Question: • $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
• $\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$
• $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
• $\frac{a}{b} = \frac{c}{d} = \frac{ad}{bc}$ ($b,c \neq 0$)
1. **Stating the problem:** We are given four fundamental fraction operations and equivalences involving fractions $\frac{a}{b}$ and $\frac{c}{d}$.
2. **Addition of fractions:** The formula for adding fractions with different denominators is
$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$
This works because we find a common denominator $bd$ and adjust numerators accordingly.
3. **Subtraction of fractions:** Similarly, subtraction is
$$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$$
Again, the common denominator is $bd$ and numerators are adjusted.
4. **Multiplication of fractions:** Multiplying fractions is straightforward:
$$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$
Multiply numerators and denominators directly.
5. **Equality of fractions:** The equality
$$\frac{a}{b} = \frac{c}{d} = \frac{ad}{bc}$$
means that two fractions are equal if and only if their cross products are equal, i.e., $ad = bc$, assuming $b,c \neq 0$.
6. **Important rules:**
- Denominators $b$ and $d$ must not be zero.
- When adding or subtracting, always find a common denominator.
- Multiplication is direct numerator and denominator multiplication.
- Equality of fractions can be checked by cross multiplication.
This set of formulas is fundamental for working with fractions in algebra and arithmetic.