Solution Manual for Mathematics All Around, 6th Edition

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INSTRUCTOR’SSOLUTIONSMANUALJAMESLAPPMATHEMATICSALLAROUNDSIXTHEDITIONThomas PirnotKutztown University of Pennsylvaniain collaboration withMargaret H.MooreUniversity of Southern Maine

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CONTENTSChapter 1Problem Solving:Strategies and Principles1.1Problem Solving.........................................................................................................................11.2Inductive and Deductive Reasoning.........................................................................................131.3Estimation................................................................................................................................18Chapter Review Exercises......................................................................................................................19Chapter Test...........................................................................................................................................21Chapter2Set Theory: Using Mathematics to Classify Objects2.1The Languageof Sets...............................................................................................................232.2Comparing Sets........................................................................................................................252.3Set Operations..........................................................................................................................282.4Survey Problems......................................................................................................................312.5Looking Deeper: Infinite Sets..................................................................................................35Chapter Review Exercises......................................................................................................................38Chapter Test...........................................................................................................................................40Chapter3Logic: The Study of What’s True or False or SomewhereinBetween3.1Statements, Connectives, and Quantifiers................................................................................433.2Truth Tables.............................................................................................................................443.3The Conditional and Biconditional..........................................................................................513.4Verifying Arguments...............................................................................................................553.5Using Euler Diagrams to Verify Syllogisms............................................................................633.6Looking Deeper: Fuzzy Logic..................................................................................................66Chapter Review Exercises......................................................................................................................68Chapter Test...........................................................................................................................................70Chapter4Graph Theory(Networks): TheMathematics of Relationships4.1Graphs, Puzzlesand Map Coloring..........................................................................................754.2The Traveling Salesman Problem............................................................................................804.3Directed Graphs.......................................................................................................................874.4Looking Deeper: Scheduling Projects Using Pert....................................................................92Chapter Review Exercises....................................................................................................................103Chapter Test.........................................................................................................................................106Chapter5Numeration Systems: Does ItMatter How We Name Numbers?5.1The Evolution of Numeration Systems..................................................................................1095.2Place Value Systems..............................................................................................................1135.3Calculating in Other Bases.....................................................................................................1225.4Looking Deeper: Modular Systems........................................................................................129Chapter Review Exercises....................................................................................................................135Chapter Test.........................................................................................................................................138Chapter6Number Theory and the Real Number System: Understanding theNumbers All Around Us6.1Number Theory......................................................................................................................1416.2The Integers............................................................................................................................1456.3The Rational Numbers...........................................................................................................1486.4The Real Number System......................................................................................................1586.5Exponents and Scientific Notation.........................................................................................1626.6Looking Deeper: Sequences...................................................................................................168Chapter Review Exercises....................................................................................................................175Chapter Test.........................................................................................................................................177

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Chapter7Algebraic Models:How Do We Approximate Reality?7.1Linear Equations....................................................................................................................1817.2Modeling with Linear Equations............................................................................................1907.3Modeling with Quadratic Equations.......................................................................................1977.4Exponential Equations and Growth........................................................................................2097.5Proportions and Variation......................................................................................................2177.6Modeling with Systems of Linear Equations and Inequalities...............................................2257.7Looking Deeper: Dynamical Systems....................................................................................246Chapter Review Exercises....................................................................................................................250Chapter Test.........................................................................................................................................257Chapter8Consumer Mathematics: The Mathematics of Everyday Life8.1Percent,Taxes, and Inflation..................................................................................................2658.2Interest....................................................................................................................................2698.3Consumer Loans.....................................................................................................................2768.4Annuities................................................................................................................................2848.5Amortized Loans....................................................................................................................2958.6Looking Deeper: Annual Percentage Rate..............................................................................302Chapter Review Exercises....................................................................................................................306Chapter Test.........................................................................................................................................311Chapter9Geometry: Ancientand Modern Mathematics Embrace9.1Lines, Angles, and Circles......................................................................................................3159.2Polygons.................................................................................................................................3209.3Perimeter and Area.................................................................................................................3269.4Volume and Surface Area......................................................................................................3409.5The Metric System and Dimensional Analysis......................................................................3489.6Geometric Symmetry and Tessellations.................................................................................3549.7Looking Deeper: Fractals.......................................................................................................359Chapter Review Exercises....................................................................................................................362Chapter Test.........................................................................................................................................364Chapter10Apportionment: How Do We Measure Fairness?10.1Understanding Apportionment...............................................................................................36910.2The Huntington–Hill Apportionment Principle......................................................................37610.3Other Paradoxes and Apportionment Methods......................................................................38410.4Looking Deeper: Fair Division..............................................................................................404Chapter Review Exercises....................................................................................................................407Chapter Test.........................................................................................................................................411Chapter11Voting: Using Mathematics to Make Choices11.1Voting Methods......................................................................................................................41711.2Defects in Voting Methods....................................................................................................42411.3Weighted Voting Systems......................................................................................................43111.4Looking Deeper: The Shapley–Shubik Index........................................................................446Chapter Review Exercises....................................................................................................................457Chapter Test.........................................................................................................................................463Chapter12Counting: Just How Many Are There?12.1Introduction to Counting Methods.........................................................................................46912.2The Fundamental Counting Principle.....................................................................................47712.3Permutations and Combinations.............................................................................................48012.4Looking Deeper: Counting and Gambling.............................................................................485Chapter Review Exercises....................................................................................................................486Chapter Test.........................................................................................................................................488

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Chapter13Probability: What Are the Chances13.1The Basics of Probability Theory...........................................................................................49113.2Complements and Unions of Events......................................................................................49813.3Conditional Probabilityand Intersections of Events..............................................................50313.4Expected Value......................................................................................................................51213.5Looking Deeper: Binomial Experiments................................................................................516Chapter Review Exercises....................................................................................................................519Chapter Test.........................................................................................................................................521Chapter14Descriptive Statistics:Making Sense of the Data14.1Organizing and Visualizing Data...........................................................................................52514.2Measures of Central Tendency...............................................................................................53014.3Measures of Dispersion..........................................................................................................53714.4The Normal Distribution........................................................................................................55014.5Looking Deeper: Linear Correlation......................................................................................556Chapter Review Exercises....................................................................................................................563Chapter Test.........................................................................................................................................566

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Section 1.1: Problem Solving1Chapter1:Problem Solving: Strategies and PrinciplesSection1.1:Problem Solving1.Drawings may vary.2.Drawings may vary.3.Drawings may vary.4.Drawings may vary.5.Answers may vary.LetHbe the hybrid automobiles,Wbe the windmill turbines, andSbe solar energy.6.Answers may vary.LetTbe Tyrion,JbeJamie,CbeCersei,DbeDaenerys, andSbeSansa.7.Answers may vary.Letsbe the dollar amount invested in stocks andbbe the dollar amount invested inbonds.8.Answers may vary.Letcbe the amount of calcium andpbe the amount of protein.9.Answers (order) may vary.Combinations would be HH, HT, TH, and TT.PennyNickelHeadsHeadsHeadsTailsTailsHeadsTailsTails10.Answers (order) may vary.Pairs would be (1,1), (1,2), (1,3), (2,1), (2,2),(2, 3), (3, 1), (3, 2), and (3, 3).11.2222232;´´´´=Graph not provided.12.6636´=13.Answers (order) may vary.Using the "Be Systematic" strategy, first list all pairs that begin withL, next allnew pairs that begin withS, etc.Pairs would beLS, LB, LE, LD, SB, SE, SD, BE, BD, ED.

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2Chapter1:Problem Solving14.Answers (order) may vary.Routes you can take are: (Begin,A,D,H,End), (Begin,A,E,H,End), (Begin,A,E,I,End), (Begin,B,E,H,End), (Begin,B,E,I,End), (Begin,B,F,I,End), (Begin,B,F,J,End),(Begin,C,F,I,End), (Begin,C,F,J,End), and (Begin,C,G,J,End).15.(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3),(4,4)16.(1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), (4,3)17.3ris the set of people who are good singers and appeared on “American Idol”.4ris the set of people whohave appeared on “American Idol” and are not good singers.18.2ris the set of people who are working to reduce global warming but not striving to reduce world hunger.4ris the set of people who are striving to reduce world hunger but not working to reduce global warming.19.35, 42, 49, 56, 63;multiplesof 720.81, 243, 729, 2187, 6561;multiplesof 321.,,,,bf cd ce cf cg22.(2,4), (2,5), (2,6), (3,1), (3, 2)23.21, 34, 55, 89, 144;sum of previous two numbers24.19, 23, 29, 31, 37; prime numbers25.Answers may vary.In how many ways can we line up three people for a picture?Let the people be labeled A, B, and C.The possible orders are ABC, ACB, BAC, BCA, CAB, and CBA.There are 6 different ways.In how many ways can we line up four people for a picture?Let the people be labeled A, B, C, and D.

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Section 1.1: Problem Solving325.(continued)The possible orders areABCD, ABDC, ACBD, ACDB, ADBC, ADCB, BACD, BADC, BCAD, BCDA,BDAC, BDCA, CABD, CADB, CBAD, CBDA, CDAB, CDBA, DABC, DACB, DBAC, DBCA, DCAB,and DCBA.There are 24 different ways.26.Answers mayvary. Ifyou guess at 2 true-false questions, how many different ways can you fill in the 2answers?12TTTFFTFFThere are4 different ways.If you guess at 3 true-false questions, how many different ways can you fill in the 3 answers?123TTTTTFTFTTFFFTTFTFFFTFFFThere are8 different ways.

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4Chapter1:Problem Solving26.(continued)If you guess at 4 true-false questions, how many different ways can you fill in the 4 answers?1234TTTTTTTFTTFTTTFFTFTTTFTFTFFTTFFFFTTTFTTFFTFTFTFFFFTTFFTFFFFTFFFFThere are16 different ways.27.Answers may vary.Using the first three letters of the alphabet, how many two-letter codes can we form ifwe are allowed to use the same letter twice?The possible codes are AA, AB,AC, BA, BB, BC, CA, CB, and CC. There are9 different codes.

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Section 1.1: Problem Solving527.(continued)Using the first five letters of the alphabet, how many two-letter codes can we form if we are allowed to usethe same letter twice?The possible codes are AA, AB, AC, AD, AE, BA, BB, BC, BD, BE, CA, CB, CC, CD, CE, DA, DB, DC,DD, DE, EA, EB, EC, ED, and EE.There are25 different codes.28.Answers may vary.A family has three children.If we list the gender of the children, how many different lists are possible?123gggggbgbggbbbggbgbbbgbbbThere are8 different lists.

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6Chapter1:Problem Solving28.(continued)A family has four children.If we list the gender of the children, how many different lists are possible?1234gggggggbggbgggbbgbgggbgbgbbggbbbbgggbggbbgbgbgbbbbggbbgbbbbgbbbbThere are16 different lists.29.Answers may vary.An electric-blue Ferrari comes with two options:run flat tiresandfront heated seats.You may buy the carwith any combination of the options (including none).How many different choices do you have?LetRberun flat tiresandFbefront heated seats. If a feature is not included, it is indicated by a “0”.If it isincluded, it is indicated by a “1”.There are4 different choices.RF00011011An electric-blue Ferrari comes with three options:run flat tires,front heated seats, andpolished rims. Youmay buy the car with any combination of the options (including none).How many different choices do youhave?LetRberun flat tires,Fbefront heated seats, andPbepolished rims. If a feature is not included,it is indicated by a “0”.If it is included, it is indicated by a “1”.There are8 different choices.RFP000001010011100101110111

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Section 1.1: Problem Solving730.Answers may vary.You have 2 different colors of paper to use and 3 different styles of font.How many different ways can youprint your resume?There are6 different ways.You have 2 different colors of paper to use and 4 different styles of font.How many different ways can youprint your resume?There are8 different ways.31.False, counterexamples may vary.Junehas 30 days.32.False, counterexamples may vary.At the time this book was written, this was false because Bill Clinton isstill alive.33.False, counterexamples may vary.13235134252,, and24444246343++=+===¹+34.False, counterexamples may vary.2295 but (9)(5)since 8125.-< -->->35.False, A is the grandfather of C.36.False, counterexamples may vary.You know your instructor and your instructor knows his/her mother, but(most likely) you don’t know your instructor’s mother.37.False, counterexamples may vary.If the price of a $10.00 item is increased by10%, its new price is $11.00.If the $11.00 item is then decreased by 10%, the new price would be $9.90, not $10.00.38.False, counterexamples may vary.If your hourly rate of $10.00 is decreased by 20%, the new rate is $8.00.If the $8.00 rate is then increased by 20%, the new rate would be $9.60, not $10.00.

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8Chapter1:Problem Solving39.Explanations may vary.These two sequences do not give the same results.The question here is equivalentto asking if the algebraic statement,225(5)xx+=+, is true.If we let1,x=we have a counterexample.()?22?21515156636+=++=¹Hence, the statement is false.40.Explanations may vary.These two sequences do not give the same results.The question here is equivalentto asking if the algebraic statement,222()xyxy-=-, is true.If we let1x=and3,y=we have a counterexample.()()?222?2131319284-=--= --¹Hence, the statement is false.41.Explanations may vary.These two sequences do give the same results.The question here is equivalent toasking if the algebraic statement,,333xyxy+=+is true.If you think of dividing by 3asequivalent tomultiplying by1 3,then you can use the distributive property to prove this statement.()111333333xyxyxyxy+=+=+=+42.Explanations may vary.These two sequences do give the same results.The question here is equivalent toasking if the algebraic statement,()555,xyxy× +× =+×is true.The proof of this statement is equivalent toproving the distributive property.43.Answers may vary.5 is a number;{ }5is a number with braces around it.Moreover, 5 is a singly listedelement, while{ }5is a set that contains the single element 5.44.Answers may vary.The secondAhas a prime()¢on it.Moreover,Ais a set (which is a subset of someuniversal set),UandA¢is the complement of setArelative to some universal set.U45.Answers may vary.One is uppercase and the other is lowercase. Moreover,Uusually denotes theuniversal set, and the lower case letters are usually elements that appear in some set.46.Answers may vary.{ }are different from( ).Moreover,{}1, 2is the set that contains only the elements1 and 2.()1, 2is the interval that contains all real numbers between 1 and 2 (not including 1 and 2themselves).Note: Another interpretation of()1, 2is being a point in a plane. However, with{}1, 2in thissame problem, the interpretation should be relative to sets.47.Answers may vary.The order of the numbers is different.Moreover, if you interpret()2, 3and()3, 2asintervals, they are both expressing the same set of numbers.The standard, however, is to express intervals(),a bin such a way thatab<(assumingaandbare real numbers).Note: Another interpretation of()2, 3and()3, 2is as points in the plane, and these two points would be different.They would be pointsthat are reflected over the line.yx=The context of this question, however, is relative to intervals.

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Section 1.1: Problem Solving948.The symbol on the right is the number zero; the symbol on the left is not a number.49.–52.No solution provided.53.Answers may vary.LetObe the age of the older building andYbe the age of the younger building.Guesses forOandYGoodPointsWeak Points100, 221Sum is 321.The older is more thantwice the younger.110, 211Sum is 321.The older is less thantwice the older.107, 214All conditions satisfied.We have the solution.The buildings are 107 and 214 years old.54.Answers may vary.LetSbe the shortest piece,Mbe the middle size piece, andLbe the longest piece.Guesses for,S,MandLGoodPointsWeak Points3, 8, and 24The longest is 3 times the middle size andthe shortest is 5 less than the middle size.Total length is less than40.5, 10, and 30The longest is 3 times the middle size andthe shortest is 5 less than the middle size.Total length is morethan 40.4, 9, and 27All conditions satisfied.We have the solutionThe three pieces should be 4, 9, and 27 inches long.55.Answers may vary.LetTbe the number of times Tom Brady threw a touchdown pass,Pbe the numberof times Phillip Rivers threw a touchdown pass, andAbe the number of times Aaron Rodgers threw atouchdown pass.Guesses for,T,PandAGoodPointsWeak Points30, 24, and 22Pis 2 more thanAand 6 less than.TSum is less than 94.40, 34, and 32Pis 2 more thanAand 6 less than.TSum is more than 94.36, 30, and 28All conditions satisfied.We have the solution.Brady had 36 touchdowns.56.Answers may vary.LetJbe the number of home runs hit byJoc Pederson,Tbe the number of home runshit byTodd FrazierandPbe the number of home runs hit by Prince Fielder.Guesses for,J,TandPGoodPointsWeak Points30, 29 and 7Jis 1 more thanTandJis23 morethan.PSum is less than 72.40, 39 and 17Jis 1 more thanJandJis23 morethan.PSum is more than 72.32, 31 and 9All conditions satisfied.We have the solution.Pedersonhad 32 hits.

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10Chapter1:Problem Solving57.Answers may vary.LetAbe the amount invested at 8%andBbe the amount invested at 6%.Guesses forAandBGoodPointsWeak Points$3,000 and $5,000Sum is $8,000.Return is less than$550.$4,000 and $4,000Sum is $8,000.Return is more than$550.$3,500 and $4,500All conditions satisfied.We have the solutionHeather invested$3,500 at 8% and $4,500 at 6%.58.Answers may vary.LetAbe the amount invested at 11% andBbe the amount invested at 8%.Guesses forAandBGoodPointsWeak Points$7,000 and $2,000Sum is $9,000.Return is less than$936.$7,500 and $1,500Sum is $9,000.Return is more than$936.$7,200 and $1,800All conditions satisfied.We have the solution.Carlosinvested$7,200 at11% and $1,800 at8%.59.Answers may vary.LetAbe the number of administrators,Sbe the number of students, andFbe thenumber of faculty members.Guesses for,A,SandFGoodPointsWeak Points10, 2, and 7Ais 5 timesSandFis 5 more than S.There are less than 26people.20, 4, and 9Ais 5 timesSandFis 5 more than S.There are more than 26people.15, 3, and 8All conditions satisfied.We have the solution.There are 3 students.60.Answers may vary.LetSbe the number of senior citizens,Ybe the number of young adults, andMbethe number of middle-aged adults.Guesses for,S,YandMGoodPointsWeak Points18, 6, and 9Sis 3 timesYandMis one-half.SThere are less than 55people.36, 12, and 18Sis 3 timesYandMis one-half.SThere are more than 55people.30, 10, and 15All conditions satisfied.We have thesolution.There are 30 senior citizens.61.LCHPL, LCPHL, LHCPL, LHPCL, LPCHL, LPHCL62.987654321362,880´´´´´´´´=63.PN, PD, PQ, ND, NQ, DQ

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Section 1.1: Problem Solving1164.PND, PNQ, PDQ, NDQ, PNDQ65.The possible schedules are given in the table below.MathEnglishSociologyArt History91112109121011109121110111291291011121110966.The possible schedules are given in the table below.MathEnglishSociologyArt History91112109121011129101167.The possible schedules are given in the table below.MathEnglishArt History91110912109121110911101191012910121112910129111211912111068.The possible schedules are given in the table below.MathEnglishSociologyArt History91112109121110109121110111291012119129111069.79; The top and bottom rows will both have 21 tiles, the middle row will have one tile, and the remaining18 rows will have 2 tiles each.2211118279´+ ´+´=70.48; The top and bottom rows will both have 1 tile, and the remaining 23 rows will have 2 tiles each.2123248´+´=

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12Chapter1:Problem Solving71.You will pay more,in total, over the twoyearsif the 8% raise occurs first. As an example, if tuition is $100and the 8% raise occurs first, you will pay()1000.08 100$108+=thefirstyearand()1080.05 108+$113.40=the second year,for a total of $221.40. If the 5% raise occurs first, you will pay()1000.05 100$105+=thefirstyearand()1050.08 105$113.40+=the second year,for a total of$218.40.Notethatin either case, you would pay the same tuition during the second year.72.The board would not follow its mandate.As an example, if tuition is $100, you will pay()1000.02 100$102+=after the first increase,()1020.03 102$105.06+=after the second increase, and()105.060.05 105.06$110.31+=after the third increase, which is an approximately 10.3% increase.73.–76.Answers will vary.77.666216´´=78.1212121,728´´=79.(41, 43), (59, 61); Pairs of sequential primes that differ by 2.80.(23, 31), (29, 37); The first numbers are the primes in order,the second numbers come from taking thesecond prime number after the first.81.There are a total of55squares.11´2544´422´1655´133´982.a)12b)6c)10011´1612´1224´341´422´913´834´232´633´414´421´1242´344´123´631´843´283.As you look at the intersections, you’ll see that there is a pattern as to how many routes can be created asyou leave the Hard Rock Cafe on the way to The Cheesecake Factory.To choose direct routes, you mustalways be traveling down and/or to the right. The numbers indicate how many ways there are to get to theintersection below and to the right of the number.For example, the “2” below and tothe right of the HardRock Cafeindicates there are two ways to arrive at that intersection, one by going right, then down, andanother by going down, and then right. There are a total of 252 possible routes.

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Section 1.2: Inductive and Deductive Reasoning1384.As you look at the intersections, you’ll see that there is a pattern as to how many routes can be created asyou leave the Hard Rock Cafe on the way to Baja Fresh and as you leave Baja Fresh on the way to TheCheesecake Factory.To choose direct routes, you must always be traveling down and/or to the right. Thenumbers indicate how many ways there are to get to the intersection below and to the right of the number.Since there are 10 ways to go from the Hard Rock Cafeto Baja Fresh and 10 ways to go from the BajaFresh to The Cheesecake Factory , there are1010100´=possible routes.Section1.2:Inductive and Deductive Reasoning1.inductive2.deductive3.deductive4.inductive5.inductive6.deductive7.deductive8.inductive9.inductive10.deductive11.1612.3213.9614.1,21515.2116.0.10101017.18.19.A blue face in a green box followed by a redface in a blue box.20.A red face in a blue box followed by a blueface in a green box.21.Hint: Think of prime numbers.23571113

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14Chapter1:Problem Solving22.Hint: Think of the Fibonacci sequence.23.313+or511+24.513+or711+25.317+or713+26.323,+719,+or1313+27.566712345, 12345622´´++++=+++++=28.24681056, 2468101267++++=´+++++=´29.1357925, 135791136++++=+++++=30.1111151111116,122334455661223344556677++++=+++++=´´´´´´´´´´´31.1491630+++=32.14916253691+++++=33.a)The total of all the numbers in the square is12316136.++++=b)The total of the numbers for each row, column, and diagonal would be136 434.=c)One can deduce the missing numbers to yield the following.34.a)The total of all the numbers in the square is12316136.++++=b)The total of the numbers for each row, column, and diagonal would be136 434.=c)One can deduce the missing numbers to yield the following.35.36.

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Section 1.2: Inductive and Deductive Reasoning1537.Yes; there are many ways to do this. Try to rewrite the puzzle in various ways with some numbers or cluesmissing to see if you can still solve it.38.Yes; there are many ways to do this. Try to rewrite the puzzle in various ways with some numbers or cluesmissing to see if you can still solve it.39.Adriana (political issues), Caleb (solar power), Ethan (waterconservation), and Julia (recycling)40.Jessica (third), Serena (second), Andre (fourth), and Emily (first)41.36644633;Reversing the number in the first pair yieldsthe relationshipGGAGLLGA ,66463364soG, A, and Lcorrespond to6, 4, and3,respectively. Assigning these values to the letters in the secondwordyieldsLLGAAGGL.33644663Reversing the order of the resulting number leads to the final valueof36644633.42.leader; Froofrug Merduc represents “Where is your”43.986763; reverse the numbers and delete one number from any pair of numbers.44.20; the numbers are found by repeatedly adding 6 then subtracting 2.45.Answers may vary.Possible responses include that Sharifa would begin by visiting one of the six branchesand then visit the five original cities in 120 ways.The total number of ways she could make her visits is6120720´=ways. The same reasoning would lead to77205040´=ways for seven cities.46.Reasonswillvary.47.(d) step 348.49.By looking at examples, inductive reasoning leads us to make conjectures which we then try to prove.

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16Chapter1:Problem Solving50.Consider examples;Look for patterns;Make a conjecture;Use deductivereasoning toprove the statement.51.–54.Answers will vary.55.53, 107, 213; Next term is the previous term plus twice the term before the previous term.56.5461, 21845, 87381; Next term is four times the previous term plus one.57.47, 76, 123; Next term is sum of two previous terms.58.150, 298, 598; Next term is the previous term plus twice the term before the previous term.59.There are a total of261220++=squares.11´1222´633´260.There are a total of38152450+++=squares.11´2422´1533´844´361.a)There are a total of60rectangles.11´1212´913´614´321´822´623´424´231´432´333´234´1b)There are a total of210rectangles.11´2441´1212´1842´913´1243´614´644´321´2051´822´1552´623´1053´424´554´231´1661´432´1262´333´863´234´464´1

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Section 1.2: Inductive and Deductive Reasoning1762.Answers may vary.One can count the rectangles in a systematic way like in Exercise 61.One can see thatthere is a pattern to the counting.Firstly, one needs to realize that the sum of the firstnnatural numbers is()1123....2n nn+++++=In Exercise 61, the first four natural numbers occurred in the last four entries.Looking above these entries, you can see multiples of these numbers.One can view the sum of all theentries as()()()()()()() ()123421234312344123451234612346 74 5123456123421 10210.22++++×++++×++++×++++×++++×+++××=+++++×+++=×=×==In general, the number of rectangles of all types would be()()1122m mn n++×for anmn´rectangle.Forthe case of a106´rectangle, we would have()()1010166155 211,15522×+×+×=×=rectangles of all types.63.The base is a64´rectangle.If you consider the diagram below as the base, we have6424´=baseballs.In order to build the next level, we are looking for the number of places in which four baseballs (squares)meet.There are5315´=such places.For the next level we would have428´=meeting places.For the last level we would have313´=meeting places.This yields a total of6453423124158350´+´+´+´=+++=baseballs.64.Using the same reasoning as in Exercise 63, we have7564534231´+´+´+´+´=3524158385++++=baseballs.65.–66.No solution provided.67.In this trick, you will always get the resultthree.a)Call the number.nb)3nc)39n+d)()3339333nnn++==+e)()333nnnn+-=+-=68.In this trick, you will always get a result that isthe number that you started with.a)Call the number.nb)5nc)520n+d)()54520455nnn++==+e)()4444nnn+-=+-=

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18Chapter1:Problem SolvingSection 1.3: Estimation1.204019040290+++=2.20301090150+++=3.351520-=4.1107040-=5.51680´=6.10001515,000´=7.18 / 36=8.2100/ 7030=9.0.180080´=10.0.00140004´=11.9%10000.09100090´=´=12.20%7000.20700140´=´=13.456120´´=miles;The estimate is larger than the exact answer.The exact answer is 111 miles.14.18$1.00$18.00;´=The estimate is larger than the exact answer.The exact answer is $16.92.15.325/ 506.5=more hours, 7:30PM;The estimate is earlier than the actual time.The actual answer is 7:50PM.16.$60.00 /15$4.00=per gallon;The estimate is less than the actual answer.The actual answer is $3.89 pergallon.17.$120.0015%$120.000.15$18.00;´=´=The estimate is more than the exact answer.The exact answer is$17.77.18.$75.00/ 3$25.00;=The estimate is less than the exact answer.The exact answer is $25.46.19.()()3$3.004$1.50$3.00$9.00$6.00$3.00$18.00;´+´+=++=The estimate is larger than the exactanswer.The exact answer is $17.20.20.$1,4005%$1,4000.05$70.00;´=´=The estimated total is$1,400.00$70.00$1,470.00.+=Theestimate is less than the exact answer.The exact answer is$1,389.00$83.34$1,472.34.+=21.It seems safe.Alicia probably weighs less than 200 pounds, so that leaves2,3002002,100-=pounds forthe 21 students.They probably each weigh less than 100 pounds.22.NFL linemen usually weigh at least 300 pounds.So the eight linemen would weigh at least83002, 400´=pounds, which is above the elevator’s capacity.23.$40,0004%$40,0000.04$1,600;´=´=The estimate is larger than the exact answer.The exact answer is$1324.40.24.()2551512515140;´+=+=42,000140300¸=weeks;The estimate is less than the actual answer.Theactual answer is326weeks.25.1,0001,0001=times greater;The estimate is very close to the actual answer.The actual answer is 996.67times greater.26.4004100 =times greater;The estimate is very close to the actual answer.The actual answer is 4.01 timesgreater.27Her total expenses are about $100 per month;100 / 714 12$14$168;»´=The estimate is larger than theactual answer.The actual answer is $163 (you round to nearest dollar on deductions).28.$20,0000.025$500, $50020%$5000.20$100;´=´=´=The estimate is less than the actual answer.Theactual answer is $104 (you round to nearest dollar on deductions).

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Chapter Review Exercises1929.Answers will vary.Actual answers are:Male high school graduates; $44,300;Females with associatesdegrees; $42,00030.Answers will vary. Actual answer is $17,200.31.college and professional degrees, high school graduates32.Answers will vary. Actual answer is $47,300.33.category 234.500;0.212,309485´»35.response 236.139;0.062309139´»37.Estimated answers may vary.Actual value is 69.3%.38.Slightly decreases.39.2014;Estimated answers may vary.Actual value is 84.8%.40.almost six times41.Estimated answers may vary.The exact answer is21650.404874.66,´=or$874.66 billion.42.Estimated answers may vary.The exactanswer is21650.432935.28,´=or$935.28 billion.43.Estimated answers may vary.The exactanswer is21650.072155.88´=or$155.88billion.44.Estimated answers may vary.The exactanswer is2165(0.4040.432)21650.8361809.94,´+=´=or$1,809.94 billion.45.Estimated answers may vary.The actual number of immigrants was705,3610.302213,019.´»46.Estimated answers may vary.The actual number of immigrants was705,3610.13293,108.´»47.Estimated answers may vary.The actual percentage is130,661/ 705,3610.185,»or18.5%.48.Estimated answers may vary.41,034 / 705,3610.058,»or5.8%.49.–54.Answers will vary.55.Answers may vary.The amount of lawn that needs to be fertilized is represented by the size of the lot, lessthe non-grassy areas such as the garden, drivewayand house.If you divide the grassy area into rectangles,you get96 16996 3065 2818 6516, 2242,8801,8201,17010,354×-×-×-×=---=square feet.They need10,354 50002.07,»orslightly over two bags of fertilizer.96 16996 3065 2818 6516, 2242,8801,8201,17010,354×-×-×-×=---=56.Answers may vary.For our estimate we will not consider areas such as windows and doorsthat may not bepainted.They need to cover exactly()7.75 2 18.52 117.75 59457.25×+×=×=square feet twice fora total of914.5 square feet.They will need to purchase914.5 2004.57,»or 5gallons of paint (no partial gallons).57.–60.No solution provided.ChapterReview Exercises1.Understand the problem;devise a plan;carry out your plan;check your answer.2.An example that shows a conjecture is false.

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20Chapter1:Problem Solving3.10;Let A, C, L, R, and T represent Amber, Chris, Lawrence,Remy,and Travis.The pairs are AC, AL, AR, AT, CL, CR, CT, LR, LT, and RT.4.Answers may vary.At a restaurant, you have 2 appetizers, 3 entrees, and 2 desserts.How many differentmeals can you choose if you select one appetizer, one entrée, and one desert?There are twelve different meals possible.5.Answers may vary.LetPbe the number of hours Picaboo worked as a stock person andIbe the numberof hours she worked as a ski instructor.Guesses forPandIGoodPointsWeak Points9 and 11Sum is 20.Amount earned is lessthan $141.20.7 and 13Sum is 20.Amount earned is morethan $141.20.8 and 12All conditions satisfied.We have thesolution.6.false;Counterexamples may vary.13235134252,, and24444246343++=+===¹+7.No solution provided.8.Answers may vary.Possible answers include: Inductive reasoning is the process of drawing a generalconclusion by observing a pattern in specific instances.In deductive reasoning, we use accepted facts andgeneral principles to arrive at a specific conclusion.9.a)deductiveb)inductive10.a)27; Add five to the previous number.b)47; Add the previous two numbers.

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Chapter Test2111.12.543,+741,+1137,+1731,+or1929+13.In this trick, you will always get twice the number you started with.a)Call the number.nb)8nc)812n+d)()4 238122344nnn++==+e)()2332332nnn+-=+-=14.a)46,000b)28,00015.a)21060150-=b)61590´=16.Estimated time left to travel would be150 503=hours, arriving at 7:00 PM.17.a)Estimated answers may vary. The actual difference is 60 milligrams.b)a little more than twice as much.c)approximately fourd)the gourmet coffeeChapter Test1.Answers will vary.2.a)true; adding fractions with like denominatorsb)false;3334545¹++3.Answers may vary.LetWbe the number ofWiiSportssold,Sbe thenumber ofSuper Mario Brotherssold, andPbe the number ofPokémonssold.Guesses for,W,SandPGoodPointsWeak Points36, 29, and 20Super Mario Brotherssold 9 more thanPokémonand 7 less thanWii Sports.Total number is lessthan 118.56, 49, and 40Super Mario Brotherssold 9 more thanPokémonand 7 less thanWii Sports.Total number is morethan 118.47, 40, and 31All conditions satisfied.We have the solution.47millionWii Sports,40millionSuper Mario Brothers, and31millionPokémonsweresold.4.a)4,320b)2,2805.a)36,000b)36,5006.If two terms are similar but sounds slightly different, they usually do not mean exactly the same thing.7.Answers may vary.Possible answers include: Inductive reasoning is the process of drawing a generalconclusion by observing a pattern in specific instances.In deductive reasoning, we use accepted facts andgeneral principles to arrive at a specific conclusion.

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22Chapter1:Problem Solving8.a)inductiveb)deductive9.Mathematical ideas can be understood verbally, graphically, and through examples.10.Answers will vary. One estimate is()$600$20012$8004$3,200.3+æö×=× =ç÷èøThe true value is $3,220.11.2222222112, 226, 6315, 15431, 31556, 56692, and 927141+=+=+=+=+=+=+=12.cde, cdf, cdg13.14.75360,+=134760,+=194160,+=233760,+=or293160+=15.False;suppose the laptop costs $1,000.After the 10% discount, the laptop would cost $900.If that price isincreased by 10%, the laptop would cost $990, not $1,000.16.In this trick, you will always get twice theoriginalnumber.a)Call the number.nb)4nc)440n+d)()2 22044022022nnn++==+e)()22020220202nnn+-=+-=

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Section 2.1: The Language of Sets23Chapter2:Set Theory: Using Mathematics to Classify ObjectsSection2.1:The Language of Sets1.{}10, 11,12,13,14,152.{},, , ,f g h i j3.{}17, 18, 19, 20, 21, 22, 23, 24, 254.{}4, 3, 2, 1, 0, 1, 2, 3, 4----5.{}4, 8, 12, 16,20, 24, 286.{}7, 9, 11, 13, 15, 17, 197.{}Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday8.{}New Hampshire, New Jersey, New Mexico, New York9.Æ10.{}11, 13, 15, 17, 19, 21, 23, 2511.Æ12.{}Alaska, Hawaii13.Answers may vary. Possible answers include{}:is a multiple of 3 between 3 and 12 inclusive .xx14.{}:is a color of the rainbowxx15.{}28,29,30,3116.{}1,2,3,. . .---17.{}January, February, March, April, May, June, July, August, September, October, November, December18.{}:is a sign of the Zodiacxx19.Answers may vary. Possible answers include{}101, 102, 103,. . . .20.Answers may vary. Possible answers include{}:is an even natural number .xx21.Answers may vary. Possible answers include{}:is an even natural number between 1 and 101 .xx22.Answers may vary. Possible answers include{}3, 6, 9, 12, 15,. . . .23.well defined24.well defined25.not well defined26.not well defined27.not well defined28.well defined29.well defined30.not well defined31.Ï32.Î33.Î34.Ï35.Ï36.Ï37.Î38.Î39.Ï40.Ï41.Î42.;Ï{}Floridais a subset, not an element.43.644.1145.046.4847.448.549.2 elements;{} {}1, 2 , 1, 2, 350.4 elements;{ }{ }1 ,, 0,0Æ

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24Chapter 2:Set Theory51.1 element;{}{}Æ52.4 elements;{ } { } { } {}1 ,2 ,3 , 1, 2, 353.finite54.finite55.infinite56.finite57.Answers may vary. Possible answers include 4.5.58.Answers may vary. Possible answers include the King(Queen) of England.59.Answers may vary. Possible answers include Sony.60.Answers may vary. Possible answers include a frog.61.Answers may vary. Possible answers include Angela Merkel.62.Answers may vary. Possible answers include Kia.63.Answers may vary. Possible answers include Sunday.64.Answers may vary. Possible answers include California.65.{}:is a humanities electivexx66.{}:does not satisfy a world culture requirementxx67.{}History012, History223, Geography115, Anthropology11168.{}History012, English010, English220, Psychology20069.{}AZ, FL, GA, LA, NJ, NM, TX, VAL=70.{}CA, MN, NY, PAG=71.{}:is a state with the price of gasoline above $2.35xx72.{}:is a state with price of gasoline below $2.10xx73.{}jogging, jumping ropeM=74.{}calisthenics, leisure cycling, slow walkingL=75.{}:burns less than 140 calories per one-half hourxx76.{}:burns at least 300 calories per one-half hourxx77.Answers will vary.78.When the set is too large or to complicated to list all the elements.79.Æis the empty set, it contains no elements.{}Æis not empty, it contains 1 element,.Æ80.a)nstands for the word.numberb)Astands for the set A. Set names are always capitalized.c)()n Ais thenumber of elements in set A.81.–84.Answers will vary.85.If the barber shaves himself, then he (the barber) does not shave himself. If the barber does not shavehimself, then he (the barber) does shave himself.Conclusion: This is a paradox.

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Section 2.2:ComparingSets2586.If the sentence is true, then the sentence is false. If the sentence is false, then the sentence is true.Conclusion: This is a paradox.87.If,SSÎthen.SSÏIf,SSÏthen.SSÎConclusion: This is a paradox.Section2.2:Comparing Sets1.These two sets are equal. They have the same elements arranged in a different order.2.These two sets are not equal. The first set is the set of vowels. The second set contains “b” along withother letters. Since “b” is not a vowel, these two sets cannot be equal.3.These two sets are not equal. The second set contains (infinitely many) elements that don’t appear in thefirst set.4.These two sets are equal. They are both{}3, 4, 5, 6, 7, 8, 9,10 .5.These two sets are equal.They are both{}1, 3, 5,, 99 .6.These two sets are not equal. The first set contains the first 5 multiples of 3 that are counting numbers.The second set contains all multiples of 3 that are counting numbers.7.These two sets are equal. Common sense dictates that nobody born before 1800 should be living.8.These two sets are not equal. The null set contains no elements. The set,{},Æcontains one element,namely the null set.9.true; All the elements of the first set are understood to be elements of the second set and moreover the firstset is not equal to the second set.10.true; All the elements of the first set are also elements of the second set.11.false; The letter “y” is an element of the first set and not an element of the second set.12.false; The setonthe left and the set on the right are equal. They both represent the set{},, ,.r u t hThefirst set cannot be apropersubset of the second set.13.true; The null set is a subset of all sets.14.false; Although the null set is a subset of all sets, it is not apropersubset because it is equal to itself.15.The first set is equivalent to the second set because they both have the same number of elements.16.The first set is not equivalent to the second set. The first set has 6 elements while the second set isunderstood to have 11 elements.17.The first set is equivalent to the second set because they both have that same number of elements, namely4.18.The first set is not equivalent to the second set. The first set has 7 elements while the second set has 6elements.19.The first set is not equivalent to the second set. The first set has 0 elements while the second set has 1element.20.The first set is equivalent to the second set. They both have 1 element.21.The first set is equivalent to the second set. They both have 8 elements.22.The first set is not equivalent to the second set. The first set has 26 elements while the second set has 24elements.23.The first set is not equivalent to the second set. The first set has 366 elements while the second set has 365elements.24.The first set is equivalent to the second set. The starting number of players for both teams is the same.

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26Chapter 2:Set Theory25.{}1, 2,{}1, 3,{}2, 326.{} {} {} {} {} {}1, 2 , 1, 3 , 1, 4 , 2, 3 , 2, 4 , 3, 427.{} {} {} {}1, 2, 3 , 1, 2, 4 , 1, 3, 4 , 2,3,428.{}1, 2,3 ,{}1, 2,4 ,{}1, 2,5 ,{}1, 3, 4 ,{}1, 3,5 ,{}1, 4,5 ,{}2, 3, 4 ,{}2, 3,5 ,{}2, 4,5 ,{}3, 4,529.There are5232=subsets and52131-=proper subsets.30.There are72128=subsets and721127-=proper subsets.31.{};Carmen, Frank, IvanaT VT==32.Answers may vary. Note:theboxedvalues have the same cardinality as.SThe set of students majoring inart.A=The set of students that are involved in drama.D=()( )( )()()()()4,6,2,3,2,3, and2n Un Ln Sn Vn An Tn D=======33.The set of lowerclassmen;L=Note:theboxedvalue indicates the set with the largest cardinality.()( )( )()()()()4,6,2,3,2,3, and2n Un Ln Sn Vn An Tn D=======34.Answers may vary. Note:theboxedvalues are the sets with the smallest cardinality.The set of studentsthat are science majors.S=The set of students that are art majors.A=The set of students that areinvolved in drama.D=()( )( )()()()()4,6,2,3,2,3, and2n Un Ln Sn Vn An Tn D=======35.6264=36.122=37.4216=38.4216=39.72128=40.9;92512=41.82256=42.10;1021024=43.744.645.{}5P,10P, 25D46.{}1S, 25S, 50D47.{}5P,10P, 25Dor{}5P,10P, 25S48.{}5P,10S, 50D49.850.a)branch 2b)branch 751.He didn’t understand that the order of elements in a set does not matter.52.He didn’t understand that repetition of elements in a set does not matter.53.a)25 is not a power of2.b)He confused25with52 .c)5232=54.There are2nways to flipncoins and also to answer annquestion true-false test.55.Answers will vary.56.3021,073,741,824=57.Over 34 years;1,073,741,8241,073,741,82434.0536524606031,536,000=»´´´

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