chriscarson said:
Here in the U.S. we are taught to add and multiply fractions at about age eight.
For ##\frac{a}{b} \times \frac{c}{d}## there are many ways to evaluate the result and get the same right answer. The canonical "right" way is to:
1. Divide a by b giving the result ##\frac{a}{b}##.
2. Divide c by d giving the result ##\frac{c}{d}##.
3. Multiply those two results together.
Another way is to:
1. Multiply a by c giving the result ##ac##.
2. Multiply b by d giving the result ##bd##.
3. Divide the result from 1) by the result from 2).
It is a rule of arithmetic and of real algebra that$$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$
One rule of algebra that they didn't teach us until age 16 or so was that putting two variable names side by side ("juxtaposition") is a notation that conventionally means "multiply them together".
[As it turns out my first exposure to the juxtaposition notation was in a standardized test -- they tested on it before my school had ever presented the content. It is over 40 years later now and that incident still cheeses me off]