Question

How do you find a midpoint of two numbers ?

Answer 1

Find their average: take their sum, and divide it by 2.

Explanation:

The midpoint `m` of two numbers `a` and `b` is, by definition, a number that is the same distance from `a` as it is from `b`.

Using this, we can get a formula for the midpoint `m` in terms of `a` and `b`.

Picture `a` and `b` as two points on a number line, with `a` to the left of `b`. Now, `m` must be between them, so `m` is greater than `a` but less than `b`. So if the distance from `a` to `m` is the same as the distance from `m` to `b`, we can set these two distances to be equal, like this:

`m-a=b-m`

This equation says: "The difference between `m` and `a` is the same as the difference between `b` and `m`."

Now, all we do is rearrange this to equation to solve for `m`:

`m-a color(green)(" "+ m)=b"          "` (add `m` to both sides)

`"       "2m"      "=bcolor(green)(" "+a)"   "` (add `a` to both sides)

`"        "m"       "=(b+a)/color(red)2"   "` (divide both sides by 2)

And there it is—the formula for the midpoint! This is often called the average of the two numbers `a` and `b`, and written as

`m=(a+b)/2`

Example:

What is the midpoint of 3 and 5? Well, the answer is intuitively 4, but with the formula, we get:

`m=(3+5)/2=8/2=4`

which matches our intuition.

Answer 2

The method Geoff K gave also works with a combination of negative and positive numbers

Explanation:

Suppose we have `-3 and +3` then the mid point is `(-3+3)/2=0`

`" "`or if you prefer you can write this as `(+3-3)/2=0`

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Suppose we have `-3 and +5` then the mid point is

`(-3+5)/2=2/2=1`