H SERIES. Percentage Basics. Curriculum Ready.

Transcription

1 Prntg Bsis Curriulum Ry

2 Copyrigt 009 P Lrning. All rigts rsrv. First ition print 009 in Austrli. A tlogu ror or tis ook is vill rom P Lrning Lt. ISBN Ownrsip o ontnt T mtrils in tis rsour, inluing witout limittion ll inormtion, txt, grpis, vrtismnts, nms, logos n tr mrks (Contnt) r prott y opyrigt, tr mrk n otr intlltul proprty lws unlss xprssly init otrwis. You must not moiy, opy, rprou, rpulis or istriut tis Contnt in ny wy xpt s xprssly provi or in ts Gnrl Conitions or wit our xprss prior writtn onsnt. Copyrigt Copyrigt in tis rsour is own or lins y us. Otr tn or t purposs o, n sujt to t onitions prsri unr, t Copyrigt At 968 (Ct) n similr lgisltion wi pplis in your lotion, n xpt s xprssly utoris y ts Gnrl Conitions, you my not in ny orm or y ny mns: pt, rprou, stor, istriut, print, isply, prorm, pulis or rt rivtiv works rom ny prt o tis rsour; or ommrilis ny inormtion, prouts or srvis otin rom ny prt o tis rsour. Wr opyrigt lgisltion in lotion inlus rmunrt sm to prmit utionl institutions to opy or print ny prt o t rsour, w will lim or rmunrtion unr tt sm wr worksts r print or potoopi y trs or us y stunts, n wr trs irt stunts to print or potoopy worksts or us y stunts t sool. A workst is pg o lrning, sign or stunt to writ on using n ink pn or pnil. Tis my l to n inrs in t s or utionl institutions to prtiipt in t rlvnt sm. Pulis P Lrning Lt For mor opis o tis ook, ontt us t: Dsign P Lrning Lt Altoug vry prution s n tkn in t prprtion o tis ook, t pulisr n utors ssum no rsponsiility or rrors or omissions. Nitr is ny liility ssum or mgs rsulting rom t us o tis inormtion ontin rin.

3 Prntgs llow us to ompr t siz o on quntity in rltion to notr! Giv two xmpls wr you v sn prntgs us or mntion? Writ wt you tink it mnt in t sitution you sw it ing us. Q Wn wit ligt ntrs glss prism, t ligt is split up (isprs) into irnt olors. Dsri in s mny irnt lr wys s possil, ow mu o t isprs wit ligt is grn. Wit ligt R Orng Yllow Grn Blu Inigo Violt Work troug t ook or grt wy to o tis Mtltis P Lrning Lt

4 ow os it work? Propr rtions to prntgs Prnt () or vry (pr) 00 (nt) In rtion trms, tis mns t nomintor is qul to 00. ` or vry Sin t vlus r ompr to 00, t wol mount is rprsnt y 00 For mny prntg lultions, you will n to ng rtions into prntgs. So, just writ s n quivlnt rtion wit nomintor o 00! r r som xmpls involving propr rtions n prntgs: (i) 00 Prnt signs mns or vry 00 (ii) ' 00 ' Divi numrtor n nomintor y 00 Equivlnt rtion wit 00 in nomintor Numrtor < nomintor < 00 (iii) 0 0 # # Multiply numrtor n nomintor y 00 Equivlnt rtion wit 00 in nomintor Numrtor < nomintor < 00 (iv) # # Multiply numrtor n nomintor y Equivlnt rtion wit 00 in nomintor 00 Numrtor nomintor 00 Mtltis P Lrning Lt

5 ow os it work? Your Turn Propr rtions to prntgs Writ ts rtions s prntg..../.../ 0... PROPER FRACTIONS TO PERCENTAGE Writ ts prntgs s rtion in simplst orm Simplst Form Simplst Form Simplst Form g i Simplst Form Simplst Form Simplst Form j 8 k l 68 Simplst Form Simplst Form Simplst Form Writ ts s quivlnt rtions wit nomintor o 00 n tn s prntg. 6 6 ' 80 ' ' 00 ' 0 ' 00 ' 00 Mtltis P Lrning Lt

6 ow os it work? Your Turn Propr rtions to prntgs Writ ts s quivlnt rtions wit nomintor o 00 n tn s prntg. # # # 0 # # # Writ ts rtions s n quivlnt prntg (sow your working) g i 00 Mtltis P Lrning Lt

7 ow os it work? Your Turn Propr rtions to prntgs r r two otr lultor-rinly mtos (i) Multiply t rtion y 00 n tn simpliy. Multiply rtion y 00 n prntg sign : ` # 00j 00 0 Simpliy t rtion to gt prntg (ii) Sin rtion numrtor ' nomintor, w n lso o tis lultion: (numrtor ' nomintor) # 00 : ^^ ' 0 # Prntg vlu Us your lultors to in t mting prntg is. Drw strigt lins joining tir lips to solv tis: Wt is t nm givn to t puntution mrk / us in morn prntg signs? X S 0 m R z Mtltis z O E S P Lrning Lt 0 60 x R U 8 x Q U m S D L F X z I 6 m 0 x

8 ow os it work? Dimls n prntgs To ng prntg to iml, just ivi y 00: Writing t prntg in rtion orm sows wy w ivi y 00 or t iml ' ` 00. Prntg s rtion Frtion s iml Tis rt sows ow w onvrt twn imls n prntgs: Mov iml point sps rigt 0.0. # Diml Prntg ' Mov iml point sps lt Lt s look t two mor xmpls moving in opposit irtions. Writ ts s tir quivlnt prntg or iml: (i) ' 00 Divi y 00 to gt quivlnt iml Mov iml point sps lt Equivlnt iml (ii) ^0. # 00 Multiply y 00 to gt quivlnt prntg Mov iml point sps rigt Equivlnt prntg 6 Mtltis P Lrning Lt

9 ow os it work? Your Turn Dimls n prntgs Us ll o t wors low to sri ow you writ prntg s n quivlnt iml. iml point pls mov ivi y /.../0... DECIMALSANDPERCENTAGES * * * Writ o ts prntgs s tir quivlnt iml. 9 Us ll o t wors low to sri ow you writ iml s n quivlnt prntg. iml point pls mov multipli y Writ ts imls s tir quivlnt prntg Mtltis P Lrning Lt

10 Frtion Diml ow os it work? Your Turn Como tim: Dimls, rtions n prntgs W n omin wt w know now to ng vlus rom rtions to imls to prntgs n k. Writ two quivlnt vlus in simplst orm or o ts: (i) (ii) ' Equivlnt iml 0.8 ^0.8 # Equivlnt iml ` ` Equivlnt simplst rtion Equivlnt simplst rtion Rmmr: Us smllst pl vlu or nging imls to rtions. Fill in t gps wit quivlnt vlus in simplst orm: Prntg Mtltis P Lrning Lt

11 ow os it work? Your Turn Como tim: Dimls, rtions n prntgs DECIMALS, FRACTIONS AND PERCENTAGES *.../.../ 0... Writ t two otr quivlnt vlus in simplst orm or o ts: Writing two otr quivlnt vlus in simplst orm or o ts: Mtltis P Lrning Lt 9

12 ow os it work? Frtionl n iml prntgs Somtims t quivlnt rtion wit nomintor o 00 s iml numrtor. ' ' 00 Initil rtion Equivlnt rtion (00 in nomintor). 00 # # Rmmr: orrt rtions v wol numrs only in t numrtor n nomintor. Simply writ t nswr s t mting iml n rtion/mix numr. `. 00 iml mix numr r r som xmpls involving propr rtions n prntgs: (i) 00 ' 00 ' or Equivlnt rtion wit 00 in nomintor Diml n quivlnt rtion orm (ii) 00 ' 00 ' Equivlnt rtions wit 00 in nomintor 6. or 6 Diml n quivlnt mix numr orm Multiply or ivi y 00 to ng vlus twn quivlnt iml prntgs n imls. (iii)...' 00 Divi y 00, mov iml point pls lt 0. (iv) # 00.6 or Multiply y 00, mov iml point pls rigt Diml n quivlnt mix numr orm 0 Mtltis P Lrning Lt

13 ow os it work? Your Turn Frtionl n iml prntgs Writ ts s prntg in iml n propr rtion orm Diml Propr rtion Diml Propr rtion 00 0 Diml Propr rtion Diml Propr rtion Writ ts s prntg in iml n mix numr orm Diml Mix numr Diml Mix numr Diml Mix numr Diml Mix numr Diml Mix numr Diml Mix numr g Diml Mix numr Diml Mix numr Mtltis P Lrning Lt

14 ow os it work? Your Turn Frtionl n iml prntgs Writ ts prntgs s quivlnt imls FRACTIONAL.../.../ 0... PERCENTAGES AND DECIMAL Writ ts imls s quivlnt prntgs Writ ts s quivlnt iml n mix numr prntgs Diml Mix numr Diml Mix numr Diml Mix numr Diml Mix numr Diml Mix numr Diml Mix numr g Diml Mix numr Diml Mix numr Mtltis P Lrning Lt

15 Wr os it work? Prntg o n mount 0 # 0 00 Using t quivlnt rtion Clult 0 o 0 'o' mns # or 0. 0 # Using t quivlnt iml r r two wor qustions involving prntg mounts: (i) 60 pis o isuit rums wr spilt on t loor. I 80 o t rums wr suk up y vuum lnr on t irst swp, ow mny isuit rums wr suk up? 80 o # Multiply y quivlnt rtion (or iml) Simpliy rtion Alwys nswr wor prolms wit sttmnt ` 8 isuit rums wr suk up in t irst swp. Sin t wol mount in prntg trms 00, w n us tis t to simpliy som prolms. (ii) During n 8 our work y, J Un spns 90 o t tim working. T rst o t tim s is tking rks t. ow mny ours os s spn y oing non-work tivitis? J Un spns 90 o t tim working. ` s spns o t tim oing non-work tivitis ` 0 o # ours Multiply y quivlnt iml (or rtion) or 0.8 ours 60 minuts 8 minuts ` J Un spns 0.8 ours (8 minuts) y oing non-work tivitis. You oul lso v oun 90 irst, tn sutrt tt tim rom 8 ours. Mtltis P Lrning Lt

16 Wr os it work? Your Turn Prntg o n mount Complt t stps or ts prntg lultions: o 0 # 0 o /.../ 0... PERCENTAGE OF AN AMOUNT ** PERCENTAGE OF AN AMOUNT 0 o 6 ' # 6 o 0 00 # 6 # Clult ts prntg mounts using quivlnt rtions only (sow ll working). 0 o 60 0 o 8 o 6 o 8 60 o 0 80 o 0 g 0 o 0 o 9 Mtltis P Lrning Lt

17 Wr os it work? Your Turn Prntg o n mount Clult ts prntg mounts using quivlnt imls only (sow ll working). 0 o 0 0 o 0 0 o o 80 o o 8 g o 0 9 o Solv ts prntg prolms, sowing ll working. At n Angls n Dvils ny rss prty, 0 o t popl tr wr rss s ngls. ow mny popl wor ngl outits t t prty? o 00 GB (gigyts) r riv on lptop is ill wit Qin s musi olltion. ow mny gigyts os r musi tk up in storg? Mtltis P Lrning Lt

18 Wr os it work? Your Turn Prntg o n mount Solv ts prntg prolms, sowing ll working. Justin wrot lov pom ontining totl o 0 wors. I t wor lov rprsnt 8 o t wors us, ow mny wors wr not t wor lov in Justin s pom? Aoring to rnt zoo stuy, 8 o ll t 800 ruit-ting nimls sm to prr t tst o nns ovr t tst o ppls. ow mny nimls prrr t tst o ppls? 6 Try ts trikir ons to rn n wsom pssport stmp..../.../ 0... * A*W*E*S*O*M*E * A*W*E*S*O*M*E A pnk mixtur ontins sl-rising lour, ggs n 6 uttr. T rst o t mixtur ontins milk. ow mny millilitrs (ml) o milk r tr in 0 ml o t pnk mixtur? int: lult t totl prntg o non-lour ingrints irst A st o ln sls urrntly s grms on t vist si. I 0 o tis wigt ns to mov to t otr si to mk t sl ln, wt is t totl mount o wigts on t sls? int: in ow mu ns to mov n tn wt it mns to in ln 6 Mtltis P Lrning Lt

19 Wr os it work? Prntgs grtr tn 00 Now you v sn propr rtions s prntgs, wt ppns wn you v impropr rtions? For impropr rtions, t prntg vlu will grtr tn 00. So i you v 0 mor tn wt you strt wit, you now v 0 o t strting mount. Ts xmpls sow tt t mto us is just t sm s or prntg vlus smllr tn 00 (i) Impropr rtions n mix numrs: ' 00 00' Equivlnt rtion wit 00 in nomintor Numrtor > nomintor > 00 wol (00) + 0 ^ # 0 # Equivlnt impropr rtion Equivlnt rtion wit 00 in nomintor Mix numr > Equivlnt impropr rtion Simplii mix numr (ii) Dimls witout ling zros: 0 0 ' Divi y 00 to gt quivlnt iml Mov iml point sps lt Equivlnt iml, no ling zro. No ling zro mns t numr in ront o t iml point is not 0.. ^. ' 00.. Mov iml point sps rigt No ling zro, > 00. Mtltis P Lrning Lt

20 Wr os it work? Your Turn Prntgs grtr tn 00 Cng impropr rtions to tir quivlnt prntg. # 0 0 # 00 Prntg ' 0 00 ' Prntg Equivlnt rtion Equivlnt rtion Cng ts mix numrs to impropr rtions n tn tir quivlnt prntgs Prntg 00 Prntg Impropr rtion Frtion Impropr rtion Frtion 0 0 Cng ts prntgs to tir simplii quivlnt mix numr Impropr rtion Mix numr Simplii Impropr rtion Mix numr Simplii Mtltis P Lrning Lt

21 Wr os it work? Your Turn Prntgs grtr tn 00 Cng o ts to tir quivlnt iml or prntg.. ' 00. # g 00. Como tim: Us t sm mtos to ng ts rtionl n iml prntgs tt r grtr tn 00.. s iml 0. s iml 6 s iml 86 s iml.0 s iml prntg.0 s iml prntg. g. s mix numr prntg.90 s mix numr prntg i.00 s mix numr prntg j.00 s mix numr prntg Mtltis P Lrning Lt 9

22 Wr os it work? Rpting iml prntgs Rpting imls soul trt just lik trminting imls wn nging to prntgs. Mov iml point sps rigt o 0... #00 Diml Prntg.... o ' Mov iml point sps lt It is usully nir to writ rpting imls s mix numr prntg wr possil. `. o Tis tl sows tt tr is pttrn or writing simpl rpting imls s rtions: 0.o 0.o 0.o 0.o 0.o 0.6o 0.o 0.8o 8 0.9o r r som mor xmpls wit sligt irns involving rpting imls. (i) (ii) Convrt o to its quivlnt prntg vlu o Rmmr: Cng so t rpting pttrn strts rom t iml point. ^ # o 9 Convrt 0.06 o o to its quivlnt prntg vlu 006. o o T numr o '9s' in t nomintor quls t numr o igits in t rpting pttrn. ^ # o o 6 99 Multiply t iml y 00 Rpting iml prntg Prntg orm rom t tl Multiply t iml y 00 Rpting iml prntg Mix numr orm rom t tl 0 Mtltis P Lrning Lt

23 Wr os it work? Your Turn Rpting iml prntgs Us tis tl to lp wn nswring ts qustions... REPEATING DECIMAL PERCENTAGES....../.../ o 0.o 0.o 0.o 0.o 0.6o 0.o 0.8o 8 0.9o Writ ts in tir quivlnt rtion orm:. o. o 6. o. oo 8.08 oo 0. o o Us tis tl to lp writ ts rpting imls s tir quivlnt mix numr prntg: o o 00. o 006. o 008. o 0. o g 0. o 098. o i 06. o Writ ts mix numr prntgs s rpting imls: Mtltis P Lrning Lt

24 Wt ls n you o? On mount s prntg o notr Wn ompring vlus su s sors out o totl mount, t rsults r otn givn s prntg. Sor o 6 out o 0 'out o' mns 'ovr' Frtion 6 0 itr or Equivlnt rtion 08. Equivlnt iml 80 Prntg r r w mor vri xmpls. 6 # 00 or 6 ' 0 # 00 0 I using lultor, r r two wys to o it Ts xmpls sow tt t mto us is just t sm s or prntg vlus smllr tn 00 (i) Clult wt prntg mount is out o 0 witout using lultor. out o 0 0 # 0 # Equivlnt rtion wit 00 in nomintor T lultor soul us or mor omplx lultions. (ii) Dnn sor. mrks out o possil 0 in rnt omptition. Clult r prntg sor.. out o 0 ^^. ' 0 # o 8 Equivlnt mix numr prntg ` Dnn s prntg sor in t omptition ws 8 Prntgs n lso roun o. (iii) Writ s prntg out o 9, roun to iml pls. out o 9 ^^ ' 9 # ^to imlpls Rmmr: I t nxt igit is or mor, roun up. Mtltis P Lrning Lt

25 Wt ls n you o? On mount s prntg o notr Clult ts prntg mounts, sowing ll you working: **** ONE **** AMOUNT AS A PERCENTAGE OF ANOTER.../.../ 0... out o 0 0 out o 8 out o 00 0 out o 00 Clult ts prntg mounts, lving your nswrs s mix numr prntgs: out o out o out o 0 0. out o 0 Clult ts prntg mounts, rouning your nswrs to iml pls: out o 9 out o 8.9 out o.8 out o Mtltis P Lrning Lt

26 Wt ls n you o? Your Turn On mount s prntg o notr Gr ws top o r sin lss wn s orrtly spll 8 out o 0 sintii trms in tst. Wt prntg o t trms i s spll orrtly? Luk m 6 goo psss uring on l o ootll mt in wi possssion o t ll tims. Wt prntg o is possssions rsult in goo psss ing omplt? 6 Am ts r pttrn ovrs to oos rom wn purs nw moil pon: Wt prntg o Am s ois strip ovrs? int: ompr ow mny ovrs r strip wit ow mny ovrs tr r in totl Wn wit ligt ntrs glss prism, t ligt is split up (isprs) into irnt olors. Aoring to t igrm, wt prntg o t isprs ligt is grn, roun to iml pls? Wit ligt R Orng Yllow Grn Blu Inigo Violt Mtltis P Lrning Lt

27 Wt ls n you o? Your Turn On mount s prntg o notr 8 A.6 L jug o mix oril rink ontins 0.8 L o lvor syrup. Wt prntg o t mix oril rink is syrup lvor, roun to iml pl? 9 In on prtiulr song t onrt, t rytm guitrist or T Ligtning Lizrs uss o t 9 irnt ors ply y r trougout t ntir onrt. Wt prntg o t ors ply r us in tt on song, roun to iml pls? 0 Niol k t sp s vill on mmory stik n oun tt so r,.6 GB o t 6 GB totl mmory n us. Wt prntg o t mmory stik storg sp s Niol lry us? Writ your nswr s mix numr prntg. Wt mix numr prntg rprsnts t numr o lk xgons in tis pitur? A g ontins oolts iniviully wrpp in irnt olor oils. Tr r pink, 6 grn n 8 r. Wt prntg o t oolts r wrpp in pink oil? Writ nswr s mix numr. Mtltis P Lrning Lt

28 Wt ls n you o? Your Turn On mount s prntg o notr r is wt w o i w wnt to in t rmining prntg mounts: Twnty iv o t tirty popl wo wr rusing or t trin sussully ugt it. Wt prntg o t popl miss gtting to t trin on tim? ` 6 out o 0 ugt t trin ` ^0 - out o 0 miss t trin 6 ^' 0 # 6.6 o 6 Diml prntg Mix numr prntg t popl rusing or t trin miss gtting to it on tim. Tr stunts i not omplt tir Mi omwork o wting TV nws rost. Wt prntg o stunts omplt tir omwork i tr r twnty iv stunts in t lss? I tirty iv o t sixty svn popl t mting vot ys on ision, wt prntg vot no? Answr urt to iml pl. A movi inm is sowing: our ilrn s, tr tion vntur, two romnti omis, n iv rm movis uring on wk. Wt prntg o t movis r not romnti omis, roun to iml pl? 6 9 o t 00 supportrs t sports gm wr rrking or t tm tt vntully won. T rmining spttors ll support t tm tt lost. Wt prntg o t supportrs lt t gm isppoint tt tir tm lost? Writ nswr s mix numr. 6 Mtltis P Lrning Lt

29 Wt ls n you o? Rltion tim Rlting on t work ovr witin tis ooklt: Wt usul skills v you gin y lrning out prntgs? Writ out on or two wys you tink you oul pply prntgs to rl li sitution. I you isovr or lrnt out ny sortuts to lp wit prntgs or som otr ool ts, jot tm own r: Mtltis P Lrning Lt

30 Ct St r is summry o som importnt tings to rmmr or prntg sis Propr rtions to prntgs Prnt () or vry (pr) 00 (nt) ` or vry Dimls n prntgs Trminting 0.0. Rpting # Diml Prntg '00 0. o Diml #00 '00 Prntg. o Frtionl n iml prntgs ' ' 00 Initil rtion Equivlnt rtion (00 in nomintor). 00 # # Prntg o n mount Clult 0 o 0 'o' mns # 0 # 0 00 or Using t quivlnt rtion 0. 0 # Using t quivlnt iml On mount s prntg o notr Sor o 6 out o 0 Frtion 6 0 itr or Equivlnt rtion 08. Equivlnt iml 6 # 00 or 6 ' 0 # 00 0 I using lultor 80 Prntg 8 Mtltis P Lrning Lt

31 Answrs Propr rtions to prntgs Dimls n prntgs i 0 j 9 0 k 9 0 l T iml is multipli y 00, so t iml point is mov pls to t rigt.. g Como tim: Dimls, rtions n prntgs. ti g 99 i Fr on D im l P r n tg z U S O E 0 S z R Diml Frtion O L I D U S m z x Prntg 0 Frtion 0. Diml 0. Diml Dimls n prntgs Prntg Prntg Frtion 0.8 Diml 8 Prntg P Lrning Lt 0.0 Diml 0.6 Diml 9 Frtion Mtltis Frtion. 8. T prntg vlu is ivi y 00, so t iml point is mov pls to t lt. Diml Prntg 0 Frtion Prntg S R Q U x X 6 x S D L F X m m 0 I Prntg 0. Diml Prntg 0.0 Diml Prntg 9

32 Answrs Frtionl n iml prntgs Frtionl n iml prntgs. 0. Diml Propr rtion. 0.. Diml Mix numrl 0. Diml Propr rtion Diml Mix numrl 08. Diml Propr rtion Diml Mix numrl 0. Diml Propr rtion Diml Mix numrl.. Diml. Diml 6. Diml. Diml Mix numrl Mix numrl Mix numrl Mix numrl g Diml Diml Mix numrl Diml Mix numrl 0.. Diml Mix numrl Mix numrl g. Diml. Diml 9. 9 Diml Mix numrl Mix numrl Mix numrl. Prntg o n mount o 0 0 o # # Diml Mix numrl ' # 0 o # ' # o # g Mtltis P Lrning Lt

33 Answrs Prntg o n mount Prntgs grtr tn g Impropr Frtion rtion 0 Prntg. 8 o t popl wr rss s ngls. Qin's musi olltion ills 0 GB o storg sp Impropr Frtion Prntg rtion o t 0 wors in Justin's pom r not t wor 'lov' o t 800 ruit ting nimls t t zoo prrr t tst o ppls Tr is 9. ml in t pnk mixtur Prntgs grtr tn 00 Totl mount o wigts on t sls # 96 9 g 0 0 # 0 # 00 Equivlnt rtion 0 # 0 # ' ' Equivlnt Prntg rtion ' 0 00 ' 00 9 # 0 80 # 0 00 Prntg Impropr Mix Simplii rtion numrl Impropr Mix rtion numrl ' # 00 0 Simplii ' 88 00' ' # ' # g 00 00' # 00 0 Mtltis P Lrning Lt

34 Answrs Prntgs grtr tn 00 On mount s prntg o notr.. ' (to.p.) 9. 9 ^to.p. 0. ' (to.p.) ^to.p. 6. ' Gr spll 96 o t wors orrtly in t tst. 86. ' o Luk's psss wr goo # # 6. o ovrs vill v strip pttrn. g j T olour grn mks up.9 o t isprs wit ligt..8 o t oril is syrup lvour. k On song ontins.9 o ll t irnt ors t n plys in on sow. Rurring iml prntgs Niol s lry us 8 o t mmory stik storg sp o t xgons r s lk o t oolts v pink oil o t stunts omplt tir omwork. g i 6..8 o t popl t t mting vot 'no' o 008. o. 006 o. 8. o t movis wr not romnti omis. On mount s prntg o notr o t supportrs lt t gm isppoint Mtltis P Lrning Lt

35 Answrs Mtltis P Lrning Lt

36 Nots Mtltis P Lrning Lt

37

38