Question

A vessel contains a mixture of milk and water in the ratio of 5:3 respectively. how much of the mixture must be siphoned off and replaced with water, so that the mixture may be half milk and half water
a.\[\dfrac{1}{7}\]
b.\[\dfrac{1}{4}\]
c.\[\dfrac{1}{5}\]
d.\[\dfrac{1}{3}\]

Answer 1

Hint: Assume the quantity of the mixture of 8 liters. Let the quantity of mixture which is siphoned off and replaced with water be x liters. Quantity of milk taken out when x liters of mixture is siphoned off will be \[\dfrac{5x}{8}\] . Quantity of milk in the new mixture will be \[5-\dfrac{5x}{8}\] . Quantity of water taken out when x liters of mixture is siphoned off \[\dfrac{3x}{8}\] . Quantity of milk in the new mixture will be \[3-\dfrac{3x}{8}+x\] . The quantity of milk is equal to the quantity of water. Now, solve it further.

Complete step-by-step answer:
Let the quantity of mixture be 8 liters
Quantity of milk in the mixture = 5 liters.
Quantity of water in the mixture = 3 liters.
As the mixture contains milk and water, so the quantity of the mixture is the summation of the quantity of milk and water.
The total quantity of the mixture = 5+3 = 8 liters.
Let the quantity be x liters of the mixture which is siphoned off and replaced with water.
Quantity of milk taken out when x liters of mixture is siphoned off = \[\dfrac{5x}{8}\] .
Quantity of milk in the new mixture = \[5-\dfrac{5x}{8}\] ……………………..(1)
Quantity of water taken out when x liters of the mixture are siphoned off = \[\dfrac{3x}{8}\] .
Also, x liters of water are added in the mixture.
Quantity of milk in the new mixture = \[3-\dfrac{3x}{8}+x\] ……………………..(2)
As the mixture contains half water and half milk.
Quantity of milk = Quantity of water
\[\begin{align}
  & \Rightarrow 5-\dfrac{5x}{8}=3-\dfrac{3x}{8}+x \\
 & \Rightarrow 5-3=\dfrac{5x}{8}-\dfrac{3x}{8}+x \\
 & \Rightarrow 2=\dfrac{10x}{8} \\
 & \Rightarrow \dfrac{16}{10}=x \\
 & \Rightarrow x=\dfrac{8}{5} \\
\end{align}\]
So, part of the mixture which is replaced = \[\dfrac{8}{5}\times \dfrac{1}{8}=\dfrac{1}{5}\] .
Hence, the correct option is (C).

Note: We can also solve this question in another way.
The final fraction of milk = \[\dfrac{1}{2}\] .
The initial fraction of milk = \[\dfrac{5}{8}\] .
Let the fraction of mixture taken out to be f.
The final fraction of milk = initial fraction of milk(1 – the fraction of mixture taken out)
\[\begin{align}
  & \Rightarrow \dfrac{1}{2}=\dfrac{5}{8}\left( 1-f \right) \\
 & \Rightarrow \dfrac{4}{5}=1-f \\
 & \Rightarrow f=1-\dfrac{4}{5} \\
 & \Rightarrow f=\dfrac{1}{5} \\
\end{align}\]
So, part of the mixture which is replaced = \[\dfrac{1}{5}\] .