Chapter 2: Fractions and Ratios

Chapter 2: Fractions and Ratios

2.1. Introduction

• The arithmetic of fractions is foundational for algebra.
• A fraction is represented as $\frac{p}{q}$ where:
  • $p$ is the numerator.
  • $q$ is the denominator (must not be zero).

2.2. Types of Fractions

• Proper Fractions:
  • When p < q.
  • Examples: $\frac{1}{2}, \frac{3}{4}$.
• Improper Fractions:
  • When $p \geq q$.
  • Examples: $\frac{11}{8}, \frac{7}{4}, \frac{3}{3}$.
• Negative signs do not affect the classification:
  • Examples of proper fractions: $-\frac{3}{5}, -\frac{7}{21}, \frac{5}{-23}$.
  • Examples of improper fractions: $\frac{3}{-3}, -\frac{8}{2}, \frac{11}{-2}$.

2.3. Simplifying Fractions

• Definition: Expressing a fraction in its simplest form.
• To simplify:
  • Multiply or divide the numerator and denominator by the same number.
  • Example: $\frac{1}{2} = \frac{2}{4}$ and both fractions are equivalent.
• A fraction is in simplest form when no factors are common between the numerator and denominator.
  • Example: $\frac{7}{12}$ is simplest; $\frac{7}{21}$ is not (simplest form is $\frac{1}{3}$).

2.4. Arithmetic of Rational Numbers

• Adding and Subtracting Fractions:
  • Rewrite fractions with a common denominator, known as the Lowest Common Denominator (LCD).
  • Example:
  • $\frac{1}{7} + \frac{4}{7}$ stops at step 2 since they share a denominator:
    • $1 + 4 = 5$;
    • $\frac{1}{7} + \frac{4}{7} = \frac{5}{7}$.
  • If denominators are unlike, find the LCD and convert:
  • Example:
    • Add $\frac{1}{6} + \frac{3}{8}$:
    • LCD = 24. Transform to $\frac{4}{24} + \frac{9}{24} = \frac{13}{24}$.

2.4.1. Adding and Subtracting Mixed Numbers

• Steps consist of finding the LCD, rewriting fraction parts, adding fraction parts, and adding whole numbers.
• Example:
  • $3 \frac{1}{5} + 4 \frac{2}{5}$ equals $7 \frac{3}{5}$.

2.5. Fraction Multiplication

• Basic Formula:
  • Multiply two fractions: $\frac{a}{b} \times \frac{c}{f} = \frac{ac}{bf}$.
  • Example: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$.
• Multiplying Whole Numbers:
  • Convert the whole into fraction form. Example:
  • $5 \times \frac{3}{4} = \frac{5}{1} \times \frac{3}{4} = \frac{15}{4}$.
• Multiplying Mixed Numbers:
  • Convert to improper fractions before multiplication.
  • Example: $\frac{3}{2} \times \frac{3}{5} = \frac{9}{10}$.

2.6. Dividing Fractions

• To divide fractions, multiply by the reciprocal:
  • Example: $\frac{a}{b} \div \frac{c}{f} = \frac{a}{b} \times \frac{f}{c}$.
• Example:
  • $\frac{3}{5} \div \frac{6}{7} = \frac{3}{5} \times \frac{7}{6} = \frac{7}{10}$.

Exercises

• Classification of fractions as proper or improper.
• Simplifying fractions.
• Finding equivalent fractions.
• Adding/subtracting rational numbers, and mixed fractions.
• Multiplication and division of fractions with included word problem applications.