UNCORRECTED. Understanding fractions, decimals percentages

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1 Online resoures Auto-mrked hpter 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 nd Wht you will lern Understnding frtions, deimls perentges A Ftors nd multiples B Highest ommon ftor nd lowest ommon multiple C Wht re frtions? D Equivlent frtions nd simplified frtions E Mixed numerls nd improper frtions F Ordering positive nd negtive frtions G Ple vlue in deimls nd ordering deimls H Rounding deimls I Deiml nd frtion onversions J Conneting perentges with frtions nd deimls K Deiml nd perentge onversions L Frtion nd perentge onversions M Perentge of quntity N Using frtions nd perentges to ompre two quntities Unorreted th smple pges Cmridge University Press Plmer et l, Ph

2 NSW urriulum Anient Egyptin frtions The nient Egyptins used frtions over 000 yers go. The Egyptin sky god Horus ws flon-heded mn whose eyes were elieved to hve mgil heling powers. Egyptin jewellery, ornments or lothing deorted with the Eye of Horus design were regrded s good luk hrms for helth nd mgil protetion from evil. The six prts in the Eye of Horus design represent the six wys tht informtion enters the rin. These six different prts or symols represented the six frtions used y nient Egyptin mthemtis. For exmple, insted of writing, Egyptins would write, nd insted of writing 8 they would write. Eye of Horus frtion symols re found in nient Egyptin medil presriptions for mixing mgil mediine. Amzingly, modern dotors still use the Eye of Horus STRAND: NUMBER AND ALGEBRA SUBSTRAND: FRACTIONS, DECIMALS AND PERCENTAGES Outome A student opertes with frtions, deimls nd perentges. (MA NA) ( ) symolism when they write ( R x ) t the strt of presription thought (eyerow losest to rin) hering (pointing to er) smell (pointing to nose) sight (pupil of the eye) touh (leg touhing the ground) tste (urled top of whet plnt) A proportion or frtion n e written using omintion of these symols. For exmple: = nd 6 =. Whih symols would represent 8? Cn e written using the Eye of Horus symols? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

3 6 Chpter Understnding frtions, deimls nd perentges In whih digrm is one-third shded? A B C D Pre-test Whih of the following is not equivlent to one whole? 6 A B C 6 Whih of the following is not equivlent to one-hlf? A B C 9 Find: Find: 0 0 D D 0 0 d d 6 6 Tom ets hlf lok of hoolte on Mondy nd hlf of the remining lok on Tuesdy. How muh hoolte is left for Wednesdy? Find the next three terms in these numer sequenes. d 0,,,,,,,,,,,,,,,,, 6,,,,, 8 Copy nd omplete. + + = 6 = 6 d = 6 8 = Find: of $ of $60 of $ d of $6 0 Stte whether eh of the following is true or flse. of 6 = 6 6 = of 6 of 00 = d one-tenth = 00 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

4 wa Ftors nd multiples A Ftors nd multiples Numer ptterns re fsinting. Ftors nd multiples re key uilding loks for rod field known s numer theory. Mny fmous mthemtiins hve studied numer ptterns in n ttempt to etter understnd our world nd to ssist with new sientifi disoveries. Around 600 e, the Greeks uilt on the erly work of the Egyptins nd Bylonins. Thles of Miletus, the fther of Greek mthemtis, is redited for signifint dvnes in numer theory. One of his students, Pythgors of Smos, went on to eome one of the most well-known mthemtiins to hve lived. Pythgors ws primrily religious leder, ut he elieved tht the understnding of the world ould e enhned through the understnding of numer. We strt this hpter y explining the onepts of ftors nd multiples. One dozen doughnuts re generlly pked into gs with rows of doughnuts eh. Sine =, we n sy tht nd re ftors of. Purhsing multiple pks of one dozen doughnuts ould result in uying, 6, 8 or 60 doughnuts, depending on the numer of pks. These numers re known s multiples of. Let s strt: The most ftors, the most multiples Whih numer tht is less thn 00 hs the most ftors? How mny ftors re there in set of? Whih numer tht is less thn 00 hs the most multiples less thn 00? Ftors of prtiulr numer re numers tht divide extly into tht numer. For exmple: The ftors of 0 n e found y onsidering pirs of numers tht multiply to give 0, whih re 0, 0 nd. Therefore, written in sending order, the ftors of 0 re,,,, 0, 0. Every whole numer is ftor of itself nd lso is ftor of every whole numer. Multiples of prtiulr numer re numers reted y multiplying the prtiulr numer y ny whole numer. For exmple: The multiples of 0 re 0, 0, 60, 80, 00, 0, Multiples of 0 re lso 80, 000, There is n infinite numer of multiples! Given the sttements ove, it follows tht ftors re less thn or equl to the prtiulr numer eing onsidered nd multiples re greter thn or equl to the numer eing onsidered. Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

5 8 Chpter Understnding frtions, deimls nd perentges Exmple Finding ftors Find the omplete set of ftors for eh of these numers. 0 SOLUTION Exmple Listing multiples Write down the first six multiples for eh of these numers. SOLUTION EXPLANATION,,,,, 66 The first multiple is lwys the given numer. Add on the given numer to find the next multiple. Repet this proess to get more multiples., 0, 0, 0,, 0 Strt t, the given numer, nd repetedly dd to ontinue produing multiples. Exmple Finding ftor pirs Express 9 s produt of two ftors, oth of whih re greter thn 0. SOLUTION EXPLANATION 9 = Systemtilly divide 9 y numers greter thn 0 in n ttempt to find lrge ftor. Exerise A UNDERSTANDING AND FLUENCY EXPLANATION Ftors of re,,,. =, = Ftors of 0 re:,,,, 8, 0, 0, 0. 0 = 0, 0 = 0 0 = 0, 8 = 0 The lst numer you need to hek is., (½),, 6 6(½) For eh of the following numers, stte whether they re ftors (F), multiples (M) or neither (N) of the numer d 0 e 6 f g 0 h i j 600 k 0 l For eh of the following numers, stte whether they re ftors (F), multiples (M) or neither (N) of the numer 6. d e 0 f g h 8 i j 6000 k l 9 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

6 A Ftors nd multiples 9 Exmple Exmple Exmple List the omplete set of ftors for eh of the following numers. 0 d 6 e 60 f g 80 h i 8 Write down the first six multiples for eh of the following numers. 8 d e 0 f g h 00 i Fill in the gps to omplete the set of ftors for eh of the following numers. 8,,, 6, 9,,,,,,,, 8,,, 8,, 6, d 0,,,,, 6,, 0,,, 0,, 0,, 60, 6 Whih numer is the inorret multiple for eh of the following sequenes?, 6, 9,,, 8,,,, 0, 86, 9, 6,, 8, 0,,,,,, 66,, 88, 99, 0 d,,, 68, 8, 0,, 6,, 0 PROBLEM-SOLVING AND REASONING Consider the set of whole numers from to inlusive. Whih numer hs the most ftors? Whih numer hs the fewest ftors? Whih numers hve n odd numer of ftors? 8 Express eh of these numers s produt of two ftors, oth of whih re greter thn d e 6 f 9 9 Zne nd Mtt re oth keen runners. Zne tkes minutes to jog round running trk nd Mtt tkes minutes. They strt t the sme time nd keep running until they oth ross the finish line t the sme time. How long do they run for? How mny lps did Zne run? How mny lps did Mtt run? 0 Anson is prepring for his th irthdy prty. He hs invited friends nd is mking eh of them lolly g to tke home fter the prty. To e fir, he wnts to mke sure tht eh friend hs the sme numer of lollies. Anson hs totl of 00 lollies to shre mong the lolly gs. How mny lollies does Anson put in eh of his friends lolly gs? How mny lollies does Anson hve left over to et himself? Anson then deides tht he wnts lolly g for himself lso. How mny lollies will now go into eh of the lolly gs?, 8, 8 0, 8 0, After muh pleding from his silings, Anson prepres lolly gs for them lso. His sister Monique noties tht the totl numer of lolly gs is now ftor of the totl numer of lollies. d Wht re the different possile numer of siling(s) tht Anson ould hve? e How mny silings do you expet Anson hs? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

7 0 Chpter Understnding frtions, deimls nd perentges Are the following sttements true or flse? A multiple of prtiulr numer is lwys smller thn tht numer. is ftor of every even numer. is ftor of every odd numer. d A ftor is lwys greter thn or equl to the given numer. e When onsidering prtiulr numer, tht numer is oth ftor nd multiple of itself. 60 is numer with mny ftors. It hs totl of ftors nd, interestingly, it hs eh of the numers,,,,, 6 s ftor. Wht would e the smllest numer tht ould ost hving,,,,, 6, nd 8 s ftors? Wht would e the smllest numer tht ould ost hving,,,,, 6,, 8, 9 nd 0 s ftors? Express, in written form, how you n determine the smllest numer tht ontins prtiulr set of ftors. Wht numers n lim the numer 00 to e multiple? Wht re the ftors of 00? All AM rdio sttions hve frequenies tht re multiples of 9. For exmple, prtiulr rdio sttion hs frequeny of kilohertz (or khz). Find three other AM rdio sttions nd show their frequenies re, indeed, multiples of 9. Two numers re htting with one nother when one numer sks the other, Are you multiple of mine? The reply omes k, Well, I hve lwys onsidered you to e one of my ftors. Explin why this response is enough to help the first numer nswer her question. ENRICHMENT Designing some help from the omputer 6 6 Design spredsheet tht will enle user to enter ny numer etween nd 00 nd it will utomtilly list the first 0 multiples of tht numer. Design spredsheet tht will enle user to enter ny prtiulr numer etween nd 00 nd it will utomtilly list the numer s ftors. Improve your ftor progrm so tht it finds the sum of the ftors nd lso sttes the totl numer of ftors for the prtiulr numer. d Use your spredsheet progrm to help you find pir of mile numers. A pir of numers is sid to e mile if the sum of the ftors for eh numer, exluding the numer itself, is equl to the other numer. Eh numer tht mkes up the first suh pir of mile numers flls etween 00 nd 00. An exmple of non-mile pir of numers: ftor sum = = 6 6 ftor sum = = The ftor sum for 6 would need to e for the pir to e mile numers. Helpful Exel formuls INT (numer) Rounds numer down to the nerest integer (whole numer). MOD (numer, divisor) Returns the reminder fter numer is divided y its divisor. IF (logil test, vlue if true, vlue if flse) Cheks whether ondition is met nd returns one vlue if true nd nother vlue if flse. COUNTIF (rnge, riteri) Counts the numer of ells within rnge tht meet the given ondition. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

8 B Highest ommon ftor nd lowest ommon multiple B Highest ommon ftor nd lowest ommon multiple In the previous exerise, ftors nd multiples of numer were explined. Rememer tht ftors re less thn or equl to given numer nd tht multiples re greter thn or equl to given numer. There re mny pplitions in mthemtis for whih the highest ommon ftor (HCF) of two or more numers must e determined. In prtiulr, the skill of finding the HCF is required for the future topi of ftoristion, whih is n importnt spet of lger. Similrly, there re mny osions for whih the lowest ommon multiple (LCM) of two or more numers must e determined. Adding nd sutrting frtions with different denomintors requires the skill of finding the LCM. Let s strt: You provide the strting numers! ftors e.g.,,,, 6, For eh of the following nswers, you must determine possile strting numers. On ll osions, the numers involved re less thn 00. The HCF of two numers is. Suggest two possile strting numers. The HCF of three numers is. Suggest three possile strting numers. The LCM of two numers is 0. Suggest two possile strting numers. The LCM of three numers is. Suggest three possile strting numers. The HCF of four numers is. Suggest four possile numers. 6 The LCM of four numers is. Suggest four possile numers. HCF stnds for highest ommon ftor. given numer e.g. multiples e.g.,, 6, 8, As the nme suggests, it refers to the highest (i.e. lrgest) ftor tht is ommon to the numers provided in the question. For exmple: Find the HCF of nd 0. Ftors of re,,,, 6, 8, nd. Ftors of 0 re,,,, 8, 0, 0 nd 0. Therefore, ommon ftors of nd 0 re,, nd 8. Therefore, the highest ommon ftor of nd 0 is 8. Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

9 Chpter Understnding frtions, deimls nd perentges Key ides LCM stnds for lowest ommon multiple. As the nme suggests, it refers to the lowest (i.e. smllest) multiple tht is ommon to the numers provided in the question. For exmple: Find the LCM of 0 nd. Multiples of 0 re 0, 0, 60, 80, 00, 0, 0, Multiples of re,, 6, 8, 60,, 8, 96, 08, 0,, Therefore, ommon multiples of 0 nd re 60, 0, 80, Therefore, the lowest ommon multiple of 0 nd is 60. The LCM of two numers n lwys e found y multiplying the two numers together nd dividing y their HCF. For exmple: Find the LCM of 0 nd. The HCF of 0 nd is. Therefore, the LCM of 0 nd is 0 = 60. Exmple Finding the highest ommon ftor (HCF) Find the highest ommon ftor (HCF) of 6 nd 8. SOLUTION Ftors of 6 re:,,,, 6, 9,, 8 nd 6. Ftors of 8 re:,,,, 6, 8,, 6, nd 8. EXPLANATION 6 = 6, 8 = 6, = 6, 9 = 6, 6 6 = 6 8 = 8, = 8, 6 = 8, = 8, 6 8 = 8 The HCF of 6 nd 8 is. Common ftors re,,,, 6 nd, of whih is the highest. Exmple Finding the lowest ommon multiple (LCM) Find the lowest ommon multiple (LCM) of the following pirs of numers. nd 6 nd 0 SOLUTION EXPLANATION The LCM of nd is. Note tht the HCF of nd is. = The LCM of 6 nd 0 is 0. Note tht the HCF of 6 nd 0 is. The LCM of 6 nd 0 is 6 0 = 0. Multiples of 6 re 6,, 8,, 0, 6, Multiples of 0 re 0, 0, 0, 0, Unorreted th smple pges Cmridge University Press Plmer et l, Ph

10 B Highest ommon ftor nd lowest ommon multiple Exmple Exmple Exerise B UNDERSTANDING AND FLUENCY The ftors of re,,,, 6 nd, nd the ftors of 6 re,,, 8 nd 6. Wht re the ommon ftors of nd 6? Wht is the HCF of nd 6? Fill in the missing numers to find out the HCF of 8 nd 0. Ftors of 8 re,,,, nd 8. Ftors of re,,,,, 0, nd 0. Therefore, the HCF of 8 nd 0 is. The first 0 multiples of 8 re 8, 6,,, 0, 8, 6, 6, nd 80. The first 0 multiples of 6 re 6,, 8,, 0, 6,, 8, nd 60. Wht re two ommon multiples of 8 nd 6? Wht is the LCM of 8 nd 6? Fill in the missing numers to find out the LCM of 9 nd. Multiples of 9 re 9, 8,, 6,,,,, 8 nd. Multiples of re, 0,, 60,,, nd 0. Therefore, the LCM of 9 nd is. Find the HCF of the following pirs of numers. nd 8 nd nd d nd e 6 nd 0 f nd 60 g 0 nd 0 h 8 nd i 80 nd 0 j nd k nd 6 l 8 nd 6 Find the HCF of the following groups of numers. 0, 0, 0 6,, 0, 00, 8 d 8,, e,, 6 f 0, 8, Find the LCM of the following pirs of numers. nd 9 nd nd d 0 nd e nd 6 f nd 0 g nd 8 h 6 nd 9 i 0 nd 0 j nd 6 k nd l nd 8 Find the LCM of the following groups of numers.,,,,,, d,, 9 e,, 8, 0 f 6,, 8, 8, (½), 6, (½), 8, 9 9(½) 9 Find the HCF of the following pirs of numers nd then use this informtion to help lulte the LCM of the sme pir of numers. nd 0 nd nd d nd Unorreted th smple pges Cmridge University Press Plmer et l, Ph

11 Chpter Understnding frtions, deimls nd perentges PROBLEM-SOLVING AND REASONING 0,,,,,, 0 Find the LCM of nd. Find the HCF of nd. Find the LCM nd HCF of 60 nd 90. Andrew runs lps of the iruit in minutes. Bryn runs lps of the sme iruit in minutes. Chris n run lps of the sme iruit in 6 minutes. They ll strt together on the strting line nd run re tht goes for 6 minutes. Wht is the first time, fter the strt, tht they will ll ross over the strting line together? How mny lps will eh oy omplete in the re? How mny times does Bryn overtke Andrew during this re? d Prepre -minute lss presenttion involving three PowerPoint slides tht desries how you solved this prolem nd explins your nswer. Given tht the HCF of pir of different numers is 8, find the two numers: if oth numers re less thn 0 when one numer is in the 0 s nd the other is in the 0s Given tht the LCM of pir of numers is 0, find the seven possile pirs of numers. ENRICHMENT LCM of lrge groups of numers 6 6 Find the LCM of these single-digit numers:,,,,, 6,, 8, 9. Find the LCM of these first 0 nturl numers:,,,,, 6,, 8, 9, 0. Compre your nswers to prts nd. Wht do you notie? Explin. d Find the LCM of the first nturl numers. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

12 C Wht re frtions? C Wht re frtions? The word frtion omes from the Ltin word frngere, whih mens to rek into piees. Although the following sentenes re not diretly relted to the mthemtil use of frtions, they ll ontin words tht re relted to the originl Ltin term frngere nd they help us gin n understnding of extly wht frtion is. The frgile vse smshed into hundred piees when it lnded on the ground. After the window ws roken, severl frgments were found on the fl oor. She frtured her leg in two ples. The omputer ws running slowly nd needed to e defrgmented. The elderly gentlemn ws eoming very fril in his old ge. Cn you think of ny other relted sentenes? Brinstorm speifi ommon uses of frtions in everydy life. The list ould inlude ooking, shopping, sporting, uilding exmples nd more. Let s strt: Wht strength do you like your ordil? Imgine prepring severl jugs of different strength ordil. Smples ould inlude strength ordil, strength ordil, 6 strength ordil or strength ordil. 8 In eh se, desrie how muh wter nd how muh ordil is needed to mke - litre mixture. Note: litre (L) = 000 millilitres (ml). On the lel of ordil ontiner, it suggests To mke up y glss or jug: dd five prts wter to one prt ordil, ording to tste. Wht frtion of the everge is ordil? A frtion is mde up of n umertor (u p) nd denomintor (down). numertor For exmple: denomintor The denomintor tells you how mny prts the whole is divided up into. The numertor tells you how mny of the divided prts you hve seleted. The horizontl line seprting the numertor nd the denomintor is lled the vinulum. Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

13 6 Chpter Understnding frtions, deimls nd perentges Key ides We n represent frtions using re. If shpe is divided into regions of equl res, then shding ertin numer of these regions will rete frtion of the whole shpe. For exmple: Frtion shded = A proper frtion or ommon frtion is less thn whole, nd therefore the numertor must e smller thn the denomintor. For exmple: is proper frtion. An improper frtion is greter thn whole, nd therefore the numertor must e lrger thn the denomintor. For exmple: is n improper frtion. We n represent positive nd negtive frtions on numer line. This numer line shows the whole numers from to. Eh unit hs then een divided eqully into four segments, therefore reting qurters Whole numers n e represented s frtions. On the numer line ove we see tht is the sme s nd is the sme s 8. Exmple 6 Understnding the numertor nd the denomintor Into how mny piees hs the whole pizz een divided? How mny piees hve een seleted (i.e. shded)? In simplest form, when representing the shded frtion of the pizz: i wht must the denomintor equl? ii wht must the numertor equl? iii write the mount of pizz seleted (shded) s frtion. SOLUTION EXPLANATION 8 Pizz ut into 8 equl piees. of the 8 piees re shded in lue. i 8 Denomintor shows the numer of prts the whole hs een divided into. ii Numertor tells how mny of the divided prts you hve seleted. Shded frtion is the numertor over the denomintor; i.e. out of iii 8 8 divided piees. 0 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

14 C Wht re frtions? Exmple Representing frtions on numer line Represent the frtions SOLUTION 0 Exmple 8 Shding res Represent the frtion in three different wys, using squre divided into four equl regions. SOLUTION Exerise C 9 UNDERSTANDING AND FLUENCY nd 9 on numer line. EXPLANATION Stte the denomintor of this proper frtion: 9. Stte the numertor of this improper frtion:. Drw numer line strting t 0 nd mrk on it the whole numers 0, nd. Divide eh whole unit into five segments of equl length. Eh of these segments hs length of one-fifth. EXPLANATION Ensure division of squre retes four equl res. Shde in three of the four regions. 6 (½), (½), 6, (½) Group the following list of frtions into proper frtions, improper frtions nd whole numers. 0 A B C D 6 9 E F G H 99 0 I J K L 8 0 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

15 8 Chpter Understnding frtions, deimls nd perentges Exmple 6 Exmple Exmple 8 Answer the following questions for eh of the pizzs ( A to D ) drwn elow. Into how mny piees hs the whole pizz een divided? How mny piees hve een seleted (shded)? In representing the shded frtion of the pizz: i wht must the denomintor equl? ii wht must the numertor equl? iii write the mount of pizz seleted (shded) s frtion. A B C D Find the whole numers mong the following list of frtions. Hint: There re five whole numers to find. A 0 B C D 8 E 0 F G H 0 I J K L 0 M N O P 8 Represent the following frtions on numer line. nd 6 nd 6 nd d 6 nd e nd 8 f, 9 nd 6 Represent eh of these frtions in three different wys, using retngle divided into equl regions. 8 6 Write the next three frtions for eh of the following frtion sequenes.,,, 6,,, 8, 6 8, 8, 8,,, 8 e,,,,,, d, 0, 9, 8,,, 9,,,,,, f 8,, 8,,,, Unorreted th smple pges Cmridge University Press Plmer et l, Ph

16 C Wht re frtions? 9 PROBLEM-SOLVING AND REASONING 8, 8, 9,, 9, 0,, 8 Wht frtions orrespond to eh of the different shpes positioned on these numer lines? 0 6 d Wht opertion (i.e. +,, or ) does the vinulum relte to? 0 For eh of the following, stte wht frtion of the digrm is shded. d e f For eh of the following, write the frtion tht is desriing prt of the totl. After one dy of -kilometre hike, they hd ompleted kilometres. From strters, went on to finish the re. Rinfll for months of the yer ws elow verge. d One egg is roken in rton tht ontins dozen eggs. e Two plyers in the soer tem sored gol. f The lunh stop ws kilometres into the -kilometre trip. g Seven memers in the lss of 0 hve visited Austrli Zoo. h One of the r tyres is worn nd needs repling. i It rined three dys this week. Explin the logi ehind the terms proper frtion nd improper frtion. Whih digrm hs one-qurter shded? A B C D Unorreted th smple pges Cmridge University Press Plmer et l, Ph

17 0 Chpter Understnding frtions, deimls nd perentges ENRICHMENT Adjusting onentrtion A 0 -millilitre glss of ordil is mde y mixing four prts wter to one prt ordil. i Wht frtion of the glss is ordil? ii Wht mount of ordil is required? Firuz drinks 0 millilitres of the glss of ordil nd thinks it is too strong. So he fills the glss k up with 0 millilitres of pure wter. i How muh ordil is in the glss now? ii Wht frtion of the glss is ordil? Firuz drinks 0 millilitres of the glss of ordil ut he still thinks it is too strong. So, one gin, he fills the glss k up with 0 millilitres of pure wter. i How muh ordil is in the glss now? ii Wht frtion of the glss is ordil? d Lynn prefers her ordil muh stronger ompred with Firuz. When she is given glss of the ordil tht is mixed t four prts to one, she drinks 0 millilitres nd deides it is too wek. So she fills the glss k up with 0 millilitres of stright ordil. i How muh ordil is in Lynn s glss fter doing this one? ii Wht frtion of the glss is ordil? e f Like Firuz, Lynn needs to repet the proess to mke her ordil even stronger. So, one gin, she drinks 0 millilitres nd then tops the glss k up with 0 millilitres of stright ordil. i How muh ordil is in Lynn s glss now? ii Wht frtion of the glss is ordil? If Firuz ontinues diluting his ordil onentrtion in this mnner nd Lynn ontinues strengthening her ordil onentrtion in this mnner, will either of them ever reh pure wter or pure ordil? Disuss. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

18 D Equivlent frtions nd simplified frtions D Equivlent frtions nd simplified frtions Often frtions my look very different when in ft they hve the equivlent vlue. For exmple, in n AFL footll mth, hlf-time is the sme s the end of the seond qurter. We n sy tht nd re equivlent frtions. In oth situtions, the equivlent frtion of the gme hs een ompleted. Consider group of friends eting pizzs during sleepover. The pizzs re homemde nd eh person uts up their pizz s they like. Trevor uts his pizz into only two piees, Jkie uts hers into four piees, Thli uts hers into six piees nd Jred uts his into eight piees. The shded piees re the mount tht they hve eten efore it is time to strt the seond movie. By looking t the pizzs, it is ler to see tht Trevor, Jkie, Thli nd Jred hve ll eten the sme mount of pizz. We n therefore onlude tht,, 6 nd 8 re equivlent frtions. This mens tht = = 6 = 8. Let s strt: Frtion lumps Prepre lss set of frtion rds. (Two exmple sets re provided elow.) Hnd out one frtion rd to eh student. Students then rrnge themselves into groups of equivlent frtions. Set n pproprite time gol y whih this tsk must e ompleted. Repet the proess with seond set of equivlent frtion rds. Smple sets of frtion rds Clss set Trevor Jkie Trevor Jkie Thli Thli Jred Jred,,, 0 80,, 8 0,, 6, 8, 0, 9,, , 00 00, 0 0, 6, 0 0, 9, 0,, 6, 8, 0, 8, 8 Clss set, 6, 8, 0,, 6, 66, 6,, 0 0,, 8, 0, 0, 6, , 9, , 6 9, 9,,, 0 0, 6 6, Unorreted th smple pges Cmridge University Press Plmer et l, Ph

19 Chpter Understnding frtions, deimls nd perentges Key ides Equivlent frtions re frtions tht mrk the sme ple on numer line. For exmple: nd re equivlent frtions. Equivlent frtions re produed y multiplying the numertor nd denomintor y the sme numer. This numer n e ny whole numer greter thn. Equivlent frtions n lso e produed y dividing the numertor nd denomintor y the sme numer. Simplifying frtions involves writing frtion in its simplest form or esiest form or most onvenient form. To do this, the numertor nd the denomintor must e divided y their highest ommon ftor (HCF). It is mthemtil onvention to write ll nswers involving frtions in their simplest form. A frtion wll n e helpful when ompring frtions. one whole hlves thirds qurters fifths sixths eighths ninths tenths twelfths Unorreted th smple pges Cmridge University Press Plmer et l, Ph

20 D Equivlent frtions nd simplified frtions Exmple 9 Produing equivlent frtions Write four equivlent frtions for. SOLUTION = 6 = 6 9 = 8 = 0 Mny other frtions re lso possile. Other ommon hoies inlude: 0 0, 00 00, , 0 60 Exmple 0 Exmple Converting to simplest form Write these frtions in simplest form. 0 SOLUTION 0 = = = 6 = 6 EXPLANATION = = = = 9 By writing either = or etween the frtions, stte whether the following pirs of frtions re equivlent or not equivlent. 0 SOLUTION = 0 Cheking for equivlene 0 EXPLANATION Convert to ommon denomintor. = nd = 9 9 = 0 EXPLANATION The HCF of nd 0 is. Both the numertor nd the denomintor re divided y the HCF of. The HCF of nd is. The is nelled from the numertor nd the denomintor. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

21 Chpter Understnding frtions, deimls nd perentges Exmple 9 Exerise D UNDERSTANDING AND FLUENCY Whih of the following frtions re equivlent to?, 6, 0,,,, 8,, 0, 6 0 Whih of the following frtions re equivlent to 8 0? 0,, 6 0, 8 0, 6 0,,, 0, 80 00, Fill in the missing numers to omplete the following strings of equivlent frtions. = 6 = = 0 = 60 = 00 8 = = = 6 = 80 = 0 In eh of these lists of equivlent frtions, irle the frtion tht is in its simplest form., 0 0, 0, , 8, 6, 6,, 6, 0 9 d 0, 0, 6 8, Fill in the gps to redue these frtions to their simplest form. 0 i HCF = 0 ii 0 0 =. Therefore, simplest form is. i HCF = ii 8 8 = 9. Therefore, simplest form is 9. i HCF = ii 8 8 =. Therefore, simplest form is. 9 d i HCF = 9 ii =. Therefore, simplest form is. 6 Write four equivlent frtions for eh of the frtions listed. e f g 9 Find the unknown vlue to mke the eqution true. e i = = 0 = 0 f j 8 = 80 = 9 0 = 90 g k 9, 6 9(½), 0 6 0(½) 6 = 8 0 = 0 = d h d h l = 6 = = 8 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

22 D Equivlent frtions nd simplified frtions Exmple 0 Exmple 8 By writing either = or etween the frtions, stte whether the following pirs of frtions re equivlent or not equivlent. d g e h Write the following frtions in simplest form. 0 8 e f 6 i j 96 g k f i These lists of frtions re ment to ontin only frtions in their simplest form; however, there is one mistke in eh list. Find the frtion tht is not in simplest form nd rewrite it in its simplest form., 8, 9,, 6, 9, 9,,, 6 d 6 6, 9 6, 8, PROBLEM-SOLVING AND REASONING A fmily lok of hoolte onsists of rows of 6 individul squres. Tni ets 6 individul squres. Wht frtion of the lok, in simplest terms, hs Tni eten? Four people win ompetition tht llows them to reeive tnk of free petrol. Find how mny litres of petrol the drivers of these rs reeive. Ford Territory with 0-litre tnk Nissn Ptrol with 90-litre tnk Holden Commodore with 60-litre tnk d Mzd with 8-litre tnk d h l ,,, Justin, Jonn nd Jk re shring lrge pizz for dinner. The pizz hs een ut into equl piees. Justin would like of the pizz, Jonn would like of the pizz nd Jk will et whtever is remining. By onsidering equivlent frtions, determine how mny slies eh person gets served. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

23 6 Chpter Understnding frtions, deimls nd perentges J. K. Rowling s first ook, Hrry Potter nd the Philosopher s Stone, is pges long. Sm plns to red the ook in three dys, reding the sme numer of pges eh dy. How mny pges should Sm red eh dy? The frtion of the ook is equivlent to wht frtion in simplest form? By the end of the seond dy, Sm is on trk nd hs red of the ook. How mny pges of the ook is equivlent to? A frtion when simplified is written s. Wht ould the frtion hve een efore it ws simplified? Explin why the numer of nswers is infinite. ENRICHMENT Equivlent rs of musi 6 Eh piee of musi hs time signture. A ommon time signture is lled time, nd is tully referred to s Common time. Common time, or time, mens tht there re four qurter notes (or rothets) in eh r. Listed elow re the five most ommonly used musil notes. whole note (fills the whole r) semireve hlf note (fills hlf the r) minim qurter note (four of these to r) rothet eighth note (eight to r) quver sixteenth note (sixteen to r) semi-quver Write six different rs of musi in time. Crry out some reserh on other types of musil time signtures. Do you know wht the time signture 8 mens? 6 Write three different rs of musi for time signture. 8 d Wht re the musil symols for different length rests? e How does dot (or dots) written fter note ffet the length of the note? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

24 E Mixed numerls nd improper frtions E Mixed numerls nd improper frtions As we hve seen in this hpter, frtion is ommon wy of representing prt of whole numer. For exmple, prtiulr r trip my require of tnk of petrol. On mny osions, you my need whole numers plus prt of whole numer. For exmple, long interstte r trip my require tnks of petrol. When you hve omintion of whole numer nd frtion, this numer is known s mixed numerl. Let s strt: Pizz frenzy With prtner, ttempt to solve the following pizz prolem. There is more thn one nswer. At Pete s pizz shop, smll pizzs re ut into four equl slies, medium pizzs re ut into six equl slies nd lrge pizzs re ut into eight equl slies. For lss prty, the teher orders pizzs, whih the students te with ese. After the lst slie is eten, totl of 8 slies of pizz hd een eten y the students. How mny pizzs of eh size did the teher order? A numer is sid to e mixed numerl when it is mix of whole numer plus proper frtion. is mixed numerl whole proper numer frtion Improper frtions (frtions greter thn whole, where the numertor is greter thn the denomintor) n e onverted to mixed numerls or whole numers. = 6 = improper mixed improper whole frtion numerl frtion numer = rem 6 = Mixed numerls n e onverted to improper frtions. In generl, improper frtions should e written s mixed numerls, with the frtion prt written in simplest form. Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

25 8 Chpter Understnding frtions, deimls nd perentges Key ides Exmple Converting mixed numerls to improper frtions Convert to n improper frtion. SOLUTION = or A numer line helps show the different types of frtions. improper frtions 0 = = 6 = + = 6 Exmple Convert to mixed numerl. SOLUTION EXPLANATION = wholes + of whole = = + + Short-ut method: Multiply the whole numer prt y the denomintor nd then dd the numertor. + = 6 Converting improper frtions to mixed numerls Method = 8 + = 8 + = + = Method Divided y. rem. proper frtions 6 mixed numerls nd whole numers EXPLANATION = qurters = = So = = Unorreted th smple pges Cmridge University Press Plmer et l, Ph

26 E Mixed numerls nd improper frtions 9 Exmple Writing mixed numerls in simplest form Convert 0 6 Exerise E to mixed numerl in simplest form. SOLUTION 0 6 = 6 = = or 0 6 = 0 = 0 = UNDERSTANDING AND FLUENCY Eh pne of glss is of the whole window. Between whih two whole numers do the following mixed numerls lie? Work out the totl numer of piees in eh of these situtions. four pizzs ut into six piees eh 0 Lego truks, where eh truk is mde from 6 Lego piees five jigsw puzzles with piees in eh puzzle d three kes ut into eight piees eh EXPLANATION Method : Convert to mixed numerl nd then simplify the frtion prt. Method : Simplify the improper frtion first nd then onvert to mixed numerl. 9, 6, 9(½) 6, 9(½) Unorreted th smple pges Cmridge University Press Plmer et l, Ph

27 0 Chpter Understnding frtions, deimls nd perentges The mixed numerl n e represented in window shpes s Exmple = + + Represent the following mixed numerls using window shpes. A window shpe onsists of four pnes of glss. How mny pnes of glss re there in the following numer of window shpes? d d e f g Wht mixed numerls orrespond to the letters written on eh numer line? d 0 A 8 9 CD 0 B E F G HI 6 M L KJ 0 h 6 On your lultor, wht utton(s) need to e pressed to: onvert to mixed numerl? onvert to n improper frtion? Convert these mixed numerls to improper frtions. e i m 6 9 q f j n 9 r g k 8 o 8 s d u 0 v 8 00 w 6 0 h 6 l 0 0 p 8 t 9 x 0 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

28 E Mixed numerls nd improper frtions Exmple Exmple 8 Convert these improper frtions to mixed numerls. 6 e f g 9 0 i j k m n o q r s 0 u v w Convert these improper frtions to mixed numerls in their simplest form e f g 6 9 PROBLEM-SOLVING AND REASONING 0 Drw numer line from 0 to nd mrk on it the following frtions.,,,,,, Fill in the gps for the following numer ptterns.,,,,,,,,,,,,,,,, 6,,,,,,,,,,,,,,,,, 6, Four friends order three lrge pizzs for their dinner. Eh pizz is ut into eight equl slies. Simone hs three slies, Izell hs four slies, Mrk hs five slies nd Alex hs three slies. How mny pizz slies do they et in totl? How muh pizz do they et in totl? Give your nswer s mixed numerl. How mny pizz slies re left uneten? d How muh pizz is left uneten? Give your nswer s mixed numerl. d h l p t x d h , 0,,,,, 0, 9 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

29 Chpter Understnding frtions, deimls nd perentges Ptrii hs three sndwihes tht re ut into qurters nd she ets ll ut one-qurter. How mny qurters does she et? Phillip hs five sndwihes tht re ut into hlves nd he ets ll ut one-hlf. How mny hlves does he et? Crystl hs x sndwihes tht re ut into qurters nd she ets them ll ut one-qurter. How mny qurters does she et? d Byron hs y sndwihes tht re ut into thirds nd he ets ll ut one-third. How mny thirds does he et? e Feliity hs m sndwihes tht re ut into n piees nd she ets them ll. How mny piees does she et? ENRICHMENT Mixed numerl swp meet - - Using the digits, nd only one, three different mixed numerls n e written. i Write down the three possile mixed numerls. ii Find the differene etween the smllest nd highest mixed numerls. Repet prt using the digits, nd. Repet prt using the digits, nd. d Predit the differene etween the lrgest nd smllest mixed numerl when using only the digits, nd 6. Use sutrtion to see if your predition is orret. e Write down rule for the differene etween the lrgest nd smllest mixed numerls when using ny three onseutive integers. f Extend your investigtion to llow mixed numerls where the frtion prt is n improper frtion. g Extend your investigtion to produe mixed numerls from four onseutive digits. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

30 F Ordering positive nd negtive frtions F Ordering positive nd negtive frtions You lredy know how to order set of whole numers. For exmple:,,, 6,, 0 re set of six different whole numers tht you ould ple in sending or desending order. In sending order, the orret order is,, 6,, 0,. In desending order, the orret order is, 0,, 6,,. In this setion you will lern how to write different frtions in sending nd desending order. To e le to do this we need to ompre different frtions nd we do this through our knowledge of equivlent frtions (see Setion D). Rememer frtion is greter thn nother frtion if it lies to the right of tht frtion on numer line. > 0 Let s strt: The order of five As wrm-up tivity, sk five volunteer students to rrnge themselves in lphetil order, then in height order nd, finlly, in irthdy order. Eh of the five students reeives lrge frtion rd nd displys it to the lss. The rest of the lss must then ttempt to order the students in sending order, ording to their frtion rd. It is group deision nd none of the five students should move until the lss grees on deision. Repet the tivity with set of more hllenging frtion rds. To order (or rrnge) positive frtions we must know how to ompre different frtions. This is often done y onsidering equivlent frtions. If the numertors re the sme, the smllest frtion is the one with the iggest denomintor, s it hs een divided up into the most piees. For exmple: < If the denomintors re the sme, the smllest frtion is the one with the smllest numertor. For exmple: 0 < 0 To order two frtions with different numertors nd denomintors, we n use our knowledge of equivlent frtions to produe frtions with ommon denomintor nd then ompre the numertors. The lowest ommon denomintor (LCD) is the lowest ommon multiple of the different denomintors. Asending order is when numers re ordered going up, from smllest to lrgest. Desending order is when numers re ordered going down, from lrgest to smllest. Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

31 Chpter Understnding frtions, deimls nd perentges Exmple Compring frtions Ple the orret mthemtil symol (i.e. <, = or > ) in etween the following pirs of frtions to mke true mthemtil sttements. d 6 SOLUTION < > 0 > 9 Hene, >. d 6 e > 6 Hene, > 6. > f 8 6 < 8 8 e EXPLANATION f 8 Denomintors re the sme, therefore ompre numertors. Numertors re the sme. Smllest frtion hs the iggest denomintor. LCD of nd is. Produe equivlent frtions. Denomintors now the sme, therefore ompre numertors. Convert mixed numerl to n improper frtion. Denomintors re the sme, therefore ompre numertors. 0 is further to the right thn is further to the right thn Unorreted th smple pges Cmridge University Press Plmer et l, Ph

32 F Ordering positive nd negtive frtions Exmple 6 Ordering frtions Ple the following frtions in sending order.,, SOLUTION 60, 8 60, , 60, 8 60,, 8,,, 9, 0, 0, 0 0, 0, 0 0 0, 0, 0, 0, 0,,,, Exerise F UNDERSTANDING AND FLUENCY Cirle the lrgest frtion in eh of the following lists.,,,, 9,, d,,,, EXPLANATION LCD of, nd is 60. Produe equivlent frtions with denomintor of 60. Order frtions in sending order. Rewrite frtions k in originl form. Express ll frtions s improper frtions. LCD of, nd is 0. Produe equivlent frtions with denomintor of 0. Order frtions in sending order. Rewrite frtions k in originl form.,,, 8,, 6, Stte the lowest ommon multiple of the following sets of numers.,,, d 6, e, 6 f, 0 g, 6 h 8, 6 i,, j,, 6 k, 8, l, 6, Stte the lowest ommon denomintor of the following sets of frtions.,,, e 6, f 8, g,, Fill in the gps to produe equivlent frtions. = = d = e 8 = 0 6-9,, 6, 8(½) -9(½) 6, -9(½) 8 f d h = 6 6 = 8 0,, Unorreted th smple pges Cmridge University Press Plmer et l, Ph

33 6 Chpter Understnding frtions, deimls nd perentges Exmple Exmple 6 Ple the orret mthemtil symol (i.e. <, = or > ) in etween the following pirs of frtions to mke true mthemtil sttements. d e f g h i j k 9 l 6 0 m n o p 9 8 q r s t 8 6 Ple the following frtions in sending order., 8,,, e,,, g 0, 9,,, 9,, 9 d 6,, f 8, 6,, h 9,, 0,, 6 Use numer line to ple the following frtions in sending order.,,, 6, 8 Ple the following frtions in desending order, without finding ommon denomintors.,,,,, 6, 8,, 8, d, 0, 0, 00 e, 8, 9, 0 PROBLEM-SOLVING AND REASONING 9 Ple the following ke frtions in deresing order of size. A sponge ke shred eqully y four people = ke B hoolte ke shred eqully y eleven people = ke f, 9, 6, C rrot nd wlnut ke shred eqully y eight people = 8 ke 0 Four friends, Den, Dvid, Andre nd Ro, ll ompeted in mrthon. Their respetive finishing times were hours, 9, 9, 0, 0,, hours, hours nd hours. Write down the orret finishing order of the four friends. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

34 F Ordering positive nd negtive frtions Rewrite the frtions in eh set with their lowest ommon denomintor nd then write the next two frtions tht would ontinue the pttern. 9,, 9,,,,,, 6,, 6,, d,, 9,, Write frtion tht lies etween the following pirs of frtions.,,, 6 d 0, e 0, f 8 0, 8 Explin how to find frtion tht lies etween two frtions with different denomintors. Write the whole numer vlues tht n tke so tht lies etween: nd nd Thoms nd Nthn hd doughnut-eting re to see who ould et the most doughnuts in minute. Before the re strted, Thoms ut eh of his doughnuts into fifths to mke them just the right ite-size. Nthn deided to ut eh of his doughnuts into qurters efore the re. After minute of frenzied eting, the stop whistle lew. Thoms hd devoured 8 fifths of doughnut nd Nthn hd munhed his wy through qurters of doughnut. Who won the doughnut eting re? Wht ws the winning doughnut mrgin? Express your nswer in simplest form. ENRICHMENT Shdy designs 6 For eh of the digrms shown, work out wht frtion of the retngle is oloured lue. Explin how you rrived t eh of your nswers. Redrw the shpes in order from the most lue to lest lue. Design nd shde two more retngle designs. i ii iii 6 iv Unorreted th smple pges Cmridge University Press Plmer et l, Ph

35 8 Chpter Understnding frtions, deimls nd perentges G Ple vlue in deimls nd ordering deimls Key ides Some quntities hnge y whole numer mounts, suh s the numer of people in room, ut there re mny quntities tht inrese or derese ontinuously, suh s your height, weight nd ge. Often we tlk out ge s whole numer (e.g. Mike is yers old) ut, in relity, our ge is n ever-inresing (ontinuous) quntity. For exmple, if Mike is yers, months, weeks, dys, hours, 6 minutes nd seonds old, then Mike is tully.806 yers old! There re mny numers in tody s soiety tht re not whole numers. For exmple, it is unusul to uy n item in supermrket tht is simply whole numer of dollrs. The prie of lmost ll shopping items involves oth dollrs nd ents. A hoolte r my ost $.9, whih is n exmple of deiml numer. L e t s strt: Split-seond timing Orgnise students into pirs nd use digitl stopwth. Mny students will hve wth with suitle stopwth funtion. Try to stop the stopwth on extly 0 seonds. Attempt this two times. Were you le to stop it extly on 0.00 seonds? Wht ws the losest time? Try these dditionl hllenges with your prtner. Stop the wth extly on: i. seonds ii 8. seonds iii 9. seonds iv. seonds How quikly n you strt nd stop the stopwth? How urtely n you time minute without looking t the stopwth? A deiml point is used to seprte the whole numer from the deiml or frtion prt. When deling with deiml numers, the ple vlue tle must e extended to involve tenths, hundredths, thousndths et. The numer 8. mens: Hundreds Tens Units Tenths Hundredths Thousndths Whole numers 0 0 deiml point Frtions Unorreted th smple pges Cmridge University Press Plmer et l, Ph

36 G Ple vlue in deimls nd ordering deimls 9 Exmple Understnding deiml ple vlue Wht is the vlue of the digit 8 in the following numers? SOLUTION EXPLANATION 8 The vlue of 8 is 0. The 8 is in the first olumn fter the deiml point, whih is the tenths olumn. 8 The vlue of 8 is 000. The 8 is in the third olumn fter the deiml point, whih is the thousndths olumn. Exmple 8 Chnging to deimls Express eh of the following proper frtions nd mixed numerls s deimls SOLUTION EXPLANATION 0 = 0. mens seven-tenths, so put the in the tenths olumn = 0.0 mens five-hundredths, so put the in the hundredths 00 olumn. =. 00 mens units nd one-hundredths. 00 hundredths is one-tenth nd seven-hundredths. Exmple 9 Arrnging deiml numers in order Arrnge the following deiml numers in sending order (i.e. smllest to lrgest)..,.,.,.,.,. SOLUTION EXPLANATION.,.,.,.,.,. The units olumn hs higher vlue thn the tenths olumn, nd the tenths olumn hs higher vlue thn the hundredths olumn.. is igger thn. euse it hs seven-tenths, whih is igger thn three-tenths. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

37 60 Chpter Understnding frtions, deimls nd perentges Exmple Exmple 8, Exmple 8 Exmple 9 Exerise G UNDERSTANDING AND FLUENCY For the numer 8., give the vlue of the digit (s frtion): A stopwth is stopped t 6. seonds. Wht is the digit displyed in the tenths olumn? Wht is the digit displyed in the units olumn? Wht is the digit displyed in the hundredths olumn? d Is this numer loser to 6 or seonds? Wht is the vlue of the digit 6 in the following numers? d 0.69 e 6. f 8.68 g.6 h.96 Stte whether eh of the following is true or flse.. <.8. < >. d 8.09 > 8. e.8 8. f g.9 8. h..09 i 0 = 0 j 00 0 = 0 k 00 0 l Express eh of the following proper frtions s deiml e f g Express eh of the following mixed numerls s deiml. 6 0 e f 00 0 g 00 d h d 6 00 h Write the following numer phrses s deimls. seven nd six-tenths twelve nd nine-tenths thirty-three nd four-hundredths d twenty-six nd fifteen-hundredths e eight nd forty-two hundredths f ninety-nine nd twelve-thousndths PROBLEM-SOLVING AND REASONING 6, 6(½), (½) 8, 9, 9,0, 9 8 How lose re the following deiml numers to their nerest whole numer? d 6. e.999 f.99 g 0.8 h Arrnge these groups of numers in sending order (i.e. smllest to lrgest)..,.0,.,. 0.6,.06,.6, 0..8,.8,.8,.8 d 6.9, 9.6, 9.6,.96,.69 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

38 G Ple vlue in deimls nd ordering deimls 6 0 The tting verges for five retired Austrlin Criket test ptins re Adm Gilhrist.60, Steve Wugh.06, Mrk Tylor.9, Alln Border 0.6 nd Kim Hughes.. List the five plyers in desending order of tting verges (i.e. lrgest to smllest). Riky Ponting s test tting verge is.8. Where does this rnk him in terms of the retired Austrlin test ptins listed ove? The depth of river t 9.m. on six onseutive dys ws: Dy :. m Dy :.8 m Dy :.9 m Dy :. m Dy :. m Dy 6:.6 m On whih dy ws the river level highest? On whih dy ws the river level lowest? On whih dys ws the river level higher thn the previous dy?, nd re digits nd > >. Write these numers from smllest to lrgest. Note tht the dot represents the deiml point..,.,.,.,.,..,.,.,.,.,.,.,.,. Write s deimls, if is digit ENRICHMENT Different deiml omintions d Write s mny different deiml numers s you n nd ple them in sending order using: i the digits 0, nd deiml point. Eh digit must e used only one. ii the digits 0,, nd deiml point. Eh digit must e used only one. iii the digits 0,,, nd deiml point. Eh digit must e used only one. Clulte the numer of different deiml numers tht ould e produed using the digits 0,,,, nd deiml point. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

39 6 Chpter Understnding frtions, deimls nd perentges H Rounding deimls Key ides Deiml numers sometimes ontin more deiml ples thn we need. It is importnt tht we re le to round deiml numers when working with money, mesuring quntities (inluding time nd distne), or writing nswers to some division lultions. For exmple, the distne round the shool ovl might e 0.96 km, whih rounded to deiml ple is 0. km or 00 m. The rounded figure, lthough not preise, is urte enough for most pplitions. Running events re eletronilly mesured nd rounded to deiml ples. Usin Bolt hs repetedly roken his own world reords. In August 009, he set new world reord of 9.8 seonds over 00 m t the World Chmpionships in Germny, whih ws -hundredths (0.0) of seond fster thn his London Olympi Gmes (August 0) reord of 9.6 seonds. Let s strt: Rounding rinstorm In group of four, rinstorm osions when it my e useful to round or estimte deiml numers. Aim to get more thn 0 ommon pplitions. In pirs one person sttes deiml numer nd the prtner needs to stte nother deiml numer tht would llow the two numers to dd up to whole numer. Use mentl rithmeti only. Strt with deiml ple nd try to uild up to or deiml ples. Rounding involves pproximting deiml numer to fewer deiml ples. To round deiml: Cut the numer fter the required deiml ple; for exmple, round to deiml ples. To determine whether you should round your nswer up or down, onsider only the digit immeditely to the right of the speified ple. For rounding purposes this n e referred to s the ritil digit. ut.6 is the ritil digit in this exmple If the ritil digit is less thn (i.e. 0,,, or ), then you round down. This mens write the originl numer to the ple required, leving off ll other digits. This n e referred to s simply leving the numer s it is. If the ritil digit is or more (i.e., 6,, 8 or 9 ), then you round up. This mens write the originl numer to the ple required, ut inrese lst digit y. Leve off ll other digits. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

40 H Rounding deimls 6 Exmple 0 Determining the ritil digit The following deiml numers need to e rounded to deiml ples. Drw line where the numer must e ut nd then irle the ritil digit..98. SOLUTION EXPLANATION. 9 8 A line is drwn diretly fter the speified numer of deiml ples, in this se,.. The ritil digit is lwys the numer stright fter the speified numer of deiml ples. Exmple Rounding deimls to deiml ple Round eh of the following to deiml ple SOLUTION EXPLANATION. The ritil digit is 8 nd therefore the tenths olumn must e rounded up from 6 to.. The ritil digit is nd therefore the tenths olumn remins the sme, in effet rounding the originl numer down to.. Exmple Rounding deimls to different deiml ples Round eh of the following to the speified numer of deiml ples. Round 8.8 to deiml ples. Round 0.9 to deiml ples. Round.996 to deiml ples. SOLUTION EXPLANATION 8. The ritil digit is, therefore round down. 0. The ritil digit is 9, therefore round up..600 The ritil digit is 6, therefore round up. Rounding up hs resulted in digits eing rried over. Rememer to show the stted numer of deiml ples; hene, the zeros must e displyed. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

41 6 Chpter Understnding frtions, deimls nd perentges Exmple 0 Exmple Exmple, Exmple Exerise H UNDERSTANDING AND FLUENCY For eh of the following, selet the loser lterntive. Is.9 loser to. or.8? Is. loser to or? Is 8 loser to 80 or 90? d Is 6. loser to 6. or 6.8? 8, 6(½),, 8 9(½) 9(½) The following deimls need to e rounded, orret to deiml ples. Drw line where the numer must e ut nd then irle the ritil digit tht must e heked s to whether to round up or down d e f 0.9 g h.99 To round orretly to speified numer of ples, you must know whih digit is the ritil digit. Rememer: The ritil digit is lwys the digit immeditely to the right of the speified numer of ples. Stte the ritil digit in eh of the following numers. i.8 rounded to deiml ple. Critil digit = ii.8 rounded to deiml ples. Critil digit = iii.8 rounded to deiml ples. Critil digit = iv.8 rounded to the nerest whole numer. Critil digit = Stte the orret rounded numers for the numers in prts i to iv ove. Round eh of the following to deiml ple d 0.8 e 6.8 f 9.9 g. h.98 Write eh of the following, orret to deiml ples d 0.9 e.6 f g.0 h 9.89 i 6.89 j.999 k 9.99 l Round eh of the following to the speified numer of deiml ples, given s the numer in the rkets..9 ().89 ().6 (0) d 0. (0) e.86 () f 9.9 () g 9.9 () h.89 () Round eh of the following to the speified numer of deiml ples..98 ().899 () 6.9 (0) d () 8 Round eh of the following to the nerest whole numer d. e.6 f. g.6 h 0. 9 Round eh of the following mounts to the nerest dollr. $.8 $0.0 $.0 d $66.80 e $0. f $9. g $.9 h $6.9 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

42 H Rounding deimls 6 PROBLEM-SOLVING AND REASONING 0, Some wise shoppers hve the hit of rounding ll items to the nerest dollr s they ple them in their shopping sket. They n then keep running totl nd hve lose pproximtion s to how muh their finl ill will ost. Use this tehnique to estimte the ost of the following. Jenette purhses 0 items: $., $0.8, $.6, $8.99, $.0, $.0, $.90, $.9, $.8, $.99 Adm purhses items: $0., $.00, $.0, $8.90, $6.90, $.9, $.0, $.0, $., $0.0, $.9, $. Jenette s tul shopping totl is $0.8 nd Adm s is $.. How urte were Jenette s nd Adm s estimtions? Eletroni timing pds re stndrd in Ntionl Swimming ompetitions. In reent Ntionl Under 00 -metre freestyle re, Edwin reeives rounded time of.8 seonds nd Jsmine time of. seonds. If the timing pds n lulte times only to the nerest seond, wht will e the time differene etween the two swimmers? If the timing pds n lulte times only to the nerest tenth of seond, wht will e the time differene etween the two swimmers? Wht is the time differene etween the two swimmers, orret to deiml ples? d If the timing pds n mesure to deiml ples, wht would e the quikest time in whih Edwin ould hve swum the re? Using lultor, evlute.69, orret to deiml ples. Wht is the lest numer of deiml ples you need to find in the quotient to ensure tht you hve rounded orretly to deiml ples? Smr elieves should e rounded up to 0., ut Cssndr elieves it should e rounded down to 0.. Mke n rgument to support eh of their sttements, ut then show the flw in one girl s logi nd lerly indite whih girl you think is orret. ENRICHMENT Rounding with tehnology, Most lultors re le to round numers orret to speified numer of ples. Find out how to do this on your lultor nd hek your nswers to Questions nd 6. Spredsheet softwre pkges n lso round numers orret to speified numer of ples. Find out the orret syntx for rounding ells in spredsheet progrm, suh s Mirosoft Exel, nd then hek your nswers to Questions nd 8. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

43 66 Chpter Understnding frtions, deimls nd perentges I Deiml nd frtion onversions Key ides Deimls nd frtions re oth ommonly used to represent numers tht re not simply whole numers. It is importnt tht we know how to onvert deiml numer to frtion, nd how to onvert frtion to deiml numer. In the photo opposite, we n see tht: or 0. of the ke hs een ut nd or 0. of the ke remins. Let s strt: Mth my ll In pirs, nominte one student to e Frtion kid nd the other to e Deiml expert. Frtion kid strts nming some ommon frtions nd Deiml expert tries to give the equivlent deiml vlue. Strt with esy questions nd uild up to hrder ones. After 0 turns, swp round. This time Deiml expert will nme some deiml numers nd Frtion kid will ttempt to ll out the equivlent frtion. Disuss the following question in pirs: Whih is esier, onverting frtions to deimls or deimls to frtions? Converting deimls to frtions: When the denomintor is, think hlves. Exmple : = 0. Exmple : =. When the denomintor is, think qurters. Exmple : = 0. Exmple : = 0. When the denomintor is 0, think tenths; e.g. 0 = 0. seven-tenths in the tenths olumn When the denomintor is 00, think hundredths. Exmple : 00 = 0.0 the hundredths olumn Exmple : 00 = 0. When the denomintor is, think tenths; e.g. = 6 0 = 0.6 When the denomintor is ftor of 00, think hundredths. e.g. 0 = 00 = 0. When ll else fils, do division y hnd (or use lultor). e.g. 8 = 0. 6 or = 8 = 0.6 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

44 I Deiml nd frtion onversions 6 Sometimes, the deiml hs pttern tht repets forever. These re lled reurring deimls or repeting deimls. Exmple : = = 0.6 Exmple : =.88 =.8 or.8 Exmple : = or 0.8 Exmple Converting deimls to frtions Convert the following deimls to frtions in their simplest form SOLUTION = 0 0 Exmple Converting frtions to deimls Convert the following frtions to deimls. 00 EXPLANATION 0.9 = 9 thousndths SOLUTION EXPLANATION 00 = 0. hundredths 0. = hundredths, whih n e simplified further y dividing the numertor nd denomintor y the highest ommon ftor of. = 6 0 = is n equivlent frtion of, whose denomintor is power of 0. = 0.8 or Key Exerise I UNDERSTANDING AND FLUENCY Complete eh of these sttements, whih onvert ommon frtions to deimls. = 0 = 0. = = 0. = 00 = 0. d = 0 = 0.,, (½), 8 8(½) ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

45 68 Chpter Understnding frtions, deimls nd perentges Exmple Exmple Exmple Exmple Complete eh of these sttements, whih onvert deimls to frtions, in simplest form. 0. = 0 = 0. = 00 = 0.8 = 8 = d 0.6 = 6 00 = Are the following true or flse? 0. = = 0... =.. d. =.. e.666 =.. 6. f.66 =.6 Convert the following deimls to frtions in their simplest form d 8. e. f 0.0 g 9.0 h.9 Convert eh of these frtions to deimls e f g Convert the following frtions to deimls, y first hnging the frtion to n equivlent frtion whose denomintor is 0, 00 or 000. d 0 0 e 9 f g h 0 8 Convert the following frtions to deimls, y dividing the numertor y the denomintor. Use lultor to hek your nswers. d 6 e f g h 8 i j k l Copy nd omplete the following frtion deiml tles. The qurters tle (prt ) hs lredy een done for you. It s well worth trying to memorise these frtions nd their equivlent deiml vlues. hlves thirds Frtion Deiml qurters 0 0 Frtion Deiml Frtion Deiml d fifths Frtion PROBLEM-SOLVING AND REASONING 9 Arrnge the following from smllest to lrgest., 0., 8, 0., 0.99, Deiml 0 0, 0., 9, 0.8, 0.8, d h , 9 0, (½) 0, 6 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

46 I Deiml nd frtion onversions 69 0 Tn nd Lillin re trying to work out who is the etter hess plyer. They hve oth een plying hess gmes ginst their omputers. Tn hs plyed gmes nd eten the omputer times. Lillin hs plyed only gmes nd hs eten the omputer 6 times. Using lultor nd onverting the pproprite frtions to deimls, determine who is the etter hess plyer. Lillin hs time to ply nother four gmes of hess ginst her omputer. To e lssified s etter plyer thn Tn, how mny of these four gmes must she win? To estimte the thikness of one sheet of A pper, Christopher mesures the thikness of rem of pper, whih onsists of 00 sheets of A pper. He determines tht the pile is mm thik. How thik is one sheet of A pper? Express your nswer s deiml numer nd lso s frtion. Copy nd omplete the following frtion deiml tle. Frtion Deiml Comment on the trend in the deiml vlues s the denomintor inreses. Try to explin why this mkes sense. Copy nd omplete the following deiml frtion tle. Deiml Frtion Comment on the trend in the frtions s the deiml vlue inreses. Try to explin why this mkes sense. Write s reurring deiml. Write s reurring deiml. Using your lultor, find. d Is the lultor orret or inorret to disply the nswer s Explin. Write three different frtions with different denomintors tht re etween the deiml vlue of. nd.. 6 When is expressed in deiml form, find the digit in the rd deiml ple. Give reson for your nswer. ENRICHMENT 6 Design deiml gme for the lss Using the skill of onverting deimls to frtions nd vie vers, design n pproprite gme tht students in your lss ould ply. Ides my inlude vritions of Bingo, Memory or Dominoes. Try reting hllenging set of question rds Unorreted th smple pges Cmridge University Press Plmer et l, Ph

47 0 Chpter Understnding frtions, deimls nd perentges J Conneting perentges with frtions nd deimls Perentges re relted losely to frtions. A perentge is frtion in whih the denomintor is 00. Per ent is Ltin for out of 00. One dollr is equivlent to 00 ents nd entury is 00 yers. Perentges re used in mny everydy situtions. Interest rtes, disounts, test results nd sttistis re usully desried using perentges rther thn frtions or deimls euse it is esier to ompre two different results. Let s strt: Compring performne Consider these netll sores hieved y four students. Annie sores 0 gols from 0 shots (i.e. 0 out of 0 ). Bell sores 9 gols from shots. Cr sores gols from shots. Dinne sores 6 gols from 0 shots. Disuss wys to ompre the ury of their gol shooting. How might perentges e used? The hrt elow might e useful. Annie divided her 0 shots in the 0 oxes nd then shded the ones she sored. 00% 90% 80% 0% 60% 0% 0% 0% 0% 0% 0% Annie (0 shots) Bell Cr Dinne Unorreted th smple pges Cmridge University Press Plmer et l, Ph

48 J Conneting perentges with frtions nd deimls Perentges hve een used for hundreds of yers ut the symol we use tody is firly reent. The symol % mens per ent. It omes from the Ltin words per entum, whih men out of 00. For exmple: % mens out of 00 or or 00 or Perentges re useful wy to ompre frtions. For exmple: = 00 = % nd 8 = 00 = %, therefore > 8. It is importnt to understnd the reltionships nd onnetions etween frtions, deimls nd perentges. The frtion wll digrm elow shows these very lerly. perentges deimls one whole hlves thirds qurters fifths sixths eighths ninths tenths twelfths 0% 0% 0% 0% 0% 60% 0% 80% 90% Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

49 Chpter Understnding frtions, deimls nd perentges Key ides It is lso importnt to memorise the most ommonly used onversions. These re listed in the tle elow. Words Digrm Frtion Deiml Perentge one whole 00% one-hlf one-third one-qurter one-fifth one-tenth 0 one-hundredth 00 The numer fts in the tle n e used to do other onversions. For exmple: = 0. = % 9 = 0. = % =. = % 9 =. = % 0. 0% 0. or 0. % 0. % 0. 0% 0. 0% 0.0 % Unorreted th smple pges Cmridge University Press Plmer et l, Ph

50 J Conneting perentges with frtions nd deimls Exmple Using known numer ft to mke onversions Given tht = 0%, omplete the following. = 0%, so = % = 0%, so 6 = % = 0%, so = % SOLUTION EXPLANATION = 0% This should e memorised numer ft. = 0% = 60% Multiply the numer ft y. 6 = 0% 6 = 0% Multiply the numer ft y 6. = 0% = 0% Multiply the numer ft y. Exmple 6 Using memorised numer fts Convert the following frtions to deimls nd perentges. seventeen-tenths nine-qurters two-thirds SOLUTION 0 = 0 = 0.0 = 0% 00 0 = 0 =.0 = 0% 00 = = 0. = % 00 9 =. = % = 0... = % = = 66 % Exerise J UNDERSTANDING AND FLUENCY EXPLANATION This should e memorised numer ft. Multiply the numer ft y. This should e memorised numer ft. Multiply the numer ft y 9. This should e memorised numer ft. Multiply the numer ft y. 0 The perentge equivlent to one-qurter is: A % B.% C % D 0% The perentge equivlent to three-qurters is: A.% B % C % D 80% The perentge equivlent to 0. is: A 0.% B % C 0% D 00% Unorreted th smple pges Cmridge University Press Plmer et l, Ph

51 Chpter Understnding frtions, deimls nd perentges Exmple Exmple 6 Use the frtion wll (see pge ) to omplete the following. three-qurters = = 0. = % nine-tenths = = 0. = % two-fifths = = 0. = % d four-fifths = = 0. = % Complete the following. = 0%, so = % = 0%, so = % = %, so = % d = %, so = % e = 0%, so 9 = % f = 0%, so = % 6 Convert the following frtions to deimls nd perentges. three-tenths three-fifths five-qurters d four-thirds Use the frtion wll (see pge ) to write down: i six frtions tht re equivlent to 0% ii ten frtions tht re greter thn % ut less thn 0% Compring your nswers to i nd ii, whih frtion is losest to 0%? 8 Use the frtion wll (see pge ) to omplete these omputtions. Give your nswer s frtion in simplest form d + + e 6 + f Use the frtion wll (see pge ) to do these omputtions. 6 8 d e f 6 g h i 8 0 Use the frtion wll (see pge ) to omplete these omputtions d 0.6 e f g 0. h 6 0. i 0. PROBLEM-SOLVING AND REASONING,,,, 6 6 Rhel s irthdy ke is ut into two equl piees, then four equl piees, then eight equl piees. Rhel ets three piees. Wht perentge of the ke did Rhel et? Wht perentge of the ke remins? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

52 J Conneting perentges with frtions nd deimls Use the frtion wll (see pge ) to nswer the following questions. Whih is igger: three-qurters or two-thirds? Whih is igger: two-thirds or three-fifths? Wht is hlf of one-hlf? d Wht is hlf of one-qurter? e Wht frtion is extly hlfwy etween one-hlf nd one-qurter? Sophie s netll tem wins six of their first seven gmes. They hve three more gmes to ply. Wht is the highest perentge the tem n hieve? Wht is the lowest perentge the tem n hieve? Use the frtions in the frtion wll (see pge ) to solve these prolems. Two frtions with the sme denomintor dd up to one-hlf. Wht ould they e? Wht else ould they e? Write down ll the possiilities from the frtions in the frtion wll. Two frtions with different denomintors dd up to one-hlf. Wht ould they e? Wht else ould they e? Write down ll the possiilities from the frtions in the frtion wll. Three frtions with the sme denomintor dd up to one-hlf. Wht ould they e? Wht else ould they e? Write down ll the possiilities from the frtions in the frtion wll. d Three frtions with different denomintors dd up to one-hlf. Wht ould they e? Wht else ould they e? Write down ll the possiilities from the frtions in the frtion wll. e Two frtions with different denomintors dd up to one-hlf. One of them is one-tenth. Wht is the other one? Are the following sttements true or flse? Explin your nswers, using the frtion wll. 0. = 0% = % = % 9 d 0 = 90% e 8 % f > 66% 6 Whih one of the following prie redutions represents the gretest perentge disount? Explin your nswer. ENRICHMENT A frog nmed Willy Mkeit Before disount After disount A $0 $8 B $90 $60 C $00 $69 D $80 $60 Willy Mkeit is very smll frog. He is on flt ground, metre from his pond. He needs to get k to the pond ut he gets very tired when he jumps. His first jump is hlf metre. Every jump he mkes fter tht is hlf the distne of the previous jump. Use the frtion wll (see pge 8) to nswer these questions. After Willy hs mde his first jump, how fr is he from the pond? In metres, how long is his seond jump? After Willy hs mde his seond jump, how fr is he from the pond? d In metres, how long is his third jump? e Will Willy mke it to the pond? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

53 6 Chpter Understnding frtions, deimls nd perentges K Deiml nd perentge onversions Key ides Perentges give n ide of proportion. For exmple, if newspper sttes tht 000 people wnt ounil swimming pool onstruted, then we know how mny wnt pool ut we don t know wht proportion of the ommunity tht is. However, if there re 00 people in this ommunity, the newspper n stte tht 80% wnt swimming pool. This informs us tht mjority of the ommunity (i.e. 80 out of every 00 people) wnt swimming pool onstruted. Let s strt: Cretive shding Drw squre of side length 0 m nd shde extly 0% or 0. of this figure. Drw squre of side length m nd shde extly 60% or 0.6 of this figure. Drw nother squre of side length 0 m nd retively shde n ext perentge of the figure. Ask your prtner to work out the perentge you shded. Exmple Converting perentges to deimls Wht perentge is shded? The symol % mens per ent. It omes from the Ltin words per entum, whih men out of 00. For exmple: % mens out of 00 or or To onvert perentge to deiml, divide y 00. This is done y moving the deiml point ples to the left. For exmple: % = 00 = 0. To onvert deiml to perentge, multiply y 00. This is done y moving the deiml point ples to the right. For exmple: 0.6 = % = 6.% Express the following perentges s deimls. 0% 0%.% d 0.% SOLUTION EXPLANATION 0% = % = =..% = 0. Deiml point ppers to move ples to the left. d 0.% = 0.00 Deiml point ppers to move ples to the left. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

54 K Deiml nd perentge onversions Exmple 8 Converting deimls to perentges Express the following deimls s perentges SOLUTION % =.%. 00% = 0% Exerise K UNDERSTANDING AND FLUENCY Wht perentge of eh squre hs een shded? EXPLANATION Multiplying y 00 ppers to move the deiml point ples to the right. Multiply. y 00..% is equivlent to whih of the following deimls? A. B. C 0. D.0 % is equivlent to whih of the following deimls? A 0. B. C 00 D is equivlent to whih of the following perentges? 9%.9% 0.9% d 0.009% d 9 6, 9(½) 9(½) Unorreted th smple pges Cmridge University Press Plmer et l, Ph

55 8 Chpter Understnding frtions, deimls nd perentges Exmple, Exmple,d Exmple 8 Prue nswered hlf the questions orretly for test mrked out of 00. Wht sore did Prue get on the test? Wht perentge did Prue get on the test? Find the sore you would expet Prue to get if the test ws out of: i 0 ii 00 iii 0 iv d Find the perentge you would expet Prue to get if the test ws out of: i 0 ii 00 iii 0 iv 6 Fill in the empty oxes. 8% = 8 out of = 8 00 = 8 = 0 8 % = out of 00 = = 00 =. 6% = 6 00 = 6 00 = =. Express the following perentges s deimls. % % 68% d % e 6% f 9% g 00% h % i 8% j % k % l 99% 8 Express the following perentges s deimls..%.%.% d 8.% e.% f 88.8% g 0% h 0% i 0.9% j 0.0% k.0% l 0.9% 9 Express the following deimls s perentges d 0. e 0.6 f 0. g. h. i 0.0 j 0.00 k. l.00 PROBLEM-SOLVING AND REASONING 0 Ple the following vlues in order from highest to lowest. 86%, 0.%, 0.6, 0., %, %, %, 0.8 %,.,.99%, 0.0%,.8, %,., 0 0,,,,, 6,, 6, An ie-rem store is offering disount of % on orders over $. A single-soop ie-rem in one is $.0 nd doule-soop ie-rem in one is $.. A fmily of six purhses three single-soop ie-rems nd three doule-soop ie-rems. Will the fmily reeive the disount? Wht perentge will they py? Lst Sturdy, Phil spent hours of the dy in the following wy: 0. of the time ws spent sleeping, 0. ws spent plying sport nd 0. ws spent eting. The only other tivity Phil did for the dy ws wth TV. Wht perentge of the dy did Phil spend wthing TV? Wht perentge of the dy did Phil spend either sitting down or lying down? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

56 K Deiml nd perentge onversions 9 Sugrlof Reservoir hs pity of 96 giglitres. However, s result of the drought it is only % full. How mny giglitres of wter re in the reservoir? The verge dily energy intke for dolesent oys is 00 kj. The verge serving size of owl of Rie Bules with up of redued-ft milk provides 0 kj. Wht perentge of oy s dily intke is owl of Rie Bules with milk? Round your nswer to deiml ple.,, nd d re digits. Write the following deiml numers s perentges. 0.d..d d 0.0dd e.d f 0.ddd 6,, nd d re digits. Write the following perentges s deiml numers..% d% % d 0.d% e % f dd.d% Trudy sys tht it is impossile to hve more thn 00%. She supports her sttement y sying tht if you get every question orret in test, then you get 00% nd you nnot get ny more. Do you gree with Trudy s sttement? Provide four exmples of when it mkes sense tht you nnot get more thn 00%. Provide four exmples of when it is perfetly logil to hve more thn 00%. ENRICHMENT AFL ldder 8 The Austrlin Rules footll ldder hs the following olumn hedings. 8, 9 Using lultor, n you determine how the perentge olumn is lulted? Wht do you think the F nd the A olumn stnd for? In their next mth, Essendon sores points for their tem nd hs 6 points sored ginst them. Wht will e their new perentge? d By how muh do Hwthorn need to win their next gme to hve perentge of 00? e If Port Adelide plys Hwthorn in the next round nd the finl sore is Port Adelide ets Hwthorn, will Port Adelide s perentge eome higher thn Hwthorn s? 9 Crete your own AFL-style ldder using spredsheet progrm. After entering the results, the progrm should utomtilly updte the points olumn nd the perentge olumn. When rrying out sort on the dt, ensure tht your progrm will utomtilly hnge ny tem s position on the ldder, if neessry. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

57 80 Chpter Understnding frtions, deimls nd perentges L Frtion nd perentge onversions Key ides We ome ross perentges in mny everydy situtions. Interest rtes, disounts, test results nd sttistis re just some of the ommon wys in whih we del with perentges. Perentges re losely relted to frtions. A perentge is nother wy of writing frtion with denomintor of 00. Therefore, 8% mens tht if something is divided into 00 piees you would hve 8 of them. Let s strt: Student rnking Five students ompleted five different mthemtis tests. Eh of the tests ws out of different numer of mrks. The results re shown elow. Your tsk is to rnk the five students in desending order, ording to their test result. Mtthew sored out of possile 0 mrks. Mengn sored 6 out of possile 0 mrks. Mri sored out of possile 0 mrks. Mrus sored out of possile 0 mrks. Meliss sored 6 out of possile 80 mrks. Chnge these test results to equivlent sores out of 00, nd stte the perentge test sore for eh student. We n write perentges s frtions y hnging the % sign to denomintor of 00 (mening out of 00 ). For exmple: % = 00 We n onvert frtions to perentges through our knowledge of equivlent frtions. For exmple: = 00 = % To onvert ny frtion to perentge, multiply y 00. For exmple: 8 00 = 8 00 = =. Therefore 8 = % =.% Common perentges nd their equivlent frtions re shown in the tle elow. It is useful to know these. Frtion Perentge 0% % % 0% % 66 % % 0% 8 0 Unorreted th smple pges Cmridge University Press Plmer et l, Ph

58 L Frtion nd perentge onversions 8 Exmple 9 Converting perentges to frtions Express these perentges s frtions or mixed numerls in their simplest form. % 6% 0% SOLUTION % = 00 6% = 6 00 = 9 = 9 0% = 0 00 = 0 0 = = Exmple 0 EXPLANATION Chnge % sign to denomintor of 00. Chnge % sign to denomintor of 00. Cnel HCF. Answer is now in simplest form. Chnge % sign to denomintor of 00. Cnel HCF. Convert nswer to mixed numerl. Converting to perentges through equivlent frtions Convert the following frtions to perentges d SOLUTION EXPLANATION 00 = % Denomintor is lredy 00, therefore simply write numer s perentge. = Require denomintor to e Therefore, multiply numertor nd denomintor y to get n equivlent frtion. = % 8 = Convert the frtion to thousndths y multiplying the 8 denomintor nd numertor y. = = 6. Convert the frtion to hundredths y dividing the denomintor 00 nd numertor y 0. = 6.% Write the frtion s perentge. d = 6 Convert the frtion to tenths y multiplying the denomintor 0 nd numertor y. = 60 Convert the frtion to hundredths y multiplying the 00 denomintor nd numertor y 0. = 60 Write the mixed numerl s n improper frtion. 00 = 60% Write the improper frtion s perentge. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

59 8 Chpter Understnding frtions, deimls nd perentges Exmple Converting to perentges y multiplying y 00% Exmple 9, Exmple 9 Convert the following frtions to perentges. 8 SOLUTION EXPLANATION 8 00 = 8 00 Exerise L = = 8 = % 8 00 = 00 = 60 = 60% UNDERSTANDING AND FLUENCY 0 Multiply y 00%. Simplify y nelling HCF. Write your nswer s mixed numer. Convert mixed numer to improper frtion. Cnel nd simplify., (½) (½) Chnge these test results to equivlent sores out of 00, nd therefore stte the perentge. out of 0 = out of 00 = % out of 0 = out of 00 = % out of 0 = out of 00 = % d out of = out of 00 = % e 80 out of 00 = out of 00 = % f 60 out of 000 = out of 00 = % Write these frtion sequenes into your workook nd write eside eh frtion the equivlent perentge vlue.,,,,,,,,, If % of students in Yer re sent due to illness, wht perentge of Yer students re t shool? If 80% of the geogrphy projet hs een ompleted, wht perentge still needs to e finished? Express these perentges s frtions in their simplest form. % % % d 9% e % f 0% g % h 88% Express these perentges s mixed numerls in their simplest form. 0% 80% % d 0% e % f 0% g 6% h 80% Unorreted th smple pges Cmridge University Press Plmer et l, Ph

60 L Frtion nd perentge onversions 8 Exmple 0 Exmple 6 Convert these frtions to perentges, using equivlent frtions d e 8 f g 0 0 h i 6 0 j k 0 0 l Convert these frtions to perentges. Chek your nswers using lultor. 8 d e f g h PROBLEM-SOLVING AND REASONING 8 A ottle of lemonde is only % full. Wht frtion of the ottle hs een onsumed? Wht perentge of the ottle hs een onsumed? Wht frtion of the ottle is left? d Wht perentge of the ottle is left? , 9, 9,, 9,, 9 A lemon trt is ut into eight equl piees. Wht perentge of the trt does eh piee represent? 0 Petrin sores 8 out of 0 on her frtions test. Wht is her perentge sore? The nutrition lel on prtiulr rnd of slied red sttes tht the verge serving size of two slies is equl to grms. It lso sttes tht there re. grms of sugr per serve. Wht perentge of the red is sugr? The Sydney Kings sketll tem hve won out of 8 gmes. They still hve two gmes to ply. Wht is the smllest nd the lrgest perentge of gmes the Kings ould win for the seson? Lee won his tennis mth with the sore 6, 6, 6. Wht frtion of gmes did he win? Wht perentge of gmes did he win? Sott nd Penny hve just tken out home lon, with n interest rte of %. Write this interest rte s frtion. Write eh of the following perentges s frtions. % 8 % % d % ENRICHMENT Lottery reserh 6 6 Condut reserh on mjor lottery ompetition. If possile: Find out, on verge, how mny tikets re sold eh week. Find out, on verge, how mny tikets win prize eh week. Determine the perentge hne of winning prize. d Determine the perentge hne of winning the vrious divisions. e Work out the verge profit the lottery ompetition mkes eh week. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

61 8 Chpter Understnding frtions, deimls nd perentges M Perentge of quntity Key ides A ommon pplition of perentges is to find ertin perentge of given numer. Throughout life you will ome ross mny exmples where you need to lulte perentges of quntity. Exmples inlude retil disounts, interest rtes, personl improvements, slry inreses, ommission rtes nd more. In this exerise we will fous on the mentl lultion of perentges. Let s strt: Perentges in your hed It is useful skill to e le to quikly lulte perentges mentlly. Clulting 0% or % is often good strting point. You n then multiply or divide these vlues to rrive t other perentge vlues. In pirs, using mentl rithmeti only, lulte these perentges. 0% of $0 0% of $ 0% of $60 d 0% of $90 e 0% of $00 f 0% of $0 g % of $80 h % of $0 i % of $00 j % of $0 k % of $00 l % of $60 Chek your nswers with lssmte or your teher. Design quik set of questions for lssmte. Disuss helpful mentl rithmeti skills to inrese your speed t lulting perentges. To find the perentge of numer, without using lultor:. Express the required perentge s frtion.. Chnge the of to multiplition sign.. Look for written or mentl strtegy to omplete the multiplition. % of 60 = of 60 = 60 = 60 = 0 = Useful mentl strtegies To find 0%, divide y To find 0%, divide y 0 To find %, divide y To find %, divide y 00 Using lultor, % of 60 = Unorreted th smple pges Cmridge University Press Plmer et l, Ph

62 M Perentge of quntity 8 Exmple Finding the perentge of numer Find: 0% of 0 % of 00 SOLUTION 0% of 0 = = 0 = Exmple Solving worded perentge prolem Jqueline hs sved up $0 to purhse new pir of jens. She tries on mny different pirs ut only likes two styles, Evie nd Next. The Evie jens re normlly $0 nd re on sle with % disount. The Next jens retil for $80 nd hve 0% disount for the next hours. Cn Jqueline fford either pir of jens? SOLUTION Evie jens Disount = % of $0 = 00 0 = $.0 Sle prie = $0 $.0 = $.0 Next jens Disount = 0% of $80 = = $ Sle prie = $80 $ = $8 Jqueline n fford the Next jens. EXPLANATION Write % s frtion. Cnel nd simplify. Mentl rithmeti: 0% of 0 = Hene, 0% of 0 =. Multiply y to get 0%. % of 00 = Write % s frtion. = Cnel nd simplify. = 60 Mentl rithmeti: 0% of 00 = 0, % of 00 = 0 % is hlf of 0%. Hene, % of 00 = 60. Multiply y to get %. EXPLANATION Clulte the disount on the Evie jens. Find % of $0. Find the sle prie y sutrting the disount. Clulte the disount on the Next jens. Find 0% of $80. Find the sle prie y sutrting the disount. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

63 86 Chpter Understnding frtions, deimls nd perentges Exmple Exerise M UNDERSTANDING AND FLUENCY Copy nd omplete the following sentenes. Finding 0% of quntity is the sme s dividing the quntity y. Finding % of quntity is the sme s dividing the quntity y. Finding 0% of quntity is the sme s dividing the quntity y. d Finding 00% of quntity is the sme s dividing the quntity y. e Finding 0% of quntity is the sme s dividing the quntity y. f Finding % of quntity is the sme s dividing the quntity y. Without lulting the ext vlues, determine whih lterntive (i or ii) hs the highest vlue. i 0% of $00 ii % of $00 i % of $ ii 0% of $ i % of $0 000 ii % of $900 d i 88% of $ ii 8% of $ Find the following perentges, using mentl strtegy. Chek your nswers with lultor. 0% of 0 0% of 60 0% of 0 d 0% of 90 e % of 0 f % of 8 g % of 00 h 80% of 0 i % of 80 j % of 00 k % of 880 l % of 900 m % of 00 n % of 00 o % of 00 p 9% of 00 Find: 0% of 80 0% of 00 0% of 60 d 00% of e % of f % of g 6% of 0 h 000% of 0 Without using lultor, mth the questions with their orret nswer. Question Answer 0% of $00 $8 0% of $0 $6 0% of $80 $0 0% of $60 $ 0% of $00 $ % of $00 $0 0% of $0 $0 0% of $60 $ % of $6000 $60 0% of $88 $9, 6(½) 6(½) 6 Without using lultor, find: 0% of $0 0% of 0 millimetres % of 60 kilogrms d % of 00 tonnes e 0% of 0 minutes f 80% of 00 entimetres g % of 0 grms h % of hetres i 0% of 0 seonds Unorreted th smple pges Cmridge University Press Plmer et l, Ph

64 M Perentge of quntity 8 PROBLEM-SOLVING AND REASONING, 8, 8,, 0, 6 8 Exmple Hrry sored 0% on his perentges test. If the test is out of 0 mrks, how mny mrks did Hrry sore? 8 Gre wnts to purhse new top nd hs $0 to spend. She relly likes red top tht ws originlly pried t $ nd hs 0% disount tiket on it. At nother shop, she lso likes striped hoody, whih osts $. There is 0% off ll items in the store on this dy. Cn Gre fford either of the tops? 9 In student survey, 80% of students sid they reeived too muh homework. If 00 students were surveyed, how mny students limed tht they reeived too muh homework? 0 % of teengers sy their fvourite fruit is wtermelon. In survey of 8 teengers, how mny students would you expet to write wtermelon s their fvourite fruit? At Gldesrook College, 0% of students wlk to shool, % of students th puli trnsport nd the reminder of students re driven to shool. If there re 00 students t the shool, find how mny students: wlk to shool th puli trnsport re driven to shool Anthe hs just reeived % slry inrese. Her wge efore the inrese ws $000 per week. How muh extr money does Anthe reeive due to her slry rise? Wht is Anthe s new slry per week? How muh extr money does Anthe reeive per yer? Sm hs hours of free time efore dinner is redy. He spends % of tht time plying omputer gmes, 0% plying his drums, 0% plying outside nd 0% reding ook. How long does Sm spend doing eh of the four different tivities? Wht perentge of time does Sm hve remining t the end of his four tivities? Sm must set the tle for dinner, whih tkes minutes. Does he still hve time to get this done? Gvin mows 60% of the lwn in 8 minutes. How long will it tke him to mow the entire lwn if he mows t onstnt rte? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

65 88 Chpter Understnding frtions, deimls nd perentges Find: 0% of (0% of 00 ) 0% of (0% of 000 ) % of (% of 8000 ) d 80% of (0% of 00 ) 6 Write survey question for the students in your lss, suh s Wht is your fvourite olour? or How mny dys lst week did you th us to shool?. Provide five different nswer options. Survey your lss nd lulte the perentge of students who hose eh option. Use setor grph (i.e. pie hrt) to disply your findings. Whih is lrger: 60% of 80 or 80% of 60? 8 Tom does the following lultion: 0. Wht perentge of 0 does he find? ENRICHMENT Wning interest 9 9 When someone loses interest or motivtion in tsk, they n e desried s hving wning interest. Jill nd Louise re enthusisti puzzle mkers, ut they grdully lose interest when tkling very lrge puzzles. Jill is ttempting to omplete 000 -piee jigsw puzzle in weeks. Her interest drops off, ompleting 00 fewer piees eh week. i How mny piees must Jill omplete in the first week to ensure tht she finishes the puzzle in the -week period? ii Wht perentge of the puzzle does Jill omplete during eh of the weeks? iii Wht is the perentge tht Jill s interest wnes eh week? Louise is ttempting to omplete n piee jigsw puzzle in weeks. Her interest wnes t onstnt rte of % per week. i Wht perentge of the puzzle must Louise omplete in the first week to ensure she finishes the puzzle in the -week period? ii Reord how mny piees of the puzzle Louise ompletes eh week nd the orresponding perentge of the puzzle. iii Produe tle showing the umultive numer of piees ompleted nd the umultive perentge of the puzzle ompleted over the -week period. Unorreted th smple pges Cmridge University Press Plmer et l, Ph

66 N Using frtions nd perentges to ompre two quntities 89 N Using frtions nd perentges to ompre two quntities Sometimes we wnt to know the proportion of ertin quntity ompred to given totl or nother quntity. This my e done using frtion, perentge or rtio. The Erth s surfe, for exmple, is out 0% oen. So, the proportion of lnd ould e written s 0% (s perentge) or (s frtion). The 0 rtio of lnd to oen ould e desried s 0 prts of lnd to 0 prts of oen. Alterntively, the rtio ould e expressed s prts of lnd to prts of oen. Let s strt: Tdpole proportion Sientists Hugh nd Jk tke seprte smples of tdpoles, whih inlude green nd rown tdpoles, from their lol wter hnnels. Hugh s smple ontins green tdpoles nd rown tdpoles, wheres Jk s smple ontins green tdpoles nd 08 rown tdpoles. Find the proportion of green tdpoles in eh of Hugh nd Jk s smples. Use oth frtions nd perentges to ompre the proportions. Whih smple might e used to onvine the lol ounil tht there re too mny rown tdpoles in the wter hnnels? To express one quntity s frtion of nother: Frtion = mount totl To express one quntity s perentge of nother, find n equivlent frtion in whih the denomintor is 00. This ide works well when the denomintor is,,, 0, 0, or 0. A rtio ompres prts of totl. This n lso e done using lultor. 00 = 0 So = 0% Red frtion = = 0 = 0 00 Red perentge = 0 00 = 0% Rtio of red prts to ll prts = : Rtio of red to yellow = : Key ides Unorreted th smple pges Cmridge University Press Plmer et l, Ph

67 90 Chpter Understnding frtions, deimls nd perentges Exmple Expressing s proportion Express the following s oth frtion nd perentge of the totl. $0 out of totl of $00 green duks out of totl of 0 duks SOLUTION Exmple Using rtios A glss of ordil is prt syrup to 9 prts wter. Express the mount of syrup s frtion of the totl. Express the mount of wter s perentge of the totl. SOLUTION EXPLANATION Frtion = 0 00 = 0 = Write the given mount over the totl. Then simplify the frtion. Perentge = 0 00 = 0 = 0% Convert to hundredths. 00 Frtion = 0 = There is totl of green duks out of totl of 0. Simplify the frtion. Perentge = = 8 0 = 80 Convert to hundredths. 00 = 80% EXPLANATION Frtion = There is totl of 0 prts, inluding prt syrup. 0 Perentge = 9 0 = 90 = 90% There is totl 9 prts wter in totl of 0 prts = wter syrup Unorreted th smple pges Cmridge University Press Plmer et l, Ph

68 N Using frtions nd perentges to ompre two quntities 9 Exmple Exmple Exerise N UNDERSTANDING AND FLUENCY Note: The numers in this exerise hve een hosen refully so tht mentl strtegies my e used, rther thn lultor. This squre shows some oloured tringles nd some white tringles. How mny tringles re oloured? How mny tringles re white? Wht frtion of the totl is oloured? d Wht perentge of the totl is oloured? e Wht frtion of the totl is white? f Wht perentge of the totl is white? A frmer s pen hs two lk sheep nd eight white sheep. How mny sheep re there in totl? Wht frtion of the sheep re lk? Wht frtion of the sheep re white? d Wht perentge of the sheep re lk? e Wht perentge of the sheep re white? Express the following s oth frtion nd perentge of the totl. Chek your nswers using lultor. 0 out of totl of 00 out of totl of $0 out of totl of $0 d $60 out of totl of $80 e kg out of totl of 0 kg f g out of totl of 8 g g L out of totl of L h 0 ml out of totl of 00 ml Write eh oloured re s oth frtion nd perentge of the totl re. d e f A jug of lemonde is mde up of prts of lemon juie to 8 prts of wter. Express the mount of lemon juie s frtion of the totl. Express the mount of lemon juie s perentge of the totl. 6 A mix of onrete is mde up of prt of ement to prts of snd. Express the mount of ement s frtion of the totl. Express the mount of ement s perentge of the totl. Express the mount of snd s frtion of the totl. d Express the mount of snd s perentge of the totl. 6, (½), (½),, 6 (½) Unorreted th smple pges Cmridge University Press Plmer et l, Ph

69 9 Chpter Understnding frtions, deimls nd perentges A pir of soks is mde up of prts of wool to prt of nylon. Express the mount of wool s frtion of the totl. Express the mount of wool s perentge of the totl. Express the mount of nylon s frtion of the totl. d Express the mount of nylon s perentge of the totl. PROBLEM-SOLVING AND REASONING 8 Gillin pys $80 tx out of her inome of $600. Wht perentge of her inome does she keep? 9 Over summer, dm s wter volume redues from 0 meglitres to meglitres. Wht frtion of the wter in the dm hs een lost? 0 Express the following s frtion nd perentge of the totl. 0 ents of $ dys out of weeks entimetres removed from totl length of metres d seonds tken from world reord time of minutes e 80 grms of totl of 9 kilogrms f 00 entimetres from totl of 0.6 kilometres Of 0 students, 0 ply sport nd ply musil instrument, with some of these students plying oth sport nd musil instrument. Two students do not ply ny sport or musil instrument. Wht frtion of the students ply oth sport nd musil instrument? Wht perentge of the students ply musil instrument ut not sport? 8, 9, 9,, 0, An orhrd of 80 pple trees is tested for diseses. 0 of the trees hve light disese, 6 hve rown rot disese nd some trees hve oth. A totl of 8 trees hve neither light nor rown rot. Wht perentge of the trees hs oth diseses? Wht frtion of the trees hs light ut does not hve rown rot? For reent lss test, Ross sored out of 60 nd Mleish sored out of 00. Use perentges to show tht Ross otined the higher mrk.??? musi sport? Unorreted th smple pges Cmridge University Press Plmer et l, Ph

70 N Using frtions nd perentges to ompre two quntities 9 The pries of two rs re redued for sle. A hth pried t $0 000 is now redued y $000 nd WD pried t $ is now redued y $ 800. Determine whih r hs the lrgest perentge redution, giving resons. A yellow sports drink hs 0 grms of sugr dissolved in fluid nd weighs totl of 0 grms. A lue sports drink hs grms of sugr dissolved in fluid nd weighs totl of 00 grms. Whih sports drink hs the lest perentge of sugr? Give resons. 6 A room ontins girls nd oys. Write n expression using the pronumerls nd for the frtion of: i oys in the room ii girls in the room Write n expression using the pronumerls nd for the perentge of: i oys in the room ii girls in the room A mixture of dough hs prts of flour to prts of wter. Write n expression for the frtion of flour. Write n expression for the perentge of wter. ENRICHMENT Trnsport turmoil 8 8 A lss survey of 0 students revels tht the students use three modes of trnsport to get to shool: ike, puli trnsport nd r. All of the students used t lest one of these three modes of trnsport in the pst week. Twelve students used r to get to shool nd did not use ny of the other modes of trnsport. One student used ll three modes of trnsport nd one student used only ike for the week. There were no students who used oth ike nd r ut no puli trnsport. Five students used oth r nd puli trnsport ut not ike. Eight students used only puli trnsport. Use this digrm to help nswer the following. How mny students used oth ike nd puli trnsport ut not r? ike Wht frtion of the students used ll three modes of trnsport? Wht frtion of the students used t lest one mode of trnsport, inluding ike? r d Wht frtion of the students used t lest one mode of trnsport, inluding puli trnsport? e Wht perentge of students used puli trnsport nd r during the week? f Wht perentge of students used puli trnsport or r or oth during the week? puli trnsport Unorreted th smple pges Cmridge University Press Plmer et l, Ph

71 9 Chpter Understnding frtions, deimls nd perentges Egyptin frtions Investigtion The frtions in the nient Egyptin Eye of Horus were used for dividing up food nd lnd, s well s portions of mediine. They re lled unitry frtions euse ll the numertors re. Clerly, the nient Egyptins hd no lultors or preise mesuring instruments; nevertheless, y repetedly dividing quntity in hlf, the frtions,, 8, 6 or were omined to estimte ny other frtion. Imgine tht you re n nient Egyptin ker nd wish to shre your lst three loves of red eqully etween four people. First, you ut two loves in hlf nd give hlf lof to eh of your four ustomers. You hve one lof remining nd you n ut tht into qurters (i.e. hlf nd then hlf gin). /8 /6 / / / So eh of your four ustomers now reeives hlf lof nd one-qurter of lof, whih is ( of lof. ) /6 Unorreted th smple pges Cmridge University Press Plmer et l, Ph