RETURN TO MAIN MENU ver-increasing computer calculation speed used for games such as Tomb Raider

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2 E vr-incrasing computr calculation spd usd for gams such as Tomb Raidr mans that Lara Croft outprforms any charactr from th past. Th first succssful high-spd lctronic digital computr, ENIAC (lctronic numrical intgrator and computr), was usd from 1946 to It had vacuum tubs, took up 170 squar mtrs of floor spac, and probably took all day to prform th sam numbr of calculations as th programmrs us to mak Lara blink. Evn modrn day non-scintific calculators outprform ENIAC and all of this is don using only two digits, 0 and 1! Rcntly, Japan s Earth Simulator suprcomputr brok th spd rcord to prform 40 trillion calculations a scond. What would Lara mak of that? hi.com.au 1

3 Worksht R1.1 Worksht R1.2 Worksht R1.3 Worksht R1.4 Worksht R1.5 Worksht R1.6 Prpar for this chaptr by attmpting th following qustions. If you hav difficulty with a qustion, click on th Rplay Worksht icon on your Maths Zon CD or ask your tachr for th Rplay Worksht. 1 Writ down th answr to ths tabls qustions as quickly as possibl. (a) 7 8 (b) 9 9 (c) 4 6 (d) () 4 7 (f) 8 9 (g) 12 8 (h) 9 7 (i) Choos th corrct answr. What is th plac valu of th rd numbr? (a) A ight B thousands C tns D ighty (b) A tn thousands B two C hundrds D hundrd thousands (c) 3490 A units B tns C hundrds D zro 3 St out ths calculations in your normal way and work out th answrs. (a) (b) (c) St out ths calculations in your normal way and work out th answrs. (a) (b) (c) St out ths calculations in your normal way and work out th answrs. (a) (b) (c) St out ths calculations in your normal way and work out th answrs. (a) (b) (c) Babylonian systm binary systm Chins systm dcimal systm Egyptian systm stimat Hindu Arabic systm magic squar magic sum ordr of oprations Roman systm rounding 2 MATHS ZONE 7

4 Th systm of numbrs w us is calld th Hindu Arabic systm. W count using th bas of tn. Our numbr systm startd around BC in India and was brought to Europ in th Middl Ags by th Arabs. It s now usd all ovr th world, but many othr numbr systms wr onc usd. Th oldst counting machin known is a yar-old wolf bon found in a country thn known as Czchoslovakia, in It had 55 notchs in two rows dividd into groups of fiv. Th ag of th bon mans that counting machins wr invntd vry arly in human history. Many popl usd counting systms basd on parts of thir bodis. In th nintnth cntury, som Torrs Strait islandrs countd by touching parts of thir bodis Th Egyptian systm Th ancint Egyptians of around 3000 BC drw complx symbols, or hiroglyphics, for thir numbrs. Th symbols could b placd from lft to right or from right to lft or from top to bottom Torrs Strait systm on stick two sticks an arch a coil of rop a lotus flowr a bnt rd a tadpol a gni Exampls: 23 is is 1 whol NUMBERS 3

5 Th Roman systm Around 300 BC, th Romans usd lttrs as thir symbols for numbrs. Th Roman systm also mad it asir to writ numbrs lik 4 and 9. How? I II III IV V VI VII VIII IX X L C D M Exampls: 14 is XIV 39 is XXXIX 76 is LXXVI 2646 is MMDCXLVI Look at th following: 4 is IV 40 is XL 9 is IX 90 is XC A lttr bfor a highr valu lttr mans w tak it away from th highr valu. But 49 is not IL. Instad w hav to writ XLIX for 49, XL for 40 and IX for 9. On of th ruls in th Roman systm is that you can only rduc th highr valu by th nxt smallr valu. Also, you ar not allowd to rduc by 5, 50 or 500. Th Babylonian systm About 2100 BC, th ancint Babylonians dvlopd a systm which was unusual bcaus it was not basd on tns. S if you can work out what th Babylonian systm is basd on. All thir numbrs ar wdg shapd bcaus thy wrot by prssing wdg-shapd sticks into damp clay Exampls: 32 is 74 is 130 is is 315 is MATHS ZONE 7

6 Th modrn Chins numbr systm In China th Hindu Arabic systm has bn adoptd for many uss, but many Chins popl still us thir charactr systm. Th Chins systm is also basd on tns. Th Chins writ from top to bottom and writ two symbols for ach numbr gratr than 9. Ths two symbols show how many lots of tn, on hundrd tc. ar ndd. Th numbr 24 is writtn as two, tn and four. 132 is writtn as on, hundrd, thr, tn, two, but 14 is just writtn as tn, four, not on, tn, four. Thr is a Chins charactr for zro but bcaus a circl is oftn usd that is what w will us hr. A simplifid vrsion of th charactrs is: Exampls: 19 is 270 is 8005 is xrcis 1.1 Numbr systms Cor Prparation: Prp Zon Q2 1 Writ out ths numbrs using th Egyptian numbr systm. (a) 12 (b) 14 (c) 600 (d) 500 () 24 (f) 31 (g) 172 (h) 253 (i) (j) (k) (l) Writ out ths numbrs using th Roman numbr systm. (a) 13 (b) 12 (c) 20 (d) 30 () 19 (f) 29 (g) 2341 (h) 3132 (i) 629 (j) 439 (k) 3646 (l) 2466 (m) 1980 (n) 1959 (o) 1999 (p) Writ out ths numbrs using th Babylonian numbr systm. (a) 20 (b) 40 (c) 45 (d) 56 () 63 (f) 62 (g) 75 (h) 84 (i) 102 (j) 91 (k) 144 (l) 162 hi.com.au Tstr 1 whol NUMBERS 5

7 4 Writ out ths numbrs using th modrn Chins numbr systm. (a) 4 (b) 37 (c) 126 (d) 270 () 18 (f) 823 (g) 1053 (h) Writ down what numbr systm ach numbr blow is writtn in and what th numbr is in th Hindu Arabic systm. (a) XI (b) (c) CXLIII (d) Worksht C1.1 () (f) (g) (h) CCXCII (i) (j) (k) (l) (m) (n) (o) DCIV (p) (q) MMDCLXIV (r) (s) MMMCDXXVIII (t) 6 Writ ach of ths Hindu Arabic numbrs in: (i) Egyptian numbrs (ii) Roman numbrs (iii) Babylonian numbrs (iv) Chins numbrs (a) 15 (b) 65 (c) 92 (d) 300 () 199 (f) 236! Extnsion 7 (a) Which on of th thr ancint numbr systms is still usd? (b) Whr hav you sn this numbr systm usd? (c) What do th Babylonian, Chins and Hindu Arabic systms hav that th othr two don t hav? (d) Which of th fiv numbr systms usually taks th longst tim to writ down? () Which usually taks th shortst tim to writ down? (f) In which of th fiv numbr systms is th position of th numbrs not important? (g) Why do you think most numbr systms wr basd on fivs or tns? (h) Nam somthing that you dal with vry day which is basd on sixtis. (i) Why do you think th Hindu Arabic systm is now th on usd around th world? 8 Choos a numbr btwn 1500 and 2000 and writ it using ach of th modrn Chins, Egyptian, Babylonian and Roman numbr systms. Worksht A1.1 6 MATHS ZONE 7

8 Copy ths Roman numrals and writ thm in Hindu Arabic form, thn arrang th lttrs in th ordr shown by th corrsponding answrs to find th cartoon caption. XIX E CM O CLI Y DII U CCXI A MDLX S CDXXI N MCDIX H CCCLXIV T DCXXVIII G MDCCXLII W MCMXXIV B CDXXVIII R MMDCXLV I MLXXIX D whol NUMBERS 7

9 In primary school you would no doubt hav spnt many hours practising th four basic oprations of mathmatics addition, subtraction, multiplication and division. In scondary school you ar xpctd to b abl to apply ths basic skills to situations whr th qustion is hiddn in a littl story. In qustions lik this you nd to rad through th whol qustion first and thn dcid which opration you will nd to us to find th answr. workd xampl 1 On 1 January th population of Summrtown was During th yar, 1987 popl did, 1245 babis wr born, 4324 popl lft th town, and 3876 movd in. (a) Find th total numbr of popl who did or lft th town. (b) Find th total numbr of popl who wr born or movd into th town. (c) Find th total numbr of popl in th town at th nd of th yar. (d) Clarly xprss th total chang in population. Stps (a) Add togthr th popl who did and th popl who lft. (b) Add togthr th popl who wr born and th popl who movd into th town. (c) 1. Subtract from th original population th total who lft. 2. Add to this rsult th total of babis born and popl who movd in. (d) 1. Th final population is lowr than th original population so w will nd to subtract th nw from th old. 2. Writ a short statmnt to answr th qustion. Solutions (a) (b) (c) (d) Th final population is 1190 lss than th original. 8 MATHS ZONE 7

10 xrcis 1.2 Whol numbr problms Cor Prparation: Prp Zon Q1 and Th highst mountain in th world, masurd from sa lvl, is th Himalayan pak of Mount Evrst. It is 8848 m abov sa lvl. If w masur mountains which start undr th ocan, th highst mountain in th world from bas to tip is Mauna Ka on th island of Hawaii. Its total hight is m, of which 4205 m is abov sa lvl. (a) How much highr is Mauna Ka than Mount Evrst? (b) How much of Mauna Ka is blow sa lvl? (c) If w don t count th part of Mauna Ka which is undr watr, how much highr is Mount Evrst? 2 Th two longst rivrs in th world ar th Amazon (6448 km) and th Nil (6670 km). Th longst rivr in Australia is th Darling (2739 km). Animation 1 whol NUMBERS 9

11 (a) How much longr is th Nil than th Amazon? (b) How much longr is th Nil than th Darling? (c) How much longr is th Amazon than th Darling? 3 Th following tabl shows a numbr of invntors and whn thy mad thir famous invntions. Invntion Invntor Yar Calculator Blais Pascal ( ) 1642 Thrmomtr Gabril Fahr nhit ( ) 1714 Parachut Louis Lnormand ( ) 1783 Camra Nicphor Nipc ( ) 1822 Tlgraph Samul Mors ( ) 1837 Tlphon Alxandr Graham Bll ( ) 1876 Car ngin Gottlib Daimlr ( ) and Karl Bnz ( ) 1887 Radio Guglilmo Mar coni ( ) 1895 Tlvision John Logi Bair d ( ) 1925 Lasr Thodor Maiman (1927 ) 1960 (a) How long aftr th tlgraph was th tlphon invntd? (b) How long aftr radio was tlvision invntd? (c) How old was Gabril Fahrnhit whn h invntd th thrmomtr? (d) How long bfor th lasr was th camra invntd? () Gottlib Daimlr and Karl Bnz invntd th car ngin indpndntly. How old was ach man whn h invntd it? (f) What ag did Blais Pascal liv to? (g) Who livd longr, Nicphor Nipc or Louis Lnormand, and by how much? 10 MATHS ZONE 7

12 (h) Who out of all th invntors cam up with thir invntion at th youngst ag? (i) Draw a timlin from 1600 to 2000 to rprsnt th information in th tabl. 4 If a car uss 8 litrs of ptrol for vry 100 km that it travls, how many litrs would it us for a trip of: (a) 900 km (b) 1200 km (c) 1500 km (d) 1050 km () 725 km? 5 Harvy Scoop Robrts, a journalist with th Monthly Farm Nws, can typ 50 words a minut. How long dos it tak him to typ an articl of 1800 words? 6 Kathy and hr brothr Hrbrt both hav papr rounds. Thy hav to start and finish at th shop. Th routs of both thir rounds and th lngths of ach strt sction thy covr ar shown on th following diagram. (a) How long is Kathy s papr round rout? (b) Whos papr round rout is longr, Kathy s or Hrbrt s, and by how much? (c) How long would Kathy s papr round rout b if Hrbrt was sick and sh had to do his as wll? (d) If Kathy could go straight hom aftr hr papr round and not hav to go back to th shop, approximatly how long would hr papr round b? () If Kathy wnt straight to school aftr hr papr round, approximatly how far would sh travl? Animation Kathy s rout 398 m Hrbrt s rout Hom 243 m Shop 632 m 647 m 526 m 212 m 647 m 435 m 605 m 412 m 281 m School 435 m 360 m 292 m 1 whol NUMBERS 11

13 7 Arni th body buildr stands on a st of scals whil h is holding two 3 kg dumb-blls. Th scals show a wight of 102 kg. How much dos Arni wigh? Extnsion 8 Stavros wants to buy som marmalad. On jar in th suprmarkt is 250 g and costs $1.25; anothr is 500 g and costs $2.25. Which on is bttr valu? 9 Littl Lucy is btwn 3 and 4 wks old. Giv thr possibl valus for hr ag in minuts. 10 Juls Vrn wrot about travlling around th world in 80 days. About how many wks is that? 11 Th lngth of a painting including th fram is 85 cm. If th fram is 6 cm wid all th way around, what is th lngth of th unframd painting? 12 MATHS ZONE 7

14 12 Th Pizza Pit-Stop mploys fiv popl. Th two cooks work 36 hours ach pr wk for $12 an hour, and th thr waitrs work 30 hours ach pr wk for $11 an hour. What dos th Pizza Pit-Stop pay its fiv mploys in total pr wk? 13 Wndy is training to b an Olympic swimmr. Evry morning sh swims 3600 m in a 50 m pool. How many laps is that? 14 For th numbrs 108 and 9, find th: (a) sum (b) diffrnc (c) product (d) quotint 15 What is th sum of 42 and 76 addd onto th product of 42 and 76? 16 What is th rsult whn th diffrnc btwn 9864 and 8 is addd onto th quotint of 9864 and 8? 1 whol NUMBERS 13

15 What numbr am I? I am a two-digit numbr gratr than 50. Th product of my digits is not 12, but 12 gos into it xactly. Th sum of my digits is odd. Th sum of my digits is lss than 13. What am I? What am I if I am lss than 50? Brak it into smallr stps. First, find what th product of th digits could b, givn that 12 gos into it. Magic squars hav fascinatd popl for ovr 4000 yars. A magic squar is a squar of numbrs in which vry row, column and diagonal adds up to th sam total. This total is calld th magic sum. Thy wr calld magic bcaus popl usd to bliv th squars had mystical powrs. On of th simplst magic squars is a 3 3 squar using th digits 1 to 9, whr all th totals ar 15. Th on shown hr is calld th Lo-Shu magic squar and according to lgnd was brought by a turtl from th rivr Lo to th Chins Empror Lu around 2200 BC MATHS ZONE 7

16 xrcis 1.3 Magic squars Prparation: Prp Zon Q3 and 4 Cor 1 Copy ths magic squars into your book and find th missing numbrs. (a) (b) (c) (d) () (f) hi.com.au Worksht C1.2 Brak it into smallr stps. Find th total of on lin bfor filling in th numbrs. 2 (a) Compar your answr to Qustion 1(a) with th Lo-Shu magic squar on pag 14. Somthing was addd to ach of th numbrs in th Lo-Shu squar to gt your answr. What was it? (b) What can b don to th Lo-Shu magic squar to gt your answr to Qustion 1(b)? (c) What can b don to th Lo-Shu magic squar to gt Qustion 1(c)? (d) What can b don to th Lo-Shu magic squar to gt Qustion 1(d)? () What can b don to th Lo-Shu magic squar to gt Qustion 1()? (f) What can b don to th Lo-Shu magic squar to gt Qustion 1(f)? (g) If you add 8 to ach of th numbrs in th Lo-Shu squar, what will b th magic sum of th nw squar? (h) If you add 100 to ach of th numbrs in th Lo-Shu squar, what will b th magic sum of th nw squar? Extnsion 3 Magic squars wr first invstigatd in Europ during th fiftnth cntury. Th Grman artist Albrcht Dürr ( ) mad a woodcut calld Mlancholy which includs a 4 4 magic squar (shown on th nxt pag). (a) Th yar in which Dürr mad th woodcut appars in th magic squar. What is it? How old was h whn h mad Mlancholy? (b) What is th magic sum of th Mlancholy squar? (c) What do th four numbrs that mak up th 2 2 squar in th top lft-hand cornr of th Mlancholy squar add up to? (d) How many othr 2 2 squars can you find insid th Mlancholy squar that add up to th magic sum? 1 whol NUMBERS 15

17 () Mak a 4 4 magic squar of your own by adding 2 to ach of th numbrs in th Mlancholy squar. What is th magic sum? (f) If you mad a nw magic squar by adding 5 to ach of th numbrs in th Mlancholy squar, what would th magic sum b? (g) Mak a nw magic squar by swapping th first column of th Mlancholy squar with th fourth column. What is th magic sum of th nw squar? (h) Work out th following 4 4 (i) Complt th following magic squar and thn s if you 4 4 magic squar and can work out what was don to thn s how it was formd th Mlancholy squar to gt it. from your answr to part (h) magic sum = magic sum = MATHS ZONE 7

18 4 Th grat Amrican thinkr and invntor Bnjamin Franklin cratd a numbr of magic squars. On of th most famous was an 8 8 squar. (a) What is th magic sum of Franklin s squar? (b) Look at th half-rows, that is, th first four numbrs in ach row and th last four numbrs in ach row. What do you notic? (c) What do you notic about th half-columns? (d) Look at th 4 4 squars in ach of th four cornrs. Ar thy magic squars? () Look at any 2 2 squar insid Franklin s squar and add up th four numbrs. Do th sam for anothr 2 2 squar. What do you notic? 5 Construct two 3 3 magic squars using any from Qustion 1 as a bas. Homwork 1.1 Worksht A1.2 Worksht E1.1 Numbr pyramids To gt ach numbr in th pyramid, add th two numbrs blow it whol NUMBERS 17

19 Cross-numbr totals This cross-numbr puzzl has bn filld out using th numbrs 1 to 5 xactly onc so that th total along ach lin is nin. Whn you ar trying to solv a cross-numbr total, us pncil to writ in th numbrs so you can rub thm out if ncssary xrcis 1.4 Numbr pyramids and cross-numbr totals Prparation: Prp Zon Q3 and 4 Cor 1 Copy th following numbr pyramids into your book and fill in th missing numbrs. (a) (b) (c) 34 (d) Copy th following numbr pyramids into your book and fill in th missing numbrs. (a) (b) (c) 14 (d) Can you find a pattrn which hlps you solv ths? 3 Mak up thr numbr pyramids of your own and giv thm to a frind to work out. 18 MATHS ZONE 7

20 4 Copy this cross-numbr total into your book thr tims. (a) Us th numbrs 1 to 6 xactly onc so that ach lin adds up to 9. (b) Us th numbrs 1 to 6 xactly onc so that ach lin adds up to 10. (c) Us th numbrs 1 to 6 xactly onc so that ach lin adds up to 11. (d) Onc you hav found an answr in ach cas, how is it possibl to find othr answrs fairly asily? 5 Copy ths cross-numbr totals into your book and try to solv thm. (a) Us th numbrs 1 to 7 xactly onc so that ach lin adds up to 12. (b) Us th numbrs 1 to 9 xactly onc so that ach lin adds up to 15. (c) Us th numbrs 1 to 11 xactly onc so that ach lin adds up to 18. (d) Us th numbrs 1 to 13 xactly onc so that ach lin adds up to th sam thing. To find what th cross-numbr puzzl has to add up to, look at th prvious qustions and s if you can find a pattrn. Extnsion 6 (a) Look at your answrs to Qustion 5. Can you find a pattrn that will giv you a quick way of finding which numbr gos in th middl? Which numbr would go in th middl of a cross-numbr puzzl of th numbrs from 1 to 15? What about for 1 to 23? (b) Can you find any othr pattrn to mak it asir to solv this sort of cross-numbr total? What is it? 7 Copy this cross-numbr total into your book and us th numbrs from 1 to 7 xactly onc so that ach lin has th sam total. Worksht E1.2 1 whol NUMBERS 19

21 Estimating Popl mak stimats of things all th tim: Th distanc from hr to th shops is about 2 kilomtrs. I had about 30 popl at my party. Th SCG holds around popl. Rounding to th first digit Thr ar 700 studnts at my school. Is that 692 or 704? Ar thr xactly 700 studnts at Al s school? Th figur is probably diffrnt from th actual numbr of studnts in th school du to rounding. Th xact numbr of studnts in th school may b 692 or 704. In fact it could hav bn any numbr from 650 to MATHS ZONE 7

22 Rounding to th first digit Whn rounding, follow th rul if you r caught right in th middl, go up. For xampl, 750 rounds to 800 (not 700) and 3500 rounds to 4000 (not 3000). workd xampl 2 Round th following numbrs to th first digit. (a) 361 (b) 2050 (c) 8 Stps (a) Look at th scond digit. Bcaus it is 5 or gratr, rais th first digit by on and rplac th following digits with zros. (b) Look at th scond digit. Bcaus it is lss than 5, kp th first digit and rplac th following digits with zros. (c) Bcaus thr is only on digit this numbr is alrady roundd to th first digit. Th symbol mans approximatly qual to. Solutions (a) (b) (c) 8 Estimating multiplication Oftn you don t nd to know th xact answr to a multiplication problm. An stimat that you can do in your had or with vry littl work will oftn do. workd xampl 3 Estimat Stps Solution 1. Round off both numbrs to th first digit Multiply th first digits. 4 5 = Count th numbr of zros altogthr in thr 0s (two in 400 and on in 50) stp Writ down th numbr from stp 2 and put th numbr of zros from stp 3 aftr it. 5. So is approximatly This is writtn as whol NUMBERS 21

23 Estimating division Division problms can also b stimatd using rounding. workd xampl 4 Estimat Stps Solution 1. Round off both numbrs to th first digit Cancl off th sam numbr of zros on both sids. 3. Do th simpl division ) = So is approximatly 175. This is writtn as xrcis 1.5 Estimating and rounding Prparation: Prp Zon Q1, 2, 5 and 6 Cor 1 Round ths numbrs to th first digit. (a) 68 (b) 74 (c) 483 (d) 4846 () 3723 (f) 619 (g) (h) (i) 970 (j) 6 (k) (l) (m) 10 (n) 9643 (o) 9510 (p) Which of th fiv stimats in ach cas do you think is th closst? Don t try to count thm. (a) Roughly how many popl ar thr in th pictur? Worksht C1.3 Tstr A 2000 B C 7000 D MATHS ZONE 7

24 (b) Approximatly how far is it dirctly from Uluru to Alic Springs? A 63 km B 789 km C 350 km D 40 km Scal 1 : km Hrmannsburg N Alic Springs Kata Tjuta Uluru Erldunda South Australia (c) About how many dots ar thr in this squar? Try to us a pattrn. A 7000 B 1000 C D 500 (d) Roughly how many stars ar thr in this pictur? A 50 B 300 C 3000 D whol NUMBERS 23

25 () (i) How many straight lins ar thr in this pictur? A 1000 B 200 C 7000 D (ii) Try to work out a way of finding th xact numbr of lins without counting thm all. 3 Us rounding to th first digit to find th stimats for ths multiplications. (a) (b) (c) (d) () (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) In ach cas choos th bst stimat from th altrnativs givn. Don t do th actual multiplication. (a) A 210 B 2100 C 2163 D (b) A 2400 B 1500 C D (c) A B C D (d) A B C D () A B C D Animation Worksht C MATHS ZONE 7

26 5 (a) Work out stimats for: (i) (ii) (b) What do you notic about your rsults for part (a)? (c) (i) Work out th xact valus for th two multiplications in part (a). (ii) How far off ar th stimats in ach cas? (iii) Which on was th closr stimat? Why do you think this is? 6 (a) Work out stimats for: (i) (ii) (b) (i) Work out th xact valus for th two multiplications in part (a). (ii) How far off ar th stimats in ach cas? (iii) Which on was th closr stimat? Why do you think this is? 7 Us rounding to th first digit to find stimats for ths quotints. (a) (b) (c) (d) () (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r) In ach cas choos th bst stimat from th altrnativs givn. Don t do th actual division. (a) A 900 B 1600 C D 400 (b) A 100 B 1000 C D 10 (c) A 350 B 200 C 700 D 50 (d) A 740 B 2000 C 330 D 220 () A 100 B 10 C D 300 You can t gt a numbr roundr than m! Extnsion 9 (a) Estimat: (i) (ii) (b) Dos it mak mor sns to lav 11 and 12 as thy ar than to round thm to 10? Why or why not? 10 Writ down thr pairs of numbrs that whn roundd to th first digit thn multiplid togthr giv: (a) 3200 (b) 200 (c) (d) Copy th following tabl and complt it, using rounding to th first digit. Us your calculator to work out th xact answr aftr you v found th roundd answr. Worksht C1.5 1 whol NUMBERS 25

27 Qustion Roundd qustion Roundd answr Exact answr (a) (b) (c) (d) () (f) Without using your calculator, dcid which of th altrnativ answrs givn is th bst stimat. Chck your answr by using your calculator. (a) A 6700 B C D (b) A 1400 B 567 C D (c) A 1100 B C D (d) A B 4000 C D () A 1200 B C D (f) A 7000 B 70 C D 700 (g) A 2500 B C D (h) A 8000 B 780 C 67 D Qustions Homwork 1.2 Worksht A1.3 Max, Minh, Al and Polly all workd out th following problm (8 3) 1 2 Max s answr was 14. Minh s answr was 6. Al s answr was 8. Polly s answr was 18. Why wr thr so many diffrnt answrs? Obviously w can t hav diffrnt popl gtting diffrnt answrs in mathmatics. Popl hav to agr on th ruls about working things out. That s why mathmaticians cam up with som ruls about th ordr in which to do th four oprations. 26 MATHSZONE 7

28 Ths ar calld th ordr of opration ruls. Th ordr of opration ruls 1 Always do th parts in brackts first. 2 Do th multiplication and division nxt, in ordr from lft to right. 3 Do th addition and subtraction nxt, in ordr from lft to right. Who had th corrct answr to th problm at th start of this sction: Max, Minh, Al or Polly? workd xampl 5 Simplify Stps 1. Do multiplication and division in th ordr in which thy appar. 2. Do addition and subtraction in th ordr in which thy appar. Solution = = = 27 4 = 23 workd xampl 6 Simplify (2 + 2) 3. Stps Solution 1. Do th brackts first (2 + 2) 3 = Do multiplication and division in th ordr in which thy appar. 3. Do addition and subtraction in th ordr in which thy appar. = = = = 9 xrcis 1.6 Ordr of oprations Prparation: Prp Zon Q1 Cor 1 Simplify. (a) (b) (c) (d) () (f) Intractiv Tstr 1 whol NUMBERS 27

29 (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) 9 (10 7) 3 (r) 24 (7 + 5) 6 (s) (5 4) (t) (10 4) (u) (v) (w) (5 1) (x) (17 2) Stat TRUE or FALSE for th following. (a) For w would do first. (b) For w would do 9 4 first. (c) For w would do 12 3 first. (d) For w would do 60 5 first. () For (3 + 5) 10 w would do first. (f) For w would do 6 2 first. (g) simplifis to 8. (h) simplifis to 3. 3 (a) What would you do first and what would you do scond in ach of ths qustions? (i) (4 2) 6 (2 2) (ii) ( ) 6 3 (iii) [(7 + 9) 2] 4 (iv) {[(21 17) 2] + 10} 1 (b) Simplify ach of th statmnts in part (a). 4 Put brackts into ths statmnts, whr ncssary, to mak thm tru. (a) = 36 (b) = 30 (c) = 25 (d) = 14 () = 10 (f) = 25 (g) = 6 (h) = 1 (i) = 5 (j) = 2 (k) = 0 (l) = 4 (m) = 13 (n) = 14 (o) = 2 (p) = 2 5 Rplac ach * with on of th four oprators (+,,, ) to mak th quation tru. (a) * 3 = 9 (b) 15 6 * 2 = 12 (c) 5 * 3 8 = 7 (d) 9 * = 13 () 14 8 * 6 = 0 (f) * 3 = 10 (g) 7 * 5 * 6 = 29 (h) 14 * 3 * 2 = 15 (i) (5 * 9) * 7 = 2 (j) (24 * 6) * 10 = 3 (k) 8 * 5 * 2 6 = 12 (l) 12 * * 9 = MATHS ZONE 7

30 6 Rplac ach * with ithr, or = to mak th statmnt tru. (a) 6 (4 2) 3 * (6 4) 2 3 (b) (1 + 4) 20 5 * 1 + (4 20) 5 (c) 8 + (5 3) 2 * (3 2) (d) * ( ) 10 () * 9 (2 + 0) (f) 36 6 (3 3) * Worksht C1.6 Rmmbr mans is lss than and mans is gratr than. Extnsion 7 Copy and complt th following using ach of th numbrs 1, 3, 4, 7 xactly onc for ach qustion. (a) * * * + * = 6 (b) ( * * ) * + * = 2 (c) * ( * * ) * = 9 (d) * + * * * = 6 () ( * + * ) ( * * ) = 20 (f) ( * * ) ( * * ) = 6 (g) ( * + * * ) * = 42 (h) * ( * + * * ) = 20 (i) * [( * + * ) * ] = 6 (j) [( * * ) * ] + * = 8 8 Mak up fiv qustions similar to thos in Qustion 7, and giv thm to a partnr to work out. Thn chck to s if your partnr is right. 8 Worksht C1.7 Th four 4s puzzl Using th numbr 4 xactly four tims, togthr with any of th four oprators (+,,, ) and brackts if you nd thm, s if you can mak thm qual th numbrs 0 to 9. Copy down th following. Th way to gt 5 has bn don for you. Thr is mor than on way in many cass. 0 = 5 = ( ) 4 1 = 6 = 2 = 7 = 3 = 8 = 4 = 9 = In a small group, s how many ways up to 100 you can find with just four 4s. Now for our nxt numbr It s usful to rmmbr som basics. For xampl, 4 4 = 1 and 4 4 = 0. hi.com.au 1 whol NUMBERS 29

31 Th multiplication targt gam What numbr do you hav to multiply th arrow by to gt a numbr within th targt rang? Exampl: Arrow: 53; Targt: Try 53 6 = 318: outsid th rang. Try 53 5 = 265: on targt. This has takn you two gusss to gt on targt. Us th following points systm to kp scor. Numbr of gusss On Two Thr Four Fiv or mor Points For th xampl abov you would gt four points. Do ths targt qustions and kp track of your total scor. What is th bst possibl scor? (a) Arrow: 38; Targt: (b) Arrow: 16; Targt: (c) Arrow: 29; Targt: (d) Arrow: 51; Targt: () Arrow: 47; Targt: (f) Arrow: 72; Targt: (g) Arrow: 824; Targt: (h) Arrow: 731; Targt: Worksht T MATHS ZONE 7

32 Maths is mad asir whn w know som stratgis to mak computations simplr. In this sction w will covr som of ths stratgis but thr ar many mor. You may alrady know som. Mntal and othr non-calculator stratgis usually involv daling with asir numbrs than thos obvious in th qustion, for xampl multiplying or adding numbrs to form multipls of tn. workd xampl 7 Us an appropriat mntal stratgy to hlp simplify ach of th following. (a) (b) Stps (a) 1. Look for ways to form asy numbrs. 2. Prform ths calculations first. Complt th qustion. (b) 1. Look for ways to form asy numbrs. 2. Prform ths calculations first. Complt th qustion. Solutions (a) = (2 5) 13 = = 130 (b) = 32 + (13 + 7) = = 52 Notic that it s asir if w mak tns first. workd xampl 8 Us an appropriat stratgy to hlp simplify ach of th following. (a) 9 15 (b) Stps (a) 1. Whn multiplying by 9 it is oftn asir to multiply by 10 first. 2. This givs on lot of 15 too many, so thn subtract 15 from th answr. (b) 1. Whn dividing by larg numbrs it can hlp to brak th numbr up into factors. Solutions (a) = = 135 So 9 15 = 135 (b) 18 = whol NUMBERS 31

33 2. Divid by ach of ths factors, on aftr th othr = = = 9 3. Writ th answr = 9 xrcis 1.7 Tutorial Mntal and non-calculator maths stratgis Cor 1 Us an appropriat stratgy to hlp simplify ach of th following. (a) (b) (c) (d) () (f) (g) (h) (i) (j) (k) (l) Us an appropriat stratgy to hlp simplify ach of th following. (a) 9 17 (b) 49 6 (c) 31 4 (d) 19 8 () (f) 61 7 (g) (h) 8 19 (i) 99 7 (j) (k) 91 7 (l) Us an appropriat stratgy to simplify ach of th following. (a) (b) (c) (d) () (f) (g) (h) (a) (i) Doubl ight, thn doubl your answr. What do you gt? (ii) Complt this statmnt: Doubling a numbr twic is th sam as multiplying th numbr by. (b) Simplify th following by doubling twic. (i) 13 4 (ii) 27 4 (iii) 32 4 (iv) 54 4 Extnsion Prparation: Ex Try to com up with a stratgy to hlp find th following. Writ down your answr and xplain what you did to find it. Not: Thr may b a rang of suitabl stratgis to choos from. (a) 15 8 (b) (c) Blow ar som mistaks studnts mad on a tst and how thy mad thm. Writ what ach studnt has don incorrctly and what th answr should b. (a) 21 7 = 161. Kat: I multiplid 7 by 20, and this gav m on lss lot of 21 than I ndd. So thn I addd 21. (b) 35 3 = 140. Sam: I doubld 35 thn doubld my answr to gt 140. (c) = 209. Lah: I first subtractd 56 to gt back to 200, and thn addd th rmaining 9. (d) = 214. Josph: I got 20 lots of 10 and addd 7 lots of MATHS ZONE 7

34 7 Us stratgis to find th following. (a) (b) (c) (d) () (f) (g) (h) (i) (j) (k) (l) Qustions Our usual counting systm is th dcimal systm. It rprsnts numbrs writtn to bas 10. Thr ar som numbr systms that us othr bass. On xampl is th binary systm (bas 2). This systm is commonly usd in computrs and lctronics. In lctronics thr ar only two signals that can b snt: an lctrical currnt (on) or no lctrical currnt (off). Th dcimal systm has tn digits (0 to 9) that can b usd in ach plac. In th binary systm ach plac can only b filld by on of th two digits 0 or 1. Look at this tabl, showing plac valus for th numbr 1101 in th dcimal systm. Hundrd-thousands Tn-thousands Thousands 1000 Hundrds 100 Tns 10 Units Th numbr 1101 mans 1 lot of 1000, 1 lot of 100, no lots of 10 and 1 unit. You can s that ach plac valu is worth tn tims mor than th plac to its right. Now compar plac valus for 1101 in th binary systm: You can s that ach plac valu is worth two tims mor than th plac to its right. Th numbr 1101 in th binary systm is writtn as (1101) 2. So (1101) 2 as a dcimal numbr is 1 lot of 8, 1 lot of 4, no lots of 2 and 1 unit = = 13 in bas 10. workd xampl 9 Writ (11001) 2 as a numbr in dcimal form. Stps Solution 1. Multiply ach digit by its plac valu. (11001) 2 = whol NUMBERS 33

35 2. Calculat th answr. = = 25 workd xampl 10 Writ 75 as a numbr in binary form. Stps 1. Find th largst binary plac valu numbr that is lss than 75. Binary plac valu numbrs ar 1, 2, 4, 8, 16, 32, 64, 128, Find th rmaindr. 2. Find th largst binary plac valu numbr that is lss than th rmaindr, thn find th rmaindr. Rpat until you gt to a rmaindr of 1 or Writ th dcimal numbr as th sum of binary plac valu numbrs. 4. As in th dcimal systm, w don t start a numbr with a zro. So writ 1 for th first binary plac usd thn put a 1 or 0 for ach plac following, down to th units. Solution = = = 1 75 = = = ( ) 2 xrcis 1.8 Extnsion: Numbr bass othr than 10 Prparation: Prp Zon Q2 4, Ex Writ ths binary numbrs in dcimal form. (a) (100) 2 (b) (110) 2 (c) (1001) 2 (d) (10) 2 () (101) 2 (f) (1101) 2 (g) (111) 2 (h) (11) 2 (i) (1100) 2 (j) (1011) 2 (k) (1111) 2 (l) (10001) 2 (m) (11010) 2 (n) (110011) 2 (o) (101111) 2 (p) (111111) 2 2 Writ ths dcimal numbrs in binary form. (a) 8 (b) 15 (c) 27 (d) 36 () 41 (f) 53 (g) 65 (h) 90 3 Writ vry dcimal numbr from 1 to 16 in binary form. 4 Choos th corrct answr. Th dcimal numbr 125 writtn in binary form is: A ( ) 2 B ( ) 2 C ( ) 2 D ( ) 2 34 MATHS ZONE 7

36 5 Each of ths numbrs is on lss than a binary plac valu: 1, 3, 7, 15, 31, 63. (a) Writ ach of ths numbrs in binary form. (b) What do all of ths numbrs hav in common whn writtn in binary form? (c) Starting with 9, list th first six dcimal numbrs that ar on lss than a plac valu numbr in th dcimal systm. 6 Anothr numbr systm is calld th octal systm. In this systm you ar only allowd to us th numbrs 0 to 7, and ach plac valu is 8 tims that of th prvious plac. (a) Copy and complt this list of numbrs, which shows th plac valu in th octal systm: 4096,, 64, 8, (b) Copy and complt th following tabl, which convrts numbrs from dcimal form to octal form. Dcimal numbr Octal systm plac valu Octal numbr (i) = = = = 5 (35) 8 (ii) = = 0 8 = 1 = ( ) 8 (iii) = 0 64 = 64 8 = 1 = (121) 8 (iv) = 0 64 = 0 8 = 1 = (1003) 8 (v) = 0 64 = 8 = 1 1 = ( ) 8 (vi) = 0 64 = 8 = 1 = ( ) 8 (vii) = 1 64 = 64 8 = 0 1 = 0 ( ) 8 7 Choos thr dcimal numbrs btwn 300 and 400 and writ thm using th octal systm. Homwork 1.3 Do ths in your had as quickly as you can and writ down th answrs. Tim targt: 2 minuts $ $ What tim is it 5 hours and 40 minuts aftr 9.17 a.m.? 8 What is on-third of $7.50? 9 How many odd numbrs ar thr btwn 20 and 36? 10 Find th missing numbr: 816, 408,, whol NUMBERS 35

37 Calculating th Grat Wall Th Grat Wall of China Could th Grat Wall of China hav bn built without th hlp of som sort of calculating dvic? Nowadays, computrs and calculators ar usd in building dsign, but how did popl in ancint tims do thir sums whn rally larg and complicatd numbrs wr involvd? 36 MATHS ZONE 7

38 Most of th arly numbr systms wr awkward to us for anything but simpl rcording. Carrying th on and othr asy tchniqus w us on papr just didn t work for th numbr systms of th ancint Grks, Romans and Chins. Thy ndd othr mthods. Th first aid was th sand abacus whr figurs wr drawn with a pointd objct onto a flat surfac covrd in sand and rasd with a fingr. Th sand abacus volvd into th lin abacus, whr countrs wr movd across lins drawn on a tabl. A skilld usr of an abacus can prform complicatd arithmtic vry quickly, which is why th abacus is still usd in som countris today. Th bad abacus was dvlopd in th lat Middl Ags in China, whr it was calld th suan-pan. Ths involv two sts of bads moving on paralll strings. Th first st contains fiv bads and allows counting from 1 to 5, and th scond st has only two bads rprsnting 5 or 10. Bads ar countd by moving thm from th outsid towards th middl bam. Th far right column rprsnts th ons, th nxt column th tns, th nxt, th hundrds and so on Th abacus at right shows th numbr = 1573 Qustions 1 What do th following abacuss show? (a) (b) (c) (d) 2 Draw abacuss that show th following numbrs. (a) 341 (b) 64 (c) 79 (d) 843 () 6492 (f) hi.com.au Rsarch Crat an annotatd postr that rflcts th history of mchanical calculators right up until th twntith cntury. Includ Napir s bons, Schickard s machin, th Pascalin and th Diffrnc Engin. 1 whol NUMBERS 37

39 Summary Copy and complt th following summary of this chaptr using th words and phrass from th list. A word or phras may b usd mor than onc. 1 Th numbr systm w us is calld th. 2 Th of numbrs is still usd, oftn on watch facs. 3 Th sum of on lin in a is calld th. 4 On mthod w can us to giv an for th valu of a calculation is by th numbrs to th first digit. 5 If thr ar no brackts involvd, thn according to th w must do division bfor addition. Ky words Babylonian systm binary systm Chins systm dcimal systm Egyptian systm stimat Hindu Arabic systm magic squar magic sum ordr of oprations Roman systm rounding Qustions 1 Put th following oprations in th ordr thy should b don according to ordr of oprations: multiplication, subtraction, brackts, addition, division. 2 Jams wrot 950 as LM using Roman numrals. Explain why this is not corrct and writ 950 corrctly using Roman numrals. 3 Whn w us stimation w can call th answr an stimat. This is th noun form. Writ a sntnc using stimat in vrb form. Notic how w say (pronounc) it diffrntly. 4 You hav com across th following words in this chaptr: product, odd, prim and powr. Each of ths words also has a non-mathmatical maning. Us ach word in a short sntnc to show its non-mathmatical maning. 5 Dscrib som bnfits of th Hindu Arabic numbr systm. 6 Mak at last 10 words of four lttrs or mor from th lttrs in stimat. 7 Arrang ach of th fiv counting systms in th list abov in ordr dpnding on how many lttrs thy hav in thir titl, starting from th on with th fwst lttrs. Worksht L1.1 Worksht L MATHS ZONE 7

40 FAQs Do I only us ordr of oprations if th qustion asks m to? No, you must always us ordr of oprations. It is a mathmatical rul. Do all calculators do ordr of oprations automatically? Most nw calculators do, but som oldr ons don t. It is bst to chck if your calculator dos by putting in an asy qustion that would gt a diffrnt answr if ordr of oprations wasn t followd; for xampl, should giv an answr of 9 if th ordr of oprations is followd. Cor 1 Look back at th symbols on pags 3 5. Writ out ths numbrs in: (i) th Egyptian numbr systm (ii) th Roman numbr systm (iii) th Babylonian numbr systm (iv) th modrn Chins numbr systm. (a) 54 (b) 146 (c) 238 (d) Look back at th symbols on pags 3 5. Writ out ths numbrs in th Hindu Arabic numbr systm. (a) CCCXL (b) MMCDLXXIII (c) (d) () (f) (g) (h) 3 Round off ths numbrs to th first digit. (a) 528 (b) 189 (c) 2500 (d) Us rounding to th first digit to stimat ths products. (a) (b) (c) Us rounding to th first digit to stimat ths quotints. (a) (b) (c) whol NUMBERS 39

41 6 Us rounding to th first digit to stimat ths, and thn us your calculator to work out how far off your stimat was from th xact answr. (a) (b) (c) Find: (a) 9 (2 + 1) 2 (b) (3 8) (c) (d) () (13 5 2) + (20 10) (f) [5 (9 + 1)] 3 8 Choos th corrct answr. In th calculation of 2 [30 (4 1)] + 6 th first opration to do is: A + B C D 9 Us an appropriat stratgy to hlp simplify ach of th following. (a) (b) (c) Extnsion 10 Copy and complt ths magic squars. (a) 13 (b) Copy and complt ths numbr pyramids. (a) (b) Put brackts into ths statmnts, whr ncssary, to mak thm tru. (a) = 3 (b) = 7 13 Rplac ach * with on of th four oprators (+,,, ) to mak th statmnt tru. (a) 9 * 7 * 3 = 30 (b) 16 * 4 2 * 2 = MATHS ZONE 7

42 14 Writ ths binary numbrs in dcimal form. (a) (110) 2 (b) (1001) 2 (c) (11111) 2 (d) (101101) 2 15 Writ ths dcimal numbrs in binary form. (a) 11 (b) 29 (c) 34 (d) St out ths calculations in your normal way and work out th answrs. (a) (b) (c) List th numbrs you gt if you count by nins, starting at 30 and nding at Copy and complt ach of th following by writing or btwn th givn numbrs. (a) (b) (c) Copy and complt th following by finding th pattrn. (a) 9, 15, 22, 30,,, (b) 1, 3, 7, 15,,, 5 Calculat: (a) (b) (c) Find th missing numbr that maks ach of th following tru. (a) + 7 = 24 (b) 4 = 15 3 (c) 7 0 = 15 7 Prform th following divisions. (a) (b) (c) List all numbrs that 8 gos into that ar gratr than 70 and lss than Simplify: (a) (b) (c) 8 (15 5) 10 Which whol numbr ar th following dcimals closst to? (a) 1.9 (b) (c) Calculat: (a) (b) (c) If Cln purchass 3 tops at $15.50 ach, how much chang will sh rciv from a $50 not? Worksht R1.7 Worksht R1.8 Worksht R1.9 Worksht R1.10 Worksht R1.11 Worksht R1.12 Worksht R1.13 Worksht R1.14 Worksht R1.15 Worksht R1.16 Worksht R1.17 Worksht R1.18 Assignmnt 1 1 whol NUMBERS 41