# Exercises: Properties of Real Numbers (1.7)

## Exercises: Use the Commutative and Associative Properties

Instructions: For questions 1–4, use the associative property to simplify.

1. $3(4x)$

Solution

$12x$

2. $4(7m)$

3. $(y+12)+28$

Solution

$y+40$

4. $(n+17)+33$

## Exercises: Use the Commutative and Associative Properties

Instructions: For questions 5–26, simplify.

5. $\frac{1}{2}+\frac{7}{8}+\left(-\frac{1}{2}\right)$

Solution

$\frac{7}{8}$

6. $\frac{2}{5}+\frac{5}{12}+\left(-\frac{2}{5}\right)$

7. $\frac{3}{20}\times\frac{49}{11}\times\frac{20}{3}$

Solution

$\frac{49}{11}$

8. $\frac{13}{18}\times\frac{25}{7}\times\frac{18}{13}$

9. $-24.7\times\frac{3}{8}$

Solution

$-63$

10. $-36\times11\times\frac{4}{9}$

11. $\left(\frac{5}{6}+\frac{8}{15}\right)+\frac{7}{15}$

Solution

$1\frac{5}{6}$

12. $\left(\frac{11}{12}+\frac{4}{9}\right)+\frac{5}{9}$

13. $17(0.25)(4)$

Solution

$17$

14. $36(0.2)(5)$

15. $\left[2.48(12)\right](0.5)$

Solution

$14.88$

16. $\left[9.731(4)\right](0.75)$

17. $7(4a)$

Solution

$28a$

18. $9(8w)$

19. $-15(5m)$

Solution

$-75m$

20. $-23\left(2n\right)$

21. $12\left(\frac{5}{6}p\right)$

Solution

$10p$

22. $20\left(\frac{3}{5}q\right)$

23. $43m+(-12n)+(-16m)+(-9n)$

Solution

$27m+(-21n)$

24. $-22p+17q+(-35p)+(-27q)$

25. $\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h$

Solution

$\frac{5}{4}g+\frac{1}{2}h$

26. $\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b$

27. $6.8p+9.14q+\left(-4.37p\right)+\left(-0.88q\right)$
Solution

$2.43p+8.26q$

28. $9.6m+7.22n+\left(-2.19m\right)+\left(-0.65n\right)$

## Exercises: Use the Identity and Inverse Properties of Addition and Multiplication

Instructions: For questions 29–32, find the additive inverse of each number.

29.

a. $\frac{2}{5}$
b. $4.3$
c. $-8$
d. $-\frac{10}{3}$

Solution

a. $-\frac{2}{5}$
b. $-4.3$
c. $8$
d. $\frac{10}{3}$

30.

a. $\frac{5}{9}$
b. $2.1$
c. $-3$
d. $-\frac{9}{5}$

31.

a. $-\frac{7}{6}$
b. $-0.075$
c. $23$
d. $\frac{1}{4}$

Solution

a. $\frac{7}{6}$
b. $0.075$
c. $-23$
d. $-\frac{1}{4}$

32.

a. $-\frac{8}{3}$
b. $-0.019$
c. $52$
d. $\frac{5}{6}$

## Exercises: Use the Identity and Inverse Properties of Addition and Multiplication

Instructions: For questions 33–36, find the multiplicative inverse of each number.

33.

a. $6$
b. $-\frac{3}{4}$
c. $0.7$

Solution

a. $\frac{1}{6}$
b. $-\frac{4}{3}$
c. $\frac{10}{7}$

34.

a. $12$
b. $-\frac{9}{2}$
c. $0.13$

35.

a. $\frac{11}{12}$
b. $-1.1$
c. $-4$

Solution

a. $\frac{12}{11}$
b. $-\frac{10}{11}$
c. $-\frac{1}{4}$

36.

a. $\frac{17}{20}$
b. $-1.5$
c. $-3$

## Exercises: Use the Properties of Zero

Instructions: For questions 37–44, simplify.

37. $\frac{0}{6}$

Solution

$0$

38. $\frac{3}{0}$

39. $0\div\frac{11}{12}$

Solution

$0$

40. $\frac{6}{0}$

41. $\frac{0}{3}$

Solution

$0$

42. $0\times\frac{8}{15}$

43. $\left(-3.14\right)\left(0\right)$

Solution

$0$

44. $\frac{\frac{1}{10}}{0}$

## Exercises: Mixed Practice

Instructions: For questions 45–58, simplify.

45. $19a+44-19a$

Solution

$44$

46. $27c+16-27c$

47. $10(0.1d)$

Solution

$d$

48. $100(0.01p)$

49. $\frac{0}{u-4.99}$, where $u\ne 4.99$

Solution

$0$

50. $\frac{0}{v-65.1}$, where $v\ne 65.1$

51. $0\div\left(x-\frac{1}{2}\right)$, where $x\ne \frac{1}{2}$

Solution

$0$

52. $0\div\left(y-\frac{1}{6}\right)$, where $x\ne \frac{1}{6}$

53. $\frac{32-5a}{0}$, where $32-5a\ne 0$

Solution

undefined

54. $\frac{28-9b}{0}$, where $28-9b\ne 0$

55. $\left(\frac{3}{4}+\frac{9}{10}m\right)\div0$ where $\frac{3}{4}+\frac{9}{10}m\ne 0$

Solution

undefined

56. $\left(\frac{5}{16}n-\frac{3}{7}\right)\div0$ where $\frac{5}{16}n-\frac{3}{7}\ne 0$

57. $15\times\frac{3}{5}(4d+10)$

Solution

$36d+90$

58. $18\times\frac{5}{6}(15h+24)$

## Exercises: Simplify Expressions Using the Distributive Property

Instructions: For questions 59–94, simplify using the distributive property.

59. $8(4y+9)$

Solution

$32y+72$

60. $9(3w+7)$

61. $6(c-13)$

Solution

$6c-78$

62. $7(y-13)$

63. $\frac{1}{4}\left(3q+12\right)$

Solution

$\frac{3}{4}q+3$

64. $\frac{1}{5}(4m+20)$

65. $9(\frac{5}{9}y-\frac{1}{3})$

Solution

$5y-3$

66. $10\left(\frac{3}{10}x-\frac{2}{5}\right)$

67. $12\left(\frac{1}{4}+\frac{2}{3}r\right)$

Solution

$3+8r$

68. $12\left(\frac{1}{6}+\frac{3}{4}s\right)$

69. $r(s-18)$

Solution

$rs-18r$

70. $u(v-10)$

71. $(y+4)p$

Solution

$yp+4p$

72. $(a+7)x$

73. $-7(4p+1)$

Solution

$-28p-7$

74. $-9(9a+4)$

75. $-3(x-6)$

Solution

$-3x+18$

76. $-4(q-7)$

77. $-(3x-7)$

Solution

$-3x+7$

78. $-(5p-4)$

79. $16-3(y+8)$

Solution

$-3y-8$

80. $18-4(x+2)$

81. $4-11(3c-2)$

Solution

$-33c+26$

82. $9-6(7n-5)$

83. $22-(a+3)$

Solution

$-a+19$

84. $8-(r-7)$

85. $(5m-3)-(m+7)$

Solution

$4m-10$

86. $(4y-1)-(y-2)$

87. $5(2n+9)+12(n-3)$

Solution

$22n+9$

88. $9(5u+8)+2(u-6)$

89. $9(8x-3)-(-2)$

Solution

$72x-25$

90. $4(6x-1)-(-8)$

91. $14(c-1)-8(c-6)$

Solution

$6c+34$

92. $11(n-7)-5(n-1)$

93. $6(7y+8)-(30y-15)$

Solution

$12y+63$

94. $7(3n+9)-(4n-13)$

## Exercises: Everyday Math

Instructions: For questions 95–98, answer the given everyday math word problems.

95. Insurance co-payment. Carrie had to have $5$ fillings done. Each filling cost $80$. Her dental insurance required her to pay $20\%$ of the cost as a copay. Calculate Carrie’s copay:

a. First, by multiplying $0.20$ by $80$ to find her copay for each filling and then multiplying your answer by $5$ to find her total copay for $5$ fillings.
b. Next, by multiplying $\left[5(0.20)\right](80)$
c. Which of the properties of real numbers says that your answers to parts (a), where you multiplied $5\left[(0.20)(80)\right]$ and (b), where you multiplied $\left[5(0.20)\right](80)$, should be equal?

Solution

a. $80$
b. $80$

96. Cooking time. Helen bought a $24$-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 $20$ minutes per pound cooking time. Calculate the length of time needed to roast the turkey:

a. First, by multiplying $24\times20$ to find the total number of minutes and then multiplying the answer by $\frac{1}{60}$ to convert minutes into hours.
b. Next, by multiplying $24\left(20\times\frac{1}{60}\right)$.
c. Which of the properties of real numbers says that your answers to parts (a), where you multiplied $\left(24\times20\right)\frac{1}{60}$, and (b), where you multiplied $24\left(20\times\frac{1}{60}\right)$, should be equal?

97. Buying by the case. Trader Joe’s grocery stores sold a bottle of wine they called “Two Buck Chuck” for $1.99$. They sold a case of $12$ bottles for $23.88$. To find the cost of 12 bottles at $1.99$, notice that 1.99 is $2-0.01$.

a. Multiply $12(1.99)$ by using the distributive property to multiply $12(2-0.01)$.
b. Was it a bargain to buy “Two Buck Chuck” by the case?

Solution

a. $23.88$
b. no, the price is the same

98. Multi-pack purchase. Adele’s shampoo sells for $3.99$ per bottle at the grocery store. At the warehouse store, the same shampoo is sold as a $3$ pack for $10.49$. To find the cost of $3$ bottles at $3.99$, notice that $3.99$ is $4-0.01$.

a. Multiply $3(3.99)$ by using the distributive property to multiply $3(4-0.01)$.
b. How much would Adele save by buying $3$ bottles at the warehouse store instead of at the grocery store?

## Exercises: Writing Exercises

Instructions: For questions 99–102, answer the given writing exercises.

Solution

100. What is the difference between the additive inverse and the multiplicative inverse of a number?

101. Simplify $8\left(x-\frac{1}{4}\right)$ using the distributive property and explain each step.

Solution

102. Explain how you can multiply $4(5.97)$ without paper or calculator by thinking of $5.97$ as $6-0.03$ and then using the distributive property.