Question

A barrel contains a mixture of wine and water in the ratio 3 : 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture in the barrel becomes 1 : 1?
(a) $\dfrac{1}{4}$
(b) $\dfrac{1}{3}$
(c) $\dfrac{3}{4}$
(d) $\dfrac{2}{3}$

Answer 1

Hint: If we interpret the ratio, we get 3x litres of wine and x litres of water present. Let us take the amount of mixture taken out to be 4p litres. The amount of water and wine contained in the removed mixture is in the same ratio as they are present in the mixture, that is if 3 parts of wine is removed, 1 part of water will be remove, so the amount of water removed is p and amount of wine removed is 3p and the new water content is x-p+4p=x+3p. Now take the ratio of left wine and water and equate it with 1:1. Finally, to get the fraction divide 4p by 4x to get the answer.

Complete step by step solution:
It is given that a barrel contains a mixture of wine and water in the ratio 3 : 1. SO, we can say that the amount of wine is 3x and the amount of water in the barrel is x. So, the total quantity of mixture present is 4x.
Now we will consider that 4p litres of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture in the barrel becomes 1 : 1. Also, the amount of water and wine in the removed mixture is in the same ratio as the composition of the mixture, i.e., 3 parts of wine is removed and 1 part of water which is 3p litres of wine and p litres of water is removed. As the same amount of water is added as the amount of mixture removed, we can say that p litres of water was taken out and 4p litres of water is added.
$\text{New water content}=x-p+4p=\left( x+3p \right)\text{ litres}$
\[\text{New wine content}=\left( 3x-3p \right)\text{ litres}\]
As the ratio between new contents of wine and water is 1:1, both have equal contents. So, if we equate their contents, we get
$x+3p=3x-3p$
$\Rightarrow 6p=2x$
$\Rightarrow 3p=x........(i)$
Now to find the fraction of mixture substituted, divide the amount of substituted mixture by total amount of mixture, i.e., 4p by 4x. On doing so, we get
$\text{Fraction of mixture substituted}=\dfrac{4p}{4x}=\dfrac{p}{x}$
Putting the value of x from equation (i).
$\text{Fraction of mixture substituted}=\dfrac{p}{3p}=\dfrac{1}{3}$
Hence, the answer to the above question is option (b).

Note: The things to be pointed out is: as some amount of mixture is substituted, the total quantity of the mixture is not changed. The other thing to keep in mind is that the fraction of mixture which is being removed and substituted will have the same composition as the whole mixture, as we consider the mixture to be a homogenous mixture until mentioned.