SOLUTION: 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 wate

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Question 41096: 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 ?
Answer by psbhowmick(878) About Me  (Show Source):
You can put this solution on YOUR website!
VERY GOOD QUESTION.

If you are from India then you can refer to the Algebra book by K.C.Nag or an equally good book by K.P.Basu for these type of problems. You will learn lot from them. Now, let's come to the solution.

Let the volume of the vessel = x litres.
As 3%2F8th part of it is full of water, so volume of water = %283%2F8%29%2Ax litres.
As 5%2F8th part of it is full of syrup, so volume of syrup = %285%2F8%29%2Ax litres.

Let us suppose that the volume of mixture drawn = y litres = volume of water added.
In y litres of mixture drawn, there is %283%2F8%29%2Ay litres of water and %285%2F8%29%2Ay litres of syrup.
So after drawing y litres of mixture, the mixture in the vessel contains:
%283%2F8%29%2Ax+-+%283%2F8%29%2Ay = %283%2F8%29%2A%28x+-+y%29 litres of water and
%285%2F8%29%2Ax+-+%285%2F8%29%2Ay = %285%2F8%29%2A%28x+-+y%29 litres of syrup.

Now, y litres of water is added.
So, the mixture in the vessel contains:
3%28x+-+y%29%2F8+%2B+y = %283x%2B5y%29%2F8 litres of water and
5%28x+-+y%29%2F8 litres of syrup.

[Check: Total volume of mixture finally in the vessel = 5%28x+-+y%29%2F8+%2B+%283x%2B5y%29%2F8 = x litres = volume of vessel. This happens because the volume of mixture taken out from the vessel has been substituted by an equal volume of water.]

Now, finally the volume of water in the vessel = volume of syrup in the vessel.
So 5%28x+-+y%29%2F8+=+%283x%2B5y%29%2F8
or 5(x - y) = 3x + 5y
or 2x = 10y
or y = %281%2F5%29%2Ax

Thus, volume of mixture to be taken out and subsequently replaced by water is 1%2F5th part of the volume of the vessel.
In other words 1%2F5th of the mixture in the vessel is to be replaced by water.

I feel very happy to solve this type of problem with my own method. You won't find any approach as mine in any text book available in the market.