Geometric and Arithmetic Means

Date: 5/10/96 at 13:7:36
From: Sid Himes
Subject: Geometric and Arithmetic Means  

Dear Dr. Math,

My son is a junior at Harborcreek High School in Erie, PA.  The 
following problem was part of a homework assignment (that has already 
been turned in). The textbook did not supply any solutions, at least 
none that I could find. Please give me an idea on how to tackle this 
problem:

Find the values of two numbers whose sum is 20 and whose:
 a) geometric mean is 8
 b) arithmetic mean is 8

Many thanks from a concerned father.

Date: 11/14/96 at 21:52:00
From: Doctor Robert
Subject: Re: Geometric and Arithmetic Means

The geometric mean between two numbers x and y is sqrt(xy). So, if the 
two numbers must add to 20, let one of them be x.  The other one must 
be 20-x.  Now the geometric mean is sqrt(x(20-x)) = 8.

Squaring both sides of this equation:  x(20-x) = 64
                                -x^2 + 20x -64 = 0 

Solving this for x you get x = 4, 16

The two numbers are 4 and 16 or, if you wish, 16 and 4.

The arithmetic mean is nothing but the average.  Again, the two 
numbers are x and 20-x and their average is 8:

                                  (x + 20-x)/2 = 8
                                          20/2 = 8. 

This statement is never true. Therefore, there are no numbers which 
add to 20 and have an arithmetic mean of 8. 

If you think about it, any two numbers which add to 20 would have to 
have an arithmetic mean of 10.

-Doctor Robert,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/