Essential Fraction Rules
Basic Definitions
Fraction = Numerator ÷ Denominator
Proper Fraction: numerator < denominator
Improper Fraction: numerator ≥ denominator
Mixed Number: whole number + proper fraction
Equivalent Fractions
a/b = (a×k)/(b×k) where k ≠ 0
Example: 2/3 = 4/6 = 6/9 = 8/12
Simplification
Divide numerator and denominator by their GCD
GCD(12, 18) = 6, so 12/18 = 2/3
Comparing Fractions
- Cross Multiplication: a/b vs c/d → compare a×d with b×c
- Common Denominator: Convert to same denominator, compare numerators
- Decimal Conversion: Convert both to decimals and compare
Fraction Operations
Addition and Subtraction
Same Denominator: a/c ± b/c = (a±b)/c
Different Denominators: a/b ± c/d = (ad±bc)/(bd)
Example 1:
2/5 + 1/5 = 3/5
Example 2:
1/3 + 1/4 = 4/12 + 3/12 = 7/12
Multiplication
a/b × c/d = (a×c)/(b×d)
Example:
2/3 × 3/4 = 6/12 = 1/2
With Whole Numbers:
5 × 2/3 = 10/3 = 3⅓
Division
a/b ÷ c/d = a/b × d/c = (a×d)/(b×c)
Example:
2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3
Remember:
"Keep, Change, Flip"
Fraction Conversions
Fraction to Decimal
Divide numerator by denominator
3/4 = 3 ÷ 4 = 0.75
Decimal to Fraction
- Count decimal places (n)
- Write decimal as fraction with denominator 10ⁿ
- Simplify by dividing by GCD
0.25 to Fraction:
0.25 = 25/100 = 1/4
0.125 to Fraction:
0.125 = 125/1000 = 1/8
Percentage Conversions
Percentage to Fraction: x% = x/100
Fraction to Percentage: (a/b) × 100%
Common Fraction-Decimal-Percentage Equivalents
1/2 = 0.5 = 50%
1/3 ≈ 0.333... = 33.33%
1/4 = 0.25 = 25%
1/5 = 0.2 = 20%
1/8 = 0.125 = 12.5%
3/4 = 0.75 = 75%
Worked Examples
Example 1: Adding Fractions with Different Denominators
Problem: 2/3 + 1/4 = ?
Solution:
• Find LCD of 3 and 4: LCD = 12
• Convert fractions: 2/3 = 8/12, 1/4 = 3/12
• Add: 8/12 + 3/12 = 11/12
Answer: 11/12
Example 2: Multiplying Mixed Numbers
Problem: 2½ × 1⅓ = ?
Solution:
• Convert to improper fractions: 2½ = 5/2, 1⅓ = 4/3
• Multiply: 5/2 × 4/3 = 20/6
• Simplify: 20/6 = 10/3 = 3⅓
Answer: 3⅓
Example 3: Converting Repeating Decimal to Fraction
Problem: Convert 0.666... to a fraction
Solution:
• Let x = 0.666...
• Multiply by 10: 10x = 6.666...
• Subtract: 10x - x = 6.666... - 0.666...
• Solve: 9x = 6, so x = 6/9 = 2/3
Answer: 2/3
Example 4: Real-World Application
Problem: A recipe calls for 2¾ cups of flour. If you want to make 1½ times the recipe, how much flour do you need?
Solution:
• Convert to improper fractions: 2¾ = 11/4, 1½ = 3/2
• Multiply: 11/4 × 3/2 = 33/8
• Convert back: 33/8 = 4⅛
Answer: 4⅛ cups of flour