Specific gravity of silver is 11 and specific gravity of iron is 8. Which one among the following is the approximate relative density of silver with respect to iron?

1. Understanding Density and the Mass-Volume Relationship (basic)

To understand the physical world, we must first look at how matter is put together. Every object around you, from a pebble to a planet, is defined by two fundamental properties: it has mass (the amount of matter in it) and it occupies volume (the space it takes up). However, mass and volume alone don't tell the whole story. To understand why some objects feel "heavy for their size" while others feel "light," we use the concept of Density.

Density is defined as the mass present in a unit volume of a substance. Think of it as a measure of how tightly packed the particles are within a material. Mathematically, we express it as:

Density = Mass / Volume

According to Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p. 140, the density of a substance is an intrinsic property. This means it does not depend on the shape or the total size of the object. For instance, a small iron nail and a massive iron girder have the exact same density because they are made of the same material. However, density can change with temperature and pressure. When substances are heated, particles generally move further apart (increasing volume), which typically decreases density.

The strength of interparticle attractions also plays a vital role in determining density. In solids, particles are usually held closely together by strong forces, leading to high density. As the distance between particles increases—such as when a solid turns into a gas—the density drops significantly Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p. 101, 107. To measure this, we use the SI unit kilogram per cubic metre (kg/m³), though in a laboratory, you will often see grams per cubic centimetre (g/cm³) or grams per millilitre (g/mL) Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p. 141.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.140, 141; Science, Class VIII. NCERT (Revised ed 2025), Particulate Nature of Matter, p.101, 107

2. Archimedes' Principle and Buoyancy (basic)

Have you ever noticed how you feel lighter while swimming in a pool, or how a heavy stone is easier to lift as long as it stays underwater? This isn't magic—it’s the result of a fascinating physical phenomenon called buoyancy. When any object is placed in a liquid, it experiences an upward force known as upthrust or buoyant force. This force acts in direct opposition to gravity, which is constantly trying to pull the object down toward the center of the Earth Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76-77.

The breakthrough in understanding how much force a liquid exerts came from the Greek scientist Archimedes. He discovered that the magnitude of this upward force is exactly equal to the weight of the liquid that the object displaces. Imagine dropping a metal block into a full cup of water; the water that spills over the edge is the "displaced liquid." If you weigh that spilled water, you’ll know the exact strength of the buoyant force pushing up on the block Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76. This principle explains why a massive iron ship can float: its shape allows it to displace a huge volume of water, creating enough upward force to balance its immense weight.

Whether an object sinks or floats depends on the outcome of a "tug-of-war" between two forces: gravitational force (weight) and buoyant force. If the object is denser than the liquid, it won't displace enough liquid to match its own weight, and it will sink. Conversely, if the buoyant force is equal to the object's weight, the object will float Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76. This relationship is summarized in the table below:

Scenario	Comparison of Forces	Outcome
Sinking	Gravitational Force > Buoyant Force	The object moves downward to the bottom.
Floating	Gravitational Force = Buoyant Force	The object stays at the surface or suspended.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.76; Science, Class VIII. NCERT (Revised ed 2025), Exploring Forces, p.77

3. Units, Dimensions, and Unitless Quantities (intermediate)

To master mechanics, we must first understand how we describe the 'physical nature' of objects. Density is a fundamental property defined as the mass present in a unit volume of a substance Science, Class VIII. NCERT(Revised ed 2025), Chapter 9, p.140. Because it is a derived quantity (Mass divided by Volume), its units are also derived. In the International System of Units (SI), we use kilogram per cubic metre (kg/m³), though in laboratory settings, you will often encounter grams per cubic centimetre (g/cm³) or grams per millilitre (g/mL) Science, Class VIII. NCERT(Revised ed 2025), Chapter 9, p.141. While the numerical value of density changes depending on the units you choose, the underlying dimensions—the physical essence of Mass [M] and Length [L⁻³]—remain constant.

However, science often requires us to compare materials directly without getting bogged down in specific measurement systems. This is where unitless quantities come into play. A prime example is Relative Density (also known as Specific Gravity). It is defined as the ratio of the density of a substance to the density of a reference substance (usually water at 4°C). Because you are dividing a density by a density (e.g., kg/m³ divided by kg/m³), the units cancel out completely, leaving you with a pure number.

These dimensionless numbers are incredibly powerful for comparison. For instance, if the specific gravity of silver is 11 and that of iron is 8, we can instantly determine that silver is 1.375 times denser than iron (11 / 8 = 1.375) without needing to know the actual mass or volume of either sample. This principle of ratios is a recurring theme in the UPSC syllabus, as it allows for a standardized way to express the 'heaviness' or 'concentration' of materials across different scientific contexts.

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

4. Practical Applications: Hydrometers and Lactometers (intermediate)

To understand how we measure the purity of liquids like milk or the concentration of battery acid, we must look at the practical application of relative density (also known as specific gravity). As we’ve noted, relative density is a dimensionless ratio comparing a substance's density to that of a reference, usually water Science, Class VIII NCERT (Revised ed 2025), Chapter 9, p.141. The tools we use for these measurements are the Hydrometer and its specialized cousin, the Lactometer.

A Hydrometer operates on the principle of flotation (Archimedes' Principle). It is typically a glass instrument with a weighted bulb at the bottom to keep it upright and a narrow stem with a scale. When placed in a liquid, the hydrometer sinks until the weight of the displaced liquid equals its own weight. In a denser liquid, the upward buoyant force is stronger, so the hydrometer does not need to displace as much volume and thus floats higher. In a less dense liquid, it sinks deeper. This allows us to read the specific gravity directly from the scale on the stem.

The Lactometer is a specific type of hydrometer designed to test the purity of milk. Fresh milk has a specific density range; however, when milkmen add water to increase volume, the density of the mixture decreases. By floating a lactometer in a sample, we can instantly tell if it has been watered down. While chemical changes, such as Lactobacillus bacteria converting lactose into lactic acid to form curd Science, Class VIII NCERT (Revised ed 2025), Chapter 2, p.22, affect the milk's pH and taste, the lactometer remains the primary physical tool for checking initial adulteration.

Instrument	Primary Use	Physics Principle
Hydrometer	Measuring relative density of various liquids (e.g., alcohol, battery acid).	Floats higher in denser liquids; sinks deeper in lighter liquids.
Lactometer	Checking the purity/density of milk to detect water adulteration.	Calibrated specifically for the density range of pure milk.

Sources: Science, Class VIII NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.141; Science, Class VIII NCERT (Revised ed 2025), Chapter 2: The Invisible Living World: Beyond Our Naked Eye, p.22

5. Pressure in Fluids and Pascal's Law (intermediate)

Concept: Pressure in Fluids and Pascal's Law

6. Relative Density and Specific Gravity (exam-level)

When we talk about how "heavy" a material is, we often intuitively compare it to something we know well—like water. In physics, this comparison is formalized as Relative Density or Specific Gravity. At its simplest, relative density tells us how many times denser a substance is compared to a reference material, usually water at 4°C. For instance, if an aluminum block has a density of 2.7 g/cm³, and water has a density of 1 g/cm³, we say the relative density of aluminum is 2.7 Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.141.

One of the most important characteristics of relative density is that it is a dimensionless quantity—it has no units. Because it is a ratio of two similar quantities (Density of Substance / Density of Reference), the units like g/cm³ or kg/m³ cancel each other out Science, Class VIII. NCERT (Revised ed 2025), Chapter 9, p.141. This makes it incredibly useful for scientists and engineers to compare materials across different systems of measurement without getting bogged down in unit conversions.

Feature	Density	Relative Density (Specific Gravity)
Definition	Mass per unit volume.	Ratio of substance density to reference density.
Units	kg/m³ or g/cm³	None (Unitless)
Dependency	Depends on volume and mass.	Depends on the nature of the substance and temperature.

In nature, these density differences are the engines behind massive physical systems. In our oceans, temperature and salinity dictate density; cold, salty water is denser and tends to sink, while warmer, fresher water rises Physical Geography by PMF IAS, Ocean Movements Ocean Currents And Tides, p.487. This principle of "denser sinks, lighter rises" isn't just a classroom concept—it drives the global ocean currents that regulate our planet's climate.

Sources: Science, Class VIII. NCERT (Revised ed 2025), Chapter 9: The Amazing World of Solutes, Solvents, and Solutions, p.141; Physical Geography by PMF IAS, Ocean Movements Ocean Currents And Tides, p.487

7. Solving the Original PYQ (exam-level)

You’ve already mastered the idea that specific gravity is simply the ratio of a substance's density to that of water. In this question, the UPSC expands on that fundamental building block by asking for the relative density of one solid compared directly to another. Remember, because specific gravity is a dimensionless ratio, it serves as a universal comparison tool. This question tests your ability to apply the principle of proportionality—moving beyond standard water-based references to evaluate how many times "heavier" one material is than another based on the same volume.

To solve this, identify your target and your reference. Since we need the density of silver with respect to iron, silver (11) becomes our numerator and iron (8) our denominator. The calculation is straightforward: 11 / 8 = 1.375. When we look at the options provided, (A) 1.4 is the most accurate approximation. This confirms that silver is roughly 1.4 times as dense as iron. This logic follows the conversion principles outlined in Science, Class VIII, NCERT (Revised ed 2025), which emphasizes that relative density is a unitless quantity used for comparison.

UPSC often includes distractor options to catch students who rush through the arithmetic or swap the formula. Option (B) 0.7 is the result of dividing iron by silver (8/11), a classic inverse trap. Option (C) 3.0 is a subtraction trap, representing the absolute difference between the two values (11 - 8) rather than their ratio. By focusing on the multiplicative relationship inherent in density problems, you can confidently eliminate these errors and arrive at the correct result.