From a survey at our school, it was found that

- all children who like mathematics solve the Mathematical Olympiad,
- $90\%$ of children who do not like mathematics do not solve the Mathematical Olympiad,
- $46\%$ of children solve the Mathematical Olympiad.

What percentage of children in our school like mathematics?

(L. Hozová)
The Mathematical Olympiad (MO) is solved by all children who like mathematics, but only one-tenth of those who do not like mathematics.

Let's say $x \%$ of children like mathematics, and thus solve MO. Then $(46-x) \%$ of children do not like mathematics and also solve MO. The total number of children who do not like mathematics is ten times more, i.e., $10(46-x)\%$. Altogether, we have

$$
x + 10(46 - x) = 100
$$

from which simple manipulations yield $x = 40$.

The number of children who like mathematics is $40 \%$.

Notes. Different notation leads to different manipulations. Suppose $y \%$ of children do not like mathematics. Then $\frac{9}{10} y \%$ of children do not solve MO. This corresponds to $54 \%$ of children $(100 - 46 = 54)$. Altogether, we have

$$
\frac{9}{10} y = 54
$$

from which simple manipulations yield $y = 60$.

The given relationships can be illustrated using a Venn diagram as follows:

![](https://cdn.mathpix.com/cropped/2024_04_17_5ebc08916757a9a5715ag-2.jpg?height=345&width=668&top_left_y=1792&top_left_x=700)

Evaluation. 2 points for auxiliary notation and partial observations; 2 points for the result; 2 points for the quality of the commentary.