Why can we say "half a [something]" but not "quarter a [something]"?
Solution 1:
"Half a cup" sounds like "Half of a cup" So I imagine any word that ends in f and would normally require the word "of" for it to make sense, have "of" left out because it sounds like it's still there. (I can't think of any others examples so I'm still unsure).
"Quarter of a cup"