### Basic Reciprocal Rules **Converting equations with reciprocals:** $\frac{1}{x} = \frac{1}{y} \implies x = y$ **Key insight:** If two reciprocals are equal, the original values must be equal. ### Simplifying Complex Fractions **Dividing fractions:** $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$ **Example:** $\frac{\frac{1}{x}}{\frac{1}{y}} = \frac{1}{x} \cdot \frac{y}{1} = \frac{y}{x}$ **Multiple terms in denominator:** $\frac{\frac{1}{x}}{\frac{1}{y} + \frac{1}{z}} = \frac{\frac{1}{x}}{\frac{z + y}{yz}} = \frac{1}{x} \cdot \frac{yz}{y + z} = \frac{yz}{x(y + z)}$ ### Finding Common Denominators **Adding/subtracting fractions:** $\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy}$ $\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}$ **Three or more terms:** $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{yz + xz + xy}{xyz}$ ### Cross-Multiplication Technique **When you have:** $\frac{a}{b} = \frac{c}{d} \implies ad = bc$ **Solving for a variable:** $\frac{1}{x} = \frac{a}{b} \implies b = ax \implies x = \frac{b}{a}$ ### Isolating Variables in Reciprocal Equations **Type 1: Simple reciprocal** $\frac{1}{x} = k \implies x = \frac{1}{k}$ **Type 2: Reciprocal sum/difference** $\frac{1}{x} + \frac{1}{y} = k$ $\frac{y + x}{xy} = k \implies y + x = kxy$ $y = kxy - x = x(ky - 1)$ $x = \frac{y}{ky - 1}$ **Alternative approach:** $\frac{1}{x} = k - \frac{1}{y} = \frac{ky - 1}{y}$ $x = \frac{y}{ky - 1}$ ### Compound Fraction Simplification **Multiply by LCD (Least Common Denominator):** For $\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x} - \frac{1}{y}}$, multiply top and bottom by $xy$: $\frac{\frac{1}{x} + \frac{2}{y}}{\frac{3}{x} - \frac{1}{y}} \cdot \frac{xy}{xy} = \frac{y + 2x}{3y - x}$ ### Reciprocal Substitution Trick **When dealing with $\frac{1}{x}$ repeatedly, substitute $u = \frac{1}{x}$:** $\frac{1}{x^2} + \frac{3}{x} + 2 = 0$ Let $u = \frac{1}{x}$: $u^2 + 3u + 2 = 0$ Solve for $u$, then $x = \frac{1}{u}$. ### Rationalising Denominators with Fractions **Eliminate nested fractions:** $\frac{1}{a + \frac{1}{b}} = \frac{1}{\frac{ab + 1}{b}} = \frac{b}{ab + 1}$ **General pattern:** $\frac{1}{a + \frac{b}{c}} = \frac{c}{ac + b}$ ### Partial Fraction Decomposition Shortcut **For simple cases:** $\frac{1}{(x-a)(x-b)} = \frac{1}{b-a}\left(\frac{1}{x-a} - \frac{1}{x-b}\right)$ **Verify by adding back:** $\frac{1}{x-a} - \frac{1}{x-b} = \frac{(x-b) - (x-a)}{(x-a)(x-b)} = \frac{b-a}{(x-a)(x-b)}$ ### Harmonic Mean Connection **Reciprocal average pattern:** $\frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{2xy}{x + y}$ This is the harmonic mean of $x$ and $y$. ### Quick Simplification Tips 1. **Always look for common factors** before expanding 2. **Multiply by conjugates** when you have $(a + b)$ in denominator and want $(a - b)$ 3. **Factor before canceling** to avoid missing simplifications 4. **When solving for $x$ in $\frac{1}{x}$ equations**, take reciprocal of both sides if possible 5. **LCD method is foolproof** but can be tedious; look for shortcuts first