Multiplying Fractions by Whole Numbers

What Does It Mean?

Multiplying a fraction by a whole number means adding that fraction multiple times.

Example: 3 × 1/4 means "1/4 + 1/4 + 1/4"

Think of it as: "I want 3 groups of 1/4"

Visual Model: Repeated Addition

Problem: 4 × 1/5

Picture it as adding: 1/5 + 1/5 + 1/5 + 1/5 = 4/5

[■][□][□][□][□]  ← 1/5
[■][□][□][□][□]  ← 1/5
[■][□][□][□][□]  ← 1/5
[■][□][□][□][□]  ← 1/5
─────────────
[■][■][■][■][□]  = 4/5 total

Answer: 4 × 1/5 = 4/5 ✓

The Simple Method

Rule: Multiply the whole number by the numerator (top number)

Steps:

1. Multiply whole number × numerator
2. Keep the denominator the same
3. Simplify if needed

Example: 5 × 2/3

Step 1: Multiply: 5 × 2 = 10 Step 2: Keep denominator: 3 Step 3: Write answer: 10/3 = 3 1/3

Answer: 5 × 2/3 = 10/3 = 3 1/3 ✓

Step-by-Step Examples

Example 1: 3 × 1/4

• Multiply: 3 × 1 = 3
• Keep denominator: 4
• Answer: 3/4 ✓

Example 2: 6 × 2/5

• Multiply: 6 × 2 = 12
• Keep denominator: 5
• Answer: 12/5 = 2 2/5 ✓

Example 3: 4 × 3/8

• Multiply: 4 × 3 = 12
• Keep denominator: 8
• Simplify: 12/8 = 3/2 = 1 1/2 ✓

Writing Whole Numbers as Fractions

Another way to think about it:

Any whole number can be written as a fraction with denominator 1.

• 3 = 3/1
• 5 = 5/1
• 7 = 7/1

Then multiply numerators and denominators:

Example: 4 × 2/3

• Write as: 4/1 × 2/3
• Multiply tops: 4 × 2 = 8
• Multiply bottoms: 1 × 3 = 3
• Answer: 8/3 = 2 2/3 ✓

(Same answer, different method!)

Real-World Examples

Recipes: You're making 3 batches of cookies. Each batch needs 2/3 cup of sugar.

• 3 × 2/3 = 6/3 = 2 cups of sugar ✓

Running: You run 3/4 mile each day for 5 days.

• 5 × 3/4 = 15/4 = 3 3/4 miles total ✓

Ribbon: You need 4 pieces of ribbon, each 5/8 yard long.

• 4 × 5/8 = 20/8 = 2 4/8 = 2 1/2 yards ✓

Converting to Mixed Numbers

When numerator > denominator:

Example: 7 × 3/4 = 21/4

Convert to mixed number:

• How many 4s in 21? → 5 (with remainder 1)
• 21 ÷ 4 = 5 R1
• Write as: 5 1/4

Answer: 21/4 = 5 1/4 ✓

Using Arrays and Area Models

Problem: 3 × 2/5

Visual:

Row 1: [■■][□□□]  ← 2/5
Row 2: [■■][□□□]  ← 2/5
Row 3: [■■][□□□]  ← 2/5
        ────
Total:  6 shaded out of 15 parts
But we're counting fifths!
3 groups of 2/5 = 6/5

Answer: 3 × 2/5 = 6/5 = 1 1/5 ✓

Number Lines

Adding 1/3 four times:

0 ―――⅓―――⅔―――1―――1⅓―――
     ↑    ↑   ↑    ↑
     1st  2nd 3rd  4th

4 × 1/3 = 4/3 = 1 1/3 ✓

Simplifying Your Answer

Always check if you can simplify:

Example: 6 × 2/4

• Multiply: 6 × 2 = 12
• Denominator: 4
• Answer: 12/4
• Simplify: 12 ÷ 4 = 3
• Final: 3 ✓

Or simplify before multiplying:

• 2/4 = 1/2
• 6 × 1/2 = 6/2 = 3 ✓
• Same answer!

Multiplying by Numbers Greater Than 1

Pattern to notice:

• 2 × 1/3 = 2/3 (smaller than 1)
• 3 × 1/3 = 3/3 = 1 (equals 1)
• 4 × 1/3 = 4/3 = 1 1/3 (greater than 1)

As the whole number gets bigger, the answer gets bigger!

Order Doesn't Matter

Commutative property works with fractions:

• 3 × 2/5 = 6/5
• 2/5 × 3 = 6/5

Same answer either way! ✓

Practice Strategy

Before solving:

1. Identify the whole number
2. Identify the fraction
3. Multiply whole number × numerator
4. Keep denominator the same

After solving:

1. Can you simplify?
2. Is it an improper fraction? (Convert to mixed number)
3. Does your answer make sense?

Common Mistakes

❌ Multiplying the denominator too

• WRONG: 3 × 2/4 = 6/12
• RIGHT: 3 × 2/4 = 6/4 = 3/2 = 1 1/2

❌ Forgetting to simplify

• 4 × 2/8 = 8/8 → Should simplify to 1

❌ Not converting improper fractions

• Leave answer as 7/3 instead of 2 1/3

✅ Only multiply the numerator ✅ The denominator stays the same ✅ Always simplify ✅ Convert improper fractions to mixed numbers

Mental Math Tricks

Multiplying by small numbers:

Times 2: Double the numerator

• 2 × 3/5 = 6/5 ✓

Times 3: Triple the numerator

• 3 × 2/7 = 6/7 ✓

Times 4: Quadruple the numerator

• 4 × 1/8 = 4/8 = 1/2 ✓

Word Problems

Problem 1: Each person gets 3/4 cup of juice. How much juice for 5 people?

• 5 × 3/4 = 15/4 = 3 3/4 cups ✓

Problem 2: A recipe calls for 2/3 teaspoon of vanilla. You're making 6 batches.

• 6 × 2/3 = 12/3 = 4 teaspoons ✓

Problem 3: Each book is 5/8 inch thick. How tall is a stack of 8 books?

• 8 × 5/8 = 40/8 = 5 inches ✓

Check Your Work

Method 1: Use repeated addition

• 4 × 2/5 should equal 2/5 + 2/5 + 2/5 + 2/5 = 8/5 ✓

Method 2: Draw a picture

• Shade the fraction multiple times
• Count total shaded parts

Method 3: Does it make sense?

• 6 × 1/2 should be 3 (half of 6)
• 6 × 1/2 = 6/2 = 3 ✓ Correct!

Key Takeaways

✅ Whole number × numerator ✅ Denominator stays the same ✅ Simplify when possible ✅ Convert improper fractions to mixed numbers ✅ Think of it as repeated addition

Remember: Multiplying by a whole number makes the fraction bigger (you're adding it that many times)!