Origins
For the Romans, thinking was counting.βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ Latin ratio meant 'reckoning' and 'reason' simultaneously β the same word for balancing accounts and weighing arguments. English split the inheritance: ratio got the numbers, reason got the logic. But they are the same word.
Latin rΔrΔ« meant 'to reckon, to calculate, to think'. Its past participle ratus gave rise to ratio β initially a business term for a financial reckoning. Cicero expanded it into philosophy, using ratio to translate the Greek logos β 'word, reason, proportion'. For Cicero, ratio was the human capacity for ordered thought.
When English borrowed ratio in the 17th century, it took only the mathematical sense: the quantitative relation between two numbers. The broader meanings had already arrived centuries earlier through French. Reason came from Old French raison (from ratio). Rational came through Latin ratiΕnΔlis. Rate came from Medieval Latin rata, short for prΕ ratΔ parte ('according to a fixed proportion').