ration n.

*rē-, Proto-Indo-European root meaning "to reason, count;" a variant of PIE root *ar-, also arə-, "to fit together." 

It forms all or part of: Alfred; arraign; arithmetic; Conrad; dread; Eldred; Ethelred; hatred; hundred; kindred; logarithm; Ralph; rate (n.) "estimated value or worth;" rathskeller; ratify; ratio; ration; read; reason; rede; rhyme; riddle (n.1) "word-game;" rite; ritual.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit radh- "to succeed, accomplish;" Greek arithmos "number, amount;" Latin reri "to consider, confirm, ratify," ritus "rite, religious custom;" Old Church Slavonic raditi "to take thought, attend to;" Old Irish im-radim "to deliberate, consider;" Old English rædan "to advise, counsel, persuade; read;" Old English, Old High German rim "number;" Old Irish rim "number," dorimu "I count."

1630s, in theological writing, "reason, rationale," from Latin ratio "a reckoning, account, a numbering, calculation," hence also "a business affair; course, conduct, procedure," also in a transferred sense, of mental action, "reason, reasoning, judgment, understanding, that faculty of the mind which forms the basis of computation and calculation." This is from rat-, past-participle stem of reri "to reckon, calculate," also "to think, believe" (from PIE root *re- "to think, reason, count").

Latin ratio often was used to represent or translate Greek logos ("computation, account, esteem, reason") in works of philosophy, though the range of senses in the two do not overlap (ratio lacks the key "speech, word, statement" meaning in the Greek word; see Logos).

The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos). The general or extended sense of "corresponding relationship between things not precisely measurable" is by 1808.