Mixed Arithmetic Practice: 6th Grade Comprehensive Guide

Master mixed arithmetic operations: step-by-step methods, definitions, and practical applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Order of Operations
Exercise 1
Solve: 24 ÷ 4 + 3 × (8 - 5) - 2²
Definition:

Order of operations: The sequence in which mathematical operations should be performed: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

PEMDAS method:
  1. Parentheses: Solve expressions in parentheses first
  2. Exponents: Calculate powers and roots
  3. Multiplication and Division: From left to right
  4. Addition and Subtraction: From left to right
24 ÷ 4 + 3 × (8 - 5) - 2²
→
24 ÷ 4 + 3 × 3 - 4
→
6 + 9 - 4
→
11
Parentheses
8-5=3
Exponent
2²=4
MD left-right
24÷4=6, 3×3=9
AS left-right
6+9-4=11
Step 1: Solve parentheses first

8 - 5 = 3, so expression becomes: 24 ÷ 4 + 3 × 3 - 2²

Step 2: Calculate exponents

2² = 4, so expression becomes: 24 ÷ 4 + 3 × 3 - 4

Step 3: Perform multiplication and division (left to right)

24 ÷ 4 = 6, then 3 × 3 = 9, so expression becomes: 6 + 9 - 4

Step 4: Perform addition and subtraction (left to right)

6 + 9 = 15, then 15 - 4 = 11

Result: 11
Final answer:

24 ÷ 4 + 3 × (8 - 5) - 2² = 11

Applied rules:

• PEMDAS: Order of operations sequence

• Left-to-right: For operations of equal precedence

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

2 Mixed Operations
Exercise 2
Calculate: (15 + 9) × 2 - 48 ÷ 6 + 7
Definition:

Mixed operations: Mathematical expressions containing multiple types of arithmetic operations that must be solved following the order of operations.

(15 + 9) × 2 - 48 ÷ 6 + 7
→
24 × 2 - 48 ÷ 6 + 7
→
48 - 8 + 7
→
47
Parentheses
15+9=24
MD left-right
24×2=48, 48÷6=8
AS left-right
48-8+7=47
Step 1: Solve parentheses first

15 + 9 = 24, so expression becomes: 24 × 2 - 48 ÷ 6 + 7

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

24 × 2 = 48, then 48 ÷ 6 = 8, so expression becomes: 48 - 8 + 7

Step 3: Perform addition and subtraction (left to right)

48 - 8 = 40, then 40 + 7 = 47

Step 4: Verify the result

Check: (15 + 9) × 2 - 48 ÷ 6 + 7 = 24 × 2 - 8 + 7 = 48 - 8 + 7 = 47 ✓

Result: 47
Final answer:

(15 + 9) × 2 - 48 ÷ 6 + 7 = 47

Applied rules:

• PEMDAS: Parentheses first, then multiplication/division, then addition/subtraction

• Left-to-right: Within same precedence operations

• Sequential processing: Complete each operation level before moving on

3 Complex Expression
Exercise 3
Evaluate: 36 ÷ (2 + 4) × 3 - 5² + 8 × 2
Definition:

Complex expression: A mathematical statement containing multiple operations, parentheses, and exponents requiring careful application of the order of operations.

36 ÷ (2 + 4) × 3 - 5² + 8 × 2
→
36 ÷ 6 × 3 - 25 + 8 × 2
→
6 × 3 - 25 + 16
→
18 - 25 + 16
→
9
Parentheses
2+4=6
Exponent
5²=25
MD left-right
36÷6=6, 6×3=18, 8×2=16
AS left-right
18-25+16=9
Step 1: Solve parentheses first

2 + 4 = 6, so expression becomes: 36 ÷ 6 × 3 - 5² + 8 × 2

Step 2: Calculate exponents

5² = 25, so expression becomes: 36 ÷ 6 × 3 - 25 + 8 × 2

Step 3: Perform multiplication and division (left to right)

36 ÷ 6 = 6, then 6 × 3 = 18, then 8 × 2 = 16, so expression becomes: 18 - 25 + 16

Step 4: Perform addition and subtraction (left to right)

18 - 25 = -7, then -7 + 16 = 9

Result: 9
Final answer:

36 ÷ (2 + 4) × 3 - 5² + 8 × 2 = 9

Applied rules:

• PEMDAS: Strict order of operations sequence

• Left-to-right: For operations of equal precedence

• Negative handling: Careful attention to signs in calculations

Key Rules and Methods for Mixed Arithmetic
P → E → MD → AS
PEMDAS Order
Parentheses
(), [], {}
Group operations
Exponents
aⁿ, √a
Powers & roots
Multiplication/Division
×, ÷
Same precedence
Key definitions:

Order of operations: The standardized sequence for solving mathematical expressions: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

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

Mixed operations: Mathematical expressions containing multiple types of arithmetic operations.

Precedence: The priority level of mathematical operations determining the order of execution.

Grouping symbols: Parentheses, brackets, and braces used to group operations that should be performed first.

Mixed arithmetic methodology:
  1. Scan the expression: Identify all operations and grouping symbols
  2. Apply PEMDAS: Follow the order of operations sequence
  3. Work left to right: For operations of equal precedence
  4. Check each step: Verify calculations as you proceed
  5. Verify final answer: Double-check the result
Tip 1: Always solve expressions inside parentheses first.
Tip 2: Calculate exponents before multiplication or division.
Tip 3: Perform multiplication and division from left to right.
Tip 4: Perform addition and subtraction from left to right.
Common errors: Ignoring order of operations, performing operations in wrong sequence, miscalculating exponents.
Success strategies: Following PEMDAS strictly, showing all work, double-checking calculations.
Essential arithmetic principles:

• Precedence hierarchy: Operations have different priority levels

• Left-to-right rule: Same precedence operations go left to right

• Grouping override: Parentheses override natural precedence

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

a ÷ b × c = (a ÷ b) × c
Division/Multiplication Left-to-Right
a + b - c = (a + b) - c
Addition/Subtraction Left-to-Right
Solution: Exercises 4 to 5
4 Nested Parentheses
Exercise 4
Solve: 48 ÷ [2 × (3 + 5)] + 6 × (7 - 4) - 2³
Definition:

Nested parentheses: Mathematical expressions with parentheses inside other parentheses, requiring solving from the innermost to outermost.

48 ÷ [2 × (3 + 5)] + 6 × (7 - 4) - 2³
→
48 ÷ [2 × 8] + 6 × 3 - 8
→
48 ÷ 16 + 18 - 8
→
3 + 18 - 8
→
13
Innermost
3+5=8, 7-4=3
Next level
2×8=16
Exponent
2³=8
MD left-right
48÷16=3, 6×3=18
AS left-right
3+18-8=13
Step 1: Solve innermost parentheses first

3 + 5 = 8 and 7 - 4 = 3, so expression becomes: 48 ÷ [2 × 8] + 6 × 3 - 2³

Step 2: Solve the next level of grouping

2 × 8 = 16, so expression becomes: 48 ÷ 16 + 6 × 3 - 2³

Step 3: Calculate exponents

2³ = 8, so expression becomes: 48 ÷ 16 + 6 × 3 - 8

Step 4: Perform multiplication and division (left to right)

48 ÷ 16 = 3, then 6 × 3 = 18, so expression becomes: 3 + 18 - 8

Step 5: Perform addition and subtraction (left to right)

3 + 18 = 21, then 21 - 8 = 13

Result: 13
Final answer:

48 ÷ [2 × (3 + 5)] + 6 × (7 - 4) - 2³ = 13

Applied rules:

• Nested parentheses: Solve from innermost to outermost

• PEMDAS: Maintain order of operations at each level

• Left-to-right: For operations of equal precedence

5 Multiple Exponents
Exercise 5
Calculate: (10 - 3)² × 2 + 48 ÷ 4 - 3³ + 5 × (6 - 2)
Definition:

Multiple exponents: Mathematical expressions containing more than one power operation, all of which must be calculated before multiplication/division.

(10 - 3)² × 2 + 48 ÷ 4 - 3³ + 5 × (6 - 2)
→
7² × 2 + 48 ÷ 4 - 27 + 5 × 4
→
49 × 2 + 12 - 27 + 20
→
98 + 12 - 27 + 20
→
103
Parentheses
10-3=7, 6-2=4
Exponents
7²=49, 3³=27
MD left-right
49×2=98, 48÷4=12, 5×4=20
AS left-right
98+12-27+20=103
Step 1: Solve parentheses first

10 - 3 = 7 and 6 - 2 = 4, so expression becomes: 7² × 2 + 48 ÷ 4 - 3³ + 5 × 4

Step 2: Calculate all exponents

7² = 49 and 3³ = 27, so expression becomes: 49 × 2 + 48 ÷ 4 - 27 + 5 × 4

Step 3: Perform multiplication and division (left to right)

49 × 2 = 98, then 48 ÷ 4 = 12, then 5 × 4 = 20, so expression becomes: 98 + 12 - 27 + 20

Step 4: Perform addition and subtraction (left to right)

98 + 12 = 110, then 110 - 27 = 83, then 83 + 20 = 103

Result: 103
Final answer:

(10 - 3)² × 2 + 48 ÷ 4 - 3³ + 5 × (6 - 2) = 103

Applied rules:

• PEMDAS: Parentheses first, then all exponents

• Left-to-right: For operations of equal precedence

• Sequential processing: Complete each operation level before moving on

Comprehensive Guide: Mixed Arithmetic Practice
(a + b) × c - d ÷ e + f = ?
Mixed Operations Structure
Key definitions:

Order of operations: The standardized sequence for solving mathematical expressions: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

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

Mixed operations: Mathematical expressions containing multiple types of arithmetic operations.

Precedence: The priority level of mathematical operations determining the order of execution.

Grouping symbols: Parentheses, brackets, and braces used to group operations that should be performed first.

Nested parentheses: Parentheses inside other parentheses, solved from innermost to outermost.

Left-to-right rule: For operations of equal precedence, perform them in order from left to right.

Complete mixed arithmetic methodology:
  1. Scan the expression: Identify all operations, parentheses, and exponents
  2. Solve parentheses: Start with innermost and work outward
  3. Calculate exponents: Evaluate all power and root operations
  4. Perform multiplication and division: From left to right
  5. Perform addition and subtraction: From left to right
  6. Verify the result: Double-check each step and final answer
Tip 1: Always work from the innermost parentheses outward in nested expressions.
Tip 2: Calculate all exponents before doing any multiplication or division.
Tip 3: Remember that multiplication and division have equal precedence and are performed left to right.
Tip 4: Similarly, addition and subtraction have equal precedence and are performed left to right.
Tip 5: Write out each step to avoid calculation errors and make it easier to check your work.
Common errors: Ignoring order of operations, performing operations in wrong sequence, miscalculating exponents, forgetting left-to-right rule for same precedence operations.
Success strategies: Following PEMDAS strictly, showing all work, double-checking calculations, practicing regularly.
Key concepts: Operation precedence, sequential processing, grouping symbols, left-to-right rule.
Essential arithmetic principles:

• Precedence hierarchy: Operations have different priority levels (P > E > MD > AS)

• Left-to-right rule: Same precedence operations go left to right

• Grouping override: Parentheses, brackets, and braces override natural precedence

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

• Verification: Always check your work by reviewing each step

• Consistency: Apply the same rules to all expressions regardless of complexity

PEMDAS = Parentheses → Exponents → Multiplication/Division → Addition/Subtraction
Order of Operations Sequence
a ÷ b × c = (a ÷ b) × c ≠ a ÷ (b × c)
Division/Multiplication Precedence
a + b - c = (a + b) - c ≠ a + (b - c)
Addition/Subtraction Precedence

Questions & Answers

Question: I always forget which comes first: multiplication or division in PEMDAS. Can you clarify?

Answer: Great question! In PEMDAS, multiplication and division have the same precedence level. They are NOT separate levels - they're grouped together as "MD".

This means you don't do all multiplication before all division. Instead, you perform multiplication and division from LEFT TO RIGHT as they appear in the expression.

For example:

  • In 12 ÷ 4 × 3, you do division first (12 ÷ 4 = 3), then multiplication (3 × 3 = 9)
  • In 8 × 6 ÷ 2, you do multiplication first (8 × 6 = 48), then division (48 ÷ 2 = 24)

The same rule applies to addition and subtraction (AS) - they have equal precedence and are performed left to right.

So remember: Parentheses, then Exponents, then Multiplication/DIVISION (left to right), then Addition/SUBTRACTION (left to right).

Question: How can I help my child practice mixed arithmetic effectively?

Answer: Effective mixed arithmetic practice requires a structured approach:

  1. Start simple: Begin with expressions containing only 2-3 operations
  2. Build gradually: Add more operations and complexity over time
  3. Use visual aids: Color-code different operation levels or use arrows to show order
  4. Practice regularly: Consistent short sessions are better than infrequent long ones
  5. Encourage showing work: Have your child write out each step to avoid errors
  6. Review mistakes: Go through incorrect answers to understand where the error occurred

Create a "PEMDAS" reference card for your child to use while practicing. Consider using online math games and worksheets that provide immediate feedback.

Make it engaging by timing simple problems or turning practice into a game. The key is consistent practice with gradual progression in difficulty.

Remind your child that mistakes are part of learning - the important thing is to understand why an answer was incorrect and how to fix it.

Question: What's the difference between brackets [] and parentheses () in order of operations?

Answer: Brackets [] and parentheses () serve the same mathematical purpose in order of operations - they indicate that the operations inside should be performed first. They have identical precedence.

The main difference is visual and organizational:

  • Parentheses () are used for the innermost grouping in most expressions
  • Brackets [] are often used for the outermost grouping when parentheses are already used inside
  • Braces {} are sometimes used for the outermost grouping in complex expressions

For example, in the expression 2 × [3 + (4 - 1)], you would solve the parentheses first: (4 - 1) = 3, then the brackets: [3 + 3] = 6, then the multiplication: 2 × 6 = 12.

The order of solving is based on the nesting level, not the type of symbol. Always work from the innermost grouping outward, regardless of whether it's parentheses, brackets, or braces.

Some regions or textbooks may have slight preferences, but mathematically, they all indicate that the enclosed operations take priority.