12-B FRACTIONS AND DECIMALS

Transcription

1 -B Frctions nd Decimls. () If ll four integers were negtive, their product would be positive, nd so could not equl one of them. If ll four integers were positive, their product would be much greter thn ny of them (even = ). Therefore, the integers must include 0, in which cse their product is 0. The lrgest set of four consecutive integers tht includes 0 is 0,,,.. () Since = ( x ) y = xy, then xy =. The only positive integers whose product is re nd. Their verge is + =. -B FRACTIONS AND DECIMALS Severl questions on the SAT involve frctions nd/or decimls. In this section we will review ll of the importnt fcts on these topics tht you need to know for the SAT. Even if you re using clcultor with frction cpbilities, it is essentil tht you review ll of this mteril thoroughly. (See Chpter for discussion of clcultors tht cn perform opertions with frctions.) When whole is divided into n equl prts, ech prt is clled one-nth of the whole, written s. For exmple, n if pizz is cut (divided ) into eight equl slices, ech slice is one-eighth of the pizz; dy is divided into equl hours, so n hour is one-twenty-fourth of dy; nd n inch is one-twelfth of foot. If Sm slept for hours, he slept for five-twenty-fourths of dy. If Tom bought eight slices of pizz, he bought eighteighths of pie. If Joe s shelf is 0 inches long, it mesures thirtytwelfths of foot. 0 0 Numbers such s,, nd, in which one integer is written over second integer, re clled frctions. The center line is the frction br. The number bove the br is clled the numertor, nd the number below the br is the denomintor. CAUTION: The denomintor of frction cn never be 0. A frction such s, in which the numertor is less thn the denomintor, is clled proper frction. Its vlue is less thn. 0 A frction such s, in which the numertor is more thn the denomintor, is clled n improper frction. Its vlue is greter thn. A frction such s, in which the numertor nd denomintor re the sme, is lso n improper frction, but it is equl to. It is useful to think of the frction br s symbol for division. If three pizzs re divided eqully mong eight people, ech person gets of pizz. If you ctully use your clcultor to divide by, you get = 0.. Key Fct B Every frction, proper or improper, cn be expressed in deciml form (or s whole number) by dividing the numertor by the denomintor. For exmple: 0 = 0. = 0. = 0.6 = 0. 6 = 00 =. = =. 6 Note: Any number beginning with deciml point cn be written with 0 to the left of the deciml point. In fct, some clcultors will express s., wheres others will print 0.. On the SAT, never do long division to convert frction to deciml. Use your clcultor. Unlike the exmples bove, when most frctions re converted to decimls, the division does not terminte fter two, three, or four deciml plces; rther it goes on forever with some set of digits repeting itself. = = 0... = =... On the SAT, you do not need to be concerned with this repetition. On grid-in problems you just enter s much of the number s will fit in the grid; nd on multiple-choice questions, ll numbers written s decimls terminte. Although on the SAT you will hve occsion to convert frctions to decimls (by dividing), you will not hve to convert decimls to frctions.

2 6 Reviewing Mthemtics Compring Frctions nd Decimls Key Fct B To compre two decimls, follow these rules: Whichever number hs the greter number to the left of the deciml point is greter: since >,.00 >.6; nd since > 0,. > 0.. (Recll tht, if deciml hs no number to the left of the deciml point, you my ssume tht 0 is there, so. >.). If the numbers to the left of the deciml point re equl (or if there re no numbers to the left of the deciml point), proceed s follows:. If the numbers do not hve the sme number of digits to the right of the deciml point, dd zeros t the end of the shorter one until the numbers of digits re equl.. Now, compre the numbers, ignoring the deciml point itself. For exmple, to compre. nd., dd 0 t the end of., forming.0. Now, thinking of them s whole numbers, compre the numbers, ignoring the deciml point: Key Fct B 0 >.0 >.. To compre two frctions, use your clcultor to convert them to decimls. Then pply KEY FACT B. This lwys works. For exmple, to compre nd, write = 0... nd = 0.. Since 0. > 0., >. CALCULATOR HINT You cn lwys use your clcultor to compre two numbers: frctions, decimls, or integers. By KEY FACT A, > b mens b is positive, nd < b mens b is negtive. Therefore, to compre two numbers, just subtrct them. For exmple,.. =.00. >.,.. =.0. <., = <, 6 ( ) = 6 >. Key Fct B When compring frctions, there re three situtions in which it is fster not to use your clcultor to convert frctions to decimls (lthough, of course, tht will work).. The frctions hve the sme positive denomintor. Then the frction with the lrger numertor is greter. Just s $ re more thn $, nd books re more thn books, tenths is more thn tenths: > The frctions hve the sme numertor. Then, if the denomintors re positive, the frction with the smller denomintor is greter. If you divide cke into five equl pieces, ech piece is lrger thn piece you would get if you divided the cke into 0 equl pieces: >, nd 0 similrly >. 0. The frctions re so fmilir or esy to work with tht you lredy know the nswer. For exmple, > nd >. 0 Key Fct B KEY FACTS B, B, nd B pply to positive decimls nd frctions. Clerly, ny positive number is greter thn ny negtive number. For negtive decimls nd frctions, use KEY FACT A, which sttes tht, if > b, then < b. > < nd. >.. <. Exmple. Which of the following lists the frctions,,, nd in order from lest to gretest? 0 (A), (B),,,,, 0 0 (C),,, (D),,, 0 0 (E),, 0, Solution. On your clcultor convert ech frction to deciml, writing down the first few deciml plces: = 0.666, = 0.6, = 0.66, nd = It is now esy to order the decimls: 0.6 < 0.66 < 0.60 < The nswer is,,, (B). 0

3 -B Frctions nd Decimls Equivlent Frctions If Bill nd Al shred pizz, nd Al Bill te nd Al te, they hd Al exctly the sme mount of the Bill Al pizz. We express this ide by Al sying tht nd re equivlent frctions: tht is, they hve the exct sme vlue. Note: If you multiply both the numertor nd the denomintor of by, you get ; nd if you divide both the numertor nd the denomintor of by, you get. This illustrtes the next KEY FACT. Key Fct B6 Two frctions re equivlent if multiplying or dividing both the numertor nd the denomintor of the first frction by the sme number gives the second frction. Consider the following two cses.. Are nd equivlent? There is number 0 tht, when multiplied by gives, nd there is number tht, when multiplied by, gives 0. By KEY FACT B6, if these numbers re the sme, the frctions re equivlent. They re the sme number: = nd = 0.. Are nd equivlent? Since =, but, the frctions re not equivlent. Alterntively, =, but. To determine whether two frctions re equivlent, convert them to decimls by dividing. For the frctions to be equivlent, the two quotients must be the sme. Exmple Which of the following is NOT equivlent to? A frction is in lowest terms if no positive integer greter thn is fctor of both the numertor nd the denomintor. For exmple, is in lowest terms, since 0 no integer greter thn is fctor of both nd 0; but is not in lowest terms, since is fctor of both nd. Key Fct B Every frction cn be reduced to lowest terms by dividing the numertor nd the denomintor by their gretest common fctor (GCF). If the GCF is, the frction is lredy in lowest terms. s tht hve frction cpbility either reduce utomticlly or hve key to reduce. On regulr clcultor, see whether some prime cn divide evenly into both the numertor nd the denomintor. On the SAT, if none of,,,, or works, the frction cnnot be reduced. Helpful Hints Keep these two fcts in mind:. On grid-in problems you should never reduce frction, such s, tht fits in grid; just enter it: 6 /. If frction cn t fit, such s, don t 6 try to reduce it; use your clcultor to convert it into 6 deciml: =..., nd enter.. 6 You should reduce frction only if you re positive you cn do so in or seconds (fster thn you could divide on your clcultor), without mking mistke Exmples of esy reductions re =, =, 00 0 = On multiple-choice questions, on the other hnd, the frction choices re lmost lwys in lowest terms, so you my hve to reduce your nswer to see which choice is correct. If you cn t do this esily in your hed, use your clcultor either to reduce the frction or to convert it to deciml, nd then use your clcultor gin to check the five choices. Solution. Since = 0.6, just check ech choice until you find the one tht is NOT equl to 0.6. Ech 60 0 of,,, nd is equl to 0.6. Only (E) 6 does not equl 0.6 = 06.. Exmple. For ny positive integer n, n! mens the product of ll 6! the integers from to n. Wht is the vlue of?! 6

4 Reviewing Mthemtics Solution. Assume tht you don t see the esy wy to do this. On your clcultor quickly multiply (or use the! key if you hve one): 6! = 6 = 0,! = 6 = 0,0. 0 You re now fced with reducing. Don t do it. 0, 0 0 Use your clcultor to divide: = Now 0, 0 test the choices, strting with C: = 0., which is too lrge. Eliminte C s well s D nd E, which re even lrger, nd try A or B. In fct, = Choice A 6 is correct. Here s the esy solution: 6!! 6 = = =. 6 6 This solution tkes only few seconds, but the clcultor solution is simple enough nd cn surely be done in less thn minute. Arithmetic Opertions with Decimls On the SAT, ll deciml rithmetic (including whole numbers) tht you cn t esily do in your hed should be done on your clcultor. This shortcut sves time nd voids creless errors. If you know tht = nd tht.. =., fine; but if you re not sure, use your clcultor rther thn your pencil. You should even use your clcultor to multiply if there s ny chnce tht you would get 0. insted of 0.0 s the nswer. You should not hve to use your clcultor to multiply or divide ny deciml number by power of 0, becuse multiplying nd dividing by 0 or 00 or 000 is clcution you should be ble to do esily in your hed. Helpful Hint Any whole number cn be treted s deciml: =.0. Key Fct B To multiply ny deciml or whole number by power of 0, move the deciml point s mny plces to the right s there re 0 s in the power of 0, filling in with 0 s if necessry. Key Fct B To divide ny deciml or whole number by power of 0, move the deciml point s mny plces to the left s there re 0 s in the power of 0, filling in with 0 s if necessry. On the SAT, you never hve to round off deciml nswers. On grid-ins just enter the number, putting in s mny digits fter the deciml point s fit. For exmple, enter. s. nd. s.. However, you do hve to know how to round off, becuse occsionlly there is question bout tht procedure. Key Fct B0. 0 =.. 00 =. 000 = 0 0 = 0 00 = 00,000,000 =,000, = = = =. 00 = 0.,000,000 = To round off deciml number to ny plce, follow these rules, which re fully explined with exmples in the tble below. Keep ll of the digits to the left of the specified plce. In tht plce, keep the digit if the next digit is <, nd increse tht digit by if the next digit is. (Note: incresed by is 0: put down the 0 nd crry the.) If there re still digits to the left of the deciml point, chnge them to 0 s nd eliminte the deciml point nd everything tht follows it. If you re t or beyond the deciml point, stop: don t write ny more digits. 6 6

5 -B Frctions nd Decimls For exmple, here is how to round off.6 to ny plce. Round to the Nerest: Procedure Answer thousnd The digit in the thousnds plce 000 is ; since the next digit () is, increse the to ; fill in the plces to the left of the deciml point with 0 s. hundred The digit in the hundreds plce is 00 ; keep everything to the left of it, nd keep the since the next digit () is < ; fill in 0 s to the left of the deciml point. ten The digit in the tens plce is ; 0 keep everything to the left of it, nd increse the to since the next digit () is ; fill in 0 s to the left of the deciml point. one The digit in the ones plce is ; keep everything to the left of it, nd keep the since the next digit () is < ; there re no more plces to the left of the deciml point, so stop. tenth The digit in the tenths plce is ;. keep everything to the left of it, nd increse the to since the next digit () is ; you re beyond the deciml point, so stop. hundredth The digit in the hundredths plce.0 is ; keep everything to the left of it, nd, since the next digit (6) is, increse the to 0; put down the 0 nd crry into the tenths plce: 0. becomes 0.0; since you re beyond the deciml point, stop. Exmple. When,0 is rounded off to the nerest thousnd, how mny digits will be chnged? (A) 0 (B) (C) (D) (E) Solution. When,0 is rounded off to the nerest thousnd, digits re chnged:,000 (D). Arithmetic Opertions with Frctions Key Fct B To multiply two frctions, multiply their numertors nd multiply their denomintors. = =. Key Fct B To multiply frction by ny other number, write tht number s frction whose denomintor is. Tctic π = = π = = B Before multiplying frctions, reduce. You my reduce by dividing ny numertor nd ny denomintor by common fctor. Exmple. Express the product in lowest terms. 6 Solution. If you just multiply the numertors nd denomintors (with clcultor, of course), you get 60, which is nuisnce to reduce. Also, dividing on 6 your clcultor won t help, since your nswer is supposed to be frction in lowest terms. It is better to use TACTIC B nd reduce first: 6 = =. Tctic B When problem requires you to find frction of number, multiply. Exmple 6. If of the 0 sophomores t Adms High School re girls, nd of the girls ply on tem, how mny sophomore girls do NOT ply on tem? Solution. There re 0 = 00 sophomore girls. Of these, 00 = ply on tem. Then, 00 = do not ply on tem. How should you multiply mentlly, you should: 0 0 0? If you cn do this = 00. π

6 0 Reviewing Mthemtics The next step, however, requires you to multiply by 00, nd more likely thn not you don t immeditely see tht 00 divided by is or tht times equls : =. 00 For ny step tht you cn t do instntly, you should use your clcultor: CALCULATOR HINT ( ) 0 ( ) =. If you re going to use your clcultor on problem, don t bother reducing nything. Given the choice of multiplying 0 or 0, you would prefer the second option, but for your clcultor the first one is just s esy. The reciprocl of ny nonzero number, x, is the number b. The reciprocl of the frction is the frction. x b Key Fct B To divide ny number by frction, multiply the number by the reciprocl of the frction = = 0 = = Helpful Hint Even if you hve clcultor with frction cpbility, be sure to review the rules in KEY FACTS B B. Some clcultions re esier without clcultor. Exmple. In the met deprtment of supermrket, 00 pounds of chopped met ws divided into pckges, ech of which weighed pound. How mny pckges were there? Solution = =. Key Fct B To dd or subtrct frctions with the sme denomintor, dd or subtrct the numertors nd keep the denomintor. + = = = nd = =. 0 To dd or subtrct frctions with different denomintors, first rewrite the frctions s equivlent frctions with the sme denomintor. + = + =. 6 Note: The esiest denomintor to find is the product of the denomintors (6 =, in this exmple), but the best denomintor to use is the lest common denomintor, which is the lest common multiple (LCM) of the denomintors ( in this cse). Using the lest common denomintor minimizes the mount of reducing tht is necessry to express the nswer in lowest terms. Key Fct B If b is the frction of the whole tht stisfies some property, then does not stisfy it. is the frction of the whole tht Exmple. In jr, of the mrbles re red, re white, nd re blue. Wht frction of the mrbles re neither red, white, nor blue? Solution. The red, white, nd blue mrbles constitute of the totl, so of the mrbles re neither red, white, nor blue. Exmple. Ali te of cke nd Json te of it. Wht frction of the cke ws still uneten? Exmple 0. Ali te of cke nd Json te of wht ws left. Wht frction of the cke ws still uneten? CAUTION: Be sure to red questions crefully. In Exmple, Json te of the cke. In Exmple 0, however, he te only of the tht ws left fter Ali hd her piece. He te of the cke. b = = 0 0 = 0 0 = 0 = 6 0

7 -B Frctions nd Decimls Solution. + = + = of the cke ws eten, nd = ws uneten. Solution 0. + = + = = of the cke ws eten, nd the other ws uneten. Arithmetic Opertions with Mixed Numbers A mixed number is number, such s, tht consists of n integer followed by frction. The mixed number is n bbrevition for the sum of the integer nd the frction; so is n bbrevition for +. Every mixed number cn be written s n improper frction, nd every improper frction cn be written s mixed number: 6 = + = + = + = 6 nd = + = + =. Key Fct B6 To write mixed number s n improper frction, or n improper frction s mixed number, follow these rules:. To write mixed number s n improper frction, multiply the whole number () by the denomintor (), dd the numertor (), nd write + the sum over the denomintor (): =.. To write n improper frction s mixed number, divide the numertor by the denomintor; the quotient () is the whole number. Plce the reminder () over the denomintor to form the frctionl prt :. CAUTION: You cn never grid in mixed number. You must chnge it to n improper frction or deciml. Key Fct B To dd mixed numbers, dd the integers nd lso dd the frctions. + = ( + ) + + = + + = + =. + = ( + ) + + = + + = + = + = + + = Key Fct B To subtrct mixed numbers, subtrct the integers nd lso subtrct the frctions. If, however, the frction in the second number is greter thn the frction in the first number, you first hve to borrow from the integer prt. For exmple, since >, you cn t subtrct until you borrow from the : = + = ( + ) + = + + = + Now, you hve = = ( ) + = + =. Key Fct B To multiply or divide mixed numbers, chnge them to improper frctions. CAUTION: Be wre tht is not ; rther: = 6 = =. = + = + = + = 6.. All rithmetic opertions on mixed numbers cn be done directly on clcultors with frction cpbility; there is no need to chnge the mixed numbers to improper frctions or to borrow.

8 Reviewing Mthemtics Complex Frctions + 6 A complex frction is frction, such s, tht hs one or more frctions in its numertor or denomintor or both. 6 nd : or write () () = =, Key Fct B0 There re two wys to simplify complex frction:. Multiply every term in the numertor nd denomintor by the lest common multiple of ll the denomintors tht pper in the frction.. Simplify the numertor nd the denomintor, nd divide. To simplify + 6, multiply ech term by, the LCM of = = =. 6 Remember tht, on the SAT, if you ever get stuck on frction problem, you cn lwys convert the frctions to decimls nd do ll the work on your clcultor. Exercises on Frctions nd Decimls Multiple-Choice Questions. A French clss hs boys nd girls. Boys re wht frction of the clss?. For how mny integers,, between 0 nd 0 is it true tht,, nd re ll in lowest terms? (A) (B) (C) (D) (E). Wht is the vlue of the product? Billy won some goldfish t the stte fir. During the first week, of them died; nd during the second week, of those still live t the end of the first week died. Wht frction of the originl goldfish were still live fter weeks? is the verge (rithmetic men) of nd wht number? If of number is, wht is of tht number? (A) 6 (B) (C) (D) (E). Wht frctionl prt of week is hours?. of is equl to of wht number? (A) (B) (C) (D) (E). Which of the following is less thn?

9 Exercises on Frctions nd Decimls 0. Which of the following is (re) greter thn x when x =? I. II. III. x x + x x + x (A) I only (B) I nd II only (C) I nd III only (D) II nd III only (E) I, II, nd III. Which of the following sttements is true? (A) < < (C) < < (E) < < (B) < < (D). If = 0., which of the following is (re) less thn? I. II. III. (A) None (B) I only (C) II only (D) III only (E) II nd III only. Let, b, c, nd d be the result of rounding off.6 to the nerest thousnd, hundred, ten, nd one, respectively. Which of the following sttements is true? (A) d < c < b < (B) d < c < < b (C) < d < c < b (D) c < d < b < (E) < c < d < b. For wht vlue of x does (. 6)(. ) = (0.6)(.)? x (A) 0.00 (B) 0.0 (C) 0. (D) 0 (E) 00. For the finl step in clcultion, Pul ccidentlly divided by 000 insted of multiplying by 000. Wht should he do to his nswer to correct it? (A) Multiply it by 000. (B) Multiply it by 00,000. (C) Multiply it by,000,000. (D) Squre it. (E) Double it. Grid-in Questions 6. One dy t Centrl High School, of the students were bsent, nd of those present went on field trip. If the number of students stying in school ws 0, how mny students re enrolled t Centrl High?. Wht is possible vlue of x if < <? x. Wht is the vlue of + +?

10 Reviewing Mthemtics. If = nd b =, wht is the vlue of? b 0. If A = {,, }, B = {,, }, nd C is the set consisting of ll the frctions whose numertors re in A nd whose denomintors re in B, wht is the product of ll of the numbers in C? Answer Key. A. C. A. C. B 6. E. E. A. D 0. B. B. C. E. D. C / or < x <.6 0. / or. / 6 or