Another way to think of "reicht" is its English cognate, "reaches". If you think metaphorically, you can see how "reaches" could be equivalent to "suffice" or "be enough". Compare it to the idiomatic use of "stretch" in "we'll have to stretch the beer so there'll be enough for everyone".
In English, "sixth" can mean two things:
1) The piece you get if you cut something into six pieces ("fraction", i.e. half, third, quarter, etc.); or
2) The position you come in a race if you cross the finish line after five other people ("ordinal number", i.e. first, second, third, fourth, etc.).
Sechstel is the fraction. The ordinal number is sechste. Generally the ending -stel is used for fractions and -te for ordinal numbers.
The cardinal numbers are eins, zwei, drei, vier, fuenf, sechs ...
There is a German word "(das) Teil" meaning "part". I expect that's where the ending "-tel" comes from, slightly changed. I wondered what the gender of the noun "Sechstel" is so I've looked it up and it's neuter. That suggests that I'm right about fractions like "das Sechstel" and "das Teil" being related.