Sunday, April 5, 2009

The thrilling conclusion to "Do you remember SAT math?"

Two weeks ago, I posted two SAT-style Quantitative Comparison questions. This week, I am posting the solutions (sorry, I know it's a bit late).

If we let the radius of the inner circle be r and the length of segment BC be s such that the radius of the outer circle is r+s, then we can express all the quantities that we are interested in in terms of r and s.

The length of arc CE is:
¼(2π(r+s)) = ½πr + ½πs

The length of arc BD plus the length of segment DE is:
¼(2πr) + s = ½πr + s

The length of segment CB plus arc BD plus the length of segment DE is:
s +¼(2πr) + s = ½πr + 2s

When comparing each pair of terms, the ½πr cancel each other out. Because s is a length, we know that s>0, which means that ½πs is always greater than s, but is always less than 2s. In terms of answering the quantitative comparison questions, this means the answer to the first question must be A and the answer to the second question must be B.

Put another way, if you are walking around a 90 degree arc (such as from point C to point E), it will always be faster to walk around the outer arc than to walk to an inner arc, traverse it, and return to the outer arc. What is particularly interesting is that this is true regardless of the radius of the inner arc.

Now before you start taking the outside edge of every curve, note that this only holds true for arcs less than a certain size. For example, what if it you were traversing a semicircle instead?

The length of the outer arc would be:
½(2π(r+s)) = πrs

The length of the two straightaways plus the inner arc would be:
s +½(2πr) + s = πr + 2s

Because πs is greater than 2s, it makes more sense to walk to the inner arc when walking around a semicircle! So where is the breaking point? This can be determined by letting x be the fraction of the circle to traverse and setting the two path lengths equal to one another and solving for x:

x(2π(r+s)) = x(2πr) + 2s
xr+2πxs = 2πxr + 2s
xs = 2s
πx = 1
x = 1 / π

Plugging that into Google Calculator, we see that x is ~0.318, so once you're walking more than a third of the way around a circle, it makes more sense to move to the inner arc.

I thought that this would make a great SAT question for several reasons. First, and most importantly, this is a quantitative comparison question that does not have any numbers, yet its answer is not D! I'm pretty sure that most students hated geometry (so sad!), so when they come to something like this, it's pretty easy to throw up one's hands and declare the question unsolvable. Of course such a student would be incorrect...

Also, this question is resistant to some common SAT-solving techniques. For example, for geometry questions where the figure is to scale (such as this one), The Princeton Review will tell you to make small marks on your answer sheet so you can use it as a ruler, but because this problem involves arcs, that is of little help. Another common SAT tip is to redraw the diagram, exaggerating r or s, but if you play around with that, you can probably convince yourself that there are cases where one quantity is larger than the other, but it's hard to say conclusively that it holds true for arbitrary values of r and s.

That means the only ones who will be answering the question correctly are those who can do the math. Isn't that how it should be?


  1. The ultimate goal of online psychology research paper writing services is to provide Psychology Assignment Writing Services and psychology research paper services since most psychology coursework writing service students lack time to complete their custom psychology coursework writing services.

  2. Students find Human Resource Writing Services as being of great assistance since they are able to complete their human resource assignment writing services and human resource research paper writing services on time.

  3. This is interesting article..Thanks for sharing this wonderful information. 온라인카지노

  4. Ahaa, its good discussion about this post here at this webpage, I have read all that, so at this time me also commenting
    here. 온라인카지노

  5. 카지노사이트 What's up it's me, I am also visiting this website daily, this website
    is genuinely nice and the visitors are in fact sharing fastidious thoughts.

  6. 토토사이트 Useful info. Lucky me I found your web site
    by chance, and I am surprised why this twist of fate
    didn't came about in advance! I bookmarked it.

  7. 스포츠토토 I'm extremely impressed with your writing skills as well as with the
    layout on your blog. Is this a paid theme or did you modify it yourself?

    Either way keep up the nice quality writing, it's rare
    to see a great blog like this one these days.

  8. Thanks for the blog loaded with so many information. Stopping by your blog helped me to get what I was looking for.

  9. I like your post. It is good to see you verbalize from the heart and clarity on this important subject can be easily observed..

  10. This is an excellent post I seen thanks to share it. It is really what I wanted to see hope in future you will continue for sharing such a excellent post.

  11. Уour blog providеd us useful information to work on. Үou have done a marvelous job! 바카라사이트