# Exercises: Solve Equations using the Division and Multiplication Properties of Equality (3.2)

## Exercises: Solve Equations Using the Division and Multiplication Properties of Equality

Instructions: For questions 1-36, solve each equation using the Division and Multiplication Properties of Equality and check the solution.

1. $8x=56$

Solution

$x=7$

2. $7p=63$

3. $-5c=55$

Solution

$c=-11$

4. $-9x=-27$

5. $-809=15y$

Solution

$y=-\frac{809}{15}$

6. $-731=19y$

7. $-37p=-541$

Solution

$p=-\frac{541}{37}$

8. $-19m=-586$

9. $0.25z=3.25$

Solution

$z=13$

10. $0.75a=11.25$

11. $-13x=0$

Solution

$x=0$

12. $24x=0$

13. $\frac{x}{4}=35$

Solution

$x=140$

14. $\frac{z}{2}=54$

15. $-20=\frac{q}{-5}$

Solution

$q=100$

16. $\frac{c}{-3}=-12$

17. $\frac{y}{9}=-16$

Solution

$y=-144$

18. $\frac{q}{6}=-38$

19. $\frac{m}{-12}=45$

Solution

$m=-540$

20. $-24=\frac{p}{-20}$

21. $-y=6$

Solution

$y=-6$

22. $-u=15$

23. $-v=-72$

Solution

$v=72$

24. $-x=-39$

25. $\frac{2}{3}y=48$

Solution

$y=72$

26. $\frac{3}{5}r=75$

27. $-\frac{5}{8}w=40$

Solution

$w=-64$

28. $24=-\frac{3}{4}x$

29. $-\frac{2}{5}=\frac{1}{10}a$

Solution

$a=-4$

30. $-\frac{1}{3}q=-\frac{5}{6}$

31. $-\frac{7}{10}x=-\frac{14}{3}$

Solution

$x=\frac{20}{3}$

32. $\frac{3}{8}y=-\frac{1}{4}$

33. $\frac{7}{12}=-\frac{3}{4}p$

Solution

$p=-\frac{7}{9}$

34. $\frac{11}{18}=-\frac{5}{6}q$

35. $-\frac{5}{18}=-\frac{10}{9}u$

Solution

$u=\frac{1}{4}$

36. $-\frac{7}{20}=-\frac{7}{4}v$

## Exercises: Solve Equations that Require Simplification

Instructions: For questions 37-46, solve each equation requiring simplification.

37. $100-16=4p-10p-p$

Solution

$p=-12$

38. $-18-7=5t-9t-6t$

39. $\frac{7}{8}n-\frac{3}{4}n=9+2$

Solution

$n=88$

40. $\frac{5}{12}q+\frac{1}{2}q=25-3$

41. $0.25d+0.10d=6-0.75$

Solution

$d=15$

42. $0.05p-0.01p=2+0.24$

43. $-10(q-4)-57=93$

Solution

$q=-11$

44. $-12(d-5)-29=43$

45. $-10(x+4)-19=85$

Solution

$x=-\frac{72}{5}$

46. $-15(z+9)-11=75$

## Exercises: Mixed Practice

Instructions: For questions 47-65, solve each equation.

47. $\frac{9}{10}x=90$

Solution

$x=100$

48. $\frac{5}{12}y=60$

49. $y+46=55$

Solution

$y=9$

50. $x+33=41$

51. $\frac{w}{-2}=99$

Solution

$w=-198$

52. $\frac{s}{-3}=-60$

53. $27=6a$

Solution

$a=\frac{9}{2}$

54. $-a=7$

55. $-x=2$

Solution

$x=-2$

56. $z-16=-59$

57. $m-41=-14$

Solution

$m=27$

58. $0.04r=52.60$

59. $63.90=0.03p$

Solution

$p=2130$

60. $-15x=-120$

61. $84=-12z$

Solution

$y=-7$

62. $19.36=x-0.2x$

63. $c-0.3c=35.70$

Solution

$c=51$

64. $-y=-9$

65. $-x=-8$

Solution

$x=8$

## Exercises: Translate to an Equation and Solve

Instructions: For questions 66-85, translate to an equation and then solve.

66. $187$ is the product of $-17$ and $m$.

67. $133$ is the product of $-19$ and $n$.

Solution

$\begin{array}{rcl}133&=&-19n\\n&=&-7\end{array}$

68. $-184$ is the product of $23$ and $p$.

69. $-152$ is the product of $8$ and $q$.

Solution

$\begin{array}{rcl}-152&=&8q\\q&=&-19\end{array}$

70. $u$ divided by $7$ is equal to $-49$.

71. $r$ divided by 12 is equal to $-48$.

Solution

$\begin{array}{rcl}\frac{r}{12}&=&-48\\r&=&-576\end{array}$

72. $h$ divided by $-13$ is equal to $-65$.

73. $j$ divided by $-20$ is equal to $-80$.

Solution

$\begin{array}{rcl}\frac{j}{-20}&=&-80\\j&=&1,600\end{array}$

74. The quotient $c$ and $-19$ is $38$.

75. The quotient of $b$ and $-6$ is $18$.

Solution

$\begin{array}{rcl}\frac{b}{-6}&=&18\\b&=&-108\end{array}$

76. The quotient of $h$ and $26$ is $-52$.

77. The quotient $k$ and $22$ is $-66$.

Solution

$\frac{k}{22}=-66;k=-1,452$

78. Five-sixths of $y$ is $15$.

79. Three-tenths of $x$ is $15$.

Solution

$\begin{array}{rcl}\frac{3}{10}x&=&15\\x&=&50\end{array}$

80. Four-thirds of $w$ is $36$.

81. Five-halves of $v$ is $50$.

Solution

$\begin{array}{rcl}\frac{5}{2}v&=&50\\v&=&20\end{array}$

82. The sum of nine-tenths and $g$ is two-thirds.

83. The sum of two-fifths and $f$ is one-half.

Solution

$\begin{array}{rcl}\frac{2}{5}+f&=&\frac{1}{2}\\f&=&\frac{1}{10}\end{array}$

84. The difference of $p$ and one-sixth is two-thirds.

85. The difference of $q$ and one-eighth is three-fourths.

Solution

$\begin{array}{rcl}q-\frac{1}{8}&=&\frac{3}{4}\\q&=&\frac{7}{8}\end{array}$

## Exercises: Translate and Solve Applications

Instructions: For questions 86-93, translate into an equation and solve.

86. Kindergarten. Connie’s kindergarten class has $24$ children. She wants them to get into $4$ equal groups. How many children will she put in each group?

87. Balloons. Ramona bought $18$ balloons for a party. She wants to make $3$ equal bunches. How many balloons did she use in each bunch?

Solution

$6$ balloons

88. Tickets. Mollie paid $36.25$ for $5$ movie tickets. What was the price of each ticket?

89. Shopping. Serena paid $12.96$ for a pack of $12$ pairs of sport socks. What was the price of pair of sport socks?

Solution

$1.08$

90. Sewing. Nancy used $14$ yards of fabric to make flags for one-third of the drill team. How much fabric, would Nancy need to make flags for the whole team?

91. MPG. John’s SUV gets $18$ miles per gallon (mpg). This is half as many mpg as his wife’s hybrid car. How many miles per gallon does the hybrid car get?

Solution

$36$ mpg

92. Height. Aiden is $27$ inches tall. He is $\frac{3}{8}$ as tall as his father. How tall is his father?

93. Real estate. Bea earned $11,700$ commission for selling a house, calculated as $\frac{6}{100}$ of the selling price. What was the selling price of the house?

Solution

$195,000$

## Exercises: Everyday Math

Instructions: For questions 94-95, solve the given everyday math word problems.

94. Commission. Every week Perry gets paid $150$ plus $12\%$ of his total sales amount. Solve the equation $840=150+0.12(a-1250)$ for $a$, to find the total amount Perry must sell in order to be paid $840$ one week.

95. Stamps. Travis bought $9.45$ worth of $49$-cent stamps and $21$-cent stamps. The number of $21$-cent stamps was $5$ less than the number of $49$-cent stamps. Solve the equation $0.49s+0.21(s-5)=9.45$ for $s$, to find the number of $49$-cent stamps Travis bought.

Solution

$15$ $49$-cent stamps

## Exercises: Writing Exercises

Instructions: For questions 96-97, answer the given writing exercises.

96. Frida started to solve the equation $-3x=36$ by adding $3$ to both sides. Explain why Frida’s method will not solve the equation.

97. Emiliano thinks $x=40$ is the solution to the equation $\frac{1}{2}x=80$. Explain why he is wrong.

Solution