PEER TUTOR SLIDES BOOTCAMP ORDER OF OPERATIONS

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COURSE: JUST 3900
INTRODUCTORY STATISTICS
FOR CRIMINAL JUSTICE
Boot camp: Order of Operations
Peer Tutor Slides
Instructor:
Mr. Ethan W. Cooper, Lead Tutor
© 2013 - - PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE
WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS,
PLEASE EMAIL MR. COOPER AT THE FOLLWING: [email protected]
Order of Operations
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Many equations only contain one mathematical equation.
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However, most of the equations in JUST 3900 contain
two or more operations.
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Example: 2 + 2
Example: 22 + 4 ∗ 32 /2
2
In such equations it is important to perform the
operations in the proper order. Simply working from left
to right will not yield the correct answer.
LEFT TO RIGHT: WRONG!!!
ORDER OF OPERATIONS: CORRECT!!!
Order of Operations
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Operations should be performed in the proper order and
can be remembered by the following acronym:
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Please
Excuse
My
Dear
Silly
Aunt
Sally
(P)
(E)
(M)
(D)
(S)
(A)
(S)
Parentheses
Exponents
Multiplication
Division
Summation
Addition
Subtraction
This operation was not taught
in high school.
Order of Operations: Then and
Now
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High School
Please
()
Excuse
X2
My Dear (*) (÷)
Aunt Sally (+) (-)
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



JUST 3900
Please
()
Excuse
X2
My Dear (*) (÷)
Silly
Ʃ
Aunt Sally (+) (-)
Order of Operations
Left to Right
 22 + 4 ∗ 32 /2
 4 + 4 ∗ 32 /22
 8 ∗ 32 /22
 242 /22
 576/22
 2882
 82944
2
Order of Operations
 22 + 4 ∗ 32 /2 2
 22 + 4 ∗ 9/2 2
 22 + 36/2 2
 22 + 18 2
 4 + 324
 328
Order of Operations:
Parentheses
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Any operations set off by parentheses should be done
first.
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2∗ 3+4
2∗7
14
Sometimes there are multiple operations within a set off
parentheses. In these cases, we still follow the proper
Exponents
order.
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2 ∗ 32 + 4 ∗ 5
2∗ 9+4∗5
2 ∗ 9 + 20
2 ∗ 29
58
Multiplication
Addition
Order of Operations:
Parentheses
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Often times, you’ll come across equations written like
this:
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10+2
5+1
In such cases, we must solve the numerator, first and
the denominator, second.
10+2
5+1
12

5+1
12

6
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
2
It’s as if it were written:
10 + 2 / 5 + 1
We do what’s in parentheses,
first, working from left to right.
Order of Operations:
Exponents
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
After performing all operations written in parentheses,
we move on to exponents:
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22 + 4
4+4
8
Order of Operations:
Exponents
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
When there are multiple exponents within an equation,
complete them in order from left to right before moving
on to the next operation:
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22 + 43 + 54
4 + 43 + 54
4 + 64 + 54
4 + 64 + 625
68 + 625
693
Order of Operations:
Exponents
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
There are times when a set of parentheses are raised to
the nth power. In these cases, perform the operations in
parentheses first, working from left to right:
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22 + 43 − 5 + 3 2
4 + 43 − 5 + 3 2
4 + 43 − 5 + 3 2
4 + 64 − 5 + 3 2
68 − 5 + 3 2
68 − 8 2
68 − 64
4
Order of Operations:
Multiplication and Division
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
After parentheses and exponents, multiplication and
division are performed, in order, from left to right:
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2 ∗ 2 + 3/3
4 + 3/3
4+1
5
Here, we multiply, first.
Order of Operations:
Multiplication and Division
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Another example:
This time we divide, first.
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20
+
5
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4+6∗3+
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4 + 18 +
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4 + 18 + 3 − 10
22 + 3 − 10
25 − 10
15
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21
− 10
7
21
− 10
7
6∗3+
21
7
− 10
Remember two things:
1) Multiplication and division should be done
after parentheses and exponents.
2) Multiplication and division should be done,
in order, from left to right.
Order of Operations:
Multiplication and Division
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Put it all together:
25
18
+
5
3
25
18
+
5
3
18
5+
3
2
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5∗3
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152 −
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225 −
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225 − 5 + 6
225 − 11
214
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−
Order of Operations:
Summation
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Summation is an operation not usually covered in high
school algebra classes, but it will be important to know
for JUST 3900.
Summation involves summing a set of numbers.
Because we’re using data sets in statistics, summation
will become an important operation.
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Values for X: 1, 2, 3, 4, 5
 =1+2+3+4+5
 =3+3+4+5
 =6+4+5
 = 10 + 5
 = 15
Order of Operations:
Summation
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Using the same set of numbers:
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2
2
2
2
2
2
= 12 + 22 + 32 + 42 + 52
= 1 + 4 + 9 + 16 + 25
= 5 + 9 + 16 + 25
= 14 + 16 + 25
= 30 + 25
= 55
Order of Operations:
Summation
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
And again….
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2
2
2
2
2
2
= 1+2+3+4+5
= 3+3+4+5 2
= 6+4+5 2
= 10 + 5 2
= 15 2
= 225
2
Order of Operations:
Addition and Subtraction
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Please Excuse My Dear Silly Aunt Sally (PEMDSAS)
Just like multiplication and division, addition and
subtraction are performed in order from left to right.
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6+5−1
11 − 1
10
Or
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6−5+1
1+1
2
Order of Operations
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Let’s put it all together and solve the following equation:
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3 + 5 − 10 + 2
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8 − 10 + 2
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8 − 12
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2
2
6
3
−42 ∗ 5 + 2
−42 ∗ 7
16 ∗ 7
112
∗ 5+
∗ 5+
∗ 5+
−42 ∗ 5 +
2
6
3
6
3
6
3

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