Two concentric circles form a ring (annulus). If the area of the ring equals two-thirds of the area of the larger circle, determine the ratio of circumferences C1 : C2 (outer : inner).

Introduction / Context:
This Data Sufficiency item concerns an annulus (a ring) formed by two concentric circles. We must decide whether the statements are sufficient to obtain the ratio of circumferences C1:C2 of the outer and inner circles.

Given Data / Assumptions:

• Concentric means they share the same center.
• Let outer radius be R and inner radius be r (R > r).
• Area of ring = area(outer) − area(inner) = πR^2 − πr^2.
• Statement I: circles are concentric.
• Statement II: ring area is 2/3 of the area of the larger circle.

Concept / Approach:
The circumference C is proportional to radius: C = 2π * radius, so C1:C2 = R:r. The area relation in Statement II directly links R and r.

Step-by-Step Solution:
From Statement II: πR^2 − πr^2 = (2/3) * πR^2 ⇒ r^2 = (1/3) * R^2.Hence r = R / √3 (positive radii assumed).Therefore C1:C2 = (2πR):(2πr) = R:r = √3:1.Statement I only asserts concentricity; without a quantitative link, it is insufficient.

Verification / Alternative check:
Plugging r = R/√3 back gives ring area fraction 1 − (1/3) = 2/3 of the larger circle, confirming consistency.

Why Other Options Are Wrong:
Statement I alone gives no numbers; both-together is unnecessary since II alone suffices; claiming insufficiency contradicts the direct derivation.

Common Pitfalls:
Confusing area ratio with circumference ratio; forgetting that circumference ratio equals radius ratio.

Final Answer:
Statement II alone is sufficient (C1:C2 = √3:1).