Etymology
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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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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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cheap (adj.)

"low in price, that may be bought at small cost," c. 1500, ultimately from Old English noun ceap "traffic, a purchase," from ceapian (v.) "to trade, buy and sell," probably from early Germanic borrowings of Latin caupo "petty tradesman, huckster, peddler," cauponari "to haggle" (see chapman). Compare, from the same borrowing, German kaufen "to buy," Old Norse kaupa "to bargain, barter," Gothic kaupon "to traffic, trade."

The sense evolution is from the noun meaning "a barter, a purchase" to "a purchase as rated by the buyer," hence the adjectival meaning "inexpensive," the main modern sense, via Middle English phrases such as god chep "favorable bargain" (12c., a translation of French a bon marché).

Sense of "lightly esteemed, common" is from 1590s (compare similar evolution of Latin vilis). The meaning "low in price" was represented in Old English by undeor, literally "un-dear" (but deop ceap, literally "deep cheap," meant "high price").

The word also was used in Old English for "market" (as in ceapdæg "market day"), a sense surviving in place names Cheapside, East Cheap, etc. To do something on the cheap "with very little expense" is from 1859. Cheap shot originally was U.S. football jargon for a head-on tackle; extended sense "unfair hit" in politics, etc. is by 1970.

German billig "cheap" is from Middle Low German billik, originally "fair, just," with a sense evolution via billiger preis "fair price," etc.

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