Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

1.10E: Exercises

  • Page ID
    30355
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Practice Makes Perfect

    Use the Commutative and Associative Properties

    In the following exercises, use the associative property to simplify.

    Exercise 1

    \(3(4x)\)

    Answer

    \(12x\)

    Exercise 2

    \(4(7m)\)

    Exercise 3

    \((y+12)+28\)

    Answer

    \(y+40\)

    Exercise 4

    \((n+17)+33\)

    In the following exercises, simplify.

    Exercise 5

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

    Answer

    \(\frac{7}{8}\)

    Exercise 6

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

    Exercise 7

    \(\frac{3}{20} \cdot \frac{49}{11} \cdot \frac{20}{3}\)

    Answer

    \(\frac{49}{11}\)

    Exercise 8

    \(\frac{13}{18} \cdot \frac{25}{7} \cdot \frac{18}{13}\)

    Exercise 9

    \(-24 \cdot 7 \cdot \frac{3}{8}\)

    Answer

    \(-63\)

    Exercise 10

    \(-36 \cdot 11 \cdot \frac{4}{9}\)

    Exercise 11

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

    Answer

    \(1 \frac{5}{6}\)

    Exercise 12

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

    Exercise 13

    \(17(0.25)(4)\)

    Answer

    \(17\)

    Exercise 14

    \(36(0.2)(5)\)

    Exercise 15

    \([2.48(12)](0.5)\)

    Answer

    \(14.88\)

    Exercise 16

    \([9.731(4)](0.75)\)

    Exercise 17

    \(7(4a)\)

    Answer

    \(28a\)

    Exercise 18

    \(9(8w)\)

    Exercise 19

    \(-15(5m)\)

    Answer

    \(-75m\)

    Exercise 20

    \(-23(2n)\)

    Exercise 21

    \(12(\frac{5}{6}p)\)

    Answer

    \(10p\)

    Exercise 22

    \(20(\frac{3}{5}q)\)

    Exercise 23

    \(43 m+(-12 n)+(-16 m)+(-9 n)\)

    Answer

    \(27m+(-21n)\)

    Exercise 24

    \(-22p+17q+(-35p)+(-27q)\)

    Exercise 25

    \(\frac{3}{8} g+\frac{1}{12} h+\frac{7}{8} g+\frac{5}{12} h\)

    Answer

    \(\frac{5}{4}g+\frac{1}{2}h\)

    Exercise 26

    \(\frac{5}{6} a+\frac{3}{10} b+\frac{1}{6} a+\frac{9}{10} b\)

    Exercise 27

    \(6.8 p+9.14 q+(-4.37 p)+(-0.88 q)\)

    Answer

    \(2.43p+8.26q\)

    Exercise 28

    \(9.6 m+7.22 n+(-2.19 m)+(-0.65 n)\)

    Use the Identity and Inverse Properties of Addition and Multiplication

    In the following exercises, find the additive inverse of each number

    Exercise 29

    1. \(\frac{2}{5}\)
    2. \(4.3\)
    3. \(-8\)
    4. \(-\frac{10}{3}\)
    Answer
    1. \(-\frac{2}{5}\)
    2. \(-4.3\)
    3. \(8\)
    4. \(\frac{10}{3}\)

    Exercise 30

    1. \(\frac{5}{9}\)
    2. \(2.1\)
    3. \(-3\)
    4. \(-\frac{9}{5}\)

    Exercise 31

    1. \(-\frac{7}{6}\)
    2. \(-0.075\)
    3. \(23\)
    4. \(\frac{1}{4}\)
    Answer
    1. \(\frac{7}{6}\)
    2. \(0.075\)
    3. \(-23\)
    4. \(-\frac{1}{4}\)

    Exercise 32

    1. \(-\frac{8}{3}\)
    2. \(-0.019\)
    3. \(52\)
    4. \(\frac{5}{6}\)

    In the following exercises, find the multiplicative inverse of each number.

    Exercise 33

    1. \(6\)
    2. \(-\frac{3}{4}\)
    3. \(0.7\)
    Answer
    1. \(\frac{1}{6}\)
    2. \(-\frac{4}{3}\)
    3. \(\frac{10}{7}\)

    Exercise 34

    1. \(12\)
    2. \(-\frac{9}{2}\)
    3. \(0.13\)

    Exercise 35

    1. \(\frac{11}{12}\)
    2. \(-1.1\)
    3. \(-4\)
    Answer
    1. \(\frac{12}{11}\)
    2. \(-\frac{10}{11}\)
    3. \(-\frac{1}{4}\)

    Exercise 36

    1. \(\frac{17}{20}\)
    2. \(-1.5\)
    3. \(-3\)

    Use the Properties of Zero

    In the following exercises, simplify.

    Exercise 37

    \(\frac{0}{6}\)

    Answer

    \(0\)

    Exercise 38

    \(\frac{3}{0}\)

    Exercise 39

    \(0 \div \frac{11}{12}\)

    Answer

    \(0\)

    Exercise 40

    \(\frac{6}{0}\)

    Exercise 41

    \(\frac{0}{3}\)

    Answer

    \(0\)

    Exercise 42

    \(0 \cdot \frac{8}{15}\)

    Exercise 43

    \((-3.14)(0)\)

    Answer

    \(0\)

    Exercise 44

    \(\frac{\frac{1}{10}}{0}\)

    Mixed Practice

    In the following exercises, simplify.

    Exercise 45

    \(19 a+44-19 a\)

    Answer

    \(44\)

    Exercise 46

    \(27 c+16-27 c\)

    Exercise 47

    \(10(0.1 d)\)

    Answer

    \(1d\)

    Exercise 48

    \(100(0.01 p)\)

    Exercise 49

    \(\frac{0}{u-4.99}, \text { where } u \neq 4.99\)

    Answer

    \(0\)

    Exercise 50

    \(\frac{0}{v-65.1}, \text { where } v \neq 65.1\)

    Exercise 51

    \(0 \div\left(x-\frac{1}{2}\right), \text { where } x \neq \frac{1}{2}\)

    Answer

    \(0\)

    Exercise 52

    \(0 \div\left(y-\frac{1}{6}\right), \text { where } y \neq \frac{1}{6}\)

    Exercise 53

    \(\frac{32-5 a}{0}, \text { where } 32-5a \neq 0\)

    Answer

    undefined

    Exercise 54

    \(\frac{28-9 b}{0}, \text { where } 28-9b \neq 0\)

    Exercise 55

    \(\left(\frac{3}{4}+\frac{9}{10} m\right) \div 0 \text { where } \frac{3}{4}+\frac{9}{10}m \neq 0\)

    Answer

    undefined

    Exercise 56

    \(\left(\frac{5}{16} n-\frac{3}{7}\right) \div 0 \text { where } \frac{5}{16} n-\frac{3}{7} \neq 0\)

    Exercise 57

    \(15 \cdot \frac{3}{5}(4 d+10)\)

    Answer

    \(36d+90\)

    Exercise 58

    \(18 \cdot \frac{5}{6}(15 h+24)\)

    Simplify Expressions Using the Distributive Property

    In the following exercises, simplify using the distributive property.

    Exercise 59

    \(8(4 y+9)\)

    Answer

    \(32y+72\)

    Exercise 60

    \(9(3 w+7)\)

    Exercise 61

    \(6(c-13)\)

    Answer

    \(6c-78\)

    Exercise 62

    \(7(y-13)\)

    Exercise 63

    \(\frac{1}{4}(3 q+12)\)

    Answer

    \(\frac{3}{4}q+3\)

    Exercise 64

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

    Exercise 65

    \(9\left(\frac{5}{9} y-\frac{1}{3}\right)\)

    Answer

    \(5y-3\)

    Exercise 66

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

    Exercise 67

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

    Answer

    \(3+8r\)

    Exercise 68

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

    Exercise 69

    \(r(s-18)\)

    Answer

    \(rs-18r\)

    Exercise 70

    \(u(v-10)\)

    Exercise 71

    \((y+4) p\)

    Answer

    \(yp+4p\)

    Exercise 72

    \((a+7) x\)

    Exercise 73

    \(-7(4 p+1)\)

    Answer

    \(-28p-7\)

    Exercise 74

    \(-9(9 a+4)\)

    Exercise 75

    \(-3(x-6)\)

    Answer

    \(-3x+18\)

    Exercise 76

    \(-4(q-7)\)

    Exercise 77

    \(-(3 x-7)\)

    Answer

    \(-3x+7\)

    Exercise 78

    \(-(5 p-4)\)

    Exercise 79

    \(16-3(y+8)\)

    Answer

    \(-3y-8\)

    Exercise 80

    \(18-4(x+2)\)

    Exercise 81

    \(4-11(3 c-2)\)

    Answer

    \(-33c+26\)

    Exercise 82

    \(9-6(7 n-5)\)

    Exercise 83

    \(22-(a+3)\)

    Answer

    \(-a+19\)

    Exercise 84

    \(8-(r-7)\)

    Exercise 85

    \((5 m-3)-(m+7)\)

    Answer

    \(4m-10\)

    Exercise 86

    \((4 y-1)-(y-2)\)

    Exercise 87

    \(5(2 n+9)+12(n-3)\)

    Answer

    \(22n+9\)

    Exercise 88

    \(9(5 u+8)+2(u-6)\)

    Exercise 89

    \(9(8 x-3)-(-2)\)

    Answer

    \(72x-25\)

    Exercise 90

    \(4(6 x-1)-(-8)\)

    Exercise 91

    \(14(c-1)-8(c-6)\)

    Answer

    \(6c+34\)

    Exercise 92

    \(11(n-7)-5(n-1)\)

    Exercise 93

    \(6(7 y+8)-(30 y-15)\)

    Answer

    \(12y+63\)

    Exercise 94

    \(7(3 n+9)-(4 n-13)\)

    Everyday Math

    Exercise 95

    Insurance copayment 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:

    1. 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.
    2. Next, by multiplying [5(0.20)](80)
    3. Which of the properties of real numbers says that your answers to parts (a), where you multiplied 5[(0.20)(80)] and (b), where you multiplied [5(0.20)](80), should be equal?
    Answer
    1. $80
    2. $80
    3. answers will vary

    Exercise 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:

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

    Exercise 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.

    1. Multiply 12(1.99) by using the distributive property to multiply 12(2−0.01).
    2. Was it a bargain to buy “Two Buck Chuck” by the case?
    Answer
    1. $23.88
    2. no, the price is the same

    Exercise 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.

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

    Writing Exercises

    Exercise 99

    In your own words, state the commutative property of addition.

    Answer

    \(Answers may vary\)

    Exercise 100

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

    Exercise 101

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

    Answer

    \(Answers may vary\)

    Exercise 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.

    Self Check

    ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “use the commutative and associative properties,” “use the identity and inverse properties of addition and multiplication,” “use the properties of zero,” and “simplify expressions using the distributive property.” The rest of the cells are blank.

    ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

    • Was this article helpful?