Solving Problems Using Order of Operations: 5 Detailed Exercises

Master order of operations: parentheses, exponents, multiplication, division, addition, subtraction through these 5 comprehensive exercises with step-by-step solutions.

Exercise Solutions 1 to 3
1 Basic Addition and Subtraction
Exercise 1
Solve: 8 + 3 - 2
Definition:

Order of Operations: The sequence in which mathematical operations should be performed to get the correct answer.

Order of Operations Rule:

When operations are at the same level (addition and subtraction), perform them from left to right.

Step 1: Perform operations from left to right
Step 2: 8 + 3 = 11
Step 3: 11 - 2 = 9
Step 1: Perform addition first (left to right)

8 + 3 = 11

Step 2: Then perform subtraction

11 - 2 = 9

8 + 3 - 2 = 9
Final answer:

8 + 3 - 2 = 9

Applied rules:

• Left-to-right rule: When operations are the same level, work from left to right

• Sequential processing: Perform one operation at a time

• Accuracy: Check each step before proceeding

2 Multiplication and Division
Exercise 2
Solve: 12 ÷ 4 × 3
Definition:

Multiplication and Division: These operations have the same precedence level and are performed from left to right.

Step 1: Perform operations from left to right
Step 2: 12 ÷ 4 = 3
Step 3: 3 × 3 = 9
Step 1: Perform division first (left to right)

12 ÷ 4 = 3

Step 2: Then perform multiplication

3 × 3 = 9

12 ÷ 4 × 3 = 9
Final answer:

12 ÷ 4 × 3 = 9

Applied rules:

• Same precedence: Multiplication and division have equal priority

• Left-to-right rule: Work from left to right when operations have equal precedence

• Sequential processing: Complete each operation before moving to the next

3 With Parentheses
Exercise 3
Solve: (6 + 2) × 3
Definition:

Parentheses: Operations inside parentheses must be performed first.

Step 1: Solve operations inside parentheses first
Step 2: 6 + 2 = 8
Step 3: 8 × 3 = 24
Step 1: Solve what's inside the parentheses first

(6 + 2) = 8

Step 2: Then perform the multiplication

8 × 3 = 24

(6 + 2) × 3 = 24
Final answer:

(6 + 2) × 3 = 24

Applied rules:

• Parentheses first: Always solve operations inside parentheses before anything else

• Hierarchy: Parentheses have the highest precedence in order of operations

• Substitution: Replace parentheses with their calculated value

Order of Operations Rules and Methods
🧮 PEMDAS Rule
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
Key Definitions:

PEMDAS: Acronym for the order of operations (Please Excuse My Dear Aunt Sally)

Parentheses: Operations inside parentheses are done first

Exponents: Powers and roots are done next

Multiplication/Division: Done left to right, same precedence

Addition/Subtraction: Done left to right, same precedence

Order of Operations Sequence:
  1. Parentheses: Solve everything inside parentheses first
  2. Exponents: Calculate any powers or roots
  3. Multiplication/Division: Work from left to right
  4. Addition/Subtraction: Work from left to right
Tip 1: Always solve operations inside parentheses first.
Tip 2: Multiplication and division have equal priority - work left to right.
Tip 3: Addition and subtraction have equal priority - work left to right.
Tip 4: Rewrite the expression after each step to stay organized.
Common Errors: Performing operations in the wrong order, forgetting parentheses, mixing up left-to-right rule.
Key Skills: Following sequence, left-to-right processing, substitution, calculation.
Essential Rules to Remember:

• Parentheses first: Always solve what's inside parentheses before other operations

• Equal precedence: Multiplication/division are equal, addition/subtraction are equal

• Left to right: When operations have equal precedence, work from left to right

• Sequential processing: Complete each step before moving to the next

• Organization: Keep your work neat and organized to avoid mistakes

PEMDAS Order

🧮
1
P - Parentheses
2
E - Exponents
3
MD - Multiplication/Division (left to right)
4
AS - Addition/Subtraction (left to right)

Worksheet Preview

1. (8 + 4) ÷ 2 = _____

2. 15 - 6 ÷ 2 = _____

3. 3 × (5 - 2) = _____

Exercise Solutions 4 to 5
4 Mixed Operations
Exercise 4
Solve: 10 + 6 ÷ 2 - 3
Definition:

Mixed Operations: Problems containing multiple types of operations that must be solved following the order of operations.

Step 1: Division comes before addition and subtraction
Step 2: 6 ÷ 2 = 3
Step 3: Now solve left to right: 10 + 3 - 3
Step 4: 10 + 3 = 13
Step 5: 13 - 3 = 10
Step 1: Identify operations with higher precedence

Division (÷) comes before addition (+) and subtraction (-)

Step 2: Perform division first

6 ÷ 2 = 3

Step 3: Rewrite the expression

10 + 3 - 3

Step 4: Perform addition and subtraction from left to right

10 + 3 = 13, then 13 - 3 = 10

10 + 6 ÷ 2 - 3 = 10
Final answer:

10 + 6 ÷ 2 - 3 = 10

Applied rules:

• Precedence hierarchy: Division before addition/subtraction

• Left-to-right for equal precedence: Addition and subtraction from left to right

• Substitution: Replace calculated parts with their values

5 Complex Expression
Exercise 5
Solve: (15 - 5) × 2 + 8 ÷ 4
Definition:

Complex Expression: A mathematical expression containing multiple operations and parentheses.

Step 1: Solve operations inside parentheses first
Step 2: 15 - 5 = 10
Step 3: Next, perform multiplication and division (left to right)
Step 4: 10 × 2 = 20, then 8 ÷ 4 = 2
Step 5: Finally, perform addition
Step 6: 20 + 2 = 22
Step 1: Solve parentheses first

(15 - 5) = 10

Expression becomes: 10 × 2 + 8 ÷ 4

Step 2: Perform multiplication and division (left to right)

10 × 2 = 20

8 ÷ 4 = 2

Expression becomes: 20 + 2

Step 3: Perform addition

20 + 2 = 22

(15 - 5) × 2 + 8 ÷ 4 = 22
Final answer:

(15 - 5) × 2 + 8 ÷ 4 = 22

Applied rules:

• Complete hierarchy: Parentheses → Multiplication/Division → Addition/Subtraction

• Left-to-right for equal precedence: When operations have equal precedence

• Sequential substitution: Replace calculated parts with their values

Comprehensive Guide: Order of Operations
🧮 PEMDAS = Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
The Order of Operations Sequence
Key definitions:

Order of Operations: The sequence in which mathematical operations should be performed

PEMDAS: Memory device for remembering the order (Please Excuse My Dear Aunt Sally)

Precedence: The priority level of mathematical operations

Complete Order of Operations:
  1. Parentheses: Solve all operations inside parentheses first
  2. Exponents: Calculate any powers or roots
  3. Multiplication/Division: Perform from left to right (equal precedence)
  4. Addition/Subtraction: Perform from left to right (equal precedence)
Tip 1: Always start by identifying and solving parentheses first.
Tip 2: When multiplication and division appear together, work left to right.
Tip 3: When addition and subtraction appear together, work left to right.
Tip 4: Rewrite the expression after each major step to stay organized.
Tip 5: Check your work by substituting back or solving step by step.
Common errors: Performing operations in the wrong order, forgetting parentheses, not following left-to-right rule for equal precedence operations.
Key skills: Identifying operation precedence, sequential processing, left-to-right execution, substitution.
Essential formulas and rules:

• PEMDAS sequence: Parentheses → Exponents → Multiplication/Division → Addition/Subtraction

• Equal precedence rule: Multiplication and division have equal priority

• Left-to-right rule: For operations of equal precedence, work from left to right

• Parentheses first: Always solve what's inside parentheses before other operations

• Sequential substitution: Replace calculated parts with their values

Order of Operations Hierarchy

🧮
PEMDAS Steps
P
Parentheses
E
Exponents
MD
×/÷
AS
+/−
Visual Summary: Operation Precedence
Parentheses
Exponents
Multiplication/Division
Addition/Subtraction
Practice Makes Perfect! Remember PEMDAS for every problem.
Comprehensive Summary: Order of Operations
Key Definitions

Order of Operations: The sequence in which mathematical operations should be performed

PEMDAS: Acronym for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction

Precedence: The priority level of mathematical operations

Equal Precedence: Operations with the same priority level (like multiplication/division)

Left-to-Right Rule: For operations with equal precedence, work from left to right

Core Rules

• Parentheses first: Always solve operations inside parentheses first

• Exponents second: Calculate powers and roots next

• Multiplication/Division third: Equal precedence, left to right

• Addition/Subtraction fourth: Equal precedence, left to right

• Sequential processing: Complete each step before moving to the next

Step-by-Step Methods

1. Identify: Look for parentheses, exponents, multiplication/division, addition/subtraction

2. Sequence: Follow PEMDAS order (Parentheses → Exponents → MD → AS)

3. Left to Right: For equal precedence operations, work from left to right

4. Substitute: Replace calculated parts with their values

5. Continue: Repeat until the entire expression is solved

Examples

Simple: 8 + 3 - 2 = 9 (left to right)

With Multiplication: 12 ÷ 4 × 3 = 9 (left to right)

With Parentheses: (6 + 2) × 3 = 24 (parentheses first)

Mixed: 10 + 6 ÷ 2 - 3 = 10 (division first, then left to right)

Complex: (15 - 5) × 2 + 8 ÷ 4 = 22 (follow entire sequence)

Tips & Tricks

• Remember PEMDAS with "Please Excuse My Dear Aunt Sally"

• Circle operations in the order they should be performed

• Rewrite the expression after each major step

• Check your work by substituting back or solving step by step

• Practice with different types of expressions regularly

Key Takeaways

• Order of operations ensures consistent results in mathematical expressions

• Parentheses always come first in the sequence

• Multiplication and division have equal precedence

• Addition and subtraction have equal precedence

• Left-to-right rule applies to equal precedence operations

Questions & Answers

Question: Why do we have to follow the order of operations? Can't we just solve from left to right all the time?

Answer: Great question! We follow the order of operations to ensure everyone gets the same answer for the same problem. Look at this example:

If we have 5 + 3 × 2:

  • Following order of operations: 3 × 2 = 6, then 5 + 6 = 11
  • If we went left to right: 5 + 3 = 8, then 8 × 2 = 16

We'd get different answers! The order of operations creates a universal rule so everyone solves the same problem the same way. It's like having traffic rules - everyone needs to follow the same system to avoid chaos.

Think of it as a recipe - if you don't follow the steps in order, you won't get the right result!

Question: I get confused when multiplication and division are in the same problem. Which one do I do first?

Answer: Multiplication and division have the same priority level, so you work from left to right! The one that appears first in the expression is the one you do first.

For example:

  • In 12 ÷ 4 × 3: Division comes first (left to right), so 12 ÷ 4 = 3, then 3 × 3 = 9
  • In 8 × 6 ÷ 2: Multiplication comes first (left to right), so 8 × 6 = 48, then 48 ÷ 2 = 24

Remember: multiplication and division are "equals" in priority, so you just go in the order you see them from left to right. The same rule applies to addition and subtraction!

Practice tip: Circle the operations in the order you'll perform them to keep track.

Question: What if there are parentheses inside parentheses? What do I do then?

Answer: When you have parentheses inside parentheses, you work from the inside out - solve the innermost parentheses first!

For example, in 3 × (2 + (4 - 1)):

  1. First, solve the innermost parentheses: (4 - 1) = 3
  2. Now the expression is: 3 × (2 + 3)
  3. Next, solve the remaining parentheses: (2 + 3) = 5
  4. Finally: 3 × 5 = 15

Think of it like peeling an onion - start with the innermost layer and work your way out. The principle is still the same: parentheses come first, and when you have nested parentheses, the innermost ones have priority.

Some problems might use different types of brackets [ ] to make it clearer, but the rule is the same: innermost first!