Multiplying and Dividing Fractions

Multiplying Fractions

The Rule

To multiply fractions: Multiply the numerators (tops) Multiply the denominators (bottoms)

Formula: a/b × c/d = (a × c)/(b × d)

Basic Example

Problem: 1/2 × 1/3 = ?

Step 1: Multiply numerators: 1 × 1 = 1 Step 2: Multiply denominators: 2 × 3 = 6 Step 3: Write answer: 1/6

Answer: 1/6 ✓

Visual: If you take 1/2 of 1/3, you get 1/6!

More Examples

Example 1: 2/3 × 3/4

2 × 3 = 6 (numerators) 3 × 4 = 12 (denominators)

Answer: 6/12 = 1/2 (simplified) ✓

Example 2: 3/5 × 2/7

3 × 2 = 6 5 × 7 = 35

Answer: 6/35 ✓ (can't simplify)

Simplifying Before You Multiply

Smart trick: Cancel common factors BEFORE multiplying!

Example: 2/3 × 3/4

Notice: 3 appears in numerator AND denominator Cancel the 3s:

2/3 × 3/4 = 2/4 = 1/2 ✓

This makes the math easier!

Example: 4/5 × 5/8

Cancel the 5s: 4/5 × 5/8 = 4/8 = 1/2 ✓

Multiplying with Whole Numbers

To multiply a fraction by a whole number: Put the whole number over 1

Example: 3 × 2/5

Write as: 3/1 × 2/5

3 × 2 = 6 1 × 5 = 5

Answer: 6/5 = 1 1/5 ✓

Multiplying Mixed Numbers

Step 1: Convert mixed numbers to improper fractions Step 2: Multiply Step 3: Convert back to mixed number (if needed)

Example: 1 1/2 × 2 1/3

Step 1: Convert

• 1 1/2 = 3/2
• 2 1/3 = 7/3

Step 2: Multiply 3/2 × 7/3 = 21/6

Step 3: Simplify 21/6 = 7/2 = 3 1/2 ✓

Dividing Fractions

The Rule: Keep, Change, Flip (KCF)

To divide fractions:

1. Keep the first fraction
2. Change ÷ to ×
3. Flip the second fraction (reciprocal)

Then multiply!

What's a Reciprocal?

The reciprocal flips a fraction upside down:

• Reciprocal of 2/3 is 3/2
• Reciprocal of 5/1 is 1/5
• Reciprocal of 4/7 is 7/4

Trick: Swap the numerator and denominator!

Basic Example

Problem: 1/2 ÷ 1/4 = ?

Step 1: Keep: 1/2 Step 2: Change: ÷ becomes × Step 3: Flip: 1/4 becomes 4/1

Now multiply: 1/2 × 4/1 = 4/2 = 2 ✓

Answer: 2 ✓

Think: How many 1/4s fit in 1/2? Two! ✓

More Examples

Example 1: 3/4 ÷ 1/2

Keep-Change-Flip: 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2 ✓

Example 2: 2/3 ÷ 4/5

Keep-Change-Flip: 2/3 × 5/4 = 10/12 = 5/6 ✓

Example 3: 5/6 ÷ 2/3

Keep-Change-Flip: 5/6 × 3/2 = 15/12 = 5/4 = 1 1/4 ✓

Dividing by a Whole Number

Put the whole number over 1, then flip it!

Example: 3/4 ÷ 2

Step 1: Write 2 as 2/1 Step 2: Keep-Change-Flip: 3/4 × 1/2 Step 3: Multiply: 3/8 ✓

Dividing with Mixed Numbers

Example: 2 1/2 ÷ 1 1/4

Step 1: Convert to improper fractions

• 2 1/2 = 5/2
• 1 1/4 = 5/4

Step 2: Keep-Change-Flip 5/2 × 4/5

Step 3: Cancel and multiply 5/2 × 4/5 = 4/2 = 2 ✓

Why Does "Flip and Multiply" Work?

Think about: 1/2 ÷ 1/4

Question: How many 1/4s are in 1/2?

Visual: Two 1/4 pieces fit perfectly into 1/2!

When you flip and multiply (1/2 × 4/1 = 2), you get the same answer! ✓

Word Problems

Multiplication Problem

Problem: A recipe needs 2/3 cup of sugar. You want to make 1/2 of the recipe. How much sugar?

Solution: 2/3 × 1/2 = 2/6 = 1/3 cup ✓

Division Problem

Problem: You have 3/4 yard of ribbon. You need to cut it into pieces that are 1/8 yard each. How many pieces?

Solution: 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6 pieces ✓

Common Mistakes

❌ Mistake 1: Multiplying denominators when dividing Wrong: 1/2 ÷ 1/4 = 1/8 ✗ Right: 1/2 ÷ 1/4 = 1/2 × 4/1 = 2 ✓

❌ Mistake 2: Forgetting to flip the second fraction Wrong: 3/4 ÷ 1/2 = 3/4 × 1/2 ✗ Right: 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2 ✓

❌ Mistake 3: Flipping the wrong fraction Wrong: 1/2 ÷ 3/4 = 2/1 × 3/4 ✗ Right: 1/2 ÷ 3/4 = 1/2 × 4/3 ✓

Multiplication vs. Division

Multiplication makes fractions smaller (usually):

• 1/2 × 1/3 = 1/6 (smaller than both!)

Division makes fractions bigger (usually):

• 1/2 ÷ 1/4 = 2 (bigger than both!)

Quick Reference

Multiplying Fractions:

• Multiply tops
• Multiply bottoms
• Simplify

Dividing Fractions:

• Keep first fraction
• Change ÷ to ×
• Flip second fraction
• Multiply
• Simplify

Tips for Success

✅ Always simplify your final answer ✅ Look for common factors to cancel before multiplying ✅ Convert mixed numbers to improper fractions first ✅ Remember: KCF (Keep, Change, Flip) for division ✅ Check: Does my answer make sense?

Practice Checklist

When multiplying: ☐ Multiply numerators ☐ Multiply denominators ☐ Simplify the answer

When dividing: ☐ Keep the first fraction ☐ Change ÷ to × ☐ Flip the second fraction ☐ Multiply ☐ Simplify the answer

Summary

Multiplying fractions: Multiply across (tops and bottoms)

Dividing fractions: Keep-Change-Flip, then multiply!

Remember: Division is just multiplication by the reciprocal! ✓