Question

A vessel contains a mixture of milk and water in ratio 14 : 25. 5 liters of mixture is taken out from the vessel and 2.5 liters of pure water and 5 liters of pure milk are added to the mixture. If resultant mixture contains 20 percent water, what was the initial quantity of mixture in the vessel before replacement in liters?

Answer 1

Hint: If the initial quantity of the mixture was x liters, according to the conditions the present quantity will be (x – 5 + 2.5 + 5) liters. Find out 20 percent of the present quantity then equate this with the present amount of water in the mixture to form an equation and solve it for x.

Complete step-by-step answer:
Let us assume that the initial quantity of the mixture in the vessel before replacement was x liter.
It is given in the question that the ratio of milk and water at the initial stage was 14 : 25.
Therefore, in x liter of mixture there was $\dfrac{14x}{\left( 14+25 \right)}=\dfrac{14x}{39}$ liter milk and $\dfrac{25x}{39}$ liter water.
5 liters of mixture is taken out.
Therefore, in 5 liter of mixture there is $\dfrac{14\times 5}{39}=\dfrac{70}{39}$ liter milk and $\dfrac{25\times 5}{39}=\dfrac{125}{39}$ liter water.
According to the question 2.5 liters of pure water and 5 liter of pure milk are added to the mixture.
Therefore, the present quantity of the mixture is:
$x-5+2.5+5=x+2.5$ liter.
Present quantity of water is $\left( \dfrac{25x}{39}-\dfrac{125}{39}+2.5 \right)$ liters.
  As per the question the present quantity of mixture contains 20 percent water. Therefore,
$\left( x+2.5 \right)\times \dfrac{20}{100}=\dfrac{25x}{39}-\dfrac{125}{39}+2.5......(1)$
Now we will solve equation (1) to get the value of x.
$\Rightarrow \dfrac{x+2.5}{5}=\dfrac{25x-125+97.5}{39}$
By cross multiplying we will get,
$\Rightarrow 39\left( x+2.5 \right)=5\left( 25x-125+97.5 \right)$
$\Rightarrow 39x+97.5=125x-625+487.5$
Now take all the terms with x on the left hand side and all the constant terms on the right hand side,
$\Rightarrow 39x-125x=-625-97.5+487.5$
$\Rightarrow -86x=-235$
Cancelling out the negative term with from both sides and dividing by 86, we will get:
$\Rightarrow x=\dfrac{235}{86}$
$\Rightarrow x=2.732$
Therefore, the initial quantity of the mixture was 2.732 liters.

Note: When some quantity of mixture is taken out both the quantity of water and milk will reduce as per the ratio. Generally we make mistakes here.