Componendo property in ratio: If a : b = b : c (i.e., a/b = b/c), then which of the following equals a^4 : b^4?

Introduction / Context:
From a : b = b : c, it follows that a/b = b/c, giving a key identity: b^2 = a c. The question asks you to express a^4 : b^4 in another equivalent form using this relationship. Such manipulations are common in algebraic ratio problems.

Given Data / Assumptions:

• a : b = b : c ⇒ a/b = b/c.
• All variables are positive for ratio interpretation.

Concept / Approach:
From a/b = b/c, cross multiplication gives a c = b^2. Then manipulate a^4 : b^4 by expressing b^4 in terms of a and c: b^4 = (b^2)^2 = (a c)^2 = a^2 c^2.

Step-by-Step Solution:
Given a c = b^2. Compute b^4 = (b^2)^2 = (a c)^2 = a^2 c^2. Therefore, a^4 : b^4 = a^4 : (a^2 c^2) = a^2 : c^2.

Verification / Alternative check:
Try sample values satisfying a : b = b : c. Let a = 1, b = 2, c = 4 (since 1/2 = 2/4). Then a^4 : b^4 = 1 : 16. Also a^2 : c^2 = 1 : 16, confirming the identity.

Why Other Options Are Wrong:

• ac : b^2 equals 1 : 1 under the same condition, not a^4 : b^4.
• c^2 : a^2 is the inverse.
• b^2 : ac again gives 1 : 1.
• ab : c^2 has no direct equivalence here.

Common Pitfalls:
Forgetting that b^2 = a c and thus b^4 = a^2 c^2; mixing up which terms square or invert leads to incorrect options.

Final Answer:
a^2 : c^2