The ratio of the weights of two solid spheres is 8 : 27, and the ratio of their weights per cubic centimetre (densities) is 8 : 1. What is the ratio of the radii of the two spheres?

Introduction / Context:
This question combines ideas from ratio and basic geometry of spheres. The weight of a solid sphere is proportional to its volume multiplied by the density of its material. Volume of a sphere is proportional to the cube of its radius. Given the ratio of weights and the ratio of densities, we are asked to find the ratio of the radii. This type of problem appears in aptitude tests where understanding proportional relationships in geometry is important.

Given Data / Assumptions:

• Weight ratio of the two spheres W1 : W2 = 8 : 27.
• Density ratio (weight per cubic centimetre) rho1 : rho2 = 8 : 1.
• Let radii of the spheres be r1 and r2.
• Weights are given by W = density * volume.

Concept / Approach:
For a sphere, volume V is proportional to r^3. Hence weight W is proportional to rho * r^3. Using proportionality, W1 / W2 = (rho1 * r1^3) / (rho2 * r2^3). We are given the ratio of W1 : W2 and rho1 : rho2, so we can substitute those and solve for r1^3 : r2^3, and then for r1 : r2 by taking cube roots. This is a standard method whenever radius and weight are linked through volume and density.

Step-by-Step Solution:
Let W1 and W2 be weights and rho1 and rho2 be densities.Given W1 : W2 = 8 : 27 and rho1 : rho2 = 8 : 1.For spheres, W is proportional to rho * r^3, so W1 / W2 = (rho1 * r1^3) / (rho2 * r2^3).Substitute ratios: 8 / 27 = (8 * r1^3) / (1 * r2^3).So 8 / 27 = 8 * r1^3 / r2^3.Divide both sides by 8: (8 / 27) / 8 = r1^3 / r2^3.This gives r1^3 / r2^3 = 1 / 27.Take cube roots on both sides: r1 / r2 = cube root of (1 / 27) = 1 / 3.Therefore, the ratio of radii r1 : r2 = 1 : 3.

Verification / Alternative check:
Assume r1 = 1 unit and r2 = 3 units.Then r1^3 = 1 and r2^3 = 27.Using rho1 : rho2 = 8 : 1, let rho1 = 8k and rho2 = k for some k.Then W1 = 8k * 1 = 8k and W2 = k * 27 = 27k.So W1 : W2 = 8k : 27k = 8 : 27, which matches the given weight ratio.

Why Other Options Are Wrong:
Ratios such as 2 : 3, 3 : 1 or 3 : 2 would give r1^3 : r2^3 as 8 : 27, 27 : 1 or 27 : 8. When these are combined with rho1 : rho2 = 8 : 1, they do not produce W1 : W2 = 8 : 27. Only r1 : r2 = 1 : 3 leads to the correct result when substituted back into the proportional relationship.

Common Pitfalls:
Some students mistakenly assume that the ratio of radii is directly 8 : 27 or confuse radius with volume. Others forget that volume of a sphere depends on the cube of radius and not on the square. Always set up the proportional relationship W proportional to rho * r^3 and carefully isolate the ratio of r1 to r2 from the given ratios.

Final Answer:
The ratio of the radii of the two spheres is 1 : 3.