Mixed Operations: Fractions - Sixth Grade
Complete Notes & Formulas
1. All Four Operations with Fractions
Quick Reference Chart
| Operation | Key Rule | Example |
|---|---|---|
| Addition | Find LCD, add numerators | 1/4 + 1/6 = 5/12 |
| Subtraction | Find LCD, subtract numerators | 5/6 − 1/4 = 7/12 |
| Multiplication | Multiply numerators & denominators | 2/3 × 4/5 = 8/15 |
| Division | Keep-Change-Flip (multiply by reciprocal) | 3/4 ÷ 2/5 = 1⅞ |
Summary of Formulas
Addition: a/b + c/d = (a×d + c×b)/(b×d) → simplify
Subtraction: a/b − c/d = (a×d − c×b)/(b×d) → simplify
Multiplication: a/b × c/d = (a×c)/(b×d) → simplify
Division: a/b ÷ c/d = a/b × d/c → simplify
2. Order of Operations with Fractions (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)
Key Point: PEMDAS works the same for fractions as for whole numbers!
Remember: Multiplication/Division have equal priority (left to right)
Remember: Addition/Subtraction have equal priority (left to right)
Steps for Fractions
Step 1: Convert all mixed numbers to improper fractions
Step 2: Simplify inside parentheses first
Step 3: Calculate any exponents
Step 4: Do multiplication and division (left to right)
Step 5: Do addition and subtraction (left to right)
Step 6: Simplify final answer
3. Evaluate Numerical Expressions
Example 1: Multiple Operations
Problem: Evaluate 1/2 + 1/3 × 1/4
Step 1: Identify operations (addition and multiplication)
Step 2: Multiplication FIRST (before addition)
1/3 × 1/4 = 1/12
Step 3: Now add
1/2 + 1/12
LCD = 12
6/12 + 1/12 = 7/12
Answer: 7/12
Example 2: With Parentheses
Problem: Evaluate (1/2 + 1/4) × 2/3
Step 1: Parentheses FIRST!
1/2 + 1/4 = 2/4 + 1/4 = 3/4
Step 2: Now multiply
3/4 × 2/3 = 6/12 = 1/2
Answer: 1/2
Example 3: Division and Subtraction
Problem: Evaluate 5/6 − 1/2 ÷ 1/3
Step 1: Division FIRST (before subtraction)
1/2 ÷ 1/3 = 1/2 × 3/1 = 3/2
Expression becomes: 5/6 − 3/2
Step 2: Now subtract
LCD = 6
5/6 − 9/6 = −4/6 = −2/3
Answer: −2/3
Example 4: Complex Expression
Problem: Evaluate 1/4 + 2/3 × 3/4 − 1/6
Original: 1/4 + 2/3 × 3/4 − 1/6
Step 1: Multiplication first
2/3 × 3/4 = 6/12 = 1/2
→ 1/4 + 1/2 − 1/6
Step 2: Addition and subtraction (left to right)
LCD = 12
3/12 + 6/12 − 2/12 = 7/12
Answer: 7/12
4. Choosing the Right Operation
Keywords Guide
Addition (+): Total, sum, combined, altogether, in all, more than
Subtraction (−): Difference, how much more, left, remaining, less than, fewer
Multiplication (×): Of, product, times, each, per, at this rate
Division (÷): Split, share equally, per, each, how many, quotient
Decision Flowchart
1. Read the problem carefully
2. Identify what you need to find
3. Look for keywords
4. Determine the operation
5. Solve and check if answer makes sense
5. Word Problems: All Operations
Example 1: Addition
Problem: Sarah painted 2/5 of a fence on Monday and 1/3 on Tuesday. What fraction of the fence did she paint in total?
Keyword: "in total" → Addition
2/5 + 1/3
LCD = 15
6/15 + 5/15 = 11/15
Answer: 11/15 of the fence
Example 2: Subtraction
Problem: A pizza is 7/8 full. If 1/2 is eaten, what fraction remains?
Keyword: "remains" → Subtraction
7/8 − 1/2 = 7/8 − 4/8 = 3/8
Answer: 3/8 of the pizza
Example 3: Multiplication
Problem: A recipe calls for 2/3 cup of sugar. If making 3/4 of the recipe, how much sugar is needed?
Keyword: "of" → Multiplication
3/4 of 2/3 = 3/4 × 2/3
= 6/12 = 1/2
Answer: 1/2 cup
Example 4: Division
Problem: A 3/4-pound bag of flour is divided equally into 6 containers. How much flour in each container?
Keyword: "divided equally" → Division
3/4 ÷ 6 = 3/4 × 1/6
= 3/24 = 1/8
Answer: 1/8 pound each
Example 5: Multi-Step Problem
Problem: Tom has 5/6 yard of ribbon. He uses 1/3 yard for one project and 1/4 yard for another. How much ribbon is left?
Step 1: Find total used: 1/3 + 1/4
LCD = 12 → 4/12 + 3/12 = 7/12
Step 2: Subtract from original: 5/6 − 7/12
10/12 − 7/12 = 3/12 = 1/4
Answer: 1/4 yard left
6. Common Mistakes to Avoid
Mistake 1: Ignoring Order of Operations
Expression: 1/2 + 1/4 × 2
❌ Wrong: 1/2 + 1/4 = 3/4, then 3/4 × 2 = 3/2
✓ Correct: 1/4 × 2 = 1/2, then 1/2 + 1/2 = 1
Mistake 2: Wrong Operation for "Of"
Problem: Find 1/2 of 3/4
❌ Wrong: 1/2 + 3/4 (addition)
✓ Correct: 1/2 × 3/4 = 3/8 (multiplication)
Mistake 3: Forgetting to Simplify
Expression: 2/3 × 3/4
❌ Wrong: 6/12 (not simplified)
✓ Correct: 6/12 = 1/2 (simplified)
7. Problem-Solving Strategy
For Expressions
✓ Convert mixed numbers to improper fractions
✓ Circle parentheses first
✓ Underline multiplication/division operations
✓ Work step by step following PEMDAS
✓ Simplify after each step
✓ Always simplify final answer
For Word Problems
✓ Read problem twice
✓ Highlight keywords
✓ Identify what you're finding
✓ Choose correct operation(s)
✓ Solve step by step
✓ Check if answer makes sense
Quick Reference: Operation Priority
| Order | Operation | With Fractions |
|---|---|---|
| 1st | Parentheses | (1/2 + 1/4) |
| 2nd | Exponents | (1/2)² |
| 3rd | Multiply/Divide (L→R) | 1/2 × 3/4 ÷ 2/3 |
| 4th | Add/Subtract (L→R) | 1/2 + 1/4 − 1/8 |
💡 Important Tips to Remember
✓ PEMDAS applies to fractions just like whole numbers
✓ Convert mixed numbers to improper fractions first
✓ Parentheses ALWAYS first - no exceptions
✓ Multiply/Divide before Add/Subtract
✓ Equal priority: M/D work left to right, A/S work left to right
✓ "Of" means multiply in fraction problems
✓ Always simplify your final answer
✓ Show your work step by step
✓ Check reasonableness - does answer make sense?
✓ Practice with different operations to master all skills
🧠 Memory Tricks & Strategies
PEMDAS Phrase:
"Please Excuse My Dear Aunt Sally"
Operation Chooser:
"Keywords are clues, choose operations you'll use!"
Addition/Subtraction:
"Find LCD, then you proceed!"
Multiplication:
"Multiply straight across, no common denominator loss!"
Division:
"Keep-Change-Flip, makes division a breeze to zip!"
Order Priority:
"Parentheses first, multiply/divide next, add/subtract last - that's the test!"
Master Mixed Fraction Operations! ➕ ➖ ✖️ ➗ 🎯
Remember PEMDAS and practice daily!