Aptitude - Alligation or Mixture - Discussion

Correct answer for the question asked "how much of the *mixture* must be drawn off" is 8/5 Or 1.6 (which consists of some part or proportion of water and syrup).

> while the c option given is the correct answer of "what is the proportion of syrup in the mixture drawn off"=1/5.
> while proportion of water in the mixture drawn off = 0.6/3 OR 1/5 (i am proving these below).

Proof:

As given in the question that total quantity (qty) of mixture in a vessel is 8 liters <3 parts of water + 5 parts of syrup>.
Now, we can find through it total ratio of water and syrup in the mixture by this way;

3/8*100 = 37.5% water.
5/8*100 = 62.5% syrup.

Let "x" is the quantity of that mixture drawn off.
& "y" is the quantity of water replaced.

As quantity of mixture drawn off is equal to quantity of replaced water, therefore;
=> x = y .........(euq-1).

Now, if we subtract proportion of water(3x/8) in the mixture drawn off from total quantity contain in the vessel (3) and then add quantity of water replaced which is "y" then it will be equal to half quantity of resulting mixture means last mixture.

"
=> 3 - 3x/8 + y = 1/2 * 8.
=> 3 -3x/8 + y = 4.
=> (-3x+8y)/8 = 4-3.
=> (8y-3x)/8 = 1.
=> 8y-3x = 8........ (equ-2).
now I will put value of "y" from equ-1 in equ-2 to solve for value of "x".

=> 8x - 3x = 8.
=> 5x = 8.
=> x = 8/5 ( this is showing that total quantity of mixture drawn off in liters is 8/5 OR 1.6 ).
Now, I will find out ratio or proportion of water and syrup in the mixture drawn off means in this total quantity=1.6.

Proportion of water in the mixture drawn off =3/8 * 1.6 = 0.6 ltrs.
Proportion of syrup in the mixture drawn off = 5/8 * 1.6 = 1 ltrs OR (1.6 - 0.6).

Total quantity of syrup drawn of is 1 out of 5 which in terms of proportion will be write as 1/5.
Quantity of water drawn of is 0.6 out of 3 which in terms of proportion will be = 0.6/3 OR 1/5.
Now, total quantity of syrup after drawn off = 5 - 1 = 4 liters (half of total quantity 8 liters).

Like wise quantity of water after drawn off and replaced with water = 3 - 0.6 + 1.6 = 4 liters (half of total quantity 8 liters).

water : syrup=3:5
requeired 1:1
3:5::a:b=1;1
3b=1
5a=1
a=1/5.

The Formula is /2*[Big value -small value/ Big value].

When a mixture is drawn, both syrup and water is drawn equally.

So, if 1 part from water and 1 part of syrup(total 2 parts) is drawn, the ratio of 3:5 becomes 2:4. When replaced wit 2 parts of water, gives the ratio 4:4 i..e 1:1.

Therefore mixture drawn is 2/8 or 1/4.

Can anyone tell me, how to find a total mixture here?

Simply;

(3/8) - (3x/8) + x = (4/8);
This will give x = 1/5.

It is not Volume of vessel.

It's a Volume of mixture at initial.

3Lwater + 5L syrup = 8L mix.
Now 1 unit of mix will also have 3 parts water and 5 parts syrup,
Let 1 unit be 800 ml for easy calculation.
And in 800ml we have 300ml water + 500ml syrup
Now, remove 1 unit from the total, you will have a mix of 7.2L.
removed 1unit = 800ml in 7.2L × 5/8 = 4.5 L syrup left.
And add 800ml of water in its place.
New ratio of mix 4.5l syrup: 2.7 old +0.8 newly added
4.5:3.5 we need 4:4 for both equal
Remove 1 more unit from the mix.
7.2 - 0.8 = 6.4L.
Syrup left 6.4L × 5/8 = 4.0L
Add water in place of 800ml
Now you have 4L syrup: 4L water.

Replacement of syrup with water = 800ml × 2 = 1.6L.

Now, how much replacement from the original 8L in ratio?
So, 1.6/8 total replacement reduces to 1/5(ans).

Can you give a shortcut for this?

I did not get last Step. Could you please explain how it was?

So, part of the mixture replaced =

(8/5) * (1/8) =1/5.