6E Introducing rates 6F Ratios and rates and the unitary method 6G Solving rate problems (Extending) 6H Speed (Extending)

Transcription

1 Chpter 6 Rtios nd rtes Wht you will lern 6A Introduing rtios 6B Simplifying rtios 6C Dividing quntity in given rtio (Extending) 6D Sle drwings 6E Introduing rtes 6F Rtios nd rtes nd the unitry method 6G Solving rte prolems (Extending) 6H Speed (Extending) Vitorin urriulum NUMBER AND ALGEBRA Rel numers Solve rnge of prolems involving rtes nd rtios, inluding distne-time prolems for trvel t onstnt speed, with nd without digitl tehnologies (VCMNA277) ISBN Greenwood et l. 2016

2 Online resoures Chpter pre-test Video demonstrtions of ll worked exmples Intertive widgets Intertive wlkthroughs Downlodle HOTsheets Aess to ll HOTmths Austrlin Curriulum ourses Aess to the HOTmths gmes lirry Ger rtios nd Formul 1 ring Gers re designed to trnsfer power from one moving prt to nother moving prt. Gers re wheels with teeth. The motor rottes the drive ger whih in turn rottes the driven ger. If the drive ger (e.g. with 30 teeth) mkes two revolutions for every one revolution mde y the driven ger (e.g. with 60 teeth), the ger rtio is 2:1. Riding iyle uphill with ger rtio of 3:1 mens tht the pedls rotte 3 times in order to drive the rer wheel round one. Crs nnot go fster thn the ger rtios llow. In Formul 1 ring, suh s the Austrlin Grnd Prix, the re engineers use dt from previous events to set the ger rtios. Eh Formul 1 tem must sumit up to 30 pirs of ger rtios tht they n hoose from during the re seson. Ger oxes must e removed whenever ger rtios re djusted. The world fmous MLren P1 hs 7 gers with first ger rtio of 3.98:1 giving mximum speed of 80 km/h in first ger. In seventh ger, the ger rtio is 0.69:1 nd the top speed eomes 460 km/h. ISBN Greenwood et l. 2016

3 334 Chpter 6 Rtios nd rtes 6A 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 points The rtio of eggs to utter to sugr will ffet the flvour nd 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. Key ides 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 isto.

4 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. Key ides 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 : 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 : : 36 EXPLANATION Both numers re multiplied y : : 5 3 Both numers re divided y 3.

5 336 Chpter 6 Rtios nd rtes Exerise 6A Write down the rtio of shded prts to unshded prts for eh grid. UNDERSTANDING 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 6, 7 8(½) 4 6, 7(½), (½) Exmple 1 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 FLUENCY 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

6 Numer nd Alger 337 6A Exmple 2 7 Complete eh pir of equivlent rtios. 1 : 3 = 4 : 1 : 7 = 2 : 2 : 5 = : 10 FLUENCY 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 4 : 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 : 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. PROBLEM-SOLVING 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 ll Austrlin sttes tht hve vowel to onsonnt rtio of 1 : , 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. REASONING 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. 2 : x = 6 : 2 : x = 5 : y : 2x = : 8x d 12xy : 6y = : 1

7 338 Chpter 6 Rtios nd rtes 6A Are rtios 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 ENRICHMENT 4 m 2 m 5 m 15 m Tringles se = 10 m height = 9 m d 4 m 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

8 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 Simplest form Key ides 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.

9 340 Chpter 6 Rtios nd rtes Exmple 3 Simplifying rtios Simplify the following rtios. 7 : : 200 SOLUTION 7 : : : : 4 EXPLANATION HCF of 7 nd 21 is 7. Divide oth numers y 7. HCF of 450 nd 200 is 50. Divide oth numers y 50. Exmple 4 Simplifying rtios involving frtions Simplify the following rtios. 3 5 : :11 4 SOLUTION EXPLANATION 3 : : : : 15 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. Exmple 5 Chnging quntities to the sme units 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 EXPLANATION 4mmto2m= 4 mm to 20 mm = 4 : 20 = 1:5 25 minutes to 2 hours = 25 minutes to 120 minutes = 25 : 120 = 5 : 24 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.

10 Numer nd Alger 341 Exerise 6B 1 4 3, 4 1 Write down the rtio of shded prts to unshded prts for eh of the following in simplest form. UNDERSTANDING 2 To express the rtio 4 : 16 in simplest form, you would: A multiply oth quntities y 2 B 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. A 2 : 28 B 15 : 75 C 14 : 45 D 13 : (½), 9(½) 5 9(½) 5 9(½) Exmple 3 5 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 FLUENCY 6 Simplify the following rtios. 2 : 4 : 6 12 : 21 : : 60 : 12 d 85 : 35 : 15 e 12 : 24 : 36 f 100 : 300 : 250 g 270 : 420 : 60 h 24 : 48 : 84 Exmple 4 Exmple 4 Exmple Simplify the following rtios. e 1 3 : : 1 4 Simplify the following rtios. f 1 4 : : 4 5 g 2 5 : : 2 15 d h 2 7 : : : : : 1 2 d : 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

11 342 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, 12 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 PROBLEM-SOLVING 14 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. REASONING 17 Simplify the following rtios. 2 : 4 d 5f : 24f 15x : 3y : 2 e hk : 3k f 24xyz : 60yz

12 Numer nd Alger 343 Aspet rtios 18 Aspet rtio is the reltionship etween the width nd height of the imge s displyed on sreen. 18 ENRICHMENT 6B A high-definition television sreen (left) ompred to the sme imge on n nlogue television sreen (right). d e f g 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!)

13 344 Chpter 6 Rtios nd rtes 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? Key ides 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 = = 5. $ 4 = 1 prt 2 2 Vlue of one prt = $20 5 = $4. $ 8 = 2 prts Therefore 2 prts = $8, nd 3 prts = $12.

14 Numer nd Alger 345 Using frtions to divide quntity in given rtio: Frtion of the mount required = numer in rtio 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. Key ides 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 EXPLANATION Unitry method Totl numer of prts = 9 Totl numer of prts = = prts = 54 m 9 Vlue of 1 prt = 54 m 9 = 6m 1 prt = 6 m 1 prt = 6 m prts = 24 m 5 prts= 30 m Chek numers dd to totl: = m divided in rtio of 4 : 5 is 24 m nd 30 m. Frtions method 4 9 of 54 = = 24 Frtion = numer in rtio totl numer of prts 5 9 of 54 = = 30 Chek numers dd to totl: = m divided in rtio of 4 : 5 is 24 m nd 30 m.

15 346 Chpter 6 Rtios nd rtes Exmple 7 Dividing quntity in rtio with three terms Divide $300 in the rtio of 2 : 1 : 3. SOLUTION EXPLANATION Unitry method Totl numer of prts = 6 Totl numer of prts = = 6 6 prts=$300 Vlue of 1 prt = $300 6 = $ prt = $50 1 prt=$ prts=$ prts=$150 3 $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 = = of 300 = = of 300 = = 150 Frtion = numer in rtio 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.

16 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 EXPLANATION Unitry method 2 prts = prt = prts= students ttend Birdsville College. Equivlent rtios method 123 oys : girls = 2 : 3 = 246 : students ttend Birdsville College. Rtio of oys : girls is 2 : 3 Boys hve 2 prts = 246 Vlue of 1 prt = = 123 Totl prts = = 5 prts 5 prts = = 615 Use equivlent rtios. Multiply eh quntity y 123. Totl numer of students = 246 oys girls = 615 Exerise 6C 1 4 3, 4 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? UNDERSTANDING 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?

17 348 Chpter 6 Rtios nd rtes 6C Exmple 6 Exmple 7 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 : (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 $ 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 6 8(½) FLUENCY 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? PROBLEM-SOLVING 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? Exmple 8 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? 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?

18 Numer nd Alger 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 16, 17 REASONING 6C 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? On one dy, the rtio ws 3 : 7. How mny oys nd how mny girls were sent? A fir shre Three students Rmshid, Tony nd Mri entered group Geogrphy ompetition. Tony spent 3 hours reserhing the topi. Mri spent 2 1 hours designing the poster. 2 Rmshid spent 5 hours prepring the finl presenttion. 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 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? ENRICHMENT

19 350 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 Terrigl 1 Mrrmrr Woy Woy Ntionl Bouddi NP Prk Broken By Rihmond Plm Pifi Windsor Ku-ring-gi Beh Oen Chse NP Hornsy Mon Vle Blktown Pymle St Mrys Prrmtt Mnly Ryde Port Jkson SYDNEY Bondi Liverpool Bnkstown Sydney Airport Botny By Sutherlnd Tsmn Cmpelltown Kurnell Port Hking Se Cmden Royl 31 Ntionl Prk Colo River Blue Mountins Ntionl Prk Nepen Springwood Penrith Blxlnd 32 Blue Mountins Ntionl Prk Wllhi Lke Burrgorng River Piton Brgo River Nepen Georges River Appin Hwkesury 1 Bulli River N km Sle 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 N 0 1 Sle 1: km

20 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. Key ides Multiply Sle distne Atul distne Divide Conversion of units. It is importnt to rememer how to orretly onvert mesurements of length when deling with sles. 1 km = 1000 m 1m = 100 m 1 m = 10 mm 1000 km m m mm

21 352 Chpter 6 Rtios nd rtes Exmple 9 Converting sle distne to tul distne A mp hs sle of 1 : Find the tul distne for eh sled distne. 2 m 5 mm 14.3 m SOLUTION EXPLANATION Atul distne = 2 m = m = 400 m (or 0.4 km) Sle ftor = Multiply sled distne y sle ftor. Convert nswer into sensile units. Atul distne = 5 mm Multiply sled distne y sle ftor. = mm = 100 m Atul distne = 14.3 m = m = 2.86 km Convert nswer into sensile units. Multiply sled distne y sle ftor. Shortut: 2, then Convert nswer into sensile units. 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 4550 mm SOLUTION EXPLANATION Sled distne = 50 m 500 = 5000 m 500 = 10 m (0.1 m) Sled distne = 75 m 500 = 750 mm 500 = 1.5 mm Sled distne = 4550 mm 500 = 45.5 mm 5 Sle ftor = 500 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. Divide tul distne y sle ftor. Shortut: 100, then 5 (or vie vers) = 9.1 mm

22 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 : Sle ftor = 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. Exerise 6D 1 4 3, 4 1 A mp hs sle of 1 : 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 m to: mm m km UNDERSTANDING 4 Convert 560 m to: km m mm

23 354 Chpter 6 Rtios nd rtes 6D Exmple (½), 7 5 6(½), 7, 8(½) Find the tul distne for eh of the following sled distnes. Give your nswer in n pproprite unit. Sle 1 : i 2 m ii 4 mm iii 7.3 m 5 8(½) FLUENCY Sle 1 : i 80 m ii 18.5 mm iii 1.25 m Sle 1 : 400 i 16 mm ii 72 m iii 0.03 m d Sle 1 : 300 i 5 mm ii 8.2 m iii 7.1 m e Sle 1 : 2 i 44 m ii 310 m iii 2.5 mm f Sle 1 : 5 i 4 m ii 24 m iii 155 mm g Sle 1 : 0.5 i 12 m ii 3.2 mm iii 400 m h Sle 100 : 1 i 3 m ii 11.5 km iii 81.5 m Exmple 10 6 Find the sled length for eh of these tul lengths. Sle 1 : 200 i 200 m ii 4 km iii 60 m Sle 1 : 500 i m ii 1 km iii 75 m Sle 1 : i 1350 m ii 45 km iii m d Sle 1 : i 12 km ii 1800 m iii 400 mm e Sle 1 : i 5000 km ii m iii 1250 m f Sle 1 : 10 i 23 m ii 165 m iii 6.15 m g Sle 1 : 0.1 i 30 m ii 5 mm iii 0.2 mm h Sle 200 : 1 i 1 mm ii 7.5 m iii 8.2 mm Exmple 11 7 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. 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

24 Numer nd Alger A model ity hs sle rtio of 1 : 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 : 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 PROBLEM-SOLVING 6D 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?

25 356 Chpter 6 Rtios nd rtes 6D Group the six rtios elow into pirs of equivlent rtios. 1 : : 1 20 : 1 1 : : 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 : B 1 : 1000 C 1 : 100 D 1 : REASONING 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 : 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 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? ENRICHMENT

26 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?

27 358 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 2 6 students for 1 teher 2 $7/kg $28 for 4 kg 4 $7 for 1 kg 4 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 65 m in 10 yers m in 1 yer Averge rte of growth = 6.5 m/yer. EXPLANATION Growth = = 65 m Divide oth numers y 10.

28 Numer nd Alger 359 Exerise 6E Whih of the following re exmples of rtes? A E $ runs/over B 180 ml/min C $60/h F 0.6 g/l G 200 m 2 D H /L UNDERSTANDING 2 Mth eh rte in the first olumn with its most likely rte in the seond olumn. 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 Employee s wge 90 people/dy Speed of r $2100/m 2 Cost of uilding new home 68 km/h Popultion growth 64 ets/min Resting hert rte $15/h Petrol osts Gol onversion rte Wter usge in the shower Criket tem s run rte 4(½) 4(½), 5 4 5(½) Exmple 12 4 Write eh of the following s simplified rte. 12 dys in 4 yers $180 in 6 hours e $ to purhse 9 res g revolutions in 10 minutes i 60 minutes to run 15 kilometres d f h j 15 gols in 3 gmes $17.50 for 5 kilogrms ns in 8 hours 80 mm rinfll in 5 dys 15 kilometres run in 60 minutes FLUENCY Exmple 13 5 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 Holly sved $420 over three yers. Sski grew 120 entimetres in 16 yers. The temperture dropped 5 C in 2 hours 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? 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? PROBLEM-SOLVING 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

29 360 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 memers. After five suessful yers nd two premierships, they now hve 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 memers? PROBLEM-SOLVING 12 12, 13 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? REASONING 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 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?

30 Numer nd Alger 361 Trget 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. How mny people live in your household? Aording to the Vitorin government, how mny litres of wter n your household use per dy? 14 ENRICHMENT 6E d e 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?

31 362 Chpter 6 Rtios nd rtes Progress quiz 6A 1 An exursion to Se World hs 6 tehers nd 144 students. Sixty-nine of the students re oys; the rest re girls. Write down, in simplest form, the rtio of: tehers to students oys to girls girls to the totl of people on the exursion 6A 2 Complete eh pir of equivlent rtios. 2 : 5 = : 15 : 3 = 12 : 9 24 : 8 = 12 : d 7 : 8 : 3 = 21 : : 6B 3 Simplify the following rtios. 24 : : 36 : 40 3/4 : 9 d 0.9 : 12 e 40 ents to $1.20 Ext 6C 4 Divide: $800 in the rtio of 5 : 3 $7500 in the rtio of 8 : 3 : 1 1 km in the rtio of 2 : 3 Ext 6C 5 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? 6D 6 A mp uses sle of 1 : Find the tul distne, in metres, if the distne on the mp mesures 8 mm. 6D 7 A model r hs sle of 1 : 100. Find the sle length, in entimetres, of Lexus tht is 4.32 metres long. 6D 8 Determine the sle rtio for 4mm on drwing representing n tul distne of 8 km. 6E 9 Express eh of the following s simplified rte. 180 students on 3 uses $5.60 for 4 kg 186 km in hours 6E 10 Find the verge rte for eh sitution. Thelm drove 8000 km in 50 dys Cllum sved $1250 in 6 months Ainslie grew 20 m in yers 6E 11 Who erns the most, nd y how muh, if Kelly is pid $ yer nd Todd erns $7985 eh month? 6E 12 Whih is fster 70 km/h or 21 m/s?

32 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 $ riket ts ost $ kg of wtermelon ost $7.50. For eh of the following, find the rte per 1 unit. Cr trvelled 140 km in 2 hours 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 prt = 3 motorikes Key ides 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.

33 364 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 r trip = 15 km r trips = 165 km Andy trvels 165 km. EXPLANATION Find the vlue of 1 unit y dividing oth quntities y 7. Solve the prolem y multiplying oth quntities y 11. Exmple 15 Solving rtio prolems using the unitry method The rtio of pples to ornges is 3 : 5. If there re 18 pples, how mny ornges re there? SOLUTION 3 prts = 18 pples 3 1 prt = 6 piees 5 5 prts = 30 ornges There re 30 ornges. 3 5 EXPLANATION 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. Exmple 16 Solving rte prolems using the unitry method 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 km for 1 L 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.

34 Numer nd Alger 365 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 for 60 minutes 24 for 1 minute 60 Meliss erns 24/min. 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. Exerise 6F Exmple 14 1 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? UNDERSTANDING 5, 6, 8(½) 5 7, 8(½) 5 6(½), 7, 8(½) Exmple 15 5 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? FLUENCY Exmple 16 6 Solve the following rte prolems. d 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 kiloytes of informtion in 4 seonds. How muh informtion n it proess in 15 seonds? 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?

35 366 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) FLUENCY 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? PROBLEM-SOLVING 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 , If x donuts ost $y: how muh would 1 donut ost? how muh would one dozen donuts ost? how muh would z donuts ost? REASONING 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.

36 Numer nd Alger 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. REASONING 6F Where will we meet? 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. ENRICHMENT 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.

37 368 Chpter 6 Rtios nd rtes 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. Key ides 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 $70 for hours Crefully onsider the units involved in eh question nd nswer.

38 Numer nd Alger 369 Exmple 18 Solving rte prolems Rhel n 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 EXPLANATION 74 words in 1 minute words in 15 minutes Rhel n type 1110 words in 15 minutes donuts in 15 minutes 4 donuts in 1 minute 800 donuts in 200 minutes Selling rte = 60 donuts/15 minutes Divide oth quntities y HCF of 15. Multiply oth quntities y 200. Lenne is likely to tke 3 hours nd 20 minutes to sell 800 donuts. Convert nswer to hours nd minutes. Exmple 19 Solving omintion rte prolems 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. 600 hours for 20 pools hours for 1 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 = = 20 pools filled. Simplify rte y dividing y HCF.

39 370 Chpter 6 Rtios nd rtes Exerise 6G 1, Fill in the gps. 60 km in 1 hour km in 3 7 questions in 3 minutes 70 questions in d 5 $25 in 1 hour $125 in 120 litres in 1 minute in 6 minutes UNDERSTANDING 2 Fill in the gps. 3 $36 for 3 hours 3 for 1 hour 5 for 5 hours 150 rottions in 5 minutes in 1 minute in 7 minutes Exmple 18 Exmple A ftory produes 40 plsti ottles/minute. How mny ottles n the ftory produe in 60 minutes? How mny ottles n the ftory produe in n 8 hour dy of opertion? Mrio is professionl home pinter. When pinting new home he uses n verge of 2.5 litres of pint per hour. How mny litres of pint would Mrio use in week if he pints for 40 hours? 5 A truk trvels 7 km per litre of fuel. How mny litres re needed for the truk to trvel 280 km? 6 Dniel prtises his guitr for 40 minutes every dy. How mny dys will it tke him to log up 100 hours of prtie? 7 A flywheel rottes t rte of 1500 revolutions per minute. How mny revolutions does the flywheel mke in 15 minutes? How mny revolutions does the flywheel mke in 15 seonds? How long does it tke for the flywheel to omplete revolutions? d How long does it tke for the flywheel to omplete 150 revolutions? 8 Putr is n elite rower. When trining, he hs stedy working hert rte of 125 ets per minute (pm). Putr s resting hert rte is 46 pm. How mny times does Putr s hert et during 30 minute workout? How mny times does Putr s hert et during 30 minutes of rest? If his oh sys tht he n stop his workout one his hert hs eten times, for how long would Putr need to trin? FLUENCY

40 Numer nd Alger Wht is the ost of pving drivewy tht is 18 m long nd 4 m wide, if the pving osts $35 per squre metre? 10 A sltwter swimming pool requires 2 kg of slt to e dded for every litres of wter. Jon s swimming pool is 1.5 metres deep, 5 metres wide nd 15 metres long. How muh slt will she need to dd to her pool? 9, PROBLEM-SOLVING 6G 11 The Bioni Womn gives Btmn 12 seond strt in 2 kilometre re. If the Bioni Womn runs t 5 km/min, nd Btmn runs t 3 km/min, who will win the re nd y how mny seonds? 12 At shool mp there is enough food for 150 students for 5 dys. How long would the food lst if there were only 100 students? If the food rn out fter only 4 dys, how mny students ttended the mp? Exmple Mihelle n omplete lndsping jo in 6 dys nd Dnielle n omplete the sme jo in 4 dys. Working together, in how mny dys ould they omplete the jo? REASONING 14 Three riklyers Mri, Hugh nd Ethn re ldding new home. If Mri were to work lone, the jo would tke him 8 dys to omplete. If Hugh were to work lone, the jo would tke him 6 dys to omplete, nd if Ethn were to work y himself, the jo would tke him 12 dys to omplete. If the three men work together, how long will it tke them to omplete the jo? Wht frtion of the house will eh riklyer omplete?

41 372 Chpter 6 Rtios nd rtes 6G 15 Four ns of dog food will feed 3 dogs for 1 dy. How mny ns re needed to feed 10 dogs for 6 dys? How mny dogs n e fed for 9 dys from 60 ns? For how mny dys will 40 ns feed 2 dogs? 16 If it tkes 4 workers, 4 hours to dig 4 holes, how long would it tke 2 workers to dig 6 holes? 17 Stte the units required for the nswer to eh of the following rte lultions. $205/kg 48 kg 62 s 12 m/s 500 ets 65 ets/min ( pm ) d 4000 revolutions 120 revs/min REASONING Vlue for money Soft drink n e purhsed from supermrkets in vriety of sizes. Below re the osts for four different sizes of ertin rnd of soft drink. 600 ml uddies 1.25 L ottles 2 L ottles ml ns $2.70 eh $1.60 eh $2.20 eh $6.00 per pk ENRICHMENT Find the eonomy rte (in $/L) for eh size of soft drink. Find nd ompre the ost of 30 litres of soft drink purhsed entirely in eh of the four different sizes. If you only hve $60 to spend on soft drink for prty, wht is the differene etween the gretest mount nd the lest mount of soft drink you ould uy? Assume you hve less thn $1.60 left. Most supermrkets now inlude the eonomy rte of eh item underneth the prie tg to llow ustomers to ompre vlue for money. d Crry out some reserh t your lol supermrket on the eonomy rtes of prtiulr food item with rnge of ville sizes (suh s drinks, rekfst erels, sugr, flour). Write report on your findings.

42 Numer nd Alger 373 6H Speed EXTENDING A rte tht we ome ross lmost every dy is speed. Speed is the rte of distne trvelled per unit of time. On most osions, speed is not onstnt nd therefore we re interested in the verge speed of n ojet. Averge speed is lulted y dividing the distne trvelled y the time tken. Distne trvelled Averge speed = Time tken Given the verge speed formul, we n tell tht ll units of speed must hve unit of length in the numertor, followed y unit of time in the denomintor. Therefore mm/h is unit of speed nd ould e n pproprite unit for the speed of snil! This snil s verge speed won t e high, ut it n still e lulted. The most ommon units of speed re m/s nd km/h. Let s strt: Whih is fster? With prtner, determine the fster of the two listed lterntives. d Cr A trvelling t 10 m/s Wlker C trvelling t 4 km/h Jogger E running t 1450 m/h Plne G flying t 700 km/h Cr B trvelling t 40 km/h Wlker D trvelling t 100 m/min Jogger F running t 3 m/s Plne H flying t 11 km/min Speed is mesure of how fst n ojet is trvelling. If the speed of n ojet does not hnge over time, the ojet is trvelling t onstnt speed. Cruise ontrol helps r trvel t onstnt speed. When speed is not onstnt, due to elertion or deelertion, we re often interested to know the verge speed of the ojet. Averge speed is lulted y the formul: Key ides Averge speed = Distne trvelled Time tken or s = d t

43 374 Chpter 6 Rtios nd rtes Key ides Depending on the unknown vlue, the ove formul n e rerrnged to mke d or t the sujet. The three formuls involving s, d, nd t re: s = d t d = s t t = d s d d d s t s t s t Cre must e tken with units for speed, nd on osions units will need to e onverted. The most ommon units of speed re m/s nd km/h. Exmple 20 Finding verge speed Find the verge speed in km/h of: ylist who trvels 140 km in 5 hours runner who trvels 3 km in 15 minutes SOLUTION EXPLANATION s = d t 140 km = 5h = 28 km/h Alterntive unitry method km in 5 hours 28 km in 1 hour 5 Averge speed = 28 km/h The unknown vlue is speed. Write the formul with s s the sujet. Distne trvelled = 140 km Time tken = 5h Speed unit is km/h. Write down the rte provided in the question. Divide oth quntities y 5. s = d t = 3 km 15 min = = 3 4 = 12 km/h Alterntive unitry method 4 3 km in 15 minutes 12 km in 60 minutes 4 Averge speed = 12 km/h Distne trvelled = 3 km Time tken = 15 min or 1 4 h Dividing y 1 4, is the sme s multiplying y 4 1. Write down the rte provided in the question. Multiply oth quntities y 4.

44 Numer nd Alger 375 Exmple 21 Finding the distne trvelled Find the distne trvelled y truk trvelling for 15 hours t n verge speed of 95 km/h. SOLUTION EXPLANATION d = s t = 95 km/h 15 h = 1425 km Alterntive unitry method 95 km in 1 hour km in 15 hours Truk trvels 1425 km in 15 hours. The unknown vlue is distne. Write the formul with d s the sujet. Distne unit is km. Write the rte provided in the question. Multiply oth quntities y 15. Exmple 22 Finding the time tken Find the time tken for hiker wlking t 4 km/h to trvel 15 km. SOLUTION EXPLANATION t = d s 15 km = 4 km/h = 3.75 h = 3 h 45 min Alterntive unitry method 4 km in 1 hour km in 1 4 hour km in hours The unknown vlue is time. Write the formul with t s the sujet. The time unit is h. Leve nswer s deiml or onvert to hours nd minutes h = = 45 min Express the rte s provided in the question. Divide oth quntities y 4. Multiply oth quntities y 15. It tkes 3 h 45 min to trvel 15 km.

45 376 Chpter 6 Rtios nd rtes Exmple 23 Converting units of speed Convert 72 km/h to m/s. Convert 8 m/s to km/h. SOLUTION EXPLANATION 72 km in 1 hour m in 60 minutes 1200 m in 1 minute 1200 m in 60 seonds 20 m in 1 seond  72 km/h = 20 m/s 8 m in 1 seond 8 m in 1 seond m in 1 minute m in 1 hour  28.8 km in 1 hour Express rte in kilometres per hour. Convert km to m nd hour to minutes. Divide oth quntities y 60. Convert 1 minute to 60 seonds. Divide oth quntities y 60. Shortut for onverting km/h m/s 3.6. Express rte in metres per seond. Multiply y 60 to find distne in 1 minute. Multiply y 60 to find distne in 1 hour. Convert metres to kilometres. Shortut: m/s 3.6 km/h. 8 m/s 3.6 = 28.8km/h Exerise 6H 1 4 3, 4 1 Whih of the following is not unit of speed? A m/s B km/h C m/h D L/kg E m/min 2 Distne trvelled If Averge speed = then the Distne trvelled must equl: Time tken A Averge speed Time tken B Averge speed Time tken C Time tken Averge speed UNDERSTANDING 3 If Averge speed = Distne trvelled then Time tken must equl: Time tken A Distne trvelled Averge speed B Distne trvelled Averge speed C Averge speed Distne trvelled 4 If n ojet trvels 800 metres in 10 seonds, the verge speed of the ojet is: A 8000 m/s B 800 km/h C 80 km/h D 80 m/s

46 Numer nd Alger 377 Exmple 20 5 Clulte the verge speed of: sprinter running 200 m in 20 seonds skteorder trvelling 840 m in 120 seonds r trvelling 180 km in 3 hours d truk trvelling 400 km in 8 hours e trin trvelling 60 km in 30 minutes f trm trvelling 15 km in 20 minutes 5 9(½) 5 7(½), (½) FLUENCY 6H Exmple 21 6 Clulte the distne trvelled y: ylist trvelling t 12 m/s for 90 seonds n nt trvelling t 2.5 m/s for 3 minutes ushwlker who hs wlked for 8 hours t n verge speed of 4.5 km/h d trtor ploughing fields for 2.5 hours t n verge speed of 20 km/h Exmple 22 7 Clulte the time tken y: sports r to trvel 1200 km t n verge speed of 150 km/h us to trvel 14 km t n verge speed of 28 km/h plne to fly 6900 km t onstnt speed of 600 km/h d ll moving through the ir t speed of 12 m/s to trvel 84 m Exmple 23 8 Convert the following speeds to m/s. 36 km/h 180 km/h 660 m/min d 4 km/s Exmple 23 9 Convert the following speeds to km/h. 15 m/s 2 m/s 12 m/min d 1 km/s 10 Complete the following tle. d e f Speed s Distne d Time t 50 km 2 h 30 m/s 1200 m 5 km/h 12 h 600 m 5 min 210 km/h 20 min 100 m/s 10 km

47 378 Chpter 6 Rtios nd rtes 6H 11 A plne is flying t ruising speed of 900 km/h. How fr will the plne trvel from 11:15.m. to 1:30 p.m. on the sme dy? 12 The wheels on Chrlie s ike hve irumferene of 1.5 m. When Chrlie is riding fstest, the wheels rotte t speed of five turns per seond. Wht is the fstest speed Chrlie n ride his ike, in km/h? How fr would Chrlie trvel in 5 minutes t his fstest speed? 11, PROBLEM-SOLVING 13 The k end of 160-metre-long trin disppers into 700-metre-long tunnel. Twenty seonds lter the front of the trin emerges from the tunnel. Determine the speed of the trin in m/s. 14 Ann rode her ike to shool one morning, distne of 15 km, t n verge speed of 20 km/h. It ws rining in the fternoon, so Ann deided to tke the us home. The us trip home took 30 minutes. Wht ws Ann s verge speed for the return journey to nd from shool? 15 In Berlin 2009, Jmin sprinter Usin Bolt set new 100 m world reord time of 9.58 seonds. Clulte Usin Bolt s verge speed in m/s nd km/h for this world reord. (Round the nswers orret to one deiml ple.) 16 The Ghn trin is n Austrlin ion. You n ord The Ghn in Adelide nd 2979 km lter, fter trvelling vi Alie Springs, you rrive in Drwin. (Round the nswers orret to one deiml ple.) If you ord The Ghn in Adelide on Sundy t 2:20 p.m. nd rrive in Drwin on Tuesdy t 5:30 p.m., wht is the verge speed of the trin journey? There re two mjor rest reks. The trin stops for 4 1 hours t Alie Springs nd 4 hours t Ktherine. Tking these reks into ount, 4 wht is the verge speed of the trin when it is moving?

48 Numer nd Alger Write two speeds in km/h tht re etween 40 m/s nd 45 m/s , Nin, Shnti nd Belle run 1000 m re t onstnt speed. When Nin rossed the finish line first, she ws 200 m hed of Shnti nd 400 m hed of Belle. When Shnti rossed the finish line, how fr hed of Belle ws she? 19 Julie nd Jenette enjoy finishing their 6 km morning run together. Julie runs t n verge speed of 10 km/h nd Jenette runs t n verge speed of 3 m/s. If Julie leves t 8.m., t wht time should Jenette leve if they re to finish their run t the sme time? 18, 19 REASONING 6H Speed reserh Crry out reserh to find nswers to the following questions. Light nd sound d e f Wht is the speed of sound in m/s? Wht is the speed of light in m/s? How long would it tke sound to trvel 100 m? How long would it tke light to trvel 100 km? How mny times quiker is the speed of light thn the speed of sound? Wht is Mh numer? Sperft g h i Wht is the espe veloity needed y sperft to rek free of Erth s grvittionl pull? Give this nswer in km/h nd lso km/s. Wht is the oritl speed of plnet Erth? Give your nswer in km/h nd km/s. Wht is the verge speed of spe shuttle on journey from Erth to the Interntionl Spe Sttion? ENRICHMENT Knots Wind speed nd ot speed re often given in terms of knots (kt). j k l Wht does knot stnd for? Wht is the link etween nutil miles nd system of loting positions on Erth? How do you onvert speed in knots to speed in km/h?

49 380 Chpter 6 Rtios nd rtes Investigtion Fun run investigtion Three mths tehers, Mrs M, Mr P nd Mr A, trined very hrd to ompete in Brisne Fun Run. The km route pssed through the otnil grdens, long the Brisne River to New Frm Prk, nd k gin. Their times nd verge stride lengths were: Mrs M: 41 minutes, 50 seonds: 8 strides per 9 m Mr P: 45 minutes, 21 seonds: 7 strides per 9 m Mr A: 47 minutes, 6 seonds: 7.5 strides per 9 m Copy nd omplete the following tle nd determine whih rtes re the most useful representtions of fitness. Give the nswers rounded to one deiml ple. Justify your onlusions. Running rtes Mrs M Mr P Mr A World reord 10 km runners Your fmily memer or friend Seonds per 100 m Seonds per km Metres per minute Km per hour Strides per 100 m Strides per minute Strides per hour

50 Numer nd Alger 381 Fitness investigtion 1 Using stopwth, mesure your resting hert rte in ets per minute. 2 Run on the spot for one minute nd then mesure your working hert rte in ets per minute. 3 Using stopwth, time yourself for 100 m run nd lso ount your strides. At the end, mesure your hert rte in ets per minute. Also lulte the following rtes. Your running rte in m/s, m/min nd km/h Your running rte in time per 100 m nd time per km Your rte of strides per minute, strides per km nd seonds per stride 4 Run 100 m four times without stopping nd, using stopwth, reord your umultive time fter eh 100 m. Orgnise these results into tle. Drw grph of the distne rn (vertil) vs time tken (horizontl). Clulte your running rte in m/min for eh 100 m setion. Clulte your overll running rte in m/min for the 400 m. Explin how nd why your running rtes hnged over the 400 m. 5 Try sprinting fst over mesured distne nd reord the time. Clulte your sprinting rte in eh of the following units: minutes per 100 m time per km metres per minute km per hour 6 Reserh the running rte of the fstest shooloy nd shoolgirl in Austrli. How do their sprinting rtes ompre to the running rtes of Austrlin Olympin thletes?

51 382 Chpter 6 Rtios nd rtes Prolems nd hllenges 1 This digrm is mde up of 8 equl-sized squres. Up for hllenge? If you get stuk on question, hek out the 'Working with unfmilir prolems' poster t the end of the ook to help you. How mny squres need to e shded if the rtio of shded squres to unshded squres is: 1:3? 2 : 3? 1 : 2? 2 Bottle A hs 1 L of ordil drink with ordil to wter rtio of 3 : 7. Bottle B hs 1 L of ordil drink with ordil to wter rtio of 1 : 4. The drink from oth ottles is omined to form 2 L drink. Wht is the new ordil to wter rtio? 3 Brothers Mro nd Mtthew strt riding from home into town, whih is 30 km wy. Mro rode t 10 km/h nd Mtthew took 20 minutes longer to omplete the trip. Assuming tht they oth rode t onstnt speed, how fst ws Mtthew riding? 4 If 1 person tkes 1 hour to dig 1 post hole, how long will it tke 2 people to dig 2 post holes? If 3 people tke 3 hours to mke 3 wooden trin sets, how mny trin sets n 6 people mke in 6 hours? 5 At mrket you n trde 2 ows for 3 gots or 3 gots for 8 sheep. How mny sheep re 3 ows worth? 6 Two rs trvel towrd eh other on 100 km stright streth of rod. They leve t opposite ends of the rod t the sme time. The rs speeds re 100 km/h nd 80 km/h. How long does it tke for the rs to pss eh other? Cr km/h 100 km Cr 2 80 km/h 7 A river is flowing downstrem t rte of 1 km/h. In still wter Mihel n swim t rte of 2 km/h. Mihel dives into the river nd swims downstrem then turns round nd swims k to the strting point, tking 0.5 hours in totl. How fr did he swim? 8 A fitness fnti wlks t 4 km/h for time t 1, nd then runs t 7 km/h for time t 2. He trvels totl of 26 km. If he hd run for the sme time tht he hd wlked (t 1 ) nd wlked for the sme time tht he hd run (t 2 ), then he would hve trvelled 29 km. Wht ws the totl time spent wlking nd running? 9 A top seret goo hs een developed so tht it doules in volume every 60 seonds. At midnight smll mount is pled in eker nd its growth oserved. At 1.m. the eker is full. At extly wht time did the goo fill only 1 of the eker? 8 10 A r verges 60 km/h for 20 km nd then inreses its speed verging 80 km/h for the next 20 km. Clulte the r s verge speed (to two deiml ples) for the full 40 km trip. 11 Mx trvels t 50 km/h for 10 km. For the next 10 km Mx wnts to drive fst enough to mke his verge speed 100 km/h over the 20 km trip. Advise Mx out rehing this verge speed.

52 Numer nd Alger 383 Eliminte deimls HCF 4 Reurring deimls to frtions Sme units Rtios re written in simplest form with whole numers Simplest form : : 12 2 : : : : : : minutes: 1 hours 4 25 : : 3 Equivlent rtios : 3 8 : 12 4 : 6 : is to to Rtios 4 2 Comprison of two quntities of the sme type nd sme unit Finding quntity from rtio Cows to horses in rtio 5 : 2 There re 8 horses, how mny ows? Totl numer of prts One prt Sentene nswer Totl numer of prts Eh frtion Eh mount Sentene nswer Dividing quntity in given rtio (Ext) Divide $180 in the rtio 4 : 5 Frtion method Divide $180 in the rtio 4 : = = nd 5 9 Unitry method nd = $80 = $100 $180 divided in the rtio 4 : 5 is $80 nd $ prts = $ prt = $20 1 prt = $ prts = $80 5 prts = $100 $180 divided in the rtio 4 : 5 is $80 nd $100 Chpter summry Unitry method C : H 5 : 2 2 prts = 8 horses 1 prt = 4 horses 5 prts = 20 ows Length units km m m mm Averge rtes (Ext) 720 km driven in 10 hours Averge speed = = 72 km/h Sle rtio 4 : : : : 50 Equivlent rtio method C : H = 5 : 2 C : 8 = 5 : = 20 : 8 There re 20 ows. Sle ftor Atul ojet 125 times lrger thn drwing 1 th of the 50 drwing size Simplified rtes 25 km/h mny km per one hour? $12/kg mny $ per one kg? Sle rtios Drwing length : Atul length = 1 : sle ftor times sle ftor Drwing (or mp) length divide sle ftor Rtes Atul length Atul size from model Model r 17 m long Sle 1 : 25 Sle ftor = 25 Atul r = m = 425 m = 4.25 m sme units Mp length from tul length Atul distne = 1 km Mp sle = 1 : Sle ftor = Mp length = 1 km = 1 km sme units = m = 4 m s : speed d : distne t : time d s t d s t d s t d s = t d = s t d t = s Unitry method to hnge units 1 For 18 km jog in 1 hours determine 2 verge rte in m/min 1 18 km in 1 hours km in 1 hour m in 60 min m in 1 min Averge speed = 200 m/min 1 prt 1 prt Compring two quntities with different units Unitry method with rtes 4 L pint to over 80 m 2? L to over 165 m L for 80 m L for 1 m prt for m L for 165 m 2

53 384 Chpter 6 Rtios nd rtes Chpter review 6A 6A Multiple-hoie questions 1 2 A shool hs 315 oys, 378 girls nd 63 tehers. The rtio of students to tehers is: A 11 : 1 B 1 : 11 C 5 : 6 D 6 : 5 Find the rtio of the shded re to the unshded re in this tringle. A 3 : 5 B 8 : 5 C 5 : 3 D 5 : 8 6B 6B 6C Ext 6C Ext The rtio 500 mm to 1 m is the sme s: 5 A 50 : 2 B 2500 : 1 C 2 : 5 D 5 : 2 The rtio : 3 simplifies to: 4 A 2 : 1 B 1 : 2 C 3 : 4 D 4 : 3 $750 is divided in the rtio 1 : 3 : 2. The smllest shre is: A $250 B $125 C $375 D $750 The rtio of the res of two tringles is 5 : 2. The re of the lrger tringle is 60 m 2. Wht is the re of the smller tringle? A 12 m 2 B 24 m 2 C 30 m 2 D 17 m 2 6E 7 Cllum fills his r with 28 litres of petrol t ents per litre. His hnge from $50 sh is: A $10 B $39.95 C $10.05 D $40 6F 8 45 km/h is the sme s: A 0.25 m/s B 25 m/s C 12.5 m/s D 75 m/s 6G Ext 9 A flg is reted y enlrging the shded retngle s shown. Wht is the length of the originl retngle? A B C D 20 m 30 m 40 m 13 m 20 m 120 m 80 m 6D 10 On mp, Sydney nd Melourne re mm prt. If the ities re 716 km prt, wht sle hs een used? A 1 : 5 B 1 : 5000 C 1 : D 1 :

54 Numer nd Alger 385 6A 6B Short-nswer questions 1 2 In Lo s penil se there re 6 oloured penils, 2 lk pens, 1 red pen, 5 texts, 3 led penils nd ruler. Find the rtio of: led penils to oloured penils lk pens to red pens texts to ll penils True or flse? 1 : 4 = 3 : 6 The rtio 2 : 3 is the sme s 3 : 2. The rtio 3 : 5 is written in simplest form. d 40 m : 1 m is written s 40 : 1 in simplest form. e : 2 = 5 : 8 Chpter review 6B 6B 3 4 Copy nd omplete. 4 : 50 = 2 : 1.2 : 2 = : : 4 = 1 : d 1 : : 5 = 5 : 15 : 25 Simplify the following rtios. 10 : : : 100 d 8 : 64 e 27 : 9 f 5 : 25 g 6 : 4 h 52 : 26 i 6 : 9 j 8 : 4 2 k 7 : 5 l : 2 m : 3 n 12 : 36 : B 5 Simplify the following rtios y first hnging to the sme units. 2 m : 8 mm 5 mm : 1.5 m 3 L : 7500 ml d 30 min : 1 h e 400 kg : 2 t f 6 h : 1 dy g 120 m : 1 km h 45 min : h 6C Ext 6 Divide: $80 in the rtio 7 : kg in the rtio 4 : 1 40 m in the rtio 6 : 2 d $1445 in the rtio 4 : 7 : 6 e $1 in the rtio 3 : 1 : 1 6C Ext 7 The rtio of the ost prie of TV to its retil prie is 5 : 12. If its ost prie is $480, lulte its retil prie. The rtio of Slly s height to Ben s height is 12 : 17. If the differene in their heights is 60 m, how tll is Slly? Ornge juie, pinepple juie nd guv juie re mixed in the rtio 4 : 3 : 2. If 250 ml of guv juie is used, how mny litres of drink does this mke? 6E 8 Express eh rte in simplest form. 10 km in 2 hours $650 for 13 hours 2800 km in 20 dys

55 386 Chpter 6 Rtios nd rtes Chpter review 6E 6D 9 10 Copy nd omplete. 400 km on 32 litres = km/l = L/100 km 5 grms in 2 min = g/min = g/h $1200 in 1 dy = $/dy 2 = $/h For sle of 1 : 1000, find the rel length (in metres) if the sle length is given s: m 2.7 m 140 mm 6D 11 Two ities re 50 km prt. How mny millimetres prt re they on mp tht hs sle of 1 : ? 6D 12 Find the sle ftor for the following pirs of similr figures, nd find the vlue of x. 4 m 6 m x m 3 m 4 m 5 m 9 m 6 m 15 m x m 6F 13 Copy nd omplete. $120/h = /min 6 m/s = km/h 720 km/h = m/s 6G Ext 14 A truk uses 12 litres of petrol to trvel 86 km. How fr will it trvel on 42 litres of petrol? Smir erns $67.20 for 12-hour shift. How muh will she ern for 7-hour shift? Tp 1 fills the pool in 12 hours, while tp 2 fills the sme pool in 15 hours. How long does it tke to fill this pool if oth tps re used? 6H Ext 15 Sndr drives to her mother s house. It tkes 45 minutes. Clulte Sndr s verge speed if her mother lives 48 km wy. How long does it tke Ari to drive 180 km long the freewy to work if he mnges to verge 100 km/h for the trip? How fr does Siri ride his ike if he rides t 4.5 km/h for 90 minutes?

56 Numer nd Alger 387 Extended-response questions 1 The Hrrison fmily nd the Nguyen fmily leve Wollongong t 8.m. on Sturdy morning for holidy in Melourne. The Hrrisons 17-yer-old son drives for the first 2 hours t 80 km/h. They then stop for rest of 1 1 hours. Mr Hrrison drives the rest of the wy. 2 The Nguyen fmily drives stright to Melourne with no stops. It tkes them 6 hours nd 15 minutes to drive the 627 km to Melourne. d e f At wht time did the Nguyen fmily rrive in Melourne? Clulte the verge speed of the Nguyen s r. Round your nswer to the nerest whole numer. At wht time did the Hrrisons resume their journey fter their rest stop? How mny kilometres did the Hrrisons still hve to over fter their rest rek efore rriving in Melourne? If the Hrrisons rrive in Melourne t 4:30 p.m., for how long did Mr Hrrison drive nd t wht speed? Clulte the ost of the petrol for eh fmily s trip if petrol ost 125/L nd the Hrrison r s onsumption is 36 L/100 km, while the Nguyen s r uses 40 L/100 km. Chpter review