Understanding Fractions

Understanding Fractions

• Definition: A fraction represents a part of a whole and is expressed in the form a/b, where:
  • a = numerator (part)
  • b = denominator (whole)

Types of Fractions

1. Proper Fractions:

   • Numerator is less than the denominator (e.g., 2/5).

2. Improper Fractions:

   • Numerator is greater than or equal to the denominator (e.g., 7/4).

3. Mixed Numbers:

   • A whole number combined with a proper fraction (e.g., 1 1/2).

Simplifying Fractions

• Process:
  1. Find the greatest common divisor (GCD) of the numerator and denominator.
  2. Divide both numerator and denominator by the GCD.

Equivalent Fractions

• Fractions that represent the same value even though they have different numerators and denominators (e.g., 1/2 = 2/4 = 3/6).
• To find equivalent fractions, multiply or divide the numerator and denominator by the same non-zero number.

Adding and Subtracting Fractions

1. Same Denominator:

   • Add or subtract numerators directly; keep the denominator the same.
   • Example: 1/4 + 2/4 = (1+2)/4 = 3/4.

2. Different Denominators:

   • Find a common denominator.
   • Convert fractions to equivalent fractions with the common denominator, then add/subtract.
   • Example: 1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2.

Multiplying and Dividing Fractions

1. Multiplication:

   • Multiply the numerators together and the denominators together.
   • Example: (2/3) × (3/4) = (2×3)/(3×4) = 6/12 = 1/2 (after simplification).

2. Division:

   • Multiply by the reciprocal of the second fraction.
   • Example: (2/3) ÷ (3/4) = (2/3) × (4/3) = (2×4)/(3×3) = 8/9.

Converting Fractions

• To Decimal: Divide the numerator by the denominator.
• To Percent: Convert to decimal and multiply by 100.
• To Mixed Number: Divide numerator by denominator; quotient is whole number, remainder/denominator is the fraction.

Applications

• Used in everyday calculations, such as cooking, budgeting, and measurements.
• Essential in algebra and higher mathematics for working with ratios, proportions, and equations.

Understanding Fractions

• Fractions represent a part of a whole, written as a/b, where 'a' is the numerator (part) and 'b' is the denominator (whole).

Types of Fractions

• Proper Fractions: The numerator is smaller than the denominator (e.g., 2/5).
• Improper Fractions: The numerator is greater than or equal to the denominator (e.g., 7/4).
• Mixed Numbers: A whole number combined with a proper fraction (e.g., 1 1/2).

Simplifying Fractions

• Simplify fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and denominator by the GCD.

Equivalent Fractions

• Equivalent fractions represent the same value, even with different numerators and denominators (e.g., 1/2 = 2/4 = 3/6).
• Equivalent fractions are created by multiplying or dividing the numerator and denominator by the same non-zero number.

Adding and Subtracting Fractions

• Same Denominator: Add or subtract the numerators and keep the denominator the same.
• Different Denominators: Find a common denominator, convert fractions to equivalent fractions with the common denominator, then add/subtract.

Multiplying and Dividing Fractions

• Multiplication: Multiply the numerators and the denominators to obtain the product.
• Division: Multiply the first fraction by the reciprocal of the second fraction.

Converting Fractions

• To Decimal: Divide the numerator by the denominator.
• To Percent: Convert to decimal and multiply by 100.
• To Mixed Number: Divide the numerator by the denominator; the quotient is the whole number, the remainder divided by the denominator is the fractional part.

Applications of Fractions

• Fractions are used in daily calculations, such as cooking, budgeting, and measurements.
• They are crucial in algebra and higher mathematics for working with ratios, proportions, and equations.