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In mathematics, the 'rule of three' is the method of finding the fourth term of a mathematical proportion when the first three terms are known, that is, where the first term is in proportion to the second as the third is to the unknown fourth term. To find the fourth term, multiply the second and third terms, then divide their product by the by the first term.^[1]
Using mathematical notation, with ''a'', ''b'' and ''c'' as the known three terms of the proportion, and ''x'' as the unknown fourth term to be found, the problem can be stated as
:$\{a\; over\; b\}=\{c\; over\; x\}\; .$
According to the rule of three,
:$x=\{b\; cdot\; c\; over\; a\}\; .$
To give an example, say that a car, driving at a constant speed, in 3 hours travels 90 miles. How far can the car drive in 7 hours if it maintains the same speed? Substituting the numbers for the letters using the rule of three,
:$x=\{90\; cdot\; 7\; over\; 3\}$ $=\{630\; over\; 3\}\; =\; 210$ miles.
The rule of three is based on the principle that, in a proportion, the product of the first and fourth terms (called the ''extremes'') is equal to the product of the second and third terms (called the ''means''). Or, where
:$\{a\; over\; b\}=\{c\; over\; d\}\; ,$ then $a\; cdot\; d=b\; cdot\; c\; .$
In the expressions above, ''a'' and ''d'' are the ''extremes'', and ''b'' and ''c'' are the ''means'', of the proportion.

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References

Further reading

References

1. Evans IH, ''Brewer's Dictionary of Phrase and Fable'', 14th ed., ISBN 0304340049, 1990. See "Rule of Three" under "Three".

Further reading

★ 'Dr Math', ''Rule of Three''

★ 'Dr Math', ''Abraham Lincoln and the Rule of Three''

★ ''Pike's System of arithmetick abridged : designed to facilitate the study of the science of numbers, comprehending the most perspicuous and accurate rules, illustrated by useful examples : to which are added appropriate questions, for the examination of scholars, and a short system of book-keeping.'', 1827 - facsimile of the relevant section

★ Hersee J, ''Multiplication is vexation'' - an article tracing the history of the rule from 1781