Equivalent Fractions

Definition

• Equivalent Fractions: Fractions that have different numerators and denominators but represent the same value.
• Example: 9/12 and 6/8 are equivalent because both simplify to 3/4.

Key Points

• All equivalent fractions reduce to the same simplest form.
• They are equal to the same fraction when simplified.

Examples

• Equivalent fractions for 1/5 are 5/25, 6/30, 4/20 (all simplify to 1/5).
• 1/2, 2/4, 3/6, 4/8 are equivalent fractions.

Methods to Find Equivalent Fractions

1. Multiplication:
   • Multiply both the numerator and the denominator by the same number.
   • Example: To find equivalent fractions of 3/4:
     • (\frac{3 \times 2}{4 \times 2} = \frac{6}{8})
     • (\frac{3 \times 3}{4 \times 3} = \frac{9}{12})
2. Division:
   • Divide both the numerator and the denominator by the same number.
   • Example: To simplify 72/108:
     • Divide by 2: (\frac{36}{54})
     • Further divide by common factors until simplest form like 2/3.

Methods to Identify Equivalent Fractions

• Same Denominator Method: Make denominators the same using Least Common Multiple (LCM).
• Decimal Method: Convert fractions to decimals to see if they match.
• Cross Multiplication: Cross multiply the fractions; if products are equal, they are equivalent.
• Visual Representation: Graphically compare shaded areas of equal shapes.

Practical Examples

• Checking equivalence with cross multiplication:
  • 2/12 and 3/18 are equivalent because (2 \times 18 = 3 \times 12).
• Non-equivalence example:
  • 12/20 and 22/30 are not equivalent as seen by making denominators the same or using cross multiplication.

Tips

• Multiply or divide both numerator and denominator by the same number to find equivalent fractions.
• Use charts for visual assistance.

Why Equivalent Fractions Matter

• They simplify arithmetic operations with fractions such as addition, subtraction, multiplication, and division.
• Useful in solving real-world problems involving fractions.

FAQs

• Equivalent Improper Fraction: An improper fraction that is equivalent to a given fraction, e.g., 3/2 equivalent to 9/6.
• How to Write Equivalent Fractions: Use multiplication or division techniques described above.

Example Questions

• Example 1: Find equivalent fractions for 6/8.
  • Answer: 12/16 and 3/4 by multiplying or dividing with a common factor.
• Example 2: Check if 2/6 and 3/9 are equivalent.
  • Answer: Yes, using cross multiplication or decimal method.

This summary covers the major points related to equivalent fractions, their identification, methods to find them, and their practical applications.