Students at our school travel in various ways. Those living nearby walk. The ratio of local students to commuting students is $3: 1$. Among the commuting students, the ratio of those who use public transportation and those who ride a bike or are driven by their parents is $3: 2$. Among those using public transportation, the ratio of those who take the train and those who take the bus is $7: 5$. Furthermore, the ratio of those who ride a bike to those who are driven by their parents is $5: 3$. By how many more students do commute by train compared to those who are driven by their parents, if 24 students use public transportation? How many students does our school have?

(M. Volfová)
Those who use public transportation are 24 and form 3 parts of the number of commuters. The remaining 2 parts, which belong to non-public transportation, thus correspond to 16 students $\left(\frac{2}{3}\right.$ of 24 is 16). The total number of commuters is $24+16=40$. Commuters form 1 part of all the students in the school, and there are three times as many students living at home, i.e., 120. Therefore, the total number of students is $40+120=160$.

24 commuting students using public transportation are divided into those traveling by train (7 parts) and by bus (5 parts); 14 students travel by train ( $\frac{7}{12}$ of 24 is 14) and 10 by bus ( $\frac{5}{12}$ of 24 is 10). The 16 students who use non-public transportation are divided into those who ride bicycles (5 parts) and those driven by parents (3 parts); 10 students commute by bicycle ( $\frac{5}{8}$ of 16 is 10), and 6 by car ( $\frac{3}{8}$ of 16 is 6).

Conclusion: The school has a total of 160 students, of which 8 more students travel by train than those driven by parents by car $(14-6=8)$.

Alternative solution. Sketch a line segment representing all the students in the school and divide it according to the given ratios, see the figure.

![](https://cdn.mathpix.com/cropped/2024_04_17_27d9befe2cb3ac9dc509g-2.jpg?height=342&width=1362&top_left_y=383&top_left_x=378)

Besides the ratios, the only numerical data given in the problem is that 24 students travel by train and bus. From the figure, we can deduce that the number of commuting students is $\frac{5}{3} \cdot 24=40$ and the total number of students is $4 \cdot 40=160$.

The smallest segments into which the part of the line segment corresponding to public transportation is divided represent $24: 12$ = 2 students. The part of the line segment corresponding to non-public transportation is also divided into such segments - the segments are the same because they once represent a twelfth of three parts, and once an eighth of two parts, and $\frac{3}{12}=\frac{2}{8}$. The part of the line segment corresponding to the train is 4 such segments larger than the part corresponding to the car. Therefore, 4 times 2 = 8 more students travel to school by train than by car.

Evaluation. 3 points for the total number of students; 3 points for the difference in the number of students commuting by train and by car.