We begin this section by discussing multiplication of whole numbers. The first order of business is to introduce the various symbols used to indicate multiplication of two whole numbers.
The key to understanding multiplication is held in the following statement.
Suppose, for example, that we would like to evaluate the product \(3 ·4\). Because multiplication is equivalent to repeated addition, \(3 · 4\) is equivalent to adding three fours. That is,
Thus, \(3 · 4 = 12\). You can visualize the product \(3 · 4\) as the sum of three fours on a number line, as shown in Figure 1.6.
Like addition, the order of the factors does not matter.
Thus, \(4 · 3 = 12\). Consider the visualization of \(4 · 3\) in Figure 1.7.
The evidence in Figure 1.6 and Figure 1.7 show us that multiplication is commutative. That is,
Exercises
In Exercises 1-4 use number line diagrams as shown in Figure 1.6 to depict the multiplication.
1. 2 · 4.
2. 3 · 4.
3. 4 · 2.
4. 4 · 3.
In Exercises 5-16, state the property of multiplication depicted by the given identity.
5. 9 · 8=8 · 9
6. 5 · 8=8 · 5
7. 8 · (5 · 6) = (8 · 5) · 6
8. 4 · (6 · 5) = (4 · 6) · 5
9. 6 · 2=2 · 6
10. 8 · 7=7 · 8
11. 3 · (5 · 9) = (3 · 5) · 9
12. 8 · (6 · 4) = (8 · 6) · 4
13. 21 · 1 = 21
14. 39 · 1 = 39
15. 13 · 1 = 13
16. 44 · 1 = 44
In Exercises 17-28, multiply the given numbers.
17. 78 · 3
18. 58 · 7
19. 907 · 6
20. 434 · 80
21. 128 · 30
22. 454 · 90
23. 799 · 60
24. 907 · 20
25. 14 · 70
26. 94 · 90
27. 34 · 90
28. 87 · 20
In Exercises 29-40, multiply the given numbers.
29. 237 · 54
30. 893 · 94
31. 691 · 12
32. 823 · 77
33. 955 · 89
34. 714 · 41
35. 266 · 61
36. 366 · 31
37. 365 · 73
38. 291 · 47
39. 955 · 57
40. 199 · 33
41. Count the number of objects in the array.

42. Count the number of objects in the array.

43. Count the number of objects in the array.

44. Count the number of objects in the array.

In Exercises 45-48, find the area of the rectangle having the given length and width.
45. L = 50 in, W = 25 in
46. L = 48 in, W = 24 in
47. L = 47 in, W = 13 in
48. L = 19 in, W = 10 in
In Exercises 49-52, find the perimeter of the rectangle having the given length and width.
49. L = 25 in, W = 16 in
50. L = 34 in, W = 18 in
51. L = 30 in, W = 28 in
52. L = 41 in, W = 25 in
53. A set of beads costs 50 cents per dozen. What is the cost (in dollars) of 19 dozen sets of beads?
54. A set of beads costs 60 cents per dozen. What is the cost (in dollars) of 7 dozen sets of beads?
55. If a math tutor worked for 47 hours and was paid $15 each hour, how much money would she have made?
56. If a math tutor worked for 46 hours and was paid $11 each hour, how much money would he have made?
57. There are 12 eggs in one dozen, and 12 dozen in one gross. How many eggs are in a shipment of 24 gross?
58. There are 12 eggs in one dozen, and 12 dozen in one gross. How many eggs are in a shipment of 11 gross?
59. If bricks weigh 4 kilograms each, what is the weight (in kilograms) of 5000 bricks?
60. If bricks weigh 4 pounds each, what is the weight (in pounds) of 2000 bricks?
In Exercises 61-68, which of the following four expressions differs from the remaining three?
61. \(\frac{30}{5}\), 30 ÷ 5, \(5 \longdiv { 3 0 }\), 5 ÷ 30
62. \(\frac{12}{2}\), 12 ÷ 2, \(2 \longiv{12}\), 2 ÷ 12
63. \(\frac{8}{2}\), 8 ÷ 2, \(2 \longdiv{8}\), \(8 \longdiv{2}\)
64. \(\frac{8}{4}\), 8 ÷ 4, \(4 \longdiv { 8 }\), \(8 \longdiv { 4 }\)
65. \(2 \longdiv { 14 }\), \(14 \longdiv { 2 }\), \(\frac{14}{2}\), 14 ÷ 2
66. \(9 \longdiv { 54 }\), \(54 \longdiv { 9 }\), \(\frac{54}{9}\), 54 ÷ 9
67. \(3 \longdiv { 24 }\), 3 ÷ 24, \(\frac{24}{3}\), 24 ÷ 3
68. \(3 \longdiv { 15 }\), 3 ÷ 15, \(\frac{15}{3}\), 15 ÷ 3
In Exercises 69-82, simplify the given expression. If the answer doesn’t exist or is undefined, write “undefined”.
69. 0 ÷ 11
70. 0 ÷ 5
71. 17 ÷ 0
72. 24 ÷ 0
73. 10 · 0
74. 20 · 0
75. \(\frac{7}{0}\)
76. \(\frac{23}{0}\)
77. \(16 \longdiv { 0 }\)
78. \(25 \longdiv { 0 }\)
79. \(\frac{0}{24}\)
80. \(\frac{0}{22}\)
81. \(0 \longdiv { 0 }\)
82. 0 ÷ 0
In Exercises 83-94, divide the given numbers.
83. \(\frac{2816}{44}\)
84. \(\frac{1998}{37}\)
85. \(\frac{2241}{83}\)
86. \(\frac{2716}{97}\)
87. \(\frac{3212}{73}\)
88. \(\frac{1326}{17}\)
89. \(\frac{8722}{98}\)
90. \(\frac{1547}{91}\)
91. \(\frac{1440}{96}\)
92. \(\frac{2079}{27}\)
93. \(\frac{8075}{85}\)
94. \(\frac{1587}{23}\)
In Exercises 95-106, divide the given numbers.
95. \(\frac{17756}{92}\)
96. \(\frac{46904}{82}\)
97. \(\frac{11951}{19}\)
98. \(\frac{22304}{41}\)
99. \(\frac{18048}{32}\)
100. \(\frac{59986}{89}\)
101. \(\frac{29047}{31}\)
102. \(\frac{33264}{86}\)
103. \(\frac{22578}{53}\)
104. \(\frac{18952}{46}\)
105. \(\frac{12894}{14}\)
106. \(\frac{18830}{35}\)
107. A concrete sidewalk is laid in square blocks that measure 6 feet on each side. How many blocks will there be in a walk that is 132 feet long?
108. A concrete sidewalk is laid in square blocks that measure 5 feet on each side. How many blocks will there be in a walk that is 180 feet long?
109. One boat to the island can take 5 people. How many trips will the boat have to take in order to ferry 38 people to the island? (Hint: Round up your answer.)
110. One boat to the island can take 4 people. How many trips will the boat have to take in order to ferry 46 people to the island? (Hint: Round up your answer.)
111. If street lights are placed at most 145 feet apart, how many street lights will be needed for a street that is 4 miles long, assuming that there are lights at each end of the street? (Note: 1 mile = 5280 feet.)
112. If street lights are placed at most 70 feet apart, how many street lights will be needed for a street that is 3 miles long, assuming that there are lights at each end of the street? (Note: 1 mile = 5280 feet.)
113. A concrete sidewalk is laid in square blocks that measure 4 feet on each side. How many blocks will there be in a walk that is 292 feet long?
114. A concrete sidewalk is laid in square blocks that measure 5 feet on each side. How many blocks will there be in a walk that is 445 feet long?
115. One boat to the island can take 3 people. How many trips will the boat have to take in order to ferry 32 people to the island? (Hint: Round up your answer.)
116. One boat to the island can take 4 people. How many trips will the boat have to take in order to ferry 37 people to the island? (Hint: Round up your answer.)
117. If street lights are placed at most 105 feet apart, how many street lights will be needed for a street that is 2 miles long, assuming that there are lights at each end of the street? (Note: 1 mile = 5280 feet.)
118. If street lights are placed at most 105 feet apart, how many street lights will be needed for a street that is 3 miles long, assuming that there are lights at each end of the street? (Note: 1 mile = 5280 feet.)
119. Writing articles. Eli writes an average of 4 articles a day, five days a week, to support product sales. How many articles does Eli write in one week?
120. Machine gun. A 0.50-caliber antiaircraft machine gun can fire 800 rounds each minute. How many rounds could fire in three minutes? Associated Press Times-Standard 4/15/09
121. Laps. The swimming pool at CalCourts is 25 yards long. If one lap is up and back again, how many yards has Wendell swam doing 27 laps?
122. Refrigerator wattage. A conventional refrigerator will run about 12 hours each day can use 150 Watts of power each hour. How many Watts of power will a refrigerator use over the day?
123. Horse hay. A full-grown horse should eat a minimum of 12 pounds of hay each day and may eat much more depending on their weight. How many pounds minimum would a horse eat over a year?
124. College costs. After a $662 hike in fees, Califormia residents who want to attend the University of California as an undergraduate should expect to pay $8,700 in for the upcoming academic year 2009- 2010. If the cost were to remain the same for the next several years, how much should a student expect to pay for a four-year degree program at a UC school?
125. Non-resident costs. Nonresident undergraduates who want to attend a University of California college should expect to pay about $22,000 for the upcoming academic year. Assuming costs remain the same, what can a four-year degree cost?
126. Student tax. The mayer of Providence, Rhode Island wants to tax its 25,000 Brown University students $150 each to contribute to tax receipts saying students should pay for the resources they use just like the town residents. How many dollars would the mayer generate?
127. New iceberg. A new iceberg, shaved off a glacier after a collision with another iceberg, measures about 48 miles long and 28 miles wide. What’s the approximate area of the new iceberg? Associated PressTimes-Standard 02/27/10 2 Huge icebergs set loose off Antarctica’s coast.
128. Solar panels. One of the solar panels on the International Space Station is 34 meters long and 11 meters wide. If there are eight of these, what’s the total area for solar collection?
129. Sidewalk. A concrete sidewalk is to be 80 foot long and 4 foot wide. How much will it cost to lay the sidewalk at $8 per square foot?
130. Hay bales. An average bale of hay weighs about 60 pounds. If a horse eats 12 pounds of hay a day, how many days will one bale feed a horse?
131. Sunspots. Sunspots, where the sun’s magnetic field is much higher, usually occur in pairs. If the total count of sunspots is 72, how many pairs of sunspots are there?