Alligations or Mixtures
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Q.

8 litres are drawn from a cask filled with 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 the total solution is 16:81.

How much wine did the cask hold originally?

 A.

24 litres

 B.

45 litres

 C.

49 litres

 D.

44 litres

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Solution:
Option(A) is correct

Let the quantity of the wine in the cask originally be $x$ litres.
Using formula: 

$\text{Final Amount of solute that is not replaced} =$ 
\[\text{Initial Amount} \times \left(
\dfrac{\text{Vol. after removal}}{\text{Vol. after replacing}}\right)^n \text{ ---(1)}\]
Or
$\text{Final ratio of solute not replaced to total} =$ 
\[\text{Initial ratio} \times \left(\dfrac{\text{Vol. after removal}}{\text{Vol. after replacing}}\right)^n \text{ ---(2)}\]
Considering $2^{nd}$ formula here,

Then ratio of wine to total solution in cask after 4 operations:
\begin{align*}
\Rightarrow 1 \times \left(\dfrac{x-8}{x}\right)^4&=\dfrac{16}{81}\\
\Rightarrow \dfrac{x-8}{x}&=\dfrac{2}{3}\\
\Rightarrow 3x-24&=2x\\
\Rightarrow x=24 \text{ litres.}
\end{align*}


(9) Comment(s)


Shree
 ()

kindly check the question,

If each time 8 litres are removed then we can not use successive removal method because in this problem "litres removed" remains constant, not the "percent of removal".



Kaustubh Mishra
 ()

let initial amount of wine be x

total water x-8

x-8-24/x-8+3x-8=16/81



Varun Chinthala
 ()

The question given is wrong, ratio in the cask left must be in the ratio 16:65.Kindly notify.


Aayush Mittal
 ()

The question shows 16:81 as the ratio of wine left to the solution (not to the water), which makes the ratio of wine to water as 16:65. You are correct, and so is the question. :)


Namita Kumari
 ()

Why multiplied 1 in the initial ratio....initial ratio means 1:0 which to put how I will understand



Prince
 ()

Why initial ratio is 1:1??


Vignesh
 ()

Initially, the cask only has wine. Which means 100% of the solution is wine. So, 1:1.

Brijesh Kakkad
 ()

not a 1:1 because at initial only wine and no water ....hence initial ratio become 1:0


Terra
 ()

I need a thorough explanation of this, mind if you could dig deeper on its relevance and implications., thanks!