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Weight of 1 L of milk is 1.032 kg. It contains butter fat (density 865 kg ${{m}^{-3}}$) to the extent of $4\%$ by volume. The density of fat free skimmed milk will be
A. $1038.5\text{ kg }{{\text{m}}^{-3}}$
B. $1032.2\text{ kg }{{\text{m}}^{-3}}$
C. $\text{997 kg }{{\text{m}}^{-3}}$
D. $1000.5\text{ kg }{{\text{m}}^{-3}}$

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: To solve the given question first we will find the mass of fat in $\text{1 }{{m}^{3}}$ milk. Then we will find the mass of skimmed milk. Then we will find the volume of skimmed milk. Then by using these values we will find the density of fat free skimmed milk.

Complete step by step solution:
We have been given that the weight of 1 L of milk is 1.032 kg and it contains butter fat (density 865 kg ${{m}^{-3}}$) to the extent of $4\%$ by volume.
We have to find the density of fat free skimmed milk.
Now, we know that density of skimmed milk is given as
$\text{Density of skimmed milk }=\dfrac{\text{mass of skimmed milk}}{\text{volume of skimmed milk}}$
Now, we know that skimmed milk is fat free milk so the mass of skimmed milk will be
$\text{Mass of skimmed milk = mass of milk }-\text{ mass of fat}$
Now, we have given that $\text{1 }{{m}^{3}}$ milk contains butter fat $4\%$ by volume. (density 865 kg ${{m}^{-3}}$)
So the mass of fat in $\text{1 }{{m}^{3}}$ milk will be
$\begin{align}
  & \Rightarrow volume\times density \\
 & \Rightarrow 0.04\times 865 \\
 & \Rightarrow 34.6\text{ kg} \\
\end{align}$
Now, the mass of 1 L of milk is 1032 kg.
So the mass of skimmed milk will be
$\begin{align}
  & \Rightarrow \text{Mass of skimmed milk = 1032 }-\text{ 34}\text{.6} \\
 & \Rightarrow \text{Mass of skimmed milk = 997}\text{.4 kg} \\
\end{align}$
Now, the volume of skimmed milk will be
$\begin{align}
  & \Rightarrow 1-0.04 \\
 & \Rightarrow 0.96\text{ }{{\text{m}}^{3}} \\
\end{align}$
Now, $\text{Density of skimmed milk }=\dfrac{\text{mass of skimmed milk}}{\text{volume of skimmed milk}}$
Substituting the values we will get
$\Rightarrow \text{Density of skimmed milk }=\dfrac{\text{997}\text{.4}}{0.96}$
Simplifying the above equation we will get
$\Rightarrow \text{Density of skimmed milk }=1038.5\text{ kg }{{\text{m}}^{-3}}$

Hence option A is the correct answer.

Note:
Alternatively we can simply solve the given question as by assuming that $100\text{ }{{m}^{3}}$ contains $4\text{ }{{m}^{3}}$ fat.
Then the weight of butter fat in $\text{1 }{{m}^{3}}$ milk will be
$\begin{align}
  & \Rightarrow \dfrac{4}{100}\times 865 \\
 & \Rightarrow 35\text{ kg} \\
\end{align}$
Also given the weight of 1 L of milk is 1.032 kg.
So we get the weight volume of skimmed milk as $1.0-0.04=0.96\text{ }{{\text{m}}^{3}}$
Then the density of fat free skimmed milk will be $\Rightarrow \dfrac{997}{0.96}=1038.5\text{ kg }{{\text{m}}^{-3}}$.
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