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/