Fractions: Arithmetic Review


 Ashlie Hubbard
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1 Frtions: Arithmeti Review Frtions n e interprete s rtios omprisons of two quntities. For given numer expresse in frtion nottion suh s we ll the numertor n the enomintor n it is helpful to interpret this numer s the rtio of to. Tht is one oul sy tht out of ville prts (of equl size) we hve s in the following rough igrm: In the following setions we wish to review some of the importnt fts out frtionl rithmeti. First n foremost is the onept of equivlent frtions: tht is mny frtions tully represent the sme numer. You shoul e le to onvine yourself (perhps y rwing igrms similr to the one ove) tht frtions equivlent to inlue 0 n 0 The tehnil reson for this is tht frtions will retin the sme vlue if oth the numertor n enomintor re multiplie (or ivie) y the sme mount sine tht will retin the lne we hve in the originl rtio. A rule for equivlent frtions woul look like this: So for exmple 0 n One of the resons tht equivlent frtions re so importnt is tht we woul usully prefer to express frtions in their lowest terms; tht is using the smllest possile numers for the numertor n enomintor. In the exmples we ve seen lrey we sw tht 0 n we woul prefer using sine 0 n re twie s ig s they oul e. Formlly we n sy: A frtion is in lowest terms when its numertor n enomintor hve no ommon ivisors. (exept ) From now on you shoul lwys reue frtions to lowest terms sine the extr ftors re not neee. We will prtie tht in our exmples.
2 One wy to use equivlent frtions is to ompre the size of two unlike frtions. For exmple whih of the following two frtions is greter? It s hr to tell until I onvert them into frtions with ommon enomintor. A ommon enomintor is ommon multiple of the originl enomintors; in this se the lest ommon multiple of n is. So I onvert eh frtion into n equivlent frtion with enomintor : ; Now it s esy to tell y just looking t the numertors: >. I n onlue tht Now with equivlent frtions still in min let s move on to the rithmeti of frtions. Multiplition Multiplition of frtions works stright ross whih is sometimes the wy people woul like to s well. However it only works for multiplying! Here s how it looks: Let s try n exmple: Aoring to our rule we shoul get numertor n enomintor y. 0 n then reue this to 0 y iviing Di you see nother wy to o it? You might sy Yes n t I ross out the n the? I woul sy Yes ut e utious! Sine n oth re ivisile y we n rossnel s it is often lle efore we multiply s follows: 0 The ie here is tht iviing n y we get n respetively.
3 The pyoff is tht we no longer hve to reue our nswer to lowest terms euse rossneling is relly prereuing! Use this tehnique if you like ut e sure to rossnel only etween numertors n enomintors never etween two numertors or two enomintors. Also on t forget tht exponentition is repete multiplition so we oul sy for exmple tht Division Division n multiplition s we know re relte very losely n frtions illustrte just how lose tht reltionship relly is: ivision is multiplition. We ll reiprols re the reiprol of ; reiprols re numers tht hve prout of. Exmples of n n n n n (hek to e sure!). Sometimes we ll reiprol flippe frtion n tht llows us to sy tht to ivie we n flip n multiply. Be reful though not to flip the wrong numer! It is the ivisor (the seon numer) tht gets reiprote. Here is n exmple of ivision: To o this you will hopefully e following steps similr to these: Notie tht I use ross neling fter I onverte it to multiplition not efore! Aition n Sutrtion As note erlier sometimes people woul like to frtions the wy tht they multiply frtions: stright ross. Tht s not how it works though.
4 Consier this inorret exmple: Why isn t this the orret nswer? Among the mny possile replies to tht question one stns out s the most ovious: our potentil nswer is less thn one of our originl ens! So how o we relly these frtions? The nswer rings us k to equivlent frtions: we mke the frtions similr efore we strt y reting ommon enomintor. In this se the lest ommon enomintor is so we onvert enomintor of y multiplying oth top n ottom y : Then we the two ommon frtions y omining the numertors only: to n equivlent frtion with Wht out Sutrtion? Well we know tht sutrtion n ition re relte very losely ft reminisent of the onnetion etween ivision n multiplition: Tht is sutrtion is the sme s ing the opposite numer. This is why I hve omine ition n sutrtion into one setion in this review. Now we ll o two more exmples. First The lest ommon enomintor of n is so we shoul onvert oth frtions to twentyfirsts efore we egin n then proee: Next let s try
5 First I onvert it into this ition prolem: ( the opposite!) Then I fin ommon enomintor suh s n go long like this: 0 As finl test of how well this pper works for you try to hek my nswers:.. 0
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