Question

If \(C_1\) and \(C_2\) are the circumferences of the outer and inner circles respectively, what is \(C_1 : C_2\)?
(I) The two circles are concentric.
(II) The area of the ring is \(\frac{2}{3}\) 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.

Hint:

Area of ring = (2/3) x outer area gives R:r directly from statement II alone.

Answer 1

The Correct Option is B

This can also be approached by directly assigning a convenient variable to the outer radius and expressing everything as a ratio, which avoids fractions until the very end.

1. Let the outer radius be $R$ and the inner radius be $r$. The circumferences are $C_1 = 2\pi R$ and $C_2 = 2\pi r$, so $C_1 : C_2 = R : r$ always, no matter where the circles are positioned.
2. Statement (I) only states the circles share a centre. Concentricity affects the geometry of the annular region's shape but not the relationship between the two radii, so it gives no numerical handle on $R:r$. Not sufficient alone.
3. Statement (II) states: (area of outer circle) $-$ (area of inner circle) $= \frac{2}{3}\times$ (area of outer circle). Writing this out:\[ \pi R^2 - \pi r^2 = \frac{2}{3}\pi R^2 \]
4. Cancel $\pi$ and rearrange:\[ R^2 - r^2 = \frac{2}{3}R^2 \Rightarrow r^2 = R^2 - \frac{2}{3}R^2 = \frac{1}{3}R^2 \]
5. Take the square root:\[ \frac{r}{R} = \frac{1}{\sqrt{3}} \Rightarrow \frac{R}{r} = \sqrt{3} \]
6. So $C_1 : C_2 = R : r = \sqrt{3} : 1$, fully determined using statement (II) alone. Statement (I) adds no extra numerical information beyond this.

\[\boxed{C_1 : C_2 = \sqrt{3} : 1 \text{, found from statement II alone (option 2)}}\]