2006 AMC 10B Problem 4
Circles of diameter inch and inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area?
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Key insight
The big circle is 3 times as wide, so its area is 9 times the red circle. The blue ring is that minus the red: 8 times.
Solution
1. Find each area. The red circle has diameter , so radius and area .
2. The large circle has diameter , so radius and area .
3. The blue part is the large circle with the red circle cut out of its middle. So
4. Divide blue by red:
The answer is .
Why this works
When one circle is times as wide as another, its area is times as big, because area grows with the square of the width. So you never really need the radii: the big circle is "red circles" of area, and removing the one red circle in the middle leaves . Look for ring-shaped regions as "big minus small."
Concise solution
A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.
The diameters are in ratio , so the areas of the two full circles are in ratio . Call the red area ; then the large circle has area .
The blue region is everything inside the large circle but outside the small one, so its area is .
The ratio of blue to red is .
The answer is .
Review in the AoPS books
- Introduction to Geometry, Chapter 11: Circles
- Introduction to Algebra, Chapter 6: Ratios and Percents
- Introduction to Geometry, Chapter 12: Circles and Angles
- Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
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Problem © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. Solution and commentary are original to this site.