Is this a good way of explaining weighted average?

In all the discussions I have seen of weighted average, the formula is always given for n values with n associated weights. What is nice is that weighted average can be defined solely in terms of combining two values at a time.

For example, suppose you are taking a course and 80% of your coursework so far has a grade point average of 93 and then you get a 98 on the final, which counts for the remaining 20%, what is your final grade? It makes no difference how you got the 80%. The final grade is (80×93 + 20×98)/100.

Suppose we designate a weighted average as a pair of numbers (w,v) where the w stands for weight and the v stands for value. The sum of two of these gives a simple formula. (w1,v1) + (w2,v2) = (w1+w2, (w1 v1 + w2 v2)/(w1+w2)). We get a new value with a weight that is the sum of the weights. Those of you who are mathematically inclined will see that if we add three (w,v) pairs, the formula works out to the general formula for three values, and that we can keep going on up to any n values.

Here is another example. You may recall from geometry class that the center of gravity, or centroid, of a triangle is the intersection of the medians, which meet at a point ⅔ along any median, starting at the vertex. Not very intuitive.

If the points of the triangle are P1, P2 and P3, it makes good intuitive sense that the center of gravity is (P1+P2+P3)/3, where we take separate averages of the x and y coordinates. Think of this as a weighted average of the 3 points where each point has a weight of 1.

What happens if we do the weighted average 2 at a time. We get (1,P1) + (1,P2) = (2, (P1+P2)/2). (P1+P2)/2 is the midpoint of the segment from P1 to P2. When we add P3, we are going to end up with a point between P3 and this midpoint, which is a point along the median from P3. The midpoint gets a weighting of 2 and P3 contributes its weighting of 1. If we designate the midpoint as M, we get Center of gravity = (2,M) +(1,P3) = (3, (2M+P3)/3). The 2 to 1 ratio of the weights means that the point is closer to M than to P3 by a factor of 2 to 1, which accounts for the ⅔ factor in a rather intuitive fashion.