Task B-1.4. King Hiero of Syracuse gave 16 pounds of gold and 4 pounds of silver to have a crown made from this material. When the crown was finished, it weighed 20 pounds, but the king still asked the great mathematician Archimedes to test whether a certain amount of gold had been replaced by an equal weight of silver.

Archimedes submerged the crown in water and found that it had lost $1 \frac{1}{4}$ pounds in weight. Knowing that 20 pounds of gold lose 1 pound of their weight in water, and 21 pounds of silver lose 2 pounds of their weight in water, Archimedes determined by calculation that indeed a certain amount of gold had been replaced by silver. What is this amount of gold?
## Solution.

Let $x$ be the amount of gold (in pounds) in the cast crown, and $y$ be the amount of silver in the cast crown. Then $x+y=20$.

$x$ pounds of gold lose $\frac{x}{20} \cdot 1$ pounds in water, and $y$ pounds of silver lose $\frac{y}{21} \cdot 2$ pounds.

This must total $1 \frac{1}{4}$ pounds of lost weight of the crown.

We write

$$
\frac{x}{20}+\frac{2 y}{21}=1 \frac{1}{4}
$$

We need to solve the system of equations

$$
\begin{cases}x+y & =20 \\ \frac{x}{20}+\frac{2 y}{21} & =\frac{5}{4}\end{cases}
$$

or equivalently

$$
\begin{cases}x+y & =20 \\ 21 x+40 y & =525\end{cases}
$$

The solution is $x=\frac{275}{19}=14 \frac{9}{19}, y=\frac{105}{19}=5 \frac{10}{19}$.

Therefore, $\frac{29}{19}$ pounds of gold have been replaced by silver.