Problem 4. Students from a school were supposed to go on a trip. $\frac{2}{9}$ more students registered than the planned number. Before departure, $\frac{3}{11}$ of the registered students canceled due to illness, so 5 fewer students went on the trip than the planned number. How many students went on the trip?
Solution. Let's denote the planned number of students by $x$. Then after registration, there were $x+\frac{2}{9} x=\frac{11}{9} x$. Since before departure, $\frac{3}{11}$ of the registered students canceled, i.e., $\frac{3}{11} \cdot \frac{11}{9} x=\frac{3}{9} x=\frac{1}{3} x$ of the planned number of students canceled. Therefore, the total number of students who went on the excursion is $\frac{11}{9} x-\frac{1}{3} x=\frac{8}{9} x$. The number of students who went on the excursion was 5 fewer than the planned number, so according to this, $\frac{8}{9} x=x-5$, i.e., $x=45$.

Thus, the number of students who went on the excursion is $x-5=45-5=40$.