Question

8 Litre is drawn from a cask full of wine and is then filled with water. This operation is performed three more times. The ratio of the quantity of wine now left in cask to that of water is $16:65$. How much wine did the cask hold originally?
(A) 18 Litres
(B) 24 Litres
(C) 32 Litres
(D) 42 Litres

Answer 1

Hint: Here, we will assume the quantity of wine originally in the cask to be some variable. Then we will use the formula of mixture and alligation to find the remaining wine in the cask. We will then divide the quantity of remaining wine by the original quantity of wine and equate it to the given ratio. We will solve this equation further to get the required value.

Formula Used:
$x{\left( {1 - \dfrac{y}{x}} \right)^n}$ where $x$ is the liquid from which $y$ units are taken out and is then replaced by the water n times.

Complete step-by-step answer:
Let us consider that quantity of wine originally in the cask was $x$.
Now, we know that the amount of wine drawn out from the cask is $8$ litres.
Therefore, $y = 8$
Next, we know that this operation was performed $1 + 3$ i.e. 4 times
Therefore, $n = 4$
Now we will find the remaining wine in the cask.
So, substituting all the value in the formula, we get
The remaining wine in the cask $ = x{\left( {1 - \dfrac{8}{x}} \right)^4}$ litres
Next, we are given that the ratio between the quantity of wine and water after the operation is $16:65$.
Therefore, we can form the equation as:
 $\left( {\dfrac{{x{{\left( {1 - \dfrac{8}{x}} \right)}^4}}}{x}} \right) = \dfrac{{16}}{{81}}$
Simplifying the expression, we get
$ \Rightarrow {\left( {1 - \dfrac{8}{x}} \right)^4} = \left( {\dfrac{{16}}{{81}}} \right)$
We can write above equation as
$ \Rightarrow {\left( {1 - \dfrac{8}{x}} \right)^4} = {\left( {\dfrac{2}{3}} \right)^4} \\
   \Rightarrow 1 - \dfrac{8}{x} = \dfrac{2}{3} \\ $
Taking LCM on LHS, we get
$ \Rightarrow \dfrac{{x - 8}}{x} = \dfrac{2}{3}$
On cross multiplication, we get
$ \Rightarrow 3\left( {x - 8} \right) = 2x$
Multiplying the terms, we get
$ \Rightarrow 3x - 24 = 2x$
Rewriting the expression, we get
$ \Rightarrow 3x - 2x = 24$
Subtracting the like terms, we get
$ \Rightarrow x = 24$
Hence, the original amount of wine in the cask was 24 litres.
So option (b) is correct.

Note: This question is from the section of mixture and alligation. A mixture is defined as mixing two or more things together, whereas allegation helps us to find the ratio in which the two entities have been mixed. Here, we need to keep in mind that we have to add 16 and 65 to the denominator of the fraction and not take 65. This is because 65 is the quantity of water and 81 is the quantity of both water and remaining wine. So, we need to be careful while writing the fraction.