A solid spherical ball made of iron is melted and two new balls are made whose diameters are in the ratio of 1 : 2. The ratio of the volume of the smaller new ball to the original ball is

Correct — D, 1 : 9. Diameters in the ratio 1 : 2 mean radii in the ratio 1 : 2, and the volume of a sphere is (4/3)πr³, so the two new balls have volumes in the ratio 1³ : 2³ = 1 : 8. Melting destroys the shape but not the metal, so the original ball's volume equals the sum of the two new ones — 1 + 8 = 9 parts. The smaller new ball is therefore 1 part out of 9, and the ratio asked for is 1 : 9.

• (a)1 : 3 — Adds the diameter ratio before cubing it, treating the original as 1 + 2 = 3 parts. Volume grows as the cube of a linear dimension, so the cubing has to happen first.
• (b)1 : 5 — Matches no consistent step. It sits between the uncubed 1 : 3 and the correct 1 : 9, which makes it attractive to anyone hedging between the two.
• (c)2 : 9 — Has the right total of nine parts but doubles the small ball. Once the volumes are 1 and 8, the smaller one is a single part of the nine, not two.

Every volume scales as the cube of any linear dimension, so a ratio of radii, diameters or circumferences becomes a ratio of volumes only after it is cubed. Recasting problems add a second idea: melting conserves volume, so the volumes before and after must balance. Putting the two together turns this from a mensuration calculation into two lines of arithmetic.

The trap is the word 'diameters'. A candidate who is comfortable with radii sometimes hesitates and converts, but the conversion is unnecessary — dividing both diameters by two leaves the ratio untouched. The second trap is answering the question that was not asked. The natural quantity to compute is the ratio of the two new balls, 1 : 8, and the paper offers no such option; what is wanted is the smaller new ball against the original, which is 1 : 9. Note that π and the factor 4/3 appear in every volume here and cancel in every ratio, so they never need to be written down at all.

• The volume of a sphere is (4/3)πr³, so volume varies as the cube of the radius or of the diameter.
• Diameters in the ratio 1 : 2 give volumes in the ratio 1 : 8.
• Melting and recasting conserves total volume, so the new volumes must add up to the original.
• In nine equal parts, the original ball is 9, the smaller new ball is 1 and the larger new ball is 8 — so the larger one alone is 8 : 9 of the original.
• Surface area scales as the square of a linear dimension, which is why the same two spheres have surface areas in the ratio 1 : 4.

• Cubing after adding instead of before.
• Converting diameters to radii unnecessarily, which wastes time and invites arithmetic slips.
• Reporting the ratio of the two new balls, 1 : 8, when the original ball was the comparison asked for.

Asked as a recasting item where the given ratio is linear and the ratio wanted is volumetric, with the conserved total supplying the denominator.

• practice — not a real PYQ

  A solid sphere of radius 3 cm is melted and recast into solid spheres of radius 1 cm each. How many such spheres are formed?

  • (a)3
  • (b)9
  • (c)27
  • (d)81

  Answer(c) 27 — volumes are in the ratio 3³ : 1³, so one large sphere yields 27 small ones.

• practice — not a real PYQ

  Two spheres have radii in the ratio 2 : 3. The ratio of their surface areas is

  • (a)2 : 3
  • (b)4 : 9
  • (c)8 : 27
  • (d)16 : 81

  Answer(b) 4 : 9 — surface area varies as the square of the radius, so the ratio is 2² : 3².