Evaluate: (0.04³ + 0.02³) ÷ (0.2³ + 0.1³)

Correct — A. Each number on top is 0.2 times the matching number below, so pull 0.2³ out instead of expanding.

Rewrite: 0.04 = 0.2 × 0.2 and 0.02 = 0.2 × 0.1
Factor the cubes: 0.04³ + 0.02³ = 0.2³ × (0.2³ + 0.1³)
Cancel the common bracket: quotient = 0.2³
Cube: 0.2 × 0.2 × 0.2 = 0.008

Check with direct values: 0.000064 + 0.000008 = 0.000072, and 0.008 + 0.001 = 0.009
Divide: 0.000072 ÷ 0.009 = 0.008 → option (a)

Why the others are wrong

• (b)
  0.004 — 0.004 is half the quotient: 0.004 × 0.009 = 0.000036, only half of the numerator 0.000072.
• (c)
  0.002 — 0.002 is a quarter of the quotient: 0.002 × 0.009 = 0.000018, one quarter of the numerator 0.000072.
• (d)
  0.016 — 0.016 is double the quotient: 0.016 × 0.009 = 0.000144, twice the numerator 0.000072.

When every term inside a sum of cubes is multiplied by the same factor k, the sum is multiplied by k³: (ka)³ + (kb)³ = k³(a³ + b³).

Here 0.04 and 0.02 are 0.2 × 0.2 and 0.2 × 0.1, so the numerator is exactly 0.2³ times the denominator and the quotient is 0.2³ = 0.008. Spotting the factor avoids six-decimal-place arithmetic, where one miscounted zero wrecks the answer.

Key facts

• (ka)³ + (kb)³ = k³(a³ + b³).
• 0.2³ = 0.008 and 0.1³ = 0.001.
• Cubing a decimal triples its decimal places: 0.04 has two, so 0.04³ = 0.000064 has six.

Common traps

• Miscounting decimal places in the cubes: 0.04³ is 0.000064, with six places, not 0.00064.
• Taking the common factor as 0.2 instead of 0.2³: the factor comes out of each cube, so it is cubed too.

No directly related past PYQ was found.