Question

A litre of milk weighs 1.032kg. The butterfat it contains to the extent of 4.0 percent by volume has a density of $865kg/{{m}^{3}}$. Calculate the density of the fat free skimmed milk.
$\begin{align}
  & a)1.04g/ml \\
 & b)1.04kg/ml \\
 & c)2.08kg/ {{m}^{3}} \\
 & d)1.075g/ ml \\
\end{align}$

Answer 1

Hint: Now we are given a density of butter fat. First we will convert the density into required units. Now we know that milk given is 1L and the percent of butterfat is 4 percent. Hence we can find the volume of butter fat. Now we know that mass = density × volume. Hence we will find the weight of butter fat. Now the volume of fat skimmed milk is the volume of milk – volume of fat. Similarly the weight of skimmed milk is the weight of milk – the weight of fat. Hence we can find the density of skimmed milk.

Complete step by step solution:
Now the weight of 1 litre milk is given as 1.032kg.
The density of butterfat is given to be $865kg/{{m}^{3}}$
Now first we will convert the units of density. We know that 1000 cubic meters is 1 litre.
Hence $865kg/{{m}^{3}}=0.865kg/L$
Now the butter fat is 4 percent by volume.
Hence the volume of butter fat is $\dfrac{4}{100}L=0.04L$
Now we know that weight = density × volume.
Hence the weight of butterfat is $0.865\times \dfrac{4}{100}=0.0346$
Now the weight of fat free skimmed milk is the weight of milk – weight of fats.
Hence the weight of skimmed milk is 1.032 – 0.0346.
Hence we get the weight of skimmed milk as 0.9974.
Now let us calculate the volume of skimmed milk.
The volume of skimmed milk is the volume of milk – volume of fat.
Hence the volume of skimmed milk is 1L – 0.04L = 0.96L.
Now the density of skimmed milk is $\dfrac{\text{mass of skimmed milk}\text{.}}{\text{volume of skimmed milk}\text{.}}$ , Hence we get,
$\begin{align}
  & \Rightarrow \text{d = }\dfrac{0.9974}{0.96}kg/L \\
 & \Rightarrow d=1.038kg/L \\
\end{align}$
$\begin{align}
  & \Rightarrow d=1.038\times 1000g/1000ml \\
 & \Rightarrow d=1.038g/ml \\
\end{align}$
Hence the density of the skimmed milk is $1.04g/ml$

So, the correct answer is “Option a”.

Note: Note that when solving such sums always take care of the units. If the units are not the same while performing operations then we need to convert the units such that the units become the same. For example if density is given in kg per litre then we need weight in kg and volume in litre whenever used.