1B. In a goldsmith's workshop, a mixture of $9 \mathrm{~kg}$ needs to be made in which gold and silver will be in the ratio 7:11. There are two mixtures available. In the first, the quantities of gold and silver are in the ratio $4: 5$, and in the second, in the ratio $2: 5$. How much should be taken from each mixture to obtain the new mixture in the given ratio?
Solution. Let the new alloy contain $a \mathrm{~kg}$ of gold and $b \mathrm{~kg}$ of silver. From the system

$$
\left\{\begin{array}{l}
a: b=7: 11 \\
a+b=9
\end{array}\right.
$$

we obtain that $a=3.5$ and $b=5.5$. Let $x \mathrm{~kg}$ be taken from the first alloy, and $y \mathrm{~kg}$ from the second. Then in the new alloy, there will be $\left(\frac{4}{9} x+\frac{2}{7} y\right) \mathrm{kg}$ of gold and $\left(\frac{5}{9} x+\frac{5}{7} y\right) \mathrm{kg}$ of silver, i.e.

$$
\left\{\begin{array}{l}
\frac{4}{9} x+\frac{2}{7} y=3.5 \\
\frac{5}{9} x+\frac{5}{7} y=5.5
\end{array},\right.
$$

from which we obtain that $x=5.85$ and $y=3.15$.