A density bottle weighs \(100 \mathrm{~g}\) when filled with liquid and \(80 \mathrm{~g}\) when filled with water. If the weight of an empty density bottle is \(20 \mathrm{~g}\), find the density of the liquid.

Understanding the relationship between mass and volume is crucial for grasping the concept of density. Mass is a measure of how much matter an object contains, while volume measures the space an object occupies. These two properties are fundamental components in calculating the density of a substance.

For instance, in our exercise, we determined the mass of a liquid by weighing the density bottle when full and subtracting the weight of the empty bottle. This method isolates the mass of the liquid itself. Similarly, determining the volume can be slightly more indirect. By knowing that the bottle's volume remains constant, we can use the known density of a reference substance (in this case, water) to find the bottle's volume. Essentially, mass and volume are the building blocks that set the stage for understanding and computing the density of various materials.

Density is a characteristic property that helps us identify and compare substances. It is the mass of a substance per unit volume. The concept of density is useful because it allows us to understand how tightly matter is packed in a substance, which can affect how substances interact and behave, from floating to sinking, or how they can be used in various applications.

In the practical world, the density of substances determines how they are used in construction, manufacturing, and even in day-to-day items like cooking oils and flotation devices. For our exercise, we looked at an unknown liquid's density and used water - with a known density of 1 g/cm³ - as our reference point. This comparison is a standard approach in density determination, making the calculation more manageable and relatable.

The density formula is a simple, yet powerful tool in physics and engineering: Density = Mass / Volume. This formula allows the calculation of an object's density by dividing its mass by its volume.

In our textbook exercise, we applied the density formula to find the density of the liquid by using the previously calculated mass (80 g) and volume (60 cm³). The solution demonstrates how the formula simplifies complex real-world problems into a straightforward calculation, yielding the density of the liquid as \(\frac{4}{3}\mathrm{~g/cm^3}\). This process highlights the importance of the formula in determining properties of substances that are not immediately apparent and allows us to compare them easily.