A train A, which is $600$ meters long, crosses a bridge of length $300$ meters in $18$ seconds. The speed ratio of train A to train B is $2:1$. If the length of train B is $25\%$ less than that of train A, find the time taken by train B to cross the same bridge (in seconds).

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**Solution:**

1. Calculate the speed of train A:
   - Total distance covered by train A while crossing the bridge = length of train A + length of bridge = $600 + 300 = 900$ meters.
   - Time taken = $18$ seconds.
   - Speed of train A = $\frac{900}{18} = 50$ meters/second.

2. Given the speed ratio of train A to train B is $2:1$, so speed of train B = $\frac{50}{2} = 25$ meters/second.

3. Length of train B is $25\%$ less than train A:
   - Length of train B = $600 - 0.25 	imes 600 = 600 - 150 = 450$ meters.

4. Distance to be covered by train B to cross the bridge = length of train B + length of bridge = $450 + 300 = 750$ meters.

5. Time taken by train B to cross the bridge:
   $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{750}{25} = 30 	ext{ seconds}.$$

**Answer:** $30$ seconds

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