Aptitude - Alligation or Mixture - Discussion

We need 2 parts of water to be added to make water and syrup equal then.

5 parts of water = 5 parts of syrup.
Total = 10 parts.
No of items=2 (Water, Syrup).

= 2/10.
= 1/5.

The above solution states that we first assume the quantity to be 5+3=8 for easier understanding.
And since the question directly states we need to first remove some amount from the mixture and then add the same amount. for ex : from 8 litres remove 2 , 8 - 2 and then add 2 litres of water 6 + 2 =8.

So clearly, the quantity remains the same.

Now while removing we need to keep in mind that we are removing an amount from the mixture and not individually from water and juice (because that would be insane) as it's a mixture.

Hence, if we remove the same amount we add the same amount.

The mixture has water and syrup in a ratio, and the mixture consists of 3/8 part of water (3 litres) and 5/8 (5 litres) part of syrup. If we remove x litres from the whole mixture, the part of juice and water removed will be calculated in this manner :

new Water = 3 - (3x/8).
new Syrup = 5 - (5x/8).

Now since the same x amount of water must be added to keep the mixture in the same volume,
new Water = 3 - (3x/8) + x.

The whole point of this transaction according to the question is that both quantities take up the same ratio or are equally present in the mixture,

New Water amount = New Syrup amount.

Why multiply with 1/8? Please explain to me.

Let 100 ltr mixture.

So, 5/8x 100 = 62.5 ltr syrup.
3/8x100 = 37.5 ltr water.

We need to remove syrup and add that much quantity of water so make it 1:1,
62.5- x/37.5+x = 1/1,
62.5- x = 37.5 +x.
2x = 25.
x = 12.5 (need to remove from syrup and this quantity into water),
12.5/62.5(syrup) = 1/5.

We need 2 parts of water to be added to make water and syrup equal.

5 parts of water = 5 parts of syrup.
Total = 10 parts.
No of items = 2 (Water, Syrup).
= 2/10.
= 1/5.

Actually here is a simple way of solving.

As there are total 8 parts (3 parts water and 5 parts syrup), the question is that how much quantity of water is added to make both water and syrup half each, frankly speaking, there is no need to remove water as it is to be added again so 1 part of syrup is removed and added as water, so 1 part of 5 parts is 1/5.

That's simple.

Let
The total mixture be 8 litres.
3 l is water and 5 l is syrup.

Now we need an equal quantity to be replaced as filled.
so let the quantity be x.
5-x/3+x = 1/1.
x=1.
Part that x forms the whole quantity
= 1/5.

Let 100 ltr mixture.

So 5/8x 100=62.5 ltr syrup.
3/8x100= 37.5 ltr water.

We need to remove syrup and add that much quantity of water so make it 1:1
62.5- x/37.5+x=1/1,
62.5- x =37.5 +x.
2x =25.
x = 12.5 (need to remove from syrup and this quantity into water),
12.5/62.5(syrup) = 1/5.

Let's say after removing some part of the mixture.

The ratio of the remaining mixture is the same i.e. 3:5 for water to syrup.
Now consider this mixture is 8L in total (3+5).
Now once we add 2L of water in the mixture the ratio becomes 5:5 which was desired.
5+5 =10.

So we can say that even in the original mixture the quantity was 10L as 2L was removed from that.
(2L mixture removed and 2L water added).
This satisfies the condition.
Now,
The question is how much of the mixture is to be removed and replaced.
2L/10L i.e.1/5.
1/5 will be the answer.

Anyone, Please explain in the simplest way to understand.