1. **Stating the problem:** We want to understand the fundamental properties of fractions. 2. **Definition:** A fraction represents a part of a whole and is written as $\frac{a}{b}$ where $a$ is the numerator and $b$ is the denominator, with $b \neq 0$. 3. **Property 1: Equivalent Fractions** Multiplying or dividing numerator and denominator by the same nonzero number does not change the value of the fraction. Example: $$\frac{a}{b} = \frac{a \times k}{b \times k}$$ where $k \neq 0$. 4. **Property 2: Simplification** A fraction can be simplified by dividing numerator and denominator by their greatest common divisor (GCD). Example: $$\frac{a}{b} = \frac{\cancel{d} \times a'}{\cancel{d} \times b'} = \frac{a'}{b'}$$ where $d = \gcd(a,b)$. 5. **Property 3: Addition and Subtraction** To add or subtract fractions, find a common denominator: $$\frac{a}{b} \pm \frac{c}{d} = \frac{a \times d}{b \times d} \pm \frac{c \times b}{d \times b} = \frac{ad \pm cb}{bd}$$ 6. **Property 4: Multiplication** Multiply numerators and denominators directly: $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$ 7. **Property 5: Division** Dividing by a fraction is multiplying by its reciprocal: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$ These properties help in manipulating and understanding fractions clearly.