Addition, subtraction, multiplication, and division

Addition and subtraction of fractions requires a common denominator. Often the easiest one to find is the product of the denominators:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{cb}{bd}=\frac{ad + bc}{bd}$$

$$\frac{a}{b}-\frac{c}{d}=\frac{ad}{bd}-\frac{cb}{bd}=\frac{ad - bc}{bd}$$

Multiplication and division are gotten somewhat more easily:

$$\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$$

and

$$\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\frac{a}{b}\frac{d}{c}=\frac{ad}{bc}$$

Use of parentheses in these situations is very helpful- put them in whenever you need them.

These operations will frequently be combined in one step:

$$\frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\frac{x-(x+h)}{h x (x+h)}$$

Fill in any steps in your notes that you need to see how the algebra worked.

Another typical example:

$$\frac{\left(\frac{x-1}{x^2+2}\right)}{\left(\frac{x-1}{(x^2+1)(x-3)}\right)}=\frac{(x^2+1)(x-3)}{x^2+2}$$