Recently I came across the weighed average. I get how it works on a technical level. Maybe I'm a bit thick, but somehow I can't get a "feel" for it on an intuitive level. Let's just say we have two points on a line:
|-----*-----------*--------->
0 A B
Now, we want to put a point $M$ somewhere between $A, B$ inclusive.
- If we want to put it at exactly at $A$, we can say $M = 1 \times A + 0 \times B$
- If we want to put it at exactly at $B$, we can say $M = 0 \times A + 1 \times B$
That seems kind of obvious. What seems to somehow surprise me is, that for example, if we want to put it exactly between $A, B$, we can say $M = 0.5 \times A + 0.5 \times B$
Somehow I have a hard time grokking that this will put the point $M$ at the middle.
The distance from $A$ to $B$ is $B-A$. If we start at $A$, and then walk half that distance, we should end up precisely at the midpoint $M$ between them. After all, that's what "midpoint" means. Remembering that "walking along the number line" means adding the distance walked to the starting point, we get $$ M = A + \frac12(B-A) = A + \frac{B-A}{2}\\ = \frac{2A+B-A}{2} = \frac{A+B}{2} = \frac A2 + \frac B2 $$