# 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. [latex]8x=56[/latex]**

**Solution**

[latex]x=7[/latex]

**2. [latex]7p=63[/latex]**

**3. [latex]-5c=55[/latex]**

**Solution**

[latex]c=-11[/latex]

**4. [latex]-9x=-27[/latex]**

**5. [latex]-809=15y[/latex]**

**Solution**

[latex]y=-\frac{809}{15}[/latex]

**6. [latex]-731=19y[/latex]**

**7. [latex]-37p=-541[/latex]**

**Solution**

[latex]p=-\frac{541}{37}[/latex]

**8. [latex]-19m=-586[/latex]**

**9. [latex]0.25z=3.25[/latex]**

**Solution**

[latex]z=13[/latex]

**10. [latex]0.75a=11.25[/latex]**

**11. [latex]-13x=0[/latex]**

**Solution**

[latex]x=0[/latex]

**12. [latex]24x=0[/latex]**

**13. [latex]\frac{x}{4}=35[/latex]**

**Solution**

[latex]x=140[/latex]

**14. [latex]\frac{z}{2}=54[/latex]**

**15. [latex]-20=\frac{q}{-5}[/latex]**

**Solution**

[latex]q=100[/latex]

**16. [latex]\frac{c}{-3}=-12[/latex]**

**17. [latex]\frac{y}{9}=-16[/latex]**

**Solution**

[latex]y=-144[/latex]

**18. [latex]\frac{q}{6}=-38[/latex]**

**19. [latex]\frac{m}{-12}=45[/latex]**

**Solution**

[latex]m=-540[/latex]

**20. [latex]-24=\frac{p}{-20}[/latex]**

**21. [latex]-y=6[/latex]**

**Solution**

[latex]y=-6[/latex]

**22. [latex]-u=15[/latex]**

**23. [latex]-v=-72[/latex]**

**Solution**

[latex]v=72[/latex]

**24. [latex]-x=-39[/latex]**

**25. [latex]\frac{2}{3}y=48[/latex]**

**Solution**

[latex]y=72[/latex]

**26. [latex]\frac{3}{5}r=75[/latex]**

**27. [latex]-\frac{5}{8}w=40[/latex]**

**Solution**

[latex]w=-64[/latex]

**28. [latex]24=-\frac{3}{4}x[/latex]**

**29. [latex]-\frac{2}{5}=\frac{1}{10}a[/latex]**

**Solution**

[latex]a=-4[/latex]

**30. [latex]-\frac{1}{3}q=-\frac{5}{6}[/latex]**

**31. [latex]-\frac{7}{10}x=-\frac{14}{3}[/latex]**

**Solution**

[latex]x=\frac{20}{3}[/latex]

**32. [latex]\frac{3}{8}y=-\frac{1}{4}[/latex]**

**33. [latex]\frac{7}{12}=-\frac{3}{4}p[/latex]**

**Solution**

[latex]p=-\frac{7}{9}[/latex]

**34. [latex]\frac{11}{18}=-\frac{5}{6}q[/latex]**

**35. [latex]-\frac{5}{18}=-\frac{10}{9}u[/latex]**

**Solution**

[latex]u=\frac{1}{4}[/latex]

**36. [latex]-\frac{7}{20}=-\frac{7}{4}v[/latex]**

**Exercises: Solve Equations that Require Simplification**

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

**37. [latex]100-16=4p-10p-p[/latex]**

**Solution**

[latex]p=-12[/latex]

**38. [latex]-18-7=5t-9t-6t[/latex]**

**39. [latex]\frac{7}{8}n-\frac{3}{4}n=9+2[/latex]**

**Solution**

[latex]n=88[/latex]

**40. [latex]\frac{5}{12}q+\frac{1}{2}q=25-3[/latex]**

**41. [latex]0.25d+0.10d=6-0.75[/latex]**

**Solution**

[latex]d=15[/latex]

**42. [latex]0.05p-0.01p=2+0.24[/latex]**

**43. [latex]-10(q-4)-57=93[/latex]**

**Solution**

[latex]q=-11[/latex]

**44. [latex]-12(d-5)-29=43[/latex]**

**45. [latex]-10(x+4)-19=85[/latex]**

**Solution**

[latex]x=-\frac{72}{5}[/latex]

**46. [latex]-15(z+9)-11=75[/latex]**

**Exercises: Mixed Practice**

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

**47. [latex]\frac{9}{10}x=90[/latex]**

**Solution**

[latex]x=100[/latex]

**48. [latex]\frac{5}{12}y=60[/latex]**

**49. [latex]y+46=55[/latex]**

**Solution**

[latex]y=9[/latex]

**50. [latex]x+33=41[/latex]**

**51. [latex]\frac{w}{-2}=99[/latex]**

**Solution**

[latex]w=-198[/latex]

**52. [latex]\frac{s}{-3}=-60[/latex]**

**53. [latex]27=6a[/latex]**

**Solution**

[latex]a=\frac{9}{2}[/latex]

**54. [latex]-a=7[/latex]**

**55. [latex]-x=2[/latex]**

**Solution**

[latex]x=-2[/latex]

**56. [latex]z-16=-59[/latex]**

**57. [latex]m-41=-14[/latex]**

**Solution**

[latex]m=27[/latex]

**58. [latex]0.04r=52.60[/latex]**

**59. [latex]63.90=0.03p[/latex]**

**Solution**

[latex]p=2130[/latex]

**60. [latex]-15x=-120[/latex]**

**61. [latex]84=-12z[/latex]**

**Solution**

[latex]y=-7[/latex]

**62. [latex]19.36=x-0.2x[/latex]**

**63. [latex]c-0.3c=35.70[/latex]**

**Solution**

[latex]c=51[/latex]

**64. [latex]-y=-9[/latex]**

**65. [latex]-x=-8[/latex]**

**Solution**

[latex]x=8[/latex]

**Exercises: Translate to an Equation and Solve**

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

**66. [latex]187[/latex] is the product of [latex]-17[/latex] and [latex]m[/latex].**

**67. [latex]133[/latex] is the product of [latex]-19[/latex] and [latex]n[/latex].**

**Solution**

[latex]\begin{array}{rcl}133&=&-19n\\n&=&-7\end{array}[/latex]

**68. [latex]-184[/latex] is the product of [latex]23[/latex] and [latex]p[/latex].**

**69. [latex]-152[/latex] is the product of [latex]8[/latex] and [latex]q[/latex].**

**Solution**

[latex]\begin{array}{rcl}-152&=&8q\\q&=&-19\end{array}[/latex]

**70. [latex]u[/latex] divided by [latex]7[/latex] is equal to [latex]-49[/latex].**

**71. [latex]r[/latex] divided by 12 is equal to [latex]-48[/latex].**

**Solution**

[latex]\begin{array}{rcl}\frac{r}{12}&=&-48\\r&=&-576\end{array}[/latex]

**72. [latex]h[/latex] divided by [latex]-13[/latex] is equal to [latex]-65[/latex].**

**73. [latex]j[/latex] divided by [latex]-20[/latex] is equal to [latex]-80[/latex].**

**Solution**

[latex]\begin{array}{rcl}\frac{j}{-20}&=&-80\\j&=&1,600\end{array}[/latex]

**74. The quotient [latex]c[/latex] and [latex]-19[/latex] is [latex]38[/latex].**

**75. The quotient of [latex]b[/latex] and [latex]-6[/latex] is [latex]18[/latex].**

**Solution**

[latex]\begin{array}{rcl}\frac{b}{-6}&=&18\\b&=&-108\end{array}[/latex]

**76. The quotient of [latex]h[/latex] and [latex]26[/latex] is [latex]-52[/latex].**

**77. The quotient [latex]k[/latex] and [latex]22[/latex] is [latex]-66[/latex].**

**Solution**

[latex]\frac{k}{22}=-66;k=-1,452[/latex]

**78. Five-sixths of [latex]y[/latex] is [latex]15[/latex].**

**79. Three-tenths of [latex]x[/latex] is [latex]15[/latex].**

**Solution**

[latex]\begin{array}{rcl}\frac{3}{10}x&=&15\\x&=&50\end{array}[/latex]

**80. Four-thirds of [latex]w[/latex] is [latex]36[/latex].**

**81. Five-halves of [latex]v[/latex] is [latex]50[/latex].**

**Solution**

[latex]\begin{array}{rcl}\frac{5}{2}v&=&50\\v&=&20\end{array}[/latex]

**82. The sum of nine-tenths and [latex]g[/latex] is two-thirds.**

**83. The sum of two-fifths and [latex]f[/latex] is one-half.**

**Solution**

[latex]\begin{array}{rcl}\frac{2}{5}+f&=&\frac{1}{2}\\f&=&\frac{1}{10}\end{array}[/latex]

**84. The difference of [latex]p[/latex] and one-sixth is two-thirds.**

**85. The difference of [latex]q[/latex] and one-eighth is three-fourths.**

**Solution**

[latex]\begin{array}{rcl}q-\frac{1}{8}&=&\frac{3}{4}\\q&=&\frac{7}{8}\end{array}[/latex]

**Exercises: Translate and Solve Applications**

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

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

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

**Solution**

[latex]6[/latex] balloons

**88. Tickets. Mollie paid [latex]$36.25[/latex] for [latex]5[/latex] movie tickets. What was the price of each ticket?**

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

**Solution**

[latex]$1.08[/latex]

**90. Sewing.** **Nancy used [latex]14[/latex] 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 [latex]18[/latex] 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**

[latex]36[/latex] mpg

**92. Height. Aiden is [latex]27[/latex] inches tall. He is [latex]\frac{3}{8}[/latex] as tall as his father. How tall is his father?**

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

**Solution**

[latex]$195,000[/latex]

**Exercises: Everyday Math**

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

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

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

**Solution**

[latex]15[/latex] [latex]49[/latex]-cent stamps

**Exercises: Writing Exercises**

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

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

**97. Emiliano thinks [latex]x=40[/latex] is the solution to the equation [latex]\frac{1}{2}x=80[/latex]. Explain why he is wrong.**

**Solution**

Answers will vary.