Enforcement Officer / Accounts Officer (EO/AO) 2020 recruitment test, advertisement 421. Transcribed from the Commission's own booklet (series A, NMJX-U-GTA); answer as per the official UPSC final answer key. This question is in Part B. Answer last reviewed 24 August 2026.

A hollow closed cube of side 10 cm, each weighing 200 gm, is made. This cube is placed in water with a horizontal face. How many cm of its height sinks in water if the density of water is 1 gm per cm^3 ?

(a)1 cm
(b)1·5 cm
(c)2 cm
(d)2·5 cm

Answer

Why

Correct — C, (c) 2 cm. The step that decides this item is Archimedes' principle in the form it takes for a floating body: an object floats at the depth at which the weight of the water it pushes aside equals its own weight. So the quantity to compute is not the cube's volume but the volume of the part below the waterline, and because the cube is set down with a face horizontal, that part is a flat box measuring 10 cm by 10 cm by the sunk height h. Its volume is 100h cubic centimetres, and since water has a density of 1 gm per cm^3, the water displaced weighs 100h grams. Setting that equal to the weight of the cube, 200 gm, gives 100h = 200 and h = 2 cm. The arithmetic can be done without algebra at all: 200 gm of water occupies 200 cubic centimetres, and spreading 200 cubic centimetres over a footprint of 100 square centimetres gives a depth of 2 cm. That is the whole question, and notice which measurement carried it — the area of the horizontal face, not the cube's own volume of 1,000 cubic centimetres, which never enters. The law of flotation gives the same answer from the other direction and is worth seeing: the fraction of a floating body that is submerged equals the ratio of its average density to the liquid's. This cube weighs 200 gm and occupies 1,000 cubic centimetres of space, so its average density is 0·2 gm per cm^3, one-fifth that of water; one-fifth of its 10 cm height is 2 cm. That is also why the word 'hollow' is in the stem — a solid iron cube of this size would weigh several kilograms and sink, and it is the empty space inside that brings the average density below water's and lets it float, which is the same reason a steel ship floats. On the printed wording: the stem reads 'A hollow closed cube of side 10 cm, each weighing 200 gm', and 'each' sits oddly against a single cube. The 200 gm is the weight of the cube as a whole, which is what the keyed answer of 2 cm follows from; read as 200 gm per face, the six faces would come to 1,200 gm and the same equation would give 12 cm, deeper than the cube is tall and matching no option on the paper.

Why the others are wrong

(a)1 cm — 1 cm of sunk height would displace 10 x 10 x 1 = 100 cubic centimetres of water, weighing 100 gm, and a floating body displaces its own weight — so this depth belongs to a cube weighing 100 gm, half of the 200 gm the stem states. The relation here is linear and worth stating once, because it makes every option on the item testable at sight: with a footprint of 100 square centimetres in water of unit density, the sunk depth in centimetres is simply the weight in grams divided by 100. A candidate who has that in mind reads the option list as a list of weights — 100 gm, 150 gm, 200 gm, 250 gm — and picks the one the question actually gave.
(b)1·5 cm — 1·5 cm would displace 150 cubic centimetres of water, weighing 150 gm, so it is the answer for a cube weighing 150 gm rather than 200 gm. There is no natural misreading of this stem that produces it, which makes it a filler in a well-designed set: the four options span the answer on both sides so that a candidate who has the method but slips in the arithmetic still finds somewhere to land. Note the typography — the booklet prints the decimal point as a raised middle dot, as it does in the last option and elsewhere on the paper, and it is the Commission's house style rather than a printing accident.
(d)2·5 cm — 2·5 cm would displace 250 cubic centimetres of water, weighing 250 gm, which is 50 gm more than the cube weighs; the cube would have to be pushed down to sit that deep and would rise again when released. This is the closest wrong option and the one to be careful about if the arithmetic is done hurriedly. It is also a good place to notice the sanity check that flotation problems always allow: the sunk depth cannot exceed the height of the body, and the fraction submerged must equal the ratio of the body's average density to the liquid's. Here 2·5 cm out of 10 cm would mean an average density of a quarter that of water, which would require the cube to weigh 250 gm, and it does not.

Concept

Two statements do all the work in flotation problems and both are worth holding exactly. Archimedes' principle says that a body immersed wholly or partly in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. The law of flotation adds that a body that floats does so in equilibrium, so the buoyant force exactly balances its weight — which means a floating body displaces a weight of fluid equal to its own weight, no more and no less. Put together they yield the fraction that a general question is usually built on: the proportion of a floating body's volume that lies below the surface equals the ratio of the body's average density to the density of the liquid. A body of average density 0·8 gm per cm^3 floats in water with four-fifths of its volume submerged; a body denser than the liquid cannot float at all. The word 'average' is what makes a hollow object interesting. Iron has a density around 7·8 gm per cm^3 and a solid iron object sinks, but a hollow one encloses air, and the mass spread over the whole enclosed volume can easily come out below 1 gm per cm^3. That is the principle a ship's hull works on, and it is what the stem is pointing at when it specifies a hollow closed cube. The last piece is dimensional care: 1 gm per cm^3 for water is the same thing as 1,000 kg per m^3, and mixing the two systems is the surest way to lose a question of this kind.

This is the last question of the paper and it sits in the quantitative block while being, in substance, a physics question — a reminder that the blocks on an EPFO EO/AO paper are thematic rather than strict, and that the general science strand can surface anywhere. It is also unusually well designed for a final item: the calculation is one line, but only after a candidate has decided what to calculate, and the wrong instinct — to work with the cube's volume of 1,000 cubic centimetres, which is the number the stem most conspicuously supplies — leads nowhere. The habit that saves the mark is to write down the balance in words before any numbers: weight of the body equals weight of the water displaced. Everything then follows, including which dimensions matter. One point about the printed stem should be noticed rather than argued with. It says the cube is 'a hollow closed cube of side 10 cm, each weighing 200 gm', and 'each' has no plural to attach to. The card reproduces the sentence as printed and solves the item the way the key does, with 200 gm as the weight of the whole cube; that reading is the only one that produces an answer on the option list at all, which is itself the strongest evidence for it.

Key facts

Archimedes' principle: a body immersed in a fluid experiences an upward force equal to the weight of the fluid displaced. For a body that floats, that force balances its weight, so it displaces exactly its own weight of fluid.
The fraction of a floating body's volume that is submerged equals the ratio of the body's average density to the density of the liquid, and a body denser than the liquid cannot float in it at all.
The working here: the submerged part is a box 10 cm by 10 cm by h, of volume 100h cubic centimetres; in water of density 1 gm per cm^3 that displaces 100h grams; setting 100h = 200 gives h = 2 cm.
The same answer without algebra: 200 gm of water occupies 200 cubic centimetres, and spread over the cube's 100 square centimetre footprint that is a depth of 2 cm — the horizontal face's area is what decides the item, not the cube's volume.
Cross-check by density: 200 gm spread through 1,000 cubic centimetres is an average density of 0·2 gm per cm^3, one-fifth that of water, so one-fifth of the 10 cm height sinks, which is 2 cm.
The word 'hollow' is load-bearing. Iron has a density of about 7·8 gm per cm^3 and a solid iron cube of this size would sink; the enclosed air is what brings the average density below water's, which is the principle a steel ship floats on.
The stem prints 'a hollow closed cube of side 10 cm, each weighing 200 gm', with 'each' standing against a single cube; the 200 gm is the weight of the whole cube, which is the reading the keyed answer of 2 cm follows from.

A floating body displaces its own weight — so the footprint decides, not the volume

What actually has to be computed→Not the cube’s volume, but the volume of the part BELOW the waterline. Because the cube is set down with a face horizontal, that part is a flat box measuring 10 cm by 10 cm by the sunk height h, of volume 100h cubic centimetres.

Archimedes, in the form a floating body takes→An object floats at the depth at which the weight of the water it pushes aside equals its own weight. Water has a density of 1 gm per cm^3, so the water displaced weighs 100h grams. Setting 100h = 200 gives h = 2 cm.

The same thing without algebra→200 gm of water occupies 200 cubic centimetres, and spreading 200 cubic centimetres over a footprint of 100 square centimetres gives a depth of 2 cm. Notice which measurement carried the question — the area of the horizontal face, not the cube’s own volume of 1,000 cubic centimetres, which never enters.

Cross-check by density→The fraction of a floating body that is submerged equals the ratio of its average density to the liquid’s. This cube weighs 200 gm and occupies 1,000 cubic centimetres of space, so its average density is 0·2 gm per cm^3, one-fifth that of water — and one-fifth of its 10 cm height is 2 cm.

Why ‘hollow’ is load-bearing→Iron has a density of about 7·8 gm per cm^3, and a solid cube of this size would weigh several kilograms and sink. The enclosed air is what brings the AVERAGE density below water’s and lets it float, which is the same principle a steel ship floats on. ‘Average’ is the word that makes a hollow object interesting at all.

Every option is testable at sight — and one caution on the wording→The relation is linear: with a footprint of 100 square centimetres in water of unit density, the sunk depth in centimetres is the weight in grams divided by a hundred. So 1 cm belongs to a cube of 100 gm, 1·5 cm to one of 150 gm, and 2·5 cm to one of 250 gm — fifty grams more than this cube weighs. The depth can also never exceed the height, 10 cm here. On the wording: the stem reads ‘each weighing 200 gm’, and ‘each’ sits oddly against a single cube — the 200 gm is the weight of the whole cube, which is the reading the keyed answer follows from.

Study next

Archimedes' principle and the law of flotation, including problems that give the fraction submerged and ask for the density of the body or of the liquidDensity and relative density, and the conversion between gm per cm^3 and kg per m^3 that numerical questions on the topic depend onPressure in fluids — the variation of pressure with depth, Pascal's law, and the hydraulic pressSurface area and volume of the cube, cuboid, cylinder and sphere, since flotation questions are as much mensuration as physicsEveryday applications the same principle explains — why a ship floats and a needle sinks, how a submarine changes its depth, and how a hydrometer measures the density of a liquid

Common traps

Working with the cube's whole volume of 1,000 cubic centimetres. Only the submerged part displaces water, and for a cube resting with a face horizontal that part is the base area multiplied by the sunk depth.
Forgetting that the object is hollow and reasoning that iron sinks. The average density of a hollow closed body, its mass spread over the whole enclosed volume, is what decides whether it floats.
Mixing the units of density. Water is 1 gm per cm^3 and also 1,000 kg per m^3, and a calculation that carries centimetres in one place and metres in another goes wrong by a factor of a million.
Checking neither limit at the end. The sunk depth cannot exceed the height of the body, and the fraction submerged must match the ratio of the densities — two checks that between them catch almost every slip in this topic.

Flotation and density reach EPFO EO/AO papers in two ways. In the physics block they appear as statements to be judged correct or incorrect, testing whether a candidate can state Archimedes' principle precisely. In the quantitative block they appear as numerical items like this one, where a regular solid floats in water and either the sunk depth, the fraction submerged or the density is asked for. The numbers are always chosen to come out clean, water is always taken as 1 gm per cm^3, and the difficulty is placed in deciding which volume to use rather than in the arithmetic. Expect the same principle in questions about ships, icebergs and hydrometers, and the companion mensuration items on volume and surface area in the same block.

Related PYQs

EPFO_EOAO_2020_Q49Open & attempt →

Which one of the following statements regarding force is correct ?

(a) A positive force implies attractive nature.
(b) A negative force implies repulsive nature.
(c) A positive force can be both attractive and repulsive in nature.
(d) A negative force implies attractive nature.

Answer(d) A negative force implies attractive nature.

The physics item on statements about force, from the paper's science block. Both turn on stating a mechanical principle exactly rather than approximately, and this one shows what that precision buys when a number has to come out of it.

EPFO_EOAO_2020_Q117Open & attempt →

If W1 and W2 are weights of two solid iron balls of radii 1/2 metre and 1/3 metre respectively, then which one of the following is equal to W1 : W2 ?

(a) 8 : 27
(b) 27 : 8
(c) 4 : 16
(d) 16 : 4

Answer(b) 27 : 8

The item on the weights of two iron balls, three questions earlier. Both link weight to volume through a shared density; there the density cancels because the material is the same, here it is the average density of a hollow body that decides how deep it sits.

Practice

practice — not a real PYQ
A cubical wooden block of side 20 cm floats in water with 5 cm of its height standing above the surface. What is the density of the material of the block, in gm per cm^3 ?
- (a)0.25
- (b)0.50
- (c)0.75
- (d)1.00
Answer(c) 0.75
practice — not a real PYQ
A solid floats in water with three-fourths of its volume submerged. What fraction of its volume will be submerged when the same solid floats in a liquid whose density is 1.5 times that of water ?
- (a)One-third
- (b)One-half
- (c)Two-thirds
- (d)Three-fourths
Answer(b) One-half

Why