$\frac{a}{b}+\frac{c}{d}=\frac{a\cdot d}{b\cdot d}+\frac{c\cdot b}{b\cdot d}=\frac{a\cdot d+c\cdot b}{b\cdot d}$

$\frac{a}{b}-\frac{c}{d}=\frac{a\cdot d}{b\cdot d}-\frac{c\cdot b}{b\cdot d}=\frac{a\cdot d-c\cdot b}{b\cdot d}$

$a\cdot \frac{b}{c}=\frac{a\cdot b}{c}$

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

$\frac{a}{b}:\frac{c}{d}=\frac{a·d}{b·c}=\frac{ad}{bc}$

$\frac{a}{b}:c=\frac{a:c}{b}$    or   : $\frac{a}{b}:c=\frac{a}{b\cdot c}$

$a:\frac{b}{c}=\frac{a\cdot c}{b}=\frac{a\cdot c}{b}$

Simplifying and Expanding Fractions

You simplify a fraction by dividing both the numerator and the denominator by the same number.

Adding Two Fractions (+)

Start by finding a common denominator, which is a number that both denominators divide into. This can be done by multiplying the two denominators together.
When you find a common denominator, you are effectively expanding the fractions to have the same denominator.
Remember to multiply both the numerator and the denominator when expanding the fractions.

When two fractions have the same denominator, you can add the numerators and keep the denominator the same.

Subtracting Two Fractions (-):

You must find a common denominator, a number that both denominators divide into.

I start by expanding both fractions using the denominator of the opposite fraction. Remember that when you expand a fraction, you must multiply both the numerator and the denominator by the same number.
When two fractions have the same denominator and need to be subtracted, subtract the numerators and keep the denominator the same.