The organizer of the exhibition "I build, you build, we build" divided the exhibition into two parts. Since he was interested in the visitors' reactions to the exhibition, every visitor filled out a simple questionnaire upon leaving. The following interesting facts emerged from it:

- $96 \%$ of visitors who liked the first part also liked the second part,
- $60 \%$ of visitors who liked the second part also liked the first part,
- $59 \%$ of visitors did not like either the first part or the second part.

What percentage of all visitors stated that they liked the first part of the exhibition?

(M. Petrová)
Let $n$ be the number of all people who visited the exhibition, $p$ the number of visitors who liked the first part of the exhibition, and $d$ the number of visitors who liked the second part of the exhibition. We are looking for a relationship between $n$ and $p$, from which we can easily derive the answer to the question.

Express the number of visitors who liked both parts: from the first condition, this is $0.96 p$, and from the second condition, it is $0.6 d$. Clearly, we have:

$$
\begin{aligned}
0.96 p & =0.6 d \\
96 p & =60 d \\
8 p & =5 d \\
1.6 p & =d
\end{aligned}
$$

From this, we can express the number of people in each group in terms of $p$. The number of people who liked:

- the first part but not the second part is $p-0.96 p=0.04 p$,
- the second part but not the first part is $d-0.6 d=0.4 d=0.4 \cdot 1.6 p=0.64 p$,
- both the first and second parts is $0.96 p$.

By summing these, we find the number of people who liked at least one part of the exhibition:

$$
0.04 p+0.64 p+0.96 p=1.64 p .
$$

According to the third condition in the problem, $0.59 n$ visitors did not like either part of the exhibition; thus, $0.41 n$ visitors liked at least one part of the exhibition. This number also corresponds to $1.64 p$. We set up the equation and further simplify it:

$$
\begin{aligned}
0.41 n & =1.64 p, \\
41 n & =164 p, \\
n & =4 p, \\
0.25 n & =p .
\end{aligned}
$$

From this, it follows that $25 \%$ of all visitors stated that they liked the first part of the exhibition.

The relationships between the discussed numbers of visitors can be illustrated, for example, as follows:

![](https://cdn.mathpix.com/cropped/2024_04_17_b0839f028217e6476178g-5.jpg?height=531&width=926&top_left_y=862&top_left_x=565)

Evaluation. 2 points for expressing $d=1.6 p$ or an analogous relationship; 2 points for expressing the number of visitors who liked at least one part in terms of the number of visitors who liked the first part (or for an analogous insight); 2 points for the final $25 \%$.