COMPLEX FRACTIONS. section. Simplifying Complex Fractions


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1 58 (66) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours FIGURE FOR EXERCISE Selling. George sells one gzine suscription every 0 inutes, wheres Theres sells one every inutes. Write rtionl epression for the nuer of gzine suscriptions tht they will sell when working together for one hour Pinting. Hrry cn pint his house y hiself in 6 dys. His wife Judy cn pint the house y herself in dys. Write rtionl epression for the portion of the house tht they pint when working together for dys Driving. Jon drove for 00 iles t one speed nd then incresed her speed y 5 iles per hour nd drove 00 dditionl iles. Write rtionl epression for her totl trvel tie hours Running. Willrd jogged for iles t one speed nd then douled his speed for n dditionl ile. Write rtionl epression for his totl running tie. 7 hours GETTING MORE INVOLVED 99. Discussion. Eplin why frctions ust hve coon denointors for ddition ut not for ultipliction. 00. Discussion. Find ech infinite su nd eplin your nswer. ) ) COMPLEX FRACTIONS In this section In this section we will use the techniques of Section 6. to siplify cople frctions. As their ne suggests, cople frctions re rther essylooking epressions. Siplifying Cople Frctions Siplifying Epressions with Negtive Eponents Applictions Siplifying Cople Frctions A cople frction is frction tht hs rtionl epressions in the nuertor, the denointor, or oth. For eple,,, nd re cople frctions. In the net eple we show two ethods for siplifying cople frction.
2 6. Cople Frctions (67) 59 E X A M P L E clcultor closeup You cn use clcultor to find the vlue of cople frction. A cople frction without vriles 5 Siplify. Method A For this ethod we perfor the coputtions of the nuertor nd denointor seprtely nd then divide: Method B For this ethod we find the LCD for ll of the frctions in the cople frction. Then we ultiply the nuertor nd denointor of the cople frction y the LCD. The LCD for the denointors,,, nd 5 is 60. So we ultiply the nuertor nd denointor of the cople frction y 60: , , 60 5 In ost cses Method B of Eple is the fster ethod for siplifying cople frctions, nd we will continue to use it. E X A M P L E A cople frction with vriles Siplify. The LCD of,, nd is. Multiply the nuertor nd denointor y : helpful hint When students see ddition or sutrction in cople frction, they often convert ll of the frctions to the se denointor. This is not wrong, ut it is not necessry. Siply ultiplying ever frction y the LCD eliintes the denointors of the originl frctions. ( ) ( ) ( ) ( ) ( ) ( ) 8 Distriutive property
3 60 (68) Chpter 6 Rtionl Epressions E X A M P L E study tip Your ood for studying should tch the ood in which you re tested. Being too reled during studying will not tch the incresed level of ctivtion you ttin during test. Likewise, if you get too tensedup during test, you will not do well ecuse your testtking ood will not tch your studying ood. More coplicted denointors Siplify. Becuse 9 ( )( ) nd 6 9 ( ), the LCD is ( ) ( ). Multiply the nuertor nd denointor y the LCD: ( ) ( ) 9 ( )( ) ( ) ( ) ( ) ( ) 6 9 ( ) ( )( ) Siplify. ( ) ( )( ) ( )( ) Fctor out. ( )[ ( )] ( ) ( ) ( )( 5 ) Siplifying Epressions with Negtive Eponents Consider the epression. Using the definition of negtive eponents, we cn rewrite this epression s cople frction: The LCD for the cople frction is. Note tht could lso e otined fro the ses of the epressions with the negtive eponents. To siplify the cople frction, we could use Method B s we hve een doing. However, it is not necessry to rewrite the originl epression s cople frction. The net eple shows how to use Method B with the originl epression. E X A M P L E A cople frction with negtive eponents Siplify the cople frction. Multiply the nuertor nd denointor y, the LCD of the frctions. Reeer tht 0. ( ) ( ) () () Distriutive property () () 6
4 6. Cople Frctions (69) 6 E X A M P L E 5 helpful hint In Eples, 5, nd 6 we re siplifying the epressions without first reoving the negtive eponents to gin eperience in working with negtive eponents. Of course, ech epression with negtive eponent could e rewritten with positive eponent nd then the cople frction could e siplified s in Eples nd. A cople frction with negtive eponents Siplify the cople frction. If we rewrote,,, nd, then the denointors would e,,, nd. So the LCD is. If we ultiply the nuertor nd denointor y, the negtive eponents will e eliinted: ( ) ( ) Distriutive property 0 0 Note tht the positive eponents of re just lrge enough to eliinte ll of the negtive eponents when we ultiply. The net eple is not ectly cople frction, ut we cn use the se technique s in the previous eple. E X A M P L E 6 More negtive eponents Eliinte negtive eponents nd siplify p p q. If we ultiply the nuertor nd denointor y pq, we will eliinte the negtive eponents: p p q ( p p q ) p q pq p q p pq p q pq p q pq Applictions The net eple illustrtes how cople frctions cn occur in prole. E X A M P L E 7 An ppliction of cople frctions Estside Eleentry hs the se nuer of students s Westside Eleentry. Onehlf of the students t Estside ride uses to school, nd twothirds of the students t Westside ride uses to school. Onesith of the students t Estside re fele, nd onethird of the students t Westside re fele. If ll of the fele students ride the uses, then wht percentge of the students who ride the uses re fele? To find the required percentge, we ust divide the nuer of feles who ride the uses y the totl nuer of students who ride the uses. Let the nuer of students t Estside.
5 6 (60) Chpter 6 Rtionl Epressions Becuse the nuer of students t Westside is lso, we hve the totl nuer of students who ride the uses nd 6 the totl nuer of fele students. Becuse ll of the fele students ride the uses, we cn epress the percentge of riders who re fele y the following rtionl epression: 6 Multiply the nuertor nd denointor y 6, the LCD for,, nd 6: % 7 7 So % of the students who ride the uses re fele. WARMUPS True or flse? Eplin.. The LCM for,, 6, nd is 6. Flse. The LCM for,, nd 6 is 6 6. True. The LCD is the LCM of the denointors. True. 5 6 True 5. ( ) Flse 6. ( ) Flse 7. 5 Flse 8. for ny rel nuer. Flse 9. To siplify, ultiply the nuertor nd denointor y. True 0. To siplify 5, 5 ultiply the nuertor nd denointor y 5. True 6. EXERCISES Reding nd Writing After reding this section, write out the nswers to these questions. Use coplete sentences.. Wht is cople frction? A cople frction is frction tht contins frctions in the nuertor, denointor, or oth.. Wht re the two ethods for siplifying cople frctions? One ethod is to perfor the opertions in the nuertor nd then in the denointor, nd then divide the results. The other ethod is to ultiply the nuertor nd the denointor y the LCD for ll of the frctions.
6 6. Cople Frctions (6) 6 Siplify ech cople frction. Use either ethod. See Eple Siplify the cople frctions. Use Method B. See Eple. n n n n y n y w t 0... w t n y wt n y 8t w n y w t 6 z 6 z z z z y y y y y y 6 y y y 6 y y Siplify ech cople frction. See Eples y.. y y y (y )( y ) w w w w w w w w 6w w w y y.. y y ( y y )( ) y y Siplify. See Eples w y 6. z y yz wz wy wz n 0. n n n
7 6 (6) Chpter 6 Rtionl Epressions y ) y ( y ( y ) 5. ( ) y y Use clcultor to evlute ech cople frction. Round nswers to four decil plces. If your clcultor does frctions, then lso find the frctionl nswer , , , 6.79, 0 Solve ech prole. See Eple Rcil lnce. Clrksville hs three eleentry schools. Northside hs onehlf s ny students s Centrl, nd Southside hs twothirds s ny students s Centrl. Onethird of the students t Northside re Africn Aericn, threefourths of the students t Centrl re AfricnAericn, nd onesith of the students t Southside re AfricnAericn. Wht percent of the city s eleentry students re AfricnAericn? 7.% 60. Eplosive sitution. All of the eployees t Ace Eplosives re in either developent, nufcturing, or sles. Onefifth of the eployees in developent re woen, onethird of the eployees in nufcturing re woen, nd onehlf of the eployees in sles re woen. Use the ccopnying figure to deterine the percentge of workers t Ace who re woen. Wht percent of the woen t Ace re in sles? 8.%, 65.% Developent Distriution of Eployees t Ace Eplosives Mnufcturing Sles FIGURE FOR EXERCISE Averge speed. Mry drove fro Clrksville to Leesville t 5 iles per hour (ph). At Leesville she discovered tht she hd forgotten her purse. She ieditely returned to Clrksville t 55 ph. Wht ws her verge speed for the entire trip? (The nswer is not 50 ph.) 9.5 ph 6. Averge price. On her wy to New York, Jenny spent the se ount for gsoline ech of the three ties tht she filled up. She pid 99.9 cents per gllon the first tie, 09.9 cents per gllon the second tie, nd 9.9 cents per gllon the third tie. Wht ws the verge price per gllon to the nerest tenth of cent for the gsoline tht she ought? 09. cents per gllon FIGURE FOR EXERCISE 6
8 6.5 Solving Equtions Involving Rtionl Epressions (6) 65 GETTING MORE INVOLVED 6. Coopertive lerning. Write stepystep strtegy for siplifying cople frctions with negtive eponents. Hve clsste use your strtegy to siplify soe cople frctions fro Eercises Discussion. ) Find the ect vlue of ech epression. i) ii) 5 8, 8 ) Eplin why in ech cse the ect vlue ust e less thn. The denointor is lrger thn the nuertor in the first frction. 65. Coopertive Lerning. Work with group to siplify the cople frction. For wht vlues of is the cople frction undefined?, 0,,, In this section 6.5 SOLVING EQUATIONS INVOLVING RATIONAL EXPRESSIONS Mny proles in lger re odeled y equtions involving rtionl epressions. In this section you will lern how to solve equtions tht hve rtionl epressions, nd in Section 6.6 we will solve proles using these equtions. Multiplying y the LCD Etrneous Roots Proportions Multiplying y the LCD To solve equtions hving rtionl epressions, we ultiply ech side of the eqution y the LCD of the rtionl epressions. E X A M P L E helpful hint Note tht it is not necessry to convert ech frction into n equivlent frction with coon denointor here. Since we cn ultiply oth sides of n eqution y ny epression we choose, we choose to ultiply y the LCD. This tctic eliintes the frctions in one step nd tht is good. An eqution with rtionl epressions Solve 6. The LCD for the denointors, 6, nd is : 6 6 Multiply ech side y. Distriutive property Divide out the coon fctors. 0 Check in the originl eqution. The solution set is.
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