Scientists were letting rats into a maze and observed whether they would reach the goal. They found that $56\%$ of the black rats and $84\%$ of the white rats reached the goal. The ratio of the number of black and white rats at the goal was $1:2$.

What was the ratio of the number of black and white rats at the start?

(M. Petrová)
Let's denote the number of black rabbits at the start as $x$, and the number of white rabbits at the start as $y$. According to the problem, $0.56 x$ black rabbits and $0.84 y$ white rabbits reached the finish line, and it is given that $0.56 x: 0.84 y = 1: 2$. We need to find the ratio $x: y$.

The above equality can be written as

$$
\frac{0.56 x}{0.84 y} = \frac{1}{2}
$$

which is equivalent to

$$
\frac{x}{y} = \frac{1}{2} \cdot \frac{84}{56} = \frac{42}{56} = \frac{3}{4}
$$

The ratio of black to white rabbits at the start was $3: 4$.

Scoring suggestion. 3 points for deriving the initial equality or a similar relationship; 3 points for expressing the ratio $x: y$.

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