Aptitude - Alligation or Mixture - Discussion

Wine:water = 16 : 65
Total =16 + 65 = 81.
Now wine: total = 16:81.

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?

x^4 = 81
=>x=3
y^4=16
=>y=2
x - y
= 3 - 2
= 1.

1 ratio means 8 litre.
so, 3 ratio means 24 litre.

Only wine 8 liters
and 3 times operation.
So, 8 * 3=24.

@All

Here is the explanation for Why it was divided by x?

Here the answer,
16/81 is a wine part of total volume initial x.
So that's why :(16/81)(x) --->step 2
And x goes LHS to denominator x.

For those who don't understand why it is divided by x?

Here in 16/81, 81 can be taken as total mixture are 81y is the total capacity of the cask. So x is the total wine that was kept at first which is nothing but the total capacity of the cask.

Let x be initial volume(in lit) of wine.
By formula final amount of wine after n(=4) iteration = x{1 - 8/x}^4 ---->1.

Given the final ratio of wine to water is 16:65.
So the volume of wine in the final iteration is {16/(16+65)}x -------> 2.

Equate 1 and 2,
And we get x=24.

Thanks all.

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