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

Step 1: Set up the relation between circumference and radius.
For any circle, circumference is proportional to radius: C = 2πr. So if R is the radius of the outer circle and r is the radius of the inner circle, then \(C_1 : C_2 = R : r\).
Step 2: Examine statement (I) alone.
Statement (I) only says the two circles are concentric, that is, they share the same centre. This tells us nothing about how large one radius is compared to the other; R and r could be in any ratio at all. So statement (I) alone is not sufficient.
Step 3: Examine statement (II) alone.
Statement (II) says the area of the ring (the region between the two circles) is 2/3 of the area of the greater (outer) circle. Area of ring = πR² - πr². This equals (2/3)πR². So πR² - πr² = (2/3)πR².
Step 4: Solve statement (II) for the radius ratio.
Divide through by πR²: 1 - (r/R)² = 2/3, so (r/R)² = 1/3, giving r/R = 1/√3, or equivalently R/r = √3.
Step 5: Convert radius ratio to circumference ratio.
Since \(C_1 : C_2 = R : r = \sqrt{3} : 1\), statement (II) alone completely determines the ratio of the circumferences, regardless of whether the circles are concentric or not, because the area formula πr² only depends on the radius, not on the position of the circle.
Step 6: Conclude.
Statement (I) alone is not sufficient, but statement (II) alone is sufficient to find \(C_1 : C_2\). This matches option (2).