Fractions and Decimals - Sixth Grade

Complete Notes & Formulas

1. Write Fractions in Lowest Terms

What Does "Lowest Terms" Mean?

A fraction is in lowest terms (or simplified form) when the numerator and denominator have no common factors except 1.

To Simplify: Divide numerator and denominator by their GCF

a/b ÷ GCF(a,b) = simplified fraction

Method: Using GCF

Example: Simplify 24/36

Step 1: Find GCF of 24 and 36

GCF(24, 36) = 12

Step 2: Divide both by GCF

24 ÷ 12 = 2

36 ÷ 12 = 3

Answer: 2/3

2. Least Common Denominator (LCD)

What is LCD?

LCD is the Least Common Multiple (LCM) of the denominators

LCD = LCM(denominators)

Example: Find LCD

Problem: Find LCD of 1/4 and 3/6

Denominators: 4 and 6

LCM(4, 6) = 12

LCD = 12

Convert fractions:

1/4 = 3/12 (multiply by 3/3)

3/6 = 6/12 (multiply by 2/2)

3. Improper Fractions ↔ Mixed Numbers

Definitions

Improper Fraction: Numerator ≥ Denominator (e.g., 7/3)

Mixed Number: Whole number + fraction (e.g., 2⅓)

Convert Improper → Mixed

Steps:

1. Divide numerator by denominator

2. Quotient = whole number

3. Remainder = new numerator

4. Keep same denominator

Example: Convert 17/5 to mixed number

17 ÷ 5 = 3 remainder 2

Whole number: 3

Fraction part: 2/5

Answer: 3⅖

Convert Mixed → Improper

Formula:

(Whole × Denominator + Numerator) / Denominator

Example: Convert 4⅔ to improper fraction

Step 1: Multiply: 4 × 3 = 12

Step 2: Add: 12 + 2 = 14

Step 3: Write over denominator: 14/3

Answer: 14/3

4. Convert Fractions to Decimals

Method

Divide the numerator by the denominator

a/b = a ÷ b

Types of Decimals

Type	Example Fraction	Decimal
Terminating	3/4	0.75
Repeating	1/3	0.333... or 0.3̄

Examples

Example 1: Convert 3/8 to decimal

3 ÷ 8 = 0.375

Answer: 0.375 (terminating)

Example 2: Convert 5/6 to decimal

5 ÷ 6 = 0.8333...

Answer: 0.83̄ (repeating)

5. Convert Decimals to Fractions

Steps for Terminating Decimals

Step 1: Write decimal as fraction over 1

Step 2: Multiply by 10, 100, or 1000 to remove decimal

Step 3: Simplify to lowest terms

Place Value Chart

0.5 = 5/10 = 1/2

0.25 = 25/100 = 1/4

0.125 = 125/1000 = 1/8

Example

Problem: Convert 0.36 to a fraction

Step 1: 0.36 = 36/100 (2 decimal places = hundredths)

Step 2: Find GCF(36, 100) = 4

Step 3: Simplify: 36÷4 / 100÷4 = 9/25

Answer: 9/25

6. Repeating Decimals ↔ Fractions

Notation

Bar notation: 0.333... = 0.3̄

Multiple digits: 0.272727... = 0.2̄7̄

Common Repeating Decimals

Fraction	Decimal
1/3	0.3̄
2/3	0.6̄
1/9	0.1̄
5/9	0.5̄
1/6	0.16̄

Pattern for Ninths

When denominator is 9: The numerator repeats!

4/9 = 0.4̄, 7/9 = 0.7̄, 8/9 = 0.8̄

7. Decimals ↔ Mixed Numbers

Decimal → Mixed Number

Example: Convert 3.75 to mixed number

Step 1: Whole number = 3

Step 2: Decimal part: 0.75 = 75/100 = 3/4

Answer: 3¾

Mixed Number → Decimal

Example: Convert 5⅖ to decimal

Step 1: Whole number = 5

Step 2: Convert fraction: 2/5 = 2 ÷ 5 = 0.4

Step 3: Combine: 5 + 0.4 = 5.4

Answer: 5.4

8. Put Fractions, Decimals & Mixed Numbers in Order

Strategy 1: Convert All to Decimals

Example: Order from least to greatest: 0.6, 5/8, 2/3

Convert to decimals:

0.6 = 0.6

5/8 = 5 ÷ 8 = 0.625

2/3 = 2 ÷ 3 = 0.666...

Compare: 0.6 < 0.625 < 0.666...

Answer: 0.6, 5/8, 2/3

Strategy 2: Convert All to Fractions

Example: Order: 1.5, 1⅓, 7/4

Convert to fractions with common denominator:

1.5 = 3/2 = 18/12

1⅓ = 4/3 = 16/12

7/4 = 21/12

Compare: 16/12 < 18/12 < 21/12

Answer: 1⅓, 1.5, 7/4

9. Understand Fractions as Division

Key Concept

a/b = a ÷ b

The fraction bar means division!

Word Problem Example 1

Problem: 6 pizzas are shared equally among 8 people. How much pizza does each person get?

Think: 6 divided by 8

6 ÷ 8 = 6/8 = 3/4

Answer: Each person gets 3/4 of a pizza

Word Problem Example 2

Problem: A ribbon is 15 meters long. It is cut into 4 equal pieces. How long is each piece?

15 ÷ 4 = 15/4 = 3¾

Answer: Each piece is 3¾ meters or 3.75 meters

Quick Reference: Conversions

Conversion	Method
Fraction → Decimal	Divide numerator by denominator
Decimal → Fraction	Place over power of 10, simplify
Improper → Mixed	Divide, quotient = whole, remainder = numerator
Mixed → Improper	(Whole × Denom) + Num / Denom
Simplify Fraction	Divide by GCF

💡 Important Tips to Remember

✓ Simplify fractions by dividing by GCF

✓ LCD = LCM of the denominators

✓ The fraction bar means division

✓ Improper fractions have numerator ≥ denominator

✓ To compare: Convert to same form (all decimals or all fractions)

✓ Repeating decimals: Use bar notation (0.3̄)

✓ Terminating decimals end; repeating decimals go forever

✓ For ninths: Numerator repeats (4/9 = 0.4̄)

✓ Check your work by converting back

✓ Always simplify final answers

🧠 Memory Tricks & Strategies

Simplifying Fractions:

"GCF is the key to simplicity!"

Improper to Mixed:

"Divide and Conquer: Quotient is whole, remainder on top!"

Mixed to Improper:

"MAD: Multiply, Add, Denominator"

(Multiply whole × denom, Add numerator, over Denominator)

Fraction to Decimal:

"Top divided by bottom!"

Decimal to Fraction:

"Say it, write it, simplify it!"

(0.25 = "25 hundredths" = 25/100 = 1/4)

Comparing:

"Same form, easy compare!"

Repeating Decimals:

"Put a bar over what repeats!"

Master Fractions & Decimals! 🔢 ➗ 📊

Practice conversions daily - they're essential for all future math!