I'm new here, so kindly bear with me if this question is too trivial for the forum. I tried working it out but to no avail.
A solution contain wine and water in ratio 2:1. Out of it 36 litre solution is replaced by 15 litre water. Again 26 litre solution is replaced by 26 litres of water. Now wine to water ratio becomes 8:7. Find the difference between the quantity of water in the initial and the final solution.
Also, why does the ratio stay the same every time we take out liquid from a mixture of liquids. I mean not in this case because water is added here. I mean is there a proof of the intuitive fact?
Let Wine = $2x$ and Water = $x $
So, total volume = $3x$ Litre
Step $1$ : $36$ litre solution is replaced by $15$ litre water.
Wine $= 2x - 24 = 2(x-12)$
Water $= x - 12 + 15 = x + 3 $
Total Volume $= 3x - 21 = 3(x - 7)$
Step $2$ : Now $26$ litre Solution is replaced by $26$ litre water
Wine $= 2(x - 12) - \frac{2(x - 12)}{3(x - 7)} \cdot 26 = \frac{2(x-12)(3x-47)}{3(x-7)}$
Water $ = x + 3 - \frac{(x + 3)}{3(x - 7)} \cdot 26 + 26 = \frac{(3x² + 40x - 687)}{3(x - 7)}$
Now we know the ratio should be $8:7$ So, we end up with $$\Rightarrow \frac{2(3x^2 -83x + 564)}{3x^2 + 40x - 687} =\frac{8}{7}$$
$$\Rightarrow \frac{3x^2 -83x + 564}{3x^2 + 40x - 687} = \frac{4}{7}$$
$$\Rightarrow 21x^2 - 581x + 3948 = 12x^2 + 160x - 2748$$
$$\Rightarrow 9x^2 - 741x + 6696 = 0$$
$$\Rightarrow 3x^2 - 247x + 2232 =0$$
$$\Rightarrow (3x - 31) (x - 72) = 0 $$
Therefore, $x = \frac{31}{3}$ or $x = 72$
$x = \frac{31}{3}$ is not possible as we are supposed to take out $36$ litres initially.
Hence, $x = 72$
Initial water volume = $72$
Final water volume $= \frac{(3x² + 40x - 687)}{3(x - 7)}$ , putting $x = 72$ gives $91$
Hence, the difference between water quantity in the initial and final solution is $19$ Litre