Aptitude - Alligation or Mixture - Discussion

Thanks all.

How 16:65 is converted to 16:81? Please explain me.

Here, x is just used for simplification.

Suppose we write the equation as:
[x({1-(8/x)}^4)]/32 = 16/65 it is correct.

But, it will be very difficult to simplify because of the fifth power in the resulting equation that we have to solve.

So instead, we take the ratio of wine to total liquid in the end. We can find the total liquid by simply adding 16+65, ie, wine=water, so this is not an issue. This way, we can eliminate the x outside bracket in the numerator and find the answer very quickly.

Hope it's clear.

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

In this statement, where did the extra "x" in the denominator come from? Also when the ratio is 16:65, then how is it changed to 16:81?

See the formula is well understood a(1-b/a)^n. Now even after adding and removing water we have the total amount o be the same that is x.

So we have 16/81 of x = This is the part of wine left that should be equal to the formula.

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.