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

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!

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 $$I=\dfrac{A+B}{2}$$.

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.