If p = 0.8 and q = 0.02, find the value of ((4p)³+(4q)³)⁄(p³+q³).

Correct — D.
Expand the cubes: (4p)³ = 64p³ and (4q)³ = 64q³
Take out the common factor: 64p³ + 64q³ = 64(p³ + q³)
Divide by the denominator: 64(p³ + q³) ÷ (p³ + q³) = 64

Check with the values: (3.2³ + 0.08³) ÷ (0.8³ + 0.02³)
= 32.768512 ÷ 0.512008 = 64
So the value is 64 → option (d)

Why the others are wrong

• (a)
  56 — Each cube carries 4³ = 64, not 56: (4p)³ = 64p³ and (4q)³ = 64q³. The ratio is 64 whatever values p and q take.
• (b)
  37 — 37 bears no relation to 4³. Taking 64 out of each cube gives 64(p³ + q³), which cancels against the denominator to leave 64.
• (c)
  59 — 59 falls short of 4³ = 64. Multiplying p and q both by 4 multiplies each cube, and so their sum, by exactly 64.

Scaling rule: multiply every variable in a sum of cubes by k and the sum is multiplied by k³, so (kp)³ + (kq)³ = k³(p³ + q³).

Here k = 4, so the ratio is 4³ = 64 for any p and q with p³ + q³ ≠ 0. The given decimals matter only if you miss the common factor.

The same idea works for squares, with factor k², and for any power: take out the common factor first.

Key facts

• (kp)³ + (kq)³ = k³(p³ + q³)
• 4³ = 64
• 0.8³ = 0.512 and 0.02³ = 0.000008

Common traps

• Cubing 3.2 and 0.08 by hand and slipping a decimal place, when the ratio never depends on p and q
• Writing (4p)³ as 4p³ or 12p³ instead of 64p³

No directly related past PYQ was found.