EPFO EO/AO 2023 General Ability Test · Question 55 of 120

Correct — C, (c) Total volume will remain the same and total surface area will increase. The whole item can be settled with arithmetic, and the arithmetic is worth doing in full because it shows exactly how much the surface area increases. First, read the stem carefully: the eight smaller cubes each have the same side AS EACH OTHER, not the same side as the original. Eight is two cubed, so the parent cube of side ten centimetres is halved in each of its three directions and each small cube has a side of five centimetres. Volume. The original cube holds ten times ten times ten, that is 1000 cubic centimetres. Each small cube holds five times five times five, that is 125 cubic centimetres, and eight of them hold 1000 cubic centimetres. Unchanged, and necessarily so — cutting rearranges matter without creating or destroying any of it. Surface area. The original cube has six faces of a hundred square centimetres each, that is 600 square centimetres. Each small cube has six faces of twenty-five square centimetres each, that is 150 square centimetres, and eight of them present 1200 square centimetres. The total surface area has exactly doubled. The doubling can be seen without multiplying anything. Three cuts are needed, one through the middle in each direction. Every cut exposes two new faces, each of area a hundred square centimetres, so the three cuts add six such faces — 600 square centimetres — to the 600 the cube already had. The surface that appears is precisely the surface that was interior before. This is the general result behind the question: subdividing a solid never changes its volume and always increases its total surface area, so the ratio of surface to volume rises. Here it goes from 0.6 to 1.2 per centimetre. That the solid is an ice cube is not decoration — it is the reason the fact matters, because melting happens at the surface, and crushed ice melts far faster than the same mass in one block.

• (a)Total volume will increase and total surface area will decrease. — Wrong on both counts, and it inverts the one relationship the question is built on. Volume cannot increase: no ice has been added, and the eight pieces together occupy exactly what the parent occupied. Surface area cannot decrease either, because every cut can only expose material that was previously enclosed. Reading the option as a description of what happens when ice MELTS does not save it — melting reduces volume, since water is denser than ice, and in any case the stem describes cutting rather than melting.
• (b)Total volume will decrease and total surface area will increase. — Half right, and it is the option that costs most marks. The surface area does increase, and a candidate who has grasped the real point of the question may stop reading there. But nothing is lost in the cutting: the eight pieces contain 1000 cubic centimetres of ice between them, exactly what the single cube contained. The instinct that volume falls seems to come from imagining sawdust or melt-water at the cut; the question is a geometrical one, and geometry loses nothing.
• (d)Total volume will increase and total surface area will remain the same. — Wrong on both counts in the other direction. Volume is conserved rather than increased, and the surface area is exactly doubled rather than preserved. The option would describe a solid that had been stretched without being cut — and even then the surface would change. It is the least tempting of the four, and its function in the set is to fill the fourth cell of the grid of possible combinations.

For a cube of side a, the volume is a cubed and the total surface area is six a squared. Because the two grow at different rates, the ratio of surface area to volume — six divided by a — falls as the object gets bigger and rises as it gets smaller. Subdividing a solid exploits that: the material is unchanged, but it is redistributed into smaller pieces, each with a high surface-to-volume ratio, and the totals add up to a much larger exposed surface. Every process that happens AT a surface therefore speeds up. Crushed ice cools a drink faster than a block of the same mass; powdered sugar dissolves faster than a lump; a catalyst is used finely divided so that more of it is available to react; a finely divided combustible dust can explode where the solid merely burns; food is chewed before it is digested; and the same principle explains why the absorptive surfaces of living bodies are folded and branched — the villi of the intestine, the alveoli of the lung, the root hairs of a plant. The ratio also explains a fact about heat: a small animal has far more surface per unit of mass than a large one, so it loses heat faster and must eat proportionately more to maintain its temperature.

This item sits in the general science block rather than the quantitative one, and the placement is a hint about what is being tested — not the mensuration, which is a school formula, but whether the candidate knows why surface area matters in physical and biological processes. The Commission has framed it around ice because melting is the process everyone has watched. The habit rewarded is doing the small computation rather than trusting an impression: the numbers here take fifteen seconds and settle the question completely, whereas the intuition that something must be lost in cutting is exactly the intuition that fails.

• For a cube of side a, volume is a cubed and total surface area is six a squared.
• A cube of side ten centimetres has a volume of 1000 cubic centimetres and a surface area of 600 square centimetres.
• Dividing it into eight equal cubes gives each a side of five centimetres, since eight is two cubed.
• The eight small cubes together still occupy 1000 cubic centimetres — volume is conserved under subdivision.
• Their total surface area is 1200 square centimetres, exactly double the original.
• Each of the three cuts exposes two new faces of a hundred square centimetres, adding 600 square centimetres in all.
• The surface-to-volume ratio rises from 0.6 to 1.2 per centimetre.
• Processes that occur at a surface — melting, dissolving, burning, catalysis, absorption — are faster when the same material is finely divided.

• Reading 'each with same side' as the same side as the original cube rather than as each other.
• Stopping at the first half of an option that is correct and ignoring the second half.
• Importing melting into a question that only describes cutting.
• Assuming that because the pieces are smaller, the total of something must have got smaller.

This idea reaches EO/AO papers in two shapes — a computation on a subdivided or reassembled solid, as here, and an application question asking why powdered or crushed material behaves differently from a block. Both are answered from the same pair of formulae plus the observation that cutting conserves volume and creates surface, so it is worth rehearsing the arithmetic once until it is automatic.

No directly related past PYQ was found.

• practice — not a real PYQ

  A cube of side 6 cm is cut into 27 identical smaller cubes. The total surface area of the smaller cubes taken together is :

  • (a)216 square cm
  • (b)432 square cm
  • (c)648 square cm
  • (d)1296 square cm

  Answer(c) 648 square cm

• practice — not a real PYQ

  Crushed ice cools a drink faster than a single block of ice of the same mass mainly because the crushed ice has :

  • (a)A lower melting point
  • (b)A greater total surface area exposed to the liquid
  • (c)A greater total volume
  • (d)A higher latent heat of fusion

  Answer(b) A greater total surface area exposed to the liquid