Question

A vessel is filled with a liquid, 3 parts of which are water and 5 parts syrup. How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
$\left( a \right)\dfrac{1}{5}$
$\left( b \right)\dfrac{3}{7}$
$\left( c \right)\dfrac{4}{5}$
$\left( d \right)\dfrac{3}{{10}}$

Answer 1

Hint: In this particular question use the concept that if there are p amount of water and q amount of syrup in the mixture so the ratio of water to mixture is p:q, and use the concept that if we have to take out some amount of mixture say x amount, so the amount of water and amount of syrup we take out is $\dfrac{{px}}{{p + q}},\dfrac{{qx}}{{p + q}}$ respectively so use these concepts to reach the solution of the question.

Complete step by step answer:
Given data:
A vessel is filled with a liquid, 3 parts of which are water and 5 parts syrup.
So the ratio of water to mixture is 3:5 in the mixture.
Now we have to find out how much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup.
Let the amount of mixture is drawn is x amount.
So the amount of water and amount of syrup we take out is $\dfrac{{3x}}{{3 + 5}} = \dfrac{{3x}}{8},\dfrac{{5x}}{{3 + 5}} = \dfrac{{5x}}{8}$ respectively.
Now x amount of mixture is replaced with water.
So the total amount of water in the mixture = $\dfrac{{{\text{amount of water in original mixture}}}}{{{\text{total amount of mixture}}}} - $Amount of water which we took out from the mixture + amount of water which was replaced.
So the total amount of water in the mixture = $\dfrac{3}{8} - \dfrac{{3x}}{8} + x$
So the total amount of water in the mixture = $\dfrac{3}{8} + \dfrac{{5x}}{8}$............ (1)
Now the total amount of syrup in the mixture = $\dfrac{{{\text{amount of water in original mixture}}}}{{{\text{total amount of mixture}}}} - $ Amount of syrup which we took out from the mixture
So the total amount of syrup in the mixture = $\dfrac{5}{8} - \dfrac{{5x}}{8}$.............. (2)
Now it is given that after replacement both amounts are equal in the mixture.
So equate equation (1) and (2) we have,
$ \Rightarrow \dfrac{3}{8} + \dfrac{{5x}}{8} = \dfrac{5}{8} - \dfrac{{5x}}{8}$
Now simplify it we have,
$ \Rightarrow \dfrac{{10x}}{8} = \dfrac{2}{8}$
$ \Rightarrow 10x = 2$
$ \Rightarrow x = \dfrac{2}{{10}} = \dfrac{1}{5}$
So this is the amount of mixture which we have to take out.

So, the correct answer is “Option A”.

Note: Whenever we face such types of questions the key concept we have to remember is that the total amount of water in the mixture = $\dfrac{{{\text{amount of water in original mixture}}}}{{{\text{total amount of mixture}}}} - $Amount of water which we took out from the mixture + amount of water which was replaced, and the total amount of syrup in the mixture = $\dfrac{{{\text{amount of water in original mixture}}}}{{{\text{total amount of mixture}}}} - $ Amount of syrup which we took out from the mixture, then equate these two equations as above and simplify we will get the required answer.