A printing company produces a board game that requires two types of figures: dogs and cats; these are made by cutting them out of cardboard paper. In each set, the ratio of cats to dogs is $5: 3$. To minimize waste, they print two types of cardboard: one type has 28 dogs and 28 cats on a sheet, the other has 42 cats. In what ratio should they print the two types of cardboard so that after cutting, the ratio of the two types of figures is exactly as required?
Solution. Let the number of printed sheets from the mixed sheet be $x$, and the number of printed sheets from the only-cats sheet be $y$. Then the number of cut-out dogs is $28 x$, and the number of cut-out cats is $28 x + 42 y$. The ratio of the number of the two animal figures is $5: 3$, so $\frac{28 x + 42 y}{28 x} = \frac{5}{3}$. From this, with some rearrangements, $\frac{x}{y} = \frac{9}{4}$. To achieve the desired number of pieces, the correct ratio of the number of printed sheets from the mixed and only-cats sheets is therefore $9: 4$.