1.3 Fractions

Fractions are a fundamental part of algebra, helping us express parts of a whole and perform calculations with precision. They're essential for everything from basic arithmetic to complex problem-solving in math and science.

Understanding fractions involves simplifying, performing basic operations, and following the order of operations. We'll also explore how fractions work in algebraic expressions and learn about different types of fractions and proportions.

Understanding Fractions

Simplification of fractions

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• Find the greatest common factor (GCF) of the numerator and denominator
  • Largest number that divides both without leaving a remainder (12 and 18)
• Divide both numerator and denominator by their GCF to simplify
  • $\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$ reduces the fraction to lowest terms

Basic operations with fractions

• Addition and subtraction require a common denominator
  • Find the least common denominator (LCD), smallest number divisible by all denominators (12 for $\frac{1}{3}$ and $\frac{1}{4}$)
  • Convert fractions to equivalent ones with LCD ($\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{4} = \frac{3}{12}$)
  • Add or subtract numerators, keep denominator the same ($\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$)
• Multiplication involves multiplying numerators and denominators separately
  • $\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} = \frac{1}{2}$ simplifies the result
• Division requires multiplying by the reciprocal of the second fraction
  • Reciprocal flips numerator and denominator ($\frac{3}{4}$ becomes $\frac{4}{3}$)
  • $\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9}$ simplifies the result

Order of operations for fractions

• PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
• Simplify fractions within the expression first
• Perform multiplication and division from left to right
• Perform addition and subtraction from left to right
  • $\frac{1}{2} + \frac{1}{3} \times \frac{3}{4} = \frac{1}{2} + \frac{1 \times 3}{3 \times 4} = \frac{1}{2} + \frac{1}{4} = \frac{2}{4} + \frac{1}{4} = \frac{3}{4}$ follows order of operations

Working with Fractions in Algebra

Fractions in algebraic expressions

• Substitute given values for variables
• Simplify fractions within the expression
• Apply order of operations to evaluate
  • If $x = 2$ and $y = 3$, evaluate $\frac{x}{y} + \frac{y}{x}$:
    1. $\frac{2}{3} + \frac{3}{2}$ substitutes values
    2. $\frac{2 \times 2}{3 \times 2} + \frac{3}{2} = \frac{4}{6} + \frac{3}{2}$ simplifies first fraction
    3. $\frac{2}{3} + \frac{3}{2} = \frac{4}{6} + \frac{9}{6} = \frac{13}{6}$ converts to common denominator and adds

Conversions between fraction types

• Improper fraction to mixed number:
  1. Divide numerator by denominator
  2. Quotient becomes whole number, remainder becomes numerator ($\frac{17}{4} = 4 \frac{1}{4}$)
• Mixed number to improper fraction:
  1. Multiply whole number by denominator, add numerator
  2. Result becomes numerator, keep denominator ($3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$)
• Fraction to decimal:
  • Divide numerator by denominator ($\frac{3}{4} = 3 \div 4 = 0.75$)
• Decimal to fraction:
  1. Write as fraction with denominator power of 10 ($0.6 = \frac{6}{10}$)
  2. Simplify if possible ($\frac{6}{10} = \frac{3}{5}$)

Types of Fractions and Proportions

• Rational numbers are expressed as fractions where both numerator and denominator are integers (e.g., $\frac{3}{4}$, $\frac{-5}{2}$)
• Common fractions have whole number numerators and denominators (e.g., $\frac{1}{2}$, $\frac{3}{4}$)
• Complex fractions contain fractions in the numerator, denominator, or both (e.g., $\frac{\frac{1}{2}}{\frac{3}{4}}$)
• Proportions are equations stating that two ratios are equal (e.g., $\frac{a}{b} = \frac{c}{d}$)
  • Cross multiplication can be used to solve proportions by multiplying the numerator of each fraction by the denominator of the other (ad = bc)