Averages with Examples

Date: 10/17/2001 at 22:47:29
From: Eric Grootendorst
Subject: Averages

Hi,

My Math 10 teacher has explained to us three ways to figure out 
averages. Not to calculate the average of, say, 3,5,8,6 = 5.5, but 
kind of like weighted averages, but without the weight. I have read 
5 or 6 of your articles but still don't understand. To make your job 
easier, I will include some of the questions that I am stuck on. 

If m pens are bought at n dollars each, and n pens are bought at m 
dollars each, write an expression for the average cost per pen.

In a group of men and women, the average age is 31. If the men's ages 
average 35 years and the women's ages average 25, what is the ratio of 
the number of men to women?

Mark has taken 7 tests and his average is 60%. What would he have to 
average on the next three tests to raise his overall average to 70%?

Thanks, this will really help me!

Date: 10/18/2001 at 10:12:18
From: Doctor Ian
Subject: Re: Averages

Hi Eric,

Thanks for providing specific questions! That's always more helpful 
than a general plea to 'explain' some topic. 

>If m pens are bought at n dollars each, and n pens are bought at m 
>dollars each, write an expression for the average cost per pen.

In this problem, you can start from the definition of average: 
Add up the individual values, and divide by the number of values. 
In this case, the average would look like
  
       m times               n times
   ________________      ________________
  /                \    /                \
  $n + $n + ... + $n  + $m + $m + ... + $m
  -----------------------------------------
                   m + n

We could rewrite this as

                 n times
              ________________
             /                \
  (m * $n) + $m + $m + ... + $m
  ------------------------------
          m + n

Can you see how to come up with the final expression? 

>In a group of men and women, the average age is 31. If the men's ages 
>average 35 years and the women's ages average 25, what is the ratio 
>of the number of men to women?

Again, it's easiest to start from the definition of average, rather 
than try to memorize some special rules or definitions. (Are you 
seeing a pattern here?) If there are m men, and the average age is 35 
years, then that's the same thing as having m men whose ages are 
exactly 35:

       m * 35
  35 = ------
         m

Similarly for the women:

       w * 25
  25 = ------
         w

So the average of the men and women together will be

       (m * 35) + (w * 25)
  31 = -------------------
              m + w

which should look somewhat familiar. 

To find the ratio of men to women, you want to fool around with this 
equation to end up with something that looks like 

  m/w = ...

I'll leave that for you to do. (But feel free to write back if you get 
stuck.) 

>Mark has taken 7 tests and his average is 60%. What would he have to 
>average on the next three tests to raise his overall average to 70%?

Starting once again from the definition of average, this looks like

        7 times
   ________________
  /                \
  60 + 60 + ... + 60 + x + x + x
  ------------------------------ = 70
            7 + 3

Or, to look at it another way, to average 70 points for 10 tests, he 
needs a total of 700 points. Right now, he has 7 * 60 = 420. Which 
means that he needs to make up 280 points in 3 tests. What would he 
have to average to do that? 

In the end, there is really only one way to compute an average, which 
is to add up the values and divide by the number of values. Once you 
have a firm grasp on that idea, you can always use it as a starting 
point, and you'll rarely get into trouble. Once you can write down 
what both sides of the equation have to look like, you can start 
looking for clever ways to compute whatever information is missing. 

I hope this helps. Write back if you have more questions, about this 
or anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/