The phrase 'on the money' meaning exactly right or dead center, can be traced back to a competition in Archery. In this competition, the coin would be laid on a target a certain distance away (perhaps a radial target as used in the modern Olympics, or even just the ground) and competitors would take turns trying to shoot their arrows closest to the coin, with the closest the winner. Hence, right on the money, hitting the coin, would be exactly correct as you could get no closer!
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Originally, money came from the Latin moneta, a surname of Juno, in whose temple at Rome money was coined.