"in proportion," from Medieval Latin pro rata (parte) "according to the calculated (share)," from pro "for, in accordance with" (see pro-) + rata, ablative singular of ratus, past participle of reri "to count, reckon" (see rate (n.)).
word-forming element meaning "forward, forth, toward the front" (as in proclaim, proceed); "beforehand, in advance" (prohibit, provide); "taking care of" (procure); "in place of, on behalf of" (proconsul, pronoun); from Latin pro (adv., prep.) "on behalf of, in place of, before, for, in exchange for, just as," which also was used as a first element in compounds and had a collateral form por-.
Also in some cases from cognate Greek pro "before, in front of, sooner," which also was used in Greek as a prefix (as in problem). Both the Latin and Greek words are from PIE *pro- (source also of Sanskrit pra- "before, forward, forth;" Gothic faura "before," Old English fore "before, for, on account of," fram "forward, from;" Old Irish roar "enough"), extended form of root *per- (1) "forward," hence "in front of, before, toward, near," etc.
The common modern sense of "in favor of, favoring" (pro-independence, pro-fluoridation, pro-Soviet, etc.) was not in classical Latin and is attested in English from early 19c.
early 15c., "estimated value or worth, proportional estimation according to some standard; monetary amount; a proportional part," from Old French rate "price, value" and directly from Medieval Latin rata (pars) "fixed (amount)," from Latin rata "fixed, settled," fem. past participle of reri "to reckon, think" (from PIE root *re- "to reason, count").
Meaning "degree of speed" (properly ratio between distance and time) is attested from 1650s. Currency exchange sense of "basis of equivalence upon which one currency is exchanged for another" is recorded by 1727. Meaning "fixed public tax assessed on property for some local purpose" is by 1712.
First-rate, second-rate, etc. are 1640s, from British Navy division of ships into six classes based on size and strength. Phrase at any rate originally (1610s) meant "at any cost," hence "positively, assuredly." weakened sense of "at least" is attested by 1760. Rate-payer "one who is assessed and pays a local tax" is by 1845.