AVERAGE OF TWO SIGNED NUMBERS

A teacher reports an average grade on a test.
You read about the average number of calories burned per hour for your favorite exercise.
What do these figures mean?
The purpose of this section is to discuss the concept, the computation, and some important properties of averaging.

To average two numbers means to add the numbers together, and then divide by [beautiful math coming... please be patient] $\,2\,$. Thus:
[beautiful math coming... please be patient] $$ \text{the average of } \,a\, \text{ and } \,b\, \text{ is } \,\frac{a+b}{2} $$ Averaging two different numbers always gives the number exactly halfway between, as illustrated below:

average of two numbers

In this web exercise, you will compute averages of two numbers, where the numbers can be [beautiful math coming... please be patient] $\,-10, -9, \ldots, 9, 10\,$.
You should be able to do this exercise without a calculator!
It is good practice with mental arithmetic, and will reinforce your skills with addition of signed numbers.

There are two key ideas to keep in mind:

Clearly, the formula [beautiful math coming... please be patient] $\,\frac{a+b}2\,$ gives some number;
but how do we know that the number given by this formula is really, always, halfway between [beautiful math coming... please be patient] $\,a\,$ and [beautiful math coming... please be patient] $\,b\,$?
Although repeated trials (with lots of different numbers) is pretty convincing, it is of course impossible to check every pair of real numbers.
To see a proof, read the text!

EXAMPLES:
Question: What is the average of [beautiful math coming... please be patient] $\,-3\,$ and $\,-5\,$?
Solution: Visualize [beautiful math coming... please be patient] $\,-3\,$ and $\,-5\,$ on the number line.
What number is halfway between?
Answer:   $-4$
Question: What is the average of [beautiful math coming... please be patient] $\,7\,$ and $\,10\,$?
Solution: Visualize [beautiful math coming... please be patient] $\,7\,$ and $\,10\,$ on the number line.
What number is halfway between?
Answer:   [beautiful math coming... please be patient] $8.5$
Question: What is the average of [beautiful math coming... please be patient] $\,-7\,$ and $\,9\,$?
Solution: These numbers are too far apart to easily visualize halfway between.
Do the arithmetic:   [beautiful math coming... please be patient] $\frac{-7+9}2 = \frac{2}2 = 1$
Answer:   $1$
Master the ideas from this section
by practicing both exercises at the bottom of this page.

When you're done practicing, move on to:
Average of Signed Numbers

 
 

In this exercise, you must write your answers in decimal form (as needed).
That is, write (say) $\,2.5\,$, not $\,5/2\,$.

    
(an even number, please)
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10
AVAILABLE MASTERED IN PROGRESS

(MAX is 10; there are 10 different problem types.)