Mario completed 30 hours in an extra math course. On the days he had classes, the duration was only 1 hour, which occurred exclusively in the morning or in the afternoon. Additionally, there were 20 afternoons and 18 mornings without classes during the course period.

a) On how many days were there no classes?

b) How many days did the course last?
Solution

We can construct a diagram with two intertwined sets. Set $T$, representing afternoons without classes, and set $M$, representing mornings without classes. Thus, the intersection $(T \cap M)$, which has $x$ elements, represents the full days without classes.

![](https://cdn.mathpix.com/cropped/2024_05_01_09dbad5281c800d666d6g-19.jpg?height=452&width=581&top_left_y=1979&top_left_x=817)
a) The subset $(T-M)$ represents the days with classes only in the morning, and the subset $(M-T)$ represents the days with classes only in the afternoon. Therefore, we fill in the diagram with the information that there were $(20-x)$ mornings with classes and $(18-x)$ afternoons with classes, totaling 30 hours of the course, so

$$
\begin{aligned}
20-x+18-x & =30 \\
-2 x & =30-38 \\
2 x & =8 \\
x & =4 \text { days without classes. }
\end{aligned}
$$

b) There are 30 days with classes and 4 without, that is, the duration was

$$
30+4=34 \text { days. }
$$

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