Elementary and Intermediate Algebra 5th

Alan S. Tussy, R. David Gustafson

Chapter 1

An Introduction to Algebra - all with Video Answers

Educators

Section 6

Multiplying and Dividing Real Numbers; Multiplication and Division Properties

Problem 1

Fill in the blanks.
The answer to a multiplication problem is called a_____. The answer to a division problem is called a _____.

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Charles C.

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Problem 2

Fill in the blanks.
The ______ property of multiplication states that changing the order when multiplying does not affect the answer.

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Charles C.

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Problem 3

Fill in the blanks.
The _____ property of multiplication states that changing the grouping when multiplying does not affect the answer.

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Charles C.

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Problem 4

Fill in the blanks.
Division of a nonzero number by 0 is _____.

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Charles C.

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Problem 5

Fill in the blanks.
a. The product or quotient of two numbers with like signs is _____.
b. The product or quotient of two numbers with unlike signs is _____.

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Problem 6

Fill in the blanks.
a. The product of an even number of negative numbers is _____.
b. The product of an odd number of negative numbers is _____.

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Problem 7

Fill in the blanks.
a. $\frac{-9}{3}=-3$ because _____ $\cdot$ _____ = _____
b. $\frac{0}{8}=0$ because _____ $\cdot$ _____ = _____

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Charles C.

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Problem 8

Fill in the blanks.
Complete each property of multiplication.
a. $a \cdot b=b$ $\cdot$ _____
b. $(a b) c=$ _____
c. $0 \cdot a=$ _____
d. $1 \cdot a=$ _____
e. $a\left(\frac{1}{a}\right)=$ _____
f. $-1 \cdot a=$ _____

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Charles C.

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Problem 9

Fill in the blanks.
Complete each property of division.
a. $\frac{a}{1}=$_____
b. $\frac{a}{a}=$_____
c. $\frac{0}{a}=$_____
d. $\frac{a}{0}$ is _____

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Problem 10

Which property justifies each statement?
a. $-5(2 \cdot 17)=(-5 \cdot 2) 17$
b. $-5\left(-\frac{1}{5}\right)=1$
c. $-5 \cdot 2=2(-5)$
d. $-5(1)=-5$
e. $-5 \cdot 0=0$

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Problem 11

For each multiplication or division, just determine the sign of the answer
a. $-\frac{19}{37}\left(-\frac{51}{75}\right)$
b. $\frac{45.568}{-2.56}$
c. $-8.2(-4.1)(-6)(-9.3)(-1.5)$

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Charles C.

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Problem 12

Use multiplication to check this division: $\quad \frac{-29.4}{7}=-4.1 .$ Is the answer correct?

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Problem 13

Complete each statement using the given property.
a.) 5 $\cdot$ 8= _____ Commutative property of multiplication
b.) -2(6 $\cdot$ 9)= _____ Associative property of multiplication
c.) 5(____)=1 Inverse property of multiplication
d.)_____ (-20)=-20 \quad Multiplication property of 1

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Problem 14

Complete the table.
$$
\begin{array}{|c|c|c|}
\hline & {\text { Opposite }} & {\text { Reciprocal }} \\
\hline \text { Number } & {\text { (additive inverse) }} & {\text { (multiplicative inverse) }} \\
\hline 2 & {} \\
\hline-\frac{4}{5} & {} \\
\hline 1.75 & {} \\
\hline
\end{array}
$$

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Problem 15

Let POS stand for a positive number and NEG stand for a negative number. Determine the sign of each result, if possible.
a. $\mathrm{POS} \cdot \mathrm{NEG}$
b. $\mathrm{POS}+\mathrm{NEG}$
c. $\mathrm{POS}-\mathrm{NEG}$
d. $\frac{\mathrm{POS}}{\mathrm{NEG}}$

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Problem 16

Let POS stand for a positive number and NEG stand for a negative number. Determine the sign of each result, if possible.
a. $\mathrm{NEG} \cdot \mathrm{NEG}$
b. $\mathrm{NEG}+\mathrm{NEG}$
c. $\mathrm{NEG}-\mathrm{NEG}$
d. $\frac{\mathrm{NEG}}{\mathrm{NEG}}$

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Problem 17

Write each sentence using symbols.
The product of negative four and negative five is twenty.

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Problem 18

Write each sentence using symbols.
The quotient of sixteen and negative eight is negative two.

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Problem 19

Multiply. See Example 1.
$$
4(-1)
$$

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Problem 20

Multiply. See Example 1.
$$
6(-1)
$$

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Charles C.

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Problem 21

Multiply. See Example 1.
$$
-2 \cdot 8
$$

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Charles C.

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Problem 22

Multiply. See Example 1.
$$
-3 \cdot 4
$$

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Charles C.

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Problem 23

Multiply. See Example 1.
$$
12(-5)
$$

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Charles C.

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Problem 24

Multiply. See Example 1.
$$
(-9)(11)
$$

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Charles C.

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Problem 25

Multiply. See Example 1.
$$
3(-22)
$$

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Charles C.

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Problem 26

Multiply. See Example 1.
$$
-8 \cdot 9
$$

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Charles C.

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Problem 27

Multiply. See Example 1.
$$
1.2(-0.4)
$$

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Charles C.

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Problem 28

Multiply. See Example 1.
$$
(-3.6)(0.9)
$$

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Charles C.

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Problem 29

Multiply. See Example 1.
$$
\frac{1}{3}\left(-\frac{3}{4}\right)
$$

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Charles C.

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Problem 30

Multiply. See Example 1.
$$
\left(-\frac{3}{4}\right)\left(\frac{4}{5}\right)
$$

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Charles C.

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Problem 31

Multiply. See Example 2.
$$
(-1)(-7)
$$

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Charles C.

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Problem 32

Multiply. See Example 2.
$$
(-2)(-5)
$$

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Charles C.

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Problem 33

Multiply. See Example 2.
$$
(-6)(-9)
$$

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Charles C.

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Problem 34

Multiply. See Example 2.
$$
(-8)(-7)
$$

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Charles C.

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Problem 35

Multiply. See Example 2.
$$
-3(-3)
$$

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Charles C.

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Problem 36

Multiply. See Example 2.
$$
-1(-1)
$$

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Charles C.

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Problem 37

Multiply. See Example 2.
$$
63(-7)
$$

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Charles C.

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Problem 38

Multiply. See Example 2.
$$
43(-6)
$$

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Charles C.

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Problem 39

Multiply. See Example 2.
$$
-0.6(-4)
$$

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Charles C.

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Problem 40

Multiply. See Example 2.
$$
-0.7(-8)
$$

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Charles C.

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Problem 41

Multiply. See Example 2.
$$
\left(-\frac{7}{8}\right)\left(-\frac{2}{21}\right)
$$

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Charles C.

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Problem 42

Multiply. See Example 2.
$$
\left(-\frac{5}{6}\right)\left(-\frac{2}{15}\right)
$$

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Charles C.

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Problem 43

Multiply. See Example 3.
$$
3.3(-4)(-5)
$$

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Charles C.

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Problem 44

Multiply. See Example 3.
$$
(-2.2)(-4)(-5)
$$

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Charles C.

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Problem 45

Multiply. See Example 3.
$$
-2(-3)(-4)(-5)(-6)
$$

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Charles C.

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Problem 46

Multiply. See Example 3.
$$
-9(-7)(-5)(-3)(-1)
$$

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Charles C.

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Problem 47

Multiply. See Example 3.
$$
(-41)(3)(-7)(-1)
$$

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Charles C.

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Problem 48

Multiply. See Example 3.
$$
56(-3)(-4)(-1)
$$

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Charles C.

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Problem 49

Multiply. See Example 3.
$$
(-6)(-6)(-6)
$$

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Charles C.

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Problem 50

Multiply. See Example 3.
$$
(-5)(-5)(-5)
$$

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Charles C.

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Problem 51

Find the reciprocal of each number. Then find the product of the given number and its reciprocal. See Example 4.
$$
\frac{7}{9}
$$

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Charles C.

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Problem 52

Find the reciprocal of each number. Then find the product of the given number and its reciprocal. See Example 4.
$$
-\frac{8}{9}
$$

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Charles C.

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Problem 53

Find the reciprocal of each number. Then find the product of the given number and its reciprocal. See Example 4.
$$
-13
$$

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Charles C.

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Problem 54

Find the reciprocal of each number. Then find the product of the given number and its reciprocal. See Example 4.
$$
\frac{1}{8}
$$

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Charles C.

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Problem 55

Divide. See Example 5.
$$
-30 \div(-3)
$$

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Charles C.

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Problem 56

Divide. See Example 5.
$$
-12 \div(-2)
$$

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Charles C.

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Problem 57

Divide. See Example 5.
$$
-6 \div(-2)
$$

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Charles C.

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Problem 58

Divide. See Example 5.
$$
-36 \div(-9)
$$

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Charles C.

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Problem 59

Divide. See Example 5.
$$
\frac{85}{-5}
$$

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Charles C.

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Problem 60

Divide. See Example 5.
$$
\frac{-84}{7}
$$

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Charles C.

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Problem 61

Divide. See Example 5.
$$
\frac{-110}{-110}
$$

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Charles C.

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Problem 62

Divide. See Example 5.
$$
\frac{-200}{-200}
$$

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Charles C.

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Problem 63

Divide. See Example 5.
$$
\frac{-10.8}{1.2}
$$

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Charles C.

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Problem 64

Divide. See Example 5.
$$
\frac{-13.5}{-1.5}
$$

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Charles C.

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Problem 65

Divide. See Example 5.
$$
\frac{0.5}{-100}
$$

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Charles C.

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Problem 66

Divide. See Example 5.
$$
\frac{-1.7}{10}
$$

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Charles C.

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Problem 67

Divide. See Example 5.
$$
-\frac{1}{3} \div \frac{4}{5}
$$

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Charles C.

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Problem 68

Divide. See Example 5.
$$
-\frac{2}{3} \div \frac{7}{8}
$$

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Charles C.

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Problem 69

Divide. See Example 5.
$$
-\frac{9}{16} \div\left(-\frac{3}{20}\right)
$$

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Charles C.

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Problem 70

Divide. See Example 5.
$$
-\frac{4}{5} \div\left(-\frac{8}{25}\right)
$$

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Charles C.

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Problem 71

Perform the operations.
$$
\frac{0}{150}
$$

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Problem 72

Perform the operations.
$$
\frac{0}{-12}
$$

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Charles C.

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Problem 73

Perform the operations.
$$
\frac{-17}{0}
$$

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Charles C.

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Problem 74

Perform the operations.
$$
\frac{225}{0}
$$

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Charles C.

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Problem 75

Perform the operations.
$$
\frac{24}{-6}
$$

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Charles C.

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Problem 76

Perform the operations.
$$
\frac{-78}{6}
$$

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Charles C.

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Problem 77

Perform the operations.
$$
\frac{17}{-17}
$$

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Charles C.

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Problem 78

Perform the operations.
$$
\frac{-24}{24}
$$

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Charles C.

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Problem 79

Perform the operations.
$$
(-2)(-2)(-2)(-2)
$$

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Charles C.

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Problem 80

Perform the operations.
$$
(-3)(-3)(-3)(-3)
$$

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Charles C.

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Problem 81

Perform the operations.
$$
-3(-4)(0)
$$

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Charles C.

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Problem 82

Perform the operations.
$$
15(0)(-22)
$$

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Charles C.

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Problem 83

Perform the operations.
$$
\frac{-23.5}{5}
$$

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Charles C.

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Problem 84

Perform the operations.
$$
\frac{-337.8}{6}
$$

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Charles C.

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Problem 85

Perform the operations.
$$
-5.2 \cdot 100
$$

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Charles C.

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Problem 86

Perform the operations.
$$
-1.17 \cdot 1,000
$$

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Charles C.

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Problem 87

Perform the operations.
$$
\frac{1}{2}\left(-\frac{1}{3}\right)\left(-\frac{1}{4}\right)
$$

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Charles C.

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Problem 88

Perform the operations.
$$
\frac{1}{3}\left(-\frac{1}{5}\right)\left(-\frac{1}{7}\right)
$$

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Charles C.

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Problem 89

Perform the operations.
$$
\frac{550}{-50}
$$

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Charles C.

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Problem 90

Perform the operations.
$$
\frac{440}{-20}
$$

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Charles C.

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Problem 91

Perform the operations.
$$
-3 \frac{3}{8} \div\left(-2 \frac{1}{4}\right)
$$

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Charles C.

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Problem 92

Perform the operations.
$$
-3 \frac{4}{15} \div\left(-2 \frac{1}{10}\right)
$$

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Charles C.

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Problem 93

Perform the operations.
$$
7.2(-2.1)(-2)
$$

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Charles C.

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Problem 94

Perform the operations.
$$
4.6(-5.4)(-2)
$$

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Charles C.

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Problem 95

Perform the operations.
$$
\frac{1}{2}\left(-\frac{3}{4}\right)
$$

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Charles C.

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Problem 96

Perform the operations.
$$
\frac{1}{3}\left(-\frac{5}{16}\right)
$$

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Charles C.

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Problem 97

Perform the operations.
$$
-\frac{16}{25} \div \frac{64}{15}
$$

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Charles C.

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Problem 98

Perform the operations.
$$
-\frac{15}{16} \div \frac{25}{8}
$$

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Charles C.

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Problem 99

Perform the operations.
$$
\frac{-24.24}{-0.8}
$$

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Charles C.

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Problem 100

Perform the operations.
$$
\frac{-55.02}{-0.7}
$$

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Charles C.

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Problem 101

Perform the operations.
$$
-1 \frac{1}{4}\left(-\frac{3}{4}\right)
$$

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Charles C.

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Problem 102

Perform the operations.
$$
-1 \frac{1}{8}\left(-\frac{3}{8}\right)
$$

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Charles C.

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Problem 103

Look Alikes . . .
a. $2.7+(-0.9)$
b. $2.7-(-0.9)$
c. $2.7(-0.9)$
d. $\frac{2.7}{-0.9}$

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Charles C.

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Problem 104

Look Alikes . . .
a. $-\frac{5}{3}+\left(-\frac{9}{25}\right)$
b. $-\frac{5}{3}-\left(-\frac{9}{25}\right)$
c. $-\frac{5}{3}\left(-\frac{9}{25}\right)$
d. $-\frac{5}{3} \div\left(-\frac{9}{25}\right)$

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Charles C.

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Problem 105

Use the associative property of multiplication to find each product.
$$
-\frac{1}{2}(2 \cdot 67)
$$

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Problem 106

Use the associative property of multiplication to find each product.
$$
\left(-\frac{5}{16} \cdot \frac{1}{7}\right) 7
$$

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Charles C.

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Problem 107

Use the associative property of multiplication to find each product.
$$
-0.2(-10 \cdot 3)
$$

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Problem 108

Use the associative property of multiplication to find each product.
$$
-1.5(-100 \cdot 4)
$$

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Problem 109

Over a 5 -year period, the value of a $\$ 200,000$ lot fell at a uniform rate to $\$ 160,000 .$ What signed number indicates the amount of depreciation per year?

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Problem 110

The ocean liner Queen Mary cost $\$ 22,500,000$ to build in $1936 .$ The ship was purchased by the city of Long Beach, California, in 1967 for $\$ 3,450,000 .$ It now serves as a convention center. What signed number indicates the annual average depreciation of the ship over the 31 -year period from 1936 to $1967 ?$ Round to the nearest dollar.

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Problem 111

In a lab, the temperature of a fluid was decreased $6^{\circ}$ per hour for 12 hours. What signed number indicates the change in temperature?

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Problem 112

A health care provider for a company estimates that 75 hours per week are lost by employees suffering from stress-related illness. In one year, how many hours are lost? Use a signed number to answer.

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Problem 113

As a result of a diet, Tom has been steadily losing $4 \frac{1}{2}$ pounds per month.
a. Which expression below can be used to determine how much heavier Tom was 8 months ago?
i. $-4 \frac{1}{2} \cdot 8$
ii. $-4 \frac{1}{2}(-8)$
iii. $4 \frac{1}{2}(-8)$
iv. $-4 \frac{1}{2}-8$
b. How much heavier was Tom 8 months ago?

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Charles C.

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Problem 114

The temperature on Pluto gets as low as $-386^{\circ} \mathrm{F}$ This is twice as low as the lowest temperature reached on Jupiter. What is the lowest temperature on Jupiter?

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Problem 115

The instructions on a container of antifreeze state, "A 50/50 mixture of antifreeze and water protects against freeze-ups down to $-34^{\circ} \mathrm{F},$ while a $60 / 40 \mathrm{mix}$ protects against freeze-ups down to one and one-half times that temperature." To what temperature does the $60 / 40$ mixture protect?

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Problem 116

For 2010 , the net income of Rite Aid Corporation (the drugstore chain) was about $-\$ 508$ million. The previous year, the company's net income was even worse, by a factor of about $5.75 .$ What signed number represents Rite Aid's net income in $2009 ?$

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Problem 117

In the 2009 income statement for Delta Air Lines below, numbers within parentheses represent a loss. Complete the statement given these facts. The second quarter loss was about 6.4 times the first quarter loss. The fourth quarter loss was about 5 times the second quarter loss. The third quarter loss was about $\frac{5}{16}$ of the fourth quarter loss.
CAN'T COPY THE CHART

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Charles C.

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Problem 118

The formula $=A 1^{*} B 1^{*} C 1$ in cell D1 of the spreadsheet instructs the computer to multiply the values in cells $\mathrm{Al}, \mathrm{Bl},$ and $\mathrm{Cl}$ and to print the result in place of the formula in cell D1. (The symbol " represents multiplication.)
What value will be printed in the cell D1? What values will be printed in cells $\mathrm{D} 2$ and $\mathrm{D} 3 ?$
CAN'T COPY THE CHART

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Charles C.

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Problem 119

Physics. An oscilloscope displays electrical signals as wavy lines on a screen. By switching the magnification dial to $\times 2$, for example, the height of the "peak" and the depth of the "valley" of a graph will be doubled. Use signed numbers to indicate the height and depth of the display for each setting of the dial.
a. normal
b. $\times 0.5$
c. $\times 1.5$
d. $\times 2$
CAN'T COPY THE IMAGE

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Charles C.

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Problem 120

Light. Water acts as a selective filter of light. In the illustration, we see that red light waves penetrate water only to
a depth of about 5 meters. How many times deeper does
a. yellow light penetrate than red light?
b. green light penetrate than orange light?
c. blue light penetrate than yellow light?
CAN'T COPY THE GRAPH

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Charles C.

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Problem 121

Explain why $\frac{16}{0}$ is undefined.

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Charles C.

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Problem 122

The commutative property states that changing the order when multiplying does not change the answer. Are the following activities commutative? Explain.
a. Washing a load of clothes; drying a load of clothes
b. Putting on your left sock; putting on your right sock

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Problem 123

What is wrong with the following statement?
A negative and a positive is a negative.

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Problem 124

If we multiply two different numbers and the answer is 0 , what must be true about one of the numbers? Explain your answer.

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Problem 125

$$
\text { Add: }-3+(-4)+(-5)+4+3
$$

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Charles C.

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Problem 126

Write $-3-(-5)$ as addition of the opposite.

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Problem 127

Find $\frac{1}{2}+\frac{1}{4}+\frac{1}{3}$. Answer in decimal form.

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Problem 128

Which integers have an absolute value equal to $45 ?$

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Problem 129

If the product of five numbers is negative, how many of them could be negative? Explain.

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Charles C.

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Problem 130

Suppose $a$ is a positive number and $b$ is a negative number. Determine whether the given expression is positive or negative.
a. $-a(-b)$
b. $\frac{-a}{b}$
c. $\frac{-a}{a}$
d. $\frac{1}{b}$

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