Given √(0.04 × 0.4 × a) = 0.4 × 0.04 × √b, determine the ratio a / b.

Introduction / Context:
This problem involves equating radicals and linear terms with decimals. By squaring both sides appropriately, we can isolate the ratio a / b exactly.

Given Data / Assumptions:

• √(0.04 × 0.4 × a) = 0.4 × 0.04 × √b.
• All quantities are real and positive.

Concept / Approach:
Square both sides to remove square roots, then solve for a in terms of b. Take care with decimal multiplication to avoid errors.

Step-by-Step Solution:
Left side: √(0.04 × 0.4 × a) = √(0.016 a). Right side: 0.4 × 0.04 × √b = 0.016 √b. Square both sides: 0.016 a = (0.016)^2 b = 0.000256 b. Thus, a / b = 0.000256 / 0.016 = 0.016.

Verification / Alternative check:
Replace a = 0.016 b in the original: LHS = √(0.016 * 0.016 b) = √(0.000256 b) = 0.016 √b = RHS, confirming the ratio.

Why Other Options Are Wrong:
1.60 and 0.16 are off by factors of 100 or 10 due to decimal mishandling. “None of these” is wrong because 0.016 matches exactly.

Common Pitfalls:
A common mistake is failing to square the decimal on the right side or misplacing decimal points in 0.016^2. Carefully compute 0.016 × 0.016 = 0.000256.

Final Answer:
0.016