Multiply Fractions and Whole Numbers | Fifth Grade
Complete Notes & Formulas
1. Multiply Fractions by Whole Numbers I (Basic Multiplication)
Definition: To multiply a fraction by a whole number, multiply the numerator by the whole number and keep the denominator the same.
📐 Formula (Method 1 - Direct Multiplication):
a/b × n = (a × n)/b
Multiply only the numerator by the whole number
📐 Formula (Method 2 - Convert Whole Number):
a/b × n = a/b × n/1 = (a × n)/(b × 1)
Write whole number as n/1, then multiply fractions
✏️ Example 1: 2/5 × 3
Method 1: Multiply numerator by whole number
2/5 × 3 = (2 × 3)/5 = 6/5 = 1 1/5
Method 2: Convert whole number to fraction
2/5 × 3/1 = (2 × 3)/(5 × 1) = 6/5 = 1 1/5
Answer: 6/5 or 1 1/5
✏️ Example 2: 3/4 × 8
3/4 × 8 = (3 × 8)/4 = 24/4 = 6
Answer: 6
2. Multiply Fractions by Whole Numbers II (Advanced/Simplification)
Definition: After multiplying, always simplify the result. Convert improper fractions to mixed numbers when appropriate.
📝 Steps with Simplification:
- Multiply the numerator by the whole number
- Keep the denominator the same
- Simplify the fraction (divide by GCF if possible)
- Convert to mixed number if improper fraction
✏️ Example 1: 5/6 × 4
Step 1: Multiply numerator: (5 × 4)/6 = 20/6
Step 2: Simplify by dividing by GCF (2): 20/6 = 10/3
Step 3: Convert to mixed number: 10/3 = 3 1/3
Answer: 3 1/3
✏️ Example 2: 3/8 × 12
Step 1: Multiply: (3 × 12)/8 = 36/8
Step 2: Simplify by dividing by GCF (4): 36/8 = 9/2
Step 3: Convert: 9/2 = 4 1/2
Answer: 4 1/2
3. Multiply Fractions by Whole Numbers: Word Problems
Definition: Apply fraction multiplication to solve real-world problems. Look for key words like "of," "times," or "groups of."
📝 Steps to Solve Word Problems:
- Read the problem carefully
- Identify the fraction and whole number
- Determine which operation is needed (usually multiply)
- Set up the multiplication problem
- Solve and simplify
- Write answer with appropriate units
✏️ Example 1: Pizza Problem
A pizza is cut into 8 equal slices. John ate 3/8 of a pizza. If he wants to eat that amount from 4 pizzas, how many slices will he eat?
Solution:
3/8 × 4 = (3 × 4)/8 = 12/8 = 3/2 = 1 1/2
1 1/2 pizzas = 1.5 × 8 slices = 12 slices
Answer: 12 slices
✏️ Example 2: Distance Problem
Sarah walks 2/3 of a mile each day. How far does she walk in 5 days?
Solution:
2/3 × 5 = (2 × 5)/3 = 10/3 = 3 1/3
Answer: 3 1/3 miles
4. Multiply Fractions and Whole Numbers: Sorting
Definition: Organize multiplication problems by their products, comparing which expressions give larger or smaller results.
🔑 Sorting Rules:
- Product < Whole Number: When fraction < 1 (multiplying makes smaller)
- Product > Whole Number: When fraction > 1 (multiplying makes larger)
- Product = Whole Number: When fraction = 1
✏️ Example: Sort by Product Size
Sort these from smallest to largest product:
A) 1/4 × 8
B) 1/2 × 8
C) 3/4 × 8
Solutions:
A) 1/4 × 8 = 8/4 = 2
B) 1/2 × 8 = 8/2 = 4
C) 3/4 × 8 = 24/4 = 6
Order: A < B < C (2 < 4 < 6)
5. Fractions of a Number I (Basic Concept)
Definition: Finding a fraction "of" a number means multiplying the fraction by that number. The word "of" means multiply.
"OF" = MULTIPLY (×)
Finding 1/2 of 10 = 1/2 × 10
📐 Two Methods to Find Fraction of a Number:
Method 1: Divide then Multiply
• Divide the number by the denominator
• Multiply the result by the numerator
Method 2: Multiply then Divide
• Multiply the number by the numerator
• Divide the result by the denominator
✏️ Example: Find 2/3 of 12
Method 1:
Divide: 12 ÷ 3 = 4
Multiply: 4 × 2 = 8
Method 2:
2/3 × 12 = (2 × 12)/3 = 24/3 = 8
Answer: 8
6. Fractions of a Number: Word Problems
Definition: Real-world problems involving finding a fractional part of a quantity.
✏️ Example 1: Classroom Problem
There are 24 students in a class. 3/4 of them wore blue shirts. How many students wore blue shirts?
Solution:
Find 3/4 of 24
3/4 × 24 = (3 × 24)/4 = 72/4 = 18
Answer: 18 students
✏️ Example 2: Money Problem
Jake has $60. He spends 2/5 of his money on books. How much did he spend?
Solution:
Find 2/5 of 60
2/5 × 60 = (2 × 60)/5 = 120/5 = 24
Answer: $24
7. Fractions of a Number II (Advanced Applications)
Definition: More complex problems involving multiple steps, larger numbers, or finding fractions of fractions.
💡 Advanced Strategies:
- Use mental math when possible (e.g., 1/2 of 50 = 25)
- Look for common factors to simplify before multiplying
- Work with larger numbers systematically
- Break complex problems into smaller steps
✏️ Example 1: Large Number
Find 3/5 of 100
Solution:
3/5 × 100 = (3 × 100)/5 = 300/5 = 60
Answer: 60
✏️ Example 2: Multi-Step Problem
A school has 240 students. 2/3 are girls. Of the girls, 3/4 play sports. How many girls play sports?
Solution:
Step 1: Find number of girls: 2/3 × 240 = 160 girls
Step 2: Find girls who play sports: 3/4 × 160 = 120
Answer: 120 girls play sports
Quick Reference Chart
| Concept | Formula | Example |
|---|---|---|
| Basic Multiplication | a/b × n = (a × n)/b | 2/5 × 3 = 6/5 |
| With Simplification | Multiply → Simplify → Convert | 3/8 × 12 = 36/8 = 9/2 = 4 1/2 |
| Fraction of a Number | "of" means × (multiply) | 2/3 of 12 = 2/3 × 12 = 8 |
💡 Key Formulas:
Method 1
Multiply numerator only: a/b × n = (a×n)/b
Method 2
Convert to fraction: a/b × n/1
Division Method
Divide by denominator, then multiply by numerator
Simplify Always
Reduce to lowest terms
🔑 Key Tips for Success:
- When multiplying fraction by whole number, only multiply the numerator
- The denominator stays the same when multiplying by whole numbers
- Always simplify your answer to lowest terms
- Convert improper fractions to mixed numbers in final answers
- Remember: "of" means multiply (×)
- In word problems, look for key phrases like "of," "times," or "groups of"
- Check your answer: multiply back to verify
- When finding a fraction of a number, you can divide first or multiply first
📚 Fifth Grade Multiply Fractions and Whole Numbers - Complete Study Guide
Master these concepts for math excellence! ✨