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### 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.

or anything else.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Statistics

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