Rate
late Middle English (expressing a notion of ‘estimated value’): from Old French, from medieval Latin rata (from Latin pro rata parte (or portione ) ‘according to the proportional share’), from ratus ‘reckoned’, past participle of reri .
wiktionary
From Middle English rate, from Old French rate, from Medieval Latin rata, from Latinprō ratā parte(“according to a fixed part”), from ratus(“fixed”), from rērī(“think, deem, judge, estimate", originally "reckon, calculate”).
From Middle English raten(“to scold, chide”), from Old Norse hrata(“to refuse, reject, slight, find fault with”), from Proto-Germanic *hratōną(“to sway, shake”), from Proto-Indo-European *krad-(“to swing”). Cognate with Swedish rata(“to reject, refuse, find fault, slight”), Norwegian rata(“to reject, cast aside”), Old English hratian(“to rush, hasten”).
etymonline
rate (n.)
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.
rate (v.1)
"to scold, chide vehemently, rebuke," late 14c., raten, probably from Old French rateir, variant of reter "to impute blame, accuse, find fault with," from Latin reputare "to count over, reflect," in Vulgar Latin, "to impute, blame," from re- "repeatedly" (see re-) + putare "to judge, suppose, believe, suspect" (originally "to clean, trim, prune," from PIE root *pau- (2) "to cut, strike, stamp"). Related: Rated; rating.
Old French reter also was borrowed into Middle English as retten "to blame" (c. 1300); also "to attribute, impute" (late 14c.), "to consider, think about" (late 14c.).
rate (v.2)
"estimate the worth or value of, reckon by comparative estimation," mid-15c., raten, from rate (n.). Intransitive sense of "have a certain value, rank, or standing" is from 1809; specifically as "have high value" by 1928. Related: Rated; rating.
