The densities of three liquids are D, 2D and 3D. What will be the density of the resulting mixture if equal volumes of the three liquids are mixed?

1. Foundations of Matter: Mass, Volume, and Density (basic)

Welcome to your first step in mastering mechanics! To understand how the physical world works, we must start with the building blocks of matter. Everything around us—from the air we breathe to the chair you are sitting on—is matter. By definition, matter is anything that possesses mass (the amount of substance) and occupies volume (the space it takes up) Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p. 140.

While mass and volume tell us "how much" of something we have, Density tells us how tightly that matter is packed. It is defined as the mass present in a unit volume of a substance. Mathematically, we express it as:

Density = Mass / Volume

The units we use depend on what we are measuring. In the International System of Units (SI), we use kilograms per cubic metre (kg/m³). However, for everyday liquids in a lab, you will often see grams per millilitre (g/mL) or grams per cubic centimetre (g/cm³) Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p. 141. An important characteristic of density is that it is intrinsic; it doesn't change based on the shape or size of the object. A drop of gold and a gold brick have the same density!

However, density is not set in stone—it changes with temperature. Generally, when you heat a substance, its particles move apart and spread out, causing the volume to increase while the mass remains the same. Because the same mass is now spread over a larger space, the density decreases Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p. 147. This is exactly why hot air balloons rise: the hot air inside is less dense than the cool air outside.

Property	Definition	Changes with Temperature?
Mass	The quantity of matter in an object.	No
Volume	The space occupied by an object.	Yes (usually increases with heat)
Density	Mass per unit volume.	Yes (usually decreases with heat)

Finally, when we mix different substances, the density of the mixture is the total mass divided by the total volume. Interestingly, if you mix equal volumes of two different liquids, the resulting density is simply the arithmetic average of their individual densities. We also use a concept called Relative Density, which is a unitless number comparing a substance's density to that of water Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p. 141.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140, 141, 146, 147

2. Units and Measurement in Basic Mechanics (basic)

In basic mechanics, understanding density is crucial because it tells us how tightly matter is packed within a specific space. Defined mathematically as the mass per unit volume (Density = Mass / Volume), it is an intrinsic property of a substance. This means whether you have a small drop of water or a whole bucketful, the density remains the same because the ratio of mass to volume is constant Science, Class VIII, Chapter 9, p.140. While density is independent of an object's shape or size, it can be influenced by external factors like temperature and pressure, particularly in gases where particles can be compressed or expanded significantly.

To measure density accurately, we look at the standard units of its components. The SI unit of mass is the kilogram (kg), and the SI unit of volume is the cubic metre (m³). Therefore, the SI unit for density is kg/m³. However, in laboratory settings or for everyday liquids, we often use more convenient units like grams per millilitre (g/mL) or grams per cubic centimetre (g/cm³, also written as 1 cc) Science, Class VIII, Chapter 9, p.141. It is helpful to remember that for liquids, 1 litre (L) is equivalent to 1 cubic decimetre (dm³), and 1 mL is exactly 1 cm³ Science, Class VIII, Chapter 9, p.143.

When we deal with mixtures of different substances, such as mixing two or more liquids that do not react with each other, we calculate the resultant density by looking at the total system. The combined density is simply the total mass of all components divided by the total volume they occupy. Interestingly, if you mix equal volumes of liquids with different densities, the density of the final mixture is the simple arithmetic mean (average) of the individual densities. For example, mixing equal volumes of a liquid with density D and another with density 3D results in a mixture density of 2D (calculated as [D + 3D] / 2).

Sources: Science, Class VIII, Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class VIII, Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.141; Science, Class VIII, Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.143

3. Relative Density and Specific Gravity (intermediate)

Concept: Relative Density and Specific Gravity

4. Archimedes' Principle and Buoyancy (intermediate)

At its heart, buoyancy is the upward force exerted by a fluid (liquid or gas) that opposes the weight of an immersed object. Have you ever noticed that a mug feels significantly lighter when it is submerged in a bucket of water compared to when it is in the air? This phenomenon occurs because the water is pushing back up against the mug. This upward force is known as the buoyant force or upthrust Science, Class VIII NCERT, Exploring Forces, p.76. While gravity pulls the object toward the center of the Earth, the fluid exerts pressure in all directions, but the pressure at the bottom of the object is greater than at the top, resulting in a net upward force.

The magnitude of this force was first quantified by the Greek scientist Archimedes. Archimedes' Principle states that when an object is fully or partially immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces Science, Class VIII NCERT, Exploring Forces, p.76. This principle is the foundation for understanding why massive iron ships float while a small pebble sinks. Whether an object sinks or floats depends on the balance between its weight and this buoyant force:

Scenario	Force Comparison	Outcome
Weight > Buoyant Force	Object is heavier than the displaced fluid.	The object sinks.
Weight = Buoyant Force	Object's weight equals weight of displaced fluid.	The object floats.

Crucially, density (mass per unit volume) dictates these interactions. An object will float if its average density is less than the density of the fluid. When dealing with mixtures of liquids, the overall density is calculated by dividing the total mass by the total volume Science, Class VIII NCERT, The Amazing World of Solutes, Solvents, and Solutions, p.140. For instance, if you mix equal volumes of three liquids with densities D, 2D, and 3D, the resulting density is the arithmetic mean (2D). This new mixture density will determine whether a specific object continues to float or begins to sink based on the new weight of the fluid it displaces.

Sources: Science, Class VIII NCERT (Revised ed 2025), Exploring Forces, p.76; Science, Class VIII NCERT (Revised ed 2025), The Amazing World of Solutes, Solvents, and Solutions, p.140

5. Anomalous Expansion and Density Variables (exam-level)

At its most fundamental level, density is the measure of how much matter is packed into a specific space. It is defined as the mass of a substance per unit volume (Density = Mass / Volume). In most materials, density is inversely proportional to temperature. When you heat a substance, its particles gain kinetic energy and move further apart, causing the volume to increase while the mass remains constant. This result is a decrease in density, which explains why hot air rises—it is less dense than the cooler air surrounding it Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.147. Conversely, pressure typically increases density, though this effect is significant only in gases; solids and liquids are nearly incompressible, meaning their density changes under pressure are usually negligible Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.148.

When we deal with mixtures of liquids that do not react chemically, we calculate the resulting density by looking at the total mass divided by the total volume. A critical shortcut to remember for your exams: if you mix equal volumes of different liquids, the density of the resulting mixture is simply the arithmetic mean (average) of the individual densities. For example, mixing equal volumes (V) of three liquids with densities D, 2D, and 3D results in a total mass of 6DV and a total volume of 3V. Dividing these gives a mixture density of 2D Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.140.

However, nature provides a fascinating exception known as the Anomalous Expansion of Water. While most substances contract and become denser as they cool, water only follows this rule until it reaches 4 °C. At this specific temperature, water reaches its maximum density. As it cools further from 4 °C down to 0 °C, water actually begins to expand. This happens because the water molecules begin to form a structured, cage-like lattice that occupies more space. This expansion makes ice less dense than liquid water, which is why ice floats and why aquatic life can survive in frozen lakes—the densest water (at 4 °C) sinks to the bottom, while the ice forms an insulating layer on top Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.148.

Feature	Most Substances	Water (0 °C to 4 °C)
Effect of Cooling	Volume decreases, Density increases	Anomalous: Volume increases, Density decreases
Phase Change (Solid)	Usually sinks in its liquid form	Floats (Ice is less dense than water)

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.147; Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.148

6. Mathematics of Mixing: Resultant Density (exam-level)

To understand the mathematics of mixing, we must first return to the fundamental definition of density: it is the amount of mass contained within a unit volume of a substance. Mathematically, this is expressed as Density = Mass / Volume. As we know from Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.140, while the density of a substance is independent of its shape or size, it is a crucial characteristic that tells us how "heavy" a substance is for its size. For instance, if an object has a mass of 27 g and a volume of 10 cm³, its density is 2.7 g/cm³ Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.141.

When we mix multiple liquids together (assuming they do not react chemically and their volumes are additive), the resultant density of the mixture is determined by the ratio of the total mass to the total volume. You cannot simply add densities together; instead, you must find the sum of all individual masses and divide it by the sum of all individual volumes. Since Mass = Density × Volume, the formula for the mixture density (ρ_mix) of three liquids is:

ρ_mix = (m₁ + m₂ + m₃) / (V₁ + V₂ + V₃) = (d₁V₁ + d₂V₂ + d₃V₃) / (V₁ + V₂ + V₃)

A very useful shortcut arises when we mix equal volumes of liquids. Suppose we mix three liquids, each with volume V, but with different densities: d, 2d, and 3d. The total mass becomes (dV + 2dV + 3dV) = 6dV. The total volume is (V + V + V) = 3V. Dividing the total mass by the total volume gives us 6dV / 3V = 2d. Interestingly, this result is exactly the arithmetic mean of the individual densities ( (d + 2d + 3d) / 3 = 2d ).

Mixing Condition	Mathematical Result
Equal Volumes	The resultant density is the Arithmetic Mean of the individual densities.
Equal Masses	The resultant density is the Harmonic Mean of the individual densities.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140; Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.141

7. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental relationship between mass, volume, and density, this question asks you to apply those building blocks to a composite system. The key insight here, as highlighted in Science, Class VIII, NCERT, is that density is an intensive property, but mass and volume are additive. To find the density of a mixture, you cannot simply add the individual densities together; you must find the total mass and divide it by the total volume of the entire system.

Let’s walk through the logic: if we assume each liquid has an equal volume V, their individual masses—using the formula Mass = Density × Volume—would be DV, 2DV, and 3DV. Adding these together gives a total mass of 6DV. Since we mixed three equal volumes, the total volume is 3V. Dividing the total mass (6DV) by the total volume (3V) yields the correct answer: (C) 2D. Notice a shortcut for your exam kit? When volumes are equal, the resulting density is simply the arithmetic mean of the individual densities: (1D + 2D + 3D) / 3 = 2D.

UPSC often includes "trap" options to test your conceptual clarity. Option (A) 6D is a classic error where a student simply sums the density values but forgets to divide by the total volume (the n parts). Option (D) 3D is a distractor for those who might incorrectly assume the mixture takes on the property of the densest component. Always remember: a mixture's density must fall between the minimum and maximum values of its components; it can never be equal to the highest density unless the other volumes are zero.