## Task 2 - 070812

At what mass ratio of 10 percent and 30 percent salt solution do you obtain a 25 percent salt solution after mixing? (The percentages are based on mass.)
A 10%, 30%, and 25% salt solution $a, b$, and $c$ of 1 unit (unit quantity) can be represented as follows, where $s$ is the proportion of salt and $w$ is the proportion of water:

$$
a=0.1 s+0.9 w \quad ; \quad b=0.3 s+0.7 w \quad ; \quad c=0.25 s+0.75 w
$$

If the 25% solution $c$ consists of a mixture of $k$ parts of the 10% solution and $l$ parts of the 30% solution, this can be expressed as the following equation:

$$
\begin{aligned}
c & =k \cdot a+l \cdot b \\
0.25 s+0.75 w & =k \cdot(0.1 s+0.9 w)+l \cdot(0.3 s+0.7 w) \\
0.25 s+0.75 w & =(0.1 k+0.3 l) \cdot s+(0.9 k+0.7 l) \cdot w
\end{aligned}
$$

The comparison of coefficients then means:

$$
0.25=0.1 k+0.3 l \quad ; \quad 0.75=0.9 k+0.7 l
$$

This results in, after multiplying the first equation by 9 and then subtracting the second equation from it: $1.5=2 l$ or $l=0.75$. Substituting this into the first equation yields: $0.25=0.1 k+0.225$ or $k=0.25$.

Thus, the desired ratio $k: l$ is as follows: $0.25: 0.75$ or $1: 3$.