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Half, third, quarter... what is a fifth and sixth part called?

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I really don't think that there is a name for fifths and sixths beyond just that --- fifth and sixth. The reason that it may be useful to have names from half, third, and quarter is because we can easily visually look at something and identify what 1/2, 1/3 or 1/4 of that something would look like. Whereas if we break the whole into more and more parts, in our minds the fractions' physical meanings are not readily visible. There really doesn't seem to be a point to having additional names like that.

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I've heard quint and sextuplet but not necessarily for fractions

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