Question 31373: okay... the teacher wants to know why this problem does NOT have a solution...
How many gallons of a 10% and a 20% solution should be mixed to obtain a 30% solution?
Answer by Fermat(136)   (Show Source):
You can put this solution on YOUR website! When you mix two solution, the concentration of the mixture will always be somewhere in between the concentration of the two original solutions.
Proof.
Suppose you mix together X litres of 10% solution and Y litres of 20% solution.
let X be the amount (in litres) of the 10% soln
let Y be the amount (in litres) of the 20% soln
concentration = amount of solute divided by amount of solution
e.g. if you have 100 litres of 10% solution, then 10% of that, 10 litres, is solute.
So we have 0.1X litres of solute from the 10% solution and 0.2Y litres of solute from the 20% solution.
Total volume, of solution, is (X+Y) litres
Therefore concentration of mixture is C = (amount of solute) over (amount of solution)
C = (0.1X + 0.2Y)/(X+Y)
C = (0.1X + 0.1Y + 0.1Y)/(X+Y)
C = (0.1X + 0.1Y)/(X+Y) + 0.1Y/(X+Y)
C = 0.1 + 0.1Y/(X+Y)
C = 0.1 + 0.1{Y/(X+Y)}
Now the fraction X/(X+Y) will always be less than one, which means the max concetration will be 0.1 + 0.1 = 0.2 = 20%. (This requires X to be zero)
You could never get more than 20%.
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