Why can the average/midpoint of two numbers be described as the sum of the numbers divided by two?

Question

Say I have two numbers, A and B. The "average" or the midpoint of the two numbers is given by $$\frac{A+B}{2}$$ My question is, why does this formula work?

Intuitively, I can derive the equation as follows. The "midpoint" of the two numbers can be given by: $$A+\frac{B-A}{2}$$ $$=\frac{2A}{2}+\frac{B-A}{2}$$ $$=\frac{A+B}{2}$$ But why? Why is it when you add two numbers and divided it by two you get its "midpoint"? I can't seem to find a way to intuitively visualize this. Please help, thank you!

Answer 1

Let's have this visualization:

$\mathtt{|----|------+------|----|}\\ O\hphantom{----}A\hphantom{------}\ \ I\hphantom{------}\ \ \,B\hphantom{---}A+B$

• $I$ is the midpoint of $[A,B]$
• but $\operatorname{dist}(O,A)=\operatorname{dist}(B,A+B)=a$

Thus $I$ is also the midpoint of $[O,A+B]$.

And since you agree it is $I=O+\dfrac{(A+B)-O}2$ and that $O$ can be identified to the zero of point addition, then $I=\dfrac{A+B}{2}$.

Answer 2

Suppose I have two measuring cups. One has $A$ ounces of water and the other has $B$ ounces. Also suppose that $B$ is bigger than $A$ and I want the same amount of liquid in each cup.

So I can pour some of the $B$ cup into the $A$ cup until the amounts are equal. How much is in each cup? I have a total of $A+B$ ounces and they are equally divided into 2 cups. So each cup has $(A+B)/2$ ounces.