deiml ples. 9:48 ¼ 40 fthe first deiml ple is, so round upg 9:48 ¼ 9: fthe seond deiml ple is 4, so round downg 9:48 ¼ 9: fthe third deiml ple is 8, so round upg EXERCISE C Write the following in expnded frtionl form: :2 :02 :024 d 9:0909 e 0:082 2 Write the following in deiml form: 4+ 0 d e Stte the vlue of the digit in the following: f :24 d 0:08 e 0: Write s frtions in simplest form: 0:2 0: 0:4 d 0:04 e 0:02 f 0:008 g 0:62 h 0:8

12 6 REAL NUMBERS (Chpter ) Ple these deimls on numer line: 0:4, 0:, :, : 0:, 0:, 0:6, :2, :4 0:, 0:, 0:6, 0:9 6 Round to the nerest whole numer: i 6:26 ii :4 iii 8:2 iv 99:6 Round orret to one deiml ple: i 4:2 ii 6:9 iii :4 iv 6:8 Round orret to two deiml ples: i :42 ii :96 iii 2:649 iv 8:99 Round 0:94 to one deiml ple. Round 0:94 to two deiml ples, nd then the result to one deiml ple. Disuss your results with your lss. D OPERATIONS WITH DECIMAL NUMBERS ADDING AND SUBTRACTING DECIMAL NUMBERS To dd or sutrt deiml numers, we write the numers under one nother so tht the deiml points line up. We then dd or sutrt s we do with whole numers. Exmple 2 The zero t the end of : is dded so the numers hve the sme numer of deiml ples. Find :+9:26. + :0 9:26 24:6 Exmple Find: 4:62 :0 8 0: : 62 : 0 : : : 06 : 294 Ple the deiml points diretly under one nother nd sutrt s for whole numers. We insert :000 fter the 8 so we hve the sme numer of deiml ples in oth numers. All of the deiml points must line up.

13 REAL NUMBERS (Chpter ) EXERCISE D. Find: 2 Find: 2: + 4: 4:9 + 6: 4:2 + 8:64 d 9:86 + : e :2 + : + 4:8 f 2: + 4:8 + :9 :84 :22 9:8 :6 :8 : d 4:8 e 4 9:22 f 26:02 8:8 For n interstte holidy, Tony hs pked suitse weighing :6 kg, nd kpk weighing 6: kg. Find the totl weight of Tony s luggge. 4 In long jump ompetition, Smnth jumped :2 m. This ws : m further thn Theres s jump. How fr did Theres jump? At the supermrket, Mrgret ought tu of mrgrine for $2:49, rton of milk for $:86, nd ottle of sunsreen for $4:90. How muh did Mrgret spend? How muh hnge will she reeive from $20 note? MULTIPLYING AND DIVIDING BY POWERS OF 0 When multiplying y power of 0, we shift the deiml point to the right. When dividing y power of 0, we shift the deiml point to the left. For exmple: 0:8 0 = :8 0:8 00=8 0:8 0 = 0:08 00:8 00 = 0:008 MULTIPLICATION AND DIVISION OF DECIMAL NUMBERS To multiply deiml numers, we multiply the numers s though they were whole numers, then divide y the pproprite power of 0. Exmple 4 Find 0: 0:. We otin whole numers y multiplying 0: y 0, nd 0: y 00, then lne y dividing y (0 00). 0: 0: = ( 0) ( 00) =( ) (0 00) =4 000 = 0:04 fshifting the deiml ples leftg

14 8 REAL NUMBERS (Chpter ) To divide deiml numers, we write the division s frtion, then multiply the numertor nd denomintor y the sme power of 0 to mke the denomintor whole numer. We then perform the division. Exmple Find: 2: 0: 0:002 0:08 2: 0: 2: 0 = 0: 0 = 2 =9 0:002 0:08 0: = 0:08 00 = 0:2 8 =0:02 0 : : EXERCISE D.2 Find: 2 Find: Find: : 0 0: : d 0: e 00 f :2 000 g 0:022 0 h 0: :8 0: 9 0:9 0:4 d :2 0: e 0:6 0: f 0:08 0: g 0: 2 h 0: ( 0:4) i ( 0:9) ( 0:8) :6 0:2 4 0:8 0:6 0:0 d 2 0:0 e 0:0 0:2 f 0:006 0:0 g 0:0008 0:0 h : 0: i ( 4:2) ( 0:) 4 Use your lultor to evlute: : : 40:8 6: 9:+6: 4:6 d 6: 8 :6 e 2 + 6:28 0: 2 Domini drnk 2:2 litres of wter eh dy for 8 dys. How muh wter did he drink in this time? 6 A mother duk weighs 2:8 kg. Her y dukling weighs 0: kg. How mny times hevier is the mother thn her y? Eh kilogrm of eef ontins 0:8 grms of holesterol. How mny grms of holesterol re in 0:20 kg serving of eef? f 2:4 2 9: 2

15 REAL NUMBERS (Chpter ) 9 8 A drinks mhine ontins 90 litres of wter. How mny 0: litre ups of wter n e filled from the mhine? 9 Bernrd uys 4 ottles of wter osting $2: eh, nd fruit rs osting $: eh. How muh money hs Bernrd spent ltogether? Investigtion Division y zero When we divide y positive numers greter thn, we notie tht the result is lwys smller thn the originl numer we divided. For exmple, when we divide 2 y 2, the result 6 is smller thn the originl numer 2. In this investigtion we onsider wht hppens when we divide y positive numers smller thn, nd wht hppens s this numer gets loser nd loser to zero. Wht to do: Evlute: 2 0 d 00 2 Evlute: 2 0:4 2 0: 2 0:0 d 2 0:00 Copy nd omplete: As the numer we re dividing y gets smller nd smller, the result gets... 4 Copy nd omplete: i Sine ii iii Sine Sine In iii ove, we re sying tht if Do you gree with this dedution? Wht n we onlude from? 6 =, 2 =:::::: 2 20 =4, 4=:::::: 2 =, 0 = :::::: is equl to some numer, then 0=2. E RATIONAL NUMBERS A rtionl numer is numer tht n e written in the form, where nd re integers nd 6= 0. Proper frtions suh s 2, improper frtions suh s 4, nd mixed numers suh s 2 = 2 re ll exmples of rtionl numers. Whole numers suh s nd re lso rtionl, sine = nd =.

16 60 REAL NUMBERS (Chpter ) DECIMAL REPRESENTATION OF RATIONAL NUMBERS We n express ny rtionl numer in deiml form. For exmple, we n write 2 s 0:. However, when we onvert rtionl numers to deimls, the result my either e terminting deiml or reurring deiml. TERMINATING DECIMALS A terminting deiml hs only finite numer of non-zero digits fter the deiml ple. For exmple, 4:26 is terminting deiml, s it finishes or termintes fter deiml ples. When written in lowest terms, rtionl numer will onvert to terminting deiml if its denomintor hs no prime ftors other thn 2 or. For exmple, 20 hs denomintor 20 = 22, whih hs only 2 nd s its prime ftors. So, 20 onverts to terminting deiml. In ft, 20 = 0:. We n onvert frtions like these to terminting deimls y first writing the frtion so its denomintor is power of 0. Exmple 6 Write the following in deiml form: 2 8 = 2 2 = 6 0 =0:6 2 = = =0:28 8 = = =0:62 RECURRING DECIMALS If you enter into your lultor, your lultor will give the nswer =0:. Your lultor n only show finite numer of digits, ut the series of s fter the deiml tully ontinues forever. This is n exmple of reurring deiml. Reurring deimls repet the sme sequene of digits without stopping. A rtionl numer in lowest terms will onvert to reurring deiml denomintor hs t lest one prime ftor other thn 2 or. if its

17 REAL NUMBERS (Chpter ) 6 8 For exmple, =0: :::: is reurring deiml sine the denomintor hs the prime ftor. We indite reurring deiml with line over the repeted digits. For exmple, =0: nd 8 =0:2. There my e some non-repeting digits in the deiml efore the repeting digits strt. For exmple, =0:8 :::: =0:8. 6 Exmple Write s deimls: 9 9 =0: :::: =0: =0:444 :::: =0:4 0 : :::: 9 : :::: 0 : 444:::: : :::: Some deimls tke long time to reur. For exmple, = 0: EXERCISE E Show tht the following numers re rtionl y writing them in the form, where nd re integers. p 6 d 4 e 0: f 2 4 g p 49 p 9 h 0:0 2 Without onverting them, stte whether these rtionl numers will onvert to terminting deiml or reurring deiml. e f 6 Use your lultor to hek your nswers to 2. 4 Convert these frtions to terminting deimls: Convert these frtions to reurring deimls: g d 2 d 6 e 0 e 2 d 40 h 8 f 8

18 62 REAL NUMBERS (Chpter ) 6 Use your lultor to write these frtions s reurring deimls: 6 Disussion d 22 9 Oliver noties tht hs prime ftor of in the denomintor, so he thinks it will onvert to reurring deiml. However, when he enters it into his lultor, he finds 9 tht =0:6. Cn you explin Oliver s mistke? We hve seen tht 0: =. Wht n we sy out 0:9? e 2 8 We hve seen tht when rtionl numer is written in lowest terms, if its denomintor only hs 2 nd s its prime ftors then it will onvert to terminting deiml. For exmple: =0:2 termintes fter 2 deiml ples 4 =0: termintes fter deiml ple. 0 If we re given rtionl numer suh s these, is there wy to predit how mny deiml ples its terminting deiml will hve? Wht to do: Copy nd omplete the tle elow: Rtionl numer Denomintor Ftoristion of denomintor Numer of 2s in ftoristion Numer of s in ftoristion Deiml Numer of deiml ples 0 2 0: :

19 REAL NUMBERS (Chpter ) 6 2 Cn you see how to predit the numer of deiml ples in the terminting deiml? Use your nswer to 2 to predit the numer of deiml ples in: d 9 2 Chek your nswers using your lultor. 4 Find rtionl numer whih termintes fter deiml ples. Chek your nswer using your lultor. F IRRATIONAL NUMBERS An irrtionl numer is numer tht nnot e written in the form, where nd re integers. p 99 An exmple of n irrtionl numer is 2 ¼ : ::::. The frtion 0 =:4428 is sometimes used to pproximte p 2, ut it is not ext. Another exmple of n irrtionl numer is ¼ ¼ :4 9 ::::, red s pi. In the Opening Prolem, Qi-Zheng ws lulting the surfe re of irulr pond with rdius m. His fmily m helped him with different pproximtions, ut the ext nswer he wnted ws ¼ m 2. Interestingly, the distne round the p m2 oundry of the pond is tully 2¼ m! 2p m Together, the rtionl numers nd irrtionl numers mke up the set of ll rel numers whih n e pled on numer line. rtionl numers -~`` p ~ Qw 0.` ~`2 2 p ~ Qw irrtionl numers DECIMAL REPRESENTATION OF IRRATIONAL NUMBERS In the previous setion, we sw tht rtionl numer onverts to either terminting deiml or reurring deiml. The deiml representtion of n irrtionl numer neither termintes nor reurs. For exmple, the first 0 deiml ples of ¼ re: ¼ ¼ : ::::. We ould keep writing these digits forever, nd they would never terminte nor get into loop of repeting digits. A omputer hs een used to lulte ¼ to trillion deiml ples!

20 64 REAL NUMBERS (Chpter ) Investigtion Deiml representtion of p 20 p 20 is n irrtionl numer. In this investigtion we will look t the deiml representtion of p 20, nd gin n understnding s to why it will ontinue forever. p 20 is the numer whih, when squred, gives 20. Now, 4 2 =6 nd 2 =2, so p 20 is etween 4 nd. Also, 4:4 2 =9:6 nd 4: 2 =20:2, so p 20 is etween 4:4 nd 4:. Thus, p 20 ¼ 4:4 ::::. Wht to do: Show tht p 20 is etween 4:4 nd 4:48, so p 20 ¼ 4:4 ::::. 2 Find the first 6 deiml ples of p 20. Chek your nswer y evluting p 20 on your lultor. Cn you see how we ould keep finding more deiml ples indefinitely? EXERCISE F Stte whether the following numers re rtionl or irrtionl. It my help to view the numers in deiml form using your lultor. 6 p 0:2 d ¼ e p 4 f ¼ 2 g h p Ple the numers in on numer line. Suppose ertin numer is irrtionl. Are the following sttements true or flse? The negtive of the numer must lso e irrtionl. more thn the numer must lso e irrtionl. Doule the numer must lso e irrtionl. d The squre of the numer must lso e irrtionl. 4 Give n exmple of n irrtionl numer etween 0 nd. Is the sum of two irrtionl numers lwys irrtionl? Explin your nswer. Disussion ² Cn you show tht there re infinitely mny rtionl numers? ² Are there infinitely mny irrtionl numers? ² Are there more rtionl numers thn irrtionl numers?

21 REAL NUMBERS (Chpter ) 6 Review set Ple the frtions 4, 4, 4, 2 4 on numer line. 2 Express in lowest terms: d e 8 6 Find: 6 4 2:6 4 9 d 0:02 0:6 e 9 2 f 0:0028 0:4 4 Stte the vlue of the digit in: 2: :22 0:000 d 4:02 Round to two deiml ples: 6:284 0:2 0:09 6 Show tht the following numers re rtionl y writing them in the form, where nd re integers. 8 0: 2 d p 8 e 6 2 f : g 0:9 h 0: Write the following rtionl numers s terminting deimls: d 2 0 e 40 8 Use your lultor to evlute: 9:6 2:4 (:8) d e 0:042 0: f Without using your lultor, determine whether the following rtionl numers will onvert to terminting deiml or reurring deiml. i v ii vi iii vii iv viii Use your lultor to write the numers in s terminting or reurring deimls. 0 Is the produt of two rtionl numers lwys rtionl? Explin your nswer.

22 66 REAL NUMBERS (Chpter ) Betty ked lemon slie on Sturdy. She te 6 of it, nd her greedy rother te. Wht frtion of the slie hs een eten? Express the portion Betty te s frtion of the portion her rother te. 2 At Gry The Greengroer, wtermelons re sold in hlves nd qurters, s well s whole melons. Gry hs melons tht he wishes to ut into 2 qurter melons. How mny qurter melons will he hve? The sles of wtermelons one dy were: whole melons hlf melons 9 qurter melons How mny wtermelons did Gry sell in totl? Crolyn ought tumlers nd wter jug from homewres store. The tumlers ost $6:8 eh nd the wter jug ost $8:90. How muh did Crolyn spend? 4 Iridium hs density of 22:6 grms per m. Aluminium hs density of 2: grms per m. How mny times more dense is iridium thn luminium? Round your nswer to deiml ple. Prtie test A Clik on the ion to otin this printle test. PRINTABLE TEST Prtie test B Ple the deimls 0:2, 0:6, 0:4, :, : on numer line. 2 Find Determine whether the following numers re rtionl or irrtionl, giving resons for your nswers: ¼ :4 9 p 2 d 0: 4 Find the ost of uying 0:9 kg of nns pried t $2:0 per kilogrm, nd 0: kg of pers pried t $:20 per kilogrm. Find: :6 6 :6 0:6

23 REAL NUMBERS (Chpter ) 6 6 Write 9 20 in deiml form. Is this terminting or reurring deiml? Write 08 8 s frtion in lowest terms. 8 Ken hd 4 ns of pint left in his shed. After 2 pinting his house, he only hd 2 ns of pint left. How muh pint did Ken use? 9 Without using your lultor, write reurring deiml of Peter s grden eds re used to grow fruit nd vegetles. Of these, of the 4 growing spe is tken up y eggplnts. Wht frtion of Peter s grden eds is tken up y eggplnts? s Prtie test C In hmmer throw event, Brooke threw 4: m nd Bronwyn threw 8:6 m. Wht ws the omined distne of the two throws? How muh longer ws Bronwyn s throw thn Brooke s throw? d Bronwyn s shotput distne ws of her hmmer throw distne. How fr did she 4 throw the shotput? Brooke injured herself nd ould only mnge shotput distne of 6:6 m. Wht frtion of Bronwyn s shotput throw is this? Give your nswer in simplest form. 2 Write s reurring deiml. Use your nswer to to help evlute: i 0:2 0: ii 0:2 0: iii 0: 0:2 iv 0: 0:2 nd hve reurring deiml expnsions. Use these numers to give n exmple where reurring deiml divided y reurring deiml is: i reurring ii terminting. In my fmily there re four people. My fther hs n priot pie whih he divides etween us. He tkes qurter of the pie for himself, nd then gives my mother third of wht is left. My sister gets hlf of wht is left fter tht, nd I get ll of the rest. How muh pie is left fter my fther hs tken his shre? How muh of the pie does my mother get? How muh of the pie does my sister get? d Is my fther s method of shring fir? Explin your nswer.

24 68 REAL NUMBERS (Chpter ) 4 Mrs Austen is hving troule with her printer. It printed the first qurter of her ook in red, the next sixth in lue, the next seventh in red gin, nd ll of the rest in lk. How muh of the ook hs lk printing? If the ook hs 22 pges, wht pge numers re in lue? Mrs Austen deides tht she would like to keep one of the olours, nd reprint the rest of the ook in tht olour. Whih olour should she hoose to use the lest mount of pper? Eenezer ought some presents for his friends. He pid $4:0, $4, $:04, $:4, $4:4, nd $:4. i List his purhses from the hepest to the most expensive. ii Wht ws the differene in prie etween the hepest nd the most expensive item? iii How muh did he spend ltogether? Meliss ought some gs of hips for $: eh. She reeived ents hnge from $0 note. How mny gs of hips did Meliss uy?

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

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