Mixed Operations: Decimals - Sixth Grade
Complete Notes & Formulas
1. All Four Operations with Decimals
Quick Reference
| Operation | Key Rule | Example |
|---|---|---|
| Addition | Line up decimal points | 3.45 + 2.8 = 6.25 |
| Subtraction | Line up decimal points | 8.7 − 3.42 = 5.28 |
| Multiplication | Count decimal places | 2.5 × 3.4 = 8.5 |
| Division | Make divisor whole | 15.75 ÷ 3.5 = 4.5 |
Choosing the Right Operation
Addition: Total, sum, combined, altogether, in all
Subtraction: Difference, how much more, left, remaining, change
Multiplication: Per, each, times, product, at this rate
Division: Each, per group, split, share, average, distribute
2. Order of Operations (PEMDAS)
What is PEMDAS?
PEMDAS
P - Parentheses ( ) [ ] { }
E - Exponents (powers, roots)
M - Multiplication (left to right)
D - Division (left to right)
A - Addition (left to right)
S - Subtraction (left to right)
Memory Trick: "Please Excuse My Dear Aunt Sally"
Important: Multiplication/Division have EQUAL priority (left to right)
Important: Addition/Subtraction have EQUAL priority (left to right)
Example 1: Basic Expression
Problem: Evaluate 12.5 + 3.2 × 4
Step 1: Multiplication first (before addition)
3.2 × 4 = 12.8
Step 2: Now add
12.5 + 12.8 = 25.3
Answer: 25.3
Wrong: 12.5 + 3.2 = 15.7, then 15.7 × 4 = 62.8 ✗
Example 2: With Parentheses
Problem: Evaluate (8.4 − 2.1) × 3.5
Step 1: Parentheses first!
(8.4 − 2.1) = 6.3
Step 2: Now multiply
6.3 × 3.5 = 22.05
Answer: 22.05
Example 3: Multiple Operations
Problem: Evaluate 20 − 4.5 × 2 + 6.3 ÷ 3
Step 1: Multiplication and Division (left to right)
4.5 × 2 = 9
6.3 ÷ 3 = 2.1
Expression becomes: 20 − 9 + 2.1
Step 2: Addition and Subtraction (left to right)
20 − 9 = 11
11 + 2.1 = 13.1
Answer: 13.1
Example 4: Complex Expression
Problem: Evaluate 15.6 ÷ (2.5 + 1.3) − 2 × 0.5
Step 1: Parentheses
(2.5 + 1.3) = 3.8
Expression becomes: 15.6 ÷ 3.8 − 2 × 0.5
Step 2: Division and Multiplication
15.6 ÷ 3.8 ≈ 4.11
2 × 0.5 = 1
Expression becomes: 4.11 − 1
Step 3: Subtraction
4.11 − 1 = 3.11
Answer: 3.11
3. Word Problems: Mixed Operations
Steps to Solve
Step 1: Read carefully and identify what you need to find
Step 2: Look for keywords to determine operations
Step 3: Write the expression or equation
Step 4: Solve using PEMDAS
Step 5: Check if answer makes sense
Example 1: Shopping Problem
Problem: Emma buys 3 notebooks at $4.75 each and 2 pens at $1.25 each. How much does she spend in total?
Step 1: Find cost of notebooks
3 × $4.75 = $14.25
Step 2: Find cost of pens
2 × $1.25 = $2.50
Step 3: Add totals
$14.25 + $2.50 = $16.75
Answer: $16.75
Example 2: Change Problem
Problem: Tom pays with a $50 bill for 4 meals at $11.25 each. How much change does he receive?
Step 1: Total cost
4 × $11.25 = $45.00
Step 2: Calculate change
$50.00 − $45.00 = $5.00
Answer: $5.00
Example 3: Division & Subtraction
Problem: A 12.6-meter rope is cut into 3 equal pieces. One piece is then shortened by 0.8 meters. How long is the shortened piece?
Step 1: Length of each piece
12.6 ÷ 3 = 4.2 meters
Step 2: After shortening
4.2 − 0.8 = 3.4 meters
Answer: 3.4 meters
Example 4: All Four Operations
Problem: Sarah has $100. She buys 5 books at $8.50 each and shares the remaining money equally with her 2 siblings. How much does each person get?
Step 1: Cost of books
5 × $8.50 = $42.50
Step 2: Money remaining
$100.00 − $42.50 = $57.50
Step 3: Share equally (3 people total)
$57.50 ÷ 3 = $19.17 (rounded)
Answer: $19.17 each
4. Evaluate Numerical Expressions
Strategy: Work Step by Step
1. Identify all operations in the expression
2. Apply PEMDAS order
3. Simplify one step at a time
4. Show your work clearly
Example: 5.5 + 8.4 ÷ 2.1 − 3 × 0.5
Original: 5.5 + 8.4 ÷ 2.1 − 3 × 0.5
Step 1: Division
8.4 ÷ 2.1 = 4
→ 5.5 + 4 − 3 × 0.5
Step 2: Multiplication
3 × 0.5 = 1.5
→ 5.5 + 4 − 1.5
Step 3: Addition (left to right)
5.5 + 4 = 9.5
→ 9.5 − 1.5
Step 4: Subtraction
9.5 − 1.5 = 8
Answer: 8
5. Common Mistakes to Avoid
Mistake 1: Working Left to Right
Expression: 6 + 4 × 2
❌ Wrong: 6 + 4 = 10, then 10 × 2 = 20
✓ Correct: 4 × 2 = 8, then 6 + 8 = 14
Mistake 2: Misaligned Decimals in Addition/Subtraction
Problem: 12.5 + 3.45
❌ Wrong: Not lining up decimals
✓ Correct: Always line up decimal points vertically
Mistake 3: Wrong Decimal Placement in Multiplication
Problem: 2.5 × 3.4
❌ Wrong: Not counting total decimal places
✓ Correct: Count 1 + 1 = 2 decimal places → 8.5
6. Problem-Solving Strategies
For Expressions
✓ Circle parentheses first
✓ Underline multiplication/division
✓ Box addition/subtraction
✓ Work through one operation at a time
✓ Rewrite the expression after each step
For Word Problems
✓ Read the problem twice
✓ Highlight or underline numbers and keywords
✓ Identify what the problem is asking for
✓ Draw a picture if helpful
✓ Check if your answer makes sense
Quick Reference: PEMDAS Order
| Order | Operation | Example |
|---|---|---|
| 1st | Parentheses | (3.5 + 2.1) = 5.6 |
| 2nd | Exponents | 2.5² = 6.25 |
| 3rd | Multiply/Divide (L→R) | 4 × 2.5 ÷ 2 |
| 4th | Add/Subtract (L→R) | 10 + 5 − 3 |
💡 Important Tips to Remember
✓ PEMDAS applies to decimals just like whole numbers
✓ Parentheses ALWAYS first - no exceptions!
✓ Multiply/Divide have equal priority - work left to right
✓ Add/Subtract have equal priority - work left to right
✓ Show your work step by step
✓ Rewrite the expression after each step
✓ In word problems, identify the question first
✓ Check reasonableness - does the answer make sense?
✓ For money, always use 2 decimal places
✓ Practice makes perfect - do many problems!
🧠 Memory Tricks & Strategies
PEMDAS Phrase:
"Please Excuse My Dear Aunt Sally"
Order of Operations:
"Parentheses are the boss, exponents next in line, multiply and divide together, add and subtract to finish fine!"
Word Problem Strategy:
"Read, Identify, Plan, Solve, Check" (RIPSC)
Equal Priority Reminder:
"MD buddies work together, AS pals stick together!"
(Multiply/Divide go left to right, Add/Subtract go left to right)
Decimal Operations:
"Same rules for decimals, don't get confused, PEMDAS works the same, just carefully used!"
Master Mixed Decimal Operations! ➕ ➖ ✖️ ➗ 🔢
Remember PEMDAS and practice every day!