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Problem 402

Geometry Difficulty 2.6 Multiple choice CEMC Gauss (Grade 7) · Canada · 2018

In the diagram, the area of the shaded middle ring is 6 times the area of the smallest circle. The area of the unshaded outer ring is 12 times the area of the smallest circle.

What fraction of the area of the largest circle is the area of the smallest circle?

Pick one

Official solution

We may begin by assuming that the area of the smallest circle is 1.

The area of the shaded middle ring is 6 times the area of the smallest circle, and thus has area 6.

The area of the unshaded outer ring is 12 times the area of the smallest circle, and thus has area 12.

The area of the largest circle is the sum of the areas of the smallest circle, the shaded middle ring, and the unshaded outer ring, or $1+6+12=19$ .

Therefore, the area of the smallest circle is $\frac{1}{19}$ of the area of the largest circle.

Note: We assumed the area of the smallest circle was 1, however we could have assumed it to have any area. For example, assume the area of the smallest circle is 5 and redo the question. What is your final answer?

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.