Question

A jeweler claims that he sells ornaments of pure gold that has a relative density of 19.3. He sold the bangle weighing 25.25gf to a person. The clever customer weighs that bangle, when immersed in pure water and finds that it weighs 23.075gf in water. Is the ornament made of pure gold?

Answer 1

In this the water displaced by the bangle is equal to the volume of the bangle. This principle is given by Archimedes. The way to find out the water displaced is to find out the loss of weight in water and after that use the density mass formula D = $\dfrac{M}{V}$, where M = mass, V = Volume, here the density of water is one gram per cubic centimeter (1g/cc). Find out the volume of water displaced which is equal to the volume of the bangle and equate it with the given mass i.e. 25.25gf, we will find the density of the bangle and then we can match it (density of the bangle) with the seller claimed density of 19.3.

Formula used:
The formula for finding out the density is given below.
$D = \dfrac{M}{V}$
Here, $D$ is the density, $M$ is the mass, and $V$ is the Volume.

Complete step by step answer:
Given:
Relative Density of bangle (${D_R}$) = $19.93$
Mass of the Bangle = $25.25gf$
Mass of the Bangle in water = $23.075gf$
The density of water = $1g/cc$ (Assume)
We need to find pure gold.
Subtract the original weight of the bangle from the weight that is observed after the bangle is immersed in water.
Loss of weight in water = $25.25gf – 23.075gf = 2.175gf$

The volume of water displaced = Density of water $\times$ Relative weight in water

$\Rightarrow V = 1 \times 2.175$ (Assume the density of water 1g/cc)
$\Rightarrow V = 2.175c{m^3}$
Now, the volume of water displaced is equal to the volume of the bangle.
We have to find out the density of the bangle. Mass of Bangle (25.25gf)
$\Rightarrow D = \dfrac{M}{V}$
$\Rightarrow D = \dfrac{{25.25}}{{2.175}}$
The density of the bangle is
$D = 11.6$

The gold is not pure because the density of the given bangle is 11.6 and the density that the jeweler sells is 19.93.

Additional Information:
Here the gold is not pure. Since the relative density of gold is given to be $19.93$. We are getting the density of the bangle $11.6$. So the density is less than the given density and hence, the gold is not pure.

Note:
Here the process is lengthy so go carefully step by step. Make sure to make a proper relationship between the water displaced and the volume of the bangle. Here, the density of water is not given so assume it as $1g/cc$.