Evaluate: (0.05³ + 0.01³) ÷ (0.25³ + 0.05³)

Correct — A. Look for a common factor before cubing anything.

Numerator: 0.05 = 5 × 0.01, so 0.05³ + 0.01³ = 0.01³ × (5³ + 1) = 0.01³ × 126
Denominator: 0.25 = 5 × 0.05, so 0.25³ + 0.05³ = 0.05³ × (5³ + 1) = 0.05³ × 126
Cancel the 126: (0.01 ⁄ 0.05)³ = (1⁄5)³ = 1⁄125
Convert: 1⁄125 = 0.008 → option (a)

Why the others are wrong

• (b)
  0.004 — 0.004 is half the answer. Multiply back: 0.004 × 0.01575 = 0.000063, half the numerator 0.000126.
• (c)
  0.006 — 0.006 × 0.01575 = 0.0000945, short of the numerator 0.000126. The ratio is exactly (1⁄5)³ = 0.008.
• (d)
  0.002 — 0.002 = 1⁄500, a quarter of 1⁄125. The cancelled ratio is (0.01⁄0.05)³ = (1⁄5)³ = 1⁄125 = 0.008.

When a³ + b³ is divided by c³ + d³ and a : b = c : d, the two sums share a factor. Write a = kb and c = kd, and the fraction becomes b³(k³ + 1) ⁄ d³(k³ + 1) = (b⁄d)³.

Here a : b = 0.05 : 0.01 = 5 : 1 and c : d = 0.25 : 0.05 = 5 : 1, so k = 5 in both.

The answer (0.01⁄0.05)³ also equals (a⁄c)³ = (0.05⁄0.25)³ = (1⁄5)³.

Key facts

• a³ + b³ = (a + b)(a² − ab + b²).
• If a : b = c : d, then (a³ + b³) ⁄ (c³ + d³) = (b⁄d)³ = (a⁄c)³.
• (1⁄5)³ = 1⁄125 = 0.008.

Common traps

• Miscounting decimal places when cubing: 0.05³ = 0.000125, six decimal places, two for each factor.
• Splitting the fraction term by term, as 0.05³⁄0.25³ + 0.01³⁄0.05³. That gives 0.008 + 0.008 = 0.016, double the answer.

No directly related past PYQ was found.