1630s, "reason, rationale," from Latin ratio "reckoning, numbering, calculation; business affair, procedure," also "reason, reasoning, judgment, understanding," from rat-, past participle stem of reri "to reckon, calculate," also "think" (from PIE root *re- "to reason, count"). Mathematical sense "relationship between two numbers" is attested from 1650s.
rat-hole
rathskeller
ratification
ratify
rating
ratio
ratiocinate
ratiocination
ration
rational
rationale