A train approaches a tunnel AB, where a cat is located at a point $$ \frac{3}{8} $$ of the total distance AB from entrance A. When the train whistles, the cat starts running. If the cat runs towards entrance A, the train catches it exactly at A. If the cat runs towards exit B, the train catches it exactly at B. Determine the ratio of the speed of the train to that of the cat.

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Let the total length of the tunnel AB be $$ x $$. The cat is initially located at point C, which is $$ \frac{3}{8}x $$ from entrance A. Thus, $$ AC = \frac{3}{8}x $$.

When the cat runs towards entrance A, it travels a distance of $$ \frac{3}{8}x $$, and the train also travels this same distance $$ AC = \frac{3}{8}x $$ in the same time. This implies that their relative speeds are such that:

$$
	ext{Speed of train} : 	ext{Speed of cat} = x : \left(\frac{3}{8}xight).
$$

Now consider when the cat runs towards exit B. The cat reaches point D, such that $$ CD = \frac{3}{8}x $$. The remaining distance $$ BD = x - AC - CD = x - \left(\frac{3}{8}x + \frac{3}{8}xight) = x - \frac{6}{8}x = \frac{x}{4} $$.

The train covers the entire length $$ AB = x $$, while the cat covers $$ BD = \frac{x}{4} $$, both in the same time. Thus, their speed ratio is:

$$
	ext{Speed of train} : 	ext{Speed of cat} = x : \left(\frac{x}{4}ight) = 4:1.
$$

Hence, the required ratio of the speed of the train to that of the cat is **4:1**.

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