I get . I'm not sure how you got to twelfths from twenty-firsts (21 being the LCD of 3 and 7)
Spoiler:
More generally lets say you have a line between and and you want to find the midpoint (we'll call this ). By definition the midpoint (c) is equidistant from either end (a and b). Thus .
With a little bit of manipulation: and . This is the same as the (arithmetic) mean of a and b.