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The diameter of the circle below is divided itno three equal lengths, and those lengths are used to make the marble-like design that splits the circle into three different-colored sections.
The ares of sections I and III are equal, but what about the area of section II? Is it less than, greater than, or equal to the area of I?
Bear in mind the following three sizes of semicircle:
The biggest semicircles in the diagram.
The medium-sized semicircles, which a length two-thirds the circle's diameter.
The smallest semicircles, with a length one-third the circle's diameter.
use these different-sized semicircles to help you solve the puzzle.
Think of the area of the smallest semicircle as one.
The diameter of the medium semicircle is twice tha tof the small semicircle, so its area will be four times as large.
The big semicircle has a diameter three times as big as the small one, so its area will be nine times as large.
To figure out the area of sections I or III, use the semicircles as follows:
Big - Medium + Small =
Replacing these with the areas we found in Hint 2 gives us:
9 - 4 + 1 = 6
This should help you figure out how to determine the area of section II.
If you're followed along with all of the previous hints, the rest is easy.
The big semicircle has an area of nine. Since it's a semicircle, the area of the whole circle is twice that, or 18. From this, subtract the size of the areas of sections I and III, and you're left with the area of II.
All right then. What is it?
Solution
Incorrect
Too bad!
There's a simple way to solve this puzzle. See if you figure it out.
Correct
Precisely!
The areas of I and II are the same! The best way to think about it is to use the three sizes of semicircles as shown above. If the ratio of the diameters is 3:2:1, then the ratio of the areas is 9:4:1.
Using the ratios as a guide, we can compare the ares of I, II, and III and conclude that they are, in fact, equal.
