Question

Make me a crown weighing 60 minae from a mixture of gold, bronze, tin, and wrought iron. Let the gold and bronze together form two-thirds of the weight, the gold and tin together three-fourths, and the gold and iron three-fifths. Tell me how much gold, tin, bronze, and iron you must use. (From the Greek Anthology by Metrodorus, ${\mathbf{6}}^{\mathbf{th}}$century A.D.)

Short answer

30.5 minae of gold, 9.5 minae of bronze, 14.5 minae of tin and 5.5 minae of iron must be used to make a crown of 60 minae.

Step-by-step solution

Represent the weight of metals gold, bronze, tin and ironin terms of system of equations.

The weight of metalscan be found using the augmented matrix.

The total weight of crown isminae.

Let G = Gold, B= Bronze, T =Tin, I =Iron

Therefore, according to question

$G+B+T+I=60.......\left(1\right)$

Gold and bronze makes up$\frac{2}{3}$ weight of the crown.

$$\begin{gathered} G+B=\frac{2}{3}\times 60 \\ \therefore G+B=40......\left(2\right) \end{gathered}$$

Gold and tin makes up$\frac{3}{4}$weight of the crown.

$$\begin{gathered} G+T=\frac{3}{4}\times 60 \\ \therefore G+T=45.....\left(3\right) \end{gathered}$$

Gold and iron makes up$\frac{3}{5}$weight of the crown.

$$\begin{gathered} G+I=\frac{3}{5}\times 60 \\ \therefore G+I=36.......\left(4\right) \end{gathered}$$

Consider the equations (1), (2)(3)and (4). The system of equations is,

$\left|\begin{array}{c}G+B+T+I=60\\ G+B=40\\ G+T=45\\ G+I=36\end{array}\right|$

Represent the above equations in terms of a matrix.

$\left|\begin{array}{ccccc}1& 1& 1& 1& 60\\ 1& 1& 0& 0& 40\\ 1& 0& 1& 0& 45\\ 1& 0& 0& 1& 36\end{array}\right|$

Perform the row operations on the matrix.

Consider the matrix.

$$\begin{gathered} \left|\begin{array}{ccccc}1& 1& 1& 1& 60\\ 1& 1& 0& 0& 40\\ 1& 0& 1& 0& 45\\ 1& 0& 0& 1& 36\end{array}\right| \\ =\left|\begin{array}{ccccc}1& 1& 1& 1& 60\\ 0& -1& 0& -1& -15\\ 0& 0& -1& -1& -20\\ 0& 0& 0& 2& 11\end{array}\right| \\ =\left|\begin{array}{ccccc}1& 0& 0& 0& \frac{61}{2}\\ 0& 1& 0& 0& \frac{19}{2}\\ 0& 0& 1& 0& \frac{29}{2}\\ 0& 0& 0& 1& \frac{11}{2}\end{array}\right| \end{gathered}$$

The values are, $G=\frac{61}{2}=30.5,B=\frac{19}{2}=9.5,T=\frac{29}{2}=14.5,I=\frac{11}{2}=5.5$

Final answer

The metal gold should weigh 30.5 minae, the metal bronze should weigh 9.5 minae, the metal tin should weigh 14.5 minae and the metal iron should weigh 5.5 minae.