Shapovalov A.V.

In a small town, there is only one tram line. It is a circular line, and trams run in both directions. There are stops called Circus, Park, and Zoo on the loop. The journey from Park to Zoo via Circus is three times longer than the journey not via Circus. The journey from Circus to Zoo via Park is half as short as the journey not via Park. Which route from Park to Circus is shorter - via Zoo or not via Zoo - and by how many times?
Let's get on the tram at the Zoo stop and travel through the Circus to the Park, and then, without leaving the tram, return to the Zoo. The second part of the journey is three times shorter than the first, meaning the first part takes up three-quarters of the full circle, and the second part takes up one-quarter. Mark the Zoo and the Park on the diagram, and somewhere on the longer arc between them, mark the Circus (see the figure). Now, on the same tram, let's travel from the Circus to the Zoo (passing the Park, as shown on the diagram).

![](https://cdn.mathpix.com/cropped/2024_05_06_c9b930b00ea53112180dg-32.jpg?height=452&width=489&top_left_y=1615&top_left_x=796)

Upon arriving at the Zoo, we will return to the Circus on the same tram, completing a full circle. The first part of the journey is half as long as the second, meaning it takes up one-third of the circle. This means that the journey from the Circus to the Park will not pass through the Zoo and will constitute \( \frac{1}{3} - \frac{1}{4} = \frac{1}{12} \) of the full circle. The journey through the Zoo, on the other hand, is \( 1 - \frac{1}{12} = \frac{11}{12} \) of the circle, which is 11 times longer.

## Answer

The path not passing through the Zoo is 11 times shorter.