Evaluate [(0.1 × 0.1 × 0.1) + (0.02 × 0.02 × 0.02)] ÷ [(0.2 × 0.2 × 0.2) + (0.04 × 0.04 × 0.04)].

Problem restatement
Compute the ratio of the two sums of cubes of small decimals.

Given data

• Left sum: 0.1³ + 0.02³
• Right sum: 0.2³ + 0.04³

Concept/Approach
Evaluate each cube exactly (powers of 10 are easy), add, then simplify the fraction by noticing a common factor.

Step-by-step calculation
Left = 0.001 + 0.000008 = 0.001008Right = 0.008 + 0.000064 = 0.008064Ratio = 0.001008 ÷ 0.008064 = 1008 ÷ 8064 = 1/8

Verification/Alternative
Both numerator and denominator are multiples of 1008: 8064 = 8 × 1008.

Common pitfalls
Rounding intermediate cubes (e.g., treating 0.02³ as 0.00002) which distorts the ratio.

Final Answer
1/8