Operations

Multiplication

Multiply straight across, then simplify.

ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}

Example:

xx+1×x+1x2=x(x+1)x2(x+1)=1x\frac{x}{x+1} \times \frac{x+1}{x^2} = \frac{x(x+1)}{x^2(x+1)} = \frac{1}{x}

Division

Flip the second fraction and multiply.

ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Example:

xx+1÷x2x+1=xx+1×x+1x2=1x\frac{x}{x+1} \div \frac{x^2}{x+1} = \frac{x}{x+1} \times \frac{x+1}{x^2} = \frac{1}{x}

Addition and Subtraction

Need a common denominator, just like regular fractions.

Example: 1x+1+2x−1\dfrac{1}{x+1} + \dfrac{2}{x-1}

Step 1: Find the common denominator

LCD=(x+1)(x−1)\text{LCD} = (x+1)(x-1)

Step 2: Rewrite each fraction

1x+1=1⋅(x−1)(x+1)(x−1)=x−1(x+1)(x−1)\frac{1}{x+1} = \frac{1 \cdot (x-1)}{(x+1)(x-1)} = \frac{x-1}{(x+1)(x-1)}

2x−1=2⋅(x+1)(x−1)(x+1)=2(x+1)(x+1)(x−1)\frac{2}{x-1} = \frac{2 \cdot (x+1)}{(x-1)(x+1)} = \frac{2(x+1)}{(x+1)(x-1)}

Step 3: Add the numerators

x−1(x+1)(x−1)+2(x+1)(x+1)(x−1)=x−1+2x+2(x+1)(x−1)\frac{x-1}{(x+1)(x-1)} + \frac{2(x+1)}{(x+1)(x-1)} = \frac{x - 1 + 2x + 2}{(x+1)(x-1)}

Step 4: Simplify

=3x+1(x+1)(x−1)= \frac{3x + 1}{(x+1)(x-1)}