Associated Topics || Dr. Math Home || Search Dr. Math

Mean Proportionals and Geometric Means

```
Date: 01/06/99 at 19:49:02
From: Mary
Subject: Mean proportional

How do you find the mean proportional of two numbers? I've never known
how to do this.
```

```
Date: 01/06/99 at 20:13:11
From: Doctor Pat
Subject: Re: Mean proportional

Mary,

The mean proportional is also called the geometric mean. One way to
define it is to say GM(a,b) = sqrt(a*b). This says "the geometric mean
of a and b is the square root of their product." Here is an example:

GM(4,9) = sqrt(4*9) = sqrt(36) = 6

So 6 is the mean proportional (geometric mean) of four and nine.

The reason it is called the mean proportional is that the question can
also be written as a proportion. We want to find a number x for 4 and
9 so that the following proportion is true:

4     x
--- = ---
x     9

If we set the cross products equal and get x^2 = 36, you can see why we
use the method above to get the same answer.

Sometimes they may ask for two mean proportionals between a pair of
numbers, and that is a lot tougher to work all the way through. Let's
do an example to show what it means.

Find two mean proportionals between 3 and 24. This means find two
numbers, call them x and y, so that:

3     x     y
--- = --- = ---
x     y     24

If you work with this a little you can find that:

24x = y^2   and
x^2 = 3y

So y = x^2/3 and y^2 = x^4/9, and so x^3 = 24*9, leading to x = 6 and
y = 12. But there is an easier way, and it will work for any number of
"mean proportionals." If we look at the sequence of numbers in a mean
proportional, we (sooner or later) notice a pattern. Here are the two
we have used so far:

4, 6, 9

and:

3, 6, 12, 24

The second pattern is easier to see. What is the rule for finding the
next number? I hope you said multiply by two, since each number is
2 times the one before. Look back at the first sequence; in that one it
was 1.5 times the one before. Mean proportionals always work that way.
So we can just look for this common ratio between numbers.

Knowing that it is there means if we want two proportionals between a
and b, we write b = a*r^3 and solve for r to find the sequence. If we
wanted three proportionals (not always whole numbers) we would write
b = a*r^4, and so on. For example, in the second part above we wanted
two mean proportionals between 3 and 24, so we use 3*r^3 = 24, which
leads us to r^3 = 8 and r = 2. Then we can write out 3, 3*2, 3*2*2,
3*2*2*2 and get all the terms. A sequence like this is called a
geometric sequence, and that is one of the reasons for calling it the
geometric mean.

Good luck,

- Doctor Pat, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Sequences, Series
Middle School Ratio and Proportion

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search