Aptitude - Alligation or Mixture - Discussion

It should be 16:81.

Explanation for (x(1 - 8/x)pow 4)/x =16/81.

In question it's given water/wine = 16/65,
We assumed x= total wine in cask,
That means total capacity of cask is x,

So looking at ratio 16/65 we can see that total
Capacity is (16+65) = 81.

(x(1 - 8/x)pow 4) is nothing but the wine replaced by water, In other word, it is the quantity of water in a final mixture of water and wine.

So, (x(1 - 8/x)pow 4)/x must be 16/81.

Let cask is full of wine = x lt.
After 4 operations wine left= x(1-8/x)^4.
Ratio of water to wine after operation = 16/65.
From this ratio part of wine present after op= 16/81.
Total amount of mixture is = x(as cask is full of x quantity).

Now,

x(1-8/x)^4.
---------------- = 16/81
X

From this x = 24.

Lets wine x.

After 1st time wine remain x-8
Now, ratio of wine:water x-8:8
So 2nd time wine removed=(x-8) * (8/x)
After 2nd time wine remain=(x-8) - (x-8) * (8/x).
= (x-8){1-8/x}
= x(1-8/x)(1-8/x)
= x(1-8/x)^2.

After 3rd time wine remain=[x(1-8/x)^2]-[x(1-8/x)^2]x/8.
= [x(1-8/x)^2]{1-8/x}
= [x(1-8/x)^2](1-8/x)
= x(1-8/x)^3 and so on;

For n operation remain=x(1-8/x)^n.

As 65:15 is wine as to water ratio is given means total solution was (65+16)=81 which was totally quantity of wine before removal of 8 liter wine that's why it is x = total =81.
Where x=initial quantity of wine.
W:T = x(1-(8/x))^4:x = 16/81.

If this process done with 5 litres only one time, and the given ratio is 361:39. What will be the answer?

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.

Thanks @Ayush.

Thanks, @Pravu.

You explained the derivation of the formula is very well.

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.