Let's analyze the problem and determine the reluctances of Path X and Path Y based on their lengths.

Reluctance (\( \text{R} \)) in a magnetic circuit is given by the formula:

\(\text{R} = \frac{l}{\mu \cdot A}\)

where:

• \( l \) is the length of the magnetic path.
• \( \mu \) is the permeability of the material (assumed to be the same for both paths as they are made of the same material).
• \( A \) is the cross-sectional area (assumed to be the same for both paths).

Given that both paths are made of the same material and have the same cross-sectional area, \(\mu \) and \(A\) are constant factors. Therefore, the reluctance is directly proportional to the length of the path.

Now, let's compare Path X and Path Y:

• Length of Path X, \( l_X = 0.1 \, \text{m} \)
• Length of Path Y, \( l_Y = 0.3 \, \text{m} \)

Since reluctance is directly proportional to the length, we can compare the reluctances:

\(\text{R}_Y = \frac{l_Y}{l_X} \cdot \text{R}_X = \frac{0.3}{0.1} \cdot \text{R}_X = 3 \cdot \text{R}_X\)

Thus, Path Y has three times the reluctance of Path X.

Let's validate other options:

• Path Y has one-third the reluctance of Path X: Incorrect, as reluctance increases with length.
• Path X has greater reluctance because shorter length increases reluctance: Incorrect, reluctance decreases with shorter length.
• Both paths have the same reluctance because they are made of the same material: Incorrect, despite the same material, different lengths mean different reluctances.

Therefore, the correct answer is that Path Y has three times the reluctance of Path X.