Evaluate the ratio: [0.5*0.5*0.5 + 0.2*0.2*0.2 + 0.3*0.3*0.3 − 3*0.5*0.3*0.2] / [0.5*0.5 + 0.2*0.2 + 0.3*0.3 − 0.5*0.2 − 0.2*0.3 − 0.5*0.3]. Use the identity for x^3 + y^3 + z^3 − 3xyz.

Introduction / Context:
This expression is a classic application of the identity x^3 + y^3 + z^3 − 3xyz = (x + y + z)(x^2 + y^2 + z^2 − xy − yz − zx). When the numerator and denominator match the two factors in this identity, the ratio collapses to x + y + z, dramatically simplifying the computation.

Given Data / Assumptions:

• x = 0.5, y = 0.2, z = 0.3.
• Numerator: x^3 + y^3 + z^3 − 3xyz.
• Denominator: x^2 + y^2 + z^2 − xy − yz − zx.

Concept / Approach:
Invoke the factorization identity directly. Since the numerator equals (x + y + z)(denominator), the entire fraction equals x + y + z (provided x + y + z ≠ 0). This avoids computing each cube and product explicitly, though you can still verify numerically if you wish.

Step-by-Step Solution:
Recognize identity: x^3 + y^3 + z^3 − 3xyz = (x + y + z)(x^2 + y^2 + z^2 − xy − yz − zx).Given denominator equals the second factor, the ratio = x + y + z.Compute x + y + z = 0.5 + 0.2 + 0.3 = 1.0.

Verification / Alternative check:
Explicit calculation confirms the same: Numerator = 0.125 + 0.008 + 0.027 − 0.09 = 0.070; Denominator = 0.25 + 0.04 + 0.09 − 0.10 − 0.06 − 0.15 = 0.07; Ratio = 0.07 / 0.07 = 1.

Why Other Options Are Wrong:

• 0.6 / 0.4 / 0.03 / 0.9: These result from misapplying the identity or arithmetic mistakes in the numerator/denominator.

Common Pitfalls:
Forgetting the exact structure of the identity; computing many decimals and rounding midstream; sign errors in −3xyz and the pairwise products in the denominator.

Final Answer:
1