19 Addition and Subtraction of Fractions

Transcription

1 19 Addition nd Subtrction of Frctions You wlked 1 of mile to school nd then 3 of mile from school to your 5 5 friend s house. How fr did you wlk ltogether? How much frther ws the second wlk thn the first? To solve the first problem, you would dd 1 nd 3. To solve the second problem, you would subtrct 1 from 3. Adding nd subtrcting frctions re the subjects of this section. Addition of Frctions Pictures nd mnipultives help understnd the process of finding the sum of two frctions. Let s consider the question of finding This sum is 5 5 illustrted in two wys in Figure The colored region nd the numberline models both show tht = Figure 19.1 The bove models suggest the following definition of frctions with like denomintors: The sum of two frctions with like denomintors is found by dding the numertors nd dividing the result by the common denomintor. Symboliclly Exmple 19.1 Find the vlue of the sum c + b c = + b. c nd write the nswer in simplest form. 1

2 Solution. By the definition bove we hve = = 8 = 1 2 = 1 2. Next we consider dding two frctions with unlike denomintors. This is done by rewriting the frctions with common denomintor. The common denomintor is referred to s the lest common denomintor nd is nothing else then the lest common multiple of both denomintors. For exmple, the lest common denomintor for 1 nd 2 is the LCM(,3), which is 12. Thus, = = = The procedure just described cn be modeled with frction strips s shown in Figure 19.2 Figure 19.2 The generl rule for dding frctions with unlike denomintors is s follows. To dd + c where b 0, d 0 nd b d : b d (1) Renme ech frction with n equivlent frction using the lest common denomintor, tht is, LCM(,b). (2) Add the new frctions using the ddition rule for like denomintors. Exmple 19.2 Find

3 Solution. The lest common denomintor is LCM(, 6, 3) = 3 = 12. Thus, = = = = = = 9 Exmple 19.3 The sum of whole number nd frction is most often written s mixed number. For exmple, = 2 3. () Express 3 2 s frction. 5 (b) Express 36 s mixed number. 7 Solution. () We hve: = = = (b) We hve 36 7 = = = Exmple 19. Compute Solution. The lest common denomintor is LCM(, 5) = 5 = 20. Thus, = = = = = Properties of Addition of Frctions The following properties of ddition cn be used to simplify computtions. Theorem 19.1 () Closure: The sum of two frctions is frction. (b) Commuttivity: Let nd c be ny frctions. Then b d b + c d = c d + b. (c) Associtivity: Let, c nd e b d f ( b d) + c + e f = ( c b + d + e f be ny frctions. Then ). 3

4 (d) Additive identity: 0 is the dditive identity since for ny frction we b hve b + 0 = b. Proof. We will prove prts () nd (b) nd leve the rest for the reder. () This follows from + c = d+bc nd the fct tht both the numertor b d bd nd denomintor re whole numbers. (b) By the definition of ddition of frctions nd the fct tht ddition nd multipliction of whole numbers re commuttive we hve Prctice Problems + c d+bc = b d bd = bc+d bd = c d + b = cb+d db Problem 19.1 If one of your students wrote = 3, how would you convince him or her 3 7 tht this is incorrect? Problem 19.2 Use the colored region model to illustrte these sums, with the unit given by circulr disc. () (b) Problem 19.3 Find using frction strips. 6 Problem 19. Represent ech of these sums with number-line digrm. () (b) Problem 19.5 Perform the following dditions. Express ech nswer in simplest form. () (b) (c) Problem 19.6 A child thinks = the nswer. 10. Use frction strips to explin why 2 10 cnnot be

5 Problem 19.7 Compute the following without clcultor. () (b) b. Problem 19.8 Compute the following without clcultor. () (b) n 5n Problem 19.9 Compute Problem Solve mentlly: () 1 8 +? = 5 8 (b) x = Problem Nme the property of ddition tht is used to justify ech of the following equtions. () = (b) + 0 = (c) ( 2 + ) = 2 + ( 3 + ) (d) is frction. 5 7 Problem Find the following sums nd express your nswer in simplest form. () (b) ,000 (c) 8 (d) Problem Chnge the following mixed numbers to frctions. () (b) (c) Problem 19.1 Use the properties of frction ddition to clculte ech of the following sums mentlly. () ( 3 + ) (b) (c) ( ( 3 5 8) )

6 Problem Chnge the following frctions to mixed numbers. () 35 3 (b) 9 6 (c) 17 5 Problem (1) Chnge ech of the following to mixed numbers: () 56 (b) (2) Chnge ech of the following to frction of the form where nd b b re whole numbers: () 6 3 (b) Problem Plce the numbers 2, 5, 6, nd 8 in the following boxes to mke the eqution true: Problem A clerk sold three pieces of one type of ribbon to different customers. One piece ws 1 yrd long, nother 2 3yd long, nd the third ws 3 1 yd. Wht 3 2 ws the totl length of tht type of ribbon sold? Problem Krl wnts to fertilize his 6 cres. If it tkes cre, how much fertilizer does he need to buy? bgs of fertilizer for ech Subtrction of Frctions You wlked 3 of mile to school nd then 7 of mile from school to your 6 6 friend s house. How much frther ws the second wlk thn the first? This problem requires subtrcting frctions with like denomintors, i.e Figure 19.3 shows how the tke-wy, mesurement, nd missing-ddend models of the subtrction opertion cn be illustrted with colored regions, the number line, nd frction strips. In ech cse we see tht 7 3 =

7 Figure 19.3 In generl, subtrction of frctions with like denomintors is determined s follows: b c b = c, c. b Now, to subtrct frctions with unlike denomintors we proceed s follows: Suppose tht c, i.e. d bc. Then b d c = d bc b d bd bd = d bc bd We summrize this result in the following theorem. Theorem 19.2 If b nd c d re ny frctions with b c d then b c d = d bc bd. Exmple 19.5 Find ech difference. () (b)

8 Solution. () Since LCM(,8) = 8 we must hve = = 3 8. (b) Since LCM(3,) = 12 we must hve Prctice Problems = (5 + 1) (2 + 3) 3 3 = ( 70 + ) ( = 7 33 = 31 = Problem Use frction strips, colored regions, nd number-line models to illustrte Problem Compute these differences, expressing ech nswer in simplest form. () (b) (c) Problem () Find the lest common denomintor of (b) Compute Problem Compute the following without clcultor. () 5 1 (b) c c Problem 19.2 Compute Problem Compute Problem Solve mentlly: ? = ) nd

9 Problem Fill in ech squre with either + sign or - sign to complete ech eqution correctly. Problem On number-line, demonstrte the following differences using the tke-wy pproch. () 5 1 (b) Problem Perform the following subtrctions. () 9 5 (b) 3 (c) Problem Which of the following properties hold for frction subtrction? () Closure (b) Commuttive (c) Associtive (d) Identity Problem Rfel te one-fourth of pizz nd Rocco te one-third of it. Wht frction of the pizz did they et? Problem You plnned to work on project for bout 1 hours tody. If you hve been 2 working on it for 1 3 hours, how much more time will it tke? Problem Mrtin bought 8 3yd of fbric. He wnts to mke skirt using 1 7 yd, pnts 8 using 2 3yd, nd vest using 1 2 yd. How much fbric will be left over? 8 3 Problem 19.3 A recipe requires 3 1c of milk. Don put in 1 3 c nd emptied the continer. 2 How much more milk does he need to put in? Problem A clss consists of 2 freshmen, 1 5 of the clss is seniors? sophomore, nd 1 10 juniors. Wht frction 9

10 Problem Slly, her brother, nd nother prtner own pizz resturnt. If Slly owns 1 nd her brother owns 1 of the resturnt, wht prt does the third prtner 3 own? 10