Two trains running in opposite directions cross a pole in 25 seconds and 32 seconds respectively. If they cross each other in 28 seconds, what is the ratio of their speeds?

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**Solution:**

1. Let the lengths of the two trains be *L_1* and *L_2*, and their speeds be *S_1* and *S_2* respectively.

2. Since the first train crosses a pole in 25 seconds, its speed is:
   $$S_1 = \frac{L_1}{25}$$

3. Similarly, the second train crosses a pole in 32 seconds, so its speed is:
   $$S_2 = \frac{L_2}{32}$$

4. When the two trains cross each other while running in opposite directions, the sum of their lengths is covered in 28 seconds at the combined speed:
   $$	ext{Time} = 28 	ext{ seconds}$$

5. The combined speed is:
   $$S_1 + S_2 = \frac{L_1 + L_2}{28}$$

6. Substitute *S_1* and *S_2* from steps 2 and 3:
   $$\frac{L_1}{25} + \frac{L_2}{32} = \frac{L_1 + L_2}{28}$$

7. Multiply both sides by the least common multiple of denominators (25 × 32 × 28) to clear fractions:
   $$28 	imes 32 L_1 + 28 	imes 25 L_2 = 25 	imes 32 (L_1 + L_2)$$

8. Calculate each term:
   - Left side: $896 L_1 + 700 L_2$
   - Right side: $800 L_1 + 800 L_2$

9. Rearranging:
   $$896 L_1 + 700 L_2 = 800 L_1 + 800 L_2$$
   $$896 L_1 - 800 L_1 = 800 L_2 - 700 L_2$$
   $$96 L_1 = 100 L_2$$

10. Simplify the ratio:
    $$\frac{L_1}{L_2} = \frac{100}{96} = \frac{25}{24}$$

11. Now, find the ratio of speeds:
    $$\frac{S_1}{S_2} = \frac{\frac{L_1}{25}}{\frac{L_2}{32}} = \frac{L_1}{L_2} 	imes \frac{32}{25} = \frac{25}{24} 	imes \frac{32}{25} = \frac{32}{24} = \frac{4}{3}$$

**Answer:** The ratio of their speeds is $4 : 3$.

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