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Contemporary Mathematics

3.3 Order of Operations

Contemporary Mathematics3.3 Order of Operations

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index
A close-up view of a notebook show a mathematical calculation. A calculator and a pen are placed by the notebook.
Figure 3.19 Calculators may automatically apply order of operations to calculations. (credit: “Precision” by Leonid Mamchenkov/Flickr, CC BY 2.0)

After completing this module, you should be able to:

  1. Simplify expressions using order of operations.
  2. Simplify expressions using order of operations involving grouping symbols.

Calculates else sure someone be rules expect explicit we what that needs to need make, we to that them what be calculate to calculated first.

You probably read that sentence and couldn't make heads or tails of it. Seems like it might concern calculations, but maybe it concerns needs? You may even be attempting to unscramble the sentence as you read it, placing words in the order you might expect them to appear in. The reason that the sentence makes no sense is that the words don't follow the order you expect them to follow. Unscrambled, the sentence was intended to be “To be sure that someone else calculates when we expect them to calculate, we need rules that make explicit what needs to be calculated first.”

Similarly, when working with math expressions and equations, if we don't follow the rules for order of operations, arithmetic expressions make no sense. Just a simple expression would be problematic if we didn't have some rules to tell us what to calculate first. For instance, 4×22+3+524×22+3+52 can be calculated in many ways. You could get 5,184. Or, you could get 80. Or, 96. The issue is that without following a set of rules for calculation, the same expression will give various results. In case you are curious, using the appropriate order of operations, we find 4×22+3+52=444×22+3+52=44.

Simplify Expressions Using Order of Operations

The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS:

E Exponents
M/D Multiplication and division
A/S Addition and Subtraction

So, what does EMDAS tell us to do? In an equation, moving left to right, we begin by calculating all the exponents first. Once the exponents have been calculated, we again move left to right, calculating the multiplications and divisions, one at a time. Multiplication and division hold the same position in the ordering, so when you encounter one or the other at this step, do it. Once the multiplications and divisions have been calculated, we again move left to right, calculating the additions and subtractions, one at a time. Additions and subtractions hold the same position in the ordering, so when you encounter one or the other at this step, do it. (You may have previously learned the order of operations as PEMDAS, with parentheses first; we will add that aspect later on.) We’ll explore this as we work an example.

Example 3.43

Using Two Order of Operations

Calculate 214×13214×13.

Your Turn 3.43

1.
Calculate 43 + 18 × 15 .

Example 3.44

Using Two Order of Operations

Calculate 4×834×83.

Your Turn 3.44

1.
Calculate 60 2 / 50 .

Example 3.45

Using Three Order of Operations

Calculate 2+32×42+32×4.

Your Turn 3.45

1.
Calculate 3 × 6 3 18 .

Even if the expression being calculated gets more complicated, we perform the operations in the order: EMDAS.

Example 3.46

Using Eight Order of Operations

Correctly apply the order of operations to compute the following:

425×6/10×32+7×23425×6/10×32+7×23.

Your Turn 3.46

1.

Correctly apply the order of operations to compute the following:

13 + 3 4 × 8 / 6 × 5 2 3 × 4 .

Example 3.47

Using Six Order of Operations

Correctly apply the rules for the order of operations to accurately compute the following:

103×53/15+56/4103×53/15+56/4.

Your Turn 3.47

1.

Correctly apply the rules for the order of operations to accurately compute the following:

12 / ( 4 ) + 8 × 9 / 12 × 2 3 24 × 25 / 10 .

Example 3.48

Using Order of Operations

Correctly apply the rules for the order of operations to accurately compute the following: (8)/2×39×24/12+9×(4)2/23(8)/2×39×24/12+9×(4)2/23.

Your Turn 3.48

1.

Correctly apply the rules for the order of operations to accurately compute the following:

3 × 4 3 × 7 + 24 / 6 × 7 2 9 / 3 × 8 .

Using the Order of Operations Involving Grouping Symbols

We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below.

P Parentheses
E Exponents
M/D Multiplication and division (division is just the multiplication by the reciprocal)
A/S Addition and subtraction (subtraction is just the addition of the negative)

As said previously, parentheses indicate that some operation or operations will be performed outside the standard order of operation rules. For instance, perhaps you want to multiply 4 and 7 before squaring. To indicate that the multiplication happens before the exponent, the multiplication is placed inside parentheses: (4×7)2(4×7)2.

This means operations inside the parentheses take precedence, or happen before other operations. Now, the first step in calculating arithmetic expressions using the order of operations is to perform operations inside parentheses first. Inside the parentheses, you follow the order of operation rules EMDAS.

Example 3.49

Prioritizing Parentheses in the Order of Operations

Correctly apply the rules for the order of operations to accurately compute the following:

(103)×53(103)×53.

Your Turn 3.49

1.

Correctly apply the rules for the order of operations to accurately compute the following:

8 ( 25 2 2 ) / 7 .

Be aware that there can be more than one set of parentheses, and parentheses within parentheses. When one set of parentheses is inside another set, do the innermost set first, and then work outward.

Example 3.50

Working Innermost Parentheses in the Order of Operations

Correctly apply the rules for order of operations to accurately compute the following:

4+2×(32(2+5)2×4)/(3+8)4+2×(32(2+5)2×4)/(3+8).

Your Turn 3.50

1.

Correctly apply the rules for the order of operations to accurately compute the following:

( 8 6 ) 2 × 100 ( ( 48 / 6 3 ) 2 4 × 7 ) .

Check Your Understanding

12.
Which operation has highest precedence?
13.
Which is performed first, exponents or addition?
14.
Calculate 2 × 3 2 5 × 8 .
15.
What is used to indicate operations that should be performed out of order?
16.
Calculate ( 4 3 ) 2 + 27 × 8 2 ÷ 6 2 .

Section 3.3 Exercises

1 .
Which operations have the lowest precedence in order of operations?
2 .
If many operations have the same precedence in an expression, in what order should the operations be performed?
3 .
Which operations have the same precedence in order of operations?
4 .
After all operations in parentheses have been performed, which operations should be performed next?
For the following exercises, perform the indicated calculation.
5 .
4 5 × 6
6 .
34 10 × 6
7 .
18 + 8 × 13
8 .
40 + 12 × 17
9 .
50 ÷ 2 + 8
10 .
72 ÷ 6 + 18
11 .
13 + 4 × 3 2
12 .
45 6 × 7 3
13 .
6 2 × 5 13 × 9 2
14 .
14 2 × 8 5 × 2 4
15 .
450 ÷ 3 2 56 ÷ 2 2
16 .
1,000 ÷ 5 3 7 × 8 4
17 .
38 × 6 4 + 5 × 18 ÷ 10
18 .
15 × 7 + 23 6 × 40 ÷ 24
19 .
600 ÷ 12 × 5 + 40 6 2
20 .
2 × 12 3 ÷ 8 × 3 4 5
21 .
10 × 6 3 ÷ 2 × 5 + 3 2 240 ÷ 8 × 9
22 .
45 ÷ 15 × 6 + 7 2 4 × 15 × 18 ÷ 12 × 3
23 .
( 4 + 3 ) × 2
24 .
6 × ( 12 + 8 )
25 .
( 14 25 ) × 2
26 .
( 86 61 ) ÷ 5
27 .
( 3 + 2 ) 3
28 .
( 17 12 ) 4
29 .
( 45 60 ) 2
30 .
( 90 101 ) 3
31 .
( 3 × 4 ) 3 ÷ 8 + 5
32 .
( 6 9 ) 2 × 5 4
33 .
4 ( 6 + 3 ) × 5 + 11
34 .
12 + ( 13 6 ) × 8 130
35 .
15 × ( 4 1 ) 2 + 5 × ( 17 + 2 × 3 ) + 4
36 .
18 × ( 12 6 ) 4 8 × ( 30 14 × 16 ) 90
37 .
4 × ( 5 + 2 × ( 6 + 8 ) ) + 10 × 3 2
38 .
9 × ( 13 12 × ( 41 32 ) ) 8 × 45
39 .
21 3 × ( 5 × ( 2 + 6 ) 3 × ( 18 11 ) + 25 ) ÷ 2
40 .
48 6 × ( 10 × ( 18 + 12 ) 25 × ( 16 9 ) + 19 ) ÷ 4
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