Divide Fractions - Sixth Grade

Complete Notes & Formulas

1. Reciprocals (Flip the Fraction)

What is a Reciprocal?

A reciprocal is a FLIPPED fraction

Numerator becomes denominator

Denominator becomes numerator

Formula

Reciprocal of a/b is b/a

Examples

Original	Reciprocal	Explanation
3/4	4/3	Flip the fraction
5	1/5	5 = 5/1, flip to get 1/5
1/8	8/1 = 8	Flip to get whole number
2⅓	3/7	Convert to 7/3, then flip

Key Fact: Any number × its reciprocal = 1

Example: 3/4 × 4/3 = 12/12 = 1

2. Dividing Fractions: Keep, Change, Flip

The Golden Rule

KEEP - CHANGE - FLIP

KEEP the first fraction

CHANGE ÷ to ×

FLIP the second fraction (reciprocal)

Formula

a/b ÷ c/d = a/b × d/c

Example: 3/4 ÷ 2/5

KEEP: Keep first fraction → 3/4

CHANGE: Change ÷ to × → 3/4 ×

FLIP: Flip second fraction → 3/4 × 5/2

Multiply:

(3 × 5)/(4 × 2) = 15/8 = 1⅞

Answer: 1⅞

3. Divide Whole Numbers by Fractions

Steps

Step 1: Write whole number as fraction (over 1)

Step 2: Apply Keep, Change, Flip

Step 3: Multiply and simplify

Example 1: 8 ÷ 1/4 (Unit Fraction)

Step 1: Write 8 as 8/1

Step 2: 8/1 ÷ 1/4

Keep-Change-Flip: 8/1 × 4/1

Multiply: (8 × 4)/(1 × 1) = 32/1 = 32

Answer: 32

Meaning: How many 1/4s are in 8? → 32 quarters

Example 2: 6 ÷ 2/3

6/1 ÷ 2/3

Keep-Change-Flip: 6/1 × 3/2

(6 × 3)/(1 × 2) = 18/2 = 9

Answer: 9

4. Divide Fractions by Whole Numbers

Steps

Step 1: Write whole number as fraction (over 1)

Step 2: Flip the whole number to get reciprocal

Step 3: Multiply and simplify

Example: 3/4 ÷ 6

3/4 ÷ 6/1

Keep-Change-Flip: 3/4 × 1/6

(3 × 1)/(4 × 6) = 3/24 = 1/8

Answer: 1/8

Recipe Example

Problem: A recipe uses 2/3 cup of flour for 4 servings. How much flour for 1 serving?

2/3 ÷ 4 = 2/3 × 1/4

= (2 × 1)/(3 × 4) = 2/12 = 1/6

Answer: 1/6 cup per serving

5. Divide Two Fractions

The Process

Dividing fractions is easier than adding!

NO need for common denominators

Just Keep-Change-Flip!

Example 1: 5/6 ÷ 2/3

Original: 5/6 ÷ 2/3

Keep-Change-Flip: 5/6 × 3/2

Multiply: (5 × 3)/(6 × 2) = 15/12

Simplify: 15/12 = 5/4 = 1¼

Answer: 1¼

Example 2: 1/2 ÷ 1/8

1/2 ÷ 1/8

Keep-Change-Flip: 1/2 × 8/1

(1 × 8)/(2 × 1) = 8/2 = 4

Answer: 4

Meaning: There are 4 eighths in one half

6. Divide Mixed Numbers

Steps

Step 1: Convert ALL mixed numbers to improper fractions

Step 2: Apply Keep-Change-Flip

Step 3: Multiply and simplify

Step 4: Convert back to mixed number if needed

Example 1: 3½ ÷ 1¼

Step 1: Convert to improper

3½ = 7/2

1¼ = 5/4

Step 2: Keep-Change-Flip

7/2 ÷ 5/4 = 7/2 × 4/5

Step 3: Multiply

(7 × 4)/(2 × 5) = 28/10 = 14/5 = 2⅘

Answer: 2⅘

Example 2: 2⅔ ÷ 4

2⅔ = 8/3

8/3 ÷ 4/1 = 8/3 × 1/4

= 8/12 = 2/3

Answer: 2/3

7. Estimate Quotients

Rounding Strategy

For mixed numbers: Round to nearest whole number

For fractions: Round to 0, ½, or 1

Then divide: Use mental math

Example: Estimate 8⅞ ÷ 2⅛

8⅞ is close to 9

2⅛ is close to 2

9 ÷ 2 ≈ 4.5

Estimate: About 4 to 5

(Actual: 8⅞ ÷ 2⅛ = 4 4/17 ≈ 4.2)

8. Word Problems with Division

Keywords

Division: Split, share equally, divide, per, each, how many

"How many ___ in ___?" → Division

Example 1: Equal Sharing

Problem: You have 6 cups of juice to share equally among 4 friends. How much does each friend get?

Keyword: "share equally" → Division

6 ÷ 4 = 6/1 × 1/4 = 6/4 = 3/2 = 1½

Answer: 1½ cups each

Example 2: How Many Groups?

Problem: A rope is 12 feet long. How many 2/3-foot pieces can be cut from it?

Keyword: "how many" → Division

12 ÷ 2/3 = 12/1 × 3/2 = 36/2 = 18

Answer: 18 pieces

Example 3: Mixed Numbers

Problem: A baker has 5¼ cups of flour. Each cookie needs ¾ cup. How many cookies can be made?

5¼ = 21/4

21/4 ÷ 3/4 = 21/4 × 4/3

= 84/12 = 7

Answer: 7 cookies

Quick Reference: Division Rules

Type	Steps	Example
Whole ÷ Fraction	Write whole as fraction, KCF	8 ÷ 1/4 = 32
Fraction ÷ Whole	Write whole as fraction, KCF	3/4 ÷ 6 = 1/8
Fraction ÷ Fraction	Keep-Change-Flip, multiply	3/4 ÷ 2/5 = 1⅞
Mixed ÷ Mixed	Convert to improper, KCF	3½ ÷ 1¼ = 2⅘