What is the value of (0.4×0.4×0.4 + 0.02×0.02×0.02) ÷ (1.2×1.2×1.2 + 0.06×0.06×0.06)?

Correct — A. Each denominator term is the matching numerator term multiplied by 3.

Pair: 1.2 = 3 × 0.4 and 0.06 = 3 × 0.02
Cube: 1.2³ = 27 × 0.4³ and 0.06³ = 27 × 0.02³
Factor: denominator = 27 × (0.4³ + 0.02³)
Divide: the bracket cancels, leaving 1⁄27 → option (a)

Why the others are wrong

• (b)
  5⁄27 — 5⁄27 is five times the true value. The denominator is exactly 27 × the numerator (1.728216 = 27 × 0.064008), so the quotient has 1 on top.
• (c)
  11⁄67 — 11⁄67 ≈ 0.164, about 4.4 times 1⁄27 ≈ 0.037. A shared factor of 3 makes the quotient exactly 1⁄3³, and 67 is not a power of 3.
• (d)
  41⁄27 — 41⁄27 is greater than 1, yet each denominator term is larger than its partner on top (1.2 > 0.4, 0.06 > 0.02). The quotient must be below 1.

Common-factor cancellation: if every denominator term is k times its numerator partner, the denominator (ka)³ + (kb)³ equals k³(a³ + b³). The bracket cancels and the value is 1⁄k³.

Pair the terms before multiplying anything: 0.4 with 1.2 and 0.02 with 0.06, both with k = 3.

Key facts

• (ka)³ + (kb)³ = k³(a³ + b³).
• 3³ = 27: a factor of 3 on every term multiplies a sum of cubes by 27.
• 1⁄27 ≈ 0.037.

Common traps

• Cancelling the factor without cubing it, which gives 1⁄3.
• Inverting the quotient: the larger sum is below the line, so the answer is 1⁄27, not 27.

No directly related past PYQ was found.