Rates and Ratios. Rates and Ratios. Curriculum Ready.

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1 Curriulum Redy

2 Copyright P Lerning. All rights reserved. First edition printed 2009 in Austrli. A tlogue reord for this ook is ville from 3P Lerning Ltd. SBN Ownership of ontent The mterils in this resoure, inluding without limittion ll informtion, text, grphis, dvertisements, nmes, logos nd trde mrks (Content) re proteted y opyright, trde mrk nd other intelletul property lws unless expressly indited otherwise. You must not modify, opy, reprodue, repulish or distriute this Content in ny wy exept s expressly provided for in these Generl Conditions or with our express prior written onsent. Copyright Copyright in this resoure is owned or liensed y us. Other thn for the purposes of, nd sujet to the onditions presried under, the Copyright At 968 (Cth) nd similr legisltion whih pplies in your lotion, nd exept s expressly uthorised y these Generl Conditions, you my not in ny form or y ny mens: dpt, reprodue, store, distriute, print, disply, perform, pulish or rete derivtive works from ny prt of this resoure; or ommerilise ny informtion, produts or servies otined from ny prt of this resoure. Where opyright legisltion in lotion inludes remunerted sheme to permit edutionl institutions to opy or print ny prt of the resoure, we will lim for remunertion under tht sheme where worksheets re printed or photoopied y tehers for use y students, nd where tehers diret students to print or photoopy worksheets for use y students t shool. A worksheet is pge of lerning, designed for student to write on using n ink pen or penil. This my led to n inrese in the fees for edutionl institutions to prtiipte in the relevnt sheme. Pulished 3P Lerning Ltd For more opies of this ook, ontt us t: Designed 3P Lerning Ltd Although every preution hs een tken in the preprtion of this ook, the pulisher nd uthors ssume no responsiility for errors or omissions. Neither is ny liility ssumed for dmges resulting from the use of this informtion ontined herein.

3 Rtes nd rtios re wys of ompring two or more quntities. Rtes nd rtios re used ll round us every dy. Reipes Speed Rte of Py The rtio of the different ingredients in reipe. The rte of distne trvelled with respet to time in r, km/h. The mount person erns per week n e expressed s rte of dollrs per hour. For exmple, $8 per hour. Cn you list some other exmples in everydy life where two or more quntities re ompred? Q Hrry nd Slly purhsed g ontining 36 ookies. They deide to shre them sed on how muh eh ontriuted to their purhse. f Hrry pid 3 times s muh money s Slly did for this purhse, how mny ookies does he keep? Work through the ook for gret wy to do this

4 How does it work? Rtios A rtio ompres two quntities in given order. For exmple, if the numer of ornges in g is twie tht of the numer of pples, this is rtio of 2 to nd is written s: 2 : Eh numer is lled term of the rtio The order in rtio is importnt. The rtio of ornges to pples is 2 :. The rtio of pples to ornges is : 2. The shpes elow re divided into equl retngulr regions. For eh region: (i) Write down the rtio of the shded region to unshded region. (ii) Write down the rtio of the shded region to the whole shpe. (i) prt is shded, while 4 prts re unshded, so the rtio of shded to unshded prts is to 4, or using rtio nottion, : 4. (ii) There re 5 equl prts in totl. ` The rtio of shded to the whole is : 5. (i) 3 prts re shded, while 2 prts re unshded, so the rtio of shded to unshded prts is 3 to 2, or using rtio nottion, 3 : 2. (ii) There re 5 equl prts in totl. ` The rtio of the shded prts to the whole shpe is 3 : 5. (i) 4 prts re shded, while 3 prts re unshded. ` The rtio of shded to unshded prts is 4 to 3, or using rtio nottion, 4 : 3. (ii) There re 7 equl prts in totl. ` The rtio of the shded prts to the whole shpe is 4 : 7. 2

5 How does it work? Rtios RATOS * RATOS * RATOS * RATOS * f there re 7 girls nd 2 oys in lssroom, then: (i) The rtio of girls to oys is: :..../.../ (ii) The rtio of oys to girls is: :. (iii) The rtio of ll students in the lssroom to the numer of oys is: :. (iv) The rtio of the numer of girls in the lssroom to the totl numer of students is: :. A shool hs 4 sketll plyers, 23 soer plyers, 3 netllers, 29 swimmers, 6 Rugy plyers, 7 tennis plyers, 2 lwn owlers, nd 8 who do not ply sport. Find: (i) The rtio of sketll plyers to rugy plyers: :. (ii) The rtio of soer plyers to swimmers: :. (iii) The rtio of rugy plyers to soer plyers: :. (iv) The rtio of tennis plyers to netllers: :. (v) The rtio of sport plyers to non-plyers: :. 2 Complete shding in the prts until they mth the rtio of ˋshded to non-shdedˊ given underneth. psst. t doesn't mtter whih ones you shde, s long s the proportion is orret. Shded to Non-shded prts = 7 : 9. Shded to Non-shded prts = 9 : 5. 3

6 How does it work? Rtios 3 (i) Write the rtios of the shded regions to the unshded regions for eh digrm elow. : : : : (ii) Write the rtios of the shded prts to the totl prts for eh digrm. : : : : 4 (i) Write down the rtio of pples to ornges in eh g. (ii) Wht is the rtio of ornges to the numer of fruit in the g? (i) : (i) : (i) : (ii) : (ii) : (ii) : 4

7 How does it work? Equivlent rtios Rtios re equivlent if they represent the sme reltive proportions etween two quntities. Using shding to demonstrte equivlent rtios These retngles re ll the sme size nd divided into different numers of equl prts. The proportion of the shded to unshded res re the sme. So the rtios of shded to unshded for eh retngle re equivlent. : 2 2 : 4 3 : 6 4 : 8 Lowest terms 2 : 4 3 : 6 4 : : 2 : 2 : : 4 3 : 6 4 : 8 Equivlent rtios re found y multiplying or dividing ll terms in the rtio y the sme numer. A rtio involving only rtionl numers is sid to e in lowest terms or in simplest form, if ll terms re integers without ny ommon ftors. - The rtio 2 : 4 is not in lowest terms, s you n still divide oth terms y 2 to otin : 2. - The rtio : 2 is in lowest terms nd nnot e simplified further. 5

8 How does it work? Equivlent rtios (i) Write the rtios of the shded to unshded prts for these digrms. = 6 : 3 = : = : = : (ii) Write eh of these rtios in lowest terms. = 2 : = : = : = : 2 Shde the next two shpes to represent equivlent rtios of shded to unshded in the first shpe. 3 (i) Write down the rtios of striped shpes to solid shpes, nd redue the rtio to simplest form. = : = : = : = : simplest form = : simplest form = : simplest form (ii) Write nother rtio equivlent to eh of the rtios ove in (i). = : = : = : 6

9 How does it work? 4 Equivlent rtios Simplify these rtios to lowest terms y dividing y gretest ommon ftors. 2 : 0 0 : 30 2 : 48?:?.../.../ *EQUVALENT RATOS *EQUVALENT RATOS d 4 : 35 e 70 : 36 f 8 : 8 5 Connet the equivlent rtios with stright line to solve the puzzle. : 2 3 : 5 A : 3 3 W 23 : 69 8 : 3 A L R 9 : 4 2 : : 4 T 64 : 6 T M 0 9 : 80 : 20 : O T E 95 : 20 5 : 0 85 : : 5 S 90 : 20 D 24 : 9 45 : 5 5 S 5 : 7 2 : 9 R E 7 : 5 23 : 23 E 8 4 : 80 : 50 L 48 :

10 How does it work? Using frtions to redue rtios in lowest terms Simplifying rtios with whole numers is similr to simplifying frtions. n ft we n write our rtios s frtions, redue the frtion, nd then onvert k to rtio nottion. Exmples of using frtions to simplify rtios prolems Redue these to lowest terms y writing s frtions to simplify first. 4 : = 8 divide top nd ottom y 4 The redued form of the rtio is then : 8. ` 4 : 32 = : 8 You n lso simplify quikly y using the frtion utton on your lultor. 42 : = 2 3 divide top nd ottom y 4 ` 42 : 28 = 3 : 2 Be reful using the lultor here s it might disply simplified s 2 ( = 2 3 ) Rtio of irles to squres. Rtio of irles to squres = 9 : = 4 3 ` 9 : 2 = 3 : 4 8

11 How does it work? Using frtions to redue rtios in lowest terms Redue these rtios to lowest terms y onverting to frtions, simplifying nd k to rtio nottion. 3 : 27 2 : = 2 30 = ` 3 : 27 = : ` 2 : 30 = : 8 : 8 d 20 : = = ` 8 : 8 = : ` 20 : 00 = : e 2 : 70 f 200 : = 52 = ` 2 : 70 = : ` 200 : 52 = : 2 Use the frtion method to redue eh of these rtios to lowest terms. Rits to rrots Lettue to slugs Spikey puffer fish ( ) to ll fish 9

12 How does it work? Rtios with deimls f the terms of rtio re terminting deimls, some simple rules llow esy simplifition to lowest terms.. Move the deiml point in the rtio y the sme numer of ples for ll terms until you hve only whole numers on oth sides. 2. Simplify the rtio to lowest form. Simplify the rtios to lowest terms 2.2 : 5.4 Both numers hve one deiml ple, nd so y moving the deiml y one ple to the right will mke oth into integers (y multiplying oth numers in the rtio y 0). 2.2 : : 5.4 = 22 : 54 = 22 2 : 54 2 = : 27 Move the deiml one to the right Divide the rtio y 2 (oth the left nd right numer of the rtio) ` 2.2 : 5.4 = : 27 This is now in lowest terms :.2 The left deiml, 0.05 onsists of two deiml ples, nd the numer on the right,.2, hs one deiml ple. Moving eh deiml twie to the right will mke oth sides into integers. This is equivlent to multiplying y : :.2 = 5 : 20 = 5 5 : 20 5 = : on oth sides of the rtio. Move the deiml y 2 ples to the right. Divide the rtio y 5 (oth the left nd right numer of the rtio) ` 0.05 :.2 = : 24 This is now in lowest terms. 0

13 RRATOS WTH DECMALS * How does it work? Rtios with deimls Simplify these rtios y moving the deiml in oth numers one ple to the right, nd then redue to lowest terms:.5 : : : Simplify these rtios y moving the deiml in oth numers two ples to the right, nd then redue to lowest terms: 0.08 : : : Simplify these rtios y moving the deiml in oth numers y three ples to the right, nd then redue to lowest terms using your lultor: : : :.84 d 6.42 : 0.60 e.24 : 4.48 f : /.../ ATOS WTH DECMALS *

14 How does it work? Rtios with frtions For rtios ontining frtion terms, we n use two methods to find the rtio in lowest integer terms.. Write their equivlent frtion with ommon denomintor, or 2. Cross multiply the denomintors. Exmples of reduing rtios involving frtions to lowest form. 2 : 3 2 : : The lowest ommon multiple of 2 nd 3 is 6 so this is good ommon denomintor. 3 6 : 2 6 Cnel the ommon denomintor 6. 3 : 2 Alterntively just ross multiply: 2 : 3 ` 3 : 2 2 : : 3 5 ` 3 : 2 s efore. 2 : : : : 3 The lowest ommon multiple of nd 5 is 5 so this is good ommon denomintor. Cnel the ommon denomintor 5. Alterntively just ross multiply: 2 : 3 5 ` 2 5 : : : 5 = 9 4 : 6 4 ` 0 : 3 s efore. Convert the mixed frtions to improper frtions. = : Mke the denomintor the sme, in this se it is 20. = : Cnel the ommon denomintor. = 45 : 24 2

15 RRATOSWTHFRACTONS * How does it work? Rtios with frtions Redue these rtios to lowest terms using the method indited, either ommon denomintor or ross multiply: 5 : 4 5 = : 2 : 7 = 4 : 4 = : 2 7 : 4 = 2 4 : d 3 6 : = 6 3 : = : = : = : = : 2 Redue these rtios to simplest form: 4 7 : : : d 5 4 : /.../ ATOS WTHFRACTONS * 3

16 How does it work? Best uys nd the unitry method Rtios n e used to deide whih produt is the est uy. Exmples using the unitry method Whih is the est uy: 50 g of heese for $24 or 70 g of heese for $35? Re-write eh rtio so the left hnd sides of the rtios mth. Then ompre the right hnd sides. 50 g: $24 = 0 g: $24 5 = 0 g: $4.80 Divided y g: $35 = 0 g: $35 7 Divided y 7. = 0 g: $5.00 The first heese osts $4.80 per 0 g, while the seond one $5 per 0 g. The first one is heper y 20. The unitry method redues one of the quntities to nd then multiplies up to find the nswer. f 9 mngos ost $40.50, how muh does eh mngo ost? 9 mngos osts $40.50 ` mngo osts $ ` eh mngo osts $4.50 Wht s the est uy, 2 penils for $6.72 or 7 penils for $9.25? The method is to find the ost of one penil nd ompre them. 2 penils for $6.72 ` 2 2 penils for $ ` penil for $ penils for $8.67 ` 7 7 penils for $ ` penil for $0.5 Therefore the seond one is slightly heper t penil for $0.5. d Cvir ws 22 g for $05 t one store nd 0 g for $55 t nother store. Convert oth of these to ost per 00 g to the nerest ent to find the est uy. 22 g for $05 ` g for $ ` g for $ ` 00 g for $ or $ g for $55 ` g for $ 55 0 or $05 0 ` g for $5.50 ` 00 g for $ The est uy is 22 g for $05. 4

17 How does it work? Best uys nd the unitry method Find the ost per single item if: 2 pples ost $6.60 pple =.../.../ g of heese osts $4.50 g heese = 7 vodos ost $5.75 vodo = d 200 m of wire osts $356 m wire = 2 Find the ost per single whole item if: Hlf pket of pegs ost $2.0 pket of pegs = 4 wtermelon ost $ kg of pst osts $.55 wtermelon = kg pst = d 0.7 m of gold osts $25 m of gold = 3 Whih re the est uys? 4 kg of fish for $24.00 or 3 sks of pottoes for $4 or 8 kg of fish for $50? 4 sks of pottoes for $7? [Hint: Multiply the first one y 2.] [Hint: Multiply the first one y 4 nd the seond one y 3 to find wht 2 sks ost in eh se.] 5 sets of guitr strings for $78.75 or d 8 loves of sourdough red for $4.60 or 7 sets of guitr strings for $0.95? 25 loves of sourdough red for $3.25? 5

18 How does it work? Best uys nd the unitry method 4 Use the unitry method to deide. Who is fster? mn who n run 20 m in 3 seonds or mn who n run 220 m in 22 seonds. [Hint: Find how fr eh n run in seond.] Whih is more expensive to uy? 8 video gmes for $62.80 or 3 video gmes for $00.23 Find the mode of trnsport tht trvels t the fster speed etween two towns if: y us you trvel 62 km in 2.7 hours or y trin you trvel 25 km in 3.32 hours. [Hint: Find out how muh eh mode trvels in hour.] 5 Find the ost 'per 00' for eh of these items: 25 g of flour osts $0.20 g = `00 g = 75 ml of lvender oil osts $7.65 ml = `00 ml = 2000 ml of milk osts $4.85 ml = `00 ml = d minutes of mhine hire osts $75.00 ml = `00 ml = 6

19 How does it work? Best uys nd the unitry method 6 Find the ost per 00 g to deide on the est uy etween: 54 g of fish for $2.60 or 750 g of rie for $5.60 or 75 g of fish for $ g of rie for $ f 60 g of orinder (A) osts $4.20, wht prie would 00 g of orinder (B) need to e less thn to e the etter uy? f 222 kg of X-soil osts $577.20, wht prie would 250 kg of Y-soil need to e more thn to eome the more expensive purhse? 7

20 How does it work? Dividing quntity in given rtio t is importnt to e le to divide quntity, suh s n mount of money, into given rtio. Exmples of dividing quntity in given rtio Two friends uy lottery tiket. One of them pys $4 towrds the tiket while the other one pys $5 towrds the tiket. They gree to split ny winnings in the sme rtio. f they win $8532 how muh does eh of them get? Totl numer of prts is: = 9 prts. ` 9 prts is $8532 ` prt is $ = $948 ` 4 prts is $948 4 = $3792 ` 5 prts is $948 5 = $4740. This questions is sking to divide the quntity $8532 into the rtio 4 : 5. One reeives $3792 nd the other reeives $4740. The rtio of girls to oys in shool is 6 : 7 nd there re 448 oys. How mny students re there in the shool? ` 7 prts is 448 ` prt is = 64 There re = 3 prts in totl. ` 3 prts is 3 64 = 832. An mount of hy ws shred etween two frmers who got 420 kg nd 700 kg. (i) Wht ws the totl mount of hy shred? Totl = 420 kg kg = 20 kg Therefore there re 832 students in the shool. (ii) n wht rtio ws the hy divided mongst the two frmers? 420 kg : 700 kg = 420 : 700 = 42 : 70 = 2 : 35 = 3 : 5 The rtio is simplified in stges here, ut it ould hve een done in one step y dividing y 40. d We n hve more thn two terms in rtio. The rtio of snd, grvel nd ement for onrete is : 2 : 4. f 24 kg of grvel re used, how muh onrete will e produed? 2 prts is 24 kg prt is 24 2 = 2 kg Totl numer of prts is = 7 7 prts is: 7 2 kg = 84 kg Therefore 84 kg of onrete re produed. 8

21 How does it work? Dividing quntity in given rtio Divide the quntities into the rtio given. $200 into the rtio : 2. Totl numer of prts is + =. *DVDNGAQUANTTYNAGVENRATO *DVDNGAQUANTTYNAGVENRATO.../.../ ` prts is $200. ` prt is $ = $. prts ` 2 prts is $. ` $200 is divided into the mounts $ nd $. 700 kg into the rtio 3 :. 360⁰ into the rtio 5 : 4. 2 Hrry nd Slly purhsed g ontining 36 ookies. They deide to shre them sed on how muh eh ontriuted to their purhse. f Hrry pid 3 times s muh money s Slly did for this purhse, how mny ookies does he keep? 9

22 How does it work? Dividing quntity in given rtio 3 An mount of money is shred etween two people who get $450 nd $650. (i) Wht ws the totl mount of the money shred? (ii) n wht rtio ws it divided mongst the two people? 4 A rrel of oil ws divided into smller rrels in the rtio x : 5. Aording to this rtio 200L is divided into 80 : 20. Find x. 5 Jon is 0 yers older thn Fion. They deide to shre $400 in the rtio of their ges. How muh do they eh get? [Hint: if Jon = x, then Fion = x 0.] 20

23 How does it work? Dividing quntity in given rtio 6 Redue these rtios to lowest terms: 2 : 2 : 4 3 : 6 : 2 4 : 4 : 4 d 2 : 2 : 4 : 8 e 0 : 5 : 35 : 20 f 6 : 5 : 9 : 27 : 8 7 Lim purhses to uy some lotto tikets with 7 friends t work. He pys $3 nd everyone else pys $. The prize is $ Write down wht everyone pys towrds the lotto tikets s rtio. [Hint. t strts : : ] How muh would Lim win if they won the gme? How muh would everyone else get? 2

24 How does it work? Dividing quntity in given rtio 8 Three whet frmers deide to split 2000kg of whet seeds etween them in the rtio 2 : 2 3 : 5 6. Simplify the rtio 2 : 2 3 : 5 6. [Hint: find ommon denomintor.] Using the nswer to prt find the totl numer of prts. How mny kilogrms of whet seed orrespond to prt? d How muh does eh frmer get to the nerest kg? e Aout 50 seeds weighing totl of.6g re plnted per squre metre. Wht is the size in squre metres of the lnd required to sow ll the seeds? 22

25 Where does it it work? Equivlent rtios with missing terms Missing terms of equivlent rtios n e found y writing oth rtios s frtions. Exmples of finding missing rtio terms Solve for x if x : 5 = 50 : 30 Write oth rtios s frtions first: x 5 = x 5 = 5 f possile, simplify frtions. method Write oth frtions with the sme denomintor: method 2 Cross multiply frtions: x 5 = x 5 = 25 5 ` x = 25 Cnel denomintors. x 5 = 25 5 x = 5 5 ` x = 25 Some questions require you to solve n eqution like this: Find the pronumerl if 3 : 2 = x : 8. Write oth rtios s frtion first: 3 2 = x = 2 x 54 = 2x = x Now solve for x y dividing oth sides y 2. ` x = 26 23

26 Where does it work? Equivlent rtios with missing terms Find the vlue of the missing terms in eh of these equivlent rtios. x : 6 = 2 : 3 = 2 3 : 5 = x : 0 = 3 3x : 4 = 30 : 4 3 : 2x = : 3 [Hint: use ross multiplition.] 24

27 EQUVALENT RATOS WTH MSSNG TERMS *** Where does it work? Equivlent rtios with missing terms Find the vlue of the missing terms in eh of these equivlent rtios. 5 5y : 20 = : : 2 = 25 : 8 7 3x : 5 = 6 : 6 8 y : 6 = 6 : y [Hint: use ross multiplition. Be reful here, there will e 2 nswers!].../.../

28 Where does it work? Sle drwings A sle drwing is n enlrged or redued version of rel life ojet. All the lengths re redued or enlrged y the sme ftor lled sle rtio. This shows how the drwing s dimensions nd the rel life dimensions re relted. The left numer in the rtio is for the sle drwing nd the right numer represents Rel Life. Exmples with sle drwings Sle Drwing : Rel Life A sle is suh tht 2m on the sle drwing represents 5m in rel life. Write this sle s rtio. As rtio the sle is 2m : 5m. Rememer tht 5 m = 500 m. ` 2m : 5 m = 2m : 500m = 2 : 500 = : 250 So the sle is : 250. Before simplifying, then units for oth terms must e the sme. The m ws nelled on eh side of the rtio. Divided oth sides y the ommon ftor 2. n rel life n ojet mesures 25m, nd the sle drwing hs sle : Find the length of the ojet on the sle drwing. 25m = 25 00m = 2500m Divide y 2000 gives 2500m 2000 =.25m So on the sle drwing the ojet hs length of.25m. m = 00m Find the sle if 20m on the sle drwing represents 0.5mm in rel life. The sle s rtio is: 20m : 0.5mm = 200mm : 0.5mm = 200 : 0.5 = 400 : m = 0mm. ` 20m = 20 # 0mm = 200mm d Find the rel life length of n ojet whose sled length is 8m on drwing with sle 2 : prts is 8m. ` prt is 4m. ` 27 prts is 27 4m = 98m. The rel life length is 98m. Beuse 2:27 = sle length : rel length 26

29 Where does it work? Sle drwings Find the sles for eh of these: m on the sle drwing represents 2m in rel life. [Hint: strt m : 2m ] 2m on the sle drwing represents m in rel life. 20 m represents km. d m represents 4 m. e 250 mm represents 250 km. 2 Look t the sle drwing of fire nt. f the sle drwing of the nt hs sle 2 :, find the length of the nt in rel life. [Hint: Let the rel life length e x.] 4.5 m 27

30 SCALE DRAWNGS * SCALE DRAWNGS * Where does it work? Sle drwings 3 Find the rel length (x) of some ojet if the sle drwing hs length 6.5 m nd the sle is given y: : 35 3 : 5 : 6 4 Find the length on the sle drwing (x) of rel life ojet of length 4 m if the sle is given y: 44 : : 9 3 : 40 5 This digrm is sle drwing of studio prtment. 5 m m Kithen 3 m Comined lounge/edroom 5 m Bthroom Sle: : 00 7 m Wht is the length of the studio prtment on the sle drwing in entimetres? Wht is the rel life length (x) of the studio prtment in metres? Find the re of the floor spe of the omined lounge/edroom re..../.../

31 Where does it work? Mps Mps re prtil exmple of sle drwings. Exmples of mp sles A mp sle is : (i) How fr is the rel life distne in kilometres if the mp distne is 5 m? Sine the sle is : then 5 m on the mp is rel life distne of: m = m = 2.5 km. (ii) f the rel life distne is 7 km, how mny entimetres is this on the mp? 7 km = 7000 m = m For eh m in rel life it will e m on the mp. ` Therefore m = 4 m Therefore on the mp the distne is 4 m. km = m. The rel life distne etween Alph Town nd Bet Town is 3 km. Alph Bet Gmm (i) Find the sle of this mp given tht eh grid unit is m. The sle is the rtio 4 grid units to 3 km. 4 m : 3 km = 4 m : m = 4 : = : (ii) Using this mp sle, find the distne etween Bet Town nd Gmm Town. The distne etween Bet nd Gmm towns on the mp is 5 grid units nd therefore 5 m. Therefore the rel life differene is: m = m 29

32 Where does it work? Mps A mp hs sle of : Wht is the rel distne (x) if two points on the mp re 2 m prt? 2 Points P, Q nd R re lotions on 3 islnds for this mp. Eh grid on the mp is m y m. P R Q m (i) f the rel life distne etween the two points P nd Q is 8 kilometres, find the sle of this mp given tht eh grid unit is m. m (ii) Using this mp sle, find the distne etween P nd R. *** MAPS *** P Q R.../.../ MAPS MAPS S ** * 30

33 Where does it work? Rtes A rte is omprison etween two quntities of different units. mgine tht you took 30 steps in 2 minutes, then your rte of wlking is 30 steps every 2 minutes. This is the sme s 5 steps every minute, whih is 5 steps per minute (written s 5 steps/minute). The forwrd slsh / mens per. divide y 5 simplify Trvelling 20 kilometres in 2 hours in r mens your rte is 60 km every hour or 60 km per hour (60 km/h). Exmples of rtes Suppose tht fruit piker piked 42 ins of fruit in 5 dys. Wht is his rte of piking fruit, expressed s ins/dy? 42 ins every 5 dys 45 5 ins every 5 5 dy ` 8.4 ins every dy 8.4 ins per dy 8.4 ins/dy Divide y 5. Simplify. Using different wording. Repled per with the slsh symol /. A ftory mkes grments t rte of 24 grments/h. How mny grments re produed in the ftory in 40 hours? 24 grments in hour ` 24 # 40 grments in 40 hours ` 960 grments in 40 hours An employee gets weekly wge of $850 per 5 dy week. Wht is his dily rte of py? $850 per 5 dys $850 5 per 5 5 dys $70 per dy $70 per dy We ould write 70 $/dy ut in prtise it is usul to put the dollr sign in front of the numer. $70/dy 3

34 ** RATES *** RATES *** RATES * Where does it work? Rtes A oy piks fruit t the rte of 2.5 ins per dy. How mny ins of fruit does he pik in 2 dys? 2 A mn works 35 hour week nd his rte of py is $7.50/h. Find his weekly inome. 3 Express the following using si rtes s indited. 920 rs pss prtiulr trffi light per dy. (Crs/hour) 224 ins of pples were piked y mn over period of 30 dys. (ins/dy to the nerest in) An employee reeives n nnul inome of $ f he only works weekdys, find his dily rte of py. ($/dy).../.../

35 Where does it work? Rtes involving flow of liquids Exmples A lrge tnk of wter holds 5000L. The wter flows out t rte of 600mL/s. (i) How muh wter flows out of the tnk in hlf n hour to the nerest litre? 600 L/s = 600 L per seond = 600 L # 60 per minute = 600 L # 60 # 30 per 30 minutes = ml = 080 L Multiply y 30 to get 30 minutes whih is hlf n hour. (ii) How long does it tke to empty this tnk to the nerest minute? Convert 600 ml (to litres) = L = 0.6 L By dividing y 000 sine L = 000 ml L 0.6 L = s = 39 minutes (nerest minute) The volume of wter in lrge wter tnk on frm is L. f it empties L over one hour, wht is the rte of flow in L/s rounded to one deiml ple? L per hour = L per 3600 s = L/s = 6.9 L/s ( d.p.) Fuel onsumption for vehiles is often expressed s rte of L/00 km. A r trvels 900 km nd uses 08 L of petrol. Clulte the fuel onsumption s rte with units L/00 km. 08 L per 900 km 08 9 L per km = 2 L/00 km Divided y 9 so it s per 00 km. d The fuel onsumption of truk is given s 24 L/00 km. f the fuel tnk of the truk hs pity of 648 L, how fr n the truk trvel on one tnk of fuel? = # 00 km = 2700 km How mny times does 24 go into 628? For eh 24 L, the truk trvels 00 km. 33

36 Where does it work? Rtes involving flow of liquids A petrol tnker dispenses fuel from hose t the rte of 50 L/min. How muh petrol is dispensed in 45 minutes? Clulte the time required to empty lod of L. 2 The volume of wter in pool is L. f it empties t rte of L in one hour, wht is the rte of flow in L/s rounded to one deiml ple? 3 A r trvels 432 km nd uses 35L of petrol. Clulte the fuel onsumption s rte with units L/00 km. Answer to d.p. 34

37 RATES NVOLVNG FLOW OF LQUDS *** Where does it work? Rtes involving flow of liquids 4 The fuel onsumption of truk is 24 L/00 km. f the fuel tnk of the truk hs pity of 648 L, how fr (x) n it trvel on one tnk of fuel? 5 There re two wter tnks on frm. The first one n hold L nd wter flows out of the nozzle t rte of 8.5 L/s. The seond one n hold L nd wter flows out of the nozzle t rte of.5 L/s. Whih tnk empties in the shortest time?.../.../

38 Where does it work? Rtes involving urreny onversions Exmple f the urreny exhnge etween US dollrs (USD) nd British pounds (GBP) is GBP/USD, (i) How muh money is $2000 (USD) in British pounds? $ USD = GBP ` $2000 USD = 2000 # GBP = Y304 GBP (ii) How muh is Y3000 (GBP) in United Sttes dollrs (USD)? ` $ USD = GBP USD = GBP `Y3000 = # USD = $ (to 2 d.p.) 6 f the urreny exhnge rte etween US dollrs (USD) nd Austrlin dollrs (AUD) is 0.96 USD/AUD. How muh money is $5000 (AUD) in US dollrs (USD)? How muh is $980 (USD) in Austrlin dollrs (AUD)? 36

39 Where does it work? Rtes involving urreny onversions 7 f the urreny exhnge rte etween British pounds (GBP) nd South Afrin rnd (ZAR) is 8.5 ZAR/GBP. How muh money is Y (GBP) in South Afrin rnd (ZAR)? How muh British pounds (GBP) will you need to exhnge for R (ZAR)? 8 Use the exhnge rte onversion tle elow to nswer these questions (to 2 d.p.) Curreny Multiplier exhnging from USD Multiplier exhnging to USD Euro (EUR) ~ 0.92 ~.096 British pound (GBP) ~ ~.533 ndin rupee (NR) ~ ~ Austrlin dollr (AUD) ~.304 ~ New Zelnd dollr (NZD) ~.398 ~ 0.76 Singpore dollr (SGD) ~ 347 ~ Jpnese yen (JPY) ~ ~ How muh money is (NR) in US dollrs (USD)? 37

40 RATES NVOLVNG CURRENCY CONVERSONS *** Where does it work? Rtes involving urreny onversions 8 How muh money is $260 (NZD) in US dollrs (USD)? How muh money is $200 (USD) in Jpnese yen (JPY)? d How muh money is $3600 (USD) in Singpore dollrs (SGD)? e Use the tle to determine whih of these mounts in different urrenies exhnge for the lrgest mount in US dollrs (USD). $ (NZD), $ (AUS), Y (GBP), $ (SGD), ) (EUR).../.../

41 Wht else n you do? Speed f r trvels t n verge speed of 60 km/h for 3 hours, then it follows tht in 3 hours the r will hve trvelled 60 km three times over. 60 km/h # 3 h = 80 km. The generl rule is: Distne = Speed # Time From this formul, the speed or time n lso e mde the sujet of the eqution: Distne Time = Speed Time Time ` Speed = Distne Time Distne Speed = Speed Time Speed ` Time = Distne Speed This n e summrized summrised using the formuls: S = D T Where: D = distne trvelled D = ST D D = ST T = D S S = verge speed T = totl time tken S = D T S T T = D S Exmples using the speed formul f trin trvels distne of 200 km in 5 hours, wht is its verge speed? S = D T Sustitute the numers into the eqution. = 200 km 5 h Keep the units so you know the units t the end. = 80 km/h f r trvels t 60 km/h for 3 hours nd 90 km/h for hour, find the verge speed of the r for the whole trip. Distne trvelled in the first 3 hours: D = ST = 60(3) = 80 km Distne trvelled in the lst hour: D = ST = 90() = 90 km Totl distne trvelled = 80 km + 90 km = 270 km Totl time = 3 hrs + hrs = 4 hrs Hene the verge speed over the whole trip is: S = D T = = 67.5 km/h 39

42 Wht else n you do? Speed For eh of these trips, find the verge speed in km/h. A ferry trvels 50 km in two hours. A smll propeller plne flies 400 km in 30 minutes. A r trvels 876 km from Mtherton to Rtito in 9 hours. 2 A trin trvels t 00 km/h for period of 3.5 hours. Find the distne trvelled y the trin. [Hint: Use D = ST ] 3 f r trvels t 80 km/h for 4 hours nd 0 km/h for nother 4 hours, find the verge speed of the r for the 8 hour trip using the following outline: Find the distne trvelled in the first 4 hours. Find the distne trvelled in the lst 4 hours. Wht s the totl distne trvelled? d Wht s the totl time period? e Find the verge speed of the r for the trip. 40

43 Wht else n you do? Speed 4 A plne flies t speed of 900 km/h nd trvels 7000 km non-stop. How long did the trip tke? (nerest minute).../.../ *** SPEED *** SPEED *** SPEED 5 A room is 6 m wide. f you run k nd forth 5 times in just 3 minutes 5 seonds, wht ws your verge speed in m/s? 6 A womn runs t speed of 6 m/s for 4 minutes nd 30 seonds. Wht distne did she run? 4

44 Wht else n you do? Speed nd different units t is importnt to ensure ll the units mth when using the speed formul. f the speed is in km/h then the time hs to e in hours nd the distne in km. Exmples How fr hs us trvelled if it goes t speed of 54 km/h for time of 40 minutes? The si formul is D = ST. The time must e expressed in terms of hours so the units ll mth. 40 minutes = hours = 2 3 hours Then: D = 54 km/h 2 3 hours The units mth. These re oth in hours. = 54 ( 2 3 )km = 36 km Whih is fster, leoprd running t 64 km/h or ird flying t 5 m/s? Convert 64 km/hinto m/s: 64 km h = m 3600 s Rememer tht h = 60 min = s = 3600 s = 7.7 m/s Therefore the leoprd is running fster thn the ird is flying. The distne the Erth trvels round the Sun in one yer is out 940 million kilometres. Assuming one yer is 365 dys, find the speed of the Erth in km/h. 365 dys = hours = s = s S = D T = km h = 30 km/s (to nerest whole km) 42

45 *SPEED AND DFFERENT UNTS*SPEED AND DFFERENT UNTS Wht else n you do? Speed nd different units How fr hs plne flown if it trvels t speed of 840 km/h for 0 minutes? 2 A shrk swims t speed of pproximtely 8 km/h. Wht is this speed in m/s? 3 Whih is fster, pddle ot doing 2 km/h or ylist trvelling t 4 m/s? [Hint: Convert 2 km/h to m/s] 4 n one orit round the Erth, the moon trvels distne of km in 27.3 dys. Show tht the speed of the moon round the Erth is 022 m/s (nerest whole)..../.../

46 Wht else n you do? Popultion growth rte Popultion Growth Rte (PGR) is the hnge in popultion over speifi time period. t is perentge of the individuls in the popultion t the eginning of tht period. The hnge in popultion over time is the popultion t the end minus the popultion t the eginning of the time period. ` Chnge in Popultion = Popultion t end of time period - Popultion t eginning of time period PGR = Chnge in popultion Popultion t eginning of time period 00% Popultion Growth Rte Multiply y 00% to mke it into perentge. Exmples of popultion growth rte lultions f the world popultion in 990 ws nd in 203 ws , find the popultion growth rte of the world popultion over this time period. Chnge in Popultion = Popultion in Popultion in 990 = = Popultion Growth Rte, PGR = % = 33.7% ( dp) f the growth rte remins the sme, lulte the world popultion in 205. So the PGR = 33.7%. The popultion t the eginning of the time period is Find 33.7% of this nd dd in to find the new popultion. 33.7% of = = ` = Add the rise in popultion to the eginning popultion to get the finl popultion. Therefore the estimted world popultion in 205 is

47 *POPULATON GROWTH RATE*POPULATON GROWTH RATE Wht else n you do? Popultion growth rte.../.../ f the Mumplegriff's popultion in 990 ws nd in 203 ws , omplete the lultion for the growth rte of Mumplegriff's popultion in this time period. Chnge in Popultion = Popultion in Popultion in 990 = - = Popultion Growth Rte, PGR = 00% % ( dp) 2 f ountry hd popultion of in the yer 2000 nd the popultion growth rte etween 2000 nd 2020 ws 37%, to find the popultion in Popultion t eginning of time period =. Find 37% of this nd dd in to find the new popultion: 37% of = 0.37 = ` Popultion in 2020: ` = 45

48 Wht else n you do? Popultion growth rte 3 The tle shows estimted popultions for severl ountries in 2000 nd 203. Complete the missing prts of the tle in the workspe provided elow for eh one indited. Country Popultion in 2000 Popultion in 203 Chnge in Popultion Popultion Growth Rte PGR (%) Austrli % ndonesi % Jpn % Chin % Vietnm % United Sttes % ndi d Ppu New Guine % Afghnistn % North Kore % South Kore % Pkistn % Ppu New Guine Popultion in 2000: Chin Popultion in 203: ndonesi Chnge in Popultion: d ndi Popultion Growth Rte: Dt resoured from: (203) 46

49 Chet Sheet Here is summry of the importnt things to rememer for rtes nd rtios Rtios A rtio ompres two quntities nd is written using olon. For exmple, 2:3. Lowest terms nd equivlent rtios A rtio n e redued to lowest terms y dividing eh side of the rtio y ommon ftors. For exmple, 3 : 6 = : 2 y dividing y 3. Two rtios re equivlent if they n e redued to the sme lowest terms. Reduing deiml rtios f oth numers in rtio re finite deimls you n redue the rtio to lowest terms y first moving the deiml point y the sme numer of times on the left nd right sides of the rtio until you hve whole numers only. Then redue further y dividing y ommon ftors. For exmple,.25 : 0.75 = 25 : 75 = 5:3. Reduing rtios involving frtions To redue rtio involving frtions, first onvert them so they hve ommon denomintor. Then nel the denomintor nd redue further. For exmple, 2 : 3 = 6 3 : 2 6 = 3 : 2. Best uys nd the unitry method To find the est uy you mke one of the numers in the rtios the sme nd then ompre. The unitry method is where you mke the one of the numers into. For exmple, 2 : = : 5 2 or 2 :. Then multiply up s required for the prolem. To divide quntity in given rtio, find the totl numer of prts, then find prt, then find the nswer y multiplying to get the required numer of prts. Sle drwings A sle drwing is n enlrged of redued version of rel life ojet. A sle is rtio whih expresses the enlrgement or redution of the sle drwing. The left numer in the rtio is for the sle drwing while the right numer represents rel life. Mps re prtil exmples of sle drwings. Rtes A rte is omprison etween two quntities suh s kilometres per hour, written km/h. The forwrd slsh / represents the word per. Fuel onsumptions is often expressed s rte L/00 km. Speed, Distne, Time Speed = Distne Time S = D T D = ST T = D S S = verge speed D = distne trvelled T = totl time tken S = D T D = ST D S T Popultion growth rte Chnge in Popultion = Finl Popultion - Popultion t eginning of time period Chnge in popultion Popultion growth rte (s %) = Popultion t eginning of time period 00% T = D S 47

50 Notes 48

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