Aptitude - Alligation or Mixture - Discussion

Intial wine = x.
After 8 units are pulled out, (x-8) units of pure wine remains.
In fractions, ((x-8)/8) = (1-(8/x)) of the initial content x remains after the first pullout.
After the second pullout, (1-(8/x)) of the previous amount is left, i.e.- (1-(8/x)) of (1-(8/x)) of x = (1-(8/x))^2 of x.
In the same way, after n such pullouts, the fraction of wine left will be (1-(8/x))^n.
And the amount of pure wine left will be x*(1-(8/x))^n.

This part is well understood. But why was the above quantity divided by x, while the question reported the ratio as that of leftover pure wine to water? I am finding it hard to understand the logic behind this. Or is it wrong?

Awesome rachit.

Can any one please explain ....How the quantity of water become x?

@Prajakt
the soln. assumes in the 1st line that the original quantity of wine be x. the question says the wine removed is replaced with water(obviously, in equal amount). Since we assume x as wine quantity, this replaced water quantity is also x.

Since each time we are adding 8 litres of water and talking away 8 litres. So total composition is same only the proportions of individual componnts will change.

So talking water as x litre is wrong.

He has calculated the ratio of wine to whole soln. !

Equation should be equated with 16/97.

Sine wine to water ratio is 16:81.

So wine to total ratio is 16:97.

Thats what we finding out actually by dividing it by x.

I agree with Sahil.

Another method:
Amt. of wine left is given by x(1-8/x)^4.
Amt. of water left is given by x- x(1-8/x)^4.

So x(1-8/x)^4/[x-x(1-8/x)^4] = 16:81

With this method, i got answers to a couple of questions which were same as this question. But unable to get the answer to this one. I think my method is perfectly alright.

@Prateek.

Here it is given that '8 liters are drawn from a cask full of wine' not 'wine and water'

so you cannot subtract i.e.,

Amt. of water left is given by x- x(1-8/x)^4.

So how can you subtract ?

Let the quantity of the wine in the cask originally be x liters.

Then, quantity of wine left in cask after 4 operations.

=[x (1- (8/8) ) ^4].

Hence, [{x (1- (8/x) ^4) ) }/x]=16/97.

=>[ (x-8) /x]^4=16/97.

=> (x-8) /x=2/3.1 (since 97^1/4=3.1).

=>x=22.5 Liters.

Answer should be 22.5 Liters.

There can be no doubt about the fact that they got the last ratio wrong in the question.

@rattle, sorry but you are right if the replacement stopped after the first round. After that when on the second round 8 litres is taken from the cask, it includes some wine and some water and then 8 litres of water in put in. This continues twice more. Sorry but we cannot assume x to be the amount of water in the cask after mixing is complete.