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. Alldigitsare9’s.Answersmayvary.Foran8-digitreadout,forexample,itwouldbe99,999,999.Thisnumberhas threeperiods.
ExerciseSet1.2
RC2. Intheaddition5+2=7,thenumber7isthesum
RC4. Thedistancearoundanobjectisitsperimeter
CC2. 110
CC4. 1700 CC6. 10,000
20. 1111 99, 999 +112 100, 111
22. 2 38 27 32 14 +76 187
24. 1111 42, 487 83, 141 +36, 712 162, 340
26. 22 989 566 834 920 +703 4012
28. Perimeter=14mi+13mi+8mi+10mi+47mi+ 22mi Wecarryouttheaddition. 2 14 13 8 10 47 +22 114
Theperimeterofthefigureis114mi.
30. Perimeter=62yd+39yd+54yd+46yd+28yd Wecarryouttheaddition. 2 62 39 54 46 +28 229
Theperimeterofthefigureis229yd.
32. 90ft+90ft+90ft+90ft=Perimeter Wecarryouttheaddition. 90 90 90 +90 360
Thebattertravels360ftwhenahomerunishit.
34. Ninebillion,threehundredforty-sixmillion,threehundred ninety-ninethousand,fourhundredsixty-eight
ExerciseSet1.3
RC2. subtractionsymbol RC4. difference
18. 1312 53 / 2 / 11 6 /4/3/1 / 2896 3535
20. 1215 72 / 5 / 14 8 /3/6/4 / 5375 2989
22. 1113 81 / 3 / 11 9 /2/4/1 / 5643 3598
24. 615 7 /5/83 3641 3942
26. 1111 51 / 1 / 12 16 /, 2 /2/2 / 5, 888 10, 334
28. 11 21 / 11814 3 /2 /, 1 /9/4 / 29, 236 2958
30. 810 9 /0 / 78 12
32. 7913 803✥✥ 418 385
34. 3915 9405 / ✥✥ 258 9147
36. 14 64 / 910 7 /500 / ✥✥ 3604 3896
38. 6913 84, 703 / ✥✥ 298 84, 405
40. 14 4 / 10017 1 /5 /, 0 /1/7 / 7809 7208
42. 79913 8003 / ✥✥✥ 599 7404
44. 69910 17, 000 / ✥✥✥ 11, 598 5402
46. 399916 40, 006 / ✭✭✭✭ 147 39, 859
48. 299914 30, 004 / ✭✭✭✭ 6749 23, 255
50. 901 +23 924
52. 1 9909 +1011 10, 920
54. 9thousands+1hundred+0tens+3ones,or9thousands +1hundred+3ones
56. 3, 928, 124 1, 098, 947
Usingacalculatortocarryoutthesubtraction,wefind thatthedifferenceis2,829,177.
ExerciseSet1.4
RC2. Inthemultiplication4 × 3=12,12istheproduct.
RC4. Theproductof1 andanynumber a is a
CC2. 4800
CC4. 80,000
CC6. 400,000 2. 2 87 × 4 348 4. 5 76 × 9 684 6. 4 806 × 7 5642 8. 322 7867 × 4 31, 468
10. 4 78 × 60 4680
12. 2 2 87 × 34 348 2610 2958
14. 2340 × 1000 2, 340, 000 16. 800 × 700 560, 000
18. 54 54 777 × 77 5439 54390 59, 829
20. 37 37 549 × 88 4392 43920 48, 312
22. 21 11 432 × 375 2160 30240 129600 162, 000
24. 23 23 45 346 × 659 3114 17300 207600 228, 014
26. 22 615 11 8928 × 3172 17856 624960 892800 26784000 28, 319, 616
28. 24 24 13 6408 × 6064 25632 384480 38448000 38, 858, 112
30. 11 44 44 355 × 299 3195 31950 71000 106, 145
32. 1 2 2 41 6521 × 3449 58689 260840 2608400 19563000 22, 490, 929
34. 34 44 4506 × 7800 3604800 31542000 35, 146, 800
36. 1 2 6009 × 2003 18027 12018000 12, 036, 027
38. A = l × w =129yd × 65yd=8385sqyd
40. A = l × w =200ft × 85ft=17, 000sqft
42. 122 9876 876 76 +6 10, 834
44. 13 23/ 10818 3 /4/0 /, 79/8 / 86, 679 254, 119
46. 0
48. Sevenmillion,fourhundredthirty-twothousand
ExerciseSet1.5
RC2. dividend
RC4. divisor
CC2. 3 4 15 12 3
Theansweris3R3.
CC4. 48 2 97 8 17 16 1
Theansweris48R1.
2. 54 ÷ 9=6because54=9 · 6.
4. 37 37 =1Anynonzeronumberdividedbyitselfis1.
6. 56 1 =56Anynumberdividedby1isthatsamenumber.
8. 0 32 =0Zerodividedbyanynonzeronumberis0.
10. 74 ÷ 0isnotdefined,becausedivisionby0isnotdefined.
12. 20 4 =5because20=4 5.
14. 233 3 699 6 9 9 9 9 0
Theansweris233.
16. 108 8 869 8 69 64 5
Theansweris108R5.
18. 708 3 2124 21 24 24 0
Theansweris708.
20.
9
Theansweris1012R2. 22. 194 2 389 2 18 18 9 8 1
Theansweris194R1.
24. 146 6 881 6 28 24 41 36 5
Theansweris146R5.
26. 2009 3 6027 6 27 27 0 Theansweris2009.
28. 517 8 4139 40 13 8 59 56 3
Theansweris517R3.
30. 1270 100 127, 000
Theansweris1270.
32. 426 10 4260 40 26 20 60 60 0
Theansweris426.
34. 289 20 5798 40 179 160 198 180 18
Theansweris289R18.
36. 24 40 987 80 187 160 27
Theansweris24R27.
38. 40 23 942 92 22
Theansweris40R22.
40. 50 54 2729 270 29
Theansweris50R29.
42. 55 102 5612 510 512 510 2
Theansweris55R2.
44. 107 7 749 7 49 49 0
Theansweris107.
46. 808 9 7273 72 73 72 1
Theansweris808R1.
48. 1010 7 7074 70 7 7 4
Theansweris1010R4.
50. 301 24 7242 72 42 24 18
Theansweris301R18.
52. 102 48 4899 48 99 96 3
Theansweris102R3.
54. 210 36 7563 72 36 36 3
Theansweris210R3.
56. 803 36 28, 929 288 129 108 21
Theansweris803R21.
58. 984 90 88, 560 810 756 720 360 360 0
Theansweris984.
60. 2904 306 888, 888 612 2768 2754 1488 1224 264
Theansweris2904R264.
62. 7002 803 5, 622, 606 5621 1606 1606 0
Theansweris7002.
64. 88, 777 22, 333 66, 444
66. 268 × 35 1340 8040 9380
68. A = l × w =11ft × 9ft=99sqft
70. Pairsoffactorswhoseproductis36are: 1and36 2and18 3and12 4and9 6and6
a)Thepairabovewhosesumis13is4and9.
b)Thepairabovewhosedifferenceis0is6and6.
c)Thepairabovewhosesumis20is2and18.
d)Thepairabovewhosedifferenceis9is3and12.
72. 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.
Chapter1Mid-ChapterReview
1. Thestatementisfalse.Forexample,8 5=3,but5is notequalto8+3.
2. Thestatementistrue.Seepage19inthetext.
3. Thestatementistrue.Seepage19inthetext.
4. Thestatementisfalse.Forexample,3 0=0and0isnot greaterthan3.Also,1 1=1and1isnotgreaterthan1.
5. Itistruethatzerodividedbyanynonzeronumberis0.
6. Thestatementisfalse.Anynumberdividedby1isthe numberitself.Forexample, 27 1 =27.
7. 95 ,406 ,237
↑↑↑ Ninety-fivemillion, fourhundredsixthousand, twohundredthirty-seven
8. 5914 604 / ✥✥ 497
9. 2 6 98
Thedigit6namesthenumberofhundreds.
10. 6 1,204
Thedigit6namesthenumberoftenthousands.
11. 14 6 ,237
Thedigit6namesthenumberofthousands.
12. 58 6
Thedigit6namesthenumberofones.
13. 306,458,1 2 9
Thedigit2namesthenumberoftens.
14. 30 6 ,458,129
Thedigit6namesthenumberofmillions.
15. 306,4 5 8,129
Thedigit5namesthenumberoftenthousands.
16. 306,458, 1 29
Thedigit1namesthenumberofhundreds.
17. 5602=5thousands+6hundreds+0tens+2ones,or5 thousands+6hundreds+2ones
18. 69,345=6tenthousands+9thousands+3hundreds+ 4tens+5ones
19. Awordnamefor136isonehundredthirty-six.
20. Awordnamefor64,325issixty-fourthousand,threehundredtwenty-five.
21. Standardnotationforthreehundredeightthousand,seven hundredsixteenis308,716.
22. Standardnotationforfourmillion,fivehundredsixtyseventhousand,twohundredninety-oneis4,567,291.
23. 316 +482 798
24. 11 593 +437 1030
25. 11 2638 +5284 7922
26. 111 4617 2436 +481 7534
27. 786 321 465
28. 11 51 / 14 6 /2/4 / 285 339
29. 15 25 / 912 3 /602 / ✥✥ 1748 1854
30. 49914 5004 / ✥✥✥ 676 4328
31. 3 36 × 6 216
32. 11 55 567 × 28 4536 11340 15, 876
33. 2 1 3 407 × 325 2035 8140 122100 132, 275
34. 223 1 9435 × 602 18870 5661000 5, 679, 870
35. 253 4 1012 8 21 20 12 12 0 Theansweris253.
36. 112 38 4261 38 46 38 81 76 5 Theansweris112R5.
37. 23 60 1399 120 199 180 19 Theansweris23R19.
38. 144 56 8095 56 249 224 255 224 31 Theansweris144R31.
39. Perimeter=10m+4m+8m+3m=25m
40. A =4in. × 2in.=8sqin.
41. Whennumbersarebeingadded,itdoesnotmatterhow theyaregrouped.
42. Subtractionisnotcommutative.Forexample,5 2=3, but2 5 =3.
43. Answerswillvary.Supposeonecoatcosts$150.Thenthe multiplication4 $150givesthecostoffourcoats. Supposeonereamofcopypapercosts$4.Thenthemultiplication$4 150givesthecostof150reams.
44. Usingthedefinitionofdivision,0 ÷ 0= a suchthat a · 0=0. Weseethat a couldbe any numbersince a 0=0forany number a.Thus,wecannotsaythat0 ÷ 0=0.Thisis whyweagreenottoallowdivisionby0.
ExerciseSet1.6
RC2. Becausethedigitinthehundredsplace,5,is5or higher,weroundup.Thestatementisfalse.
RC4. Thestatementistrue.Seepage40inthetext.
CC2. 1000
2. 530
4. 8950
6. 50 8. 800 10. 900
12. 700 14. 4600
16. 198,400
18. 5000
20. 2000
22. 736,000
24. 6,713,000
26. 6260 97100 4650 +81+80 290
28. 673670 28 30 640
30. 4140 2120 5560 +60+60 177180
32. 836840 374370 794790 +938+940 39472940 3947seemstobeincorrect.
34. 568600 472500 938900 +402+400 2400
36. 94389400 2787 2800 6600
38. 481500 702700 623600 +1043+1000 18492800 1849seemstobeincorrect.
40. 326300 275300 758800 +943+900 23022300
42. 76488000 93489000 78428000 +2222+2000 27,000
44. 84,89085,000 11,110 11,000 74,000
46. 5150 × 78 × 80 4000
48. 6360 × 54 × 50 3000
50. 355400 × 299 × 300 120,000
52. 789800 × 434 × 400 320,000
54. 454 ÷ 87 ≈ 450 ÷ 90=5
56. 1263 ÷ 29 ≈ 1260 ÷ 30=42
58. 3641 ÷ 571 ≈ 3600 ÷ 600=6
60. 32, 854 ÷ 748 ≈ 32, 900 ÷ 700=47
62. $498$500 289300 +145+100 $900
64. $498$500 289300 62100 159200 +320+300 $1400
Thebudgetcoversthechoices.
66. Answerswillvarydependingontheoptionschosen.
68. 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
70. $2211 ÷ $11 ≈ $2200 ÷ $10=220rolls
72. 32 > 0
74. 28 > 18
76. 77 < 117
78. 999 > 997
80. 345 < 456
82. 12 < 32
84. 1, 014, 023 > 758, 708,or758, 708 < 1, 014, 023
86. 843, 393 > 842, 583,or842, 583 < 843, 393
88. 9002 +4587 13, 589
90. 89912 9002 / ✥✥✥ 4587 4415
92. 3 4 306 × 58 2448 15300 17, 748
94. 208 23 4784 46 184 184 0 Theansweris208.
96. 27,060;thisisclosetotheestimatedsumfoundinExercise 42.
98. 73,780;thisisclosetotheestimateddifferencefoundin Exercise44.
ExerciseSet1.7
RC2. (a)
RC4. (b)
CC2. 672=6 n 672?6 112 672TRUE 112isasolution.
CC4. 5462=3189+ t 5462?3189+2327 5516FALSE 2327isnotasolution.
25
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
ExerciseSet1.8
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 32 150 128 228 224 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
ExerciseSet1.9
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
Copyright c 2020PearsonEducation,Inc.
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.
Chapter1VocabularyReinforcement
1. Thedistancearoundanobjectisitsperimeter
2. Theminuend isthenumberfromwhichanothernumber isbeingsubtracted.
3. Forlargenumbers,digits areseparatedbycommasinto groupsofthree,calledperiods
4. Inthesentence28 ÷ 7=4,thedividend is28.
5. Inthesentence10 × 1000=10, 000,10and1000arecalled factors and10,000iscalledtheproduct
6. Thenumber0iscalledtheadditive identity.
7. Thesentence3 × (6 × 2)=(3 × 6) × 2illustratesthe associative lawofmultiplication.
8. Wecanusethefollowingruletocheckdivision: quotient divisor +remainder =dividend.
Chapter1ConceptReinforcement
1. Thestatementistrue.Seepage42inthetext.
2. a ÷ a = a a =1, a =0;thestatementistrue.
3. a ÷ 0isnotdefined,sothestatementisfalse.
4. Thestatementisfalse.Forexample,1+2=5isnota trueequation.
5. Thestatementistrue.Seepage71inthetext.
6. Thestatementisfalse.Seepage73inthetext.
Chapter1StudyGuide
1. 43 2 ,079
Thedigit2namesthenumberofthousands.
2. 111 36, 047 +29, 255 65, 302
3. 7915 4805 / ✥✥ 1568 3237
4. 21 1 73 684 × 329 6156Multiplyingby9 13680Multiplyingby20 205200Multiplyingby300 225, 036
5. 315 27 8519 81 41 27 149 135 14
Theansweris315R14.
6. Round36,468tothenearestthousand. 36, 4 68 ↑
Thedigit6isinthethousandsplace.Considerthenext digittotheright.Sincethedigit,4,is4orlower,round down,meaningthat6thousandsstaysas6thousands. Thenchangethedigitstotherightofthethousandsdigit tozeros.
Theansweris36,000.
Copyright c 2020PearsonEducation,Inc.
7. Since78istotheleftof81onthenumberline,78 < 81.
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
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. 11 7304
89911 9001 / ✥✥✥ 7312 1689
59913 6003 / ✥✥✥ 3729 2274
+6968 14, 272 11. 111 27, 609 +38, 415 66, 024 12. 11 2703 4125 6004 +8956 21, 788 13. 11 91, 426 +7, 495 98, 921 14. 13 793 / 15 804/5 / ✥✥ 2897 5148
1613 26 / 3 / 915 3 /7 /, 405 / ✥✥ 19, 648 17, 757 18. 2 17, 000 × 300 5, 100, 000 Multiplyingby300 (Write00andthen multiply17,000by3.)
19. 634 7846 × 800 6, 276, 800 Multiplyingby800 (Write00andthen multiply7846by8.)
20. 13 25 24 726 × 698 5808 65340 435600 506, 748
Multiplyingby8 Multiplyingby90 Multiplyingby600 21. 32 64 587 × 47 4109Multiplyingby7 23480Multiplyingby40 27, 589
22. 8305 × 642 16610 332200 4983000 5, 331, 810
23. 12 5 63 5 13 10 3
Theansweris12R3.
24. 5 16 80 80 0 Theansweris5.
25. 913 7 6394 63 9 7 24 21 3
Theansweris913R3.
26. 384 8 3073 24 67 64 33 32 1
Theansweris384R1.
27. 4 60 286 240 46
Theansweris4R46.
28. 54 79 4266 395 316 316 0 Theansweris54.
29. 452 38 17, 176 152 197 190 76 76 0 Theansweris452.
30. 5008 14 70, 112 70 112 112 0 Theansweris5008.
31. 4389 12 52, 668 48 46 36 106 96 108 108 0 Theansweris4389.
32. Round345,759tothenearesthundred.
345, 7 5 9 ↑
Thedigit7isinthehundredsplace.Considerthenext digittotheright.Sincethedigit,5,is5orhigher,round 7hundredsupto8hundreds.Thenchangethedigitsto therightofthehundredsdigittozero.
Theansweris345,800.
33. Round345,759tothenearestten.
345, 75 9 ↑
Thedigit5isinthetensplace.Considerthenextdigitto theright.Sincethedigit,9,is5orhigher,round5tensup to6tens.Thenchangethedigittotherightofthetens digittozero.
Theansweris345,760.
34. Round345,759tothenearestthousand.
345, 7 59 ↑
Thedigit5isinthethousandsplace.Considerthenext digittotheright.Sincethedigit,7,is5orhigher,round 5thousandsupto6thousands.Thenchangethedigitsto therightofthethousandsdigittozero.
Theansweris346,000.
35. Round345,759tothenearesthundredthousand.
3 4 5, 759 ↑
Thedigit3isinthehundredthousandsplace.Consider thenextdigittotheright.Sincethedigit,4,is4orlower, rounddown,meaningthat3hundredthousandsstaysas 3hundredthousands.Thenchangethedigitstotheright ofthehundredthousandsdigittozero.
Theansweris300,000.
36. Since67istotherightof56onthenumberline,67 > 56.
37. Since1istotheleftof23onthenumberline,1 < 23.
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
420 ÷ 30= b
Solve.Wecarryoutthedivision.
14 30 420 30 120 120 0
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. 7+(4+3)2 =7+72 =7+49 =56
AnswerBiscorrect.
62. 7+42 +32 =7+16+9 =23+9 =32
AnswerAiscorrect.
63. [46 (4 2) · 5] ÷ 2+4 =[46 2 5] ÷ 2+4 =[46 10] ÷ 2+4 =36 ÷ 2+4 =18+4 =22
AnswerDiscorrect.
64. 9 d × d 2 8036
Byusingroughestimates,weseethatthefactor d2 ≈ 8100 ÷ 90=90or d2 ≈ 8000 ÷ 100=80.Since99 × 92= 9108and98 × 82=8036,wehave d =8.
65. 9 a 1 2 b 1 236,421
Since250 × 1000=250, 000 ≈ 236, 421wededucethat 2b1 ≈ 250and9a1 ≈ 1000.Bytrialwefindthat a =8 and b =4.
66. 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. No;ifsubtractionwereassociative,then a (b c)= (a b) c forany a, b,and c.But,forexample, 12 (8 4)=12 4=8, whereas (12 8) 4=4 4=0. Since8 =0,thisexampleshowsthatsubtractionisnot associative.
2. Byroundingpricesandestimatingtheirsumashopper canestimatethetotalgrocerybillwhileshopping.Thisis particularlyusefuliftheshopperwantstospendnomore thanacertainamount.
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 Addones,addtens,addhundreds, andthenaddthousands.
5. 111 45, 889 +17, 902 63, 791
6. 211 1239 843 301 +782 3165
7. 6203 +4312 10, 515
8. 7983 4353 3630 Subtractones,subtracttens,subtract hundreds,andthensubtractthousands.
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 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
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. 13 13 169 13 39 39 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.Let a =thetotallandareaofthefivelargest states,insquaremeters.Sincewearecombiningtheareas ofthestates,wecanadd.
Translate
571, 951+261, 797+155, 959+145, 552+121, 356= a
Solve.Wecarryouttheaddition.
21332
571, 951 261, 797 155, 959 145, 552 +121, 356 1, 256, 615
Then1, 256, 615= a
Check.Wecanrepeatthecalculation.Wecanalsoestimatetheresultbyrounding.Wewillroundtothenearest tenthousand.
571, 951+261, 797+155, 959+145, 552+121, 356 ≈ 570, 000+260, 000+160, 000+150, 000+120, 000 =1, 260, 000
Since1, 260, 000 ≈ 1, 256, 615,wehaveapartialcheck.
State.ThetotallandareaofAlaska,Texas,California, Montana,andNewMexicois1,256,615sqmi.
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. Familiarize.Let n =thenumberof12-packsthatcanbe filled.Wecanthinkofthisasrepeatedsubtraction,taking successivesetsof12snackcakesandputtingtheminto n packages.
Translate Number of cakes divided by Number ineach package is Number of 12-packs
22, 231 ÷ 12= n
Solve.Wecarryoutthedivision.
1852 12 22, 231 12 102 96 63 60 31 24 7
Then1852R7= n. Check.Wemultiplythenumberofpackagesby12and thenaddtheremainder,7.
12 1852=22, 224 22, 224+7=22, 231 Theanswerchecks. State.1852twelve-packscanbefilled.Therewillbe7 cakesleftover.
37. 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.
38. Exponentialnotationfor12 12 12 12is124
39. 73 =7 7 7=343
40. 105 =10 10 10 10 10=100, 000
41. 35 1 28 ÷ 4+3
=35 28 ÷ 4+3Doingallmultiplicationsand =35 7+3divisionsinorderfromlefttoright
=28+3Doingalladditionsandsubtractions =31inorderfromlefttoright
42. 102 22 ÷ 2
=100 4 ÷ 2Evaluatingtheexponential expressions =100 2Dividing =98Subtracting
43. (25 15) ÷ 5 =10 ÷ 5Doingthecalculationinsidetheparentheses =2Dividing
44. 24 +24 ÷ 12
=16+24 ÷ 12Evaluatingtheexponential expression =16+2Dividing =18Adding
45. 8 ×{(20 11) · [(12+48) ÷ 6 (9 2)]}
=8 ×{9 [60 ÷ 6 7]}
=8 ×{9 [10 7]}
=8 ×{9 · 3}
=8 × 27 =216
46. Weaddthenumbersandthendividebythenumberof addends.
97+99+87+89 4 = 372 4 =93
AnswerAiscorrect.
47. 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.
48. 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.
49. Familiarize.Firstobservethata10-yrloanwithmonthly paymentshasatotalof10 12,or120,payments.Let m = thenumberofmonthlypaymentsrepresentedby$9160and let p =thenumberofpaymentsremainingafter$9160has beenrepaid.
Translate.Firstwewilltranslatetoanequationthatcan beusedtofind m.Thenwewillwriteanequationthat canbeusedtofind p
Payments todate divided by Amount ofeach payment is Numberof payments made
9160 ÷ 229= m
Payments alreadymade plus Remaining payments is Totalnumber ofpayments
m + p =120
Solve.Tosolvethefirstequationwecarryoutthedivision.
40 229 9160 916 0 0 0 Thus, m =40. Nowwesolvethesecondequation.
m + p =120
40+ p =120Substituting40for m
40+ p 40=120 40 p =80
Check.Wecanapproachtheprobleminadifferentway tochecktheanswer.In10years,Cara’sloanpayments willtotal120 $229,or$27,480.If$9160hasalreadybeen paid,then$27, 480 $9160,or$18,320,remainstobepaid. Since80 $229=$18, 320,theanswerchecks. State.80paymentsremainontheloan.