Resolve a radical equation and identify the invariant ratio: If √0.05 * 0.5 * a = 0.5 * 0.05 * √b, find the value of a^2 / b.

Introduction / Context:The original statement asked for a/b, which cannot be uniquely determined from the given relation. Using the recovery-first policy, we minimally refine the target to a^2 / b, an invariant ratio fully determined by the equation. This preserves the intent and yields a solvable, well-posed task.

Given Data / Assumptions:

• Equation: √0.05 * 0.5 * a = 0.5 * 0.05 * √b.
• All quantities are real and positive for the square roots to be real.
• Goal: compute a^2 / b.

Concept / Approach:First cancel the common factor 0.5 from both sides. Then isolate a in terms of √b, and square to obtain a relation between a^2 and b. Finally, form the requested ratio.

Step-by-Step Solution:

Verification / Alternative check:Pick b = 100 for convenience: a^2 = 0.05 * 100 = 5 ⇒ a = √5. Substituting into the original equation satisfies the equality and yields a^2 / b = 5 / 100 = 0.05, confirming the result.

Why Other Options Are Wrong:0.0025 and 0.025 are smaller by factors of 20 or 2. 0.25 is too large. The derived identity fixes the ratio exactly at 0.05.

Common Pitfalls:Attempting to compute a/b directly (underdetermined), or mishandling the division 0.05 / √0.05 instead of recognizing it equals √0.05.

Final Answer:0.05