Mixed Operations | Fourth Grade

Complete Notes & Formulas

1. Add, Subtract, Multiply and Divide

Definition: Using all four basic operations together in mathematical expressions and following the correct order.

📐 Order of Operations (PEMDAS/BODMAS):

P - Parentheses/Brackets ( ): Do operations inside parentheses first
E - Exponents/Orders: Powers and roots (not common in 4th grade)
MD - Multiplication and Division: Left to right
AS - Addition and Subtraction: Left to right

✏️ Example:

Solve: 5 + (3 × 4) ÷ 2

Step 1: Parentheses first → 3 × 4 = 12

Expression becomes: 5 + 12 ÷ 2

Step 2: Division → 12 ÷ 2 = 6

Expression becomes: 5 + 6

Step 3: Addition → 5 + 6 = 11

💡 Remember:

Please Excuse My Dear Aunt Sally (PEMDAS)
Brackets, Orders, Division, Multiplication, Addition, Subtraction (BODMAS)

2. Mixed Operations Word Problems

Definition: Word problems requiring the use of multiple operations (addition, subtraction, multiplication, division) to find the solution.

📝 Steps to Solve Mixed Operation Problems:

Read the problem carefully and identify what you need to find
Identify all given numbers and what they represent
Look for keywords to determine operations needed
Decide the order of operations
Solve step by step
Check if answer makes sense

✏️ Example:

Problem: Sarah bought 3 packs of pencils with 8 pencils in each pack. She gave 12 pencils to her friend. How many pencils does she have left?

Solution:

Step 1: Find total pencils → 3 × 8 = 24 pencils

Step 2: Subtract given away → 24 - 12 = 12 pencils

Answer: 12 pencils left

3. Estimate Sums, Differences, Products & Quotients

Definition: Finding approximate answers for problems involving multiple operations by rounding first.

📝 Estimation Strategy:

Round each number to nearest 10, 100, or 1000
Perform operations with rounded numbers
Follow order of operations with estimates
Use ≈ (approximately equals) symbol

✏️ Example:

Problem: Estimate (48 × 9) + 127

Round: 48 ≈ 50, 9 stays 9, 127 ≈ 130

Estimate: (50 × 9) + 130

= 450 + 130

≈ 580

4. Multi-Step Word Problems

Definition: Problems requiring two or more operations to reach the final answer.

📋 Problem-Solving Steps:

Step 1: Read the entire problem carefully
Step 2: Identify what the question is asking
Step 3: Break the problem into smaller steps
Step 4: Solve each step in order
Step 5: Write the final answer with units

✏️ Example:

Problem: A bakery made 156 cookies. They packed them into boxes of 12. They sold 8 boxes. How many cookies are left?

Solution:

Step 1: Find cookies sold → 8 × 12 = 96 cookies

Step 2: Subtract from total → 156 - 96 = 60 cookies

Answer: 60 cookies left

5. Word Problems with Extra or Missing Information

Definition: Problems that include unnecessary information or lack essential information needed to solve.

📝 Two Types:

Type A: Extra Information

Strategy: Cross out information not needed to answer the question

Example: "Tom has 5 red marbles and 3 blue marbles. He bought 4 more red marbles. How many red marbles does he have now?"

Extra info: "3 blue marbles" (not needed)

Answer: 5 + 4 = 9 red marbles

Type B: Missing Information

Strategy: Identify what information is needed but not given

Example: "A movie ticket costs some money. John bought 3 tickets. How much did he pay?"

Missing info: Price of one ticket

Cannot solve without ticket price!

6. Guess-and-Check Problem Solving

Definition: A problem-solving strategy where you make an educated guess, check if it works, and adjust until you find the correct answer.

📋 Guess-and-Check Steps:

Step 1: Understand - Read problem and identify what to find
Step 2: Guess - Make a reasonable first guess
Step 3: Check - Test if your guess works
Step 4: Revise - If wrong, adjust guess (higher or lower)
Step 5: Repeat - Continue until answer is found

✏️ Example:

Problem: There are 20 animals (chickens and cows). They have 56 legs total. How many cows are there?

Solution using Guess-and-Check:

Guess 1: 10 cows → (10×4) + (10×2) = 40+20 = 60 legs ✗ (too many)

Guess 2: 8 cows → (8×4) + (12×2) = 32+24 = 56 legs ✓

Answer: 8 cows

💡 Tips for Smart Guessing:

Start with a middle value
Organize guesses in a table
Adjust systematically (if too high, guess lower)
Keep track of all guesses to see patterns

7. Choose Numbers with Particular Sum, Difference, Product or Quotient

Definition: Selecting specific numbers from a set that produce a target result when combined with a particular operation.

🎯 General Strategy:

Identify the target result
Determine which operation is needed
Look for number pairs/groups in the set
Test combinations systematically
Use inverse operations to check

📐 Quick Formulas:

Operation	To Find Numbers
Sum	Number 1 + Number 2 = Target
Difference	Larger - Smaller = Target
Product	Number 1 × Number 2 = Target
Quotient	Dividend ÷ Divisor = Target

✏️ Example:

Given set: {12, 15, 18, 20, 24}

Find two numbers with:

• Sum = 39 → Answer: 15 + 24 = 39

• Product = 360 → Answer: 15 × 24 = 360

• Difference = 6 → Answer: 18 - 12 = 6

8. Mentally Add and Subtract Numbers Ending in Zeros

Definition: Quick mental calculation strategies for numbers that end with zeros (10, 100, 1000, etc.).

📝 Mental Math Strategies:

Strategy 1: Add Zeros Separately

For numbers like 300 + 500:

Think: 3 + 5 = 8, then add two zeros → 800

Strategy 2: Count by Hundreds/Thousands

2,400 + 300:

Think: 24 hundreds + 3 hundreds = 27 hundreds = 2,700

Strategy 3: Adjust for Easy Numbers

498 + 150:

Think: 500 + 150 = 650, then subtract 2 → 648

✏️ Practice Examples:

Addition:

• 600 + 400 = 1,000 (6+4=10, add two zeros)

• 3,500 + 2,000 = 5,500

• 4,800 + 1,200 = 6,000

Subtraction:

• 900 - 300 = 600 (9-3=6, add two zeros)

• 7,000 - 2,500 = 4,500

• 5,400 - 1,800 = 3,600

💡 Key Tips:

Focus on the non-zero digits first
Count the zeros and add them back
Use place value understanding (hundreds, thousands)
Practice skip counting (by 10s, 100s, 1000s)

Mixed Operations Quick Reference Chart

Concept	Key Strategy/Rule
Order of Operations	PEMDAS/BODMAS: Parentheses → Multiply/Divide → Add/Subtract
Mixed Operations	Use multiple operations in correct sequence
Estimation	Round first, then calculate with ≈
Multi-Step Problems	Break into smaller steps, solve in order
Extra Information	Cross out what's not needed for the question
Missing Information	Identify what data is needed but not given
Guess-and-Check	Guess → Check → Revise → Repeat
Choose Numbers	Test combinations to reach target result
Mental Math (Zeros)	Work with non-zero digits, add zeros back