6A 6B 6C 6D 6E 6F 6G 6H Chpter Wht you will lern Introduing rtios Simplifying rtios Dividing quntity in given rtio (Extending) Sle drwings Introduing rtes Rtios nd rtes nd the unitry method Solving rte prolems (Extending) Speed (Extending) 6Rtios nd rtes Austrlin urriulum NUMBER AND ALGEBRA Rel numers Solve rnge of prolems involving rtes nd rtios, with nd without digitl tehnologies (ACMNA188)
Biyle ger rtios The iyle is renowned s the most energy-effi ient mens of humn trnsport. A iyle with rnge of gers is n extremely verstile nd enjoyle mhine. The ger rtio of iyle is the rtio of pedl rottions ompred to the rte t whih the wheels rotte. Mnufturers pply the mthemtis of ger rtios to provide ylists with the rnge of gers tht is most useful for the style of ike. Hve you ever tried to ride up steep hill in too high ger or rehed the ottom of hill while still in very low ger? A iyle rider hs omfortle rnge of pressures nd rottion rtes tht they n pply to the pedls. Gers mke it esier to sty in this omfortle rnge over wide rnge of riding onditions suh s BMX downhill ring, fst time tril for ring ikes on fl t ground or liming steep, rough trils on mountin ike. Uniyles hve only one wheel nd so hve rtio of 1:1 for pedl turns to wheel turns. Uniyles n t reh high speeds euse it is very diffi ult to rotte pedls more thn out 100 rpm; however, riding uniyle is fun nd it requires strength nd gility to remin stle, espeilly when plying uniyle hokey! Online resoures Chpter pre-test Videos of ll worked exmples Intertive widgets Intertive wlkthroughs Downlodle HOTsheets Aess to HOTmths Austrlin Curriulum ourses
334 Chpter 6 Rtios nd rtes 6A Key ides Introduing rtios Rtios re regulrly used in everydy life. They re used to show the reltionship etween two (or more) relted quntities. Here re five ommon uses of rtios: Ingredients the rtio of different ingredients in reipe (ooking, industril) Mps most mps inlude sle whih is written s rtio Sporting suess showing tem s win to loss rtio or the rtio of kiking gols to The rtio of eggs to utter to sugr will ffet the flvour nd points texture of ke. Compring size the rtio of length, re or volume of different shpes Legl requirements minimum stndrds of supervision, stff to student rtios When deling with rtios, the order in whih the rtio is written is very importnt. For exmple, tem s win : loss rtio of 5 : 2 is very different to tem s win : loss rtio of 2 : 5. Rtios ompre quntities of the sme type nd given in the sme unit. Therefore rtio is not generlly written with unit of mesurement. Let s strt: Estimting rtios Estimte the following rtios. Rtio of oys to girls in your shool Rtio of Yer 8 students to Yer 7 students in your shool Rtio of your teher s ge to your ge! Rtio of hours you spend outside to hours you spend inside Rtio of hours you re wke to hours you re sleep Rtio of prents to hildren in n verge Austrlin fmily Rtio of the length to the width of n A4 sheet of pper Disuss your nswers s lss. Rtios show the reltionship etween two (or more) relted quntities. For exmple, drink ws mde with the rtio of ordil to wter of 1 : 3. The olon : is the mthemtil symol used to represent rtios. The written rtio of : is red s the rtio of to or is to.
Numer nd Alger 335 The order in whih the quntities re written in rtio is importnt. Before rtios n e written nd simplified, the quntities must e expressed in the sme unit. If eh vlue in rtio is multiplied or divided y the sme numer n equivlent rtio is formed. For exmple, 1 : 3 nd 2 : 6 nd 10 : 30 re equivlent rtios. Exmple 1 Writing rtios A smple of mixed nuts ontins 5 shews, 12 penuts nd 2 mdmi nuts. Write down: the rtio of shews to penuts to mdmis the rtio of shews to the totl numer of nuts the rtio of penuts to other nuts SOLUTION EXPLANATION 5 : 12 : 2 shews : penuts : mdmis 5 : 19 5 shews, totl nuts 19 12 : 7 12 penuts, other nuts 7 Exmple 2 Produing equivlent rtios Complete eh pir of equivlent rtios. 4 : 9 = 16 : 30 : 15 = : 5 SOLUTION 4 4 : 9 4 16 : 36 30 :15 3 10 : 5 3 EXPLANATION Both numers re multiplied y 4. Both numers re divided y 3. Key ides
336 Chpter 6 Rtios nd rtes Exmple 1 Exerise 6A 1 3 3 1 Write down the rtio of shded prts to unshded prts for eh grid. 2 Write down the rtio of shded prts to totl prts for eh grid. 3 Express eh of the following pirs of quntities s rtio. 9 gols to 4 gols 52 litres to 17 litres 7 pottoes to 12 rrots d 3 lue mrles to 5 green mrles 4 A g ontins 5 green mrles, 7 lue mrles nd 3 yellow mrles Write down the rtio of: green mrles to yellow mrles lue mrles to totl mrles yellow to lue to green mrles d green to yellow to lue mrles 4 6, 7 8(½) 4 6, 7(½), 8 9 5 Over the pst fortnight, it hs rined on eight dys nd it hs snowed on three dys. Write down the rtio of: riny dys to snowy dys snowy dys to totl dys fine dys to riny nd snowy dys d riny dys to non-riny dys 6 In ox of 40 flvoured iy poles there were 13 green, 9 lemonde, 11 rsperry nd 7 ornge iy poles. Write down the rtio of: green iy poles to ornge iy poles rsperry iy poles to lemonde iy poles the four different flvours of iy poles d green nd ornge iy poles to rsperry nd lemonde iy poles 4 9(½) UNDERSTANDING FLUENCY
Numer nd Alger 337 Exmple 2 7 Complete eh pir of equivlent rtios. 1 : 3 = 4 : 1 : 7 = 2 : 2 : 5 = : 10 d 3 : 7 = : 21 e 5 : 10 = 1 : f 12 : 16 = 3 : g 12 : 18 = : 3 h 20 : 50 = : 25 i 4 : 7 = 44 : j 14 : 17 = 12 : k 27 : 6 = : 2 l 121 : 66 = : 6 8 Complete eh pir of equivlent rtios. 2 : 3 : 5 = 4 : : 4 : 12 : 16 = : 6 : 1 : 7 : 9 = : : 63 d 22 : 110 : 66 = 11 : : 9 Write three equivlent rtios for eh of the following rtios. 1 : 2 2 : 5 8 : 6 d 9 : 3 10 Sort the following rtios into three pirs of equivlent rtios. 2 : 5, 6 : 12, 7 : 4, 1 : 2, 4 : 10, 70 : 40 10, 11 11 13 11 During reent dry spell, it rined on only 3 dys during the month of Septemer. Wht ws the rtio of wet dys to totl dys for the month of Septemer? Write equivlent rtios for totl of 10 dys nd 100 dys. 12 On their wy to work, Andrew psses 15 sets of trffi lights nd Puline psses 10 sets of trffi lights. One morning Andrew ws stopped y 12 red trffi lights. How mny green trffi lights would Puline need to pss through to hve the equivlent rtio of red to green trffi lights s Andrew? 13 Write the rtio of vowels to onsonnts for eh of the following ple nmes. Queenslnd Cnerr Wgg Wgg d Austrli 14 Nme ny Austrlin sttes tht hve vowel to onsonnt rtio of 1 : 1. 15 Cn 10 people e divided into two groups with rtio of 1 : 2? Explin. 16 Bertrnd wins one-third of his gmes of tennis this seson. Write down his win to loss rtio. 17 Wht is the rtio of lk keys to white keys in one otve of the pino? Wht is the rtio of lk keys to white keys for the entire 88 keys of the pino? Are these equivlent rtios? 18 Write the missing expression. 15 15, 16 2 : x = 6 : 2 : x = 5 : y : 2x = : 8x d 12xy : 6y = : 1 12 14 16 18 FLUENCY PROBLEM-SOLVING REASONING 6A
338 Chpter 6 Rtios nd rtes 6A Are rtios 19 Using the dimensions provided, find the rtio of the shded re to the unshded re for eh of the following digrms. 10 m 20 m 4 m 4 m 2 m 8 m 5 m d 15 m 20 Estimte the rtio of the shded floor to the unshded floor in this photo. 21 Design your own digrm for whih the rtio of shded re to unshded re is: 1 : 3 1 : 7 30 50 19 21 Tringles se = 10 m height = 9 m ENRICHMENT
Numer nd Alger 339 6B Simplifying rtios In similr wy to frtions, rtios re simplified y dividing eh term y ommon ftor. A rtio is sid to e in its simplest form when it ontins whole numers only nd the highest ommon ftor (HCF) etween the terms in the rtio is 1. If rtio ontins frtions or deimls, it n e simplified y multiplying rther thn dividing or nelling. Let s strt: Clss rtios Look round your lssroom nd write down the following rtios: d e f g h Rtio of girls to oys Rtio of tehers to students Rtio of wering wth to not wering wth Rtio of white soks to lk soks Rtio of textooks open to textooks losed Rtio of not hving penil se to hving penil se Rtio of londe hir to rown hir to lk hir Rtio of lue eyes to rown eyes to other olour eyes Design your own rtio question for your lss or lssroom. Cn ny of your rtio nswers e simplified? Simplifying rtios A rtio is simplified y dividing oth numers in the rtio y their highest ommon ftor (HCF). For exmple, the rtio 15 : 25 n e simplified to 3 : 5. 15 : 25 5 5 3 : 5 Simplest form It is onvention to express rtios in their simplest form. Rtios in simplest form use whole numers only. If rtio is expressed with frtions, it is simplified y onverting the quntities to whole numers. This is generlly done y multiplying y the lowest ommon denomintor (LCD). Before rtios re simplified the quntities must e expressed in the sme unit. Key ides
340 Chpter 6 Rtios nd rtes Exmple 3 Simplifying rtios Simplify the following rtios. 7 : 21 450 : 200 SOLUTION 7 : 21 7 7 1 : 3 450 : 200 50 50 9 : 4 Exmple 4 Simplifying rtios involving frtions Simplify the following rtios. EXPLANATION 3 5 : 1 2 1 2 3 : 1 1 4 SOLUTION 3 : 1 10 5 2 10 6 : 5 7 5 : 12 3 4 12 28 : 15 Exmple 5 Chnging quntities to the sme units HCF of 7 nd 21 is 7. Divide oth numers y 7. HCF of 450 nd 200 is 50. Divide oth numers y 50. EXPLANATION LCD of 5 nd 2 is 10, so multiply y 10. Alterntively, 3 5 : 1 2 = 6 10 : 5 10 = 6 : 5 Convert mixed numerls to improper frtions. LCD of 3 nd 4 is 12. Multiply oth numers y 12. First hnge the quntities to the sme unit, nd then express eh pir of quntities s rtio in simplest form. 4 mm to 2 m 25 minutes to 2 hours SOLUTION 4 mm to 2 m = 4 mm to 20 mm = 4 : 20 = 1 : 5 25 minutes to 2 hours = 25 minutes to 120 minutes = 25 : 120 = 5 : 24 EXPLANATION 2 m = 20 mm One in sme unit, write s rtio. Simplify rtio y dividing y the HCF of 4. 2 hours = 120 minutes One in sme unit, write s rtio. Simplify rtio y dividing y the HCF of 5.
Numer nd Alger 341 Exmple 3 Exmple 4 Exmple 4 Exmple 5 Exerise 6B 1 Write down the rtio of shded prts to unshded prts for eh of the following in simplest form. 2 To express the rtio 4 : 16 in simplest form, you would: A multiply oth quntities y 2 B 1 4 3, 4 sutrt 4 from oth quntities C divide oth quntities y 2 D divide oth quntities y 4 3 Deide whih one of the following rtios is not written in simplest form. A 1 : 5 B 3 : 9 C 2 : 5 D 11 : 17 4 Deide whih one of the following rtios is written in simplest form. 5 A 2 : 28 B 15 : 75 C 14 : 45 D 13 : 39 Simplify the following rtios. 2 : 8 10 : 50 4 : 24 d 6 : 18 e 8 : 10 f 25 : 40 g 21 : 28 h 24 : 80 i 18 : 14 j 26 : 13 k 45 : 35 l 81 : 27 m 51 : 17 n 20 : 180 o 300 : 550 p 150 : 75 q 1200 : 100 r 70 : 420 s 200 : 125 t 90 : 75 6 Simplify the following rtios. 2 : 4 : 6 12 : 21 : 33 42 : 60 : 12 d 85 : 35 : 15 e 12 : 24 : 36 f 100 : 300 : 250 g 270 : 420 : 60 h 24 : 48 : 84 7 8 9 Simplify the following rtios. e 1 3 : 1 2 3 8 : 1 4 Simplify the following rtios. f 1 4 : 1 5 7 10 : 4 5 5 7(½), 9(½) g 2 5 : 3 4 11 10 : 2 15 d h 2 7 : 1 5 9 8 : 7 12 1 1 2 : 3 2 1 4 5 : 3 3 1 5 3 : 1 2 d 4 1 5 3 : 3 3 4 First hnge the quntities to the sme unit, nd then express eh pir of quntities s rtio in simplest form. 12 mm to 3 m 7 m to 5 mm 120 m to 1 km d 60 mm to 2.1 m e 3 kg to 450 g f 200 g to 2.5 kg g 2 tonnes to 440 kg h 1.25 L to 250 ml i 400 ml to 1 L j 20 minutes to 2 hours k 3 hours to 15 minutes l 3 dys to 8 hours m 180 minutes to 2 dys n 8 months to 3 yers o 4 dys to 4 weeks p 8 weeks to 12 dys q 50 ents to $4 r $7.50 to 25 ents 5 9(½) 5 9(½) UNDERSTANDING FLUENCY
342 Chpter 6 Rtios nd rtes 6B 10 Whih of these rtios is the simplest form of the rtio 1 2 : 2? A 2 : 1 B 1 : 4 2 C 1 4 : 1 D 1 : 1 11 Kwok ws sent from shool on 8 dys in Term 1. There were 44 shool dys in Term 1. Write the following rtios in simplest form. Rtio of dys sent to totl dys Rtio of dys present to totl dys Rtio of dys sent to dys present 12 Over the pst four weeks, the Pske fmily hd eten tkewy food for dinner on eight osions. They hd hmurgers twie, fish nd hips three times nd pizz three times. Every other night they hd home-ooked dinners. Write the following rtios in simplest form. Rtio of hmurgers to fish nd hips to pizz Rtio of fish nd hips to pizz Rtio of tkewy dinners to home-ooked dinners d Rtio of home-ooked dinners to totl dinners 13 When Lis mkes fruit sld for her fmily, she uses 5 nns, 5 pples, 2 pssionfruit, 4 ornges, 3 pers, 1 lemon (for juie) nd 20 strwerries. 10, 11 11 13 Write the rtio of the fruits in Lis s fruit sld. Lis wnted to mke four times the mount of fruit sld to tke to prty. Write n equivlent rtio tht shows how mny of eh fruit Lis would need. Write these rtios in simplest form: i nns to strwerries ii strwerries to other fruits 14 Andrew inorretly simplified 12 m to 3 mm s rtio of 4 : 1. Wht ws Andrew s mistke nd wht is the orret simplified rtio? 15 Mrih hs $4 nd Rogn hs 50 ents. To write rtio in simplest form, vlues must e written in the sme unit. First onvert oth units to dollrs, nd then express the rtio of Mrih s money to Rogn s money in simplest form. As n lterntive, onvert oth units to ents nd express the rtio in simplest form. Do you rrive t the sme simplified rtio? 16 Write two quntities of time, in different units, whih hve rtio of 2 : 5. Write two quntities of distne, in different units, whih hve rtio of 4 : 3. 17 Simplify the following rtios. 2 : 4 d 5f : 24f 14 14, 15 15x : 3y : 2 e hk : 3k f 24xyz : 60yz 11 13 15 17 PROBLEM-SOLVING REASONING
Numer nd Alger 343 Aspet rtios 18 Aspet rtio is the reltionship etween the width nd height of the imge s displyed on sreen. d e f g A high-definition television sreen (left) ompred to the sme imge on n nlogue television sreen (right). An old nlogue television hs n spet rtio of 1. 3 : 1. Write this rtio in simplest form. A high definition digitl television hs n spet rtio of 1. 7 : 1. Write this rtio in simplest form. Although these two rtios re lerly not equivlent, there is onnetion etween the two. Wht is it? The digitl television spet rtio of 1. 7 : 1 ws hosen s the est ompromise to show widesreen movies on television. Mny mjor films re presented in Pnvision, whih hs n spet rtio of 2.35 : 1. Write this rtio in simplest form. Investigte the history of spet rtio in films nd television. Reserh how formts of unequl rtios re onverted through the tehniques of ropping, zooming, letteroxing, pillroxing or distorting. Investigte spet rtios of other ommon medi. i Find the spet rtio for severl different newsppers. ii Find the spet rtio for severl ommon sizes of photogrphs. iii Find the spet rtio for piee of A4 pper ( surd is involved!) 18 ENRICHMENT 6B
344 Chpter 6 Rtios nd rtes Key ides 6C Dividing quntity in given rtio EXTENDING Rtios re losely onneted to frtions. To help solve prolems involving rtios, we n think of eh quntity in terms of numer of prts. A drink mde in the rtio of ordil to wter of 1 : 4 mens tht there is 1 prt of ordil nd 4 prts of wter. There is totl of 5 prts in the drink. In terms of rtios, it does not mtter whether the drink is 250 ml glss, 2 L ottle or 50 L urn. Chnging the size of the drink will not hnge the rtio of prts ; it will simply hnge the size of eh prt. numer of ordil prts The frtion of ordil in the drink is totl numer of prts = 1 5. numer of wter prts The frtion of wter in the drink is totl numer of prts = 4 5. Thinking out rtios in terms of prts helps us to divide quntities into prtiulr rtios. Let s strt: Shring $120 For eh sitution elow you hve $120 to shre out in the given rtios. With prtner, work out how muh eh person reeives. Sitution 1: Two people, Jk nd Pipp, rtio of 1 : 1 Sitution 2: Two people, Andrew nd Alex, rtio of 2 : 2 Sitution 3: Two people, Jess nd Teres, rtio of 2 : 3 Sitution 4: Two people, Kyle nd Kr, rtio of 3 : 5 Sitution 5: Three people, Mtt, Leos nd Djrrin, rtio of 1 : 1 : 1 Sitution 6: Three people, Christine, Prue nd Crolyn, rtio of 3 : 5 : 7 Disuss the strtegy tht you used to shre the $120. Does your strtegy work for every sitution? Think of rtio in terms of prts. A rtio of 2 : 3 hs 2 prts of one quntity for every 3 prts of nother quntity nd totl of 5 prts. Using the unitry method to divide quntity in given rtio: Find the totl numer of prts in the rtio. Find the vlue of one prt. Find the vlue of the numer of prts required in the rtio. For exmple: Shre $20 in rtio of 2 : 3. Think of shring $20 into 2 prts nd 3 prts. $ 20 = 5 prts 5 5 Totl numer of prts = 2 + 3 = 5. $4 = 1 prt 2 2 Vlue of one prt = $20 5 = $4. $8 = 2 prts Therefore 2 prts = $8, nd 3 prts = $12.
Numer nd Alger 345 Using frtions to divide quntity in given rtio: numer in rtio Frtion of the mount required = totl numer of prts Clulte the frtion of the mount for eh shre of the rtio. For exmple, shre $20 in rtio of 2 : 3. Frtions of the mount required re 2 5 nd 3 5. Therefore 2 5 of $20 = $8 nd 3 of $20 = $12. 5 To find totl quntity from given rtio: Use the onept of prts nd the unitry method to find the vlue of one prt nd therefore the vlue of the totl prts n e lulted. Or Use equivlent rtios to find the vlue of eh quntity in the rtio nd then dd the numers together to find the totl. Exmple 6 Dividing quntity in prtiulr rtio Divide 54 m in rtio of 4 : 5. SOLUTION Unitry method Totl numer of prts = 9 EXPLANATION Totl numer of prts = 4 + 5 = 9 9 9 prts = 54 m 9 Vlue of 1 prt = 54 m 9 = 6 m 1 prt = 6 m 1 prt = 6 m 4 4 5 5 4 prts = 24 m 5 prts= 30 m Chek numers dd to totl: 24 + 30 = 54 54 m divided in rtio of 4 : 5 is 24 m nd 30 m. Frtions method 4 9 of 54 = 4 9 54 1 = 24 Frtion = numer in rtio totl numer of prts 5 9 of 54 = 5 9 54 = 30 Chek numers dd to totl: 1 24 + 30 = 54 54 m divided in rtio of 4 : 5 is 24 m nd 30 m. Key ides
346 Chpter 6 Rtios nd rtes Exmple 7 Dividing quntity in rtio with three terms Divide $300 in the rtio of 2 : 1 : 3. SOLUTION Unitry method Totl numer of prts = 6 6 prts= $300 6 6 1 prt = $50 1 prt= $50 2 2 3 3 2 prts= $100 3 prts= $150 EXPLANATION Totl numer of prts = 2 + 1 + 3 = 6 Vlue of 1 prt = $300 6 = $50 $300 divided in rtio of 2 : 1 : 3 is Chek numers dd to totl: $100, $50 nd $150. $100 + $50 + $150 = $300 Frtions method 2 6 of 300 = 2 6 300 1 = 100 1 6 of 300 = 1 6 300 1 = 50 3 6 of 300 = 3 6 300 1 = 150 numer in rtio Frtion = totl numer of prts $300 divided in rtio of 2 : 1 : 3 is Chek numers dd to totl: $100, $50 nd $150. $100 + $50 + $150 = $300 An exmple of dividing quntity ( length of steel) in prtiulr rtio.
Numer nd Alger 347 Exmple 8 Finding totl quntity from given rtio The rtio of oys to girls t Birdsville College is 2 : 3. If there re 246 oys t the shool, how mny students ttend Birdsville College? SOLUTION Unitry method 2 5 2 prts = 246 1 prt = 123 5 prts= 615 2 5 615 students ttend Birdsville College. Equivlent rtios method 123 oys : girls = 2 : 3 = 246 : 369 123 615 students ttend Birdsville College. Exerise 6C EXPLANATION Rtio of oys : girls is 2 : 3 Boys hve 2 prts = 246 Vlue of 1 prt = 246 2 = 123 Totl prts = 2 + 3 = 5 prts 5 prts = 5 123 = 615 Use equivlent rtios. Multiply eh quntity y 123. Totl numer of students = 246 oys + 369 girls = 615 1 Find the totl numer of prts in the following rtios. 3 : 7 1 : 5 11 : 3 d 2 : 3 : 4 2 The rtio of girls to oys in lss is 3 : 5. Wht frtion of the lss is girls? Wht frtion of the lss is oys? 3 The digrm shows four glsses tht ontin different mounts of ordil. Wter is then dded to fill eh glss. For eh drink shown, wht is the rtio of ordil to wter? d 4 Wht frtion of eh of the drinks ove is ordil? 1 4 3, 4 UNDERSTANDING
348 Chpter 6 Rtios nd rtes 6C Exmple 6 Exmple 7 Exmple 8 5 Divide: $60 in the rtio of 2 : 3 $110 in the rtio of 7 : 4 $1000 in the rtio of 3 : 17 d 48 kg in the rtio of 1 : 5 e 14 kg in the rtio of 4 : 3 f 360 kg in the rtio of 5 : 7 g 72 m in the rtio of 1 : 2 h 40 m in the rtio of 3 : 5 i 155 m in the rtio of 4 : 1 6 Shre $400 in the rtio: 1 : 3 2 : 3 3 : 5 d 9 : 11 7 5(½), 6, 7(½) 5 8(½) Divide: $200 in the rtio of 1 : 2 : 2 $400 in the rtio of 1 : 3 : 4 12 kg in the rtio of 1 : 2 : 3 d 88 kg in the rtio of 2 : 1 : 5 e 320 kg in the rtio of 12 : 13 : 15 f $50 000 in the rtio of 1 : 2 : 3 : 4 8 Shre 600 lollies in the rtio: 1 : 9 2 : 1 : 3 2 : 5 : 5 d 12 : 7 : 8 : 3 9 Evergreen Fertiliser is mde up of the three vitl nutrients nitrogen, potssium nd phosphorus in mss rtio of 4 : 5 : 3. How muh of eh nutrient is in 1.5 kg g? 10 The ngles of tringle re in the rtio of 2 : 3 : 4. Find the size of eh ngle. 11 Three friends, Cm, Molly nd Se, shre prize of $750 in rtio of 3 : 4 : 8. How muh more money does Se reeive thn Cm? 12 Trudy nd Bell mde 80 isuits. If Trudy mde 3 isuits in the time tht Bell mde 2 isuits, how mny isuits did Trudy mke? 13 9 11 10 13 In Yer 8, the rtio of oys to girls is 5 : 7. If there re 140 girls in Yer 8, wht is the totl numer of students in Yer 8? 5 8(½) 14 A light rye red requires rtio of wholemel flour to plin flour of 4 : 3. A ker mking lrge quntity of loves uses 126 ups of plin flour. Wht is the totl mount of flour used y the ker? 15 A textook ontins three hpters nd the rtio of pges in these hpters is 3 : 2 : 5. If there re 24 pges in the smllest hpter, how mny pges re in the textook? 12 15 FLUENCY PROBLEM-SOLVING
Numer nd Alger 349 16 The rtio of the ost of shirt to the ost of jket is 2 : 5. If the jket ost $240 more thn the shirt, find the ost of the shirt nd the ost of the jket. 17 In lss of 24 students the rtio of girls to oys is 1 : 2. How mny more girls re needed to mke the rtio 1 : 1? If 4 more girls nd 4 more oys joined the lss, wht would e the new rtio? 16 16, 17 On one dy, the rtio ws 3 : 7. How mny oys nd how mny girls were sent? A fir shre 18 Three students Rmshid, Tony nd Mri entered group Geogrphy ompetition. Rmshid spent 5 hours prepring the PowerPoint presenttion. Tony spent 3 hours reserhing the topi. Mri spent 2 1 hours designing the poster. 2 Their group won seond prize in the ompetition nd reeived prize of $250. Rmshid, Tony nd Mri deide to shre the prize in rtio of 3 : 2 : 1. d e f g 16, 17 How muh money did eh student reeive? Round the nswer to nerest ent. If the prize ws divided up ording to the time spent on the projet, wht would e the new rtio? Write the rtio in whole numers. How muh money did eh student reeive with the new rtio? Round the nswer to the nerest ent. Although she spent the lest time on the projet, Mri would prefer to divide the prize ording to time spent. How muh more money did Mri reeive with the new rtio? Tony preferred tht the originl rtio remined. How muh ws Tony etter off using the originl rtio? Whih rtio would Rmshid prefer nd why? The group ended up going with rtio sed on time spent ut then rounded mounts to the nerest $10. How muh prize money did eh student reeive? 18 REASONING ENRICHMENT 6C
350 Chpter 6 Rtios nd rtes 6D Sle drwings Sle drwings re speil pplition of rtios. The purpose of sle drwing is to provide urte informtion out n ojet whih hs dimensions tht re either too lrge or too smll to e shown on pge. If sle drwing hs rtio of 1 : 1, then the drwing would e extly the sme size s the tul (rel-life) ojet. For exmple, it is not prtil to hve mp tht is the sme size s the tul distne you need to trvel, so muh smller mp ( sled drwing) is used. The mp shows sle to indite the reltionship of the mp distne to the tul distne. The sle is displyed s rtio. Three ommon trvel mps with their sles re shown elow. Wyong Colo Dhrug Tuggerh Lke Heights Ntionl The Entrne Prk Gosford Blue Mountins Terrigl Ntionl Prk 1 Mrrmrr Woy Woy Ntionl Bouddi NP Prk Broken By Rihmond Plm Pifi Windsor Ku-ring-gi Beh Oen Springwood Chse NP Mon Vle Hornsy Penrith Blktown Blxlnd Pymle 32 St Mrys Prrmtt Mnly Blue Mountins Ntionl Prk Ryde Port Jkson SYDNEY Wllhi Lke Bondi Liverpool Burrgorng Bnkstown Sydney Airport Botny By Sutherlnd Tsmn Cmpelltown Kurnell Port Hking Se Cmden Royl 31 Ntionl Piton Prk Appin N 1 0 5 10 km Brgo Sle 1:400 000 Bulli Colo River Nepen River River Nepen Georges River Hwkesury River 1 Killr Drummoyne Leihhrdt 31 Sydenhm Rokdle Roseville Newtown Sydney Airport Chtswood Lne 1 Cove St Leonrds Prrmtt R North Sydney Sydney Redfern 1 Botny Frenhs Forest Middle Hrour Mosmn P o t r Bondi Juntion J k s o n Brookvle North Hrour Rose By Bondi Rndwik Coogee Mrour Mnly Dee Why Pifi Oen Tsmn Se Sle 1:200 000 0 N 1 2 km
Numer nd Alger 351 Let s strt: Sling down nd sling up Brinstorm speifi exmples where you hve ome ross sle drwings. Think of exmples where sle drwing hs een used to show very lrge ojets or distnes, nd lso think of exmples where sle drwing hs een used to show very smll ojets. Shre your list of exmples with your prtner. As lss, mke list on the ord of speifi exmples of sle drwings. Wht re some exmples of ommon sles tht re used? A sle drwing hs extly the sme shpe s the originl ojet, ut different size. All sle drwings should indite the sle rtio. The sle on drwing is written s sle rtio: Drwing length : Atul length Drwing length represents the length on the digrm, mp or model. Atul length represents the rel length of the ojet. A sle rtio of 1 : 100 mens the tul or rel lengths re 100 times greter thn the lengths mesured on the digrm. A sle rtio of 20 : 1 mens the sled lengths on the model re 20 times greter thn the tul or rel lengths. It is helpful if sles egin with 1. Then the seond numer in the rtio n e referred to s the sle ftor. Sle rtios tht do not strt with 1 n e onverted using equivlent rtios. To onvert sled distne to n tul distne you multiply y the sle ftor. To onvert n tul (rel) distne to sled distne you divide y the sle ftor. Sle distne Multiply Divide Conversion of units. It is importnt to rememer how to orretly onvert mesurements of length when deling with sles. 1 km = 1000 m Atul distne 1 m = 100 m 1 m = 10 mm 1000 km m m mm 1000 100 10 100 10 Key ides
352 Chpter 6 Rtios nd rtes Exmple 9 Converting sle distne to tul distne A mp hs sle of 1 : 20 000. Find the tul distne for eh sled distne. 2 m 5 mm SOLUTION Atul distne = 2 m 20 000 = 40 000 m = 400 m (or 0.4 km) Atul distne = 5 mm 20 000 = 100 000 mm = 100 m Atul distne = 14.3 m 20 000 = 286 000 m = 2.86 km EXPLANATION Exmple 10 Converting tul distne to sled distne A model ot hs sle of 1 : 500. Find the sled length for eh of these tul lengths. 50 m 75 m SOLUTION Sled distne = 50 m 500 = 5000 m 500 = 10 m (0.1 m) Sled distne = 75 m 500 = 750 mm 500 = 1.5 mm 14.3 m Sle ftor = 20 000 Multiply sled distne y sle ftor. Convert nswer into sensile units. Multiply sled distne y sle ftor. Convert nswer into sensile units. Multiply sled distne y sle ftor. Shortut: 2, then 10 000 Convert nswer into sensile units. EXPLANATION Sle ftor = 500 4550 mm Convert metres to entimetres. Divide tul distne y sle ftor. Convert nswer into sensile units. Convert entimetres to millimetres. Divide tul distne y sle ftor. Sled distne = 4550 mm 500 Divide tul distne y sle ftor. = 45.5 mm 5 Shortut: 100, then 5 (or vie vers) = 9.1 mm
Numer nd Alger 353 Exmple 11 Determining the sle ftor Stte the sle ftor in the following situtions. A length of 4 mm on sle drwing represents n tul distne of 50 m. An tul length of 0.1 mm is represented y 3 m on sled drwing. SOLUTION EXPLANATION Sle rtio = 4 mm : 50 m Write the rtio drwing length : tul length. Sle rtio = 4 : 500 Sle ftor = 125 = 4 mm : 500 mm Convert to like units. Write the sle rtio without units. = 1 : 125 Divide oth numers y 4. Rtio is now in the form 1 : sle ftor. The tul size is 125 times lrger thn the sled drwing. Sle rtio = 3 m : 0.1 mm Write the rtio drwing length : tul length. Sle rtio = 30 : 0.1 = 30 mm : 0.1 mm Convert to like units. Write the sle rtio without units. = 300 : 1 Multiply oth numers y 10. 1 = 1 : 300 Sle ftor = 1 300 Exerise 6D Divide oth numers y 300. Rtio is now in the form 1 : sle ftor. The tul size is 300 times smller thn the sled drwing. 1 A mp hs sle of 1 : 50 000. How mny entimetres in rel life does 1 m on the mp equl? 2 A mp hs sle of 1 : 100. How mny metres on the mp does 300 m in rel life equl? 3 Convert 10 000 m to: mm 4 Convert 560 m to: km m m 1 4 3, 4 km mm UNDERSTANDING
354 Chpter 6 Rtios nd rtes 6D Exmple 9 Exmple 10 Exmple 11 5 6 7 Find the tul distne for eh of the following sled distnes. Give your nswer in n pproprite unit. Sle 1 : 10 000 i 2 m ii 4 mm iii 7.3 m d e f g h Sle 1 : 20 000 Sle 1 : 400 Sle 1 : 300 Sle 1 : 2 Sle 1 : 5 Sle 1 : 0.5 Sle 100 : 1 i i i i i i i 80 m 16 mm 5 mm 44 m 4 m 12 m 3 m ii ii ii ii ii ii ii 18.5 mm 72 m 8.2 m 310 m 24 m 3.2 mm 11.5 km Find the sled length for eh of these tul lengths. Sle 1 : 200 i 200 m ii 4 km d e f g h Sle 1 : 500 Sle 1 : 10 000 Sle 1 : 20 000 Sle 1 : 250 000 Sle 1 : 10 Sle 1 : 0.1 Sle 200 : 1 i i i i i i i 10 000 m 1350 m 12 km 5000 km 23 m 30 m 1 mm 5 6(½), 7 5 6(½), 7, 8(½) ii ii ii ii ii ii ii 1 km 45 km 1800 m 750 000 m 165 m 5 mm 7.5 m Determine the sle rtio for eh of the following. A length of 2 mm on sle drwing represents n tul distne of 50 m. A length of 4 m on sle drwing represents n tul distne of 2 km. A length of 1.2 m on sle drwing represents n tul distne of 0.6 km. d A length of 5 m on sle drwing represents n tul distne of 900 m. e An tul length of 7 mm is represented y 4.9 m on sled drwing. f An tul length of 0.2 mm is represented y 12 m on sled drwing. iii iii iii iii iii iii iii iii iii iii iii iii iii iii iii 1.25 m 0.03 m 7.1 m 2.5 mm 155 mm 400 m 81.5 m 60 m 75 m 736.5 m 400 mm 1250 m 6.15 m 0.2 mm 8.2 mm 5 8(½) 8 Convert the two mesurements provided in eh sle into the sme unit nd then write the sle s rtio of two numers in simplest form. 2 m : 200 m 5 mm : 500 m 12 mm : 360 m d 4 mm : 600 m e 4 m : 5 m f 1 m : 2 km g 28 mm : 2800 m h 3 m : 0.6 mm i 1.1 m : 0.11 mm FLUENCY
Numer nd Alger 355 9 A model ity hs sle rtio of 1 : 1000. Find the tul height of skysrper tht hs sled height of 8 m. Find the sled length of trin pltform tht is 45 m long in rel life. 10 Blkottle nd Toowool re 17 m prt on mp with sle of 1 : 50 000. How fr prt re the towns in rel life? 11 Using the house plns shown elow, stte the tul dimensions of the following rooms. Bedroom 1 Fmily room Ptio 12 From the sled drwing elow, lulte the tul length nd height of the truk, giving your nswer to the nerest metre. 13 The photogrph on the right shows Jkson enjoying wlk with his dog. Jkson is 1.8 m tll in rel life. Find sle for the photogrph nd use this to estimte the height of Jkson s dog. 14 A sle length of 5 1 m is equl to n tul distne of 44 km. 2 How mny kilometres does sle length of 3 m equl? How mny kilometres does sle length of 20 m equl? How mny entimetres does n tul distne of 100 km equl? 9 11 10 13 11 14 PROBLEM-SOLVING 6D
356 Chpter 6 Rtios nd rtes 6D 15 15 17 17 19 15 Group the six rtios elow into pirs of equivlent rtios. 1 : 0.01 25 : 1 20 : 1 1 : 0.05 100 : 1 50 : 2 16 Whih of the following sles would e most pproprite if you wnted to mke sle model of: r your shool grounds Mt Kosiuszko A 1 : 10 000 B 1 : 1000 C 1 : 100 D 1 : 10 17 A mp mker wnts to mke mp of mountinous re tht is 64 km 2 (side length of 8 km). She deides to use sle of 1 : 1000. Desrie the prolem she is going to hve. 18 A sientist sys tht the imge of 1 mm long inset is mgnified y sle of 1 : 1000 through his mgnifying glss. Explin wht might e wrong with the sientist s rtio. 19 Otin mp of Austrli. Wht is the sle of your mp? Whih two pitl ities in Austrli re the furthest prt? Stte the distne. Whih two pitl ities in Austrli re losest together? Stte the distne. d Whih pitl ity is furthest wy from Sydney? Stte the distne. e Whih pitl ity is losest to Adelide? Stte the distne. f Chek the ury of your nswers on the internet. Sled drwing 20 Design sled drwing of one of the following: Your lssroom nd the ssoited furniture Your edroom nd the ssoited furniture A room in your house (throom, fmily room, grge...) Your sled drwing should fit omfortly onto n A4 pge. Mesure the dimensions of your hosen room nd the dimensions of n A4 pge, nd determine the most pproprite sle for your digrm. Show size nd lotion of doors nd windows where pproprite. Produe seond sle digrm of your room, ut show different rrngement of the furniture. Cn you produe the sle digrm on omputer softwre pkge? 20 REASONING ENRICHMENT
Numer nd Alger 357 6E Introduing rtes If you were to monitor wht you sid eh dy, you might well find tht you spek out rtes mny times per dy! A rtio shows the reltionship etween the sme type of quntities with the sme units, ut rte shows the reltionship etween two different types of quntities with different units. The following re ll exmples of rtes: Cost of petrol ws $1.45 per litre. Rump stek ws on speil for $18/kg. Dd drove to shool t n verge speed of 52 km/h. After the mth, your hert rte ws 140 ets/minute. The top redout on this hert monitor shows hert rte in ets per minute. Unlike rtios, rte ompres different types of quntities, nd so units must e shown. For exmple: The rtio of oys to girls in shool ws 4 : 5. The verge rte of growth of n dolesent oy is 12 m/yer. Let s strt: Stte the rte For eh of the following sttements, write down orresponding rte. The Lodges trvelled 400 km in 5 hours. Gry ws pid $98 for 4-hour shift t work. Feliity spent $600 on two-dy shopping spree. Mx hd grown 9 m in the lst three months. Vuong pid $37 for hlf ui metre of rushed rok. Pul yled totl distne of 350 km for the week. Wht ws the rte (in questions/minute) t whih you nswered these questions?
358 Chpter 6 Rtios nd rtes Key ides Rtes ompre quntities mesured in different units. All rtes must inlude units for eh quntity. The two different units re seprted y slsh /, whih is the mthemtil symol for per. For exmple: 20 km/h = 20 km per hour = 20 km for eh hour It is onvention to write rtes in their simplest form. This involves writing the rte for only one unit of the seond quntity. For exmple: spent $45 in 3 hours = $45 in 3 hours Non simplified rte 3 3 = $15 in 1 hour Simplified rte ($15/h) The verge rte is lulted y dividing the totl hnge in one quntity y the totl hnge in the seond quntity. For exmple: reding 400 pge ook in 4 dys Averge reding rte = 400 pges in 4 dys 4 4 = 100 pges in 1 dy Therefore verge reding rte = 100 pges/dy. Exmple 12 Writing simplified rtes Express eh of the following s simplified rte. 12 students for two tehers SOLUTION $28 for 4 kilogrms EXPLANATION 6 students/teher 12 students for 2 tehers $7/kg $28 for 4 kg 4 2 6 students for 1 teher $7 for 1 kg Exmple 13 Finding verge rtes Tom ws 120 m tll when he turned 10 yers old. He ws 185 m tll when he turned 20 yers old. Find Tom s verge rte of growth per yer etween 10 nd 20 yers of ge. SOLUTION 10 65 m in 10 yers 6.5 m in 1 yer 10 Averge rte of growth = 6.5 m/yer. EXPLANATION 4 Growth = 185 120 = 65 m Divide oth numers y 10. 2
Numer nd Alger 359 Exmple 12 Exmple 13 Exerise 6E 1 Whih of the following re exmples of rtes? A E $5.50 4.2 runs/over B 180 ml/min 2 Mth eh rte in the first olumn with its most likely rte in the seond olumn. C $60/h F 0.6 g/l G 200 m 2 3 Write typil units for eh of the following rtes. Prie of susges Typing speed d e Energy nutrition informtion f g Pin relief medition h 4 5 Write eh of the following s simplified rte. 12 dys in 4 yers $180 in 6 hours e $126 000 to purhse 9 res g 12 000 revolutions in 10 minutes i 60 minutes to run 15 kilometres Find the verge rte of hnge for eh sitution. Relm drove 6000 kilometres in 20 dys. A riket tem sored 78 runs in 12 overs. d e Russell gined 6 kilogrms in 4 yers. f 6 A dripping tp filled 9 litre uket in 3 hours. Wht ws the dripping rte of the tp in litres/hour? How long would it tke the tp to fill 21 litre uket? d f h j 1 3 3 Employee s wge D H 5 23 84/L 90 people/dy Speed of r $2100/m 2 Cost of uilding new home Popultion growth Resting hert rte Petrol osts Gol onversion rte Wter usge in the shower Criket tem s run rte 4, 5 4(½), 5 68 km/h 64 ets/min $15/h 15 gols in 3 gmes $17.50 for 5 kilogrms 36 000 ns in 8 hours 80 mm rinfll in 5 dys 15 kilometres run in 60 minutes Holly sved $420 over three yers. Sski grew 120 entimetres in 16 yers. The temperture dropped 5 C in 2 hours. 6 9 8 10 4 5(½) 7 Mrtine grew t n verge rte of 6 m/yer for the first 18 yers of her life. If Mrtine ws 50 m long when she ws orn, how tll ws Mrtine when she turned 18? 8 If 30 sld rolls were ought to feed 20 people t pini, nd the totl ost ws $120, find the following rtes. Sld rolls/person Cost/person Cost/roll 9 11 UNDERSTANDING FLUENCY PROBLEM-SOLVING
360 Chpter 6 Rtios nd rtes 6E 9 The numer of hours of sunshine ws reorded eh dy for one week in April. The results re listed. Mondy 6 hours, Tuesdy 8 hours, Wednesdy 3 hours, Thursdy 5 hours Fridy 7 hours, Sturdy 6 hours, Sundy 7 hours Find the verge numer of hours of sunshine: i per weekdy ii per weekend dy iii per week iv per dy Given the ove rtes, how mny hours of sunshine would you expet in April? 10 Hrvey finished 10 kilometre re in 37 minutes nd 30 seonds. Jques finished 16 kilometre re in 53 minutes nd 20 seonds. Clulte the running rte of eh runner. Whih runner hd fster running pe? 11 The Tungmh Footll Clu hd 12 000 memers. After five suessful yers nd two premierships, they now hve 18 000 memers. Wht hs een the verge rte of memership growth per yer for the pst 5 yers? If this memership growth rte ontinues, how mny more yers will it tke for the lu to hve 32 000 memers? 12 A r uses 24 L of petrol to trvel 216 km. Express these quntities s simplified rte in: i km/l ii L/km How n you onvert km/l to L/km? 13 The Teleonnet teleommunitions ompny hs vrile ll hrge rte for phone lls of up to 30 minutes. The hrges re 50/min for first 10 minutes, 75/min for the seond 10 minutes nd $1/min for the third 10 minutes. Find the ost of phone lls of these given lengths. i 8 minutes ii 13 minutes iii 24 minutes Wht is the verge hrge rte per minute for 30 minute ll? iv 30 minutes Connetplus, rivl teleommunitions ompny, hrges onstnt ll rte of 60/minute. d e 12 12, 13 12, 13 If you normlly mde lls tht were 15 minutes long, whih ompny hs the etter del for you? If you normlly mde lls tht were 25 minutes long, whih ompny hs the etter del for you? Wht is the length of phone ll for whih oth ompnies would hrge the sme mount? PROBLEM-SOLVING REASONING
Numer nd Alger 361 Trget 155 14 In Vitori, due to drought onditions, the stte government in 2008 urged ll residents to sve wter. The gol ws set for eh Vitorin to use no more thn 155 litres of wter per dy. d e How mny people live in your household? Aording to the Vitorin government, how mny litres of wter n your household use per dy? Perform some experiments nd lulte the following rtes of wter flow. i Shower rte (L/min) ii Wshing mhine (L/lod) iii Hose (L/min) iv Toilet (L/flush, L/hlf flush) v Running tp (L/min) vi Drinking wter (L/dy) vii Dishwsher (L/wsh) viii Wter for food preprtion (L/dy) Estimte the verge dily rte of wter usge for your household. Ask your prents for reent wter ill nd find out wht your fmily household wter usge rte ws for the pst three months. Before the inititive, Vitorins were using n verge of 164 litres/dy/person. Twelve months fter the inititive, Vitorins were using 151 litres/dy/person. f g How muh wter per yer for the stte of Vitori does this sving of 13 litres/dy/person represent? Wht is the rte t whih your fmily is hrged for its wter? 14 ENRICHMENT 6E
362 Chpter 6 Rtios nd rtes 38pt 6A 38pt 6A 38pt 6B 38pt 6C Ext 38pt 6C Ext 38pt 6D 38pt 6D 38pt 6D 38pt 6E 38pt 6E Progress quiz 1 2 3 4 5 6 7 8 9 10 An exursion to Se World hs 6 tehers, 144 students, 69 oys the rest girls. Write down, in simplest form, the rtio of: tehers to students oys to girls girls to the totl of people on the exursion Complete eh pir of equivlent rtios. 2 : 5 = : 15 : 3 = 12 : 9 24 : 8 = 12 : d 7 : 8 : 3 = 21 : : Simplify the following rtios. 24 : 36 24 : 36 : 40 3/4 : 9 d 0.9 : 12 e 40 ents to $1.20 Divide: $800 in the rtio of 5 : 3 $7500 in the rtio of 8 : 3 : 1 1 km in the rtio of 2 : 3 Tony nd Vi shre their ompny s profits in the rtio of 15 : 7. Tony s shre totls $1200. How muh profit did the ompny mke? Wht is Vi s shre? A mp uses sle of 1 : 40 000. Find the tul distne, in metres, if the distne on the mp mesures 8 mm. A model r hs sle of 1 : 100. Find the sle length, in entimetres, of Lexus tht is 4.32 metres long. Determine the sle rtio for 4mm on drwing representing n tul distne of 8 km. Express eh of the following s simplified rte. 180 students on 3 uses $5.60 for 4 kg 186 km in 2 1 2 hours 38pt 6E 11 Find the verge rte for eh sitution. Thelm drove 8000 km in 50 dys Ainslie grew 20 m in 2 1 2 yers Cllum sved $1250 in 6 months Who erns the most, nd y how muh, if Kelly is pid $96 570 yer nd Todd erns $7985 eh month? 38pt 6E 12 Whih is fster 70 km/h or 21 m/s?
Numer nd Alger 363 6F Rtios nd rtes nd the unitry method The onept of solving prolems using the unitry method ws introdued in Chpter 3. The unitry method involves finding the vlue of one unit. This is done y dividing the mount y the given quntity. One the vlue of one unit is known, multiplying will find the vlue of the numer of units required. Let s strt: Finding the vlue of 1 unit? For eh of the following, find the vlue of 1 unit or 1 item. 8 sketlls ost $200. 4 riket ts ost $316. 5 kg of wtermelon ost $7.50. For eh of the following, find the rte per 1 unit. Cr trvelled 140 km in 2 hours. 1000 L of wter leked out of the tnk in 8 hours. $51 ws the prie pid for 3 kg of John Dory fish. For eh of the following, find the vlue of 1 prt. Rtio of ooks to mgzines red ws 2 : 5. Milli hd red 14 ooks. Rtio of pink to red flowers is 7 : 11. A totl of 330 red flowers re in loom. Rtio of girls to oys is 8 : 5. There re 40 girls in group. The unitry method involves finding the vlue of one unit nd then using this informtion to solve the prolem. When deling with rtios, find the vlue of 1 prt of the rtio. For exmple: Rtio of rs to motorikes is 35 : 2 nd there re 6 motorikes. 2 prts = 6 motorikes 2 2 1 prt = 3 motorikes When deling with rtes, find the vlue of the rte per 1 unit. For exmple: Pedro erned $64 for 4 hour shift t work. Therefore, wge rte = $64 per 4 hours = $16 per hour = $16/h. One the vlue of one prt or the rte per one unit is known, the vlue of ny numer of prts or units n e found y multiplying. The tehnique of dividing nd/or multiplying vlues in suessive one-step lultions n e pplied to the onept of onverting rtes from set of given units to different set of units. Key ides
364 Chpter 6 Rtios nd rtes Exmple 14 Reviewing the unitry method Andy trvels 105 km in 7 identil r trips from home to shool. How fr would she trvel in 11 suh r trips? SOLUTION 7 r trips = 105 km 7 7 1 r trip = 15 km 11 11 11 r trips = 165 km Andy trvels 165 km. EXPLANATION Exmple 15 Solving rtio prolems using the unitry method Find the vlue of 1 unit y dividing oth quntities y 7. Solve the prolem y multiplying oth quntities y 11. The rtio of pples to ornges is 3 : 5. If there re 18 pples, how mny ornges re there? SOLUTION 3 5 3 prts = 18 pples 1 prt = 6 piees 5 prts = 30 ornges There re 30 ornges. 3 5 EXPLANATION Exmple 16 Solving rte prolems using the unitry method Apples = 3 prts, Ornges = 5 prts Need to find the vlue of 1 prt. To find 5 prts multiply the vlue of 1 prt y 5. A truk uses 4 L of petrol to trvel 36 km. How fr will it trvel if it uses 70 L of petrol? SOLUTION Rte of petrol onsumption 36 km for 4 L 4 4 9 km for 1 L 70 70 630 km for 70 L Truk will trvel 630 km on 70 L. EXPLANATION Find the petrol onsumption rte of 1 unit y dividing oth quntities y 4. Solve the prolem y multiplying oth quntities y 70.
Numer nd Alger 365 Exmple 14 Exmple 15 Exmple 16 1 Exmple 17 Converting units using the unitry method Meliss works t the lol supermrket nd erns $57.60 for 4 hour shift. How muh does she ern in /min? SOLUTION Wge = $57.60 for 4 hours 4 $14.40 for 1 hour 4 1440 for 60 minutes 60 60 24 for 1 minute Meliss erns 24/min. Exerise 6F EXPLANATION Write down Meliss s wge rte. Find the rte of $ per 1 hour. Convert $ to ents nd hours to minutes. Divide rte y 60 to find rte of ents per minute. 1 4 4 If 8 kg of hiken fillets ost $72, how muh would 3 kg of hiken fillets ost? 2 If one dozen tennis lls ost $9.60, how muh would 22 tennis lls ost? 3 If three pirs of soks ost $12.99, how muh would 10 pirs of soks ost? 4 If 500 g of mine met osts $4.50, how muh would 4 kg of mine met ost? 5 6 Solve the following rtio prolems. The required stff to student rtio for n exursion is 2 : 15. If 10 tehers ttend the exursion, wht is the mximum numer of students who n ttend? The rtio of ommerils to tul show time for prtiulr TV hnnel is 2 : 3. How mny minutes of tul show were there in 1 hour? A retngle hs length nd width dimensions in rtio of 3 : 1. If prtiulr retngle hs length of 21 m, wht is its width? d Wlter nd Willim hve height rtio of 7 : 8. If Willim hs height of 152 m, how tll is Wlter? Solve the following rte prolems. d 5, 6, 8(½) 5 7, 8(½) A tp is dripping t rte of 200 ml every 5 minutes. How muh wter drips in 13 minutes? A professionl footller sores n verge of 3 gols every 6 gmes. How mny gols is he likely to sore in full seson of 22 gmes? A snil trvelling t onstnt speed trvels 400 mm in 8 minutes. How fr does it trvel in 7 minutes? A omputer proessor n proess 500 000 kiloytes of informtion in 4 seonds. How muh informtion n it proess in 15 seonds? 5 6(½), 7, 8(½) UNDERSTANDING FLUENCY 7 Leonie, Spener nd Mkenzie hve just won prize. They deide to shre it in the rtio of 4 : 3 : 2. If Spener reeives $450, how muh do Leonie nd Mkenzie reeive, nd wht ws the totl vlue of the prize?
366 Chpter 6 Rtios nd rtes 6F Exmple 17 8 Convert the following rtes into the units given in the rkets. $15/h (/min) $144/h (/s) 3.5 L/min (L/h) d 20 ml/min (L/h) e 0.5 kg/month (kg/yer) f 120 g/dy (kg/week) g 60 g/ (kg/$) h $38/m (/mm) i 108 km/h (m/s) j 14 m/s (km/h) 9 The Mighty Ots rekfst erel is sold in oxes of three different sizes: smll (400 g) for $5.00, medium (600 g) for $7.20, lrge (750 g) for $8.25 Find the vlue of eh ox in $/100 g. Wht is the hepest wy to uy minimum of 4 kg of the erel? 10 Gemm runs 70 metres in 8.4 seonds. If she mintins the sme verge speed, in wht time will she run 100 metres? 11 Zn s hir grew 6 m in 5 months. Find Zn s verge rte of hir growth in m/month nd in m/yer. How long would it tke for Zn s hir to grow 30 m? 12 Mri n pint 15 m 2 in 20 minutes. Wht is the rte t whih Mri pints in m 2 /h? Wht re n Mri pint in 20 hours? Mri must pint 1000 m 2 in 20 hours. Find the rte t whih she will need to pint in m 2 /min. 13 If x donuts ost $y: how muh would 1 donut ost? how muh would one dozen donuts ost? how muh would z donuts ost? 9, 10 10 12 14 13, 14 14 A tringle hs side lengths in rtio of 19 : 22 : 17. If the shortest side is 17 m, find the lengths of the other two sides nd the perimeter of the tringle. A tringle hs side lengths in rtio of 3 : 5 : 4. If the longest side is 35 m, find the lengths of the other two sides nd the perimeter of the tringle. 10 12 13 15 FLUENCY PROBLEM-SOLVING REASONING
Numer nd Alger 367 15 In frwy glxy, thriving lien olony uses the following units: For money they hve puks nd pks: 1 puk (pu) = 1000 pk (p) For length they hve doits nd minidoits: 1 doit (D) = 80 minidoits (md) Polynute rope is pried t 4 pu/d. Find the ost of the rope in terms of p/md. Where will we meet? 16 Phil lives in Perth nd his friend Werner lives in Sydney. The distne, y rod, etween their two houses is 4200 km (rounded to the nerest 100 km). Phil deides to drive to Sydney nd Werner deides to drive to Perth. They leve home t the sme time nd trvel the sme route, ut in opposite diretions. Phil drives t onstnt speed of 75 km/h nd Werner drives t onstnt speed of 105 km/h. Will they meet on the rod t spot loser to Sydney or loser to Perth? How long will it tke Phil to trvel to Sydney? How long will it tke Werner to trvel to Perth? d Stte the lotion of eh friend fter they hve een driving for 15 hours. e At wht lotion (distne from Sydney nd/or Perth) will they meet? When they meet, Phil deides to hnge into Werner s r nd they drive k to Werner s home t n verge speed of 105 km/h. f How long did it tke Phil to trvel to Sydney? g Design similr prolem for two friends trvelling t different onstnt speeds etween two different pitl ities in Austrli. 16 REASONING ENRICHMENT 6F
368 Chpter 6 Rtios nd rtes Key ides 6G Solving rte prolems EXTENDING One thing tht is ertin in life is tht hnge is inevitle. We re onstntly interested in rtes of hnge nd how things hnge over period of time. We re lso regulrly fed with prolems involving speifi rtes. Strong rithmeti skills nd knowing whether to multiply or divide y the given rte llows mny rte prolems to e solved quikly nd urtely. Over the next few dys, keep reord of ny rtes you oserve or experiene. Look out for the slsh / sign nd listen for the per word. Let s strt: Estimte the rte For eh of the following sttements, estimte orresponding rte. Commeril rte: The numer of ommerils on TV per hour Typing rte: Your typing speed in words per minute Lughing rte: The numer of times teenger lughs per hour Growth rte: The verge growth rte of hild from 0 to 15 yers of ge Running rte: Your running pe in metres per seond Homework rte: The numer of sujets with homework per night Clpping rte: The stndrd rte of udiene lpping in lps per minute Thnkyou rte: The numer of opportunities to sy thnk you per dy Compre your rtes. Whih rte is the highest? Whih rte is the lowest? Disuss your nswers. When rte is provided, hnge in one quntity implies tht n equivlent hnge must our in the other quntity. For exmple: Ptrik erns $20/hour. How muh will he ern in 6 hours? $20 for 1 hour 6 6 $120 for 6 hours For exmple: Ptrik erns $20/hour. How long will it tke him to ern $70? $20 for 1 hour 1 1 3 2 3 $70 for 3 1 2 2 hours Crefully onsider the units involved in eh question nd nswer.
Numer nd Alger 369 Exmple 18 Solving rte prolems Rhel n touh type t 74 words/minute. How mny words n she type in 15 minutes? Lenne works in donut vn nd sells on verge 60 donuts every 15 minutes. How long is it likely to tke her to sell 800 donuts? SOLUTION 74 words in 1 minute 15 15 1110 words in 15 minutes Rhel n type 1110 words in 15 minutes. 15 200 60 donuts in 15 minutes 4 donuts in 1 minute 800 donuts in 200 minutes Lenne is likely to tke 3 hours nd 20 minutes to sell 800 donuts. 15 200 Exmple 19 Solving omintion rte prolems EXPLANATION 74 15 370 740 1110 Selling rte = 60 donuts/15 minutes Divide oth quntities y HCF of 15. Multiply oth quntities y 200. Convert nswer to hours nd minutes. Three wter hoses from three different tps re used to fill lrge swimming pool. The first hose lone tkes 200 hours to fill the pool. The seond hose lone tkes 120 hours to fill the pool nd the third hose lone tkes only 50 hours to fill the pool. How long would it tke to fill the pool if ll three hoses were used? SOLUTION In 600 hours: hose 1 would fill 3 pools hose 2 would fill 5 pools hose 3 would fill 12 pools Therefore in 600 hours the three hoses together would fill 20 pools. Filling rte = 600 h/20 pools 20 20 30 h/pool It would tke 30 hours to fill the pool if ll three hoses were used. EXPLANATION LCM of 200, 120 nd 50 is 600, Hose 1 = 600 h 200 h/pool = 3 pools Hose 2 = 600 h 120 h/pool = 5 pools Hose 3 = 600 h 50 h/pool = 12 pools Together = 3 + 5 + 12 = 20 pools filled. Simplify rte y dividing y HCF.