Multiplying Fractions

For Elementary Students

What Does Multiplying Fractions Mean?

When you multiply a fraction by another number, you're finding part of something.

Think about it like this: "1/2 of 6" means "multiply 1/2 × 6" — you're finding half of 6!

Multiplying a Fraction by a Whole Number

Example: 1/4 × 8 = ?

Think: "What is 1/4 of 8?"

Divide 8 into 4 equal groups:

• 8 ÷ 4 = 2

Each group has 2, and you take 1 group.

Answer: 2

Or use multiplication: (1 × 8) / 4 = 8/4 = 2

Picture It!

Example: 1/3 × 6 = ?

Draw 6 circles:

⭕ ⭕ ⭕ ⭕ ⭕ ⭕

Divide into 3 groups:

Group 1: ⭕ ⭕
Group 2: ⭕ ⭕
Group 3: ⭕ ⭕

1/3 of 6 circles = 2 circles

So 1/3 × 6 = 2

Multiplying Two Fractions

The Rule: Multiply the tops, multiply the bottoms!

top/bottom × top/bottom = (top × top)/(bottom × bottom)

Example: 1/2 × 1/3 = ?

• Tops: 1 × 1 = 1
• Bottoms: 2 × 3 = 6
• Answer: 1/6

What does this mean? Half of one-third equals one-sixth!

Another Example

Example: 2/3 × 1/4 = ?

• Tops: 2 × 1 = 2
• Bottoms: 3 × 4 = 12
• Answer: 2/12

Simplify: 2/12 = 1/6 (divide both by 2)

Remember!

Multiplying fractions is easier than adding them — no common denominator needed!

Just multiply across: top × top, bottom × bottom!

For Junior High Students

The Multiplication Rule

To multiply fractions, multiply numerators together and denominators together.

Formula: a/b × c/d = (a × c)/(b × d)

Example: 2/3 × 4/5

• Numerators: 2 × 4 = 8
• Denominators: 3 × 5 = 15

Answer: 8/15

Note: Unlike addition/subtraction, you do NOT need a common denominator!

More Examples

Example 1: 3/7 × 2/5

• (3 × 2)/(7 × 5) = 6/35

Example 2: 1/4 × 3/8

• (1 × 3)/(4 × 8) = 3/32

Multiplying a Whole Number by a Fraction

Step 1: Write the whole number as a fraction over 1

Step 2: Multiply

Example: 5 × 2/3

Rewrite 5 as 5/1:

• 5/1 × 2/3 = (5 × 2)/(1 × 3) = 10/3

Convert to mixed number: 10/3 = 3 1/3

Example: 4 × 3/8

• 4/1 × 3/8 = 12/8 = 3/2 = 1 1/2

Understanding What It Means

"1/2 of 10" means "1/2 × 10"

• 1/2 × 10 = 10/2 = 5

Important: Multiplying by a fraction less than 1 makes the number smaller!

• 1/4 × 20 = 5 (smaller than 20)
• 1/10 × 100 = 10 (smaller than 100)

Simplifying Before Multiplying (Cross-Canceling)

You can simplify before multiplying to keep numbers small!

Example: 4/9 × 3/8

Before multiplying, notice:

• 4 and 8 share a factor of 4
• 3 and 9 share a factor of 3

Simplify diagonally:

• 4 ÷ 4 = 1 and 8 ÷ 4 = 2
• 3 ÷ 3 = 1 and 9 ÷ 3 = 3

New problem: 1/3 × 1/2 = 1/6

Much easier!

Multiplying Mixed Numbers

Step 1: Convert mixed numbers to improper fractions

Step 2: Multiply

Step 3: Convert answer back to mixed number if needed

Example: 1 1/2 × 2 2/3

Convert:

• 1 1/2 = 3/2
• 2 2/3 = 8/3

Multiply:

• 3/2 × 8/3 = 24/6 = 4

Answer: 4

Another Mixed Number Example

Example: 2 1/4 × 1 1/3

Convert:

• 2 1/4 = 9/4
• 1 1/3 = 4/3

Multiply:

• 9/4 × 4/3 = 36/12 = 3

Answer: 3

Multiplying Three or More Fractions

Just multiply all the tops, then all the bottoms!

Example: 1/2 × 2/3 × 3/4

• Numerators: 1 × 2 × 3 = 6
• Denominators: 2 × 3 × 4 = 24
• Answer: 6/24 = 1/4 (simplified)

Word Problems

Example: "A recipe needs 2/3 cup of sugar. You want to make half the recipe. How much sugar?"

Think: "Half of 2/3" = 1/2 × 2/3

• (1 × 2)/(2 × 3) = 2/6 = 1/3 cup

Example: "A rope is 5/6 meter long. You cut it into 1/2. How long is each piece?"

• 5/6 × 1/2 = 5/12 meter

Real-Life Uses

Cooking: "Half of 3/4 cup" = 1/2 × 3/4 = 3/8 cup

Sharing: "1/3 of 12 cookies" = 1/3 × 12 = 4 cookies

Measurement: "3/4 of a 1/2 foot" = 3/4 × 1/2 = 3/8 foot

Checking Your Answer

Use division to check!

If 2/3 × 3/4 = 6/12 = 1/2, then:

• 1/2 ÷ 2/3 should equal 3/4 ✓

Practice