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Is there a country that uses the term “dollar” for currency without also using “cents” as fractional monetary units?
Going through the list of economies that currently use the dollar, all of them list cents as a fractional unit. In Hong Kong and Taiwan, the 1/100 fractional unit is still called a cent, but it's no longer in circulation in coin form and only finds use in financial markets or electronic payments.  In countries like Malaysia, the word "sen" is used as the translation of the word "cent", even though the word for the actual currency, "ringgit", isn't a translation of the word "dollar".  A similar situation occurs in Panama. The local currency is called the balboa, and it's priced on par (1:1) with the US dollar. US banknotes are also accepted as legal tender, and Panamanians sometimes use the terms balboa/dollar interchangeably. The 1/100 subdivision of the balboa is the centésimo, which is merely a translation of cent. Like Malaysia, the fractional unit is called "cent" (or a translation) but the main unit isn't merely a translation of the word "dollar." On a historical note, the Spanish Dollar was subdivided into 8 reales in order to match the German thaler (the word that forms the basis for the English word "dollar").