SECTION 2.5 Dividing Integers & Order of Operations

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UNIT TWO: Prealgebra in a Technical World
2.5 Dividing Integers & Order of Operations
SWBAT
1. Divide integers.
2. Write exact quotients using mixed numbers.
3. Use order of operations to simplify integer expressions.
4. Solve applied problems using integers.
Dividing Integers
Once people had started multiplying, they developed an operation for finding a missing
factor. We call that operation division. Another name for the missing factor is the quotient.
The quotient is the result when we divide one number by another.
3
For instance, we know that 2) 6 , because 2 ∙ 3 = 6.
What about −2) − 6 ? Since (−2) + (−2) + (−2) = −6, the answer is 3. So, 3 is the
missing factor in the multiplication (−2) ∙ ? = −6. The good news is that when the signs are
the same, just as in multiplication, the answer in division is positive.
RULE: To divide two numbers with the same sign, divide their absolute values.
The quotient is positive.
What about division problems with different signs? To divide 2) − 6 we check for the
number that makes this multiplication true: 2 ∙ ? = −6. In this equation, the missing factor
must be −3. Similarly, when we divide −2) 6 , we are looking for the missing factor that
makes (−2) ∙ ? = 6 true and that number is −3.
The rule of signs for division matches the rule for multiplication:
RULE: To divide two numbers with different signs, divide their absolute values.
The quotient is negative.
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SECTION 2.5 Dividing Integers & Order of Operations
If we use (+) to represent any positive number and (−) to represent any negative number,
then:
Multiplication
Division
(+) ∙ (+) = (+)
(+) ÷ (+) = (+)
(−) ∙ (−) = (+)
(−) ÷ (−) = (+)
(+) ∙ (−) = (−)
(+) ÷ (−) = (−)
(−) ∙ (+) = (−)
(−) ÷ (+) = (−)
 Check Point 1
Tell whether the quotient is positive (+) or negative(−), then divide.
Division
Sign
Quotient
Division
a.
𝟏𝟒 ÷ (−𝟐)
d. (−𝟓𝟒) ÷ (−𝟔)
b.
(−𝟒𝟎) ÷ 𝟖
e.
c.
(−𝟑𝟐) ÷ (−𝟖)
Sign
Quotient
𝟐𝟕 ÷ (−𝟗)
f. (−𝟖𝟏) ÷ (−𝟗)
RULE: 𝟎 ÷ 𝒂 = 𝟎, and 𝒂 ÷ 𝟎 is undefined.
Dividing into 0 is different from dividing by 0. We can divide into 0, 0 ÷ 4 = 0, because 4 ∙ 0 =
0. (If I have no cookies and four people, each person can have 0 cookies.)
However, we cannot divide by 0, 4 ÷ 0 = ____ does not have an answer. The
missing factor in 0 · ? = 4 , does not exist. We say that dividing by zero is
“undefined.”
Order of Operations
The order of operations is used for all real numbers. In this section we practice both
using the correct order of operations and simplifying integer expressions.
UNIT TWO: Prealgebra in a Technical World
Example 1: 5 + (−4)2 + 21 ÷ (−15 + 8)
Think it through:
5 + (−4)2 + 21 ÷ (−15 + 8) Parentheses first
5 + (−4)2 + 21 ÷ (−7)
Exponents
5 + 16 + 21 ÷ (−7)
Multiplication and Division left to right
5 + 16 + −3
Addition and Subtraction left to right
21 + −3
Addition (and we subtract to find the sum!)
ANSWER: 𝟓 + (−𝟒)𝟐 + 𝟐𝟏 ÷ (−𝟏𝟓 + 𝟖) = 𝟏𝟖
Example 2: Simplify 3 + (−6)2 ÷ 22 ∙ 7.
Think it through:
3 + (−6)2 ÷ 22 ∙ 7
3 + 36 ÷ 4 ∙ 7
Exponents left to right (when possible,
simplify all)
Multiplication and Division left to right
3+9∙7
Multiplication and Division left to right
3 + 63
Addition and Subtraction left to right
ANSWER: 𝟑 + (−𝟔)𝟐 ÷ 𝟐𝟐 ∙ 𝟕 = 𝟔𝟔
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SECTION 2.5 Dividing Integers & Order of Operations
Example 3: Simplify
−2+(5−1)(−3+7)
9−2
.
Think it through:
Hint: The fraction bar is treated like
parentheses. Simplify the numerator and
denominator first, and then divide.
−2+(𝟓−𝟏)(−𝟑+𝟕)
Groupings first (when possible, simplify all)
9−2
−2+(4)(4)
Multiply and Divide left to right
9−2
−2+𝟏𝟔
Add and Subtract (when possible, simplify all)
9−2
14
Divide
7
ANSWER:
−2+(5−1)(−3+7)
9−2
 Check Point 2
Simplify −6 − (−14 ÷ 7) ∙ 32
=
𝟐
UNIT TWO: Prealgebra in a Technical World
 Check Point 3
Jasmine and Joanie both simplified the expression 3 − 36 ÷ (−9 + 13) ∙ (−3)2 + 1. Jasmine’s
work is shown below. Joanie’s answer is −77, and she checked her work. Jasmine has made a
mistake. Tell what the error is and on which row Jasmine made the error.
3 − 36 ÷ (−9 + 13) ∙ (−3)2 + 1 (This is the original problem.)
(1)
3 − 36 ÷ (4) ∙ (−3)2 + 1
(2)
3 − 36 ÷ (4) ∙ 9 + 1
(3)
3 − 36 ÷ 36 + 1
(4)
(5)
3−1 +1
3 (This is Jasmine’s answer.)
a. The error occurs in Row __________.
b. Jasmine's error is __________________________________________________________
____________________________________________________________________________
____________________________________________________________________________
Writing Exact Quotients Using Mixed Numbers
The rules for signs are the same when the exact quotient in our division problem is
written as a mixed number (as an integer and a fraction). When we divide exactly, we leave no
remainders. Instead we write a fraction to complete the division.
STUDY SKILLS: If you have forgotten how to do long division, now is the time to
review. Ask your instructor for extra problems, with solutions. If you need
instructions or tutoring, use the Tutoring Center and the Web!
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SECTION 2.5 Dividing Integers & Order of Operations
Example 4 Calculate −51 ÷ 8 exactly.
Think it through:
If you have forgotten how to write exact quotients using fractions, or you have
forgotten how to use the long division algorithm, practice many of these problems until you can
divide automatically. You can find many more practice problems, with answers so that you can
check your work, at the class Web site.
Order of Operations & Applied Problems
In the last section we estimated the Fahrenheit
temperature when given temperature in degrees
Celsius. That formula is written 𝐹 ≈ 2𝐶 + 30 or 2𝐶 +
30 ≈ 𝐹 . In the table at right we have used the formula
Celsius
0°
20°
40°
to convert several Celsius readings to estimated Fahrenheit degrees.
We can put an order to the operations in this formula:
2𝐶
+ 30 ≈
𝐹
degrees Celsius → times 2 → plus 30 ≈ degrees Fahrenheit
𝟐𝑪 + 𝟑𝟎 :
2(0) + 30
2(20) + 30
2(40) + 30
Fahrenheit
≈ 30°
≈ 70°
≈ 110°
UNIT TWO: Prealgebra in a Technical World
If we reverse this order, and perform the opposite operation, we create a formula that
reverses the process and estimates the Celsius temperature when given the temperature in
degrees Fahrenheit.
degrees Fahrenheit → minus 30 →divided by 2 ≈ degrees Celsius
(𝐹 − 30)
÷ 2 ≈
𝐶
In the new table, we use this new formula to
estimate Celsius temperature when given
temperature in degrees Fahrenheit. In algebra you
will study this process of finding different formulas,
Fahrenheit
30°
70°
110°
(𝑭 − 𝟑𝟎) ÷ 𝟐 ≈ 𝑪
(30 − 30) ÷ 2
(70 − 30) ÷ 2
(110 − 30) ÷ 2
Celsius
≈ 0°
≈ 20°
≈ 40°
but here we will simply use our new formula: 𝐶 ≈ (𝐹 − 30) ÷ 2.
Example 5: Today a child dies every 15 seconds because of a water-related disease1, yet water
can easily be pasteurized. All of the viruses, bacteria, protozoa and parasites can be killed by
bringing water to just 150°F. Most of the developing world uses Celsius temperature readings.
What is the approximate temperature needed to pasteurize water in degrees Celsius?
Understand: Find the degrees Celsius for 150℉.
Plan:
Use the formula 𝐶 ≈ (𝐹 − 30) ÷ 2 with 𝐹 = 150
Solve:
Substitute 150 for 𝐹 and simplify result: 𝐶 ≈ (150 − 30) ÷ 2 = 120 ÷ 2 = 60.
Check:
Approximate temperature needed to pasteurize water is 𝟔𝟎°𝑪.
Compared with result in the table on the last page, 60°𝐶 is reasonable.
The calculations are most probably accurate.
1

Water. (2009) Web site http://water.org/facts Accessed 1 August 2009.
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SECTION 2.5 Dividing Integers & Order of Operations
 Check Point 4
A greenhouse is to be maintained between 74℉ and 78℉ to attain the best germination rate
for tomato seeds. The greenhouse venting system was made in Canada and must be set using
degrees Celsius. What is a good approximation for a Celsius temperature setting for tomato
seed germination that will make sure the temperature is within the given Fahrenheit range?
______________________________________________________________________________
______________________________________________________________________________
Sentence:______________________________________________________________________
______________________________________________________________________________
 Check Point 5
Chloe is in the National Guard and is stationed in Bagdad, Iraq. While Armed Forces Radio gives
the daily temperature in degrees Fahrenheit, Chloe reads the thermometer in her barracks.
The lowest reading she has ever seen is -2°C and the highest reading she has seen is 46°C.
What are these two readings approximately equal to in degrees Fahrenheit? ________________
______________________________________________________________________________
______________________________________________________________________________
Sentence:______________________________________________________________________
______________________________________________________________________________
UNIT TWO: Prealgebra in a Technical World
2.5 Exercise Set
Name _______________________________
Skills
Determine the sign of each quotient first, then divide. Remember, division by 0 is undefined.
1. 24 ÷ (−6)
2. −32 ÷ (−16)
3. −99 ÷ 11
4. −5 ÷ (−5)
5. −39 ÷ (−13)
6. −144 ÷ 12
7. −1 ÷ 0
8. 63 ÷ 0
9.
0
−22
10.
14
−7
11.
15
−1
12.
13.
21
−7
14.
−60
12
15.
99
−11
16. − 42
−3
18.
−56
−7
19.
63
−3
20.
−19
−1
17.
256
0
1
−1
21.
5
−5
22.
0
−34
23.
72
−9
24.
−45
−9
25.
−169
−13
26.
242
−121
27.
−18
0
28.
0
−412
Simplify using order of operation:
29. 2 − 3 ∙ 4
30.
7 − 5 ∙ (2)
31. −5 − (−2)(4)
32. −4 − (4)(3)
33.
2 − 3 ∙ 4 + 32
34. −5[(−2 + (−4)]
35. 3 − 5 ∙ 7 + 42
36.
−4[(−1) + (−5)]
37. (5 − 8)(3 − 9)
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SECTION 2.5 Dividing Integers & Order of Operations
38. (6 − 8)(2 − 3))
39.
32 (7) − 4
40. 23 (−5) − 6
41. −(4 − 6)(2)
42.
−4(6) + 24 ÷ 3
43. −(4 − 6)(−5)
44.
6(−7) − 66 ÷ (−6)
45.
12 − 62 ÷ 3(22 )
46. 2 − 102 ÷ 5(−4)
47.
−3 − 42 (4 − 8) ÷ 2
48.
4 − |4 − 8|
49. 20 ÷ 22 − 4(−5)
51.
|23 − 42 | − 92
52. 9 − |4 − 7|3
50. 3 − |8 − 9|
Fraction bars are grouping symbols. Work these carefully, step-by-step, using correct order of
operations.
53.
2 ∙ 43 − 8 ∙ 16
=
43 − 24
54. 3 − 4(5) ÷ 2 + 1
=
7 − 42
UNIT TWO: Prealgebra in a Technical World
Applications UPS
55.
What would the approximate Celsius temperature be if the Fahrenheit temperature is:
a. 10℉ ≈
b.
−6℉ ≈
c. 2℉ ≈
d.
30℉ ≈
56. What would the approximate Fahrenheit temperature be if the Celsius temperature is:
a. −5℃ ≈
b.
−28℃ ≈
c. 30℃ ≈
d.
45℃ ≈
57.
Olaf, who lived in Florida, went to visit his cousin in Denmark. He had a hard time
adjusting to the cool weather there. When he checked the temperature gauge, it read
16℃ . If his gauge in Florida was measuring the temperature, what would the
approximate temperature be in ℉?
Sentence: _______________________________________________________________
58.
Sarah has credit card slips for items she purchased this week: $15.95, $22.49, $49.24, and
5 lunch receipts each for $5.99. If her weekly paycheck was $234.79 and she had to pay
her brother back $150 that she borrowed from him, will she be able to pay off her credit
card?
Sentence: _______________________________________________________________
59.
Dirk's monthly profits at his men's store for the quarter were:
What was Dirk's average profit (or loss for the quarter)?
January: −$3,935
February: −$1,412
March: −$5,212
Sentence: _______________________________________________________________________
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SECTION 2.5 Dividing Integers & Order of Operations
60.
Susan is working in Excel, and enters $124 in Cell A1, but she doesn't have a dollar
amount for Cell B1, so she enters 0. Then she writes a formula in Cell C1 that says to
divide the number in A1 by the number in B1. Excel does not show a number in Cell C1.
Instead, Excel reads,"#DIV?0." Why did Excel give Susan this answer?
Sentence:_________________________________________________________________
61.
Bubba enters the following problem into his calculator: 200 ÷ 50. His calculator gives
him the answer "Error." Can you tell Bubba what he probably entered incorrectly?
Sentence:_________________________________________________________________
Review and Extend
In order to master order of operations, practice problems that have many steps. You will need
to copy these carefully to your own paper and work the following step by step.
62.
8 − 2 ∙ 53 − 6(37 − 102 )
−2(−3 + 5)2 ∙ (25 − 3 ∙ 5)
63. 9 − |32 − 52 | + (32 − 5)2
−|32 − 42 | + (−2)2
64.
(4 − 6)2 − (9 − 15)
10 − |2 ∙ 7 − 42 |
− 4(
)
(500 − 30 ÷ 6)(−5) + 3 ∙ (−7)
5 − 32
The goal of this chapter is to learn the laws of signs. Before calculating, you should always
know the sign of your answer. Without simplifying, write P for positive, N for negative to show
the results for the following problems.
65. −12 − 4
66. −14/7
67. −2(−4)(−7)
68.
54 + (−56)
69. −15 + (−12)
70. 5 − (−2)
71. 19 − 30
72.
(3)(−3) ÷ 3
73. (−12) ÷ (−2)
74. 4(−12)
75. −3(−4)3
76.
9 + (−3)
77. −5 + 4
78. (−3)(−20)
79. 4(−3)
−12
80.
−14 − (−15)
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