The speed of train A is 150\% more than that of train B. Train A runs at a speed of $90 ext{ km/h}$ and crosses a pole in $16$ seconds. The ratio of the lengths of trains A to B is $8:5$. Calculate the time taken by train B to cross a platform $150$ meters long (in seconds).

Question

The speed of train A is 150\% more than that of train B. Train A runs at a speed of $90 	ext{ km/h}$ and crosses a pole in $16$ seconds. The ratio of the lengths of trains A to B is $8:5$. Calculate the time taken by train B to cross a platform $150$ meters long (in seconds).

Accepted Answer

**Solution:**

1. Let the speed of train B be $x 	ext{ km/h}$. Given that train A's speed is 150\% more than train B's speed, so:
   
   $$	ext{Speed of A} = x + 1.5x = 2.5x$$

2. We know the speed of train A is $90 	ext{ km/h}$, so:
   
   $$2.5x = 90 \implies x = \frac{90}{2.5} = 36 	ext{ km/h}$$

Thus, train B's speed is $36 	ext{ km/h}$.

3. Convert train A's speed to meters per second (m/s):
   
   $$90 	imes \frac{5}{18} = 25 	ext{ m/s}$$

4. Train A crosses a pole in $16$ seconds, so the length of train A is:
   
   $$	ext{Length of A} = 	ext{Speed} 	imes 	ext{Time} = 25 	imes 16 = 400 	ext{ meters}$$

5. The ratio of lengths of trains A to B is $8:5$, so:
   
   $$\frac{	ext{Length of A}}{	ext{Length of B}} = \frac{8}{5} \implies 	ext{Length of B} = \frac{5}{8} 	imes 400 = 250 	ext{ meters}$$

6. Convert train B's speed to m/s:
   
   $$36 	imes \frac{5}{18} = 10 	ext{ m/s}$$

7. Total distance train B needs to cover to cross the platform is:
   
   $$	ext{Length of train B} + 	ext{Length of platform} = 250 + 150 = 400 	ext{ meters}$$

8. Time taken by train B to cross the platform:
   
   $$	ext{Time} = \frac{	ext{Distance}}{	ext{Speed}} = \frac{400}{10} = 40 	ext{ seconds}$$

**Answer:** 40 seconds

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