Aptitude - Alligation or Mixture - Discussion

How can did wine left ratio is 16:65 to change 16:81 please explain anybody?

If 16x/81 is written based on the LHS it converted to 2/3 then what about x.

Step 1: (x - 8)^4/x = 16/81.
Step 2: (x - 8)^4/x = (2/3)^4.
Therefore, x - 8/x = 2/3.
So, x=24.

Hi, anyone clears my doubt.

Logically, How can you remove only wine again after mixing 8 liters of water to the wine already present in the cask?

Does anybody know the formula for calculating no of liters of wine present if the solution removed from second time onwards contain both wine and the water?

I think most of you understood the solution but for those who didn't let me explain:

First of all, There is a formula which says:

Suppose container contains x of liquid from which if y units are taken out (of the whole liquid) and then replaced with water, and this process is simply repeated n no of times, then the quantity of original liquid present in the mixture now is:

x[(1- y/x )^n].

Ok , now as the question says here we need to find the original quantity of the wine, let it be x.

Now the quantity of liquid (not wine) removed is = 8 which is 'y' for the formula.

This thing is done total of 4 times.

So now the quantity of the wine present in the final mixture will be:

x[(1 - 8/x) ^4]. No problem till here.

Now as the problem says, the ratio of wine to water in the final mixture, we have calculated wine but we can't calculate the amount of water in the final mixture.
So, what to do next.

We are given the ratio of wine to water in the mixture and remember the water was being replaced each time, so total quantity of the mixture is still 'x'.

Thus, if 16/65 was the given ratio of wine to water then, 16/(16+65)

will be the ratio of wine to total mixture ie:

x[(1-8/x)^n] / x = 16/81

I hope everyone get it now.

Good explanation @Ayush. You cleared the doubt everyone had.

Let initial quantity of wine = x.
After, 4 operations, quantity of wine left = x(1 - 8/x)^4.
Also the quantity of wine in the jar = (16/81)x.
Therefore, x(1-8/x)^4 = (16/81)x.
On solving, x = 24.

Thanks, @Pravu.

You explained the derivation of the formula is very well.

Thanks @Ayush.

Suppose, the initial amount of mixture is X.

Remaining quantity of wine after four operations ( P suppose)= X*(1-8/X)^4 = P(Suppose)
The total quantity of mixture will always remain X because how much mixture we are removing same we are adding as water and quantity which is not of wine that is water so after 4 operations quantity of water in the mixture = X-P.

Given: remaining wine/water=16/65= P/(X-P)...................(1).
By solving equation (1) we will get P/X=16/81, that is the ratio of remaining wine and present water in the mixture.

So this is the explanation why he divided by X.
Further is simple as we supposed P = X * (1-8/X) ^4.
So, P/X = (1-8/X)^4 = 16/81.............(2).
Solving equation (2) we get X = 24.