Question

If C₁ and C₂ are the circumferences of the outer and inner circles respectively. What is C₁ : C₂?
(I) The two circles are concentric.
(II) The area of the ring is \(\frac 23\) the area of greater circle.

• A. Statement I alone is sufficient to answer the question.
• B. Statement II alone is sufficient to answer the question.
• C. Both statement I and II together are necessary to answer the question.
• D. Both statements I and II together are not sufficient to answer the question.

Answer 1

The Correct Option is B

The problem is to determine the ratio of the circumferences \(C_1 : C_2\) of the outer circle to the inner circle. To do this, let's evaluate the given statements individually: 

Statement I: The two circles are concentric. This information means that both circles share the same center, but it provides no information about their sizes or areas. Therefore, this statement alone is not sufficient to find the circumference ratio.

Statement II: The area of the ring is \(\frac{2}{3}\) the area of the greater circle. Let \(R\) be the radius of the greater circle (outer), and \(r\) be the radius of the smaller circle (inner). Thus, the area of the outer circle is \(\pi R^2\), and the area of the inner circle is \(\pi r^2\). The area of the ring is \(\pi R^2 - \pi r^2 = \frac{2}{3} \pi R^2\). Simplifying gives:

\(\pi R^2 - \pi r^2 = \frac{2}{3} \pi R^2\)

\(\pi R^2 - \pi r^2 = \frac{2}{3} \pi R^2\)

Expanding and simplifying:

\(\pi (R^2 - r^2) = \frac{2}{3} \pi R^2\)

\(R^2 - r^2 = \frac{2}{3} R^2\)

\(R^2 (1 - \frac{2}{3}) = r^2\)

\(\frac{1}{3}R^2 = r^2\)

\(r = \frac{\sqrt{1/3}}{1}R = \frac{R}{\sqrt{3}}\)

The circumference \(C_1 = 2\pi R\) and \(C_2 = 2\pi r\), so:

\(C_1 : C_2 = 2\pi R : 2\pi (\frac{R}{\sqrt{3}})\)

\(C_1 : C_2 = R : \frac{R}{\sqrt{3}}\)

\(C_1 : C_2 = \sqrt{3} : 1\)

Thus, Statement II is sufficient to determine the ratio of the circumferences.

Conclusion: Statement II alone is sufficient to answer the question, as it provides all needed information to find \(C_1 : C_2 = \sqrt{3} : 1\).