I first came accross the average circle while studying for the SAT's too (in a Princeton Review book). Basically it provides a visual way to solve average problems, by drawing circles. I'm not about to try to draw a bunch of circles using ASCII art, I uploaded an image I made in paint instead.
The average pie just helps you visualize the relationship they want you to understand
Total = Avg. * # of things
Avg = Total / # of things
etc....
The example problem you gave would require you to draw 3 average circles.
(If the image doesn't appear, go here to view it:
www.sonicimpulse.com/robotics/average.JPG )
Someone already worked out the math, but what the heck.
Using the 3rd average pie, you can use the relationship:
Total = Avg. * # of things
to solve
40v + 64w = 54(v+w)
40v + 64w = 54v + 54w
10w = 14v
10/14 = v/w
v/w = .714
From someone who's taken the SAT twice, I can tell you that, for whatever reason, the average pie actually does help you on the SAT. (even though it feels like a second grade thing) You quickly draw one, fill in the info you're given, and solve for what you don't know. It does help you visualize what the question is asking.
Thats all an average circle is, a visual to help you solve the problem.