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The "rule of three" was a historical method of teaching the standard solution of math problems by proportions. There are numerous websites that explain this:
http://mathworld.wolfram.com/RuleofThree.html,
https://en.wikipedia.org/wiki/Cross-multiplication,
https://math.stackexchange.com/questions/14715/the-logic-behind-the-rule-of-three-on-this-calculation.
The last from stachexchange includes an entertaining comment: "the 'rule of three' is an ancient ad-hoc mindless rote rule of inference that is best ignored". And for a modern student that is certainly true. The "rule of three" dates from the era before algebra was taught to nearly every student in general education. 

The phrase "rule of three" is perhaps difficult to track down because it's so generic. This is one of those cases where you need to be careful with your Internet search phrases. If you search on rule of three with no qualification, you will get some useful hits (on Google, in fact, the very first hit gives the answer you were looking for), but you can do much better if you put the phrase in quotes (asking search engines to look for the exact phrase) and add a few extra words. For example, you could search on the phrase "rule of three" in quotes plus some words related to the specific subject, like this:
history mathematics "rule of three".
And now your have many answers to your question.

The Internet is your friend.

Frank Reed