Etymology
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prime (n.)

"earliest canonical hour of the day" (6 a.m.), from Old English prim and Old French prime and directly from Medieval Latin prima "the first service," from Latin prima hora "the first hour" (of the Roman day), from Latin primus "first, the first, first part" (see prime (adj.)).  (In classical Latin, the noun uses of the adjective meant "first part, beginning; leading place.")

By extension, "the first division of the day" 6 a.m. to 9 a.m. (early 13c.). The sense of "beginning of a period or course of events" is from late 14c. From the notion of "the period or condition of greatest vigor in life" (by 1530s) comes the specific sense "springtime of human life" (often meaning ages roughly 21 to 28) is from 1590s. Also from 1590s as "that which is best in quality, highest or most perfect state of anything." As "a prime number," by 1530s.

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prime (v.)

"to fill, charge, load" (a weapon, before firing), 1510s, probably from prime (adj.). General sense of "perform the first operation on, prepare (something, especially wood, etc., for painting)" is from c. 1600. To prime a pump (1769) meant to pour water down the tube, which saturated the sucking mechanism and made it draw up water more readily. Related: Primed; priming.

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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.

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prime (adj.)

late 14c., "first, original, first in order of time," from Old French prime and directly from Latin primus "first, the first, first part," figuratively "chief, principal; excellent, distinguished, noble" (source also of Italian and Spanish primo), from Proto-Italic *prismos, superlative of PIE *preis- "before," from root *per- (1) "forward," hence "in front of, before, first, chief."

The meaning "of fine quality, of the first excellence" is from c. 1400. The meaning "first in rank, degree, or importance" is from 1610s in English. Arithmetical sense (as in prime number, one indivisible without a remainder except by 1) is from 1560s; prime meridian "the meridian of the earth from which longitude is measured, that of Greenwich, England," is from 1878. Prime time originally (c. 1500) meant "spring time;" the broadcasting sense of "peak tuning-in period" is attested by 1961.

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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 1825.

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interest (v.)
"cause to be interested, engage the attention of," c. 1600, earlier interesse (1560s), from the noun (see interest (n.)). Perhaps also from or influenced by interess'd, past participle of interesse.
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interest (n.)
mid-15c., "legal claim or right; a concern; a benefit, advantage, a being concerned or affected (advantageously)," from Old French interest "damage, loss, harm" (Modern French intérêt), from noun use of Latin interest "it is of importance, it makes a difference," third person singular present of interresse "to concern, make a difference, be of importance," literally "to be between," from inter "between" (see inter-) + esse "to be" (from PIE root *es- "to be"). The sense development to "profit, advantage" in French and English is not entirely clear.

The earlier Middle English word was interesse (late 14c.), from Anglo-French interesse "what one has a legal concern in," from Medieval Latin interesse "compensation for loss," noun use of Latin interresse (compare German Interesse, from the same Medieval Latin source).

Financial sense of "money paid for the use of money lent" (1520s) earlier was distinguished from usury (illegal under Church law) by being in reference to "compensation due from a defaulting debtor." Sense of "personal or selfish consideration" is from 1620s. Meaning "business in which several people are interested" is from 1670s. Meaning "curiosity, feeling that something concerns one, appreciative or sympathetic regard" is first attested 1771. Interest group is attested from 1907; interest rate by 1868.
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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.).

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prime minister 

"leading minister of a government, the chief of the cabinet or ministry," 1640s, see prime (adj.) and minister (n.). Applied to the First Minister of State of Great Britain since 1694.

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