Multiply Fractions - Sixth Grade

Complete Notes & Formulas

1. Multiply Fractions by Whole Numbers

The Golden Rule

Write whole number as a fraction (over 1)

Then multiply numerators and denominators

Formula

a/b × c = a/b × c/1 = (a × c)/(b × 1) = (a × c)/b

Steps

Step 1: Write whole number as fraction (denominator = 1)

Step 2: Multiply numerators

Step 3: Multiply denominators

Step 4: Simplify the result

Example 1: 1/8 × 5

Step 1: Write 5 as 5/1

Step 2: 1/8 × 5/1

Step 3: Multiply numerators: 1 × 5 = 5

Step 4: Multiply denominators: 8 × 1 = 8

Answer: 5/8

Example 2: 5 × 3/10

5/1 × 3/10 = (5 × 3)/(1 × 10) = 15/10

Simplify: 15/10 = 3/2 = 1½

Answer: 1½

Shortcut: You can also multiply the whole number with just the numerator!

Example: 1/3 × 15 = (1 × 15)/3 = 15/3 = 5

2. Multiply Two Fractions

The Rule

Multiply numerators × Multiply denominators

NO need to find common denominator!

Formula

a/b × c/d = (a × c)/(b × d)

Example 1: 2/3 × 4/5

Step 1: Multiply numerators: 2 × 4 = 8

Step 2: Multiply denominators: 3 × 5 = 15

Step 3: Result: 8/15

Step 4: Already in simplest form

Answer: 8/15

Example 2: 3/4 × 2/9

(3 × 2)/(4 × 9) = 6/36

Simplify: Divide by GCF(6,36) = 6

6/36 = 1/6

Answer: 1/6

Cross-Canceling (Simplify Before Multiplying)

Example: 4/9 × 3/8

Cross-cancel: 4 and 8 have GCF of 4

4÷4 = 1, 8÷4 = 2

Also: 3 and 9 have GCF of 3

3÷3 = 1, 9÷3 = 3

Now: 1/3 × 1/2 = 1/6

Answer: 1/6

3. Multiply Mixed Numbers

The Process

Convert mixed numbers to IMPROPER fractions

Then multiply as usual

Steps

Step 1: Convert mixed numbers to improper fractions

Step 2: Multiply numerators

Step 3: Multiply denominators

Step 4: Simplify and convert back to mixed number if needed

Example 1: Mixed × Whole 2⅓ × 6

Step 1: Convert 2⅓ to improper: (2×3+1)/3 = 7/3

Step 2: Write 6 as 6/1

Step 3: Multiply: 7/3 × 6/1 = 42/3

Step 4: Simplify: 42/3 = 14

Answer: 14

Example 2: Mixed × Mixed 1½ × 2⅓

Step 1: Convert both to improper

1½ = 3/2

2⅓ = 7/3

Step 2: Multiply

3/2 × 7/3 = 21/6

Step 3: Simplify

21/6 = 7/2 = 3½

Answer: 3½

4. Scaling by Fractions

Understanding Scaling

Multiply by fraction > 1: Result is GREATER

Multiply by fraction = 1: Result is SAME

Multiply by fraction < 1: Result is SMALLER

Examples

Example 1: 8 × 3/4 = 6 (smaller, because 3/4 < 1)

Example 2: 6 × 5/3 = 10 (greater, because 5/3 > 1)

Example 3: 12 × 4/4 = 12 (same, because 4/4 = 1)

Think of it: Multiplying by 1/2 means "taking half"

Multiplying by 2/1 means "doubling"

5. Multiply Three or More Fractions

The Strategy

1. Convert mixed numbers to improper fractions

2. Multiply all numerators together

3. Multiply all denominators together

4. Simplify the result

Example: 1/2 × 2/3 × 3/4

Numerators: 1 × 2 × 3 = 6

Denominators: 2 × 3 × 4 = 24

Result: 6/24

Simplify: 6/24 = 1/4

Answer: 1/4

Tip: Cross-cancel before multiplying to make calculations easier!

6. Estimate Products of Fractions

Rounding Strategy

For fractions: Round to 0, ½, or 1

For mixed numbers: Round to nearest whole number

Example 1: Estimate 7/8 × 3/5

7/8 is close to 1

3/5 is close to ½

1 × ½ = ½

Estimate: About ½

(Actual: 7/8 × 3/5 = 21/40 ≈ 0.525)

Example 2: Estimate 5⅞ × 2⅛

5⅞ is close to 6

2⅛ is close to 2

6 × 2 = 12

Estimate: About 12

(Actual: 5⅞ × 2⅛ = 12½)

7. Word Problems with Multiplication

Keywords

Multiplication: Of, product, times, each

"Of" usually means multiply (e.g., 1/2 of 10 = 1/2 × 10)

Example 1: Fraction × Whole

Problem: Sarah completed 2/3 of her 15 homework problems. How many problems did she complete?

Keyword: "of" → Multiplication

2/3 × 15 = 2/3 × 15/1 = 30/3 = 10

Answer: 10 problems

Example 2: Fraction × Fraction

Problem: A recipe needs 2/3 cup of flour. If making 1/2 of the recipe, how much flour is needed?

1/2 of 2/3 = 1/2 × 2/3

= (1 × 2)/(2 × 3) = 2/6 = 1/3

Answer: 1/3 cup

Example 3: Mixed Numbers

Problem: A garden measures 2½ meters by 1⅓ meters. What is the area?

Convert: 2½ = 5/2 and 1⅓ = 4/3

5/2 × 4/3 = 20/6 = 10/3 = 3⅓

Answer: 3⅓ square meters

Quick Reference: Fraction Multiplication

Type	Key Rule	Example
Fraction × Whole	Write whole as fraction/1	3/4 × 8 = 6
Fraction × Fraction	Multiply across	2/3 × 4/5 = 8/15
Mixed × Mixed	Convert to improper first	1½ × 2⅓ = 3½
Three or More	Multiply all together	1/2 × 2/3 × 3/4 = 1/4