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**Hint:**The question has given us all the information we need. First, find the net weight of water in the water in the first case. We all know that the density of water is $1\,g/cc$ . Use this to find the volume of the given density bottle. Then, find the net weight of mercury in the density bottle and use the volume we found out in the first case to find the density of mercury.

**Complete step by step answer:**

We will be trying to solve the question exactly as told in the hint section of the solution to the question.

Using the first case, we will find the volume of the density bottle. And then, we will use the newly-found volume of the density bottle to find the density of mercury.

Let us first find the volume of the bottle through the first case:

Weight of the empty density bottle as given in the question: ${W_b} = 25\,g$

Total weight after being completely filled by water: ${W_t} = 50\,g$

Hence, weight of the water can be found out as:

$\implies$ ${W_w} = \,{W_t} - {W_b}$

Substituting in the values, we get:

$

\implies {W_w} = 50 - 25 \\

\implies {W_w} = 25\,g \\

$

We already know that the density of water is $1\,g/cc$

Hence, volume of the density bottle is: $V = \dfrac{{25\,g}}{{1\,g/cc}}$ or $25\,cc$

Now, let us consider the second case:

Weight of the empty density bottle as given in the question: ${W_b} = 25\,g$

Total weight after being completely filled with mercury: ${W_t} = 365\,g$

Weight of mercury can be found out as:

$\implies$ ${W_m} = {W_t} - {W_b}$

Substituting in the values, we get:

$

\implies {W_m} = 365 - 25 \\

\implies {W_m} = 340\,g \\

$

We have already found out the value of volume of the given density bottle using the first case as:

$V = 25\,cc$

Using this, we can find the density of mercury as:

$\implies$ $d = \dfrac{{{W_m}}}{V}$

Substituting in the values that we found out:

$

\implies d = \dfrac{{340\,g}}{{25\,cc}} \\

\implies d = 13.6\,g/cc \\

$

Hence, option (C) is the correct option.

**Note:**A very important thing to do in such questions is to find out the density of the given density bottle or the container in which liquids or fluids are being stored. So always start solving the question by first finding out the volume of the container or density bottle. Also, always check the units of the weight and volume and tick the answer accordingly.

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