Exercises: Properties of Real Numbers (1.7)
Exercises: Use the Commutative and Associative Properties
1. [latex]3(4x)[/latex]
Solution
[latex]12x[/latex]
2. [latex]4(7m)[/latex]
3. [latex](y+12)+28[/latex]
Solution
[latex]y+40[/latex]
4. [latex](n+17)+33[/latex]
Exercises: Use the Commutative and Associative Properties
5. [latex]\frac{1}{2}+\frac{7}{8}+\left(-\frac{1}{2}\right)[/latex]
Solution
[latex]\frac{7}{8}[/latex]
6. [latex]\frac{2}{5}+\frac{5}{12}+\left(-\frac{2}{5}\right)[/latex]
7. [latex]\frac{3}{20}\times\frac{49}{11}\times\frac{20}{3}[/latex]
Solution
[latex]\frac{49}{11}[/latex]
8. [latex]\frac{13}{18}\times\frac{25}{7}\times\frac{18}{13}[/latex]
9. [latex]-24.7\times\frac{3}{8}[/latex]
Solution
[latex]-63[/latex]
10. [latex]-36\times11\times\frac{4}{9}[/latex]
11. [latex]\left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15}[/latex]
Solution
[latex]1\frac{5}{6}[/latex]
12. [latex]\left(\frac{11}{12}+\frac{4}{9}\right)+\frac{5}{9}[/latex]
13. [latex]17(0.25)(4)[/latex]
Solution
[latex]17[/latex]
14. [latex]36(0.2)(5)[/latex]
15. [latex]\left[2.48(12)\right](0.5)[/latex]
Solution
[latex]14.88[/latex]
16. [latex]\left[9.731(4)\right](0.75)[/latex]
17. [latex]7(4a)[/latex]
Solution
[latex]28a[/latex]
18. [latex]9(8w)[/latex]
19. [latex]-15(5m)[/latex]
Solution
[latex]-75m[/latex]
20. [latex]-23\left(2n\right)[/latex]
21. [latex]12\left(\frac{5}{6}p\right)[/latex]
Solution
[latex]10p[/latex]
22. [latex]20\left(\frac{3}{5}q\right)[/latex]
23. [latex]43m+(-12n)+(-16m)+(-9n)[/latex]
Solution
[latex]27m+(-21n)[/latex]
24. [latex]-22p+17q+(-35p)+(-27q)[/latex]
25. [latex]\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h[/latex]
Solution
[latex]\frac{5}{4}g+\frac{1}{2}h[/latex]
26. [latex]\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b[/latex]
Solution
[latex]2.43p+8.26q[/latex]
Exercises: Use the Identity and Inverse Properties of Addition and Multiplication
29.
a. [latex]\frac{2}{5}[/latex]
b. [latex]4.3[/latex]
c. [latex]-8[/latex]
d. [latex]-\frac{10}{3}[/latex]
Solution
a. [latex]-\frac{2}{5}[/latex]
b. [latex]-4.3[/latex]
c. [latex]8[/latex]
d. [latex]\frac{10}{3}[/latex]
30.
a. [latex]\frac{5}{9}[/latex]
b. [latex]2.1[/latex]
c. [latex]-3[/latex]
d. [latex]-\frac{9}{5}[/latex]
31.
a. [latex]-\frac{7}{6}[/latex]
b. [latex]-0.075[/latex]
c. [latex]23[/latex]
d. [latex]\frac{1}{4}[/latex]
Solution
a. [latex]\frac{7}{6}[/latex]
b. [latex]0.075[/latex]
c. [latex]-23[/latex]
d. [latex]-\frac{1}{4}[/latex]
32.
a. [latex]-\frac{8}{3}[/latex]
b. [latex]-0.019[/latex]
c. [latex]52[/latex]
d. [latex]\frac{5}{6}[/latex]
Exercises: Use the Identity and Inverse Properties of Addition and Multiplication
33.
a. [latex]6[/latex]
b. [latex]-\frac{3}{4}[/latex]
c. [latex]0.7[/latex]
Solution
a. [latex]\frac{1}{6}[/latex]
b. [latex]-\frac{4}{3}[/latex]
c. [latex]\frac{10}{7}[/latex]
34.
a. [latex]12[/latex]
b. [latex]-\frac{9}{2}[/latex]
c. [latex]0.13[/latex]
35.
a. [latex]\frac{11}{12}[/latex]
b. [latex]-1.1[/latex]
c. [latex]-4[/latex]
Solution
a. [latex]\frac{12}{11}[/latex]
b. [latex]-\frac{10}{11}[/latex]
c. [latex]-\frac{1}{4}[/latex]
36.
a. [latex]\frac{17}{20}[/latex]
b. [latex]-1.5[/latex]
c. [latex]-3[/latex]
Exercises: Use the Properties of Zero
37. [latex]\frac{0}{6}[/latex]
Solution
[latex]0[/latex]
38. [latex]\frac{3}{0}[/latex]
39. [latex]0\div\frac{11}{12}[/latex]
Solution
[latex]0[/latex]
40. [latex]\frac{6}{0}[/latex]
41. [latex]\frac{0}{3}[/latex]
Solution
[latex]0[/latex]
42. [latex]0\times\frac{8}{15}[/latex]
43. [latex]\left(-3.14\right)\left(0\right)[/latex]
Solution
[latex]0[/latex]
44. [latex]\frac{\frac{1}{10}}{0}[/latex]
Exercises: Mixed Practice
45. [latex]19a+44-19a[/latex]
Solution
[latex]44[/latex]
46. [latex]27c+16-27c[/latex]
47. [latex]10(0.1d)[/latex]
Solution
[latex]d[/latex]
48. [latex]100(0.01p)[/latex]
49. [latex]\frac{0}{u-4.99}[/latex], where [latex]u\ne 4.99[/latex]
Solution
[latex]0[/latex]
50. [latex]\frac{0}{v-65.1}[/latex], where [latex]v\ne 65.1[/latex]
51. [latex]0\div\left(x-\frac{1}{2}\right)[/latex], where [latex]x\ne \frac{1}{2}[/latex]
Solution
[latex]0[/latex]
52. [latex]0\div\left(y-\frac{1}{6}\right)[/latex], where [latex]x\ne \frac{1}{6}[/latex]
53. [latex]\frac{32-5a}{0}[/latex], where [latex]32-5a\ne 0[/latex]
Solution
undefined
54. [latex]\frac{28-9b}{0}[/latex], where [latex]28-9b\ne 0[/latex]
55. [latex]\left(\frac{3}{4}+\frac{9}{10}m\right)\div0[/latex] where [latex]\frac{3}{4}+\frac{9}{10}m\ne 0[/latex]
Solution
undefined
56. [latex]\left(\frac{5}{16}n-\frac{3}{7}\right)\div0[/latex] where [latex]\frac{5}{16}n-\frac{3}{7}\ne 0[/latex]
57. [latex]15\times\frac{3}{5}(4d+10)[/latex]
Solution
[latex]36d+90[/latex]
58. [latex]18\times\frac{5}{6}(15h+24)[/latex]
Exercises: Simplify Expressions Using the Distributive Property
59. [latex]8(4y+9)[/latex]
Solution
[latex]32y+72[/latex]
60. [latex]9(3w+7)[/latex]
61. [latex]6(c-13)[/latex]
Solution
[latex]6c-78[/latex]
62. [latex]7(y-13)[/latex]
63. [latex]\frac{1}{4}\left(3q+12\right)[/latex]
Solution
[latex]\frac{3}{4}q+3[/latex]
64. [latex]\frac{1}{5}(4m+20)[/latex]
65. [latex]9(\frac{5}{9}y-\frac{1}{3})[/latex]
Solution
[latex]5y-3[/latex]
66. [latex]10\left(\frac{3}{10}x-\frac{2}{5}\right)[/latex]
67. [latex]12\left(\frac{1}{4}+\frac{2}{3}r\right)[/latex]
Solution
[latex]3+8r[/latex]
68. [latex]12\left(\frac{1}{6}+\frac{3}{4}s\right)[/latex]
69. [latex]r(s-18)[/latex]
Solution
[latex]rs-18r[/latex]
70. [latex]u(v-10)[/latex]
71. [latex](y+4)p[/latex]
Solution
[latex]yp+4p[/latex]
72. [latex](a+7)x[/latex]
73. [latex]-7(4p+1)[/latex]
Solution
[latex]-28p-7[/latex]
74. [latex]-9(9a+4)[/latex]
75. [latex]-3(x-6)[/latex]
Solution
[latex]-3x+18[/latex]
76. [latex]-4(q-7)[/latex]
77. [latex]-(3x-7)[/latex]
Solution
[latex]-3x+7[/latex]
78. [latex]-(5p-4)[/latex]
79. [latex]16-3(y+8)[/latex]
Solution
[latex]-3y-8[/latex]
80. [latex]18-4(x+2)[/latex]
81. [latex]4-11(3c-2)[/latex]
Solution
[latex]-33c+26[/latex]
82. [latex]9-6(7n-5)[/latex]
83. [latex]22-(a+3)[/latex]
Solution
[latex]-a+19[/latex]
84. [latex]8-(r-7)[/latex]
85. [latex](5m-3)-(m+7)[/latex]
Solution
[latex]4m-10[/latex]
86. [latex](4y-1)-(y-2)[/latex]
87. [latex]5(2n+9)+12(n-3)[/latex]
Solution
[latex]22n+9[/latex]
88. [latex]9(5u+8)+2(u-6)[/latex]
89. [latex]9(8x-3)-(-2)[/latex]
Solution
[latex]72x-25[/latex]
90. [latex]4(6x-1)-(-8)[/latex]
91. [latex]14(c-1)-8(c-6)[/latex]
Solution
[latex]6c+34[/latex]
92. [latex]11(n-7)-5(n-1)[/latex]
93. [latex]6(7y+8)-(30y-15)[/latex]
Solution
[latex]12y+63[/latex]
94. [latex]7(3n+9)-(4n-13)[/latex]
Exercises: Everyday Math
95. Insurance co-payment. Carrie had to have [latex]5[/latex] fillings done. Each filling cost [latex]$80[/latex]. Her dental insurance required her to pay [latex]20\%[/latex] of the cost as a copay. Calculate Carrie’s copay:
a. First, by multiplying [latex]0.20[/latex] by [latex]80[/latex] to find her copay for each filling and then multiplying your answer by [latex]5[/latex] to find her total copay for [latex]5[/latex] fillings.
b. Next, by multiplying [latex]\left[5(0.20)\right](80)[/latex]
c. Which of the properties of real numbers says that your answers to parts (a), where you multiplied [latex]5\left[(0.20)(80)\right][/latex] and (b), where you multiplied [latex]\left[5(0.20)\right](80)[/latex], should be equal?
Solution
a. [latex]$80[/latex]
b. [latex]$80[/latex]
c. Answers will vary
96. Cooking time. Helen bought a [latex]24[/latex]-pound turkey for her family’s Thanksgiving dinner and wants to know what time to put the turkey in to the oven. She wants to allow [latex]20[/latex] minutes per pound cooking time. Calculate the length of time needed to roast the turkey:
a. First, by multiplying [latex]24\times20[/latex] to find the total number of minutes and then multiplying the answer by [latex]\frac{1}{60}[/latex] to convert minutes into hours.
b. Next, by multiplying [latex]24\left(20\times\frac{1}{60}\right)[/latex].
c. Which of the properties of real numbers says that your answers to parts (a), where you multiplied [latex]\left(24\times20\right)\frac{1}{60}[/latex], and (b), where you multiplied [latex]24\left(20\times\frac{1}{60}\right)[/latex], should be equal?
97. Buying by the case. Trader Joe’s grocery stores sold a bottle of wine they called “Two Buck Chuck” for [latex]$1.99[/latex]. They sold a case of [latex]12[/latex] bottles for [latex]$23.88[/latex]. To find the cost of 12 bottles at [latex]$1.99[/latex], notice that 1.99 is [latex]2-0.01[/latex].
a. Multiply [latex]12(1.99)[/latex] by using the distributive property to multiply [latex]12(2-0.01)[/latex].
b. Was it a bargain to buy “Two Buck Chuck” by the case?
Solution
a. [latex]$23.88[/latex]
b. no, the price is the same
98. Multi-pack purchase. Adele’s shampoo sells for [latex]$3.99[/latex] per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a [latex]3[/latex] pack for [latex]$10.49[/latex]. To find the cost of [latex]3[/latex] bottles at [latex]$3.99[/latex], notice that [latex]3.99[/latex] is [latex]4-0.01[/latex].
a. Multiply [latex]3(3.99)[/latex] by using the distributive property to multiply [latex]3(4-0.01)[/latex].
b. How much would Adele save by buying [latex]3[/latex] bottles at the warehouse store instead of at the grocery store?
Exercises: Writing Exercises
99. In your own words, state the commutative property of addition.
Solution
Answers may vary
100. What is the difference between the additive inverse and the multiplicative inverse of a number?
101. Simplify [latex]8\left(x-\frac{1}{4}\right)[/latex] using the distributive property and explain each step.
Solution
Answers may vary
102. Explain how you can multiply [latex]4($5.97)[/latex] without paper or calculator by thinking of [latex]$5.97[/latex] as [latex]6-0.03[/latex] and then using the distributive property.
