5. In the first alloy of copper and lead, the mass ratio is $1: 3$; in the second alloy, it is $1: 2$. How many grams of the first alloy should be taken to obtain 10 g of a new alloy with a mass ratio of copper to lead of $3: 7$?
Solution. Let there be x g of copper and 3x g of lead in the first alloy. In the second alloy, there are y g of copper and 2y g of lead. Then $k \cdot 4x + n \cdot 3y = 10 ; \frac{kx + n \cdot y}{k \cdot 3x + n \cdot 2y} = \frac{3}{7}$; we need to find $k \cdot 4x$ and 3ny. Let $ny = b ; kx = a$
$\cdot \frac{a + b}{3a + 2b} = \frac{3}{7} \cdot 7a + 7b = 9a + 6b ; b = 2a$ Using

$4a + 6a = 10 ; a = 1 ; b = 2$ Therefore, $k \cdot 4x = 4 ; 3$ n $= 6$

Answer: 4.