1 MCrhsA.a lCpgateetbarrl 1da1 oI Chapter 11 Notes S1 3 4 5 h.... FFFFoaiiarnntcc ddttQoo tturrhh iixeezn²: Gm t ‐oC ix spFysr fi–ionm rg2 e y8tse²x..r ² m a1n 5ind6 t2 h4iys ²t.r i no2 m. iFa iln sdq tuhaer eG. C F6 4oxf ²5 +2 _a_n__d +7 89.. 2 11.1 Fractions A fraction has the form € ab where b is the # of parts a whole is at eh gie.s _Ft_ho_er_ _ #t_h_ o_e_f f_tr_ha_o_c_st_ie_o _pn. i eces we have. a is called the ______________ d__i_v;i dbe ids cinatlloe dan d € 25 b( tahre i sp ideicveids)e. d into (the w weh coalne )il, launsdtr _a_t_e i ist tbheel onwum. b__e_r i os ft htheo nsuem pabretrs ocfo ploarretsd the € 15 € 15 € 15 € 15 € 15 Another way we can write a frac tion is € a1b where we are taking a parts of € 1b th o Tfh ae rwefhoorlee,. w e can write € 25 as 2 € 15 Also we can think of a fraction as. a quotient of a divided by b or € aTherefore, b . € 25 d Wft erhgiavah. citest iniisoo wnnths.e e t wyo _ ed rei_es_c _itt_mha_le_ak _ lsi_sna_ _gbm_ ya_e _b d ao_isv.u i2td ridn aitgvi iotdhneead ml bn oyu um5t,.b w eShrosimc whe e cf rasanacw bti eot hnsahst oc wwo men ceao sou u0 ldt.4 ec bvoeynn vloleynr gt € 34 = 0.75 € 18 = 0.125 € 12 = 0.5 3 Some fractions are repeating eg. decimals. € 13 = 0.333… = 0. € 4C Tahnin ak idnegn oofm ai nfraatcotrio onf aa sf rdaicvtiisoionn e vweer c7ea q=nu sael e0 ?th _a_t_ __ € −a −b = +( € ab ), € −a b = ‐( € ab ) and € a −b = ‐( € ab ) Fl Torhwaacett siisot ,nt esr amrse. equivalent when they can be reduced to the same fraction in € 13 = € 26 = € 4 because we can d12iv ide the numerator and denominator of € 26 by 3 to get € 1we can divide the numerator and denominator of 3 and € 4 12 by 4 to get € 1I EPnax gagmee n5pe0lre3a:, l ,s weth 2e r#e8 nh R0,e duce to floorw est term s. 3. € 1075 2025 4 11.2 Algebraic Fractions Algebraic fractions are fractions with variables in d ege.n ominator. the numerator or € x2 € 5x € 3x N ote that for x +1 € 5x W A e Sff efariggorloclh..s tmyt Rtaoho.t e r eua sdc l apculinn arcton hetxp he rne eeto r ed nTttrui u,mhee cxmsqees n uo e tai oafrndl la gf li trftoevbhoaobiweecdrrrc t eetanaaish oinuuotcends mu t fsef etrdr e hrattaermhhcocanteelttsiodi oo.dcom nr toen r smionF un rotaemao cmd tf tooloeoionrnnr rwt. ao oat tRetem hlbogsreeerietmm n nbtcvaseeruaa,tra mnmolimiinnrcdbe .s o?t fer hrt bar_ aib _tystco_eo t_dcr i __a fio_a_avs n_ncie.ds td .(oi nxdr‐g eo2 nou) o,ut t mtao c ifgonaemac ttt moorro n € 3Nx ES =exot2at 2me b t4ephcfla,ea 5t up easr,lehot h,bt h6oleaeumt,gf ,whs 7 :ot ehu, el8d oe ,mr 9igaii knea lt hfrea octriiognin caal nfr baect rioednu hcaevde, i0t acas nan doetn boem trinuaet foor5r.. 5 11.3 Adding and Subtracting Fractions In the last section we learned about algebraic fractions w Icnah nteh anids p dso e scstiibolne,. w e start by remembering how to add fra acntido nres.d uWcien kgn tohwem w e € 15 + € 35 to equal € 45 But how do we add because the fractions have common denominators. € 13 and € 12 e Fi Ttqih riuses int6v w,aw 2leee xn ac ts3h kf a=r wna 6gch.te ai( oItt ni css o awu ciltdohm a clmosom?o bn m We mo 1enu2 mld toieuprns laeotn mmoyfi a ontkhtaehet eo tthrrw sem.o tu wdletoin pforleam.c)it nioantosr isn?t o I n this case € 13 into an equivalent fraction with the denominator of 6. € 13 = € ?6 g et an eYqouui vgaelte tnhti sfr bayct miounl.t iplying the numerator and the denominator by 2 to € 12 = € ?6 Now t.h eYyo uca gne tb teh aisd dbeyd m. ultiplying the numerator and denominator by ___. € 26 + € 36 = € 56 In general, € ab + cb = (a + c) b For fractions without a common denominator, € ab + cd = ad bd + cb bd = ad + cb bd O EDxfo ac Smoeuptr l2se e 4p agrl,o l4 btohle,e 5msdes , p 5rji,n 6cei ples apply to subtracting fractions too. 6 Short S1 F4 ia.. m ‐c 148tpo( x‐l2r i 8–fQ xay x 4un‐. 5 diz) s: implif y. 25 .. x‐(²5 ‐+ 42 x) 6 . x² ‐3 1. ( 8 x + 7 .5 x )x² –+ ‐1 (13 x ‐2) 7 11.4 More on Addition and Subtraction A mixed number is a number that contains an integer eg. 5 and a _________. € 12 An Wu mrmitieexre 5ad t½ onru a miss ba__ne_r _i m_c_a_pn_r_ bo__pe_ e_w_r_ rf_ir_ta_tec ttnhi oaanns. ta h n5e _ =d_ _e_n__o_m__i_n__a tforarc. tion where the € 10 2 , € 10 2 + € 12 Note that 5 = _______ € 12 = 5 + € 12 W ege. can write an algeb. r aic expression as a fraction too. € x − 1x = = € x2 −1 Fnd egieur.nm s oWte,m grr aeiitntteo aa rt ocao rnsmd a mtrheoe en dq deuenanol,o mamdinidna aot otro rsr uo fbfo ttrx hr a te h c feti rtthswte ot ne truemmrm ebrsya bxtyo. r mWs.u hletinp tlyhien g the € (x + 2) + 1 x +1 M A Redasdku etlh tx ien + gf2r fa ricna t tci o ot i n ao n f r _a_c_t_ i t_ aoo_sn_ ahu_ _fes_ri _aan_bcg_to_ it_ovh_en_e_ . f _cr_o a_mc__tm_io__on n. deno minat or of x + 1. 8 If we want to reverse this procedure and write an algebraic at ehlgge. em Wb.r raiitce mixed number, we write the terms as separate fra cfrtiaocntiso ann ads raend uce € 8 − x 4 d Wisrtirteib utive pro apse ar tdyi foffe rdeivnicseio bne towveere na dad nituiomnb aenr da nsudb at rfraacctitoionn p (.r 2e0m1e)m. ber € 84− x4 . Simplify € 84 ED xoa Smept l2e 5pcr,of,bi, l7ecm, s9 b. as _____________ and write _______________. 9 11.5 Multiplying Fractions Mr. Jones wants to tell a fish story using photoshop. He enlarges hb iigs fiiss thh 1e ½pic ttiumrees w ahnedn t hheen is 1 d ½on teim? es again. If his fish is a 6 inch as upnicfitsuhr,e h oofw € (6• 32)• 32 = 54 4 = 27 2 inches or € 1312inches Would that be more impressive? W( eughn. elink em audldtiiptiloynin agn fdra scutbiotnrasc, wtioen c)a. n multiply n umerators and denominators, € (x + 2) x • (x − 2) 3 = (x2 − 4) E1 2 ))x aSSmiimmp pp l elliisff:yy ((=ll ooookk ffoorr ccoomm3xmmoo nn ffaaccttoorrss ffiirrsstt)) € x2 + 6x + 5 x2 + 7x +10 • x2 + 7x +12 x2 + 4x + 3 10 11.6 More on Multiplication Multiply an integer by a fraction by remembering that w ritten as a fraction with a denominator of 1. an integer can be € 5• 34 = 51• 34 = 5• 3 1• 4 = 14 M ultiply and reduce if necessary. € x • −7 x U D A rsoee t ththhe= ee s apanmastetwe bre nyrs m( taha +ek ibsn)ag(m ct he+?e d _e)_ x_=p_ _(r aecs s+i o ands + i nbt co + fr badc t)i oorn s“ Ff iOrIsLt.” t o mult iply 11 11.7 Dividing Fractions An inverted fraction is the ______________ of that fraction. A b I1w Infy6 ea t if hctmrsaea on mrbve moeiceosiup kiplsr tte oifhpricle lasmy lem 1e=c odo6 _v n afi_drte_a . 1_ tmI.h6 f ea wfstr e/ala smwsetecas snb /ftyo st re¾o c¾ k, sin etoo cft wa a=k sh1eeso2c w tofhrn mead mra heneacsyis.p fbrreoaemc nnaelu tsoma trkh beeen rm amto uav ilretai phtelai eso,d f € 1 f Smpooear,r ni offyrn y aefomr fuarem ah omaervse ei. t a i sm, yoovuie h tahvaet tiso 2d i¼vi dseec tohned nsu lmonbge ar nodf syeocuo nwdasn bt yt1o t6 hk oen fos aew cs oehncoodwns d € 214seconds 1 16 seconds/frame L Feirts’st, swimripteli f 2y t¼h iass. € 94 . Then take € 94÷ 1 16 or94•16 R educe to make multiplying easier. Look for 1co m mon factors. € 91• 41= 36 12 E1 2))x a mples: = € x4 −16 x − 2 ÷ (x3 + 2x2) 3) € x2 + 3x + 2 x ÷ (x + 2) 13 Short Quiz: 1. What value(s) can x NOT equal? 2. S 3 5 i.. m 3p(‐li4fy ² 46.. F acto r and r educe. 14 11.8 Complex Fractions Complex fraction= a fraction that has ________________________ o r denominator. in the numerator So E EWfrirvxamh aedcemrtpenyinolp ecponlo elm fa m:r 2nianp9ncal ti+entioxo g rn fh.r =ais c atm aifosron adt chce teali o nonn fub ttmehh erabe tesw ord lrooaieft r ts tes i_myn_s _eat_e s_ t m_ha_e _,_ H_he_aau_ rv_yt_egh_ e_f g_rn_aos_c e wftsri oaraoncrttosie uoi ntnnh d.i et s tch n oeum smpuelner xian t or t H E h xoeaw mti wmploeeu :i tlSd ti amwkpeel sri fetyhw er pitlea ntheits S faratucrtino nto? orbit the sun once. 15 Ep 2 5 7 8 1 adbe2.x 5 a b 5m 4p Rlee Pvireo wbl Seemt s 2 : 46 71cfc1 f