Mixed Operations: Whole Numbers | Fifth Grade

Complete Notes & Formulas

1. Order of Operations (PEMDAS/BODMAS)

Definition: When an expression has more than one operation, we must follow the order of operations to get the correct answer.

🔑 PEMDAS Rule (Order of Operations):

Letter	Stands For	What to Do
P	Parentheses	Solve operations inside ( ) first
E	Exponents	Calculate powers (5², 2³, etc.)
M	Multiplication	Work from LEFT to RIGHT
D	Division	Work from LEFT to RIGHT
A	Addition	Work from LEFT to RIGHT
S	Subtraction	Work from LEFT to RIGHT

💡 Memory Tip:

"Please Excuse My Dear Aunt Sally"

🌍 BODMAS (Used in UK, India, and other countries):

Brackets, Orders, Division, Multiplication, Addition, Subtraction

(Same as PEMDAS, just different names!)

2. Addition of Whole Numbers

Definition: Addition means combining two or more numbers to find the total or sum.

📐 Key Concepts:

Addend + Addend = Sum

Example: 234 + 567 = 801

📝 Properties of Addition:

Commutative Property: a + b = b + a (Example: 5 + 3 = 3 + 5)
Associative Property: (a + b) + c = a + (b + c)
Identity Property: a + 0 = a (Adding zero doesn't change the number)

3. Subtraction of Whole Numbers

Definition: Subtraction means taking away one number from another to find the difference.

📐 Key Concepts:

Minuend - Subtrahend = Difference

Example: 801 - 234 = 567

💡 Important Rules:

Subtract from right to left (ones, tens, hundreds, etc.)
Borrow (regroup) when needed
Check: Difference + Subtrahend = Minuend

4. Multiplication of Whole Numbers

Definition: Multiplication is repeated addition or finding the total of equal groups.

📐 Key Concepts:

Factor × Factor = Product

Example: 25 × 12 = 300

📝 Properties of Multiplication:

Commutative Property: a × b = b × a (Example: 4 × 5 = 5 × 4)
Associative Property: (a × b) × c = a × (b × c)
Identity Property: a × 1 = a
Zero Property: a × 0 = 0
Distributive Property: a × (b + c) = (a × b) + (a × c)

5. Division of Whole Numbers

Definition: Division means splitting a number into equal parts or groups.

📐 Key Concepts:

Dividend ÷ Divisor = Quotient (+ Remainder)

Example: 300 ÷ 12 = 25

💡 Important Rules:

Check: Quotient × Divisor + Remainder = Dividend
Cannot divide by zero (undefined)
Any number ÷ 1 = the number itself
Any number ÷ itself = 1

6. Solving Mixed Operations

Definition: When solving problems with multiple operations, always follow PEMDAS/BODMAS order.

✏️ Example 1: Simple Mixed Operations

Solve: 8 + 2 × 5

Solution:

Step 1: Multiplication first (2 × 5 = 10)

Step 2: Then addition (8 + 10 = 18)

Answer: 18

❌ Common Mistake: Solving left to right (8 + 2 = 10, then 10 × 5 = 50) is WRONG!

✏️ Example 2: With Parentheses

Solve: (15 - 3) × 2 + 8

Solution:

Step 1: Parentheses first (15 - 3 = 12)

Step 2: Multiplication (12 × 2 = 24)

Step 3: Addition (24 + 8 = 32)

Answer: 32

✏️ Example 3: All Four Operations

Solve: 48 ÷ 6 + 3 × 4 - 2

Solution:

Step 1: Division (48 ÷ 6 = 8)

Step 2: Multiplication (3 × 4 = 12)

Step 3: Now we have: 8 + 12 - 2

Step 4: Add from left to right (8 + 12 = 20)

Step 5: Subtract (20 - 2 = 18)

Answer: 18

✏️ Example 4: With Exponents

Solve: 5 + 3² × 2

Solution:

Step 1: Exponents (3² = 9)

Step 2: Multiplication (9 × 2 = 18)

Step 3: Addition (5 + 18 = 23)

Answer: 23

7. Word Problems: Mixed Operations

Definition: Word problems require reading carefully, identifying operations needed, and solving step by step.

📝 Steps to Solve Word Problems:

Step 1: Read - Read the problem carefully
Step 2: Identify - Find the important numbers and operations
Step 3: Plan - Decide which operations to use and in what order
Step 4: Solve - Do the calculations following PEMDAS
Step 5: Check - Does the answer make sense?

✏️ Word Problem 1: Addition & Multiplication

A bakery makes 12 trays of cookies. Each tray has 24 cookies. They sell 150 cookies. How many cookies are left?

Solution:

Step 1: Total cookies = 12 × 24 = 288 cookies

Step 2: Cookies left = 288 - 150 = 138 cookies

Answer: 138 cookies remaining

✏️ Word Problem 2: All Four Operations

Maria had 500 stamps. She bought 3 packs with 25 stamps each. Then she gave 45 stamps to her friend and divided the remaining stamps equally among 5 albums. How many stamps in each album?

Solution:

Step 1: Stamps bought = 3 × 25 = 75 stamps

Step 2: Total stamps = 500 + 75 = 575 stamps

Step 3: After giving away = 575 - 45 = 530 stamps

Step 4: Per album = 530 ÷ 5 = 106 stamps

Answer: 106 stamps per album

✏️ Word Problem 3: Shopping Problem

A store sells pencils for ₹5 each and notebooks for ₹35 each. Sarah buys 8 pencils and 4 notebooks. She pays with a ₹200 note. How much change does she get?

Solution:

Step 1: Cost of pencils = 8 × ₹5 = ₹40

Step 2: Cost of notebooks = 4 × ₹35 = ₹140

Step 3: Total cost = ₹40 + ₹140 = ₹180

Step 4: Change = ₹200 - ₹180 = ₹20

Answer: ₹20 change

✏️ Word Problem 4: Multi-Step Problem

A school has 6 classes with 32 students each. The school bought 240 books. If the books are shared equally among all students, how many books does each student get?

Solution:

Step 1: Total students = 6 × 32 = 192 students

Step 2: Books per student = 240 ÷ 192 = 1.25 books

Answer: 1 book each (48 books left over)

8. Key Words in Word Problems

Definition: Certain words in problems tell you which operation to use.

Operation	Key Words
Addition (+)	sum, total, altogether, combined, plus, increase, more than
Subtraction (-)	difference, less than, minus, decrease, left, remaining, fewer
Multiplication (×)	product, times, each, every, groups of, multiply, of
Division (÷)	quotient, divided by, shared equally, per, split, each, average

Quick Reference Chart

Operation	Symbol	Formula	Example
Addition	+	a + b = sum	45 + 23 = 68
Subtraction	-	a - b = difference	68 - 23 = 45
Multiplication	×	a × b = product	12 × 8 = 96
Division	÷	a ÷ b = quotient	96 ÷ 8 = 12