Chapter 1 - XYZ Custom Plus

6Chapter 1 Whole NumbersCWriting Numbers in WordsThe idea of place value and expanded form can be used to help write the namesfor numbers. Naming numbers and writing them in words takes some practice.Let’s begin by looking at the names of some two-digit numbers. Table 2 lists afew. Notice that the two-digit numbers that do not end in 0 have two parts. Theseparts are separated by a hyphen.Table 2Number in English number in English25 Twenty-five 30 Thirty47 Forty-seven 62 Sixty-two93 Ninety-three 77 Seventy-seven88 Eighty-eight 50 FiftyThe following examples give the names for some larger numbers. In each casethe names are written according to the place values given in Table 1.7. Write each number in words.a. 724b. 595c. 307Example 7Write each number in words.a. 452 b. 397 c. 608Solution a. Four hundred fifty-twob. Three hundred ninety-sevenc. Six hundred eight8. Write each number in words.a. 4,758b. 62,779c. 305,4409. Write each number in words.a. 707,044,002b. 452,900,008c. 4,008,002,001Example 8Write each number in words.Solutiona. 3,561 b. 53,662 c. 547,801a. Three thousand, five hundred sixty-onehNotice how the comma separatesthe thousands from the hundredsb. Fifty-three thousand, six hundred sixty-twoc. Five hundred forty-seven thousand, eight hundred oneExample 9Write each number in words.Answers7. a. Seven hundred twenty-fourb. Five hundred ninety-fivec. Three hundred seven8. a. Four thousand, seven hundredfifty-eightb. Sixty-two thousand, sevenhundred seventy-ninec. Three hundred five thousand,four hundred forty9. a. Seven hundred seven million,forty-four thousand, twob. Four hundred fifty-two million,nine hundred thousand,eightc. Four billion, eight million,two thousand, oneSolutiona. 507,034,005b. 739,600,075c. 5,003,007,006a. Five hundred seven million, thirty-four thousand, fiveb. Seven hundred thirty-nine million, six hundred thousand,seventy-fivec. Five billion, three million, seven thousand, six

10Chapter 1 Whole NumbersC Write each of the following numbers in words. [Examples 7–9]37. 29Twenty-nine38. 75Seventy-five39. 40Forty40. 90Ninety41. 573Five hundred seventy-three42. 895Eight hundred ninety-five43. 707Seven hundred seven44. 405Four hundred five45. 770Seven hundred seventy46. 450Four hundred fifty47. 23,540Twenty-three thousand, five hundred forty48. 56,708Fifty-six thousand, seven hundred eight49. 3,004Three thousand, four50. 5,008Five thousand, eight51. 3,040Three thousand, forty52. 5,080Five thousand, eighty53. 104,065,780One hundred four million, sixty-five thousand, seven hundred eighty54. 637,008,500Six hundred thirty-seven million, eight thousand, five hundred55. 5,003,040,008Five billion, three million, forty thousand, eight56. 7,050,800,001Seven billion, fifty million, eight hundred thousand, one57. 2,546,731Two million, five hundred forty-six thousand, seven hundred thirty-one58. 6,998,454Six million, nine hundred ninety-eight thousand, four hundred fifty-fourD Write each of the following numbers with digits instead of words. [Examples 10, 11]59. Three hundred twenty-five32561. Five thousand, four hundred thirty-two5,43263. Eighty-six thousand, seven hundred sixty-two86,76265. Two million, two hundred2,000,20067. Two million, two thousand, two hundred2,002,20060. Forty-eight4862. One hundred twenty-three thousand, sixty-one123,06164. One hundred million, two hundred thousand, threehundred100,200,30066. Two million, two2,000,00268. Two billion, two hundred thousand, two hundred two2,000,200,202

1.1 Problem Set11Applying the Concepts69. The illustration shows the average income of workers18 and older by education.70. Write the following numbers in words from the informationin the given illustration:Who’s in the Money?100,00080,00060,00040,00020,000$19,041$28,631$51,568$67,073$93,333Such Great HeightsTaipei 101Taipei, TaiwanPetronas Tower 1 & 2Kuala Lumpur, Malaysia1,483 ft1,670 ft Sears TowerChicago, USA1,450 ft0No H.S.DiplomaHigh SchoolGrad/ GEDBachelor’sDegreeMaster’sDegreePhDSource: U.S. Census BureauSource: www.tenmojo.comWrite the following numbers in words:a. the average income of someone with only a highschool educationTwenty-eight thousand, six hundred thirty-oneb. the average income of someone with a Ph.D.Ninety-three thousand, three hundred thirty-threea. the height in feet of the Taipei 101 building in Taipei,TaiwanOne thousand, six hundred seventyb. the height in feet of the Sears Tower in Chicago,IllinoisOne thousand, four hundred fifty71. MP 3s A new MP 3 player has the ability tohold over 125,000 songs. Write the placevalue of the 1 in the number of songs.Hundred thousandsiPodMusicPhotosExtrasSettingsMENU>>>>72. Music Downloads The top three downloaded songs forone month on Amazon.com had a combined 450,320downloads. Write the place value of the 3 in the numberof downloads.Hundreds73. Baseball Salaries According to mlb.com, Major LeagueBaseball’s 2008 average player salary was $3,173,403,representing an increase of 7% from the previous season’saverage. Write 3,173,403 in words.74. Astronomy The distance from the sun to the earth is92,897,416 miles. Write this number in expanded form.90,000,000 + 2,000,000 + 800,000 + 90,000 + 7,000 +400 + 10 + 6Average Player Salary‘07‘08 $3,173,4037% increaseThree million, one hundred seventy-three thousand, four hundred three

Introduction . . .Addition with Whole Numbers,and PerimeterThe chart shows the number of babies born in 2006, grouped together accordingto the age of mothers.Who’s Having All the BabiesUnder 20: 441,8321.2ObjectivesA Add whole numbers.B Understand the notation andvocabulary of addition.C Use the properties of addition.D Find a solution to an equation byinspection.E Find the perimeter of a figure.20–29:30–39:40–54:2,262,6941,449,039112,432Source: National Center for Health Statistics, 2006There is much more information available from the table than just the numbersshown. For instance, the chart tells us how many babies were born to mothersless than 30 years of age. But to find that number, we need to be able to do additionwith whole numbers. Let’s begin by visualizing addition on the number line.Facts of AdditionUsing lengths to visualize addition can be very helpful. In mathematics we generallydo so by using the number line. For example, we add 3 and 5 on the numberline like this: Start at 0 and move to 3, as shown in Figure 1. From 3, move 5 moreunits to the right. This brings us to 8. Therefore, 3 + 5 = 8.Start3 units 5 unitsEndInstructor NoteAlthough there are no problems inthe Problem Set that use the numberline for addition, I always use thenumber line for several examplesin class. Doing so prepares studentsfor addition with negative numbers,where the number line is used extensively.0 1 2 3 4 5 6 7 8Figure 1If we do this kind of addition on the number line with all combinations of thenumbers 0 through 9, we get the results summarized in Table 1 on the next page.We call the information in Table 1 our basic addition facts. Your success withthe examples and problems in this section depends on knowing the basic additionfacts.1.2 Addition with Whole Numbers, and Perimeter13

1.2 Addition with Whole Numbers, and Perimeter15Addition with CarryingIn Examples 1 and 2, the sums of the digits with the same place value werealways 9 or less. There are many times when the sum of the digits with the sameplace value will be a number larger than 9. In these cases we have to do what iscalled carrying in addition. The following examples illustrate this process.Example 3Add: 197 + 213 + 324Solutionvalue.We write the sum vertically and add digits with the same place19 1 7 When we add the ones, we get 7 + 3 + 4 = 14213 We write the 4 and carry the 1 to the tens column+ 32441 1 9 1 7 We add the tens, including the 1 that was carried213 over from the last step. We get 13, so we write+ 324 the 3 and carry the 1 to the hundreds column341 1 9 1 7 We add the hundreds, including the 1 that was213 carried over from the last step+ 3247343. Add.a. 375 + 121 + 473b. 495 + 699 + 978NoteNotice that PracticeProblem 3 has twoparts. Part a is similarto the problem shown in Example3. Part b is similar also, but alittle more challenging in nature.We will do this from time to timethroughout the text. If a practiceproblem contains more parts thanthe example to which it corresponds,then the additional partscover the same concept, but aremore challenging than Part a.Example 4Add: 46,789 + 2,490 + 864Solution We write the sum vertically—with the digits with the same placevalue aligned—and then use the shorthand form of addition.4. Add.a. 57,904 + 7,193 + 655b. 68,495 + 7,236 + 878 + 29 + 5m8888888 These are the numbers that have been carried4 1 6 2 , 7 2 8 1 92 , 4 9 08 6 45 0 , 1 4 3m88888888888m888888888m8888888m8888m8Write the 3; carry the 1Write the 4; carry the 2Write the 1; carry the 2Write the 0; carry the 1No carrying necessaryOnesTensHundredsThousandsTen thousandsAdding numbers as we are doing here takes some practice. Most people don’tmake mistakes in carrying. Most mistakes in addition are made in adding thenumbers in the columns. That is why it is so important that you are accurate withthe basic addition facts given in this chapter.B VocabularyThe word we use to indicate addition is the word sum. If we say “the sum of 3 and5 is 8,” what we mean is 3 + 5 = 8. The word sum always indicates addition. Wecan state this fact in symbols by using the letters a and b to represent numbers.Answers3. a. 969 b. 2,1724. a. 65,752 b. 76,643

16Chapter 1 Whole NumbersDefinitionIf a and b are any two numbers, then the sum of a and b is a + b. To find thesum of two numbers, we add them.NoteWhen mathematics isused to solve everydayproblems, theproblems are almost always statedin words. The translation of Englishto symbols is a very important partof mathematics.Table 2 gives some phrases and sentences in English and their mathematicalequivalents written in symbols.Table 2In Englishin SymbolsThe sum of 4 and 1 4 + 14 added to 1 1 + 48 more than m m + 8x increased by 5 x + 5The sum of x and yx + yThe sum of 2 and 4 is 6. 2 + 4 = 6Instructor NoteAn important part of this section,the properties of addition, beginshere. You will see that this bookdevelops the rules for arithmeticfrom an algebraic point of view.This should help those studentswho are going on to take an algebracourse.C Properties of AdditionOnce we become familiar with addition, we may notice some facts about additionthat are true regardless of the numbers involved. The first of these facts involvesthe number 0 (zero).Whenever we add 0 to a number, the result is the original number. For example,7 + 0 = 7 and 0 + 3 = 3Because this fact is true no matter what number we add to 0, we call it a propertyof 0.Property Addition Property of 0Text If we here. let a represent any number, then it is always true thata + 0 = a and 0 + a = aIn words: Adding 0 to any number leaves that number unchanged.NoteWhen we use lettersto represent numbers,as we do when wesay “If a and b are any two numbers,”then a and b are called variables,because the values they takeon vary. We use the variables aand b in the definitions and propertieson this page because wewant you to know that the definitionsand properties are true for allnumbers that you will encounter inthis book.A second property we notice by becoming familiar with addition is that the orderof two numbers in a sum can be changed without changing the result.3 + 5 = 8 and 5 + 3 = 84 + 9 = 13 and 9 + 4 = 13This fact about addition is true for all numbers. The order in which you add twonumbers doesn’t affect the result. We call this fact the commutative property ofaddition, and we write it in symbols as follows.Commutative Property of AdditionIf a and b are any two numbers, then it is always true thata + b = b + aIn words: Changing the order of two numbers in a sum doesn’t change theresult.

1.2 Addition with Whole Numbers, and Perimeter17Study SkillSAccept DefinitionsIt is important that you don’t overcomplicate definitions. When I tell my students that my nameis Mr. McKeague, they don’t ask “why?” You should approach definitions in the same way.Just accept them as they are, and memorize them if you have to. If someone asks you whatthe commutative property is, you should be able to respond, “With addition, the commutativeproperty says that if a and b are two numbers then a + b = b + a. In other words, you canchange the order of two numbers you are adding without changing the result.”Example 5Use the commutative property of addition to rewrite eachsum.a. 4 + 6 b. 5 + 9 c. 3 + 0 d. 7 + nSolution The commutative property of addition indicates that we can changethe order of the numbers in a sum without changing the result. Applying thisproperty we have:a. 4 + 6 = 6 + 4b. 5 + 9 = 9 + 5c. 3 + 0 = 0 + 3d. 7 + n = n + 7Notice that we did not actually add any of the numbers. The instructions were touse the commutative property, and the commutative property involves only theorder of the numbers in a sum.The last property of addition we will consider here has to do with sums of morethan two numbers. Suppose we want to find the sum of 2, 3, and 4. We could add2 and 3 first, and then add 4 to what we get:(2 + 3) + 4 = 5 + 4 = 9Or, we could add the 3 and 4 together first and then add the 2:2 + (3 + 4) = 2 + 7 = 9The result in both cases is the same. If we try this with any other numbers, thesame thing happens. We call this fact about addition the associative property ofaddition, and we write it in symbols as follows.5. Use the commutative propertyof addition to rewrite each sum.a. 7 + 9b. 6 + 3c. 4 + 0d. 5 + nInstructor NoteSome students will question thepurpose of problems like the onesshown in Example 5: “Why isn’t theanswer for Example 5a just 10?” Ipoint out that the purpose of thistype of problem is to help thembecome familiar with a propertyor idea that may be new to them.These are not addition problems butare problems designed to illustratesomething important that they willneed later on in the course.NoteThis discussion is hereto show why we writethe next property theway we do. Sometimes it is helpfulto look ahead to the propertyitself (in this case, the associativeproperty of addition) to see what itis that is being justified.Associative Property of AdditionIf a, b, and c represent any three numbers, then(a + b) + c = a + (b + c)In words: Changing the grouping of three numbers in a sum doesn’t changethe result.Answer5. a. 9 + 7 b. 3 + 6 c. 0 + 4d. n + 5

18Chapter 1 Whole Numbers6. Use the associative property ofaddition to rewrite each sum.a. (3 + 2) + 9b. (4 + 10) + 1c. 5 + (9 + 1)d. 3 + (8 + n)Example 6Use the associative property of addition to rewrite eachsum.a. (5 + 6) + 7 b. (3 + 9) + 1 c. 6 + (8 + 2) d. 4 + (9 + n)Solution The associative property of addition indicates that we are free toregroup the numbers in a sum without changing the result.a. (5 + 6) + 7 = 5 + (6 + 7)b. (3 + 9) + 1 = 3 + (9 + 1)c. 6 + (8 + 2) = (6 + 8) + 2d. 4 + (9 + n) = (4 + 9) + nThe commutative and associative properties of addition tell us that when addingwhole numbers, we can use any order and grouping. When adding several numbers,it is sometimes easier to look for pairs of numbers whose sums are 10, 20,and so on.7. Add.a. 6 + 2 + 4 + 8 + 3b. 24 + 17 + 36 + 13NoteThe letter n as weare using it here is avariable, because itrepresents a number. In this case itis the number that is a solution toan equation.Example 7Add: 9 + 3 + 2 + 7 + 1Solution We find pairs of numbers that we can add quickly:88n9 + 3 + 2 + 7 + 1= 10 + 10 + 2= 22D Solving Equations88n88nm8888888888888m8888We can use the addition table to help solve some simple equations. If n is used torepresent a number, then the equationn + 3 = 5will be true if n is 2. The number 2 is therefore called a solution to the equation,because, when we replace n with 2, the equation becomes a true statement:2 + 3 = 5Equations like this are really just puzzles, or questions. When we say, “Solvethe equation n + 3 = 5,” we are asking the question, “What number do we add to3 to get 5?”When we solve equations by reading the equation to ourselves and then statingthe solution, as we did with the equation above, we are solving the equation byinspection.8. Use inspection to find the solutionto each equation.a. n + 9 = 17b. n + 2 = 10c. 8 + n = 9d. 16 = n + 10Answers6. a. 3 + (2 + 9) b. 4 + (10 + 1)c. (5 + 9) + 1 d. (3 + 8) + n7. a. 23 b. 908. a. 8 b. 8 c. 1 d. 6Example 8Find the solution to each equation by inspection.a. n + 5 = 9b. n + 6 = 12c. 4 + n = 5d. 13 = n + 8Solution We find the solution to each equation by using the addition factsgiven in Table 1.a. The solution to n + 5 = 9 is 4, because 4 + 5 = 9.b. The solution to n + 6 = 12 is 6, because 6 + 6 = 12.c. The solution to 4 + n = 5 is 1, because 4 + 1 = 5.d. The solution to 13 = n + 8 is 5, because 13 = 5 + 8.

1.2 Addition with Whole Numbers, and Perimeter19EPerimeterfacts from geometry PerimeterWe end this section with an introduction to perimeter. Here we will find theperimeter of several different shapes called polygons. A polygon is a closedgeometric figure, with at least three sides, in which each side is a straight linesegment.The most common polygons are squares, rectangles, and triangles.Examples of these are shown in Figure 2.squarerectangletrianglewhslbFigure 2NoteIn the triangle, thesmall square wherethe broken line meetsthe base is the notation we use toshow that the two line segmentsmeet at right angles. That is, theheight h and the base b are perpendicularto each other; the anglebetween them is 90°.In the square, s is the length of the side, and each side has the same length.In the rectangle, l stands for the length, and w stands for the width. Thewidth is usually the lesser of the two. The b and h in the triangle are the baseand height, respectively. The height is always perpendicular to the base. Thatis, the height and base form a 90°, or right, angle where they meet.DefinitionThe perimeter of any polygon is the sum of the lengths of the sides, and itis denoted with the letter P.Example 9Find the perimeter of each geometric figure.a. b. c.36 yards24 feet23 yards9. Find the perimeter of each geometricfigure.a.15 inches37 feet24 yards24 yards7 feet12 yardsb.Solution In each case we find the perimeter by adding the lengths of all thesides.a. The figure is a square. Because the length of each side in thesquare is the same, the perimeter isP = 15 + 15 + 15 + 15 = 60 inchesc.88 inches44 yards33 inches66 yardsb. In the rectangle, two of the sides are 24 feet long, and the othertwo are 37 feet long. The perimeter is the sum of the lengths ofthe sides.77 yardsP = 24 + 24 + 37 + 37 = 122 feetc. For this polygon, we add the lengths of the sides together. Theresult is the perimeter.P = 36 + 23 + 24 + 12 + 24 = 119 yardsAnswer9. a. 28 feet b. 242 inchesc. 187 yards

20Chapter 1 Whole NumbersU s i n gCalculatorsT e c h n o l o g yFrom time to time we will include some notes like this one, which show howa calculator can be used to assist us with some of the calculations in the book.Most calculators on the market today fall into one of two categories: thosewith algebraic logic and those with function logic. Calculators with algebraiclogic have a key with an equals sign on it. Calculators with function logic donot have an equals key. Instead they have a key labeled ENTER or EXE (forexecute). Scientific calculators use algebraic logic, and graphing calculators,such as the TI-83, use function logic.Here are the sequences of keystrokes to use to work the problem shown inPart c of Example 9.Scientific Calculator: 36 + 23 + 24 + 12 + 24 =Graphing Calculator: 36 + 23 + 24 + 12 + 24 ENTGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. What number is the sum of 6 and 8?2. Make up an addition problem using the number 456 that does notinvolve carrying.3. Make up an addition problem using the number 456 that involves carryingfrom the ones column to the tens column only.4. What is the perimeter of a geometric figure?

1.2 Problem Set21Problem Set 1.2A Find each of the following sums. (Add.) [Examples 1–4]1. 3 + 5 + 7152. 2 + 8 + 6163. 1 + 4 + 9144. 2 + 8 + 3135. 5 + 9 + 4 + 6246. 8 + 1 + 6 + 2177. 1 + 2 + 3 + 4 + 5158. 5 + 6 + 7 + 8 + 9359. 9 + 1 + 8 + 22010. 7 + 3 + 6 + 420A Add each of the following. (There is no carrying involved in these problems.) [Examples 1, 2]11. 43256812. 56237913. 81179814. 37225915. 4,2813,0167,29716. 2,7491,2503,99917. 3,4823,0056,48718. 2,4967,5039,99919. 3221439620. 52134013599621. 6,2452031,0017,44922. 274,5103424,879A Add each of the following. (All problems involve carrying in at least one column.) [Examples 3, 4]23. 49166524. 852911425. 742810226. 36468227. 68219387528. 43927070929. 63819182930. 4445951,03931. 4,9635,42810,39132. 8,2917,48915,78033. 6,2059,99916,20434. 8,8889,99918,88735. 56,78998,765155,55436. 45,67887,654133,33237. 52,46858,642111,11038. 13,57997,531111,11039. 4,2968,7204,37517,39140. 5,6374817,89914,01741. 4,9944499,44914,89242. 6,8243714,85712,05243. 1234567818044. 2143658721645. 9994445552222,22046. 6464645252521,88747. 9,2456728,3412718,28548. 459,876546,78916,764

22Chapter 1 Whole NumbersBComplete the following tables.49.First Second Their50.Number Number Suma b a + b61 38 9963 36 9965 34 9967 32 9951.First Second Their52.Number Number Suma b a + b9 16 2536 64 10081 144 225144 256 400First Second TheirNumber Number Suma b a + b10 45 5520 35 5530 25 5540 15 55First Second TheirNumber Number Suma b a + b25 75 10024 76 10023 77 10022 78 100C Rewrite each of the following using the commutative property of addition. [Example 5]53. 5 + 99 + 554. 2 + 11 + 255. 3 + 88 + 356. 9 + 22 + 957. 6 + 44 + 658. 1 + 77 + 1C Rewrite each of the following using the associative property of addition. [Example 6]59. (1 + 2) + 31 + (2 + 3)60. (4 + 5) + 94 + (5 + 9)61. (2 + 1) + 62 + (1 + 6)62. (2 + 3) + 82 + (3 + 8)63. 1 + (9 + 1)(1 + 9) + 164. 2 + (8 + 2)(2 + 8) + 265. (4 + n) + 14 + (n + 1)66. (n + 8) + 1n + (8 + 1)D Find a solution for each equation. [Example 8]67. n + 6 = 10n = 468. n + 4 = 7n = 369. n + 8 = 13n = 570. n + 6 = 15n = 971. 4 + n = 12n = 872. 5 + n = 7n = 273. 17 = n + 9n = 874. 13 = n + 5n = 8B Write each of the following expressions in words. Use the word sum in each case. [Table 2]75. 4 + 9The sum of 4 and 976. 9 + 4The sum of 9 and 477. 8 + 1The sum of 8 and 178. 9 + 9The sum of 9 and 979. 2 + 3 = 5The sum of 2 and 3 is 5.80. 8 + 2 = 10The sum of 8 and 2 is 10.B Write each of the following in symbols. [Table 2]81. a. The sum of 5 and 2 5 + 2b. 3 added to 8 8 + 383. a. m increased by 1 m + 1b. The sum of m and n m + n82. a. The sum of a and 4 a + 4b. 6 more than x x + 684. a. The sum of 4 and 8 is 12. 4 + 8 = 12b. The sum of a and b is 6. a + b = 6

1.2 Problem Set23E Find the perimeter of each figure. The first four figures are squares. [Example 9]85.86.87.88.3 in. 9 in. 4 ft 2 ft12 in.36 in.16 ft8 ft89.3 yd90.1 yd10 yd5 yd26 yd12 yd91.92.5 in.6 in.4 in.10 in.7 in.12 in.18 in.26 in.EApplying the Concepts93. Classroom appliances use a lot of energy. You cansave energy by unplugging or turning of unusedappliances.94. The information in the illustration represents the numberof picture messages sent for the first nine months of theyear, in millions.Energy EstimatesAll units given as watts per hour.Ceiling fanStereoTelevisionVCR/DVD playerPrinterPhotocopierCoffee maker201251304004004001000A Picture’s Worth 1,000 Words50403021 2120101030Jan Feb Mar Apr May Jun26Jul32Aug41SepSource: dosomething.org 2008Use the information in the given illustration to find thefollowing:a. the number of watts/hour saved by unplugging aDVD player and a television150 watts/hourb. the number of watts/hour saved by unplugging aceiling fan and a coffee maker1,125 watts/hourUse the information to find the following:a. the number of picture messages sent in all ninemonths154 million picture messagesb. the number of picture messages sent in March andApril13 million picture messages

24Chapter 1 Whole Numbers95. Checkbook Balance On Monday Bob had a balance of$241 in his checkbook. On Tuesday he made a depositof $108, and on Thursday he wrote a check for $24.What was the balance in his checkbook on Wednesday?$349RECORD ALL CHARGES OR CREDITS THAT AFFECT YOUR ACCOUNTNUMBER DATE DESCRIPTION OF TRANSACTIONPAYMENT/DEBITDEPOSIT/CREDIT(-)(+)11/06 Deposit $108 001401 11/18 Postage Stamps $24 00BALANCE$241??0096. Number of Passengers A plane flying from Los Angelesto New York left Los Angeles with 67 passengers onboard. The plane stopped in Bakersfield and picked up28 passengers, and then it stopped again in Dallas where57 more passengers came on board. How many passengerswere on the plane when it landed in New York?152 passengers97. College Costs According to data from The Chronicle ofHigher Education, the most expensive college in thecountry is George Washington University inWashington, D.C. According to the university’swebsite, a student entering as a freshman during the2008 – 09 academic year can expect to pay theexpenses shown in the chart below:2008-09 Costs for Attending George Washington UniversityTuition/Fees $40,392Transportation $2,200Health Insurance $1,800Room and Board $13,600Books/Supplies $1,185Personal Expenses $3,200a. What are the total of the expenses for one year atGeorge Washington University?$62,377b. How much of these total expenses are collegerelated?$55,177c. What is the total amount for expenses that are notdirectly related to attending this college?$7,20098. Improving Your Quantitative Literacy Quantitative literacyis a subject discussed by many people involved inteaching mathematics. The person they are concernedwith when they discuss it is you. We are going to workat improving your quantitative literacy, but before we dothat we should answer the question, what is quantitativeliteracy? Lynn Arthur Steen, a noted mathematicseducator, has stated that quantitative literacy is “thecapacity to deal effectively with the quantitative aspectsof life.”a. Give a definition for the word quantitative.b. Give a definition for the word literacy.c. Are there situations that occur in your life that youfind distasteful or that you try to avoid because theyinvolve numbers and mathematics? If so, list some ofthem here. (For example, some people find the processof buying a car particularly difficult because theyfeel that the numbers and details of the financing arebeyond them.)

Rounding and Estimating1.3Introduction . . .Many times when we talk about numbers, it is helpful to usenumbers that have been rounded off, rather than exact numbers.For example, the city where I live has a population of 43,704. Butwhen I tell people how large the city is, I usually say, “The populationis about 44,000.” The number 44,000 is the original numberrounded to the nearest thousand. The number 43,704 iscloser to 44,000 than it is to 43,000, so it is rounded to 44,000.We can visualize this situation on the number line.Welcome toSan Luis ObispoFounded 1772Population 43,704ObjectivesA Round whole numbers.B Estimate the answer to a problem.FurtherCloser43,000 43,704 44,000A RoundingThe steps used in rounding numbers are given below.Strategy Rounding Whole NumbersTo summarize, we list the following steps:Step 1: Locate the digit just to the right of the place you are to round to.NoteAfter you have usedthe steps listed hereto work a few problems,you will find that the procedurebecomes almost automatic.Step 2: If that digit is less than 5, replace it and all digits to its right withzeros.Step 3: If that digit is 5 or more, replace it and all digits to its right with zeros,and add 1 to the digit to its left.You can see from these rules that in order to round a number you must be toldwhat column (or place value) to round to.Example 1Round 5,382 to the nearest hundred.Solution There is a 3 in the hundreds column. We look at the digit just to itsright, which is 8. Because 8 is greater than 5, we add 1 to the 3, and we replacethe 8 and 2 with zeros:Practice Problems1. Round 5,742 to the nearesta. hundredb. thousand5,382 is 5,400 to the nearest hundred8n8n8888nGreater than 5 Add 1 to Put zerosget 4 hereExample 2Round 94 to the nearest ten.Solution There is a 9 in the tens column. To its right is 4. Because 4 is lessthan 5, we simply replace it with 0:94 is 90 to the nearest ten8n8nLess than 5Replaced with zero1.3 Rounding and Estimating2. Round 87 to the nearesta. tenb. hundredAnswers1. a. 5,700 b. 6,0002. a. 90 b. 10025

26Chapter 1 Whole Numbers3. Round 980 to the nearesta. hundredb. thousandExample 3Round 973 to the nearest hundred.Solution We have a 9 in the hundreds column. To its right is 7, which isgreater than 5. We add 1 to 9 to get 10, and then replace the 7 and 3 with zeros:973 is 1,000 to the nearest hundred88n88n88nGreater Add 1 to Put zerosthan 5 get 10 here4. Round 376,804,909 to thenearesta. millionb. ten thousandExample 4Round 47,256,344 to the nearest million.Solution We have 7 in the millions column. To its right is 2, which is less than5. We simply replace all the digits to the right of 7 with zeros to get:47,256,344 is 47,000,000 to the nearest million8n8n88nLess than 5 Leave as is Replaced with zerosTable 1 gives more examples of rounding.Table 1rounded to the NearestOriginal Number ten Hundred thousandHouse Payments $10,200Taxes $6,137Miscellaneous $6,142Entertainment $2,142Car Expenses $4,847Savings $2,149Rule6,914 6,910 6,900 7,0008,485 8,490 8,500 8,0005,555 5,560 5,600 6,0001,234 1,230 1,200 1,000If we are doing calculations and are asked to round our answer, we do all ourarithmetic first and then round the result. That is, the last step is to round theanswer; we don’t round the numbers first and then do the arithmetic.Food $5,2965. Use the pie chart above toanswer these questions.a. To the nearest ten dollars,what is the total amountspent on food and carexpenses?b. To the nearest hundred dollars,how much is spent onsavings and taxes?c. To the nearest thousand dollars,how much is spent onitems other than food andentertainment?Example 5The pie chart in the margin shows how a family earning$36,913 a year spends their money.a. To the nearest hundred dollars, what is the total amount spenton food and entertainment?b. To the nearest thousand dollars, how much of their income isspent on items other than taxes and savings?Solution In each case we add the numbers in question and then round thesum to the indicated place.a. We add the amounts spent on food and entertainment and thenround that result to the nearest hundred dollars.Food $5,296Entertainment 2,142Total$7,438 = $7,400 to the nearest hundred dollarsAnswers3. a. 1,000 b. 1,0004. a. 377,000,000 b. 376,800,000

1.3 Rounding and Estimating27b. We add the numbers for all items except taxes and savings.House payments $10,200Food 5,296Car expenses 4,847Entertainment 2,142Miscellaneous 6,142Total$28,627 = $29,000 to the nearestthousand dollarsBEstimatingWhen we estimate the answer to a problem, we simplify the problem so that anapproximate answer can be found quickly. There are a number of ways of doingthis. One common method is to use rounded numbers to simplify the arithmeticnecessary to arrive at an approximate answer, as our next example shows.Example 6Estimate the answer to the following problem by roundingeach number to the nearest thousand.4,8721,691777+ 6,124Solution We round each of the four numbers in the sum to the nearest thousand.Then we add the rounded numbers.4,872 rounds to 5,0001,691 rounds to 2,000777 rounds to 1,000+ 6,124 rounds to + 6,00014,000We estimate the answer to this problem to be approximately 14,000. The actualanswer, found by adding the original unrounded numbers, is 13,464.Here is a practical application for which the ability to estimate can be a usefultool.Example 7On the way home from classes you stop at the local grocerystore to pick up a few things. You know that you have a $20.00 bill in yourwallet. You pick up the following items: a loaf of wheat bread for $2.29, a gallonof milk for $3.96, a dozen eggs for $2.18, a pound of apples for $1.19, and a boxof your favorite cereal for $4.59. Use estimation to determine if you will haveenough to pay for your groceries when you get to the cashier.6. Estimate the answer by firstrounding each number to thenearest thousand.a. 5,2872,561888+4,898b. 7023,9441,001+3,500NoteIn Example 6 we areasked to estimatean answer, so it isokay to round the numbers inthe problem before adding them.In Example 5 we are asked for arounded answer, meaning that weare to find the exact answer to theproblem and then round to theindicated place. In that case wemust not round the numbers in theproblem before adding.SolutionWe round the items in our grocery cart off to the nearest dollar:wheat bread for $2.29 rounds to $2.00milk for $3.96 rounds to $4.00eggs for $2.18 rounds to $2.00apples for $1.19 rounds to $1.00+ cereal for $4.59 rounds to + $5.00$14.00We estimate our total to be $14.00. Thus, $20.00 will be enough to pay for thegroceries. (The actual cost of the groceries is $14.21.)Answer5. a. $10,140 b. $8,300c. $29,0006. a. 14,000 b. 10,000

28Chapter 1 Whole NumbersDescriptive StatisticsBar ChartsThe table and chart below give two representations for the amount of caffeinein five different drinks, one numeric and the other visual.Table 2100100BeverageCaffeine(6-ounce cup)(in milligrams)8070Brewed coffee 1006050Instant coffee 7040Tea 5020Cocoa 55 40Decaffeinated coffee 4The diagram in Figure 1 is called a bar chart. The horizontal line below whichthe drinks are listed is called the horizontal axis, while the vertical line that isCaffeine (in milligrams)labeled from 0 to 100 is called the vertical axis.BrewedcoffeeInstantcoffeeTeaFigure 1CocoaDecafcoffeeGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Describe the process you would use to round the number 5,382 to thenearest thousand.2. Describe the process you would use to round the number 47,256,344 tothe nearest ten thousand.3. Find a number not containing the digit 7 that will round to 700 whenrounded to the nearest hundred.4. When I ask a class of students to round the number 7,499 to the nearestthousand, a few students will give the answer as 8,000. In what way arethese students using the rule for rounding numbers incorrectly?

1.3 Problem Set29Problem Set 1.3A Round each of the numbers to the nearest ten. [Examples 1–5]1. 42402. 44403. 46504. 48505. 45506. 73707. 77808. 75809. 45846010. 45546011. 47147012. 68068013. 56,78256,78014. 32,80732,81015. 4,5044,50016. 3,8973,900Round each of the numbers to the nearest hundred. [Examples 1–5]17. 54950018. 9541,00019. 83380020. 60460021. 89990022. 9881,00023. 10901,10024. 6,7786,80025. 5,0445,00026. 56,99057,00027. 39,60339,60028. 31,99932,000Round each of the numbers to the nearest thousand. [Examples 1–5]29. 4,6705,00030. 9,0549,00031. 9,76010,00032. 4,4444,00033. 9781,00034. 5671,00035. 657,892658,00036. 688,909689,00037. 509,905510,00038. 608,433608,00039. 3,789,3453,789,00040. 5,744,5005,745,000

30Chapter 1 Whole NumbersA Complete the following table by rounding the numbers on the left as indicated by the headings in the table. [Examples 1–5]rounded to the NearestOriginalNumber ten Hundred thousand41. 7,821 7,820 7,800 8,00042. 5,945 5,950 5,900 6,00043. 5,999 6,000 6,000 6,00044. 4,353 4,350 4,400 4,00045. 10,985 10,990 11,000 11,00046. 11,108 11,110 11,100 11,00047. 99,999 100,000 100,000 100,00048. 95,505 95,510 95,500 96,000B Estimating Estimate the answer to each of the following problems by rounding each number to the indicated placevalue and then adding. [Example 6]49. hundred750275+ 1201,20050. thousand1,891765+ 3,2236,00051. hundred472422536+ 5111,90052. hundred399601744+ 2982,00053. thousand25,3997,60118,744+ 6,29858,00054. thousand9,9998,8887,777+ 6,66634,00055. hundred9,9998,8887,777+ 6,66633,40056. ten thousand127,67572,560+ 219,065420,00057. ten thousand65,00031,00015,555+ 72,000190,00058. ten10,06110,04410,035+ 10,02540,17059. hundred20,15018,25012,350+ 30,45081,40060. hundred1,9502,8493,750+ 4,64913,200

1.3 Problem Set31Applying the Concepts61. Age of Mothers About 4 million babies were born in 2006. The chart shows the breakdown by mothers’ age and numberof babies. Use the chart to answer the following questions.Who’s Having All the Babiesa. What is the exact number of babies born in 2006?4,265,997 babiesUnder 20: 441,83220–29:30–39:40–54:Source: National Center for Health Statistics, 20062,262,6941,449,039112,432b. Using your answer from Part a, is the statement“About 4 million babies were born in 2006” correct?Yesc. To the nearest hundred thousand, how many babieswere born to mothers aged 20 to 29 in 2006?2,300,000 babiesd. To the nearest thousand, how many babies wereborn to mothers 40 years old or older?112,000 babies62. Business Expenses The pie chart shows one year’s worth of expenses for a small business. Use the chart to answer thefollowing questions.Salaries $20,761Supplies $11,456Postage $3,792Telephone $3,652All Other Expenses $8,496Rent and Utilities $7,499Car Expenses $3,205a. To the nearest hundred dollars, how much was spenton postage and supplies?$15,200b. Find the total amount spent, to the nearest hundreddollars, on rent and utilities and car expenses.$10,700c. To the nearest thousand dollars, how much wasspent on items other than salaries and rent andutilities?$31,000d. To the nearest thousand dollars, how much wasspent on items other than postage, supplies, and carexpenses?$40,000

32Chapter 1 Whole NumbersThe bar chart below is similar to the one we studied in this section. It was given to me by a friend who owns and operatesan alcohol dragster. The dragster contains a computer that gives information about each of his races. This particular racewas run during the 1993 Winternationals. The bar chart gives the speed of a race car in a quarter-mile drag race everysecond during the race. The horizontal lines have been added to assist you with Problems 63–66.63. Is the speed of the race car after 3 seconds closer to 160miles per hour or 190 miles per hour?160 miles per hour250Speed of a Race Car64. After 4 seconds, is the speed of the race car closer to 150miles per hour or 190 miles per hour?190 miles per hour65. Estimate the speed of the car after 1 second.Answers will vary, but 70 miles per hour is a good estimate.Speed (in miles per hour)20015010050012 3 4 5 6Time (in seconds)66. Estimate the speed of the car after 6 seconds.Answers will vary, but 225 miles per hour is a good estimate.67. Fast Food The following table lists the number of calories consumed by eating some popular fast foods. Use the axes inthe figure below to construct a bar chart from the information in the table.Calories in Fast FoodFoodCaloriesMcDonald’s Hamburger 270Burger King Hamburger 260Jack in the Box Hamburger 280McDonald’s Big Mac 510Burger King Whopper 630Jack in the Box Colossus Burger 940Number of calories10009008007006005004003002001000McDonald’sHamburgerBurger KingHamburgerJack in the BoxHamburgerMcDonald’sBig MacBurger KingWhopperJack in the BoxColossus Burger68. Exercise The following table lists the number of calories burned in 1 hour of exercise by a person who weighs 150pounds. Use the axes in the figure below to construct a bar chart from the information in the table.Calories Burned by a150-Pound Person in One HourActivityCaloriesBicycling 374Bowling 265Handball 680Number of calories burnedin one hour700600500400300200100Jazzercise 3400Jogging 680Skiing 544BicyclingBowlingHandballJazzerciseActivityJoggingSkiing

Introduction . . .Subtraction with Whole NumbersIn business, subtraction is used to calculate profit. Profit is found by subtractingcosts from revenue. The following double bar chart shows the costs and revenueof the Baby Steps Shoe Company during one 4-week period.1.4ObjectivesA Understand the notation andvocabulary of subtraction.B Subtract whole numbers.C Subtraction with borrowing.$12,000$10,000$8,000$6,000Costs Revenue$7,500$6,000 $6,000$5,000$6,300$8,400$7,000$10,500$4,000$2,0000Week 1 Week 2 Week 3 Week 4To find the profit for Week 1, we subtract the costs from the revenue, as follows:Profit = $6,000 − $5,000Profit = $1,000Subtraction is the opposite operation of addition. If you understand additionand can work simple addition problems quickly and accurately, then subtractionshouldn’t be difficult for you.A VocabularyThe word difference always indicates subtraction. We can state this in symbols byletting the letters a and b represent numbers.DefinitionThe difference of two numbers a and b is a − bTable 1 gives some word statements involving subtraction and their mathematicalequivalents written in symbols.Table 1In Englishin SymbolsThe difference of 9 and 1 9 − 1The difference of 1 and 9 1 − 9The difference of m and 4 m − 4The difference of x and yx − y3 subtracted from 8 8 − 32 subtracted from t t − 2The difference of 7 and 4 is 3. 7 − 4 = 3The difference of 9 and 3 is 6. 9 − 3 = 61.4 Subtraction with Whole Numbers33

34Chapter 1 Whole NumbersBThe Meaning of SubtractionWhen we want to subtract 3 from 8, we write8 − 3, 8 subtract 3, or 8 minus 3The number we are looking for here is the difference between 8 and 3, or thenumber we add to 3 to get 8. That is:8 − 3 = ? is the same as ? + 3 = 8In both cases we are looking for the number we add to 3 to get 8. The number weare looking for is 5. We have two ways to write the same statement.SubtractionAddition8 − 3 = 5 or 5 + 3 = 8For every subtraction problem, there is an equivalent addition problem. Table 2lists some examples.Table 2SubtractionAddition7 − 3 = 4 because 4 + 3 = 79 − 7 = 2 because 2 + 7 = 910 − 4 = 6 because 6 + 4 = 1015 − 8 = 7 because 7 + 8 = 15To subtract numbers with two or more digits, we align the numbers verticallyand subtract in columns.Practice Problems1. Subtract.a. 684 − 431b. 7,406 − 3,405Example 1Subtract: 376 − 241Solution We write the problem vertically, aligning digits with the same placevalue. Then we subtract in columns.376− 241 m888 Subtract the bottom number in each column135 from the number above it2. a. Subtract 405 from 6,857.b. Subtract 234 from 345.Example 2Subtract 503 from 7,835.SolutionIn symbols this statement is equivalent to7,835 − 503To subtract we write 503 below 7,835 and then subtract in columns.7 , 8 3 5− 5 0 37 , 3 3 2Answers1. a. 253 b. 4,0012. a. 6,452 b. 111m888888888888888m8888888888m88888m5 − 3 = 2 Ones3 − 0 = 3 Tens8 − 5 = 3 Hundreds7 − 0 = 7 Thousands

1.4 Subtraction with Whole Numbers35As you can see, subtraction problems like the ones in Examples 1 and 2 arefairly simple. We write the problem vertically, lining up the digits with the sameplace value, and subtract in columns. We always subtract the bottom numberfrom the top number.C Subtraction with BorrowingSubtraction must involve borrowing when the bottom digit in any column islarger than the digit above it. In one sense, borrowing is the reverse of the carryingwe did in addition.Example 3Subtract: 92 − 45Solution We write the problem vertically with the place values of the digitsshowing:3. Subtract.a. 63 − 47b. 532 − 40392 = 9 tens + 2 ones− 45 = 4 tens + 5 onesLook at the ones column. We cannot subtract immediately, because 5 is largerthan 2. Instead, we borrow 1 ten from the 9 tens in the tens column. We canrewrite the number 92 as9 tens + 2 onesm88888888m8= 8 tens + 1 ten + 2 onesmm8888= 8 tens + 12 onesNow we are in a position to subtract.NoteThe discussion hereshows why borrowingis necessary and howwe go about it. To understandborrowing you should pay closeattention to this discussion.92 = 9 tens + 2 ones = 8 tens + 12 ones− 45 = 4 tens + 5 ones = 4 tens + 5 ones4 tens + 7 onesThe result is 4 tens + 7 ones, which can be written in standard form as 47.Writing the problem out in this way is more trouble than is actually necessary.The shorthand form of the same problem looks like this:9∙ 8 12 2∙− 4 54 7m88888m8m88888888812 − 5 = 7 Ones8 − 4 = 4 TensThis shows we haveborrowed 1 ten to gowith the 2 onesThis shortcut form shows all the necessary work involved in subtraction withborrowing. We will use it from now on.Answer3. a. 16 b. 129

36Chapter 1 Whole NumbersThe borrowing that changed 9 tens + 2 ones into 8 tens + 12 ones can be visualizedwith money.=$90 $2 $80 $124. a. Find the difference of 656 and283.b. Find the difference of 3,729and 1,749.Example 4Find the difference of 549 and 187.Solution In symbols the difference of 549 and 187 is written549 − 187Writing the problem vertically so that the digits with the same place value arealigned, we have549− 187The top number in the tens column is smaller than the number below it. Thismeans that we will have to borrow from the next larger column.5∙ 4 14 m888888888888888 Borrow 1 hundred to go4∙ 9with the 4 tens− 1 8 73 6 2m8888888888m88888m9 − 7 = 2 Ones14 − 8 = 6 Tens4 − 1 = 3 HundredsThe actual work we did in borrowing looks like this:5 hundreds + 4 tens + 9 onesm8m88888888888= 4 hundreds + 1 hundred + 4 tens + 9 onesm7m888888888= 4 hundreds + 14 tens + 9 onesAnswers4. a. 373 b. 1,980Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Which sentence below describes the problem shown in Example 1?a. The difference of 241 and 376 is 135.b. The difference of 376 and 241 is 135.2. Write a subtraction problem using the number 234 that involvesborrowing from the tens column to the ones column.3. Write a subtraction problem using the number 234 in which the answeris 111.4. Describe how you would subtract the number 56 from the number 93.

1.4 Problem Set37Problem Set 1.4A Perform the indicated operation. [Examples 1, 2, 4]1. Subtract 24 from 56.323. Subtract 23 from 45.225. Find the difference of 29 and 19.107. Find the difference of 126 and 15.1112. Subtract 71 from 89.184. Subtract 97 from 98.16. Find the difference of 37 and 27.108. Find the difference of 348 and 32.316B Work each of the following subtraction problems. [Examples 1, 2]9. 975− 66331210. 480− 26022011. 904− 50140312. 657− 50715013. 9,876− 8,7651,11114. 5,008− 3,0022,00615. 7,976− 3,4324,54416. 6,980− 4706,510C Find the difference in each case. (These problems all involve borrowing.) [Example 3]17. 52 − 371518. 65 − 481719. 70 − 373320. 90 − 216921. 74 − 69522. 31 − 28323. 51 − 183324. 64 − 58625. 329 − 2349526. 518 − 4922627. 348 − 19615228. 759 − 6619829. 932− 65827430. 895− 59729831. 647− 15948832. 842− 19964333. 905− 36753834. 804− 23856635. 600− 43716336. 800− 34245837. 4,583− 2,9731,61038. 7,849− 2,9574,89239. 79,040− 32,95746,08340. 86,492− 78,5067,986

38Chapter 1 Whole NumbersAComplete the following tables.41.First Second the Difference42.Number number of a and ba b a – b25 15 1024 16 823 17 622 18 4First Second the DifferenceNumber number of a and ba b a – b90 79 1180 69 1170 59 1160 49 1143.First Second the Difference44.Number number of a and ba b a – b400 256 144400 144 256225 144 81225 81 144First Second the DifferenceNumber number of a and ba b a – b100 36 64100 64 3625 16 925 9 16AWrite each of the following expressions in words. Use the word difference in each case.45. 10 − 2The difference of 10 and 246. 9 − 5The difference of 9 and 547. a − 6The difference of a and 648. 7 − xThe difference of 7 and x49. 8 − 2 = 6The difference of 8 and 2 is 6.50. m − 1 = 4The difference of m and 1 is 4.51. What number do you subtract from 8 to get 5?352. What number do you subtract from 6 to get 0?653. What number do you subtract from 15 to get 7?854. What number do you subtract from 21 to get 14?755. What number do you subtract from 35 to get 12?2356. What number do you subtract from 41 to get 11?30AWrite each of the following sentences as mathematical expressions.57. The difference of 8 and 38 − 358. The difference of x and 2x − 259. 9 subtracted from yy − 960. a subtracted from bb − a61. The difference of 3 and 2 is 1.3 − 2 = 162. The difference of 10 and y is 5.10 −y = 563. The difference of 37 and 9x is 10.37 − 9x = 1064. The difference of 3x and 2y is 15.3x − 2y = 1565. The difference of 2y and 15x is 24.2y − 15x = 2466. The difference of 25x and 9y is 16.25x − 9y = 1667. The difference of (x + 2) and(x + 1) is 1.(x + 2) − (x + 1) = 168. The difference of (x − 2) and(x − 4) is 2.(x − 2) − (x − 4) = 2

1.4 Problem Set39DApplying the ConceptsNot all of the following application problems involve only subtraction. Some involve addition as well. Be sure to read eachproblem carefully.69. Checkbook Balance Diane has $504 in her checkingaccount. If she writes five checks for a total of $249,how much does she have left in her account?$255Instructor NoteThe instructions to the word problems on this page point to a commonfeature of this book. Students will find that not all the word problemshere involve subtraction. Also, there may be numbers given in theproblem that are not a part of the solution to the problem.70. Checkbook Balance Larry has $763 in his checkingaccount. If he writes a check for each of the three billslisted below, how much will he have left in his account?$203ItemAmountRent $418Phone $25Car repair $11771. Home Prices In 1985, Mr. Hicks paid $137,500 for hishome. He sold it in 2008 for $310,000. What is thedifference between what he sold it for and what hebought it for?$172,50072. Oil Spills In March 1977, an oil tanker hit a reef offTaiwan and spilled 3,134,500 gallons of oil. In March1989, an oil tanker hit a reef off Alaska and spilled10,080,000 gallons of oil. How much more oil was spilledin the 1989 disaster?6,945,500 gallons73. Wind Speeds On April 12, 1934, the wind speed on topof Mount Washington was recorded at 231 miles perhour. When Hurricane Katrina struck on August 28,2005, the highest recorded wind speed was 140 milesper hour. How much faster was the wind on top ofMount Washington, than the winds from HurricaneKatrina?91 miles per hour74. Concert Attendance Eleven thousand, seven hundred fiftytwopeople attended a recent concert at the Pepsi Arenain Albany, New York. If the arena holds 17,500 people,how many empty seats were there at the concert?5,748 seats

40Chapter 1 Whole Numbers75. Computer Hard Drive You purchase a new computer with320 gigabytes of hard drive capacity. (A gigabyte isroughly a billion bytes). After loading a variety of programsyou discover that you have used 147 gigabytesof your hard drive’s capacity. How much hard drivecapacity do you still have available?173 gigabytes76. State Size Alaska is the largest state in the United Stateswith an area of 663,267 square miles. Rhode Island is thesmallest state with an area of 1,545 square miles. Howmany more square miles does Alaska have when comparedto Rhode Island?661,722 square miles77. Wind Energy The bar chart below shows the states producingthe most wind energy in 2006.78. Auto Insurance Costs The bar chart below shows the citieswith the highest annual insurance rates in 2006.Wind EnergyPriciest Cities for Auto InsuranceTexasCaliforniaIowaMinnesotaWashington2,768 MW2,361 MW936 MW895 MW818 MWSource: American Wind Energy Association 2006Detroit$5,894Philadelphia$4,440Newark, N.J.$3,977Los Angeles$3,430New York City$3,3030 $1000 $2000 $3000 $4000 $5000 $6000Annual PremiumsSource: Runzheimer Internationala. Use the information in the bar chart to fill in themissing entries in the table.a. Use the information in the bar chart to fill in themissing entries in the table.State Energy (megawatts)Texas 2,768California 2,361Iowa 936Washington 818CityCost (dollars)Detroit 5,894Philadelphia 4,440Los Angeles 3,430New York City 3,303b. How much more wind energy is produced in Texasthan in California?407 MWb. How much more does auto insurance cost in Detroitthan in Los Angeles?$2,464

Introduction . . .Multiplication with Whole Numbers,and AreaA supermarket orders 35 cases of a certain soft drink. If each case contains 12cans of the drink, how many cans were ordered?12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans1.5ObjectivesA Multiply whole numbers.B Understand the notation andvocabulary of multiplication.C Identify properties of multiplication.D Find the area of squares andrectangles.E Solve equations with multiplication.12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cansTo solve this problem and others like it, we must use multiplication.Multiplication is what we will cover in this section.A Multiplying Whole NumbersTo begin, we can think of multiplication as shorthand for repeated addition. Thatis, multiplying 3 times 4 can be thought of this way:3 times 4 = 4 + 4 + 4 = 12Multiplying 3 times 4 means to add three 4’s. We can write 3 times 4 as 3 × 4, or3 ⋅ 4.Example 1Multiply: 3 ⋅ 4,000Solution Using the definition of multiplication as repeated addition, we have3 ⋅ 4,000 = 4,000 + 4,000 + 4,000= 12,000Practice Problems1. Multiply.a. 4 ⋅ 70b. 4 ⋅ 700c. 4 ⋅ 7,000Here is one way to visualize this process.+ + =$4,000 $4,000 $4,000 $12,000Notice that if we had multiplied 3 and 4 to get 12 and then attached three zeroson the right, the result would have been the same.1.5 Multiplication with Whole Numbers, and AreaAnswer1. a. 280b. 2,800c. 28,00041

42NoteThe kind of notationwe will use to indicatemultiplication willdepend on the situation. For example,when we are solving equationsthat involve letters, it is not a goodidea to indicate multiplicationwith the symbol ×, since it couldbe confused with the letter x. Thesymbol we will use to indicate multiplicationmost often in this bookis the multiplication dot.NoteWe are assuming thatyou know the basicmultiplication factsgiven in the table below. If youneed some practice with thesefacts, go to Appendix 2 at the backof the book.Basic Multiplication Facts× 1 2 3 4 5 6 7 8 91 1 2 3 4 5 6 7 8 92 2 4 6 8 10 12 14 16 183 3 6 9 12 15 18 21 24 274 4 8 12 16 20 24 28 32 365 5 10 15 20 25 30 35 40 456 6 12 18 24 30 36 42 48 547 7 14 21 28 35 42 49 56 638 8 16 24 32 40 48 56 64 729 9 18 27 36 45 54 63 72 812. Identify the products and factorsin the statement6 ⋅ 7 = 423. Identify the products and factorsin the statement70 = 2 ⋅ 5 ⋅ 7Chapter 1 Whole NumbersBNotationThere are many ways to indicate multiplication. All the following statements areequivalent. They all indicate multiplication with the numbers 3 and 4.3 ⋅ 4, 3 × 4, 3(4), (3)4, (3)(4), 4× 3If one or both of the numbers we are multiplying are represented by letters, wemay also use the following notation:Vocabulary5n means 5 times nab means a times bWe use the word product to indicate multiplication. If we say “The product of 3and 4 is 12,” then we mean3 ⋅ 4 = 12Both 3 ⋅ 4 and 12 are called the product of 3 and 4. The 3 and 4 are called factors.Table 1Example 2Identify the products and factors in the statement9 ⋅ 8 = 72Solution The factors are 9 and 8, and the products are 9 ⋅ 8 and 72.Example 3Identify the products and factors in the statement30 = 2 ⋅ 3 ⋅ 5In Englishin SymbolsThe product of 2 and 5 2 ⋅ 5The product of 5 and 2 5 ⋅ 2The product of 4 and n4nThe product of x and yxyThe product of 9 and 6 is 54 9 ⋅ 6 = 54The product of 2 and 8 is 16 2 ⋅ 8 = 16Solution The factors are 2, 3, and 5. The products are 2 ⋅ 3 ⋅ 5 and 30.C Distributive PropertyTo develop an efficient method of multiplication, we need to use what is calledthe distributive property. To begin, consider the following two problems:Problem 1 Problem 23(4 + 5) 3(4) + 3(5)= 3(9) = 12 + 15= 27 = 27Answers2. Factors: 6, 7; products: 6 ⋅ 7 and423. Factors: 2, 5, 7;products: 2 ⋅ 5 ⋅ 7 and 70The result in both cases is the same number, 27. This indicates that the originaltwo expressions must have been equal also. That is,3(4 + 5) = 3(4) + 3(5)

1.5 Multiplication with Whole Numbers, and Area43This is an example of the distributive property. We say that multiplication distributesover addition.3(4 + 5) = 3(4) + 3(5)4 + 5 4 53 times = 3 times + 3 times3(4 + 5) = 3 ⋅ 4 + 3 ⋅ 5We can write this property in symbols using the letters a, b, and c to represent anythree whole numbers.Distributive PropertyIf a, b, and c represent any three whole numbers, thena(b + c) = a(b) + a(c)Suppose we want to find the product 7(65). By writing 65 as 60 + 5 and applyingthe distributive property, we have:7(65) = 7(60 + 5) 65 = 60 + 5= 7(60) + 7(5) Distributive property= 420 + 35 Multiplication= 455 AdditionWe can write the same problem vertically like this:60 + 5× 735 m 7(5) = 35+ 420 m 7(60) = 420455This saves some space in writing. But notice that we can cut down on theamount of writing even more if we write the problem this way:Step 2: 7(6) = 42; add the 8n 6 3 5 Step 1: 7(5) = 35; write the 53 we carried to 42 to get 45 × 7in the ones column, and then carrythe 3 to the tens column455 m88888888888888888nThis shortcut notation takes some practice.Example 4Multiply: 9(43)Step 2: 9(4) = 36; add the 8n 4 2 3 Step 1: 9(3) = 27; write the 72 we carried to 36 to get 38 × 9 in the ones column, and then carrythe 2 to the tens column387m88888888888888888n4. Multiply.a. 8(57)b. 8(570)Answer4. a. 456 b. 4,560

44Chapter 1 Whole Numbers5. Multiply.a. 45(62)b. 45(620)Example 5Multiply: 52(37)Solution This is the same as 52(30 + 7) or by the distributive property52(30) + 52(7)We can find each of these products by using the shortcut method:52 5 1 2× 30 × 71,560 364NoteThis discussion is toshow why we multiplythe way we do.You should go over it in detail, soyou will understand the reasonsbehind the process of multiplication.Besides being able to domultiplication, you should understandit.The sum of these two numbers is 1,560 + 364 = 1,924. Here is a summary of whatwe have so far:52(37) = 52(30 + 7) 37 = 30 + 7= 52(30) + 52(7) Distributive property= 1,560 + 364 Multiplication= 1,924 AdditionThe shortcut form for this problem is52× 37364 m88888 7(52) = 364+ 1,560 m888 30(52) = 1,5601,924In this case we have not shown any of the numbers we carried, simply because itbecomes very messy.6. Multiply.a. 356(641)b. 3,560(641)Example 6Multiply: 279(428)Solution 279× 4282,232 m888888 8(279) = 2,2325,580 m88888 20(279) = 5,580+ 111,600 m888 400(279) = 111,600119,412U s i n gCalculatorsT e c h n o l o g yHere is how we would work the problem shown in Example 6 on a calculator:Scientific Calculator: 279 × 428 =Graphing Calculator: 279 × 428 ENTEstimatingOne way to estimate the answer to the problem shown in Example 6 is to roundeach number to the nearest hundred and then multiply the rounded numbers.Doing so would give us this:300(400) = 120,000Answers5. a. 2,790 b. 27,9006. a. 228,196 b. 2,281,960Our estimate of the answer is 120,000, which is close to the actual answer,119,412. Making estimates is important when we are using calculators; having anestimate of the answer will keep us from making major errors in multiplication.

1.5 Multiplication with Whole Numbers, and Area45Example 7A s u p e r m a r k e torders 35 cases of a certain soft drink. Ifeach case contains 12 cans of the drink,how many cans were ordered?Solution We have 35 cases, and eachcase has 12 cans. The total number of cansis the product of 35 and 12, which is 35(12):12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans12 cans7. If each tablet of vitamin C contains550 milligrams of vitaminC, what is the total number ofmilligrams of vitamin C in abottle that contains 365 tablets?1212 cans12 cans12 cans× 3560 m888888 5(12) = 60+ 360 m888888 30(12) = 36042012 cans12 cans12 cansThere is a total of 420 cans of the soft drink.Example 8Shirley earns $12 an hour for the first 40 hours she workseach week. If she has $109 deducted from her weekly check for taxes and retirement,how much money will she take home if she works 38 hours this week?Solution To find the amount of money she earned for the week, we multiply12 and 38. From that total we subtract 109. The result is her take-home pay.Without showing all the work involved in the calculations, here is the solution:8. If Shirley works 36 hours thenext week and has the sameamount deducted from hercheck for taxes and retirement,how much will she take home?38($12) = $456 Her total weekly earnings$456 − $109 = $347 Her take-home payExample 9In 1993, the governmentstandardized the way in whichnutrition information is presented on thelabels of most packaged food products.Figure 1 shows one of these standardizedfood labels. It is from a package of FritosCorn Chips that I ate the day I was writingthis example. Approximately howmany chips are in the bag, and what isthe total number of calories consumed ifall the chips in the bag are eaten?Solution Reading toward the top ofthe label, we see that there are about 32chips in one serving, and 3 servings inthe bag. Therefore, the total number ofchips in the bag is3(32) = 96 chipsNutrition FactsServing Size 1 oz. (28g/About 32 chips)Servings Per Container: 3Amount Per ServingCalories 160Calories from fat 90% Daily Value*Total Fat 10 g16%Saturated Fat 1.5g8%Cholesterol 0mg0%Sodium 160mg7%Total Carbohydrate 15g5%Dietary Fiber 1g4%Sugars 0gProtein 2gVitamin A 0% • Vitamin C 0%Calcium 2% • Iron 0%*Percent Daily Values are based on a 2,000calorie dietFigure 19. The amounts given in themiddle of the nutrition label inFigure 1 are for one serving ofchips. If all the chips in the bagare eaten, how much fat hasbeen consumed? How muchsodium?NoteThe letter g that isshown after some ofthe numbers in thenutrition label in Figure 1 standsfor grams, a unit used to measureweight. The unit mg stands formilligrams, another, smaller unit ofweight. We will have more to sayabout these units later in the book.This is an approximate number, because each serving is approximately 32 chips.Reading further we find that each serving contains 160 calories. Therefore, thetotal number of calories consumed by eating all the chips in the bag is3(160) = 480 caloriesAs we progress through the book, we will study more of the information in nutritionlabels.Answers7. 200,750 milligrams8. $3239. 30 g of fat, 480 mg of sodium

46Chapter 1 Whole Numbers10. If a 150-pound person bowlsfor 3 hours, will he or she burnall the calories consumed byeating two bags of the chipsmentioned in Example 9?Example 10The table below lists the number of calories burned in 1hour of exercise by a person who weighs 150 pounds. Suppose a 150-pound persongoes bowling for 2 hours after having eaten the bag of chips mentioned inExample 9. Will he or she burn all the calories consumed from the chips?ActivityCalories Burned in 1 Hourby a 150-Pound PersonBicycling 374Bowling 265Handball 680Jazzercize 340Jogging 680Skiing 544Solution Each hour of bowling burns 265 calories. If the person bowls for 2hours, a total of2(265) = 530 calorieswill have been burned. Because the bag of chips contained only 480 calories, allof them have been burned with 2 hours of bowling.Instructor NoteAgain, we are emphasizing the propertiesof numbers in order to helpwith the transition to algebra.More Properties of MultiplicationMultiplication Property of 0If a represents any number, thena ⋅ 0 = 0 and 0 ⋅ a = 0In words: Multiplication by 0 always results in 0.Multiplication Property of 1If a represents any number, thena ⋅ 1 = a and 1 ⋅ a = aIn words: Multiplying any number by 1 leaves that number unchanged.Commutative Property of MultiplicationIf a and b are any two numbers, thenab = baIn words: The order of the numbers in a product doesn’t affect the result.Associative Property of MultiplicationIf a, b, and c represent any three numbers, then(ab)c = a(bc)In words: We can change the grouping of the numbers in a product withoutchanging the result.Answer10. No

1.5 Multiplication with Whole Numbers, and Area47To visualize the commutative property, we can think of an instructor with12 students.4 chairs across, 3 chairs back =3 chairs across, 4 chairs backExample 11Use the commutative property of multiplication to rewriteeach of the following products:a. 7 ⋅ 9 b. 4(6)Solution Applying the commutative property to each expression, we have:11. Use the commutative propertyof multiplication to rewriteeach of the following products.a. 5 ⋅ 8b. 7(2)a. 7 ⋅ 9 = 9 ⋅ 7 b. 4(6) = 6(4)Example 12Use the associative property of multiplication to rewriteeach of the following products:a. (2 ⋅ 7) ⋅ 9 b. 3 ⋅ (8 ⋅ 2)Solution Applying the associative property of multiplication, we regroup asfollows:a. (2 ⋅ 7) ⋅ 9 = 2 ⋅ (7 ⋅ 9) b. 3 ⋅ (8 ⋅ 2) = (3 ⋅ 8) ⋅ 212. Use the associative property ofmultiplication to rewrite eachof the following products.a. (5 ⋅ 7) ⋅ 4b. 4 ⋅ (6 ⋅ 4)DFormulas for AreaFACTS FROM GEOMETRY: Formulas for AreaA square and rectangle are shown in the following figures. Note that wehave labeled the dimensions of each with variables. The formula for the areaof each object is given in terms of its dimensions.A SquaresA RectanglelsswwlsArea = s 2Area = lwThe formula for area gives us a measure of the amount of surface the objecthas.Answers11. a. 8 ⋅ 5 b. 2(7)12. a. 5 ⋅ (7 ⋅ 4) b. (4 ⋅ 6) ⋅ 4

48Chapter 1 Whole Numbers13. Find the area of the figures inExample 13, with the followingchanges:a. s = 4 feetb. l = 7 inchesw = 5 inchesExample 13Find the area of each figure.a. b.6 in.5 ft 8 in.SolutionWe use the preceding formulas to find the area. In each case, theunits for area are square units.a. Area = s 2 = (5 feet) 2 = 25 square feetb. Area = lw = (8 inches)(6 inches) = 48 square inchesESolving EquationsIf n is used to represent a number, then the equation4 ⋅ n = 12is read “4 times n is 12,” or “The product of 4 and n is 12.” This means that we arelooking for the number we multiply by 4 to get 12. The number is 3. Because theequation becomes a true statement if n is 3, we say that 3 is the solution to theequation.14. Use multiplication facts to findthe solution to each of the followingequations.a. 5 ⋅ n = 35b. 8 ⋅ n = 72c. 49 = 7 ⋅ nd. 27 = 9 ⋅ nExample 14Find the solution to each of the following equations:a. 6 ⋅ n = 24 b. 4 ⋅ n = 36 c. 15 = 3 ⋅ n d. 21 = 3 ⋅ nSolution a. The solution to 6 ⋅ n = 24 is 4, because 6 ⋅ 4 = 24.b. The solution to 4 ⋅ n = 36 is 9, because 4 ⋅ 9 = 36.c. The solution to 15 = 3 ⋅ n is 5, because 15 = 3 ⋅ 5.d. The solution to 21 = 3 ⋅ n is 7, because 21 = 3 ⋅ 7.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Use the numbers 7, 8, and 9 to give an example of the distributiveproperty.2. When we write the distributive property in words, we say“multiplication distributes over addition.” It is also true that multiplicationdistributes over subtraction. Use the letters a, b, and c to write thedistributive property using multiplication and subtraction.3. We can multiply 8 and 487 by writing 487 in expanded form as400 + 80 + 7 and then applying the distributive property. Apply the distributiveproperty to the expression below and then simplify.8(400 + 80 + 7) =4. Find the mistake in the following multiplication problem. Then work theproblem correctly.Answers13. a. 16 square feetb. 35 square inches14. a. 7 b. 9 c. 7 d. 343× 68344+ 258602

1.5 Problem Set49Problem Set 1.5A Multiply each of the following. [Example 1]1. 3 ⋅ 1003002. 7 ⋅ 1007003. 3 ⋅ 2006004. 4 ⋅ 2008005. 6 ⋅ 5003,0006. 8 ⋅ 4003,2007. 5 ⋅ 1,0005,0008. 8 ⋅ 1,0008,0009. 3 ⋅ 7,00021,00010. 6 ⋅ 7,00042,00011. 9 ⋅ 9,00081,00012. 7 ⋅ 7,00049,000A Find each of the following products. (Multiply.) In each case use the shortcut method. [Examples 4–6]13. 25× 410014. 43× 938715. 38× 622816. 45× 731517. 18× 23618. 29× 38719. 72× 201,44020. 68× 302,04021. 19× 5095022. 24× 4096023. 69× 251,72524. 27× 3697225. 11× 1112126. 12× 2125227. 97× 161,55228. 24× 3993629. 168× 254,20030. 452× 3415,368

50Chapter 1 Whole Numbers31. 728× 9166,24832. 680× 7651,68033. 698× 400279,20034. 879× 600527,40035. 111× 11112,32136. 123× 32139,48337. 532× 200106,40038. 277× 900249,30039. 856× 232198,59240. 455× 248112,84041. 976× 628612,92842. 432× 555239,76043. 2,468× 135333,18044. 2,725× 324882,90045. 24,563× 73518,053,80546. 56,728× 85248,332,25647. 44,777× 5,888263,646,97648. 33,999× 2,55586,867,445BComplete the following tables.49.First Second their50.Number number producta b ab11 11 12111 22 24222 22 48422 44 968First Second theirNumber number producta b ab25 15 37525 30 75050 15 75050 30 1,50051.First Second their52.Number number producta b ab25 10 25025 100 2,50025 1,000 25,00025 10,000 250,000First Second theirNumber number producta b ab11 111 1,22111 222 2,44222 111 2,44222 222 4,88453.First Second their54.Number number producta b ab12 20 24036 20 72012 40 48036 40 1,440First Second theirNumber number producta b ab10 12 120100 12 1,2001,000 12 12,00010,000 12 120,000

1.5 Problem Set51BWrite each of the following expressions in words, using the word product.55. 6 ⋅ 7The product of 6 and 756. 9(4)The product of 9 and 457. 2 ⋅ nThe product of 2 and n58. 5 ⋅ xThe product of 5 and x59. 9 ⋅ 7 = 63The product of 9 and 7 is 63.60. (5)(6) = 30The product of 5 and 6 is 30.BWrite each of the following in symbols.61. The product of 7 and n7 ⋅ n62. The product of 9 and x9 ⋅ x63. The product of 6 and 7 is 42.6 ⋅ 7 = 4264. The product of 8 and 9 is 72.8 ⋅ 9 = 7265. The product of 0 and 6 is 0.0 ⋅ 6 = 066. The product of 1 and 6 is 6.1 ⋅ 6 = 6BIdentify the products in each statement.67. 9 ⋅ 7 = 63Products: 9 ⋅ 7 and 6368. 2(6) = 12Products: 2(6) and 1269. 4(4) = 16Products: 4(4) and 1670. 5 ⋅ 5 = 25Products: 5 ⋅ 5 and 25BIdentify the factors in each statement.71. 2 ⋅ 3 ⋅ 4 = 24Factors: 2, 3, and 472. 6 ⋅ 1 ⋅ 5 = 30Factors: 6, 1, and 573. 12 = 2 ⋅ 2 ⋅ 3Factors: 2, 2, and 374. 42 = 2 ⋅ 3 ⋅ 7Factors: 2, 3, and 7C Rewrite each of the following using the commutative property of multiplication. [Example 11]75. 5(9)9(5)76. 4(3)3(4)77. 6 ⋅ 77 ⋅ 678. 8 ⋅ 33 ⋅ 8C Rewrite each of the following using the associative property of multiplication. [Example 12]79. 2 ⋅ (7 ⋅ 6)(2 ⋅ 7) ⋅ 680. 4 ⋅ (8 ⋅ 5)(4 ⋅ 8) ⋅ 581. 3 × (9 × 1)(3 × 9) × 182. 5 × (8 × 2)(5 × 8) × 2

52Chapter 1 Whole NumbersCUse the distributive property to rewrite each expression, then simplify.83. 7(2 + 3)7(2) + 7(3) = 3584. 4(5 + 8)4(5) + 4(8) = 5285. 9(4 + 7)9(4) + 9(7) = 9986. 6(9 + 5)6(9) + 6(5) = 8487. 3(x + 1)3x + 388. 5(x + 8)5x + 4089. 2(x + 5)2x + 1090. 4(x + 3)4x + 12E Find a solution for each equation. [Example 14]91. 4 ⋅ n = 12n = 392. 3 ⋅ n = 12n = 493. 9 ⋅ n = 81n = 994. 6 ⋅ n = 36n = 695. 0 = n ⋅ 5n = 096. 6 = 1 ⋅ nn = 6Applying the ConceptsMost, but not all, of the application problems that follow require multiplication. Read the problems carefully before tryingto solve them.97. Planning a Trip A family decides to drive their compactcar on their vacation. They figure it will require a totalof about 130 gallons of gas for the vacation. If eachgallon of gas will take them 22 miles, how long is thetrip they are planning?98. Rent A student pays $675 rent each month. How muchmoney does she spend on rent in 2 years?$16,2001 GAL./22 MI.2,860 miles1 GAL./22 MI.RENTRENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENT DUE JAN. 1RENTRENT DUE JAN. 1DUE JAN. 1DUE JAN. 1$675

1.5 Problem Set5399. Downloading Songs You receive a gift card for theApple iTunes store for $25.00 and download 18songs at $0.99 per song. How much is left on your giftcard?$7.18100. Cost of Building a Home When you consider building anew home it is helpful to be able to estimate the costof building that house. A simple way to do this is tomultiply the number of square feet under the roof ofthe house by the average building cost per square foot.Suppose you contact a builder who estimates that, onaverage, he charges $142.00 per square foot. Determinethe cost to build a 2,067 square foot house.$293,514101. World’s Busiest Airport Atlanta, Georgia is home to theworld’s busiest airport, Hartsfield-Jackson AtlantaInternational Airport. According to the FederalAviation Administration about 50 jets can land andtake off every 15 minutes which is about 200 jets anhour. About how many jets land and take off in themonth of July?148,800 jets102. Flowers It is probably no surprise that Valentine’s Dayis the busiest day of the year for florists. It is estimatedthat 214 million roses were produced for Valentine’sDay in 2007 (Source: Society of American Florists). If asingle rose costs a consumer $2.50, what was the totalrevenue for the roses produced?$535,000,000Exercise and Calories The table below is an extension of the table we used in Example10 of this section. It gives the amount of energy expended during 1 hour of variousactivities for people of different weights. The accompanying figure is a nutritionlabel from a bag of Doritos tortilla chips. Use the information from the table and thenutrition label to answer Problems 103–108.Calories Burned Through ExerciseCalories Per HourActivity 120 Pounds 150 Pounds 180 PoundsBicycling 299 374 449Bowling 212 265 318Handball 544 680 816Jazzercise 272 340 408Jogging 544 680 816Skiing 435 544 653Nutrition FactsServing Size 1 oz. (28g/About 12 chips)Servings Per Container About 2Amount Per ServingCalories 140Calories from fat 60% Daily Value*Total Fat 7g11%Saturated Fat 1g6%Cholesterol 0mg0%Sodium 170mg7%Total Carbohydrate 18g6%Dietary Fiber 1g4%Sugars less than 1gProtein 2gVitamin A 0% • Vitamin C 0%Calcium 4% •Iron 2%*Percent Daily Values are based on a 2,000calorie diet103. Suppose you weigh 180 pounds. How many calorieswould you burn if you play handball for 2 hours andthen ride your bicycle for 1 hour?2,081 calories104. How many calories are burned by a 120-lb person whojogs for 1 hour and then goes bike riding for 2 hours?1,142 calories105. How many calories would you consume if you ate theentire bag of chips?280 calories106. Approximately how many chips are in the bag?About 24 chips107. If you weigh 180 pounds, will you burn off the caloriesconsumed by eating 3 servings of tortilla chips ifyou ride your bike 1 hour?Yes108. If you weigh 120 pounds, will you burn off the caloriesconsumed by eating 3 servings of tortilla chips if youride your bike for 1 hour?No

54Chapter 1 Whole NumbersD Find the area of each figure. [Example 13]109.110.1 in.1 in.1 in 215 mm225 mm 215 mm111.112.0.75 in.1.5 in.1.125 in 26.75 cm 24.5 cm1.5 cmEstimatingMentally estimate the answer to each of the following problems by rounding each number to the indicated place and thenmultiplying.113. 750 hundred× 12 ten8,000114. 591 hundred× 323 hundred180,000115. 3,472 thousand× 511 hundred1,500,000116. 399 hundred× 298 hundred120,000117. 2,399 thousand× 698 hundred1,400,000118. 9,999 thousand× 666 hundred7,000,000Extending the Concepts: Number SequencesA geometric sequence is a sequence of numbers in which each number is obtained from the previous number by multiplyingby the same number each time. For example, the sequence 3, 6, 12, 24, . . . is a geometric sequence, starting with 3, inwhich each number comes from multiplying the previous number by 2. Find the next number in each of the following geometricsequences. The definition given here for geometric sequences is used again in a number of Extending the Concepts problems.119. 5, 10, 20, . . .40120. 10, 50, 250, . . .1,250121. 2, 6, 18, . . .54122. 12, 24, 48, . . .96

Introduction . . .Division with Whole NumbersDarlene is planning a party and would like to serve 8-ounce glasses of soda. Theglasses will be filled from 32-ounce bottles of soda. In order to know how manybottles of soda to buy, she needs to find out how many of the 8-ounce glasses canbe filled by one of the 32-ounce bottles. One way to solve this problem is withdivision: dividing 32 by 8. A diagram of the problem is shown in Figure 1.1.6ObjectivesA Understand the notation andvocabulary of division.B Divide whole numbers.C Solve applications using division.32-ounce bottle8-ounce glassesFigure 1As a division problem:As a multiplication problem:32 ÷ 8 = 4 4 ⋅ 8 = 32ANotationAs was the case with multiplication, there are many ways to indicate division. Allthe following statements are equivalent. They all mean 10 divided by 5.10 ÷ 5, ​_105 ​, 10/5, 5​ ___) 10 ​The kind of notation we use to write division problems will depend on the situation.We will use the notation 5​) 10 ​mostly with the long-division problems found___in this chapter. The notation ​_10 ​will be used in the chapter on fractions and in5later chapters. The horizontal line used with the notation ​_10 ​is called the fraction5bar.VocabularyThe word quotient is used to indicate division. If we say “The quotient of 10 and 5is 2,” then we mean10 ÷ 5 = 2 or ​_105 ​= 2The 10 is called the dividend, and the 5 is called the divisor. All the expressions,10 ÷ 5, ​_10 ​, and 2, are called the quotient of 10 and 5.51.6 Division with Whole Numbers55

56Chapter 1 Whole NumbersTable 1In Englishin SymbolsThe quotient of 15 and 3 15 ÷ 3, or ​_15 ​, or 15/33The quotient of 3 and 15The quotient of 8 and n3 ÷ 15, or ​_3 ​, or 3/15158 ÷ n, or ​_8 ​, or 8/nnx divided by 2x ÷ 2, or ​_x ​, or x/22The quotient of 21 and 3 is 7. 21 ÷ 3 = 7, or ​_213 ​= 7The Meaning of DivisionOne way to arrive at an answer to a division problem is by thinking in terms ofmultiplication. For example, if we want to find the quotient of 32 and 8, we mayask, “What do we multiply by 8 to get 32?”32 ÷ 8 = ? means 8 ⋅ ? = 32Because we know from our work with multiplication that 8 ⋅ 4 = 32, it must betrue that32 ÷ 8 = 4Table 2 lists some additional examples.Table 2DivisionMultiplication18 ÷ 6 = 3 because 6 ⋅ 3 = 1832 ÷ 8 = 4 because 8 ⋅ 4 = 3210 ÷ 2 = 5 because 2 ⋅ 5 = 1072 ÷ 9 = 8 because 9 ⋅ 8 = 72B Division by One-Digit NumbersConsider the following division problem:465 ÷ 5We can think of this problem as asking the question, “How many fives can wesubtract from 465?” To answer the question we begin subtracting multiples of 5.One way to organize this process is shown below:90 m88 We first guess that there are at least 90 fives in 465____5​) 465 ​− 450 m88 90(5) = 45015 m88 15 is left after we subtract 90 fives from 465What we have done so far is subtract 90 fives from 465 and found that 15 is stillleft. Because there are 3 fives in 15, we continue the process.

1.6 Division with Whole Numbers5790____5​) 465 ​3 m88 There are 3 fives in 15−45015− 15 m88 3 ⋅ 5 = 150 m88 The difference is 0The total number of fives we have subtracted from 465 is90 + 3 = 93We now summarize the results of our work.465 ÷ 5 = 93 which we check 9 1 3with multiplication 8n × 5465The division problem just shown can be shortened by eliminating the subtractionsigns, eliminating the zeros in each estimate, and eliminating some of the numbersthat are repeated in the problem.390 93________The shorthand 5​) 465 ​ looks like 5​) 465 ​ The arrowform for this 450 this. 45 indicates thatproblem 15 15 we bring down15 15 the 5 after0 0 we subtract.The problem shown above on the right is the shortcut form of what is called longdivision. Here is an example showing this shortcut form of long division from startto finish.m78Example 1Divide: 595 ÷ 7Solution Because 7(8) = 56, our first estimate of the number of sevens thatcan be subtracted from 595 is 80:Practice Problems1. Divide.a. 296 ÷ 4b. 2,960 ÷ 4Since 7(5) = 35, we have8 m88 The 8 is placed above the tens column____7​) 595 ​ so we know our first estimate is 8056 m88 8(7) = 5635 m88 59 − 56 = 3; then bring down the 585 m88 There are 5 sevens in 35____7​) 595 ​563535 m88 5(7) = 350 m88 35 − 35 = 0Our result is 595 ÷ 7 = 85, which we can check with multiplication:8 3 5× 7595m78m78Answer1. a. 74 b. 740

58Chapter 1 Whole NumbersDivision by Two-Digit Numbers2. Divide.a. 6,792 ÷ 24b. 67,920 ÷ 24Example 2Divide: 9,380 ÷ 35Solution In this case our divisor, 35, is a two-digit number. The process ofdivision is the same. We still want to find the number of thirty-fives we can subtractfrom 9,380.2 m88 The 2 is placed above the hundreds column______35​) 9,380 ​7 0 m88 2(35) = 702 38 m88 93 − 70 = 23; then bring down the 8We can make a few preliminary calculations to help estimate how many thirtyfivesare in 238:5 × 35 = 175 6 × 35 = 210 7 × 35 = 245Because 210 is the closest to 238 without being larger than 238, we use 6 as ournext estimate:Because 35(8) = 280, we have26 m88 6 in the tens column means this estimate is 60______35​) 9,380 ​7 02 382 10 m88 6(35) = 210280 m88 238 − 210 = 28; bring down the 0268______35​) 9,380 ​7 02 382 10280280 m88 8(35) = 2800 m88 280 − 280 = 0We can check our result with multiplication:268× 351,3408,0409,380m78m88888883. Divide.1,872 ÷ 9Example 3Divide: 1,872 by 18.SolutionHere is the first step.1 m88 1 is placed above hundred column______18​) 1,872 ​1 8 m88 Multiply 1(18) to get 180 m88 Subtract to get 0Answer2. a. 283 b. 2,830

1.6 Division with Whole Numbers59The next step is to bring down the 7 and divide again.10 m88 0 is placed above tens column. 0 is the largest number______18​) 1,872 ​ we can multiply by 18 and not go over 71 807Here is the complete problem.104______18​) 1,872 ​1 8m780 m88 Multiply 0(18) to get 07 m88 Subtract to get 7m78m88888880707272To show our answer is correct, we multiply.018(104) = 1,872Division with RemaindersSuppose Darlene was planning to use 6-ounce glasses instead of 8-ounce glassesfor her party. To see how many glasses she could fill from the 32-ounce bottle,she would divide 32 by 6. If she did so, she would find that she could fill 5 glasses,but after doing so she would have 2 ounces of soda left in the bottle. A diagramof this problem is shown in Figure 2.2 ounces left in bottle32-ounce bottle6-ounce glasses30 ounces totalFigure 2Writing the results in the diagram as a division problem looks like this:5 m88 Quotient___Divisor 88n 6​) 32 ​ m88 Dividend302 m88 RemainderAnswer3. 208

60Chapter 1 Whole Numbers4. Divide.a. 1,883 ÷ 27b. 1,883 ÷ 18Example 4Divide: 1,690 ÷ 67Solution Dividing as we have previously, we get25______67​) 1,690 ​1 34m7835033515 m88 15 is left overWe have 15 left, and because 15 is less than 67, no more sixty-sevens can be subtracted.In a situation like this we call 15 the remainder and writeThese indicate that the remainder is 1525 R 15 25​_15____________ 67 ​67​) 1,690 or 67​) 1,690 ​1 34 1 34m78m8m8350 350335 33515 15Both forms of notation shown above indicate that 15 is the remainder. The notationR 15 is the notation we will use in this chapter. The notation ​_15 ​will be useful67in the chapter on fractions.To check a problem like this, we multiply the divisor and the quotient as usual,and then add the remainder to this result:67× 253351,3401,675 m88 Product of divisor and quotientm781,675 + 15 = 1,690m8888Remainderm8888DividendU s i n gCalculatorsT e c h n o l o g yHere is how we would work the problem shown in Example 4 on a calculator:Scientific Calculator: 1690 ÷ 67 =Graphing Calculator: 1690 ÷ 67 ENTIn both cases the calculator will display 25.223881 (give or take a few digits atthe end), which gives the remainder in decimal form. We will discuss decimalslater in the book.Answer4. a. 69 R 20, or 69​_2027 ​b. 104 R 11, or 104​_1118 ​

1.6 Division with Whole Numbers61CApplicationsExample 5A family has an annual income of $35,880. How much is5. A family spends $1,872 on atheir average monthly income?12-day vacation. How muchdid they spend each day onSolution Because there are 12 months in a year and the yearly (annual)average?income is $35,880, we want to know what $35,880 divided into 12 equal parts is.Therefore we have2 990_______12​) 35,880 ​24To estimate the answer11 8to Example 5 quickly,10 8we can replace 35,8801 08with 36,000 and mentally calculate1 0836,000 ÷ 1200which gives an estimate of 3,000.Our actual answer, 2,990, is closeBecause 35,880 ÷ 12 = 2,990, the monthly income for this family is $2,990.enough to our estimate to convinceus that we have not made a majorerror in our calculation.Division by ZeroWe cannot divide by 0. That is, we cannot use 0 as a divisor in any division problem.Here’s why.Suppose there was an answer to the problem_​ 8 0 ​= ?That would mean that0 ⋅ ? = 8But we already know that multiplication by 0 always produces 0. There is nonumber we can use for the ? to make a true statement out of0 ⋅ ? = 8Because this was equivalent to the original division problem_​ 8 0 ​= ?we have no number to associate with the expression ​_8 0 ​. It is undefined.Rulem8m888888m888888888888Division by 0 is undefined. Any expression with a divisor of 0 is undefined.We cannot divide by 0.NoteAnswer5. $156

62Chapter 1 Whole NumbersGetting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.1. Which sentence below describes the problem shown in Example 1?a. The quotient of 7 and 595 is 85.b. Seven divided by 595 is 85.c. The quotient of 595 and 7 is 85.2. In Example 2, we divide 9,380 by 35 to obtain 268. Suppose we add 35 to9,380, making it 9,415. What will our answer be if we divide 9,415 by 35?3. Example 4 shows that 1,690 ÷ 67 gives a quotient of 25 with a remainderof 15. If we were to divide 1,692 by 67, what would the remainder be?4. Explain why division by 0 is undefined in mathematics.

1.6 Problem Set63Problem Set 1.6AWrite each of the following in symbols.1. The quotient of 6 and 36 ÷ 32. The quotient of 3 and 63 ÷ 63. The quotient of 45 and 945 ÷ 94. The quotient of 12 and 412 ÷ 45. The quotient of r and sr ÷ s6. The quotient of s and rs ÷ r7. The quotient of 20 and 4 is 5.20 ÷ 4 = 58. The quotient of 20 and 5 is 4.20 ÷ 5 = 4Write a multiplication statement that is equivalent to each of the following division statements.9. 6 ÷ 2 = 32 ⋅ 3 = 610. 6 ÷ 3 = 23 ⋅ 2 = 611. ​_369 ​= 49 ⋅ 4 = 3612. ​_364 ​= 94 ⋅ 9 = 3613. ​ 48 _6 ​= 814. ​ 35 _7 ​= 515. 28 ÷ 7 = 416. 81 ÷ 9 = 96 ⋅ 8 = 487 ⋅ 5 = 357 ⋅ 4 = 289 ⋅ 9 = 81B Find each of the following quotients. (Divide.) [Examples 1–3]17. 25 ÷ 518. 72 ÷ 819. 40 ÷ 520. 12 ÷ 2598621. 9 ÷ 022. 7 ÷ 123. 360 ÷ 824. 285 ÷ 5Undefined7455725. ​ 138 _6 ​26. ​ 267 _3 ​______27. 5​) 7,650 ​______28. 5​) 5,670 ​23891,5301,134______29. 5​) 6,750 ​______30. 5​) 6,570 ​_______31. 3​) 54,000 ​_______32. 3​) 50,400 ​1,3501,31418,00016,800_______33. 3​) 50,040 ​_______34. 3​) 50,004 ​16,68016,668

64Chapter 1 Whole NumbersEstimatingWork Problems 35 through 38 mentally, without using a calculator.35. The quotient 845 ÷ 93 is closest to which of thefollowing numbers?a. 10 b. 100 c. 1,000 d. 10,000a36. The quotient 762 ÷ 43 is closest to which of thefollowing numbers?a. 2 b. 20 c. 200 d. 2,000b37. The quotient 15,208 ÷ 771 is closest to which of thefollowing numbers?a. 2 b. 20 c. 200 d. 2,000b38. The quotient 24,471 ÷ 523 is closest to which of thefollowing numbers?a. 5 b. 50 c. 500 d. 5,000bMentally give a one-digit estimate for each of the following quotients. That is, for each quotient, mentally estimate theanswer using one of the digits 1, 2, 3, 4, 5, 6, 7, 8, or 9.39. 316 ÷ 28940. 662 ÷ 28941. 728 ÷ 35542. 728 ÷ 177122443. 921 ÷ 24344. 921 ÷ 44245. 673 ÷ 10946. 673 ÷ 2184263BDivide. You shouldn’t have any wrong answers because you can always check your results with multiplication.[Examples 1–3]47. 1,440 ÷ 324548. 1,206 ÷ 671849. ​_2,40149 ​4950. ​_4,60649 ​94_______51. 28​) 12,096 ​_______52. 28​) 96,012 ​_______53. 63​) 90,594 ​_______54. 45​) 17,595 ​4323,4291,438391_______55. 87​) 61,335 ​_______56. 79​) 48,032 ​________57. 45​) 135,900 ​________58. 56​) 227,920 ​7056083,0204,070

1.6 Problem Set65BComplete the following tables.59.First Second the Quotient60.Number number of a and baba_b ​100 25 4100 26 3 R 22100 27 3 R 19100 28 3 R 16First Second the QuotientNumber number of a and baba_b ​100 25 4101 25 4 R 1102 25 4 R 2103 25 4 R 3B The following division problems all have remainders. [Example 4]____61. 6​) 370 ​____62. 8​) 390 ​____63. 3​) 271 ​61 R 448 R 690 R 1____64. 3​) 172 ​57 R 1____65. 26​) 345 ​13 R 7____66. 26​) 543 ​20 R 23_______67. 71​) 16,620 ​234 R 6_______68. 71​) 33,240 ​468 R 12______69. 23​) 9,250 ​402 R 4_______70. 23​) 20,800 ​904 R 8______71. 169​) 5,950 ​35 R 35_______72. 391​) 34,450 ​88 R 42C Applying the Concepts [Example 5]The application problems that follow may involve more than merely division. Some may require addition, subtraction, ormultiplication, whereas others may use a combination of two or more operations.73. Monthly Income A family has an annual income of$42,300. How much is their monthly income?$3,52574. Hourly Wages If a man works an 8-hour shift and is paid$96, how much does he make for 1 hour?$1275. Price per Pound If 6 pounds of a certain kind of fruitcost $4.74, how much does 1 pound cost?79¢76. Cost of a Dress A dress shop orders 45 dresses for a totalof $2,205. If they paid the same amount for each dress,how much was each dress?$49

66Chapter 1 Whole Numbers77. Filling Glasses How many 32-ounce bottles of Coke willbe needed to fill sixteen 6-ounce glasses?3 bottles78. Filling Glasses How many 8-ounce glasses can be filledfrom three 32-ounce bottles of soda?12 glassessodasodapoppopsodapopthree 32-ounce bottles = ______ 8-ounce glasses79. Filling Glasses How many 5-ounce glasses can be filledfrom a 32-ounce bottle of milk? How many ounces ofmilk will be left in the bottle when all the glasses arefull?6 glasses with 2 oz left over80. Filling Glasses How many 3-ounce glasses can be filledfrom a 28-ounce bottle of milk? How many ounces ofmilk will be left in the bottle when all the glasses arefilled?9 glasses with 1 oz left over81. Boston Red Sox The annual payroll for the BostonRed Sox for the 2007 season was about $156 milliondollars. If there are 40 players on the roster what is theaverage salary per player for the Boston Red Sox?$3,900,00082. Miles per Gallon A traveling salesman kept track of hismileage for 1 month. He found that he traveled 1,104miles and used 48 gallons of gas. How many miles didhe travel on each gallon of gas?23 miles83. Milligrams of Calcium Suppose one egg contains 25milligrams of calcium, a piece of toast contains 40milligrams of calcium, and a glass of milk contains215 milligrams of calcium. How many milligrams ofcalcium are contained in a breakfast that consists ofthree eggs, two glasses of milk, and four pieces oftoast?665 mg84. Milligrams of Iron Suppose a glass of juice contains 3 milligramsof iron and a piece of toast contains 2 milligramsof iron. If Diane drinks two glasses of juice and has threepieces of toast for breakfast, how much iron is containedin the meal?12 mg85. Fitness Walking The guidelines for fitness now indicatethat a person who walks 10,000 steps daily is physicallyfit. According to The Walking Site on the Internet,it takes just over 2,000 steps to walk one mile. If thatis the case, how many miles do you need to walk inorder to take 10,000 steps?5 miles86. Fundraiser As part of a fundraiser for the Earth Dayactivities on your campus, three volunteers work to stuff3,210 envelopes with information about global warming.How many envelopes did each volunteer stuff?1,070 envelopes2,000 steps = 1 mile

Exponents and Order of OperationsExponents are a shorthand way of writing repeated multiplication. In the expression2 3 , 2 is called the base and 3 is called the exponent. The expression 2 3 is read“2 to the third power” or “2 cubed.” The exponent 3 tells us to use the base 2 as amultiplication factor three times.2 3 = 2 ⋅ 2 ⋅ 2 2 is used as a factor three timesThe expression 2 3 is equal to the number 8. We can summarize this discussionwith the following definition.1.7ObjectivesA Identify the base and exponent ofan expression.B Simplify expressions withexponents.C Use the rule for order of operations.DefinitionAn exponent is a whole number that indicates how many times the base isto be used as a factor. Exponents indicate repeated multiplication.A ExponentsIn the expression 5 2 , 5 is the base and 2 is the exponent. The meaning of theexpression is5 2 = 5 ⋅ 5 5 is used as a factor two times= 25The expression 5 2 is read “5 to the second power” or “5 squared.”Here are some more examples.Example 13 2 The base is 3, and the exponent is 2. The expression isread “3 to the second power” or “3 squared.”Example 23 3 The base is 3, and the exponent is 3. The expression isread “3 to the third power” or “3 cubed.”Practice ProblemsFor each expression, name the baseand the exponent, and write theexpression in words.1. 5 22. 2 3Example 32 4 The base is 2, and the exponent is 4. The expression isread “2 to the fourth power.”3. 1 4As you can see from these examples, a base raised to the second power is alsosaid to be squared, and a base raised to the third power is also said to be cubed.These are the only two exponents ( 2 and 3 ) that have special names. All otherexponents are referred to only as “fourth powers,” “fifth powers,” “sixth powers,”and so on.B Expressions with ExponentsThe next examples show how we can simplify expressions involving exponentsby using repeated multiplication.Example 43 2 = 3 ⋅ 3 = 9Simplify each of the following byusing repeated multiplication.4. 5 25. 9 2Example 54 2 = 4 ⋅ 4 = 161.7 Exponents and Order of OperationsAnswers1–3. See solutions section.4. 25 5. 8167

68Chapter 1 Whole Numbers6. 2 3Example 63 3 = 3 ⋅ 3 ⋅ 3 = 9 ⋅ 3 = 277. 1 4Example 73 4 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 9 ⋅ 9 = 81Example 82 4 = 2 ⋅ 2 ⋅ 2 ⋅ 2 = 4 ⋅ 4 = 168. 2 5 U s i n g T e c h n o l o g yCalculatorsHere is how we use a calculator to evaluate exponents, as we did inExample 8:Scientific Calculator: 2 x y 4 =Graphing Calculator: 2 ^ 4 ENT or 2 x y4 ENT(depending on the calculator)Finally, we should consider what happens when the numbers 0 and 1 are usedas exponents. First of all, any number raised to the first power is itself. That is, ifwe let the letter a represent any number, thena 1 = aTo take care of the cases when 0 is used as an exponent, we must use the followingdefinition:DefinitionAny number other than 0 raised to the 0 power is 1. That is, if a representsany nonzero number, then it is always true thata 0 = 1Simplify each of the followingexpressions.9. 7 1Example 95 1 = 510. 4 1Example 109 1 = 911. 9 0Example 114 0 = 112. 1 0Example 128 0 = 1C Order of OperationsThe symbols we use to specify operations, +, −, ⋅ , ÷, along with the symbols weuse for grouping, ( ) and [ ], serve the same purpose in mathematics as punctuationmarks in English. They may be called the punctuation marks of mathematics.Consider the following sentence:Bob said John is tall.Answers6. 8 7. 1 8. 329. 7 10. 4 11. 1 12. 1It can have two different meanings, depending on how we punctuate it:1. “Bob,” said John, “is tall.”2. Bob said, “John is tall.”

1.7 Exponents and Order of Operations69Without the punctuation marks we don’t know which meaning the sentence has.Now, consider the following mathematical expression:4 + 5 ⋅ 2What should we do? Should we add 4 and 5 first, or should we multiply 5 and 2first? There seem to be two different answers. In mathematics we want to avoidsituations in which two different results are possible. Therefore we follow the rulefor order of operations.DefinitionNoteOrder of Operations When evaluating mathematical expressions, we willTo help you to remember= 76 − 8 Add and subtract,13. a. 17 b. 170 14. 37= 68 left to right perform the operations in the following order:the order ofoperations you can1. If the expression contains grouping symbols, such as parentheses ( ),brackets [ ], or a fraction bar, then we perform the operations inside thegrouping symbols, or above and below the fraction bar, first.2. Then we evaluate, or simplify, any numbers with exponents.use the popular sentencePlease Excuse My Dear AuntSally, or the acronym PEMDASParentheses (or grouping)Exponents3. Then we do all multiplications and divisions in order, starting at the leftMultiplication andDivision, from left to rightand moving right.Addition and4. Finally, we do all additions and subtractions, from left to right.Subtraction, from left to rightAccording to our rule, the expression 4 + 5 ⋅ 2 would have to be evaluated bymultiplying 5 and 2 first, and then adding 4. The correct answer—and the onlyanswer—to this problem is 14.4 + 5 ⋅ 2 = 4 + 10 Multiply first,= 14 then addHere are some more examples that illustrate how we apply the rule for order ofoperations to simplify (or evaluate) expressions.Example 13Simplify: 4 ⋅ 8 − 2 ⋅ 613. Simplify.Solution We multiply first and then subtract:a. 5 ⋅ 7 − 3 ⋅ 6b. 5 ⋅ 70 − 3 ⋅ 604 ⋅ 8 − 2 ⋅ 6 = 32 − 12 Multiply first,= 20 then subtractExample 14Simplify: 5 + 2(7 − 1)14. Simplify: 7 + 3(6 + 4)Solution According to the rule for the order of operations, we must do what isinside the parentheses first:5 + 2(7 − 1) = 5 + 2(6) Inside parentheses first,= 5 + 12 then multiply,= 17 then addExample 15Simplify: 9 ⋅ 2 3 + 36 ÷ 3 2 − 815. Simplify.Solution 9 ⋅ 2 3 + 36 ÷ 3 2 − 8 = 9 ⋅ 8 + 36 ÷ 9 − 8 Exponents first,a. 28 ÷ 7 − 3b. 6 ⋅ 3 2 + 64 ÷ 2 4 − 2= 72 + 4 − 8 then multiply and divide,left to right.Answers}

70Chapter 1 Whole NumbersU s i n gCalculatorsT e c h n o l o g yHere is how we use a calculator to work the problem shown in Example 14:Scientific Calculator: 5 + 2 × ( 7 − 1 ) =Graphing Calculator: 5 + 2 ( 7 − 1 ) ENTExample 15 on a calculator looks like this:Scientific Calculator: 9 × 2 x y 3 + 36 ÷ 3 x y 2 − 8 =Graphing Calculator: 9 × 2 ^ 3 + 36 ÷ 3 ^ 2 − 8 ENT16. Simplify.a. 5 + 3[24 − 5(6 − 2)]b. 50 + 30[240 − 50(6 − 2)]Example 16Simplify: 3 + 2[10 − 3(5 − 2)]Solution The brackets, [ ], are used in the same way as parentheses. In a caselike this we move to the innermost grouping symbols first and begin simplifying:3 + 2[10 − 3(5 − 2)] = 3 + 2[10 − 3(3)]= 3 + 2[10 − 9]= 3 + 2[1]= 3 + 2= 5Table 1 lists some English expressions and their corresponding mathematicalexpressions written in symbols.Table 1in EnglishMathematical Equivalent5 times the sum of 3 and 8 5(3 + 8)Twice the difference of 4 and 3 2(4 − 3)6 added to 7 times the sum of 5 and 6 6 + 7(5 + 6)The sum of 4 times 5 and 8 times 9 4 ⋅ 5 + 8 ⋅ 93 subtracted from the quotient of 10 and 2 10 ÷ 2 − 3Study SkillSRead the Book Before Coming to ClassAs we mentioned in the Preface, it is best to have read the section to be covered in classbefore getting to class. Even if you don’t understand everything that you have read, you arestill better off reading ahead than not. The Getting Ready for Class questions at the end ofeach section are intended to give you things to look for in the reading that will be importantin understanding what is in the section.Getting Ready for ClassAfter reading through the preceding section, respond in your ownwords and in complete sentences.Answers16. a. 17 b. 1,2501. In the expression 5 3 , which number is the base?2. Give a written description of the process you would use to simplify theexpression 3 + 4(5 + 6).3. What is the first step in simplifying the expression 8 + 6 ÷ 3 − 1?4. How do you remember the correct Order of Operations?

1.7 Problem Set71Problem Set 1.7A For each of the following expressions, name the base and the exponent. [Examples 1–3]1. 4 5Base 4Exponent 52. 5 4Base 5Exponent 43. 3 6Base 3Exponent 64. 6 3Base 6Exponent 35. 8 2Base 8Exponent 26. 2 8Base 2Exponent 87. 9 1Base 9Exponent 18. 1 9Base 1Exponent 99. 4 0Base 4Exponent 010. 0 4Base 0Exponent 4BUse the definition of exponents as indicating repeated multiplication to simplify each of the following expressions.[Examples 4–12]11. 6 23612. 7 24913. 2 3814. 2 41615. 1 4116. 5 1517. 9 0118. 27 0119. 9 28120. 8 26421. 10 11022. 8 1823. 12 11224. 16 0125. 45 0126. 3 481C Use the rule for the order of operations to simplify each expression. [Examples 13–16]27. 16 − 8 + 41228. 16 − 4 + 82029. 20 ÷ 2 ⋅ 1010030. 40 ÷ 4 ⋅ 55031. 20 − 4 ⋅ 4432. 30 − 10 ⋅ 21033. 3 + 5 ⋅ 84334. 7 + 4 ⋅ 94335. 3 ⋅ 6 − 21636. 5 ⋅ 1 + 61137. 6 ⋅ 2 + 9 ⋅ 88438. 4 ⋅ 5 + 9 ⋅ 78339. 4 ⋅ 5 − 3 ⋅ 21440. 5 ⋅ 6 − 4 ⋅ 31841. 5 2 + 7 27442. 4 2 + 9 29743. 480 + 12(32) 212,76844. 360 + 14(27) 210,56645. 3 ⋅ 2 3 + 5 ⋅ 4 210446. 4 ⋅ 3 2 + 5 ⋅ 2 376

72Chapter 1 Whole Numbers47. 8 ⋅ 10 2 − 6 ⋅ 4 341648. 5 ⋅ 11 2 − 3 ⋅ 2 358149. 2(3 + 6 ⋅ 5)6650. 8(1 + 4 ⋅ 2)7251. 19 + 50 ÷ 5 22152. 9 + 8 ÷ 2 21153. 9 − 2(4 − 3)754. 15 − 6(9 − 7)355. 4 ⋅ 3 + 2(5 − 3)1656. 6 ⋅ 8 + 3(4 − 1)5757. 4[2(3) + 3(5)]8458. 3[2(5) + 3(4)]6659. (7 − 3)(8 + 2)4060. (9 − 5)(9 + 5)5661. 3(9 − 2) + 4(7 − 2)4162. 7(4 − 2) − 2(5 − 3)1063. 18 + 12 ÷ 4 − 31864. 20 + 16 ÷ 2 − 52365. 4(10 2 ) + 20 ÷ 440566. 3(4 2 ) + 10 ÷ 55067. 8 ⋅ 2 4 + 25 ÷ 5 − 3 212468. 5 ⋅ 3 4 + 16 ÷ 8 − 2 240369. 5 + 2[9 − 2(4 − 1)]1170. 6 + 3[8 − 3(1 + 1)]1271. 3 + 4[6 + 8(2 − 0)]9172. 2 + 5[9 + 3(4 − 1)]9215 + 5(4)73. ​_17 − 12 ​720 + 6(2)74. ​_11 − 7 ​8Translate each English expression into an equivalent mathematical expression written in symbols. Then simplify.75. 8 times the sum of 4 and 28(4 + 2) = 4876. 3 times the difference of 6 and 13(6 − 1) = 1577. Twice the sum of 10 and 32(10 + 3) = 2678. 5 times the difference of 12 and 65(12 − 6) = 3079. 4 added to 3 times the sum of 3 and 43(3 + 4) + 4 = 2580. 25 added to 4 times the difference of 7 and 54(7 − 5) + 25 = 3381. 9 subtracted from the quotient of 20 and 2(20 ÷ 2) − 9 = 182. 7 added to the quotient of 6 and 2(6 ÷ 2) + 7 = 1083. The sum of 8 times 5 and 5 times 4(8 ⋅ 5) + (5 ⋅ 4) = 6084. The difference of 10 times 5 and 6 times 2(10 ⋅ 5) − (6 ⋅ 2) = 38

1.7 Problem Set73Applying the ConceptsNutrition Labels Use the three nutrition labels below to work Problems 95–100.Spaghetti Canned Italian Tomatoes Shredded Romano CheeseNutrition FactsServing Size 2 oz. (56g/l/8 of pkg) dryServings Per Container: 8Amount Per ServingCalories 210Calories from fat 10% Daily Value*Total Fat 1g2%Saturated Fat 0g0%Poly unsaturated Fat 0.5gMonounsaturated Fat 0gCholesterol 0mg0%Sodium 0mg0%Total Carbohydrate 42g14%Dietary Fiber 2g7%Sugars 3gProtein 7gVitamin A 0% • Vitamin C 0%Calcium 0% • Iron 10%*Percent Daily Values are based on a 2,000calorie dietNutrition FactsServing Size 1/2 cup (121g)Servings Per Container: about 3 1/2Amount Per ServingCalories 25Total Fat 0gSaturated Fat 0gCalories from fat 0% Daily Value*0%0%Cholesterol 0mg0%Sodium 300mg12%Potassium 145mg 4%Total Carbohydrate 4g2%Dietary Fiber 1g4%Sugars 4gProtein 1gVitamin A 20% • Vitamin C 15%Calcium 4% • Iron 15%*Percent Daily Values are based on a 2,000calorie diet. Your daily values may be higheror lower depending on your calorie needs.Nutrition FactsServing Size 2 tsp (5g)Servings Per Container: 34Amount Per ServingCalories 20Total Fat 1.5gSaturated Fat 1gCholesterol 5mgSodium 70mgTotal Carbohydrate 0gFiber 0gSugars 0gCalories from fat 10% Daily Value*2%5%2%3%0%0%Protein 2gVitamin A 0% • Vitamin C 0%Calcium 4% • Iron 0%*Percent Daily Values (DV) are based on a2,000 calorie dietFind the total number of calories in each of the following meals.85. Spaghetti 1 servingTomatoes 1 servingCheese 1 serving255 calories86. Spaghetti 1 servingTomatoes 2 servingsCheese 1 serving280 calories87. Spaghetti 2 servingsTomatoes 1 servingCheese 1 serving465 calories88. Spaghetti 2 servingsTomatoes 1 servingCheese 2 servings485 caloriesThe following table lists the number of calories consumed by eating some popular fast foods. Use the table to workProblems 89 and 90.89. Compare the total number of calories in the meal inProblem 95 with the number of calories in a McDonald’sBig Mac.Big Mac has twice the calories.90. Compare the total number of calories in the meal inProblem 98 with the number of calories in a Burger Kinghamburger.Hamburger has 225 fewer calories.Calories in FoodFoodCaloriesMcDonald’s hamburger 270Burger King hamburger 260Jack in the Box hamburger 280McDonald’s Big Mac 510Burger King Whopper 630Jack in the Box Colossus burger 940

74Chapter 1 Whole NumbersExtending the Concepts: Number SequencesThere is a relationship between the two sequences below. The first sequence is the sequence of odd numbers. The secondsequence is called the sequence of squares.1, 3, 5, 7, . . . The sequence of odd numbers1, 4, 9, 16, . . . The sequence of squares91. Add the first two numbers in the sequence of oddnumbers.492. Add the first three numbers in the sequence of oddnumbers.993. Add the first four numbers in the sequence of oddnumbers.1694. Add the first five numbers in the sequence of oddnumbers.25

Chapter 1 SummaryEXAMPLEsThe numbers in brackets indicate the sections in which the topics were discussed.Place Values for Decimal Numbers [1.1]The margins of the chapter summarieswill be used for examples ofthe topics being reviewed, wheneverconvenient.The place values for the digits of any base 10 number are as follows:Trillions Billions Millions Thousands Ones1. The number 42,103,045 writtenin words is “forty-two million,one hundred three thousand,forty-five.”HundredsTensOnesHundredsTensOnesHundredsTensOnesHundredsTensOnesHundredsTensOnesThe number 5,745 written inexpanded form is5,000 + 700 + 40 + 5Vocabulary Associated with Addition, Subtraction,Multiplication, and Division [1.2, 1.4, 1.5, 1.6]The word sum indicates addition.The word difference indicates subtraction.The word product indicates multiplication.The word quotient indicates division.2. The sum of 5 and 2 is 5 + 2.The difference of 5 and 2 is5 − 2.The product of 5 and 2 is5 ⋅ 2.The quotient of 10 and 2 is10 ÷ 2.Properties of Addition and Multiplication [1.2, 1.5]If a, b, and c represent any three numbers, then the properties of addition andmultiplication used most often are:Commutative property of addition: a + b = b + aCommutative property of multiplication: a ⋅ b = b ⋅ aAssociative property of addition: (a + b) + c = a + (b + c)Associative property of multiplication: (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)Distributive property: a(b + c) = a(b) + a(c)3. 3 + 2 = 2 + 33 ⋅ 2 = 2 ⋅ 3(x + 3) + 5 = x + (3 + 5)(4 ⋅ 5) ⋅ 6 = 4 ⋅ (5 ⋅ 6)3(4 + 7) = 3(4) + 3(7)Perimeter of a Polygon [1.2]The perimeter of any polygon is the sum of the lengths of the sides, and it isdenoted with the letter P.4. The perimeter of the rectanglebelow isP = 37 + 37 + 24 + 24= 122 feet24 ft37 ftChapter 1Summary75

76Chapter 1 Whole NumbersSteps for Rounding Whole Numbers [1.3]5. 5,482 to the nearest ten is 5,480.5,482 to the nearest hundred is5,500.5,482 to the nearest thousand is5,000.1. Locate the digit just to the right of the place you are to round to.2. If that digit is less than 5, replace it and all digits to its right with zeros.3. If that digit is 5 or more, replace it and all digits to its right with zeros, and add1 to the digit to its left.Division by 0 (Zero) [1.6]6. Each expression below isundefined.Division by 0 is undefined. We cannot use 0 as a divisor in any division problem.5 ÷ 0 ​ 7 _0 ​ 4/0Order of Operations [1.7]7. 4 + 6(8 − 2)= 4 + 6(6) Inside parentheses first= 4 + 36 Then multiply= 40 Then addTo simplify a mathematical expression:1. We simplify the expression inside the grouping symbols first. Grouping symbolsare parentheses ( ), brackets [ ], or a fraction bar.2. Then we evaluate any numbers with exponents.3. We then perform all multiplications and divisions in order, starting at the leftand moving right.4. Finally, we do all the additions and subtractions, from left to right.Exponents [1.7]8. 2 3 = 2 ⋅ 2 ⋅ 2 = 85 0 = 13 1 = 3In the expression 2 3 , 2 is the base and 3 is the exponent. An exponent is a shorthandnotation for repeated multiplication. The exponent 0 is a special exponent.Any nonzero number to the 0 power is 1.

Chapter 1 ReviewThe numbers in brackets indicate the sections in which problems of a similar type can be found.1. One of the largest Pacific blue marlins was caught nearHawaii in 1982. It weighed 1,376 pounds. Write 1,376in words. [1.1]one thousand, three hundred seventy-six2. In 2003 the New York Yankees had the highest homeattendance in major league baseball. The attendancethat year was 3,465,600. Write 3,465,600 in words. [1.1]three million, four hundred sixty-five thousand, six hundredFor Problems 3 and 4, write each number with digits instead of words. [1.1]3. Five million, two hundred forty-five thousand, sixhundred fifty-two5,245,6524. Twelve million, twelve thousand, twelve12,012,0125. In 2003 the Montreal Expos had the lowest attendancein major league baseball. The attendance that year was1,025,639. Write 1,025,639 in expanded form. [1.1]1,000,000 + 20,000 + 5,000 + 600 + 30 + 96. According to the American Medical Association, in 2002,there were 215,005 female physicians practicing medicinein the United States. Write 215,005 in expandedform. [1.1]200,000 + 10,000 + 5,000 + 5Identify each of the statements in Problems 7–14 as an example of one of the following properties. [1.2, 1.5]a. Addition property of 0b. Multiplication property of 0c. Multiplication property of 1d. Commutative property of addition7. 5 + 7 = 7 + 5de. Commutative property of multiplicationf. Associative property of additiong. Associative property of multiplication8. (4 + 3) + 2 = 4 + (3 + 2)f9. 6 ⋅ 1 = 6c10. 8 + 0 = 8a11. 5 ⋅ 0 = 0b12. 4 ⋅ 6 = 6 ⋅ 4e13. 5 ⋅ (3 ⋅ 2) = (5 ⋅ 3) ⋅ 2g14. (6 + 2) + 3 = (2 + 6) + 3dFind each of the following sums. (Add.) [1.2]15. 498+ 25174916. 784+ 5981,38217. 7,384251+ 6378,27218. 4,901648+3,5929,141Chapter 1Review77

78Chapter 1 Whole NumbersFind each of the following differences. (Subtract.) [1.4]19. 789−47531420. 792−17861421. 5,908−2,7593,14922. 3,527−1,7891,738Find each of the following products. (Multiply.) [1.5]23. 8(73)58424. 7(984)6,88825. 63(59)3,71726. 49(876)42,924Find each of the following quotients. (Divide.) [1.6]27. 692 ÷ 428. 1,020 ÷ 1517368_______29. 36​) 15,408 ​428_______30. 286​) 21,736 ​76Round the number 3,781,092 to the nearest: [1.3]31. Ten3,781,09032. Hundred3,781,10033. Hundred thousand3,800,00034. Million4,000,000Use the rule for the order of operations to simplify each expression as much as possible. [1.7]35. 4 + 3 ⋅ 5 27936. 7(9) 2 − 6(4) 318337. 3(2 + 8 ⋅ 9)22238. 7 − 2(6 − 4)339. 24 ÷ 6 ⋅ 2840. 20 ⋅ 3 ÷ 12 ⋅ 21041. 4(3 − 1) 33242. 36 ÷ 9 ⋅ 3 236Write an expression using symbols that is equivalent to each of the following expressions; then simplify. [1.7]43. 3 times the sum of 4 and 63(4 + 6) = 3045. Twice the difference of 17 and 52(17 − 5) = 2444. 9 times the difference of 5 and 39(5 − 3) = 1846. The product of 5 and the sum of 8 and 25(8 + 2) = 50Find the perimeter of the shapes. [1.2]47.6 m48.5 m4 m7 ftP = 22 mP = 38 ft12 ft

Chapter 1 Test1. Write the number 20,347 in words.Twenty thousand, three hundred forty-seven2. Write the number two million, forty-five thousand, sixwith digits instead of words.2,045,0063. Write the number 123,407 in expanded form.100,000 + 20,000 + 3,000 + 400 + 7Identify each of the statements in Problems 4–7 as an exampleof one of the following properties.a. Addition property of 0b. Multiplication property of 0c. Multiplication property of 1d. Commutative property of additione. Commutative property of multiplicationf. Associative property of additiong. Associative property of multiplication4. (5 + 6) + 3 = 5 + (6 + 3) 5. 7 ⋅ 1 = 7fc6. 9 + 0 = 97. 5 ⋅ 6 = 6 ⋅ 5aeFind each of the following sums. (Add.)8. 135+ 7418769. 5,401329+ 10,65316,383Find each of the following differences. (Subtract.)10. 937− 41352411. 7,052− 3,9673,08521. Twice the sum of 11 and 72(11 + 7) = 3622. The quotient of 20 and 5 increased by 9(20 ÷ 5) + 9 = 1323. Find the perimeter.P = 46 m13 m10 m7 mThe snapshot shows the top grossing films. Use the chartto answer the following questions.Top Grossing Films of All Time:Titanic (1997)$1,835,388,188The Lord of the Rings: The Return of the King (2003)$1,051,431,553Harry Potter and the Philosopher’s Stone (2001)$966,996,609Star Wars: Episode I - The Phantom Menace (1999)$923,136,820Jurassic Park (1993)$920,067,947Source: boxofficemojo.com24. How much more did Lord of the Rings gross thanJurrasic Park?$131,363,60625. How much more did Titanic gross than Star Wars:Episode 1?$912,251,368Find each of the following products. (Multiply.)12. 9(186)1,67413. 62(359)22,258Find each of the following quotients. (Divide.)_______14. 1,105 ÷ 1315. 583​) 12,243 ​852116. Round the number 516,249 to the nearest tenthousand.520,000Use the rule for the order of operations to simplify each expressionas much as possible.17. 8(5) 2 − 7(3) 31119. 7 + 2(53 − 3)10718. 8 − 2(5 − 3)420. 3(x − 2)3x − 6Chapter 1 Test79

Chapter 1 ProjectsWhole Numbersgroup PROJECTEgyptian NumbersNumber of PeopleTime NeededEquipmentBackground310 minutesPencil and paperThe Egyptians had a fully developed numbersystem as early as 3500 b.c. They recorded verylarge numbers in the macehead of Narmer,which boasts of the spoils taken during wars,and the Book of the Dead, a collection of religioustexts. The Egyptians used a base-ten system.A special pictograph was used to representeach power of ten. Here are some pictographsused.1 10 100 1,000 10,000 100,000 1,000,000staff horseshoe rope lotus bent finger tadpole or astonishedflower frog personExampleUsually the direction of writing was from rightto left, with the larger units first. Symbols wereplaced in rows to save lateral space. Writing thenumber 132,146 in Egyptian hieroglyphics lookslike this:Express each of the given numbers in Egyptianhieroglyphics.3. 1,8424. 4,310,175132,146 =ProcedureWrite each of the following Egyptian numbersin our system.1.2.Students and Instructors: The end of each chapter in this book will have two projects. The group projects are intended tobe done in class. The research projects are to be completed outside of class. They can be done in groups or individually.Chapter 1Projects81

RESEARCH PROJECTLeopold KroneckerLeopold Kronecker (1823–1891) was a Germanmathematician and logician who thought thatarithmetic should be based on whole numbers.He is known for the quote, “God made thenatural numbers; all else is the work of man.”He was openly critical of the efforts of his contemporaries.Kronecker’s primary work was inthe field of algebraic number theory. Researchthe life of Leopold Kronecker, or discuss thework of a mathematician who was criticized byKronecker.Courtesy of Wolfram Research/National Science Foundation82Chapter 1 Whole Numbers

A Glimpse of AlgebraAt the end of most chapters of this book you will find a section like this one.These sections show how some of the material in the chapter looks when it isextended to algebra. If you are planning to take an algebra course after you havefinished this one, these sections will give you a head start. If you are not planningto take algebra, these sections will give you an idea of what algebra is like. Whoknows? You may decide to take an algebra class after you work through a few ofthese sections.In this chapter we did some work with exponents. We can use the definitionof exponents, along with the commutative property of multiplication, to rewritesome expressions that contain variables and exponents.We can expand the expression (5x) 2 using the definition of exponents as(5x) 2 = (5x)(5x)Because the expression on the right is all multiplication, we can rewrite it as(5x)(5x) = 5 ⋅ x ⋅ 5 ⋅ xAnd because multiplication is a commutative operation, we can rearrange thislast expression so that the numbers are grouped together, and the variables aregrouped together:5 ⋅ x ⋅ 5 ⋅ x = (5 ⋅ 5)(x ⋅ x)Now, because5 ⋅ 5 = 25 and x ⋅ x = x 2we can rewrite the expression as(5 ⋅ 5)(x ⋅ x) = 25x 2Here is what the problem looks like when the steps are shown together:(5x) 2 = (5x)(5x) Definition of exponents= (5 ⋅ 5)(x ⋅ x) Commutative property= 25x 2 Multiplication and definition of exponentsWe have shown only the important steps in this summary. We rewrite theexpression by (1) applying the definition of exponents to expand it, (2) rearrangingthe numbers and variables by using the commutative property, and then (3)simplifying by multiplication.Here are some more examples.Example 1Expand (7x) 2 using the definition of exponents, and thensimplify the result.Solution We begin by writing the expression as (7x)(7x), then rearranging thenumbers and variables, and then simplifying:1. Expand (3x) 2 using the definitionof exponents, and then simplifythe result.Practice Problems(7x) 2 = (7x)(7x) Definition of exponents= (7 ⋅ 7)(x ⋅ x) Commutative property= 49x 2 Multiplication and definition of exponentsAnswer1. 9x 2A Glimpse of Algebra83

84Chapter 1 Whole Numbers2. Expand and simplify: (2a) 3Example 2Expand and simplify: (5a) 3SolutionWe begin by writing the expression as (5a)(5a)(5a):(5a) 3 = (5a)(5a)(5a) Definition of exponents= (5 ⋅ 5 ⋅ 5)(a ⋅ a ⋅ a) Commutative property= 125a 3 5 ∙ 5 ∙ 5 = 125; a ∙ a ∙ a = a 33. Expand and simplify: (7xy) 2Example 3Expand and simplify: (8xy) 2SolutionProceeding as we have above, we have:(8xy) 2 = (8xy)(8xy) Definition of exponents= (8 ⋅ 8)(x ⋅ x)(y ⋅ y) Commutative property= 64x 2 y 2 8 ∙ 8 = 64; x ∙ x = x 2 ; y ∙ y = y 24. Simplify: (3x) 2 (7xy) 2Example 4Simplify: (7x) 2 (8xy) 2SolutionWe begin by applying the definition of exponents:(7x) 2 (8xy) 2 = (7x)(7x)(8xy)(8xy)= (7 ⋅ 7 ⋅ 8 ⋅ 8)(x ⋅ x ⋅ x ⋅ x)(y ⋅ y) Commutative property= 3,136x 4 y 25. Simplify: (5x) 3 (2x) 2Example 5Simplify: (2x) 3 (4x) 2SolutionProceeding as we have above, we have:(2x) 3 (4x) 2 = (2x)(2x)(2x)(4x)(4x)= (2 ⋅ 2 ⋅ 2 ⋅ 4 ⋅ 4)(x ⋅ x ⋅ x ⋅ x ⋅ x)= 128x 5Answer2. 8a 3 3. 49x 2 y 2 4. 441x 4 y 25. 500x 5

A Glimpse of Algebra Problems85Problem A Glimpse Set of 1.7 Algebra ProblemsUse the definition of exponents to expand each of the following expressions. Apply the commutative property, and simplifythe result in each case.1. (6x) 22. (9x) 236x 2 81x 23. (4x) 216x 2 100x 25. (3a) 327a 3 216a 37. (2ab) 38a 3 b 3 125a 3 b 39. (9xy) 281x 2 y 2 25x 2 y 211. (5xyz) 225x 2 y 2 z 2 49x 2 y 2 z 213. (4x) 2 (9xy) 21,296x 4 y 2 2,500x 4 y 215. (2x) 2 (3x) 2 (4x) 216. (5x) 2 (2x) 2 (10x) 2

86Chapter 1 Whole Numbers17. (2x) 3 (5x) 218. (3x) 3 (4x) 2200x 5 432x 519. (2a) 3 (3a) 2 (10a) 220. (3a) 3 (2a) 2 (10a) 27,200a 7 10,800a 721. (3xy) 3 (4xy) 222. (2xy) 4 (3xy) 2432x 5 y 5 144x 6 y 623. (5xyz) 2 (2xyz) 424. (6xyz) 2 (3xyz) 3400x 6 y 6 z 6 972x 5 y 5 z 525. (xy) 3 (xz) 2 ( yz) 426. (xy) 4 (xz) 2 ( yz) 3x 5 y 7 z 6 x 6 y 7 z 527. (2a 3 b 2 ) 2 (3a 2 b 3 ) 428. (4a 4 b 3 ) 2 (5a 2 b 4 ) 2324a 14 b 16 400a 12 b 1429. (5x 2 y 3 )(2x 3 y 3 ) 330. (8x 2 y 2 ) 2 (3x 3 y 4 ) 240x 11 y 12 576x 10 y 12