Understanding Rational Expressions 1

Transcription

1 Understnding Rtionl Epressions

2 Wht Is This Module Aout? Rtionl epressions re concepts similr to frctions. In this module, ou will lern how to simplif rtionl epressions nd how to perform opertions involving them. Knowing the concepts which re discussed in this module is importnt. The knowledge nd skills tht ou will gin here will help ou solve rel-life prolems which involve rtionl epressions. This module contins si lessons. These re: Lesson Simplifing Rtionl Epressions Using the Lws of Eponents Lesson Adding Similr Rtionl Epressions Lesson Sutrcting Similr Rtionl Epressions Lesson Simplifing Rtionl Epressions Using Fctoring Lesson Multipling Rtionl Epressions Lesson Dividing Rtionl Epressions Wht Will You Lern From This Module? After reding this module, ou should e le to: reduce the given rtionl lgeric epressions to lowest terms; dd similr lgeric epressions; sutrct similr lgeric epressions; multipl the given rtionl lgeric epressions; nd divide the lgeric epressions. Wit! Before ou stud this module, mke sure tht ou hve lred red the module Studing Polnomils.

3 Before ou proceed to Lesson, tr to solve the items given in Let s See Wht You Alred Know to determine wht ou lred know out the topics to e discussed in this module. Let s See Wht You Alred Know A. Reduce the following rtionl epressions to their simplest forms z. 00 z.. B. Add the given rtionl epressions... C. Sutrct the given rtionl epressions.. 0. D. Multipl the given rtionl epressions. c. c.

4 . E. Divide the following rtionl epressions Well, how ws it? Do ou think ou fred well? Compre our nswers with those found in the Answer Ke on pges nd to find out. If ll our nswers re correct, ver good! This shows tht ou lred know much out the topics in this module. You m still stud the module to review wht ou lred know. Who knows, ou might lern few more new things s well. If ou got low score, don t feel d. This mens tht this module is for ou. It will help ou to understnd some importnt concepts tht ou cn ppl in our dil life. If ou stud this module crefull, ou will lern the nswers to ll the items in the test nd lot more! Are ou red? You m go now to the net pge to egin Lesson.

5 LESSON Simplifing Rtionl Epressions Using the Lws of Eponents This lesson will introduce rtionl epressions to ou. It will discuss when rtionl epression is in its lowest term nd how to reduce rtionl epression to its lowest term. The knowledge nd skills ou will gin from this lesson re importnt. The will help ou understnd nd solve rel-life prolems which involve simplifing rtionl epressions. Let s Stud nd Anlze Mother wnts to cover the top of rectngulr tle with new tlecloth. She doesn t know the length of the tlecloth. However, she knows tht the rtio of the re of the old tlecloth is squre meters (m ) to its width, which is meters (m). How long should the tlecloth e? Let us help Mother nlze her prolem. Wht is the re of the tlecloth?. Wht is the width of the tlecloth?. Wht is the rtio of the re of the tlecloth to its width? You re right. The re of the tlecloth is m to the width which is m. Hence, the rtio of the re of the tlecloth to its width is. The rtio is n emple of rtionl epression. A rtionl epression is rtio of two polnomils. Wht does rtionl epression hve?.

6 A rtionl epression hs polnomil in the numertor nd polnomil in the denomintor. In, the numertor is while the denomintor is. Both nd re polnomils. This mens tht rtionl epression is similr to frction. However, the numertor nd the denomintor m not just e rel numers ut polnomils s well. Cn ou give some rtionl epressions? Here re some rtionl epressions:,, nd. Let s Lern is rtionl epression in its simplest form. This mens tht ech term in the polnomil is t its lowest. Other emples of rtionl epressions which re in their simplest forms re,, nd. Wht do ou notice out the numertors nd the denomintors of the rtionl epressions ove? Do the hve common fctors (esides nd )? The numertor nd the denomintor of ech rtionl epression do not hve n common fctor other thn nd. A rtionl epression is in its simplest form if its numertor nd denomintor hve no common fctor ecept nd. Let us recll some lws of eponents for rtionl epressions. For n rel numer not equl to 0, Let us consider the following emples: m n m mn m EXAMPLE EXAMPLE

7 EXAMPLE 0 EXAMPLE To reduce rtionl epression to its simplest form: STEP Appl the lws of eponents for ech vrile. STEP Simplif the result. Let us hve some emples. Let s Tr This EXAMPLE A jeepne runs kilometers (km) in hours (h). Reduce the rte to the lowest term. STEP Appl the lws of eponents to the vrile. Appl the lws of eponents to the vrile. STEP Simplif the result. The rte of the jeepne is. EXAMPLE STEP term. tnks cn e emptied in Appl the lws of eponents to the given vriles. h. Reduce the rte to the lowest

8 STEP Simplif the result. The rte t which the tnks re emptied is. EXAMPLE STEP Cecille cn finish tping words in h. Epress the rte in its simplest form. Appl the lws of eponents to the given vriles. STEP Simplif the result. This mens tht Cecille s rte is. Compre our nswers with mine: or EXAMPLE STEP Mnn cn run z km in z h. Reduce his rte to the lowest term. Appl the lws of eponents to the given vriles. STEP Simplif the result. This mens tht Mnn s rte is.

9 Are our nswers the sme s those written elow? STEP z z z z STEP z z z EXAMPLE 0 Mnn cn pint z houses in z ds. Reduce this rte to the lowest term. z 0 z z z z This mens tht Mnn s rte is. Let s See Wht You Hve Lerned Solve the following prolems.. Dnd cn finish wtering plnts in h. Epress her rte in its simplest form. 0. Conch cn wlk m in minutes. Epress her rte in its simplest form.. At wht rte, in the lowest term, cn Mnuel finish jo if he finishes c items in c hours?

10 0. The re of rectngulr wll decor is p q r sq. ft. Its length is r p q ft. Wht is its width? 0. Your mother cn finish reding t u w pges of ook in t u w minutes. Wht is her rte in its simplest form? Compre our nswers with those found in the Answer Ke on pge. Did ou get everthing right? If ou did, tht s ver good. You understood this lesson well. If ou missed some items, don t worr. Just review this lesson, then proceed to Lesson. Let s Rememer A rtionl epression is the rtio or quotient of two polnomils. It is in its simplest form if the numertor nd the denomintor hve no common fctor ecept nd. Some lws of eponents tht we ppl in reducing rtionl epression to the lowest term re: For n rel numer not equl to zero, () () m n m mn m To reduce frction to the lowest term, we do the following steps: STEP Appl the lws of eponents to the given vriles. STEP Simplif the result.

11 LESSON Adding Similr Rtionl Epressions This lesson will enle ou to dd similr rtionl epressions. The process is similr to tht of dding similr frctions. Knowing how to dd similr rtionl epressions is importnt. Your understnding of this topic will enle ou to solve mthemticl prolems in everd life. Let s Stud nd Anlze Look t the figure elow. Mndo nd Mnn joined volunteers orgniztion to distriute cnned goods mong the flood victims in certin re. red Ligo srdines were given to victims. green Ligo srdines were given to the sme numer of victims. How mn red Ligo srdines were given ech victim? How mn green Ligo srdines were given ech victim? Are our nswers nd? If the re, ver good! Your nswers re rtionl epressions, right? Wht do ou notice out them? Wht do the hve in common? The rtionl epressions hve the sme or common denomintor. Wht is the common denomintor? The common denomintor is. nd re emples of similr rtionl epressions. 0

12 Similr rtionl epressions re rtionl epressions with common denomintor. Suppose ou re sked to find the totl numer of srdines given to ech victim. Wht opertion re ou going to use? You will dd the two rtionl epressions nd to get the totl numer of srdines tht ech victim received. Thus, receive is. This mens tht the totl numer of srdines tht ech victim will. You were le to get our nswer dding the similr rtionl epressions. Let us lern how to dd similr rtionl epressions step step. Wht do ou do to get the numertor? You get the numertor dding the given numertors. Thus, Wht do ou notice out the denomintor? The denomintor of the nswer is the sme s the denomintors of the given epressions. The denomintors of the given rtionl epressions re oth hence, the denomintor of the nswer is lso. How do ou get the finl nswer? The finl nswer is the rtio of the numertor to the denomintor which is tht ech victim received Ligo srdines.. This mens Let s Tr This EXAMPLE Am inherited pieces of lnd. One piece of lnd hs n re of m, which she divided equll mong her children. The other hs n re of m, which she lso divided equll mong her children. Wht is the re of the piece of lnd tht Am gve to ech of her children? STEP Find the two rtionl epressions. One piece of lnd hs n re of m distriuted equll mong Am s children. Hence, the first rtionl epression is.

13 The other piece of lnd hs n re of m lso distriuted equll mong Am s children. Therefore, the second rtionl epression is. STEP STEP Add the numertors of the two epressions. 0 Cop the common denomintor nd epress the nswer in its simplest form. ()() 0 This mens tht ech child got m of lnd. EXAMPLE One cr trvels km in h. Another cr trvels in km in h. Wht is the sum of the speeds of the two crs? STEP STEP Find the two rtionl epressions. One cr trvels km in h. Hence, the speed of the first cr is. The other cr trvels km in h. Therefore, the speed of the second cr is. Add the numertors of the two epressions. ( ) ( ) STEP Cop the common denomintor nd epress the nswer in its simplest form. ( ) ( ) This mens tht sum of the two epressions is. Compre our nswer with mine. STEP STEP ( ) ( ) STEP Cn we solve similr prolem using shortcut method?

14 EXAMPLE Lett wlks km in h. Gene wlks 0 km in h. Wht is their comined rte? 0 0 ( )( ) ( )( ) Their comined rte is. EXAMPLE One cndle hs length of cm nd the other, cm. The two cndles urn in h. Tonight, the two cndles re lighted. Wht is the sum of the lengths of the cndles tht hd urned fter one hour? Did ou nswer ( )( )? If ou did, tht s ver good! Let s See Wht You Hve Lerned Answer the following prolems.. Mrs. Grci, dressmker, ought m of red cloth tht she would mke into gown for customer. The net d, her customer gve her m of lck cloth to dorn the gown. If she used ll the cloth, how mn meters did she consume in ll?

15 . A met vendor sold kg of met to Mrs. Anton nd Chvez. How mn kg of met did he sell in ll? kg of met to Mrs.. During hrvest seson, Edwin hrvested scks of mngoes nd scks of vocdos from his frm. Wht is the totl numer of scks of fruits tht he hrvested?. Mnong Lndo hd two scks of rice which hve different weights. One sck weighed kg while the other weighed weight of the two scks of rice? kg. Wht ws the totl

16 . Lou works s cook in resturnt. Tod, she cooked kg of tocino for rekfst nd cook in ll? kg of tocino for dinner. How mn kg of tocino did she Compre our nswers with those found in the Answer Ke on pges nd. Did ou get everthing right? If ou did, tht s ver good! You lerned lot from this lesson. If ou missed some items, tht s ok. Review this lesson, then proceed to Lesson. Let s Rememer Similr rtionl epressions re rtionl epressions with the sme or common denomintor. We follow these steps in dding similr epressions: STEP STEP STEP Find the rtionl epressions. Add the numertors of the rtionl epressions. Cop the common denomintor nd epress the nswer in its simplest form.

17 LESSON Sutrcting Similr Rtionl Epressions This lesson is similr to Lesson. In this lesson, we will e deling with similr rtionl epressions gin. This time, however, ou will lern to sutrct these rtionl epressions. Let s Stud nd Anlze Look t the figure given elow. Mel jogs km in h. Jen jogs fster thn Mel; she jogs km in h. Let us nlze the given sitution. Wht is Jen s jogging speed? Jen s speed is. Wht is Mel s speed? Mel s jogging speed is. Are the two rtionl epressions similr? Yes, the re. If ou re sked to determine how much fster thn Mel Jen jogs, wht mthemticl opertion will ou perform?

18 You will perform sutrction in order to nswer the question. You ll need to sutrct Mel s jogging speed from tht of Jen. We sutrct Mel s rte from Jen s rte. Jen s rte Mel s rte How do ou sutrct similr rtionl epressions? First, let us sutrct the numertors. ( ) Second, we cop the common denomintor. Thus, This mens tht Jen jogs km/h fster thn Mel does. Let s Tr This EXAMPLE STEP Mother wnts to u two rectngulr pieces of cloth to mke ed sheets. One piece of cloth should hve n re of m nd the other, m. The length of the two ed sheets is. How much wider is one ed sheet thn the other? Determine the two rtionl epressions. One ed sheet hs n re of nd length of. Hence, the width of one ed sheet is. The other ed sheet hs n re of width of the other ed sheet is. nd length of. Therefore, the STEP Sutrct the numertors of the two rtionl epressions. The numertor of the first epression is while the numertor of the second is.. Thus we hve

19 STEP Cop the common denomintor nd reduce the nswer to the lowest term. ( )( ) This mens tht one ed sheet is m wider thn the other. EXAMPLE STEP STEP A lut vendor plced lut in skets. He plced peno in skets. How mn more lut thn peno were plced in ech sket? Determine the two rtionl epressions. skets. Thus, the first epression is lut were plced in. On the other hnd, peno were plced in skets. Thus, the second epression is. Sutrct the numertors of the two rtionl epressions. The numertor of the first epression is second is while the numertor of the. Thus, we hve ( ). STEP EXAMPLE STEP Cop the common denomintor nd reduce the nswer to the lowest term. ( )( ) ( )( ) This mens tht in ech sket, there re more lut thn peno. Henr wnted to distriute his pieces of lnd mong his grndchildren. One is commercil nd the other is residentil. The commercil piece of lnd hs n re of m, which Henr divided equll mong his grndchildren. The residentil lnd hs n re of m which Henr lso divided equll mong his grndchildren. How much lrger is the commercil piece of lnd thn the residentil one tht ech grndchild received? Determine the two rtionl epressions. The commercil lnd hs n re of m which Henr divided equll mong his grndchildren. Thus the first rtionl epression is. The residentil lnd hs n re of m, which he lso divided equll mong his grndchildren. This mens tht the second rtionl epression is.

20 STEP STEP Sutrct the numertors of the two rtionl epressions. The numertor of the first epression is while the numertor of the second is. Thus we hve Cop the common denomintor nd reduce the nswer to lowest term. This mens tht ech grndchild received m more of the commercil piece of lnd thn the residentil piece. Did ou nswer the following? STEP STEP ;; first rtionl epression second rtionl epression STEP ( )( ) ( )( ) EXAMPLE In going to reltive s residence, Angel s cr trvels km in 0 h. Her sister s cr trvels slower km in 0 h. How much fster thn her sister s cr does Angel s cr trvel? 0 0 ( )( ) ( )( ) ( ) Angel s cr trvels km/h fster thn her sister s cr.

21 Let s See Wht You Hve Lerned Solve the following:. Aling Susn ought 0 ottles of vinegr to cook dinugun for her resturnt. However, she used ottles for pksiw n ngus. How mn ottles of vinegr were left for dinugun?. Mother noticed tht she hs scks of rice to deliver to Aling Jun. However, Aling Susn requested her to deliver for Aling Jun? scks of rice. How mn scks were left. A dressmker hs m of cloth. After she cut how mn meters of cloth were left? m from the sid cloth, 0

22 . Fther gve kg of rice to his sister nd kg of rice to his oungest rother. How mn more kg of rice did he give to his sister compred to wht he gve his rother?. Mrs. Fortes ought dozens of eggs nd sold of eggs were unsold? dozens. How mn dozens Compre our nswers with those found in the Answer Ke on pge. If ll our nswers re correct, tht s ver good. If ou missed some items, tht s ok. Just review this lesson, then proceed to Lesson. Let s Rememer In sutrcting rtionl similr epressions, we do the following: STEP STEP STEP Determine the two rtionl epressions. Sutrct the numertors of the two rtionl epressions. Cop the common denomintor nd reduce the nswer to the lowest term.

23 LESSON Simplifing Rtionl Epressions Using Fctoring This lesson will tech ou how to simplif other rtionl epressions which cnnot e simplified or reduced to the lowest terms using the lws of eponents. You will find out how to simplif rtionl epressions using fctoring. Let s Stud nd Anlze Jke cn run 0 km in 0 h. Epress his speed in its simplest form. Let us nlze the given sitution. How fr cn Jke run in the given time? In 0 h. Jke cn cover 0 km. Therefore, wht is Jke s speed? Jke s speed is How will ou epress this in its simplest form? Let us nlze the following steps. STEP Fctor the numertor. ( ) STEP STEP Fctor the denomintor. ( )( ) 0 Cncel the common fctor in the numertor nd in the denomintor. 0 0 ( ) ( )( ) Jke s speed cn thus e epressed s.

24 Let s Tr This Let us consider more emples. EXAMPLE STEP STEP STEP Mnn cn pint houses in ds. Reduce Mnn s speed to the lowest term. Fctor the numertor. is lred in its simplest form. Fctor the denomintor. ( )( ) Cncel the common fctor in the numertor nd in the denomintor. ( )( ) Mnn s speed cn thus e epressed s. EXAMPLE STEP STEP STEP At wht speed (in the lowest term) cn Mel finish jo if she works on items in h? Fctor the numertor. ( )( ) Fctor the denomintor. ( )( ) Cncel the common fctor in the numertor nd in the denomintor. Mel s speed is. Did ou get the following nswers? STEP ( )( ) STEP ( )( ) STEP ( )( ) ( )( ) Mel s speed is.

25 EXAMPLE The re of door is sq. ft. Its length is ft. Wht is its width? STEP STEP STEP Fctor the numertor. ( )( ) Fctor the denomintor. ( )( ) Cncel the common fctor in the numertor nd in the denomintor. This mens tht the width of the door is. Are our nswers the sme s these? STEP ( )( ) STEP ( )( ) STEP ( )( ) ( )( ) This mens tht the width is. Let s Tr This EXAMPLE Meruelle cn polish crs in h. Wht is his rte in the lowest term? M nswer: ( )( ) ( )( ) Meruelle s speed is. EXAMPLE Your nswer: Mind cn red pges in speed in the lowest term. h. Epress Mind s reding

26 Compre our nswer with mine: ( )( ) ( )( ) Mind s reding speed is. Let s See Wht You Hve Lerned Solve the following prolems.. A us runs 0 m in 0 0 h. Epress the speed of the us in its simplest form.. Wht is Pul s reding speed if he cn finish reding ( ) pges in ( ) minutes?. The re of the surfce of rectngulr tle is sq. ft. Its length is ft. Wht is its width?. Michel cn finish pinting rooms in h. Wht is his speed in the simplest form?

27 . tnks cn e filled in 0 h. Reduce the rte to the lowest term. Compre our nswers with those found in the Answer Ke on pges nd. If ou got everthing right, tht s ver good! If ou did not, don t worr. Just review this lesson, then proceed to the net lesson. Let s Rememer To reduce frction to its simplest form fctoring, follow the steps elow: STEP STEP STEP Fctor the numertor. Fctor the denomintor. Cncel the common fctor in the numertor nd in the denomintor.

28 LESSON Multipling Rtionl Epressions This lesson will tech ou how to multipl rtionl epressions. The process of multipling rtionl epressions is just the sme s tht of multipling frctions. Let s Stud nd Anlze Merc ought oes of iscuits in grocer. Ech o contins 0 How mn iscuits did she u in ll? iscuits. Let us nlze the given sitution. How mn oes of iscuits did Merc u? Merc ought oes of iscuits. How mn iscuits re there in ech o? There re 0 iscuits in ech o. Wht opertion re ou going to perform on the given rtionl epressions? You will multipl the given rtionl epressions in order to solve the prolem. But how do ou multipl rtionl epressions?

29 STEP Check if n pir of numertor nd denomintor hs common fctor. If it does, use cncelltion. Cncelltion is the process of removing the common fctor/s etween numertor nd denomintor. Looking t nd, ou cn see tht the numertor of the first 0 rtionl epression nd the denomintor of the second rtionl epression hve common fctor. Cncel the common fctor in the numertor of the first epression nd in the denomintor of the the second epression. Thus, 0 STEP Multipl the resulting numertors. From step, ou hve. B multipling the resulting numertors, ou rrive t STEP You lso hve to multipl the resulting denomintors. Then, epress the result s rtionl epression. ( ). Thus, Let s Tr This EXAMPLE STEP Olive plnted seedlings in ech plot of soil in her ckrd. If there re plots in ll, how mn seedlings did Olive plnt? Check for n common fctor. ( )( ) nd ( )( )

30 STEP Multipl the resulting numertors. From step, we hve ( )( ) B multipling the resulting numertors we rrive t ( ) STEP Multipl lso the resulting denomintors. Then, epress the result s rtionl epression..thus,. ( ) EXAMPLE STEP In prepring for irthd prt, Mriss plced glsses on ech tle. There re set in ll? Check for n common fctor. tles in ll. How mn glsses did Mriss ( )( ) ( )( ) ( ) STEP STEP Multipl the resulting numertors. From step, ( ). B multipling the resulting numertors ou will rrive t. Multipl lso the resulting denomintors. Then, epress the result s rtionl epression. The onl fctor in the denomintor is. Finll, we hve. EXAMPLE Cris nd his friends used liters of pint for putting on finishing touches on the interiors of uilding tht the constructed. If there re floors in ll, how mn liters of pint did the consume? Your nswers: STEP

31 0 STEP STEP Compre our nswers with mine: STEP STEP STEP Thus, Do ou know how to multipl rtionl epressions without nming ll the steps in the process? Consider the emple elow. EXAMPLE Suppose certin usinessmn owns triccles in his town. Ech triccle hs oundr of pesos in d. How much is the usinessmn s gross income per d? (

32 Let s See Wht You Hve Lerned Answer the following. r s. Lor plced cups on ech shelf of the kitchen cinet. If there re 0r s shelves in ll, how mn cups did Lor plce in the cinet?. Germn ought scks of cement for uilding his house. If ech sck contins kg of cement, how mn kg of cement did Germn u in ll?. A deliver o delivered gllons of ice crem to ech of the in Luzon. How mn gllons of ice crem were delivered in ll? stores

33 . Bgs re on sle in deprtment store. There re rnches of this deprtment store in Metro Mnil. Ech rnch sold gs in one week. How mn gs in ll were sold the deprtment store in one week?. Ech nk in certin town hs pproimtel customers. There re nks in ll in tht re. Wht is the pproimte totl numer of customers if ech customer trnscts with one nk onl? Compre our nswers with those found in the Answer Ke on pge. If our nswers re ll correct, tht s ver good. If not, tht s ok. Review this lesson then move on to Lesson. Let s Rememer We use the following steps in multipling rtionl epressions: STEP STEP STEP Check for n common fctor. If there re n, use cncelltion. Multipl the resulting numertors. Multipl lso the resulting denomintors. Then epress the result s rtionl epression.

34 LESSON Dividing Rtionl Epressions This lesson will discuss how to divide rtionl epressions. Your knowledge in multipling rtionl epressions will e used here. The method of cncelltion is ver necessr in this lesson. It is importnt to know how rtionl epressions re divided. There re mn rel-life prolems which involve dividing rtionl epressions. Some of these prolems re discussed here. Let s Stud nd Anlze Giselle is mking pom-poms for cheering competition she nd her friends will prticipte in. She needs of meter for ech strip of crepe pper. How mn strips cn she cut from meters of crepe pper? Let us nlze the given prolem. How mn meters re needed for ech strip of crepe pper? for ech strip of crepe pper. of meter is needed How mn meters of crepe pper re ville? There re meters ville.

35 Wht is sked for in the prolem? You re sked for the numer of strips of crepe pper tht cn e mde from meters. Wht opertion re ou going to use? If ou nswered division, ver good. You will then hve. But how do ou divide rtionl epressions? First, recll tht in division, The dividend is the numer to e divided. The divisor is the numer tht divides the dividend. Of the two rtionl epressions ove, is the dividend nd is the divisor. Do ou know wht reciprocl is? A reciprocl is the inverse of rtionl epression. Therefore, the reciprocl of is. Now ou re red to divide. STEP Get the reciprocl of the divisor. In, the divisor is. Its reciprocl is. STEP Multipl the reciprocl of the divisor the dividend.

36 STEP Follow the steps in multipling rtionl epressions. B multipling the results, ou rrive t This mens tht there will e strips of crepe pper in ll. Let s Tr This EXAMPLE Mng Hector ought sntol for his children. He hs children nd he wnted to divide the sntol equll mong his children. How mn sntol would e given to ech child? STEP Get the reciprocl of the divisor. In the given prolem, The divisor is. Its reciprocl is. STEP Multipl the reciprocl of the divisor the dividend. Thus, STEP Follow the steps in multipling rtionl epressions. We see tht in the denomintor. Similrl, in, the numertor.

37 Thus, ( ) ( ). B multipling the results, ou rrive t. This mens tht Now tr solving the following: sntol will e given to ech child. EXAMPLE Your nswers: STEP Lito nd his friends need to pck ll the cnned goods in oes. Ech o should hve the sme numer of cnned goods. How mn cnned goods should there e in ech o? STEP STEP Compre our nswers with mine. STEP Get the reciprocl of the divisor. In the given prolem,. The divisor is. Its reciprocl is.

38 STEP Multipl the reciprocl of the divisor the dividend. STEP Follow the steps in multipling rtionl epressions:. Determine the fctors of the first term.. Determine the fctors of the second term. c. Use the process of cncelltion. d. Multipl the remining terms. Cn ou now divide rtionl epressions without nming ll the steps in the process? Look t the following prolem: EXAMPLE Aling Luis ought meters of cloth for the gowns of the sponsors in her dughter s wedding. If there re sponsors, how mn meters of cloth should ech sponsor get? M nswer: ( Thus, ech sponsor will get meters of cloth.

39 Let s See Wht You Hve Lerned Solve the following prolems:. Aling Soledd ought cups of intog for her children. She hs children c c nd she wnted to divide the intog equll mong her children. How mn cups should ech child receive?. A pinter used liters of pint in pinting houses in sudivision. If he used the sme mount of pint for ech house, how much pint did he consume per house?. Cnned goods will e distriuted in remote re. There re cnned goods to e distriuted equll mong fmilies. How mn goods will e given to ech fmil?

40 . Mrth ought eggs to prepre leche fln for her son s seventh irthd. If ech continer of leche fln requires eggs, how mn continers will e used?. Susn wishes to pl different pino pieces in mn pieces should she pl ech d? ds. How Compre our nswers with those found in the Answer Ke on pges nd. Did ou get everthing right? If ou did, congrtultions. If ou did not, don t worr. Just review this lesson. Let s Rememer In division, the numer we divide is clled the dividend. The numer tht divides is clled the divisor. The reciprocl of rtionl epression is its inverse. In dividing rtionl epressions, follow these steps: STEP STEP STEP Get the reciprocl of the divisor. Multipl the reciprocl of the divisor the dividend. Follow the steps in multipling rtionl epressions. We will now check if ou rell understnd ll the topics tht we discussed in this module.

41 0 Wht Hve You Lerned? A. Epress ech of the following in its simplest form.... B. Add the given epressions C. Sutrct the second epression from the first.... D. Multipl the following epressions....

42 E. Divide the first epression the second r t r t Compre our nswers with those found in the Answer Ke on pges nd. If ou got: Congrtultions! You did ver good jo. Ver good. Go ck to the prts tht ou did not get right. 0 Good. Red gin the discussion which ou did not understnd well. 0 You should red this module gin.

43 Answer Ke A. Let s See Wht You Alred Know (pges ) A z z z z.. B... C.. ( ( D.. c c c.. c

44 E.. ( ( or ( B. Lesson Let s See Wht You Hve Lerned (pges ).. 0. c c c c. 0 r q p r q p r q p r q p. u t u t w u t w u t 0 C. Lesson Let s See Wht You Hve Lerned (pges )..

45 ... D. Lesson Let s See Wht You Hve Lerned (pges 0 ). 0. (.. ( (. E. Lesson Let s See Wht You Hve Lerned (pges )

46 . 0 F. Lesson Let s See Wht You Hve Lerned (pges ). 0 s s s r s r... ( 0. G. Lesson Let s See Wht You Hve Lerned (pges ). c c c c c. (. c

47 .. H. Wht Hve You Lerned? (pges 0 ) A.... B C... ( (

48 . D.... E t r t r t r t r t r t r. ( ( 0 0. References Si, Luc O., et l. st Centur Mthemtics, Second Yer. Quezon Cit: Phoeni Pulishing House, Inc., 000. Cpitulo, F. M. Alger: A Simplified Approch. Mnil: Ntionl Bookstore,.