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

The volume of a sphere is directly proportional to the cube of its diameter. Let the diameters of the two new balls be 1x and 2x. Their volumes are proportional to (1x)³ = 1x³ and (2x)³ = 8x³. Since the original iron ball was melted to form these two balls, the total volume of the original ball is the sum of the volumes of the new balls: 1x³ + 8x³ = 9x³. The question asks for the ratio of the volume of the smaller new ball (1x³) to the original ball (9x³). This ratio is 1x³: 9x³, which simplifies to 1: 9. This calculation relies on the principle that volume remains conserved during melting and recasting, and that for any sphere, the volume formula V = (4/3)πr³ can be expressed in terms of diameter (d) as V = (π/6)d³.