Question

A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. how much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup? solve it using allegation rule

Answer 1

Water : Syrup = 3:5

Let's assume the volume of the vessel is 80 Litres.

Water = 30 Litres

Syrup = 50 Litres

Since all the liquid taken out of the vessel is replaced by water, the total volume remains constant.

Therefore, we need to ensure there's 40 litres of water and Syrup individually.

Therefore, we need to take out 10 litres of syrup and replace it with water.

For every 8 litres taken out of the vessel, we replace 5 litres of syrup with water.

Therefore, we need to replace 16 litres, or 20% of the solution with water.

OR

1/5

Let us say that the quantity in the vessel is 8x units. It has 3x water and 5x syrup. 
Now,  let's say we withdraw 8y units of mixture are replace them with water.  When we take out 8y units of mixture, we are actually taking out 3y water and 5y syrup. We are adding pure water back into the vessel.

New quantity of syrup = 5x - 5y
New quantity of water = 3x - 3y + 8y = 3x + 5y
They are equal to each other.
=> 5x - 5y = 3x + 5y
=> 2x = 10y
=> y/x = 2/10 = 1/5
=> 8y/8x = 1/5

So, 1/5th of the mixture should be taken out and replaced with water.

Hope This Helps :)

Answer 2

Suppose the vessel initially contains 8 liters of liquid.

Let x liters of this liquid be replaced with water.

Water in new mixture = (3-3x/8+x)

Syrup in new mixture = (5-5x/8)

Then (3-3x/8+x) = (5-5x/8)

5x + 24 = 40 - 5x

10x=16

==>x=8/5

So part of mixture replaced is 8/5*1/8=1/5