Average Circles

[shyra1353]

While studying for my SATs, I came across a couple of questions that required you to use average circles to answer them. I have never heard of an average circle, and neither have the people I have asked. I ran a search on google, but didn't find anything remotely useful. So here is my question, what is an average circle, when and how do you use it, and is there a simpler way ??

For those of you wondering, the original question is:
The average (arithmetic mean) of the weight (in pounds) of a kennel of v dogs is 40, and the average weight (in pounds) of a kennel of w dogs is 64. When the weights of both kennels are combined, the average is 54. What is the value of (v/w)?
If anyone is interested, I will post the correct answer later ...

And for the record, an average circle is NOT a circle that is not too big and not too small, but is just right .. lol ...

[Astronouth7303]

hmm....
so for the first group (A), there are V values that have a mean of 40. The second group (B) has W values that have a mean of 64. So, Total(A) = V*40 and Total(B) = W*64. And (Total(A) + Totak(B))/(V + W) = 54

(system here)
{
 Total(A) = V*40
 Total(B) = W*64
 (Total(A) + Totak(B))/(V + W) = 54
}

(40V + 64W)/(V + W) = 54
40V + 64W = 54*(V + W)
40V + 64W = 54V + 54W
40V + 10W = 54V
10W = 14V
V/W = 1.4

[Chris Hibner]

Originally Posted by Astronouth7303

hmm....
so for the first group (A), there are V values that have a mean of 40. The second group (B) has W values that have a mean of 64. So, Total(A) = V*40 and Total(B) = W*64. And (Total(A) + Totak(B))/(V + W) = 54

(system here)
{
 Total(A) = V*40
 Total(B) = W*64
 (Total(A) + Totak(B))/(V + W) = 54
}

(40V + 64W)/(V + W) = 54
40V + 64W = 54*(V + W)
40V + 64W = 54V + 54W
40V + 10W = 54V
10W = 14V
V/W = 1.4

Small mistake:

10W = 14V
10/14 = V/W
V/W = 1/1.4
or
V/W = 0.7143

[shyra1353]

wow .. okay that is so much simpler than anything come up so far. thanks for your help. and it is the right answer .. (the .714)

but the question still remains .. "what is an average circle ??"

[Chris Hibner]

Originally Posted by shyra1353

wow .. okay that is so much simpler than anything come up so far. thanks for your help. and it is the right answer .. (the .714)

but the question still remains .. "what is an average circle ??"

I did a quick search on "average circle" and one thing came up with an example problem just like the one you described at the beginning of this thread. It was from some SAT preparation thing.

It's my guess that "average circle" is some fancy name that some SAT prep company came up with to describe a solution method to the type of problem that you describe. In other words, "average circle" has nothing to do with circles (in a geometric sense), but I think it is more of a touchy-feely descriptor (kind of like "the circle of life" or "your solution has come full-circle".) The "average circle" is probably like the "food pyramid", if you know what I mean.

If you give us the context surrounding "average circle", perhaps we can better help decrypt the meaning. Is it described anywhere in the test-prep documents, or did you only find it in an example problem?

I have never heard of the term before now.

-Chris

[Astronouth7303]

Me neither. I assumed that if you plot it right, it looks like a circle.

[shyra1353]

it was used in the answer to the question ... i cant access the actual answer right now, but it said something like "since the question uses the word average three times, you have three average circles." then it goes on to form the three equations Astronouth made above, but the explanation did not make complete sense.

I assumed it was a technique that the book uses, because I hadnt studied the math section yet, but when I flipped through the book it didnt mention anything about average circles, and neither did any of the online lessons.

[George1902]

When I teach my SAT prep classes, I always tell my students to remember ANT on average problems:

Average * Number of items = Total

Always find the total in average problems. Do any math associated using the totals. Convert back to average at the end if you have to.

I think the term "average circle" might be referring to converting from average to total and back again.

[mtrawls]

Originally Posted by shyra1353

it was used in the answer to the question ... i cant access the actual answer right now, but it said something like "since the question uses the word average three times, you have three average circles."

Hrm... are you sure it wasn't more like "...you have three averages, circle" (using circle as a verb, and separately from the preceding word averages)? Searching pages of google, the majority of references seem to be for circle and average just by chance appearing side by side, with no implied concept of an "average circle." I did find a reference to an average circle dealing with astronomy (probably not relevant here), and a vague reference in a neuroscience research paper (more than likely not relevant). (Oh, and the term seems to apply to eyebrow or self nipple piercings as well ... and I really *hope* those aren't relevant.)

It seems to me if the term was used in the context of the theories talked about here, it would warrant an explanation, and one you'd probably remember. Otherwise I'm tempted to say the two words were unrelated. (Perhaps, though, I ought not overestimate SAT help books?) But, of course ... the actual text would help

[Phil 33]

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.