Quantiles and Quintiles

Will anyone tell me which is the last quintile in a group with 10 numbers? Will it be the very last number, or goes above it so that between each quintile there'll be exactly 2 numbers?

The idea of quantiles is based upon the ideal of an infinitely large set of numbers, and therefore compromise is necessary to extend the concept to a finite set.  Unfortunately, there are several compromises in use that will yield different answers.  Different statistical software packages can also yield different answers based upon the same data.

coninuing my answer after finding my notes:
When based on the simplest way of defining quantiles for finite sets, the kth q-quantile has at most k/q of the n numbers ranked less while at least that proportion ranked less or equal (and consequently, at most 1-k/q of the numbers would be ranked more).

With repeated numbers in the data set, the same value may have two or more associated ranks and quantiles.  Any quantile associated with a repeated number other than the lowest ranked of the repeating set loses some validity in meaningfulness.

The smaller the set relative to q and the more the repetition of values in it, the less meaningful are any declarations of quantile values.  (infinite continuous sets don't have this trouble)

More complicated methods of compromise involve averaging or interpolation.

So, the 5th 5-quantile (quintile) has at most 5/5 (100%) of the 10 numbers ranked less, while at least 5/5 (100%) ranked less or equal.

Using the simplest way of defining compromised quantiles for finite sets, the largest number is the qth q-quantile, so then the largest number naturally is the 5th quintile.  I have read of a preference that every named quantile be an actual member of the data set.  However, many common (and more complicated) methods violate that preference.