WholeNumbers
ExerciseSet1.1
RC2. In615,702,thenumber615isinthethousandsperiod.
RC4. Thenumber721iswritteninstandard notation.
CC2. Awordnamefor42,000,000isforty-twomillion.
CC4. Awordnamefor18,000,000,000iseighteenbillion.
CC6. Awordnamefor40,000,000,000,000isfortytrillion.
2. 5tenthousands
4. 5hundredthousands
6. 9
8. 3
10. 6thousands+6hundreds+8tens+8ones
12. 1thousand+7hundreds+8tens+6ones
14. 3tenthousands+8thousands+4hundreds+5tens+ 3ones
16. 1hundredthousand+3tenthousands+5thousands+ 0hundreds+8tens+0ones,or1hundredthousand+ 3tenthousands+5thousands+8tens
18. 1billion+2hundredmillions+6tenmillions+6 millions+8hundredthousands+8tenthousands+3 thousands+5hundreds+9tens+8ones
20. 3hundredmillions+2tenmillions+3millions+9hundredthousands+9tenthousands+5thousands+5 hundreds+2tens+8ones
22. Forty-eight
24. Forty-fivethousand,ninehundredeighty-seven
26. Onehundredeleventhousand,thirteen
28. Forty-threebillion,fivehundredfiftymillion,sixhundred fifty-onethousand,eighthundredeight
30. Ninety-ninethousand,eighthundredfifty-three
32. Twohundredtwenty-sixmillion,onethousand,twohundredeighty-eight
34. 354,702
36. 17,112
38. 19,610,439
40. 700,000,000
42. 26,000,000,000
44. 200,017
46. 2,793,000,000
48. 32 > 0
50. 28 > 18
52. 77 < 117
54. 999 > 997
56. 345 < 456
58. 12 < 32
60. 1, 014, 023 > 758, 708,or758, 708 < 1, 014, 023
62. 843, 393 > 842, 583,or842, 583 < 843, 393
64. Alldigitsare9’s.Answersmayvary.Foran8-digitreadout,forexample,itwouldbe99,999,999.Thisnumberhas threeperiods.
ExerciseSet1.2
RC2. Inthesubtraction18 5=13,thenumber5isthe subtrahend
RC4. Thedistancearoundanobjectisitsperimeter
CC2. 910 100 / ✥✥ 7 93
CC4. 10 1 /0/00 400 600
9910 1000 / ✥✥✥ 999 1 2. 1521
8. 1 3654 +2700 6354
10. 1 271 +3338 3609
12. 1 280 +34, 902 35, 182
14. 111 45, 879 +21, 786 67, 665
16. 1111 99, 999 +112 100, 111 18. 1111 42, 487 83, 141 +36, 712 162, 340
20. 22 989 566 834 920 +703 4012
22. Perimeter=14mi+13mi+8mi+10mi+47mi+ 22mi
Wecarryouttheaddition.
Theperimeterofthefigureis114mi.
24. 90ft+90ft+90ft+90ft=Perimeter Wecarryouttheaddition. 90 90 90 +90 360
Thebattertravels360ftwhenahomerunishit.
26. 87 34 53
/ 5375 2989
52. 7913 803✥✥ 418 385
54. 3915 9405 / ✥✥ 258 9147
56. 14 64 / 910 7 /500 / ✥✥ 3604 3896
58. 6913 84, 703 / ✥✥ 298 84, 405
60. 14 4 / 10017 1 /5 /, 0 /1/7 / 7809 7208
62. 79913 8003 / ✥✥✥ 599 7404
64. 69910 17, 000 / ✥✥✥ 11, 598 5402
66. 399916 40, 006 / ✭✭✭✭ 147 39, 859
68. 299914 30, 004 / ✭✭✭✭ 6749 23, 255
70. Ninebillion,threehundredforty-sixmillion,threehundred ninety-ninethousand,fourhundredsixty-eight
72. Onemethodisdescribedintheanswersectioninthetext. Anothermethodis:1+100=101,2+99=101, ,50 +51=101.Thenthesumof50101’sis5050.
ExerciseSet1.3
RC2. Inthemultiplication4 × 3=12,12istheproduct.
RC4. Theproductof1 andanynumber a is a
RC6. dividend
RC8. divisor
CC2. 1000
22. 22 615 11 8928 × 3172 17856 624960 892800 26784000 28, 319, 616
24. 24 24 13 6408 × 6064 25632 384480 38448000 38, 858, 112
26. 11 44 44 355 × 299 3195 31950 71000 106, 145
28. 1 2 2 41 6521 × 3449 58689 260840 2608400 19563000 22, 490, 929
30. 34 44 4506 × 7800 3604800 31542000 35, 146, 800
32. 1 2 6009 × 2003 18027 12018000 12, 036, 027
34. A = l × w =129yd × 65yd=8385sqyd
36. A = l × w =200ft × 85ft=17, 000sqft
38. 54 ÷ 9=6because54=9 6.
40. 37 37 =1Anynonzeronumberdividedbyitselfis1.
42. 56 1 =56Anynumberdividedby1isthatsamenumber.
44. 0 32 =0Zerodividedbyanynonzeronumberis0.
46. 74 ÷ 0isnotdefined,becausedivisionby0isnotdefined.
48. 20 4 =5because20=4 5.
50. 233 3 699 6 9 9 9 9 0
Theansweris233. 52. 108 8 869 8 69 64 5
Theansweris108R5. 54. 708 3 2124 21 24 24 0
Theansweris708.
Theansweris1012R2.
Theansweris194R1.
Theansweris146R5.
62. 2009 3 6027 6 27 27 0
Theansweris2009.
64. 517 8 4139 40 13 8 59 56 3
Theansweris517R3.
66. 1270 100 127, 000 100 270 200 700 700 0 0 0
Theansweris1270.
68. 426 10 4260 40 26 20 60 60 0
Theansweris426.
70. 289 20 5798 40 179 160 198 180 18
Theansweris289R18.
72. 24 40 987 80 187 160 27
Theansweris24R27.
74. 40 23 942 92 22
Theansweris40R22.
76. 50 54 2729 270 29
Theansweris50R29.
78. 55 102 5612 510 512 510 2
Theansweris55R2.
80. 107 7 749 7 49 49 0
Theansweris107.
82. 808 9 7273 72 73 72 1
Theansweris808R1.
84. 1010 7 7074 70 7 7 4
Theansweris1010R4.
86. 301 24 7242 72 42 24 18
Theansweris301R18.
88. 102 48 4899 48 99 96 3
Theansweris102R3. Copyright c 2020PearsonEducation,Inc.
90. 210 36 7563 72 36 36 3
Theansweris210R3.
92. 803 36 28, 929 288 129 108 21
Theansweris803R21.
94. 984 90 88, 560 810 756 720 360 360 0
Theansweris984.
96. 2904 306 888, 888 612 2768 2754 1488 1224 264
Theansweris2904R264.
98. 7002 803 5, 622, 606 5621 1606 1606 0
Theansweris7002.
100. 530
102. 8950
104. 50
106. 800
108. 900
110. 700
112. 4600
114. 198,400
116. 5000
118. 2000
120. 736,000
122. 6,713,000
124. 6260 97100 4650 +81+80 290 126. 673670 28 30 640
128. 568600 472500 938900 +402+400 2400
130. 94389400 2787 2800 6600
132. 76488000 93489000 78428000 +2222+2000 27,000
134. 84,89085,000 11,110 11,000 74,000
136. 5150 × 78 × 80 4000
138. 6360 × 54 × 50 3000
140. 355400 × 299 × 300 120,000
142. 789800 × 434 × 400 320,000
144. 454 ÷ 87 ≈ 450 ÷ 90=5
146. 1263 ÷ 29 ≈ 1260 ÷ 30=42
148. 3641 ÷ 571 ≈ 3600 ÷ 600=6
150. 32, 854 ÷ 748 ≈ 32, 900 ÷ 700=47
152. $498$500 289300 +145+100 $900
154. $498$500 289300 62100 159200 +320+300 $1400
Thebudgetcoversthechoices.
156. Answerswillvarydependingontheoptionschosen.
158. a) Totalcostofattending:
$250 490=$122, 500
Totalcostofhotelrooms:
$170 · 2 · 320=$108, 800
Totalamountspent:
$122, 500+$108, 800=$231, 300
b) Totalcostofattending:
$200 500=$100, 000
Totalcostofhotelrooms:
$200 2 300=$120, 000
Totalamountspent:
$100, 000+$120, 000=$220, 000
160. 9002 +4587 13, 589
162. 13 23/ 10818 3 /4/0 /, 79/8 / 86, 679 254, 119
164. 1tenthousand+2thousands+8hundreds+4tens+ 7ones
166. Perimeter=62yd+39yd+54yd+46yd+28yd Wecarryouttheaddition. 2 62 39 54 46 +28 229
Theperimeterofthefigureis229yd.
168. Pairsoffactorswhoseproductis36are: 1and36 2and18 3and12 4and9 6and6
a)Thepairabovewhosesumis13is4and9.
b)Thepairabovewhosedifferenceis0is6and6.
c)Thepairabovewhosesumis20is2and18.
d)Thepairabovewhosedifferenceis9is3and12.
170. 34, 584, 132 ÷ 76 =4 , 386
Considertherelatedmultiplicationsentence: 4 , 386 76 =34, 584, 132 Sincetheonesdigitoftheproductis2,themissingones digitmustbeeither2or7(6 2=12and6 7=42).
Wetry2: 34, 584, 132 ÷ 762=45, 386 Weseethatthemissingonesdigitis2andthemissing thousandsdigitis5.
ExerciseSet1.4
RC2. (a)
RC4. (b)
CC2. 672=6 n 672?6 112 672TRUE 112isasolution.
CC4. 5462=3189+ t 5462?3189+2327 5516FALSE 2327isnotasolution.
2. 25
4. 8
6. t =5678+9034=14, 712
8. m =9007 5667=3340
10. z =34 · 15=510
12. w =256 ÷ 16=16
14. t =22 15=7
16. t =16 16=0
18. x =57 20=37
20. w =53 17=36
22. x = 42 6 =7
24. m = 162 9 =18
26. y = 96 4 =24
28. t = 741 3 =247
30. y =9281 8322=959
32. p =92 56=36
34. y =23 × 78=1794
36. z =133 67=66
38. w = 3404 4 =851
40. x =807 438=369
42. q =10, 534 ÷ 458=23
44. x = 6080 19 =320
46. x = 1500 20 =75
48. t =9281 8322=959
50. n =3004 1745=1259
52. n = 660 12 =55
54. x = 22, 135 233 =95
56. z =512 63=449
58. 142 9 1278 9 37 36 18 18 0
Theansweris142.
60. 334 17 5689 51 58 51 79 68 11
Theansweris334R11.
62. 342 > 339
64. 0 < 11
66. 6,375,600
68. x = 14, 332, 388 48, 916 =293
Chapter1Mid-ChapterReview
1. Thestatementisfalse.Forexample,8 5=3,but5is notequalto8+3.
2. True
3. Thestatementisfalse.Forexample,3 0=0and0isnot greaterthan3.Also,1 1=1and1isnotgreaterthan1.
4. Itistruethatzerodividedbyanynonzeronumberis0.
5. Thestatementisfalse.Anynumberdividedby1isthe numberitself.Forexample, 27 1 =27.
6. 95 ,406 ,237 ↑↑↑
Ninety-fivemillion, fourhundredsixthousand, twohundredthirty-seven
7. 5914 604 / ✥✥ 497 107
8. 2 6 98
Thedigit6namesthenumberofhundreds.
9. 6 1,204
Thedigit6namesthenumberoftenthousands.
10. 14 6 ,237
Thedigit6namesthenumberofthousands.
11. 58 6
Thedigit6namesthenumberofones.
12. 306,458,1 2 9
Thedigit2namesthenumberoftens.
13. 30 6 ,458,129
Thedigit6namesthenumberofmillions.
14. 306,4 5 8,129
Thedigit5namesthenumberoftenthousands.
15. 306,458, 1 29
Thedigit1namesthenumberofhundreds.
16. 5602=5thousands+6hundreds+0tens+2ones,or5 thousands+6hundreds+2ones
17. 69,345=6tenthousands+9thousands+3hundreds+ 4tens+5ones
18. Awordnamefor136isonehundredthirty-six.
19. Awordnamefor64,325issixty-fourthousand,threehundredtwenty-five.
20. Standardnotationforthreehundredeightthousand,seven hundredsixteenis308,716.
21. Standardnotationforfourmillion,fivehundredsixtyseventhousand,twohundredninety-oneis4,567,291.
22. Since61istotherightof16onthenumberline,61 > 16.
23. Since100istotheleftof101onthenumberline, 100 < 101.
24. Since0istotheleftof18onthenumberline,0 < 18. Copyright c 2020PearsonEducation,Inc.
25. Since380istotherightof327onthenumberline, 380 > 327.
26. 4 b =72 4 b 4 = 72 4 b =18
Thenumber18checks.Itisthesolution.
27. 45=23+ x 45 23=23+ x 23 22= x Thenumber22checks.Itisthesolution.
28. t =725 ÷ 25 t =29Doingthedivision
Thenumber29checks.Itisthesolution.
29. 3902 260= y 3642= y Doingthesubtraction
Thenumber3642checks.Itisthesolution.
30. 316 +482 798
31. 11 593 +437 1030
32. 11 2638 +5284 7922
33. 111 4617 2436 +481 7534
34. 786 321 465
35. 11 51 / 14 6 /2/4 / 285 339
36. 15 25 / 912 3 /602 / ✥✥ 1748 1854
37. 49914 5004 / ✥✥✥ 676 4328
3 36 × 6 216
11 55 567 × 28 4536 11340 15, 876
2 1 3 407 × 325 2035 8140 122100 132, 275
223 1 9435 × 602 18870 5661000 5, 679, 870 42. 253 4 1012 8 21 20 12 12 0 Theansweris253.
43. 112 38 4261 38 46 38 81 76 5
Theansweris112R5. 44. 23 60 1399 120 199 180 19
Theansweris23R19.
45. 144 56 8095 56 249 224 255 224 31 Theansweris144R31.
46. Perimeter=10m+4m+8m+3m=25m
47. A =4in. × 2in.=8sqin.
48. Round647tothenearesthundred. 6 4 7 ↑
Thedigit6isinthehundredsplace.Considerthenext digittotheright.Sincethedigit,4,is4orlower,round down,meaningthat6hundredsstaysas6hundreds.Then changethedigitstotherightofthehundredsdigittozeros. Theansweris600.
49. Round823,502tothenearestthousand. 823, 5 02 ↑
Thedigit3isinthethousandsplace.Considerthenext digittotheright.Sincethedigit,5,is5orhigher,round3 thousandsupto4thousands.Thenchangethethedigits totherightofthethousandsdigittozeros. Theansweris824,000.
50. Roundedto thenearesthundred 218900 × 865 × 200 180,000 ← Estimatedanswer
51. Byroundingpricesandestimatingtheirsumashopper canestimatethetotalgrocerybillwhileshopping.Thisis particularlyusefuliftheshopperwantstospendnomore thanacertainamount.
52. Commasseparatetheperiodsandmakethenumberseasier toread.
53. Answerswillvary.Supposeonecoatcosts$150.Thenthe multiplication4 · $150givesthecostoffourcoats. Supposeonereamofcopypapercosts$4.Thenthemultiplication$4 150givesthecostof150reams.
54. Usingthedefinitionofdivision,0 ÷ 0= a suchthat a 0=0. Weseethat a couldbe any numbersince a · 0=0forany number a.Thus,wecannotsaythat0 ÷ 0=0.Thisis whyweagreenottoallowdivisionby0.
ExerciseSet1.5
RC2. Translate.
RC4. Check.
CC2. Let x =thenumberofmilesbywhichWednesday’s driveexceededThursday’sdrive.Thenwehave500= x +125.Thecorrectansweris(b).
CC4. Let x =thetotalamountspentongroceriesinMay andJune.Thenwehave500+125= x.Thecorrect answeris(a).
2. Let w =thenumberofpoundsbywhichthewastegeneratedannuallypercapitaintheUnitedStatesexceedsthe wastegeneratedinDenmark.
Solve:1473+ w =1606 w =123lb
4. Let w =thenumberofpoundsofwastegeneratedannually percapitainIceland.
Solve: w +531=1713 w =1182lb
6. Let n =thenumberofentriesineachrow.
Solve:504 ÷ 36= n n =14entries
8. Let r =thenumberofactiverotaryoilrigsin2007.
Solve: r +687=984 r =297rigs
10. Let n =thenumberofmilesbywhichthelengthofthe NileexceedsthelengthoftheMissouri-Mississippi.
Solve:3860+ n =4135 n =275mi
12. Let p =thetotalnumberofsquaresinthepuzzle.
Solve:15 15= p p =225squares
14. Let c =thenumberofmilligramsofcaffeineina20-oz bottleofCocaCola.
Solve:25+32= c c =57milligrams
16. Let m =thenumberofminutesinaday.
Solve:60 24= m m =1440minutes
18. Let r =theamountbywhichtheaveragemonthlyrentin AtlantaexceedstheaveragemonthlyrentinIndianapolis.
Solve:905+ r =1401 r =$496permonth
20. Let r =theamountofrenteachsisterpays.
Solve:2r =936 r =$468permonth
22. Let r =theamountofrentatenantwouldpayforaonebedroomapartment,onaverage,inSeattleduringa6monthperiod.
Solve:6 · 2063= r r =$12, 378
24. Let s =thespeedlimitfortrucks.
Solve: s +10=75 s =65mph
26. Let q =thenumberofquiresinaream.
Solve:25 q =500 q =20quires
28. Let s =theamountbywhichspendingbyvisitorstothe UnitedStatesexceededspendingbyAmericanstraveling abroad.
Solve:110, 500, 000, 000+ s =153, 700, 000, 000 s =$43, 200, 000, 000
30. Let c =thetotalcostofthepurchase.
Solve:96 88= c c =$8448
32. Let w =thenumberoffullweeksthatwillpassbeforethe stationmustbeginre-airingepisodes.
Solve:5 · w =208 w =41R3,so41fullweekswillpassand3episodeswillbe shownthefollowingweekbeforepreviouslyairedepisodes arererun.
34. Let l =thenumberoflabelsoneachsheet.
Solve:25 · l =750 l =30labels
36. Let g =thenumberofgallonsrequiredfor3795miofcity driving.
Solve:3795 ÷ 23= g g =165gal
38. a) Let A =theareaofthecourt,insquarefeet.
Solve: A =94 50 A =4700squarefeet
b) Let P =theperimeterofthecourt,infeet.
Solve: P =94+50+94+50 P =288ft
c) Let a =theamountbywhichtheareaofacollegecourtexceedstheareaofahighschoolcourt, insquarefeet.
Solve:4200+ a =4700 a =500squarefeet
40. Let c =thenumberofcartonsneeded.
Solve:528 ÷ 12= c c =44cartons
42. Let m =thedistanceonthemap,ininches,betweentwo citiesthat,inreality,are2016miapart.
Solve:2016 ÷ 288= m m =7in.
Let r =thedistanceinmiles,inreality,betweentwocities thatare8in.apartonthemap.
Solve:288 8= r r =2304mi
44. Let m =thenumberofmonthsrequiredtopayoffthe loan.
Solve:7824 ÷ 163= m m =48months
46. Let n =thenumberof100’sin3500.
Solve:3500 ÷ 100= n n =35
Let t =thenumberofminutesyoumustgolf,walking,in ordertoloseonepound.
Solve: t =35 20 t =700min;wecouldalsoexpressthisas11hr,40min.
48. Let n =thenumberofnewjobsthatwillbecreatedfor marketingmanagersandaccountants.
n =19, 700+142, 400=162, 100
Let s =thenumberofnewjobsthatwillbecreatedfor salesmanagers.
Solve: s +143, 100=162, 100
s =19, 000jobs
50. Let F =thenumberofseatsinfirstclass, E =thenumber ofseatsineconomyclass,and T =thetotalnumberof seats.
Solve:3 4= F ,23 6= E ,and T = F + E F =12, E =138, T =12+138=150seats
52. Let c =thetotalcostofthe5videogames.
Solve:5 64= c c =$320
Thenlet n =thenumberof$20billsrequired.
Solve:320 ÷ 20= n n =16$20bills
54. Let b =thenewbalance.
Solve:749 34 65+123= b b =$773
56. Let l =thetotallengthofthebookshelves,infeet.
Solve:6 3= l l =18ft
Sincethetotallengthofthebookshelvesisgreaterthan 16ft,theshelvescannotbeputsidebysideonthe16-ft wall.
58. 15 85 / 912 9 /602 / ✥✥ 1843 7759
60. 147 32 4708 32 150 128
4
Theansweris147R4.
62. A = l × w =211ft × 46ft=9706sqft
64. x =81 15=66
66. Consideronestudenttakingoneclassa“student-class unit.”Thenlet s =thetotalnumberofstudent-classunits and p =thenumberofstudentstaughtbyeachinstructor.
Solve:1200 5= s,4 30= p s =6000, p =120
Nowlet n =thenumberofinstructors.
Solve:6000 ÷ 120= n n =50instructors
ExerciseSet1.6
RC2. Theexpression92 canberead“ninesquared.”
RC4. Tofindtheaverageof7,8,and9,weaddthenumbers anddividethesumby3
CC2. Division
CC4. Multiplication
2. 25
4. 133 6. 92 8. 14 10. 125 12. 64
14. 100,000 16. 64
18. (12+6)+18=18+18 =36
20. (52 40) 8=12 8 =4
22. (1000 ÷ 100) ÷ 10=10 ÷ 10 =1
24. 256 ÷ (64 ÷ 4)=256 ÷ 16 =16
26. 22 +52 =4+25 =29
28. (32 27)3 +(19+1)3 =53 +203 =125+8000 =8125
30. 23+18 20=23+360 =383
32. 10 7 4=70 4 =66 34. 90 5 5 2=90 50 =40
36. 82 8 2=64 8 2 =64 16 =48
38. 1000 ÷ 25 (15+5)=1000 ÷ 25 20 =40 20 =20
40. 3 · 8+5 · 8=24+40 =64
42. 144 ÷ 4 2=36 2 =34
44. 7 · (10 3)2 2 · (3+1)2 =7 · 72 2 · 42 =7 49 2 16 =343 32 =311
46. 62 34 ÷ 33 =36 81 ÷ 27 =36 3 =33
48. 72 +20 4 (28+9 2) =72 +20 · 4 (28+18) =72 +20 4 46 =49+20 4 46 =49+80 46 =83
50. 8 × 9 (12 8) ÷ 4 (10 7) =8 × 9 4 ÷ 4 3 =72 1 3 =68
52. 80 24 15 ÷ (7 5 45 ÷ 3) =80 24 15 ÷ (35 15) =80 24 · 15 ÷ 20 =80 16 15 ÷ 20 =80 240 ÷ 20 =80 12 =68
54. 27 ÷ 25 24 ÷ 22 =128 ÷ 32 16 ÷ 4 =4 · 16 ÷ 4 =64 ÷ 4 =16
56. 86+92+80+78 4 = 336 4 =84
58. $1025+$775+$2062+$942+$3721 5 = $8525 5 =$1705
60. 72 ÷ 6 −{2 × [9 (4 × 2)]}
=72 ÷ 6 −{2 × [9 8]}
=72 ÷ 6 −{2 × 1}
=72 ÷ 6 2 =12 2 =10
62. [92 × (6 4) ÷ 8]+[7 × (8 3)]
=[92 × 2 ÷ 8]+[7 × 5] =[184 ÷ 8]+35 =23+35 =58
64. (18 ÷ 2) ·{[(9 9 1) ÷ 2] [5 20 (7 9 2)]}
=9 ·{[(81 1) ÷ 2] [5 20 (63 2)]}
=9 ·{[80 ÷ 2] [5 20 61]}
=9 ·{40 [100 61]}
=9 ·{40 39} =9 ·{1} =9
66. 15(23 4 2)3 ÷ (3 25) =15(23 8)3 ÷ 75Multiplyinginsideparentheses
=15 · 153 ÷ 75Subtractinginsideparentheses =15 3375 ÷ 75Evaluatingtheexponential expression
=50, 625 ÷ 75Doingallmultiplicationand =675divisionsinorderfromlefttoright
68. (19 24 )5 (141 ÷ 47)2 =(19 16)5 32 =35 32 =243 9 =234
70. x =5032 4197=835
72. y = 1554 42 =37
74. t = 10, 000 100 =100
76. Let g =thetotalnumberofgallonsofgasolinepurchased.
Solve:23+24+26+25= g g =98gallons
78. 12 ÷ 4+2 3 2=3+6 2 =7Correctanswer 12 ÷ (4+2) · (3 2)=2
80. Answersmayvary.Onecorrectansweris
9 8+7 6 5 4+3 2 1=100.
ExerciseSet1.7
RC2. 6isafactor of42.
RC4. 42isamultiple of3.
RC6. Onefactorization of42is2 21.
CC2. True;30=2 15.
CC4. Thenumber1isnotprimebecauseitdoesnothave twodifferentfactors.Thestatementisfalse.
CC6. Thenumber10isnotprime(10=2 5),sothestatementisfalse.
2. 4 13 52 52 0
Theremainderis0,so13isafactorof52.
4. 42 16 680 64 40 32 8
Theremainderisnot0,so16isnotafactorof680.
6. 1,2,4,8,16
8. 1,2,3,4,6,8,12,16,24,48
10. 1,3,9
12. 1,13
14. 1,2,4,5,10,20,25,50,100
16. 1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
18. 11,22,33,44,55,66,77,88,99,110
20. 50,100,150,200,250,300,350,400,450,500
22. 5,10,15,20,25,30,35,40,45,50
24. 13,26,39,52,65,78,91,104,117,130
26. 6,12,18,24,30,36,42,48,54,60
28. 14,28,42,56,70,84,98,112,126,140
30. 6 8 48 48 0 48isdivisibleby8.
32. 1409 3 4227 3 12 12 27 27 0 4227isdivisibleby3. Copyright c
34. 25 4 102 8 22 20 2 102isnotdivisibleby4.
36. 591 7 4143 35 64 63 13 7 6 4143isnotdivisibleby7.
38. Thenumber2isprime.Ithasonlythefactors1and2.
40. Thenumber19isprime.Ithasonlythefactors1and19.
42. Thenumber27hasfactors1,3,9,and27.Itiscomposite.
44. Thenumber49hasfactors1,7,and49.Itiscomposite.
46. 2 2 2 2
48. 3 · 5
50. 2 2 2 2 2
52. 2 2 2 5
54. 2 31
56. 2 2 5 7
58. 2 · 5 · 11
60. 2 5 7
62. 2 43
64. 3 3 11
66. 2 2 11 11
68. 7 · 13
70. 2 2 2 3 3 5 5
72. 3 3 3 5 5
74. 3 3 5 11 13
76. 4millions
78. 4tenthousands
80. 34,600
82. 2,428,500
ExerciseSet1.8
RC2. Anumberisdivisibleby3ifthesumofitsdigitsis divisibleby3.Thecorrectansweris(a).
RC4. Anumberisdivisibleby5ifitsonesdigitis0or5. Thecorrectansweris(d).
RC6. Anumberisdivisibleby8ifthenumbernamedbyits lastthreedigitsisdivisibleby8.Thecorrectanswer is(h).
RC8. Anumberisdivisibleby10ifitsonesdigitis0.The correctansweris(e).
2. Because4+6+7=17and17isnotdivisibleby9,467is notdivisibleby9.
4. Theonesdigitiseven; 2+0+0+4=6and6is divisible by3.Thus2004isdivisibleby6.
6. 6120isdivisibleby5becausetheonesdigitis0.
8. Because3+2+8+6=19and19isnotdivisibleby3, 3286isnotdivisibleby3.
10. 64,091isnotdivisibleby10becausetheonesdigitisnot 0.
12. 9840isdivisibleby2becausetheonesdigitiseven.
14. Becausethenumbernamedbythelastthreedigits,106, isnotdivisibleby8,thenumber546,106isnotdivisible by8.
16. Becausethenumbernamedbythelasttwodigits,36,is divisibleby4,thenumber298,736isdivisibleby4.
18. 12,600isdivisibleby2becausetheonesdigitiseven.
Because 1+2+6+0+0=9and9is divisibleby3,then 12,600isdivisibleby3.
12,600isdivisibleby4becausethenumbernamedbyits lasttwodigits,00,isdivisibleby4.
12,600isdivisibleby5becausetheonesdigitis0.
12,600isdivisibleby2andby3(seeabove),soitisdivisibleby6.
12,600isdivisibleby8becausethenumbernamedbyits lastthreedigits,600,isdivisibleby8.
Becausethesumofthedigitsisdivisibleby9(seeabove), 12,600isdivisibleby9.
12,600isdivisibleby10becausetheonesdigitis0.
20. 2916isdivisibleby2becausetheonesdigitiseven.
Because2+9 +1+6=18and18is divisibleby3,2916 isdivisibleby3.
2916isdivisibleby4becausethenumbernamedbyits lasttwodigits,16,isdivisibleby4.
2916isnotdivisibleby5becausetheonesdigitisneither 0nor5.
2916isdivisibleby2andby3(seeabove),soitisdivisible by6.
2916isnotdivisibleby8becausethenumbernamedby itslastthreedigits,916,isnotdivisibleby8.
Becausethesumofthedigitsisdivisibleby9(seeabove), 2916isdivisibleby9.
2916isnotdivisibleby10becausetheonesdigitisnot0.
22. 25,088isdivisibleby2becausetheonesdigitiseven.
Because2+5+0+8+8=23and23isnotdivisibleby 3,then25,088isnotdivisibleby3.
25,088isdivisibleby4becausethenumbernamedbyits last2digits,88,isdivisibleby4.
25,088isnotdivisibleby5becausetheonesdigitisneither 0nor5.
25,088isnotdivisibleby6becauseitisnotdivisibleby3 (seeabove).
25,088isdivisibleby8becausethenumbernamedbyits lastthreedigits,088,isdivisibleby8.
25,088isnotdivisibleby9becausethesumofthedigits isnotdivisibleby9(seeabove).
25,088isnotdivisibleby10becausetheonesdigitisnot 0.
24. 143,507isnotdivisibleby2becausetheonesdigitisnot even.
Because1+4+3+5+0+7=20and20isnotdivisible by3,then143,507isnotdivisibleby3.
143,507isnotdivisibleby4becausethenumbernamed byitslasttwodigits,07,isnotdivisibleby4.
143,507isnotdivisibleby5becausetheonesdigitisneither0nor5.
143,507isnotdivisibleby6becauseitisnoteven.
143,507isnotdivisibleby8becausethenumbernamed byitslastthreedigits,507,isnotdivisibleby8.
143,507isnotdivisibleby9becausethesumofthedigits isnotdivisibleby9(seeabove).
143,507isnotdivisibleby10becausetheonesdigitisnot 0.
26. Thenumbersforwhichtheonesdigitisevenare56,324, 784,200,42,812,and402.Thesenumbersaredivisibleby 2.
28. Thenumbersforwhichthelasttwodigitsaredivisible by4are56,324,784,200,and812.Thesenumbersare divisibleby4.
30. Thenumbersthataredivisiblebyboth2and3are324, 42,and402.Thesenumbersaredivisibleby6.
32. Theonlynumberforwhichthesumofthedigitsisdivisible by9is324.Thisnumberisdivisibleby9.
34. Thenumbersforwhichthesumofthedigitsisdivisibleby 3are1101,313,332,111,126,876,1110,9990,and126,111.
36. Thenumbersforwhichtheonesdigitis0or5are305, 13,025,1110,64,000,and9990.
38. Thenumbersforwhichthelastthreedigitsaredivisible by8are7624,5128,and64,000.
40. Thenumbersforwhichthelasttwodigitsaredivisibleby 4are313,332,7624,876,5128,and64,000.
42. y +124=263 y =263 124=139
44. 18 t =1008 t = 1008 18 =56
46. 338= a · 26 338 26 = a 13= a
48. Let m =thenumberofminutesin72hours. Solve:60 · 72= m m =4320minutes.
50. 2520isdivisibleby2:2520=2 1260;1260isdivisibleby 2:2520=2 2 630;630isdivisibleby2:2520=2 2 2 315; 315isnotdivisibleby2,butitisdivisibleby3:2520= 2 2 2 3 105;105isdivisibleby3:2520=2 2 2 3 3 35; 35isnotdivisibleby3,butitisdivisibleby5:2520= 2 2 2 3 3 5 7.Since7isaprimenumber,thelast factorizationistheprimefactorization.
52. 1998isdivisibleby2:1998=2 · 999;999isnotdivisible by2,butitisdivisibleby3:1998=2 3 333;333is divisibleby3:1998=2 3 3 111;111isdivisibleby3: 1998=2 3 3 3 37.Since37isaprimenumber,thelast factorizationistheprimefactorization.
54. Thenumbermustbeamultipleof11.Wetrynumbersof theform11 n where n isaprimenumbergreaterthan5. Thesmallestmultiplethatmeetsthecriteriais11 11,or 121.Thisisthenumber.
ExerciseSet1.9
RC2. True
RC4. False
CC2. 20=2 2 5, 24=2 2 2 3
TheLCMis2 2 2 3 5,or120.
CC4. 36=2 · 2 · 3 · 3, 600=2 2 2 3 5 5
TheLCMis2 2 2 3 3 5 5,or1800.
CC6. 15=3 · 5, 40=2 · 2 · 2 · 5 28=2 2 7
125=5 5 5
TheLCMis2 2 2 3 5 5 5 7,or21,000.
InthissectionwewillfindtheLCMusingthelistof multiplesmethodinExercises2-20andtheprimefactorizationmethodinExercises22-50.
2. a)15isamultipleof3,soitistheLCM. c)TheLCM=15.
4. a)15isnotamultipleof10.
b)Checkmultiples:
2 15=30Amultipleof10
c)TheLCM=30.
6. a)12isnotamultipleof8.
b)Checkmultiples:
2 12=24Amultipleof8
c)TheLCM=24.
8. a)Wenoticeattheoutsetthat9and11havenocommon primefactor.Therefore,theLCMistheproductofthe twonumbers.
c)TheLCM=9 11,or99.
10. a)36isnotamultipleof24.
b)Checkmultiples:
2 36=72Amultipleof24
c)TheLCM=72.
12. a)27isnotamultipleof21.
b)Checkmultiples:
2 · 27=54Notamultipleof21
3 27=81Notamultipleof21
4 27=108Notamultipleof21
5 27=135Notamultipleof21
6 27=162Notamultipleof21
7 27=189Amultipleof21
c)TheLCM=189.
14. a)18isnotamultipleof12.
b)Checkmultiples:
2 18=36Amultipleof12
c)TheLCM=36.
16. a)45isnotamultipleof35.
b)Checkmultiples:
2 45=90Notamultipleof35
3 · 45=135Notamultipleof35
4 45=180Notamultipleof35
5 45=225Notamultipleof35
6 45=270Notamultipleof35
7 45=315Amultipleof35
c)TheLCM=315.
18. a)20isnotamultipleof18.
b)Checkmultiples:
2 20=40Notamultipleof18
3 20=60Notamultipleof18
4 20=80Notamultipleof18
5 20=100Notamultipleof18
6 · 20=120Notamultipleof18
7 20=140Notamultipleof18
8 20=160Notamultipleof18
9 20=180Amultipleof18
c)TheLCM=180.
20. a)48isnotamultipleof36.
b)Checkmultiples:
2 48=96Notamultipleof36
3 · 48=144Amultipleof36
c)TheLCM=144.
22. Notethateachofthenumbers3,5,and7isprime.They havenocommonprimefactor.Whenthishappens,the LCMisjusttheproductofthenumbers.
TheLCMis3 5 7,or105.
24. a) 6=2 3 12=2 2 3 18=2 3 3
b)TheLCMis2 · 2 · 3 · 3,or36.
26. a) 8=2 2 2 16=2 · 2 · 2 · 2 22=2 11
b)TheLCMis2 2 2 2 11,or176.
28. a) 12=2 2 3 18=2 · 3 · 3 40=2 2 2 5
b)TheLCMis2 2 2 3 3 5,or360.
30. a) 8=2 2 2 16=2 2 2 2 12=2 2 3
b)TheLCMis2 2 2 2 3,or48.
32. a) 18=2 · 3 · 3 30=2 3 5 50=2 5 5 48=2 2 2 2 3
b)TheLCMis2 · 2 · 2 · 2 · 3 · 3 · 5 · 5,or3600.
34. 16isafactorof32,sotheLCMis32.
36. 12isafactorof72,sotheLCMis72.
38. 13and14havenocommonprimefactor,sotheLCMis 13 14,or182.
40. 23and25havenocommonprimefactor,sotheLCMis 23 25,or575.
42. a) 56=2 2 2 7 72=2 2 2 3 3
b)TheLCMis2 · 2 · 2 · 3 · 3 · 7,or504.
44. a) 75=3 5 5 100=2 2 5 5
b)TheLCMis2 2 3 5 5,or300.
46. a) 22=2 11 42=2 3 7 51=3 17
b)TheLCMis2 · 3 · 7 · 11 · 17,or7854.
48. a) 625=5 5 5 5
75=3 5 5
500=2 2 5 5 5
25=5 · 5
b)TheLCMis2 2 3 5 5 5 5,or7500.
50. a)
300=2 2 3 5 5 4000=2 2 2 2 2 5 5 5
b)TheLCMis2 2 2 2 2 3 5 5 5,or12,000.
52. ThetimeittakesUranustomakeacompleterevolution, 84yr,isamultipleofthetimeittakesJupiter,12yr,so JupiterandUranusappearinthesamedirectioninthe nightskyonceevery84years.
54. Jupiter:12=2 2 3
Saturn:30=2 3 5
Uranus:84=2 2 3 7
TheLCMis2 2 3 5 7,or420.Thus,Jupiter,Saturn, andUranuswillappearinthesamedirectioninthenight skyonceevery420years.
56. 1,2,4,23,46,92
58. 1,2,4,5,10,22,44,55,110,220
60. 64 ÷ (16 ÷ 4)=64 ÷ 4=16
62. 799914 80, 004 / ✭✭✭✭ 2305 77, 699
64. 8=2 2 2 12=2 2 3
a)No;theLCMhasonemore2andoneless3.
b)No;theLCMneedsonemore2.
c)No;theLCMneedstwomore2’sandoneless3. d)Yes;theLCMhasthree2’sandone3.
Chapter1VocabularyReinforcement
1. Thedistancearoundanobjectisitsperimeter.
2. Since20=4 5,wesaythat4 5isafactorization of20.
3. Inthesentence10 × 1000=10, 000,10and1000arecalled factors and10,000iscalledtheproduct
4. Thenumber0iscalledtheadditive identity.
5. Anaturalnumberthathasexactlytwodifferentfactors, onlyitselfand1,iscalledaprime number.
6. Theleastcommonmultiple oftwonumbersisthesmallest numberthatisamultipleofbothnumbers.
7. Since20=4 5,wesaythat20isamultiple of5.
Chapter1ConceptReinforcement
1. a ÷ a = a a =1, a =0;thestatementistrue.
2. True
3. False;theaverageofthreenumbersisthesumofthenumbersdividedby3.
4. Thestatementisfalse.Forexample,1+2=5isnota trueequation.
5. Thestatementistrue.Ifonenumberisamultipleofthe other,theLCMisthelargernumber.Ifonenumberisnot amultipleoftheother,thentheLCMcontainseachprime factorthatappearsineithernumberthegreatestnumber oftimesitoccursinanyonefactorizationand,thus,is largerthanbothnumbers.
Chapter1StudyGuide
1. 43 2 ,079 Thedigit2namesthenumberofthousands.
2. Since78istotheleftof81onthenumberline,78 < 81.
3. 111 36, 047 +29, 255 65, 302
4. 7915 4805 / ✥✥ 1568 3237
5. 21 1 73 684 × 329 6156Multiplyingby9 13680Multiplyingby20 205200Multiplyingby300 225, 036
6. 315 27 8519 81 41 27 149 135 14 Theansweris315R14.
7. Round36,468tothenearestthousand.
36, 4 68 ↑
Thedigit6isinthethousandsplace.Considerthenext digittotheright.Sincethedigit,4,is4orlower,round down,meaningthat6thousandsstaysas6thousands. Thenchangethedigitstotherightofthethousandsdigit tozeros.
Theansweris36,000.
8. 24 x =864
24 x 24 = 864 24 Dividingby24 x =36
Check:24 x =864 24 36?864 864 TRUE
Thesolutionis36.
9. 63 =6 · 6 · 6=216
10. Wefindasmanytwo-factorfactorizationsaswecan.
104=1 104104=4 26
104=2 52104=8 13
Factors:1,2,4,8,13,26,52,104
11. 13 ← 13isprime 2 26 2 52 2 104
104=2 2 2 13
12. 52=2 · 2 · 13 78=2 3 13
TheLCMis2 · 2 · 3 · 13,or156.
Chapter1ReviewExercises
1. 4,67 8 ,952
Thedigit8means8thousands.
2. 1 3 ,768,940
Thedigit3namesthenumberofmillions.
3. 2793=2thousands+7hundreds+9tens+3ones
4. 56,078=5tenthousands+6thousands+0hundreds +7tens+8ones,or5tenthousands+6thousands+ 7tens+8ones
5. 4,007,101=4millions+0hundredthousands+0ten thousands+7thousands+1hundred+0tens+1one, or4millions+7thousands+1hundred+1one
6. 67 ,819 ↑↑ Sixty-seventhousand, eighthundrednineteen
7.
2 ,781 ,427 ↑↑↑ Twomillion, sevenhundredeighty-onethousand, fourhundredtwenty-seven
8. Fourhundredseventy-sixthousand, fivehundredeighty-eight ↓↓ Standardnotationis 476, 588.
9. Onebillion, fivehundredmillion, ↓↓ Standardnotationis 1, 500,000,000.
10. Since67istotherightof56onthenumberline,67 > 56.
11. Since1istotheleftof23onthenumberline,1 < 23.
12. 11 7304 +6968 14, 272 13. 111 27, 609 +38, 415 66, 024 14. 11 2703 4125 6004 +8956 21, 788 15. 11 91, 426 +7, 495 98, 921 16. 13 793 / 15 804/5 / ✥✥ 2897 5148 17. 89911 9001 / ✥✥✥ 7312 1689 18. 59913 6003 / ✥✥✥ 3729 2274 19. 1613 26 / 3 / 915 3 /7 /, 405 / ✥✥ 19, 648 17, 757
20. 2 17, 000 × 300 5, 100, 000 Multiplyingby300 (Write00andthen multiply17,000by3.)
21. 634 7846 × 800 6, 276, 800 Multiplyingby800 (Write00andthen multiply7846by8.)
22. 13 25 24 726 × 698 5808 65340 435600 506, 748
Multiplyingby8 Multiplyingby90 Multiplyingby600
23. 32 64 587 × 47 4109Multiplyingby7 23480Multiplyingby40 27, 589
24. 8305 × 642 16610 332200 4983000 5, 331, 810
25. 12 5 63 5 13 10 3
Theansweris12R3.
26. 5 16 80 80 0
Theansweris5.
27. 913 7 6394 63 9 7 24 21 3
Theansweris913R3.
28. 384 8 3073 24 67 64 33 32 1
Theansweris384R1.
29. 4 60 286 240 46
Theansweris4R46.
30. 54 79 4266 395 316 316 0
Theansweris54.
31. 452 38 17, 176 152 197 190 76 76 0
Theansweris452.
32. 5008 14 70, 112 70 112 112 0
Theansweris5008.
33. 4389 12 52, 668 48 46 36 106 96 108 108 0
Theansweris4389.
34. Round345,759tothenearesthundred.
345, 7 5 9 ↑
Thedigit7isinthehundredsplace.Considerthenext digittotheright.Sincethedigit,5,is5orhigher,round 7hundredsupto8hundreds.Thenchangethedigitsto therightofthehundredsdigittozero.
Theansweris345,800.
35. Round345,759tothenearestten.
345, 75 9 ↑
Thedigit5isinthetensplace.Considerthenextdigitto theright.Sincethedigit,9,is5orhigher,round5tensup to6tens.Thenchangethedigittotherightofthetens digittozero.
Theansweris345,760. Copyright c 2020PearsonEducation,Inc.
36. Round345,759tothenearestthousand.
345, 7 59 ↑
Thedigit5isinthethousandsplace.Considerthenext digittotheright.Sincethedigit,7,is5orhigher,round 5thousandsupto6thousands.Thenchangethedigitsto therightofthethousandsdigittozero.
Theansweris346,000.
37. Round345,759tothenearesthundredthousand.
3 4 5, 759 ↑
Thedigit3isinthehundredthousandsplace.Consider thenextdigittotheright.Sincethedigit,4,is4orlower, rounddown,meaningthat3hundredthousandsstaysas 3hundredthousands.Thenchangethedigitstotheright ofthehundredthousandsdigittozero.
Theansweris300,000.
38. Roundedto thenearesthundred
41,34841,300 +19,749+19,700 61,000 ← Estimatedanswer
39. Roundedto thenearesthundred
38,65238,700 24,549 24,500 14,200 ← Estimatedanswer
40. Roundedto thenearesthundred 396400 × 748 × 700 280,000 ← Estimatedanswer
41. 46 · n =368
46 n 46 = 368 46 n =8
Check:46 n =368 46 8?368 368 TRUE
Thesolutionis8.
42. 47+ x =92
47+ x 47=92 47 x =45
Check:47+ x =92 47+45?92 92 TRUE
Thesolutionis45.
43. 1 y =58 y =58(1 y = y ) Thenumber58checks.Itisthesolution.
44. 24= x +24 24 24= x +24 24 0= x Thenumber0checks.Itisthesolution.
45. Exponentialnotationfor4 4 4is43
46. 104 =10 10 10 10=10, 000
47. 62 =6 · 6=36
48. 8 6+17=48+17Multiplying =65Adding
49. 10 24 (18+2) ÷ 4 (9 7) =10 24 20 ÷ 4 2Doingthecalculations insidetheparentheses =240 5 2Multiplyinganddividing
=235 2Subtractingfrom =233lefttoright
50. (80 ÷ 16) × [(20 56 ÷ 8)+(8 · 8 5 · 5)] =5 × [(20 7)+(64 25)] =5 × [13+39] =5 × 52 =260
51. Weaddthenumbersanddividebythenumberofaddends. 157+170+168 3 = 495 3 =165
52. Familiarize.Let x =theadditionalamountofmoney,in dollars,Natashaneedstobuythedesk.
Translate Money available plus Additional amount is Price ofdesk
196+ x =698
Solve.Wesubtract196onbothsidesoftheequation.
196+ x =698
196+ x 196=698 196 x =502
Check.Wecanestimate. 196+502 ≈ 200+500 ≈ 700 ≈ 698 Theanswerchecks. State.Natashaneeds$502dollars.
53. Familiarize.Let b =thebalanceinToni’saccountafter thedeposit. Translate Originalbalance plusDeposit isNewbalance
406+78= b
Solve.Weaddontheleftside.
406+78= b 484= b
Check.Wecanrepeatthecalculation.Theanswer checks.
State.Thenewbalanceis$484.
54. Familiarize.Let y =theyearinwhichthecoppercontent ofpennieswasreduced.
Original year plus73yr is Yearof copperreduction
1909+73= y
Solve.Weaddontheleftside.
1909+73= y 1982= y
Check.Wecanestimate.
1909+73 ≈ 1910+70 ≈ 1980 ≈ 1982 Theanswerchecks.
State.Thecoppercontentofpennieswasreducedin1982.
55. Familiarize.Wefirstmakeadrawing.Let c =thenumberofcartonsfilled.
12ineachrow Howmanyrows?
Translate Number ofcans divided by Number percarton is Number ofcartons
228 ÷ 12= c
Solve.Wecarryoutthedivision. 19 12 228 12 108 108 0
Thus,19= c,or c =19.
Check.Wecancheckbymultiplying:12 19=228.Our answerchecks.
State.19cartonswerefilled.
56. Familiarize.Thisisamultistepproblem.Let s =the costof13stoves, r =thecostof13refrigerators,and t = thetotalcostofthestovesandrefrigerators.
Translate.
Number ofstoves times Priceper stove is Totalcost ofstoves
13 425= s
Numberof refrigerators times Priceper refrigerator is Totalcostof refrigerators
13 620= r
Costof stoves plus Costof refrigerators isTotalcost
s + r = t
Solve.Wefirstcarryoutthemultiplicationsinthefirst twoequations.
13 425= s 13 620= r
5525= s 8060= r Nowwesubstitute5525for s and8060for r inthethird equationandthenaddontheleftside.
s + r = t
5525+8060= t 13, 585= t
Check.Werepeatthecalculations.Theanswerchecks. State.Thetotalcostwas$13,585.
57. Familiarize.Let b =thenumberofbeehivesthefarmer needs.
30ineachrow Howmanyrows?
Translate Number oftrees divided by
Numberoftrees pollinatedby eachhive is Number ofhives needed
÷ 30= b
Solve.Wecarryoutthedivision.
Thus,14= b,or b =14. Check.Wecancheckbymultiplying:30 · 14=420.The answerchecks. State.Thefarmerneeds14beehives.
58. A = l w =14ft 7ft=98squareft Perimeter=14ft+7ft+14ft+7ft=42ft
59. Familiarize.Wemakeadrawing.Let b =thenumberof beakersthatwillbefilled.
20ineachrow Howmanyrows?
Translate
Amountof alcohol divided by Amount perbeaker is Numberof beakers filled
2753 ÷ 20= b
Solve.Wecarryoutthedivision. 137 20 2753 20 75 60 153 140 13
Thus,137R13= b
Check.Wecancheckbymultiplyingthenumberof beakersby137andthenaddingtheremainder,13.
137 20=2740and2740+13=2753
Theanswerchecks.
State.137beakerscanbefilled;13mLwillbeleftover.
60. Familiarize.Thisisamultistepproblem.Let b =thetotalamountbudgetedforfood,clothing,andentertainment andlet r =theincomeremainingaftertheseallotments.
Translate
Foodand clothing budget plus Entertainment budget is Totalof theseallotments
7825+2860= b
Food,clothing, andentertainment allotments plus Remaining income is Total income
b + r =38, 283
Solve.Weaddontheleftsidetosolvethefirstequation.
7825+2860= b 10, 685= b
Nowwesubstitute10,685for b inthesecondequationand solvefor r .
b + r =38, 283
10, 685+ r =38, 283
10, 685+ r 10, 685=38, 283 10, 685 r =27, 598
Check.Werepeatthecalculations.Theanswerchecks.
State.Aftertheallotmentsforfood,clothing,andentertainment,$27,598remains.
61. Wefindasmanytwo-factorfactorizationsaswecan: 60=1 6060=4 15 60=2 3060=5 12
60=3 2060=6 10
Factors:1,2,3,4,5,6,10,12,15,20,30,60
62. Wefindasmanytwo-factorfactorizationsaswecan: 176=1 · 176176=8 · 22
176=2 88176=11 16 176=4 · 44
Factors:1,2,4,8,11,16,22,44,88,176
63. 1 8=86 8=48 2 8=167 8=56 3 8=248 8=64 4 8=329 8=72 5 · 8=4010 · 8=80
64. 84 11 924 88 44 44 0
Sincetheremainderis0,924isdivisibleby11.
65. 112 16 1800 16
8
Sincetheremainderisnot0,1800isnotdivisibleby16.
66. Theonlyfactorsof37are1and37,so37isprime.
67. 1isneitherprimenorcomposite.
68. Thenumber91hasfactors1,7,13,and91,soitiscomposite.
69. 7 ← 7isprime. 5 35 2 70 70=2 5 7
70. 5 ← 5isprime. 3 15 2 30 30=2 3 5
71. 5 ← 5isprime. 3 15 3 45 45=3 3 5
72. 5 ← 5isprime. 5 25 3 75 2 150 150=2 3 5 5
73. 3 ← 3isprime. 3 9 3 27 3 81 2 162 2 324 2 648 648=2 2 2 3 3 3 3
74. 7 ← 7isprime. 5 35 5 175 5 875 3 2625 2 5250
5250=2 3 5 5 5 7
75. Anumberisdivisibleby3ifthesumofitsdigitsisdivisible by3.Thenumberswhosedigitsaddtoamultipleof3are 4344,600,93,330,255,555,780,2802,and711.
76. Anumberisdivisibleby2ifitsonesdigitiseven.Thus, thenumbers140,182,716,2432,4344,600,330,780,and 2802aredivisibleby2.
77. Anumberisdivisibleby4ifthenumbernamedbyitslast twodigitsisdivisibleby4.Thus,thenumbers140,716, 2432,4344,600,and780aredivisibleby4.
78. Anumberisdivisibleby8ifthenumbernamedbyits lastthreedigitsisdivisibleby8.Thus,thenumbers2432, 4344,and600aredivisibleby8.
79. Anumberisdivisibleby5ifitsonesdigitis0or5.Thus, thenumbers140,95,475,600,330,255,555,and780are divisibleby5.
80. Anumberisdivisibleby6ifitsonedigitisevenandthe sumofthedigitsisdivisibleby3.Thenumberswhose onesdigitsareevenaregiveninExercise73above.Of thesenumbers,theoneswhosedigitsaddtoamultipleof 3are4344,600,330,780,and2802.
81. Anumberisdivisibleby9ifthesumofitsdigitsisdivisible by9.Thenumberswhosedigitsaddtoamultipleof9are 255,555and711.
82. Anumberisdivisibleby10ifitsonesdigitis0.Thus,the numbers140,600,330,and780aredivisibleby10.
83. a)18isnotamultipleof12.
b) Checkmultiples: 2 18=36Amultipleof12
c)TheLCMis36.
84. a)45isnotamultipleof18.
b) Checkmultiples: 2 45=90Amultipleof18
c)TheLCMis90.
85. Notethat3and6arefactorsof30.Sincethelargest number,30,hastheothertwonumbersasfactors,itisthe LCM.
86. a)Findtheprimefactorizationofeachnumber.
26=2 13
36=2 2 3 3
54=2 3 3 3
b) Createaproductbywritingeachfactorthegreatest numberoftimesitoccursinanyonefactorization.
Thegreatestnumberoftimes2occursinanyone factorizationistwotimes.
Thegreatestnumberoftimes3occursinanyone factorizationisthreetimes.
Thegreatestnumberoftimes13occursinanyone factorizationisonetime.
Sincetherearenootherprimefactorsinanyofthe factorizations,theLCMis2 2 3 3 3 13,or1404.
87. 7+(4+3)2 =7+72 =7+49 =56
AnswerBiscorrect.
88. 7+42 +32 =7+16+9 =23+9 =32
AnswerAiscorrect.
89. [46 (4 2) 5] ÷ 2+4 =[46 2 5] ÷ 2+4 =[46 10] ÷ 2+4 =36 ÷ 2+4 =18+4 =22
AnswerDiscorrect.
90. 9 a 1 2 b 1 236,421 Since250 × 1000=250, 000 ≈ 236, 421wededucethat 2b1 ≈ 250and9a1 ≈ 1000.Bytrialwefindthat a =8 and b =4.
91. 13and31arebothprimenumbers,so13isapalindrome prime.
19isprimebut91isnot(91=7 13),so19isnota palindromeprime.
16isnotprime(16=2 8=4 4),soitisnotapalindrome prime.
11isprimeandwhenitsdigitsarereversedwehave11 again,so11isapalindromeprime.
15isnotprime(15=3 5),soitisnotapalindromeprime.
24isnotprime(24=2 12=3 8=4 6),soitisnota palindromeprime.
29isprimebut92isnot(92=2 · 46=4 · 23),so29isnot apalindromeprime.
101isprimeandwhenitsdigitsarereversedweget101 again,so101isapalindromeprime.
201isnotprime(201=3 · 67),soitisnotapalindrome prime.
37and73arebothprimenumbers,so37isapalindrome prime.
92. Atthebeginningofeachdaythetunnelreaches500ft 200ft,or300ft,fartherintothemountainthanitdidthe daybefore.Wecalculatehowfarthetunnelreachesinto themountainatthebeginningofeachday,startingwith Day2.
Day2:300ft
Day3:300ft+300ft=600ft
Day4:600ft+300ft=900ft
Day5:900ft+300ft=1200ft
Day6:1200ft+300ft=1500ft
Weseethatthetunnelreaches1500ftintothemountain atthebeginningofDay6.OnDay6thecrewtunnelsan additional500ft,sothetunnelreaches1500ft+500ft,or 2000ft,intothemountain.Thus,ittakes6daystoreach thecopperdeposit.
Chapter1DiscussionandWritingExercises
1. 9432=9 1000+4 100+3 10+2 1=9(999+1)+ 4(99+1)+3(9+1)+2 1=9 999+9 1+4 99+4 1+ 3 9+3 1+2 1.Since999,99,and9areeachamultiple of9,9 999,4 99,and3 9aremultiplesof9.Thisleaves 9 · 1+4 · 1+3 · 1+2 · 1,or9+4+3+2.If9+4+3+2,the sumofthedigits,isdivisibleby9,then9432isdivisible by9.
2. Findtheproductoftwoprimenumbers.
3. Answerswillvary.AnthonyisdrivingfromKansasCity toMinneapolis,adistanceof512miles.Hestopsforgas afterdriving183miles.Howmuchfarthermusthedrive?
4. Theparenthesesarenotnecessaryintheexpression 9 (4 2).Usingtherulesfororderofoperations,the multiplicationwouldbeperformedbeforethesubtraction eveniftheparentheseswerenotpresent. Theparenthesesarenecessaryintheexpression(3 4)2 ; (3 · 4)2 =122 =144,but3 · 42 =3 · 16=48.
Chapter1Test
1. 5 46,789
Thedigit5tellsthenumberofhundredthousands.
2. 8843=8thousands+8hundreds+4tens+3ones 3. 38 ,403 ,277 ↑↑↑ Thirty-eightmillion, fourhundredthreethousand, twohundredseventy-seven
4. 6811 +3178 9989
5. 111 45, 889 +17, 902 63, 791
6. 211 1239 843 301 +782 3165
7. 6203 +4312 10, 515
8. 7983 4353 3630
9. 614 297/4 / 1935 1039
10. 8917 8907 / ✥✥ 2059 6848 11. 12 12 / 916 2 /3, 06/7✥✥ 17, 892 5175 12. 567 4568 × 9 41, 112
13. 543 8876 × 600 5, 325, 600
Addones,addtens,addhundreds, andthenaddthousands.
Subtractones,subtracttens,subtract hundreds,andthensubtractthousands.
Multiplyby6hundreds(Wewrite00 andthenmultiply8876by6.) 14. 65 × 37 455 1950 2405
Multiplyingby7 Multiplyingby30 Adding
15. 678 × 788 5424 54240 474600 534, 264
16. 3 4 15 12 3
Theansweris3R3.
17. 70 6 420 42 0 0 0 Theansweris70.
18. 97 89 8633 801 623 623 0 Theansweris97.
19. 805 44 35, 428 352 228 220 8
Theansweris805R8.
20. Round34,528tothenearestthousand. 34, 5 28 ↑
Thedigit4isinthethousandsplace.Considerthenext digittotheright,5.Since5is5orhigher,round4thousandsupto5thousands.Thenchangealldigitstothe rightofthousandstozeros. Theansweris35,000.
21. Round34,528tothenearestten. 34, 52 8 ↑
Thedigit2isinthetensplace.Considerthenextdigitto theright,8.Since8is5orhigher,round2tensupto3 tens.Thenchangethedigittotherightoftenstozero. Theansweris34,530.
22. Round34,528tothenearesthundred. 34, 5 2 8 ↑
Thedigit5isinthehundredsplace.Considerthenext digittotheright,2.Since2is4orlower,rounddown, meaningthat5hundredsstaysas5hundreds.Thenchange alldigitstotherightofhundredstozero. Theansweris34,500.
23. Roundedto thenearesthundred 23,64923,600 +54,746+54,700 78,300 ← Estimatedanswer
24. Roundedto thenearesthundred 54,75154,800 23,649 23,600 31,200 ← Estimatedanswer
25. Roundedto thenearesthundred 824800 × 489 × 500 400,000 ← Estimatedanswer
26. Since34istotherightof17onthenumberline,34 > 17.
27. Since117istotheleftof157onthenumberline, 117 < 157.
28. 28+ x =74 28+ x 28=74 28Subtracting28onbothsides x =46
Check:28+ x =74 28+46?74 74 TRUE Thesolutionis46.
29. 169 ÷ 13= n Wecarryoutthedivision.
0 Thesolutionis13.
30. 38 y =532 38 y 38 = 532 38 Dividingby38onbothsides y =14
Check:38 y =532 38 14?532 532 TRUE Thesolutionis14.
31. 381=0+ a 381= a Addingontherightside Thesolutionis381.
32. Familiarize.Let s =thenumberofcaloriesinan8-oz servingofskimmilk.
Translate
Numberof caloriesin skimmilk plus Howmany morecalories is Numberof caloriesin wholemilk
s +63=146
Solve.Wesubtract63onbothsidesoftheequation.
s +63=146
s +63 63=146 63
s =83
Check.Since63caloriesmorethan83caloriesis83+63, or146calories,theanswerchecks.
State.An8-ozservingofskimmilkcontains83calories.
33. Familiarize.Let s =thenumberofstaplersthatcan befilled.Wecanthinkofthisasrepeatedsubtraction, takingsuccessivesetsof250staplesandputtingtheminto s staplers.
Translate.
Number ofstaples divided by Numberin eachstapler is Numberof staplersfilled
5000 ÷ 250= s
Solve.Wecarryoutthedivision.
20
250 5000 500 0 0 0
Then20= s.
Check.Wecanmultiplythenumberofstaplersfilledby thenumberofstaplesineachone.
20 250=5000
Theanswerchecks.
State.20staplerscanbefilledfromaboxof5000staples.
34. Familiarize.Thisamultistepproblem.Let b =thetotal costoftheblackcartridges, p =thetotalcostofthephoto cartridges,and t =thetotalcostoftheentirepurchase.
Translate.
Fortheblackinkcartridges:
Number ofblack cartridges times Priceper cartridge is Totalcost ofblack cartridges
3 15= b
Forthephotocartridges:
Number ofphoto cartridges times Priceper cartridge is Totalcost ofphoto cartridges
2 · 25= p
Forthetotalcostoftheorder:
Costof black cartridges plus Costof photo cartridges is Total costof purchase
b + p = t
Solve.Wesolvethefirsttwoequationsandthenaddthe solutions.
3 15= b 45= b
2 25= p 50= p b + p = t 45+50= t 95= t
Check.Werepeatthecalculations.Theanswerchecks.
State.Thetotalcostofthepurchasewas$95.
35. a) WewillusetheformulaPerimeter=2 length+ 2 widthtofindtheperimeterofeachpooltablein inches.WewillusetheformulaArea=length · width tofindtheareaofeachpooltable,insqin.
Forthe50in.by100in.table:
Perimeter=2 100in.+2 50in. =200in.+100in. =300in.
Area=100in. 50in.=5000sqin. Forthe44in.by88in.table:
Perimeter=2 88in.+2 44in. =176in.+88in. =264in.
Area=88in. 44in.=3872sqin. Forthe38in.by76in.table:
Perimeter=2 76in.+2 38in. =152in.+76in. =228in.
Area=76in. 38in.=2888sqin.
b) Let a =thenumberofsquareinchesbywhichthe areaofthelargesttableexceedstheareaofthe smallesttable.Wesubtracttofind a a =5000sqin. 2888sqin.=2112sqin.
36. Exponentialnotationfor12 12 12 12is124 .
37. 73 =7 7 7=343 Copyright c 2020PearsonEducation,Inc.
38. 105 =10 10 10 10 10=100, 000
39. 35 1 28 ÷ 4+3 =35 28 ÷ 4+3Doingallmultiplicationsand =35 7+3divisionsinorderfromlefttoright =28+3Doingalladditionsandsubtractions =31inorderfromlefttoright
40. 102 22 ÷ 2 =100 4 ÷ 2Evaluatingtheexponential expressions =100 2Dividing =98Subtracting
41. (25 15) ÷ 5 =10 ÷ 5Doingthecalculationinsidetheparentheses =2Dividing
42. 24 +24 ÷ 12 =16+24 ÷ 12Evaluatingtheexponential expression =16+2Dividing =18Adding
43. 8 ×{(20 11) [(12+48) ÷ 6 (9 2)]}
=8 ×{9 [60 ÷ 6 7]}
=8 ×{9 [10 7]}
=8 ×{9 · 3}
=8 × 27
=216
44. Thenumber41isprime.Ithasonlythefactors41and1.
45. Thenumber14iscomposite.Ithasthefactors1,2,7,and 14.
46. 3 ← 3isprime. 3 9 2 18 18=2 3 3
47. Weuseafactortree.
60=2 3 2 5,or2 2 3 5
48. 1784 isdivisibleby8because784 isdivisibleby8.
49. 7+8+4=19; since19isnotdivisibleby9,784isnot divisibleby9.
50. 5552 isnotdivisibleby5becausetheonesdigit(2)isnot 0or5.
51. Theonesdigit(2)iseven;thesumofthedigits2+3+2+2, or9isdivisibleby3.Thus,2322isdivisibleby6.
52. WefindtheLCMusingalistofmultiples. a)16isnotamultipleof12.
b) Checkmultiplesof16:
1 16=16Notamultipleof12
2 16=32Notamultipleof12
3 16=48Amultipleof12
TheLCM=48.
53. WewillfindtheLCMusingprimefactorizations.
a) Findtheprimefactorizationofeachnumber.
15=3 5
40=2 2 2 5
50=2 5 5
b) Createaproductbywritingfactorsthatappearin thefactorizationsof15,40,and50,usingeachthe greatestnumberoftimesitoccursinanyonefactorization.
TheLCMis2 2 2 3 5 5,or600.
54. Weaddthenumbersandthendividebythenumberof addends.
97+99+87+89 4 = 372 4 =93
AnswerAiscorrect.
55. Familiarize.Wemakeadrawing.
12 in. 8 in. 6 in.
Observethatthedimensionsoftwosidesofthecontainer are8in.by6in.Theareaofeachis8in. 6in.andtheir totalareais2 8in. 6in.Thedimensionsoftheothertwo sidesare12in.by6in.Theareaofeachis12in. 6in. andtheirtotalareais2 12in. 6in.Thedimensionsof thebottomoftheboxare12in.by8in.anditsarea is12in. 8in.Let c =thenumberofsquareinchesof cardboardthatareusedforthecontainer.
Translate.Weaddtheareasofthesidesandthebottom ofthecontainer.
2 · 8in. · 6in.+2 · 12in. · 6in.+12in. · 8in.= c
Solve.Wecarryoutthecalculation.
2 8in. 6in.+2 12in. 6in.+12in. 8in.= c 96sqin.+144sqin.+96sqin.= c 336sqin.= c
Check.Wecanrepeatthecalculations.Theanswer checks.
State.336sqin.ofcardboardareusedforthecontainer.
56. Wecanreducethenumberoftrialsrequiredbysimplifying theexpressionontheleftsideoftheequationandthen usingtheadditionprinciple.
359 46+ a ÷ 3 × 25 72 =339
359 46+ a ÷ 3 × 25 49=339
359 46+ a 3 × 25 49=339
359 46+ 25 a 3 49=339
313+ 25 a 3 49=339
264+ 25 a 3 =339
264+ 25 a 3 264=339 264 25 a 3 =75
Weseethatwhenwemultiply a by25anddivideby3, theresultis75.Bytrial,wefindthat 25 · 9 3 = 225 3 =75, so a =9.Wecouldalsoreasonthatsince75=25 3and 9/3=3,wehave a =9.