The speeds of two trains are in the ratio $3:1$, and their lengths are in the ratio $1:3$ respectively. When both trains run in opposite directions, they cross each other in $12$ seconds. What are the speeds of the two trains (in m/s)?

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**Solution:**

1. Let the speeds of the two trains be $3X$ and $1X$ respectively.

2. Let their lengths be $1Y$ and $3Y$ respectively.

3. When two trains move in opposite directions, their relative speed is the sum of their speeds:
   
   $$3X + 1X = 4X$$

4. The total distance to be covered when they cross each other is the sum of their lengths:

$$1Y + 3Y = 4Y$$

5. Given that they cross each other in $12$ seconds, using the formula for speed:

$$	ext{Speed} = \frac{	ext{Distance}}{	ext{Time}}$$

So,

$$4X = \frac{4Y}{12}$$

Simplifying,

$$4X = \frac{4Y}{12} \implies 4X = \frac{Y}{3} \implies 12X = Y$$

6. From this relation, we get:

$$\frac{X}{Y} = \frac{1}{12}$$

7. Since $X$ and $Y$ are related by this ratio but no further information is given, the exact values of speeds cannot be determined.

**Answer:** Can’t be determined

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