Question

Milk contains 5% water .What quantity of pure milk should be added to 10 litres of milk to reduce this to 2% ? \[\]
A.5 litres\[\]
B.7 litres\[\]
C.11 litres\[\]
D. 15 litres\[\]

Answer 1

Hint: We find the amount of pure milk ${{M}_{1}}$and water ${{W}_{1}}$.We assume the amount of pure milk to be added as $x$ litre and express the amount of pure milk ${{M}_{2}}$and water ${{W}_{2}}$ after addition in terms of $x$ . We use the given percentage 2% after addition to find the ratio $\dfrac{{{W}_{2}}}{{{M}_{2}}}$ and solve for $x$. \[\]

Complete step-by-step answer:
We know that the percentage in mathematics is a number or ratio expressed as a fraction of 100. If we have $a$ number of elements and there are total $b$ number of elements then we can express $a$as a percentage $p$ of $b$ using the working rule,
\[p=\dfrac{a}{b}\times 100\]
We know that when we say $p \% $ it means there are $p$ number of items out of every 100 items. When we say $p \% $ of $a$ it means there are $\dfrac{p}{a}\times 100$ number of items. \[\]
We are given in the question that the 10 litre of mixture milk and water contains 5% of water, so the amount of pure milk in 10 litres of mixture milk and water in percentage is $100-5=95 \% $.
So the amount of pure milk in the in 10 litres of mixture milk and water is 95% of 10 litre that is
\[{{M}_{1}}=\dfrac{95}{100}\times 10=9.5\text{litre}\]
The amount of water is
\[{{W}_{2}}=0.5\text{litre}\]
Let the amount of pure milk added to the mixture be $x$litre. So the new amount of pure milk is
\[{{M}_{2}}={{M}_{1}}+x=9.5+x\]
The amount of water is still the same after addition of pure milk. So we have
\[{{W}_{2}}=0.5\text{litre}\]
It is given in the question that the concentration of water in the mixture after addition is 2% which means the amount of pure milk is $100-2=98 \% $. So the ratio of water and milk after addition of pure milk is
\[\dfrac{{{W}_{2}}}{{{M}_{2}}}=\dfrac{2 \% }{98 \% }=\dfrac{0.02}{0.98}=\dfrac{0.2}{9.8}\]
We put ${{W}_{2}}$ and ${{M}_{2}}$ in amounts in litres and have
\[\begin{align}
  & \dfrac{0.5}{9.5+x}=\dfrac{0.2}{9.8} \\
 & \Rightarrow 4.9=11.9+0.2x \\
 & \Rightarrow 0.2x=3.0 \\
 & \Rightarrow x=\dfrac{3}{0.2}=15 \\
\end{align}\]
So the asked amount of pure milk is 15litres and the correct option is D.

So, the correct answer is “Option D”.

Note: We can directly find the amount of pure milk $M$ using the formula $M=V\left( \dfrac{{{p}_{1}}-{{p}_{2}}}{{{p}_{2}}} \right)$ where $V$the volume of the mixture is, ${{p}_{1}}$ is the initial percentage of pure milk before addition, ${{p}_{2}}$ is the final percentage of milk after addition.