Etymology
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*re- 

*rē-, Proto-Indo-European root meaning "to reason, count;" a variant of PIE root *ar-, also arə-, "to fit together." 

It forms all or part of: Alfred; arraign; arithmetic; Conrad; dread; Eldred; Ethelred; hatred; hundred; kindred; logarithm; Ralph; rate (n.) "estimated value or worth;" rathskeller; ratify; ratio; ration; read; reason; rede; rhyme; riddle (n.1) "word-game;" rite; ritual.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit radh- "to succeed, accomplish;" Greek arithmos "number, amount;" Latin reri "to consider, confirm, ratify," ritus "rite, religious custom;" Old Church Slavonic raditi "to take thought, attend to;" Old Irish im-radim "to deliberate, consider;" Old English rædan "to advise, counsel, persuade; read;" Old English, Old High German rim "number;" Old Irish rim "number," dorimu "I count."

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rat-race (n.)

also rat race, 1934 in reference to aviation training, from rat (n.) + race (n.1).

A rat race is ... a simple game of "follow the leader" in fighter planes. The leader does everything he can think of — Immelmanns, loops, snap rolls, and turns, always turns, tighter and tighter. [Popular Science, May 1941]

In the 1930s actual rat races of some sort are frequently mentioned among popular carnival and gambling attractions. Meaning "fiercely competitive struggle," especially to maintain one's position in work or life is by 1939. Rat-run is from 1870 in the sense of "maze-like passages by which rats move about their territory," but originally and usually in a derogatory transferred sense.

[Matthew] Milton was not, at the period we write of [c. 1811], at all in the ring ; for in the following March he was steward of a rat-race, held at a public-house in Shepherd's-market, where four of these "varmin," decorated with different coloured ribands, were started for a sweepstakes, round the clubroom, before a host of sportsmen. ["Sporting Incidents at Home and Abroad," The Sporting Review, May 1848]
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*pau- (1)

Proto-Indo-European root meaning "few, little."

It forms all or part of: catchpoll; encyclopedia; filly; foal; few; hypnopedia; impoverish; orthopedic; Paedophryne; paraffin; parvi-; parvovirus; paucity; Paul; pauper; pedagogue; pederasty; pedo-; pedophilia; poco; poltroon; pony; pool (n.2) "game similar to billiards;" poor; poulterer; poultry; poverty; puericulture; puerile; puerility; puerperal; pullet; pullulate; Punch; Punchinello; pupa; pupil (n.1) "student;" pupil (n.2) "center of the eye;" puppet; pusillanimous; putti.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit potah "a young animal," putrah "son;" Avestan puthra- "son, child;" Greek pauros "few, little," pais (genitive paidos) "child," pōlos "foal;" Latin paucus "few, little," paullus "little," parvus "little, small," pauper "poor," puer "child, boy," pullus "young animal;" Oscan puklu "child;" Old English feawe "not many, a small number," fola "young horse;" Old Norse fylja "young female horse;" Old Church Slavonic puta "bird;" Lithuanian putytis "young animal, young bird;" Albanian pele "mare."

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