When does "part" mean "quarter"?

‘Behind us in the caves of the Deep are three parts of the folk of Westfold, old and young, children and women,’ said Gamling. ‘But great store of food, and many beasts and their fodder, have also been gathered there.’

JRR Tolkien, The Lord of the Rings, Volume 2: “The Two Towers”, Book Three, Chapter VII, “Helm’s Deep”

By three parts, Gamling means “three parts of four”, or “three quarters”. Or, at least, so I assume. Why do I assume that? It’s an assumption built in so deeply that I have no idea where it comes from. Is it a correct assumption? How long has the assumption that everything has four parts (or, rather, that a part means a quarter) been around? And is it still safe to use language that way today?

It's not that the word part has any direct connection with quarter. Tolkien's usage is just a variant on the same trope that gives us...

possession is nine parts/points/tenths of the law

If Tolkien had written "four parts of the folk" (unlikely, I know), it would have meant four fifths. Once you specify how many parts you have, the implication is there's one more unaccounted for.

But I do think three is a good number for these contexts - three quarters is a familiar fraction, two is too small for any expression emphasising scale, and four fifths is getting complicated. We're much more likely to say three parts drunk or three sheets to the wind, rather than four or five.