#noxhing Continuous t 17 Contnts: A Th mn o ontinuous t [11.] B Histogrms [11.6] C Cumultiv rquny [11.7] Opning prolm Anrino ollt t or th rinll rom th lst month or 9 towns in Argntin. Th rsults r isply in th rquny tl longsi: Things to think out: ² Is th t isrt or ontinuous? ² Wht os th intrvl 6 6 r<7 tully mn? ² How n th shp o th istriution sri? ² Is it possil to lult th xt mn o th t? Rinll (r mm) Frquny 6 r<6 7 6 6 r<7 7 6 r<8 3 8 6 r<9 9 6 r<1 9 Totl 9 In Chptr 13 w sw tht ontinuous numril vril n thortilly tk ny vlu on prt o th numr lin. A ontinuous vril otn hs to msur so tht t n ror. Exmpls o ontinuous numril vrils r: Th hight o yr 1 stunts: Th sp o rs on strth o highwy: th vril n tk ny vlu rom out 1 m to m. th vril n tk ny vlu rom km/h to th stst sp tht r n trvl, ut is most likly to in th rng km/h to 1 km/h. 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk y:\haese\igcse1\ig1_17\33igcse1_17.cdr Wnsy, 8 Otor 8 9:31:1 AM PETER
3 Continuous t (Chptr 17) A THE MEAN OF CONTINUOUS DATA [11.] Continuous t is pl into lss intrvls whih r usully rprsnt y inqulitis. For xmpl, or hights in th 1s o ntimtrs w oul writ 1 6 h<1. To in th mn o ontinuous t, w us th sm mtho s or group isrt t sri in Chptr 13 stion F. Sin th t is givn in intrvls, our nswr will only n stimt o th mn. Exmpl 1 Th hights o stunts (h m) in hoky trining squ wr msur n th rsults tl: Estimt th mn hight. Stt th mol lss. Sl Tutor Hight (h m) Frquny () 13 6 h<1 1 6 h<1 1 6 h<16 1 16 6 h<17 17 6 h<18 9 18 6 h<19 3 Hight (h m) Mi-vlu (x) Frquny () x 13 6 h<1 13 7 1 6 h<1 1 8 1 6 h<16 1 1 186 16 6 h<17 16 33 17 6 h<18 17 9 17 18 6 h<19 18 3 Totl 81 P x Mn = P ¼ 81 ¼ 163 m Th mol lss is 16 6 h<17: EXERCISE 17A 1 A rquny tl or th wights o vollyll squ is givn longsi. Explin why wight is ontinuous vril. Wht is th mol lss? Explin wht this mns. Dsri th istriution o th t. Estimt th mn wight o th squ mmrs. Wight (kg) Frquny 7 6 w<8 8 6 w<8 8 6 w<9 8 9 6 w<9 7 9 6 w<1 1 6 w<1 1 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\3IGCSE1_17.CDR Mony, Otor 8 :31: PM PETER
Continuous t (Chptr 17) 3 A plnt insptor tks rnom smpl o tn wk ol plnts rom nursry n msurs thir hight in millimtrs. Th rsults r shown in th tl longsi. Wht is th mol lss? Estimt th mn hight. How mny o th slings r mm or mor? Wht prntg o th slings r twn 6 n 8 mm? I th totl numr o slings in th nursry is 87, stimt th numr whih msur: i lss thn 1 mm ii twn n 1 mm. Hight (mm) Frquny 6 h< 6 h<6 17 6 6 h<8 1 8 6 h<1 8 1 6 h<1 1 6 h<1 3 Th istns trvll to shool y rnom smpl o stunts wr: Distn ( km) 6 <1 1 6 < 6 <3 3 6 < 6 < Numr o stunts 76 87 3 Wht is th mol lss? Estimt th mn istn trvll y th stunts. Wht prntg o th stunts trvll t lst km to shool? I thr r 8 stunts in Jos s lss, stimt th numr who trvll lss thn 1 km to shool. Th tims tkn in minuts or plyrs to inish omputr gm wr: Tim (t) 6 t<1 1 6 t<1 1 6 t< 6 t< 6 t<3 3 6 t<3 Frquny 8 16 1 Wht prntg o th plyrs inish th gm in lss thn minuts? Estimt th mn tim to inish th gm. I 89 othr popl ply th gm, stimt th numr who will omplt it in lss thn minuts. B HISTOGRAMS [11.6] Whn t is ror or ontinuous vril thr r likly to mny irnt vlus. This t is thror orgnis using lss intrvls. A spil typ o grph ll histogrm is us to isply th t. A histogrm is similr to r hrt ut, to ount or th ontinuous ntur o th vril, th rs r join togthr. Column Grph Histogrm isrt t ontinuous t 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\3IGCSE1_17.CDR Tusy, 18 Novmr 8 11:7: AM PETER
36 Continuous t (Chptr 17) Consir th ontinuous t opposit whih summriss th hights o girls in yr 9. Th t is ontinuous, so w n grph it using histogrm. In this s th with o h lss intrvl is th sm, so w n onstrut rquny histogrm. Th hight o h olumn is th rquny o th lss, n th mol lss is simply th lss with th highst olumn. 3 1 rquny Hight (m) Frquny () 1 6 h<11 1 11 6 h<1 1 1 6 h<13 13 6 h<1 19 1 6 h<1 1 6 h<16 16 6 h<17 13 17 6 h<18 18 6 h<19 1 1 11 1 13 1 1 16 17 18 19 hight (m) In som situtions w my hv lss intrvls with irnt withs. Thr r oupl o rsons why this my hppn: ² W my wish to ollt smll numrs o t t th xtrmitis o our t rng. For xmpl, to mk th tl o girls hights sir to isply, w my omin th smllst thr lsss n lso th tllst two lsss: Hight (m) Frquny () 1 6 h<13 6 13 6 h<1 19 1 6 h<1 1 6 h<16 3 16 6 h<17 13 17 6 h<19 6 ² Th t my nturlly group in th ontxt o th prolm. For xmpl, post oi will hrg irnt rts pning on th wight o th prl ing snt. Th wight intrvls will proly not qully sp, ut it mks sns or th post oi to ror th numr o prls snt in h lss. So, th post oi my ollt th ollowing t o prls snt ovr wk: Mss (m kg) 6 m<1 1 6 m< 6 m< 6 m< Numr o prls 18 11 In ithr s, th histogrm w rw is not rquny histogrm. Th rquny o h lss is not rprsnt y th hight o its r, ut rthr y its r. Th hight o h r is ll th rquny nsity o th lss. Sin rquny = rquny nsity lss intrvl with, rquny nsity = rquny lss intrvl with Th mol lss is th lss with th highst rquny nsity, n so it is th highst r on th histogrm. It is not nssrily th lss with th highst rquny. 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\36IGCSE1_17.CDR Tusy, 18 Novmr 8 11:8:37 AM PETER
Continuous t (Chptr 17) 37 Exmpl Sl Tutor Th tl low shows msss o prls riv y ompny uring on wk. Mss (kg) 6 m<1 1 6 m< 6 m<3 3 6 m<6 6 6 m<1 Numr o prls 3 1 6 Drw histogrm to rprsnt th t. Fin th mol lss intrvl. Us grphis lultor to stimt th mn mss o th prls riv. Mss (m kg) Frquny CIW Frquny nsity 6 m<1 1 1 6 m< 3 1 3 6 m<3 1 1 1 3 6 m<6 6 3 6 6 m<1 1 CIW mns lss intrvl with. Histogrm showing msss o prls Th mol lss is 1 6 m<. rquny nsity 3 1 ky rprsnts prls 6 8 1 1 mss ( m kg) Mil vlu (x) Frquny () : 1: 3 : 1 : 6 8 mn ¼ :1 kg (lultor) Exmpl 3 Sl Tutor Th tims tkn or stunts to omplt ross-ountry run wr msur. Th rsults wr: Tim (t min) 6 t<3 3 6 t<6 6 6 t<31 Frquny 7 1 1 31 6 t<1 7 Drw histogrm to rprsnt th t. Fin th mol lss. Estimt th mn tim or stunts to run th vnt. Tim (t min) Frquny CIW Frquny nsity 6 t<3 7 3 9 3 6 t<6 1 3 17 6 6 t<31 1 31 6 t<1 7 1 7: 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\37IGCSE1_17.CDR Tusy, 18 Novmr 8 11:9:13 AM PETER
38 Continuous t (Chptr 17) Histogrm showing ross-ountry tims Th mol lss is 6 6 t<31. rquny nsity 1 1 ky rprsnts 1 stunts Mil vlu (x) Frquny () 1: 7 : 1 8: 1 36 7 Th mn ¼ 9: mins. 6 8 3 3 3 36 38 tim ( t min) EXERCISE 17B 1 Stunts wr sk to rw skth o thir vourit ruit. Th tim tkn (t min) ws msur or h stunt, n th rsults summris in th tl low. Tim (t min) 6 t< 6 t< 6 t<8 8 6 t<1 Frquny 1 1 3 6 Construt histogrm to illustrt th t. Stt th mol lss. Us grphis lultor to stimt th mn tim. Whn th msss o popl in Singpor itnss lu wr msur, th rsults wr: Mss (m kg) 6 m< 6 m<6 6 6 m<6 6 6 m<7 7 6 m<8 Frquny 7 6 7 3 Rprsnt this t on histogrm. Fin th mol lss. Us thnology to stimt th mn mss. 3 A group o stunts ws sk to throw sll s r s thy oul in givn irtion. Th rsults wr ror n tl. Thy wr: Distn ( m) 6 < 6 <3 3 6 < 6 < 6 <8 Frquny 1 3 3 1 Drw histogrm o th t. Trvl tims to work Wht is th mol lss? Us grphis lultor to stimt th mn istn thrown. Th histogrm shows th tims spnt trvlling to work y smpl o mploys o lrg orportion. rquny nsity Givn tht 6 popl took twn 1 n minuts to gt to work, in th smpl siz us. 1 3 6 tim ( t min) 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\38IGCSE1_17.CDR Tusy, 18 Novmr 8 11:9:1 AM PETER
Continuous t (Chptr 17) 39 C CUMULATIVE FREQUENCY [11.7] Somtims it is usul to know th numr o sors tht li ov or low prtiulr vlu. In suh situtions it is onvnint to onstrut umultiv rquny istriution tl n umultiv rquny grph to rprsnt th t. Th umultiv rquny givs running totl o th sors up to prtiulr vlu. It is th totl rquny up to prtiulr vlu. From rquny tl w n onstrut umultiv rquny olumn n thn grph this t on umultiv rquny urv. Th umultiv rqunis r plott on th vrtil xis. From th umultiv rquny grph w n in: ¾ ² th min Q ² th qurtils Q 1 n Q Ths ivi th orr t into qurtrs. 3 ² prntils Th min Q splits th t into two hlvs, so it is % o th wy through th t. Th irst qurtil Q 1 is th sor vlu % o th wy through th t. Th thir qurtil Q 3 is th sor vlu 7% o th wy through th t. Th nth prntil P n is th sor vlu n% o th wy through th t. So, P = Q 1, P = Q n P 7 = Q 3. Exmpl Th t shown givs th wights o 8 ml sktll plyrs. Construt umultiv rquny istriution tl. Rprsnt th t on umultiv rquny grph. Us your grph to stimt th: i min wight ii numr o mn wighing lss thn 83 kg iii numr o mn wighing mor thn 9 kg iv 8th prntil. Sl Tutor Wight (w kg) Frquny 6 6 w<7 1 7 6 w<7 7 6 w<8 8 8 6 w<8 16 8 6 w<9 1 9 6 w<9 19 9 6 w<1 8 1 6 w<1 3 1 6 w<11 1 11 6 w<11 1 Wight (w kg) rquny umultiv rquny 6 6 w<7 1 1 7 6 w<7 3 7 6 w<8 8 11 8 6 w<8 16 7 8 6 w<9 1 8 9 6 w<9 19 67 9 6 w<1 8 7 1 6 w<1 3 78 1 6 w<11 1 79 11 6 w<11 1 8 this is 1+ this is 1++8 this 8 mns tht thr r 8 plyrs who wigh lss thn 9 kg, so (9, 8) is point on th umultiv rquny grph. 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\39IGCSE1_17.CDR Tusy, 18 Novmr 8 11::37 AM PETER
36 Continuous t (Chptr 17) 8 7 6 umultiv rquny 68 6 Cumultiv rquny grph o sktllrs wights i ii iii iv % o 8 =, ) min ¼ 88 kg Thr r mn who wigh lss thn 83 kg. Thr r 8 6 = mn who wigh mor thn 9 kg. 8% o 8 = 68, so th 8th prntil ¼ 96 kg. 3 1 83 9 96 wight (kg) 6 7 8 9 1 11 1 min is kg Cumultiv rquny grphs r vry usul or ompring two istriutions o unqul sizs. In suh ss w us prntils on th vrtil xis. This tivly sls h grph so tht thy oth rng rom to 1 on th vrtil xis. Exmpl Sl Tutor Th hights o 1 1-yr-ol girls n 1-yr-ol oys wr msur n th rsults tl. Drw on th sm xs th Frquny (girls) Hight (h m) Frquny (oys) umultiv rquny urv or oth 1 6 h<1 th oys n th girls. 1 1 6 h<1 1 Estimt or oth th oys n th girls: 1 1 6 h<1 3 1 6 h<16 6 i th min ii th intrqurtil rng (IQR). 16 6 h<16 1 16 6 h<17 6 Compr th two istriutions. 8 17 6 h<17 3 17 6 h<18 1 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\36IGCSE1_17.CDR Mony, Otor 8 :37:38 PM PETER
Continuous t (Chptr 17) 361 CF (girls) Frq. (girls) Hight (h m) Frq. (oys) CF (oys) 1 6 h<1 1 1 1 6 h<1 1 1 3 1 1 6 h<1 3 6 3 1 6 h<16 6 6 8 16 6 h<16 1 9 1 16 6 h<17 6 16 98 8 17 6 h<17 3 19 1 17 6 h<18 1 1 =7%, so (1, 7) is point on th oys umultiv rquny grph. 1 prntils Cumultiv rquny grph o oys n girls hights 9 8 7 6 3 1 hight ( h m) 1 1 1 1 16 16 17 17 18 For girls: i min ¼ 18 m ii For oys: i min ¼ 16 m ii IQR ¼ 163: 13:7 ¼ 1 m IQR ¼ 169 18 ¼ 11 m Th two istriutions r similr in shp, ut th oys hights r urthr right thn th girls. Th min hight or th oys is 7 m mor thn or th girls. Thy r onsirly tllr. As th IQRs r nrly th sm, th spr o hights is similr or h gnr. 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk y:\haese\igcse1\ig1_17\361igcse1_17.cdr Friy, 1 Otor 8 9:7:1 AM PETER
36 Continuous t (Chptr 17) EXERCISE 17C 1 In running r, th tims (in minuts) o 16 omptitors wr ror s ollows: Drw umultiv rquny grph o th t n us it to stimt: th min tim th pproximt numr o runnrs whos tim ws not mor thn 3 minuts th pproximt tim in whih th stst runnrs omplt th ours th intrqurtil rng. Tims (min) Frquny 6 t< 18 6 t<3 3 6 t<3 37 3 6 t< 33 6 t< 19 6 t< 8 1 11 1 9 8 7 6 umultiv rquny Th umultiv rquny urv shows th wights o Sm s got hr in kilogrms. How mny gots os Sm hv? Estimt th min got wight. Any gots hvir thn th 6th prntil will go to mrkt. How mny gots will go to mrkt? Wht is th IQR or Sm s hr? 3 1 wight ( w kg) 1 3 6 7 3 Th lngths o 3 trout (l m) wr msur. Th ollowing t ws otin: Lngth (m) 36 l<3 36 l<3 36 l<36 366 l<38 386 l< 6 l< 6 l< Frquny 1 1 3 7 11 g Construt umultiv rquny urv or th t. Estimt th prntg o trout with lngth lss thn 39 m. Estimt th min lngth o trout ught. Estimt th intrqurtil rng o trout lngth n xplin wht this rprsnts. Estimt th 3th prntil n xplin wht this rprsnts. Us lultor to stimt th mn o th t. Commnt on th shp o th istriution o trout lngths. 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk y:\haese\igcse1\ig1_17\36igcse1_17.cdr Thursy, 3 Otor 8 1:19: PM PETER
Continuous t (Chptr 17) 363 Th wights o gs grown y two rothrs on sprt proprtis wr msur or omprison. Th rsults r shown in th tl: Drw, on th sm xs, umultiv rquny urvs or oth g smpls. Estimt or h rothr: i th min wight ii th IQR Compr th 6th prntil wights. Compr th two istriutions. Frquny (Aln) Wight (w grms) Frquny (John) 6 w< 3 6 w<7 6 7 6 w<8 7 8 6 w<1 6 1 6 w<11 3 11 6 w<13 totls Th tims tkn or trmprs to lim Bn Nvis wr ror n th rsults tl. Tim (t min) 17 6 t<19 19 6 t< 6 t< 6 t<3 3 6 t< Frquny 11 3 7 3 8 Construt umultiv rquny urv or th wlking tims. Estimt th min tim or th wlk. Estimt th IQR n xplin wht it mns. Guis on th wlk sy tht nyon who omplts th wlk in 3 hours 1 min or lss is xtrmly it. Estimt th numr o xtrmly it trmprs. Commnt on th shp o th istriution o wlking tims. 6 3 3 Cumultiv rquny urv o wtrmlon wight t umultiv rquny Th givn grph sris th wight o wtrmlons. Estimt th: i min wight ii IQR or th wight o th wtrmlons. Construt umultiv rquny tl or th t inluing rquny olumn. Estimt th mn wight o th wtrmlons. 1 1 wight ( w kg) 6 8 1 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_17\363IGCSE1_17.CDR Tusy, 18 Novmr 8 11:1:18 AM PETER
#noxhing 36 Continuous t (Chptr 17) Rviw st 17A 1 A rquny tl or th msss o ggs (m grms) in rton mrk g ggs is givn low. Explin why mss is ontinuous vril. Wht is th mol lss? Explin wht this mns. Estimt th mn o th t. Dsri th istriution o th t. Th sps o vhils (v km/h) trvlling long strth o ro r ror ovr 6 minut prio. Th rsults r givn in th tl longsi. Estimt th mn sp o th vhils. Fin th mol lss. Wht prntg o rivrs x th sp limit o 6 km/h? Dsri th istriution o th t. Mss (g) Sp (v km/h) Frquny 8 6 m<9 1 9 6 m< 1 6 m<1 16 1 6 m< 6 m<3 3 Frquny 6 v< 1 6 v< 6 v< 3 6 v<6 38 6 6 v<6 6 6 v<7 1 3 A sltion o msuring ottls wr xmin n thir pitis wr not. Th rsults r givn in th tl low: Cpity (C litrs) 6 C<: : 6 C<1 1 6 C< 6 C<3 3 6 C< Frquny 13 18 18 16 Drw histogrm to illustrt this inormtion. Wht is th mol lss? Th hights o plnts in il wr msur n th rsults ror longsi: Rprsnt this t on histogrm. Fin th mol lss. Estimt th mn hight o th plnts. Hight (h m) Frquny 6 h<1 11 1 6 h< 1 6 h<3 3 6 h< 1 6 h<6 18 6 6 h<1 1 Th wkly wgs o mploys in tory r ror in th tl low. Wkly wg ($w) 6w< 6w<8 86w<1 16w<16 166w< Frquny 6 1 1 Drw umultiv rquny grph to illustrt this inormtion. Us th grph to stimt: i th min wg ii th wg tht is x y % o th mploys. 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk y:\haese\igcse1\ig1_17\36igcse1_17.cdr Thursy, 3 Otor 8 1:8:8 PM PETER
Continuous t (Chptr 17) 36 6 Th umultiv rquny urv shows th tim spnt y popl in suprmrkt on givn y. Construt umultiv rquny tl or th t, using th intrvls 6 t<, 6 t< 1, n so on. Us th grph to stimt: i th min tim ii th IQR iii th 8th prntil. Copy n omplt: i 6% o th popl spnt lss thn... minuts in th suprmrkt ii 8% o th popl spnt t lst... minuts in th suprmrkt. 9 8 7 6 3 1 umultiv rquny tim ( t minuts) 1 3 6 Rviw st 17B 1 Th tl longsi summriss th msss o omsti ts hosn t rnom. Wht is th lngth o h lss intrvl? Wht is th mol lss? Fin th pproximt mn. Drw rquny histogrm o th t. From rnom sltion o 8 ts, how mny woul you xpt to wigh t lst 8 kg? Mss (m kg) Frquny 6 m< 6 m< 18 6 m<6 1 6 6 m<8 9 8 6 m<1 1 6 m<1 1 Th tl longsi summriss th st tims o 1 swimmrs who swim m. Estimt th mn tim. Wht is th mol lss? Tim ( t s) Frquny 6 t<3 3 6 t<3 17 3 6 t< 3 6 t< 9 6 t< 1 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk y:\haese\igcse1\ig1_17\36igcse1_17.cdr Thursy, 3 Otor 8 1:36:3 PM PETER
366 Continuous t (Chptr 17) 3 Th tl low isplys th istns jump y yr 1 stunts in long jump omptition: Distn ( m) 3 6 < 6 < 6 <: : 6 <6 6 6 <7 Frquny 8 16 1 9 Disply this inormtion on histogrm. Wht is th mol lss? Th histogrm longsi shows th rs o ln loks on strt. I ln loks wr twn 3 m to m in siz: onstrut rquny tl or th t stimt th mn r. rquny nsity 3 7 9 11 r (mx) Th prntg sors in tst wr ror. Th rsults wr tgoris y gnr. Drw th umultiv rquny grphs or oys n girls on th sm st o xs. Us prntils on th vrtil xis. Estimt th min n intrqurtil rng o h t st. Compr th istriutions. Frquny (oys) Prntg sor (s) Frquny (girls) 6 s<1 8 1 6 s< 1 6 s<3 8 1 3 6 s< 1 3 6 s< 1 6 s<6 6 6 s<7 1 7 6 s<8 1 8 6 s<9 9 6 s<1 3 6 In on month prio t prtiulr hospitl th lngths o nworn is wr ror. Th rsults r shown in th tl givn. Rprsnt th t on rquny histogrm. How mny is r m or mor? Wht prntg o is hv lngths in th intrvl m 6 l<3 m? Construt umultiv rquny istriution tl. Rprsnt th t on umultiv rquny grph. Us your grph to stimt th: i min lngth ii numr o is with lngth lss thn 1: m. Lngth ( l m) Frquny 8 6 l<9 1 9 6 l< 3 6 l<1 9 1 6 l< 1 6 l<3 16 3 6 l< 6 l< 6 l<6 7 9 1 7 9 1 7 9 1 7 9 1 IGCSE1 yn mgnt yllow lk y:\haese\igcse1\ig1_17\366igcse1_17.cdr Thursy, 3 Otor 8 1:37:16 PM PETER
ANSWERS 73 3 + 3 + 3y + i m m +3n 3m 3 x +6 3 x 18 6 6x 3 i x 1x 1 3 + +3 3 j k mn mp n o np m 1 + x +3 j 11x 3 EXERCISE 16D.1 1 9x x +1 6 13x 9 x +11 1 i 3x 3x 1 j 1 1 x 1 (x + 1)(x ) x +1 (x 1)(x +) 7x +8 (x 1)(x +) g x +3 x(x +1) h j (x 1) x 3 k m 7 (x 1)(x +3) o x (x + )(x ) q x +1 x(x 1)(x +1) 3(x +) x(x +3) (x 1) x + n p r 3 g 3 1 + x 3x 17 3 g 3 x 6 k +x x 11 g x +3 1 p x 7 g x +1 3 x +1 k 1 1x +17 (x + 1)(x +) 6 (x + )(x +1) 3(x +) (1 x)(x +) i m + h l 16 1 p 1 6 1 x +3 h x l 11y 6 1 y h 1 x 3 + 6 h x 7 x + l x x +6 (x + )(x ) x +x 3 l (x + 3)(x +) 17x 7 x(3x 1) x + x(x +1) x x 9 (x + 1)(x 1)(x +) s x 3 x +1 x(x + 1)(x 1) 3 6 1 1 x +3 x x 1 x g x 1 x +3 h x + x 3x + x 1 os µ = x + x, sin µ = x 1 x, tn µ = x 1 x + sin µ os µ = x 1 x x + x = x 1 x x x + = x 1 x + EXERCISE 16D. 1 +x x(x +1) (x +) x + +x x(x +1) x x +3 (x )(x +3) (x +x +) (x + )(x 3) x x g (x ) x 1 3 x x 6 x (x +1) (x + )(x 3) x +3 3(x +1) REVIEW SET 16A h x +1 x +7 1 3x 3n x 6 +3 3 19x 1 i x = or 3 ii x = 1 x + (x +) x x x + 1 x 16x i x = 1 or ii x = 3 nnot simplii x + x x 6 7 11x +1 1 x + (x ) x +1 REVIEW SET 16B x +3 x x 1 9 x 1 16x 9 1 x +1 3x + i x = 3 or 1 ii x = 1 3 x 3n x nnot simplii x + 3 11x x 3x 1 3x 13x 1 x +3 6 6 (x ) x 7 x 7 3(x ) x 8 1 x 1 EXERCISE 17A + 3 8 x +6 x(x +) 3x +1 x +1 i x = ii x = x +3 x 3x + x(x +) x 3(x +) ( ) 1 Th vril n tk ny vlu in th ontinuous rng 7 to 1. 8 6 w<9. This lss hs th highst rquny. symmtril ¼ 89: kg 6 h<6 ¼ 69:6 mm 6 o thm 3% i ¼ 7 ii ¼ 686 3 1 6 < ¼ 1:66 km 33:% 9 stunts 3:% ¼ :9 min ¼ 19 popl 7 9 1 7 9 1 7 9 1 7 9 1 IB MYP_3 ANS yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_n\73IB_IGC1_n.CDR Wnsy, 19 Novmr 8 9:1:9 AM PETER
7 ANSWERS EXERCISE 17B 1 8 6 t<1 ¼ 7:6 min 1 rquny nsity 1 6 6 m<7 1 ¼ 61: kg rquny nsity 1 m (kg) 3 6 7 8 3 3 6 < ¼ 1:1 m rquny nsity 3 1 3 mploys EXERCISE 17C 1 18 umultiv rquny 16 1 1 1 6 8 1 1 8 6 3 6 7 8 9 t (min) (m) Cumultiv rquny grph o r t tim ( t min) 3 3 3 min 8 8 min IQR = 1min 1 9 kg 8 17: kg 3 Cumultiv rquny grph o trout lngth 3 umultiv rquny 3 1 1 lngth (m) 3 3 3 36 38 7% 38:6 m 3: m 37:6 m. 3% o th lngths r lss thn or qul to this vlu. ¼ 38:3 m g ngtivly skw prntils 1 1 8 6 i Aln, ¼ 91; John, ¼ 83 ii Aln, ¼ 31; John, ¼ 9 Aln, ¼ 97; John, ¼ 89 Aln s gs r gnrlly hvir thn John s. Th spr o h t st is out th sm. umultiv rquny 16 1 1 1 8 6 ¼ 1 min 17 1 min. This is th lngth o tim in whih th mil % o th t lis. ¼ symmtril 6 i kg ii : kg ¼ : kg Cumultiv rquny grph o g wight t John Aln 7 8 1 11 13 wight ( w grms) Cumultiv rquny grph o Bn Nvis lim t 18 17 19 3 tim (min) Wight (w grms) Frq. Cum. Frq. 6 w<1 1 1 1 6 w< 3 6 w<3 8 3 6 w< 1 6 w< 8 8 6 w<6 6 3 6 6 w<7 3 37 7 6 w<8 39 8 6 w<9 1 7 9 1 7 9 1 7 9 1 7 9 1 IB MYP_3 ANS yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_n\7IB_IGC1_n.CDR Friy, 1 Novmr 8 9::6 AM PETER
ANSWERS 7 REVIEW SET 17A 1 Th vril n tk ny vlu in th ontinuous rng 8 6 m<3 grms. 6 m<1 ¼ :8 grms slightly ngtivly skw ¼ :9 km/h 6 v<6 ¼ :3% symmtril 3 : 6 C<1 rquny nsity 3 1 1 3 pity ( C litrs) Histogrm o msss o ts rquny rquny nsity 1 1 6 8 1 1 ¼ 39:1 s 3 6 t< 3 Histogrm o long jump t 3 1 377 o thm m (kg) 6 h<3. ¼ 3: m rquny nsity umultiv rquny 1. 1. 6 8 1 3 1 1 i ¼ $9 ii ¼ $1 6 Tim (t min) Frq. Cum. Frq. 6 t< 6 t<1 3 1 6 t<1 1 1 6 t< 1 6 t< 6 t<3 1 3 6 t<3 6 3 6 t< 1 7 6 t< 6 76 6 t< 8 REVIEW SET 17B Cumultiv rquny grph o wg t hight ( h m) 8 1 16 wg ( $ ) 1 kg 6 m< ¼ :76 i min ii 1 min iii 37 min i 7 ii 18 (m) 3 6 7 6 <: Ar (A m ) Frquny ¼ 71 m 3 6 A< 6 A<6 6 6 A<7 3 7 6 A<8 8 6 A<11 Prntils UE point CF (oys) CF (girls) B G 1 3:1 : 3 8:1 3:3 3 1 1:6 1: 3 1:9 18:3 7 37 6:9 3:8 6 1 6 78:1 1:7 7 1 1 9:6 8 8 1 11 96:9 93:3 9 16 117 1: 97: 1 16 1 1: 1: prntils 1 8 6 Cumultiv rquny grph o tst sors oys girls 1 3 6 7 8 9 1 sors (%) For oys: mium ¼, IQR ¼ 19 For girls: mium ¼ 9, IQR ¼ As th girls grph is urthr to th right o th oys grph, th girls r outprorming th oys. Both istriutions r ngtivly skw. 7 9 1 7 9 1 7 9 1 7 9 1 IB MYP_3 ANS yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_n\7IB_IGC1_n.CDR Thursy, Novmr 8 :17:8 PM PETER
76 ANSWERS 6 Histogrm o lngths o nworn is 7 is 7% o thm rquny 1 1 8 6 Uppr n point Cumultiv rquny 9 1 1 13 3 3 39 3 8 6 rquny i ¼ :1 m ii 18 is EXERCISE 18A lngth (m) Cumultiv rquny grph o lngths o nworns 3 3 1 1 8 9 1 3 6 lngth (m) 1 x =:8 x ¼ 1:13 x ¼ :76 x =8 x =8:7 x =:8 x ¼ 3:18 3 6 m k =1: 8m 8m tru ls,.g., m m EXERCISE 18B.1 1 All o ths igurs hv tringls whih n shown to quingulr n thror r similr. For xmpl, in, CBD is similr to CAE s thy shr n qul ngl t C n CBD = C AE =9 o. EXERCISE 18B. 1 x =: x =:8 x ¼ 3:7 x =9:6 x =11: x = g x ¼ 6:67 h x =7 i x =7: EXERCISE 18C 1 7 m 7: m 1:8 m 3 :67 m 1: m 1: m 6 9 sons 7 ¼ 117 m 8 113 m 9 SU =: m, BC =8: m No, th ll s ntr is ¼ 11 m on th D si o C. EXERCISE 18D 1 x =18 x =6 x = x ¼ :38 k = m n m r A : r B =1:16 3 k =: 1 m 8 m ED ¼ 1: V =8 V =: x ¼ :6 x =8 6 67 m 3 7 m 68 m 3 8 6 m 9 k = 3 1 8 m 3 8 m 1 No. Compring pitis, k ¼ 1:37 Compring lngths, k =1:6 Ths vlus shoul th sm i th ontinrs r similr. 11 : m 16 : 6 : 1 REVIEW SET 18A 1 n x ¼ 1:71 x ¼ 1:83 3 x =:8 Hint: Crully show tht tringls r quingulr, giving rsons. x ¼ 6:7 x = p 6 ¼ :9 6 A =7 x ¼ 8:1 7 x =1 y =3 8 ¼ 66:7 m wi 9 k = :99 m : m 3 REVIEW SET 18B 1 8m 8m tru ls,.g., Thy r not similr. Compring lngths; k = 16 1 = 3 Compring withs; k = 1 8 = 3 n 3 6= 3. 6 FG =: m 7 n n 1 m 3m 1\Qw_ 1\Qw_ 3 8 No, ny two quingulr tringls r similr. 3 1 m 9m m k = 7 9 : 3 x =3 x = x =1 B AC = NMC =9 o givng ]C is ommon to oth ) s ABC n MNC r quingulr, i.., similr. x 8 = 6 1 ) x = 8 =3: 1 6: m x = x ¼ :7 6 x =3:6 y =6: 7 Hint: Explin rully, with rsons, why thy r quingulr. CD =7: m : m 8 p 13 ¼ 7:1 m y 3 p 13 ¼ 1:8 m 9 68 m 3 CHALLENGE 1 ¼ 17:1 m 3:7 m m 7 9 1 7 9 1 7 9 1 7 9 1 IB MYP_3 ANS yn mgnt yllow lk Y:\HAESE\IGCSE1\IG1_n\76IB_IGC1_n.CDR Thursy, Novmr 8 :19:17 PM PETER