For the Romans, thinking was counting. Latin ratio meant 'reckoning' and 'reason' simultaneously — 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 rērī meant 'to reckon, to calculate, to think'. Its past participle ratus gave rise to ratio — initially a business term for a financial reckoning. Cicero expanded it into philosophy, using ratio to translate the Greek logos — '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 ratiōnālis. Rate came from Medieval Latin rata, short for prō ratā parte ('according to a fixed proportion').
Ration is the most revealing sibling. A ration is a calculated portion — an amount determined by ratio. During wartime rationing, authorities divided supplies by ratio: so much per person, calculated and measured. The word reveals its ancestry perfectly.
Ratify also belongs. To ratify a treaty is to 'make it reckoned' — to confirm it by calculation and judgement. Even the modern internet sense of 'getting ratioed' preserves the Latin mathematics: your replies outnumber your likes by an unfavourable ratio.