The most common and most likely answer to this crossword clue is the 6 plus 8 letter word PROPER FRACTION
I've seen this crossword clue in The Daily Telegraph
P
R
O
P
E
R
F
R
A
C
T
I
O
N
Buddy explains!
The straight part
We know that "half, for example" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case half, for example = proper fraction.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.