Question

A piece of gold is cut into two halves. If the density of gold before cutting is $d$ then the density of each half after cutting will be $\dfrac{d}{2}$ . Explain if this statement is true or false.

Answer 1

Density of a body is defined as the ratio of the mass of the body to its volume. When the gold piece is cut into halves, the mass of the gold piece and its volume gets halved. But the ratio of the mass and volume of each half will remain the same.

Formula used:
-The density of a body is given by, $d = \dfrac{m}{V}$ where $m$ is the mass of the body and $V$ is the volume of the body.

Complete step by step answer.
Step 1: Describe the problem at hand and mention the points of importance.
The problem at hand mentions a piece of gold being cut into two halves. It is given that the density of the gold piece before cutting is $d$ .
Since the gold is cut into two halves, the densities of both halves must be equal i.e., ${d_1} = {d_2}$ .
We need to check whether the density of each half is equal to $\dfrac{d}{2}$ or not.
Step 2: Find the densities of each half.
As the gold piece is cut into halves, the mass of each half will be half of the original mass. If $m$ is the mass of the gold piece before cutting then, then the mass of each half will be $\dfrac{m}{2}$ .
Similarly, the volume of each half will be half of the original volume. If $V$ is the volume of the gold piece before cutting then, then the volume of each half will be $\dfrac{V}{2}$ .
The density of the gold piece before cutting is given by, $d = \dfrac{m}{V}$ .
Then the density of any one of the half can be expressed as ${d_1} = \dfrac{{\left( {\dfrac{m}{2}} \right)}}{{\left( {\dfrac{V}{2}} \right)}} = \dfrac{{2m}}{{2V}} = \dfrac{m}{V}$
The above expression suggests that the densities of both halves are equal to the original density i.e., $d = {d_1} = {d_2} = \dfrac{m}{V}$

So, the given statement is false.

Note: Density of the gold piece is an intensive property i.e., it does not depend on the size of the gold piece. An increase or decrease in the amount of the gold piece, cutting into two halves for an instance will only cause a change in its mass. The density can only be changed by changing the pressure or temperature.